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On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions 18 Jan 2006 Ambedkar Dukkipati [email protected] Department of Computer Science and Automation Indian Institute of Science Bangalore-560012India Narasimha Murty Department of Computer Science and Automation Indian Institute of Science Bangalore-560012India Shalabh Bhatnagar [email protected] Department of Computer Science and Automation Indian Institute of Science Bangalore-560012India On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions 18 Jan 2006arXiv:cs/0601080v1 [cs.IT] PACS numbers: ‡ Corresponding author 2 Though Shannon entropy of a probability measure P , defined as − X dP dµ ln dP dµ dµ on a measure space (X, M, µ), does not qualify itself as an information measure (it is not a natural extension of the discrete case), maximum entropy (ME) prescriptions in the measure-theoretic case are consistent with that of discrete case. In this paper, we study the measure-theoretic definitions of generalized information measures and discuss the ME prescriptions. We present two results in this regard: (i) we prove that, as in the case of classical relative-entropy, the measuretheoretic definitions of generalized relative-entropies, Rényi and Tsallis, are natural extensions of their respective discrete cases, (ii) we show that, ME prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case.PACS numbers: ‡ Corresponding author § Counting or cardinality measure µ on a measurable space (X, M), when is X is a finite set and M = 2 X , is defined as µ(E) = #E, ∀E ∈ M. Introduction Shannon measure of information was developed essentially for the case when the random variable takes a finite number of values. However, in the literature, one often encounters an extension of Shannon entropy in the discrete case to the case of a one-dimensional random variable with density function p in the form (e.g [1,2]) S(p) = − +∞ −∞ p(x) ln p(x) dx . This entropy in the continuous case as a pure-mathematical formula (assuming convergence of the integral and absolute continuity of the density p with respect to Lebesgue measure) resembles Shannon entropy in the discrete case, but can not be used as a measure of information. First, it is not a natural extension of Shannon entropy in the discrete case, since it is not the limit of the sequence finite discrete entropies corresponding to pmf which approximate the pdf p. Second, it is not strictly positive. Inspite of these short comings, one can still use the continuous entropy functional in conjunction with the principle of maximum entropy where one wants to find a probability density function that has greater uncertainty than any other distribution satisfying a set of given constraints. Thus, in this use of continuous measure one is interested in it as a measure of relative uncertainty, and not of absolute uncertainty. This is where one can relate maximization of Shannon entropy to the minimization of Kullback-Leibler relative-entropy (see [3, pp. 55]). Indeed, during the early stages of development of information theory, the important paper by Gelfand, Kolmogorov and Yaglom [4] called attention to the case of defining entropy functional on an arbitrary measure space (X, M, µ). In this respect, Shannon entropy of a probability density function p : X → R + can be written as, S(p) = − X p(x) ln p(x) dµ . One can see from the above definition that the concept of "the entropy of a pdf" is a misnomer: there is always another measure µ in the background. In the discrete case considered by Shannon, µ is the cardinality measure § [1, pp. 19]; in the continuous case considered by both Shannon and Wiener, µ is the Lebesgue measure cf. [1, pp. 54] and [5, pp. 61, 62]. All entropies are defined with respect to some measure µ, as Shannon and Wiener both emphasized in [1, pp.57, 58] and [5, pp.61, 62] respectively. This case was studied independently by Kallianpur [6] and Pinsker [7], and perhaps others were guided by the earlier work of Kullback [8], where one would define entropy in terms of Kullback-Leibler relative entropy. Unlike Shannon entropy, measure-theoretic definition of KL-entropy is a natural extension of definition in the discrete case. In this paper we present the measure-theoretic definitions of generalized information measures and show that as in the case of KL-entropy, the measure-theoretic definitions of generalized relative-entropies, Rényi and Tsallis, are natural extensions of their respective discrete cases. We discuss the ME prescriptions for generalized entropies and show that ME prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case, which is true for measure-theoretic Shannon-entropy. Rigorous studies of the Shannon and KL entropy functionals in measure spaces can be found in the papers by Ochs [9] and by Masani [10,11]. Basic measure-theoretic aspects of classical information measures can be found in [7,12,13]. We review the measure-theoretic formalisms for classical information measures in § 2 and extend these definitions to generalized information measures in § 3. In § 4 we present the ME prescription for Shannon entropy followed by prescriptions for Tsallis entropy in § 5. We revisit measure-theoretic definitions of generalized entropic functionals in § 6 and present some results. Measure-Theoretic definitions of Classical Information Measures Discrete to Continuous Let p : [a, b] → R + be a probability density function, where [a, b] ⊂ R. That is, p satisfies p(x) ≥ 0, ∀x ∈ [a, b] and b a p(x) dx = 1 . In trying to define entropy in the continuous case, the expression of Shannon entropy was automatically extended by replacing the sum in the Shannon entropy discrete case by the corresponding integral. We obtain, in this way, Boltzmann's H-function (also known as differential entropy in information theory), S(p) = − b a p(x) ln p(x) dx .(1) But the "continuous entropy" given by (1) is not a natural extension of definition in discrete case in the sense that, it is not the limit of the finite discrete entropies corresponding to a sequence of finer partitions of the interval [a, b] whose norms tend to zero. We can show this by a counter example. Consider a uniform probability distribution on the interval [a, b], having the probability density function p(x) = 1 b − a , x ∈ [a, b] . The continuous entropy (1), in this case will be S(p) = ln(b − a) . On the other hand, let us consider a finite partition of the the interval [a, b] which is composed of n equal subintervals, and let us attach to this partition the finite discrete uniform probability distribution whose corresponding entropy will be, of course, S n (p) = ln n . Obviously, if n tends to infinity, the discrete entropy S n (p) will tend to infinity too, and not to ln(b − a); therefore S(p) is not the limit of S n (p), when n tends to infinity. Further, one can observe that ln(b − a) is negative when b − a < 1. Thus, strictly speaking continuous entropy (1) cannot represent a measure of uncertainty since uncertainty should in general be positive. We are able to prove the "nice" properties only for the discrete entropy, therefore, it qualifies as a "good" measure of information (or uncertainty) supplied by an random experiment. The "continuous entropy" not being the limit of the discrete entropies, we cannot extend the so called nice properties to it. Also, in physical applications, the coordinate x in (1) represents an abscissa, a distance from a fixed reference point. This distance x has the dimensions of length. Now, with the density function p(x), one can specify the probabilities of an event [c, d) ⊂ [a, b] as d c p(x) dx, one has to assign the dimensions (length) −1 , since probabilities are dimensionless. Now for 0 ≤ z < 1, one has the series expansion − ln(1 − z) = z + 1 2 z 2 + 1 3 z 3 + . . . ,(2) it is necessary that the argument of the logarithm function in (1) be dimensionless. Hence the formula (1) is then seen to be dimensionally incorrect, since the argument of the logarithm on its right hand side has the dimensions of a probability density [14]. Although Shannon [15] used the formula (1), he does note its lack of invariance with respect to changes in the coordinate system. In the context of maximum entropy principle Jaynes [16] addressed this problem and suggested the formula, S ′ (p) = − b a p(x) ln p(x) m(x) dx ,(3) in the place of (1), where m(x) is a prior function. Note that when m(x) is probability density function, (3) is nothing but the relative-entropy. However, if we choose m(x) = c, a constant (e.g [17]), we get S ′ (p) = S(p) − ln c , where S(p) refers to the continuous entropy (1). Thus, maximization of S ′ (p) is equivalent to maximization of S(p). Further discussion on estimation of probability density functions by ME-principle in the continuous case can be found in [18,17,19]. Prior to that, Kullback [8] too suggested that in the measure-theoretic definition of entropy, instead of examining the entropy corresponding to only on given measure, we have to compare the entropy inside a whole class of measures. Classical information measures Let (X, M, µ) be a measure space. µ need not be a probability measure unless otherwise specified. Symbols P , R will denote probability measures on measurable space (X, M) and p, r denote M-measurable functions on X. An M-measurable function p : X → R + is said to be a probability density function (pdf) if X p dµ = 1. In this general setting, Shannon entropy S(p) of pdf p is defined as follows [20]. Definition 2.1. Let (X, M, µ) be a measure space and M-measurable function p : X → R + be pdf. Shannon entropy of p is defined as S(p) = − X p ln p dµ ,(4) provided the integral on right exists. Entropy functional S(p) defined in (4) can be referred to as entropy of the probability measure P , in the sense that the measure P is induced by p, i.e., P (E) = E p(x) dµ(x) , ∀E ∈ M .(5) This reference is consistent because the probability measure P can be identified a.e by the pdf p. Further, the definition of the probability measure P in (5), allows us to write entropy functional (4) as, S(p) = − X dP dµ ln dP dµ dµ ,(6) since (5) implies ¶ P ≪ µ, and pdf p is the Radon-Nikodym derivative of P w.r.t µ. Now we proceed to the definition of Kullback-Leibler relative-entropy or KL-entropy for probability measures. Definition 2.2. Let (X, M) be a measurable space. Let P and R be two probability measures on (X, M). Kullback-Leibler relative-entropy KL-entropy of P relative to R is defined as I(P R) =        X ln dP dR dP if P ≪ R , +∞ otherwise.(7) The divergence inequality I(P R) ≥ 0 and I(P R) = 0 if and only if P = R can be shown in this case too. KL-entropy (7) also can be written as I(P R) = X dP dR ln dP dR dR .(8) Let the σ-finite measure µ on (X, M) such that P ≪ R ≪ µ. Since µ is σ-finite, from Radon-Nikodym theorem, there exists a non-negative M-measurable functions p : X → R + and r : X → R + unique µ-a.e, such that P (E) = E p dµ , ∀E ∈ M ,(9) Say p and r are two pdfs and P and R are corresponding induced measures on measurable space (X, M) such that P and R are identical, i.e., E p dµ = E r dµ, ∀E ∈ M. Then we have p a.e = r and hence − X p ln p dµ = − X r ln r dµ. ¶ If a nonnegative measurable function f induces a measure ν on measurable space (X, M) with respect to a measure µ, defined as and ν(E) = E f dµ, ∀E ∈ M then ν ≪ µ. ConverseR(E) = E r dµ , ∀E ∈ M .(10) The pdfs p and r in (9) and (10) (they are indeed pdfs) are Radon-Nikodym derivatives of probability measures P and R with respect to µ, respectively, i.e., p = dP dµ and r = dR dµ . Now one can define relative-entropy of pdf p w.r.t r as follows + . Definition 2.3. Let (X, M, µ) be a measure space. Let M-measurable functions p, r : X → R + be two pdfs. The KL-entropy of p relative to r is defined as I(p r) = X p(x) ln p(x) r(x) dµ(x) ,(11) provided the integral on right exists. As we have mentioned earlier, KL-entropy (11) exist if the two densities are absolutely continuous with respect to one another. On the real line the same definition can be written as I(p r) = R p(x) ln p(x) r(x) dx , which exist if the densities p(x) and r(x) share the same support. Here, in the sequel we use the convention ln 0 = −∞, ln a 0 = +∞ forany a ∈ R, 0.(±∞) = 0.(12) Now we turn to the definition of entropy functional on a measure space. Entropy functional in (6) is defined for a probability measure that is induced by a pdf. By the Radon-Nikodym theorem, one can define Shannon entropy for any arbitrary µcontinuous probability measure as follows. Definition 2.4. Let (X, M, µ) be a σ-finite measure space. Entropy of any µ-continuous probability measure P (P ≪ µ) is defined as S(P ) = − X ln dP dµ dP .(13) Properties of entropy of a probability measure in the Definition 2.4 are studied in detail by Ochs [9] under the name generalized Boltzmann-Gibbs-Shannon Entropy. In the literature, one can find notation of the form S(P |µ) to represent the entropy functional in (13) viz., the entropy of a probability measure, to stress the role of the measure µ (e.g [9,20]). Since all the information measures we define are with respect to the measure µ on (X, M), we omit µ in the entropy functional notation. By assuming µ as a probability measure in the Definition 2.4, one can relate Shannon entropy with Kullback-Leibler entropy as, Note that when µ is not a probability measure, the divergence inequality I(P µ) ≥ 0 need not be satisfied. A note on the σ-finiteness of measure µ. In the definition of entropy functional we assumed that µ is a σ-finite measure. This condition was used by Ochs [9], Csiszár [22] and Rosenblatt-Roth [23] to tailor the measure-theoretic definitions. For all practical purposes and for most applications, this assumption is satisfied. (See [9] for a discussion on the physical interpretation of measurable space (X, M) with σ-finite measure µ for entropy measure of the form (13), and of the relaxation σ-finiteness condition.) By relaxing this condition, more universal definitions of entropy functionals are studied by Masani [10,11]. S(P ) = −I(P µ) .(14) Interpretation of Discrete and Continuous Entropies in terms of KL-entropy First, let us consider discrete case of (X, M, µ), where X = {x 1 , . . . , x n }, M = 2 X and µ is a cardinality probability measure. Let P be any probability measure on (X, M). Then µ and P can be specified as follows. µ: µ k = µ({x k }) ≥ 0, k = 1, . . . , n, n k=1 µ k = 1 , and P : P k = P ({x k }) ≥ 0, k = 1, . . . , n, n k=1 P k = 1 . The probability measure P is absolutely continuous with respect to the probability measure µ if µ k = 0 implies P k = 0 for any k = 1, . . . n. The corresponding Radon-Nikodym derivative of P with respect to µ is given by dP dµ (x k ) = P k µ k , k = 1, . . . n . The measure-theoretic entropy S(P ) (13), in this case, can be written as S(P ) = − n k=1 P k ln P k µ k = n k=1 P k ln µ k − n k=1 P k ln P k . If we take referential probability measure µ as a uniform probability distribution on the set X, i.e. µ k = 1 n , we obtain S(P ) = S n (P ) − ln n , where S n (P ) denotes the Shannon entropy of pmf P = (P 1 , . . . , P n ) and S(P ) denotes the measure-theoretic entropy in the discrete case. Now, lets consider the continuous case of (X, M, µ), where X = [a, b] ⊂ R, M is set of Lebesgue measurable sets of [a, b], and µ is the Lebesgue probability measure. In this case µ and P can be specified as follows. µ: µ(x) ≥ 0, x ∈ [a, b], ∋ µ(E) = E µ(x) dx, ∀E ∈ M, b a µ(x) dx = 1 , and P : P (x) ≥ 0, x ∈ [a, b], ∋ P (E) = E P (x) dx, ∀E ∈ M, b a P (x) dx = 1 . Note the abuse of notation in the above specification of probability measures µ and P , where we have used the same symbols for both measures and pdfs. The probability measure P is absolutely continuous with respect to the probability measure µ, if µ(x) = 0 on a set of a positive Lebesgue measure implies that P (x) = 0 on the same set. The Radon-Nikodym derivative of the probability measure P with respect to the probability measure µ will be dP dµ (x) = P (x) µ(x) . Then the measure-theoretic entropy S(P ) in this case can be written as S(P ) = − b a P (x) ln P (x) µ(x) dx . If we take referential probability measure µ as a uniform distribution, i.e. µ( x) = 1 b−a , x ∈ [a, b], then we obtain S(P ) = S [a,b] (P ) − ln(b − a) , where S [a,b] (P ) denotes the Shannon entropy of pdf P (x), x ∈ [a, b] (1) and S(P ) denotes the measure-theoretic entropy in the continuous case. Hence, one can conclude that measure theoretic entropy S(P ) defined for a probability measure P on the measure space (X, M, µ), is equal to both Shannon entropy in the discrete and continuous case case up to an additive constant, when the reference measure µ is chosen as a uniform probability distribution. On the other hand, one can see that measure-theoretic KL-entropy, in discrete and continuous cases are equal to its discrete and continuous definitions. Further, from (14) and (15), we can write Shannon Entropy in terms Kullback-Leibler relative entropy S n (P ) = ln n − I(P µ) . Thus, Shannon entropy appearers as being (up to an additive constant) the variation of information when we pass from the initial uniform probability distribution to new probability distribution given by P k ≥ 0, n k=1 P k = 1, as any such probability distribution is obviously absolutely continuous with respect to the uniform discrete probability distribution. Similarly, by (14) and ( Therefore, the continuous entropy or Boltzmann H-function S(p) may be interpreted as being (up to an additive constant) the variation of information when we pass from the initial uniform probability distribution on the interval [a, b] to the new probability measure defined by the probability distribution function p(x) (any such probability measure is absolutely continuous with respect to the uniform probability distribution on the interval [a, b]). Thus, KL-entropy equips one with unitary interpretation of both discrete entropy and continuous entropy. One can utilize Shannon entropy in the continuous case, as well as Shannon entropy in the discrete case, both being interpreted as the variation of information when we pass from the initial uniform distribution to the corresponding probability measure. Also, since measure theoretic entropy is equal to the discrete and continuous entropy upto an additive constant, ME prescriptions of measure-theoretic Shannon entropy are consistent with discrete case and the continuous case. Measure-Theoretic Definitions of Generalized Information Measures We begin with a brief note on the notation and assumptions used. We define all the information measures on the measurable space (X, M), and default reference measure is µ unless otherwise stated. To avoid clumsy formulations, we will not distinguish between functions differing on a µ-null set only; nevertheless, we can work with equations between M-measurable functions on X if they are stated as valid as being only µ-almost everywhere (µ-a.e or a.e). Further we assume that all the quantities of interest exist and assume, implicitly, the σ-finiteness of µ and µ-continuity of probability measures whenever required. Since these assumptions repeatedly occur in various definitions and formulations, these will not be mentioned in the sequel. With these assumptions we do not distinguish between an information measure of pdf p and of corresponding probability measure P -hence we give definitions of information measures for pdfs, we use corresponding definitions of probability measures as well, when ever it is convenient or required -with the understanding that P (E) = E p dµ, the converse being due to the Radon-Nikodym theorem, where p = dP dµ . In both the cases we have P ≪ µ. First we consider the Rényi generalizations. Measure-theoretic definition of Rényi entropy can be given as follows. S α (p) = 1 1 − α ln X p(x) α dµ(x) ,(18) provided the integral on the right exists and α ∈ R, α > 0. The same can be defined for any µ-continuous probability measure P as S α (P ) = 1 1 − α ln X dP dµ α−1 dP .(19) On the other hand, Rényi relative-entropy can be defined as follows. Definition 3.2. Let p, r : X → R + be two pdfs on measure space (X, M, µ). The Rényi relative-entropy of p relative to r is defined as I α (p r) = 1 α − 1 ln X p(x) α r(x) α−1 dµ(x) ,(20) provided the integral on the right exists and α ∈ R, α > 0. The same can be written in terms of probability measures as, I α (P R) = 1 α − 1 ln X dP dR α−1 dP = 1 α − 1 ln X dP dR α dR ,(21) whenever P ≪ R; I α (P R) = +∞, otherwise. Further if we assume µ in (19) is a probability measure then S α (P ) = I α (P µ) . Tsallis entropy in the measure theoretic setting can be defined as follows. Definition 3.3. Tsallis entropy of a pdf p on (X, M, µ) is defined as S q (p) = X p(x) ln q 1 p(x) dµ(x) = 1 − X p(x) q dµ(x) q − 1 ,(23) provided the integral on the right exists and q ∈ R and q > 0. ln q in (23) is referred to as q-logarithm and is defined as ln q x = x 1−q − 1 1 − q (x > 0, q ∈ R). The same can be defined for µ-continuous probability measure P , and can be written as S q (P ) = X ln q dP dµ −1 dP .(24) The definition of Tsallis relative-entropy is given below. Definition 3.4. Let (X, M, µ) be a measure space. Let p, r : X → R + be two probability density functions. The Tsallis relative-entropy of p relative to r is defined as I q (p r) = − X p(x) ln q r(x) p(x) dµ(x) = X p(x) q r(x) q−1 dµ(x) − 1 q − 1 (25) provided the integral on right exists and q ∈ R and q > 0. The same can be written for two probability measures P and R, as I q (P R) = − X ln q dP dR −1 dP ,(26) whenever P ≪ R; I q (P R) = +∞, otherwise. If µ in (24) is a probability measure then S q (P ) = I q (P µ) . Maximum Entropy and Canonical Distributions For all the ME prescriptions of classical information measures we consider set of constrains of the form X u m dP = X u m (x)p(x) dµ(x) = u m , m = 1, . . . , M ,(28) with respect to M-measurable functions u m : X → R, m = 1, . . . M, whose expectation values u m , m = 1, . . . M are (assumed to be) a priori known, along with the normalizing constraint X dP = 1. (From now on we assume that any set of constraints on probability distributions implicitly includes this constraint, which will not be mentioned in the sequel.) To maximize the entropy (4) with respect to the constraints (28), the solution is calculated via the Lagrangian: L(x, λ, β) = − X ln dP dµ (x) dP (x) − λ X dP (x) − 1 − M m=1 β m X u m (x) dP (x) − u m ,(29) where λ and β m m = 1, . . . , M are Lagrange parameters (we use the notation β = (β 1 , . . . , β M )). The solution is given by ln dP dµ (x) + λ + M m=1 β m u m (x) = 0 . The solution can be calculated as dP (x, β) = exp − ln Z(β) − M m=1 β m u m (x) dµ(x)(30) or p(x) = dP dµ (x) = e − M m=1 βmum(x) Z(β) ,(31) where the partition function Z(β) is written as Z(β) = X exp − M m=1 β m u m (x) dµ(x) .(32) The Lagrange parameters β m , m = 1, . . . M are specified by the set of constraints (28). The maximum entropy, denoted by S, can be calculated as S = ln Z + M m=1 β m u m .(33) ME prescription for Tsallis Entropy The great success of Tsallis entropy is attributed to the power-law distributions one can derive as maximum entropy distributions by maximizing Tsallis entropy with respect to the moment constraints. But there are subtilities involved in the choice of constraints one would choose for ME prescriptions of these entropy functionals. These subtilities are still part of the major discussion in the nonextensive formalism [24,25,26]. In the nonextensive formalism maximum entropy distributions are derived with respect to the constraints which are different from (28), which are used for classical information measures. The constraints of the form (28) are inadequate for handling the serious mathematical difficulties (see [27]). To handle these difficulties constraints of the form X u m (x)p(x) q dµ(x) X p(x) q dµ(x) = u m q , m = 1, . . . , M(36) are proposed. (36) can be considered as the expectation with respect to the modified probability measure P (q) (it is indeed a probability measure) defined as P (q) (E) = X p(x) q dµ −1 E p(x) q dµ .(37) The measure P (q) is known as escort probability measure. The variational principle for Tsallis entropy maximization with respect to constraints (36) can be written as L(x, λ, β) = X ln q 1 p(x) dP (x) − λ X dP (x) − 1 − M m=1 β (q) m X p(x) q−1 u m (x) − u m q dP (x) ,(38) where the parameters β (q) m can be defined in terms of true Lagrange parameters β m as β (q) m = X p(x) q dµ −1 β m , m = 1, . . . , M.(39) The maximum entropy distribution in this case can be written as p(x) = 1 − (1 − q) dx p(x) q −1 M m=1 β m u m (x) − u m q 1 1−q Z q (40) p(x) = e −( X p(x) q dµ) −1 M m=1 βm(um(x)− um q ) q Z q ,(41) where Z q = X e −( X p(x) q dµ) −1 M m=1 βm(um(x)− um q ) q dµ(x) .(42) Maximum Tsallis entropy in this case satisfies S q = ln q Z q ,(43) while corresponding thermodynamic equations can be written as ∂ ∂β m ln q Z q = − u m q , m = 1, . . . M ,(44)∂S q ∂ u m q = −β m , m = 1, . . . M ,(45) where ln q Z q = ln q Z q − M m=1 β m u m q .(46) Measure-Theoretic Definitions: Revisited It is well known that unlike Shannon entropy, Kullback-Leibler relative-entropy in the discrete case can be extended naturally to the measure-theoretic case. In this section, we show that this fact is true for generalized relative-entropies too. Rényi relative-entropy on continuous valued space R and its equivalence with the discrete case is studied by Rényi [28]. Here, we present the result in the measure-theoretic case and conclude that both measure-theoretic definitions of Tsallis and Rényi relative-entropies are equivalent to its discrete case. We also present a result pertaining to ME of measure-theoretic Tsallis entropy. We show that ME of Tsallis entropy in the measure-theoretic case is consistent with the discrete case. On Measure-Theoretic Definitions of Generalized Relative-Entropies Here we show that generalized relative-entropies in the discrete case can be naturally extended to measure-theoretic case, in the sense that measure-theoretic definitions can be defined as a limit of a sequence of finite discrete entropies of pmfs which approximate the pdfs involved. We call this sequence of pmfs as "approximating sequence of pmfs of a pdf". To formalize these aspects we need the following lemma. f n (x) = 1 µ(E n,k ) E n,k p dµ , ∀x ∈ E n,k , k = 1, . . . m(n) ,(47) where (E n,1 , . . . , E n,m(n) ) is the measurable partition corresponding to f n (the notation m(n) indicates that m varies with n). Further each f n satisfies X f n dµ = 1 . Proof. Define a sequence of simple functions {f n } as f n (x) =                  1 µp −1 ([ k 2 n , k+1 2 n )) p −1 ([ k 2 n , k+1 2 n )) p dµ , if k 2 n ≤ p(x) < k+1 2 n , k = 0, 1, . . . n2 n − 1 1 µp −1 ([n,∞)) p −1 ([n,∞)) p dµ , if n ≤ p(x),(49) Each f n is indeed a simple function and can be written as f n = n2 n −1 k=0 1 µE n,k E n,k p dµ χ E n,k + 1 µF n Fn p dµ χ Fn ,(50) where E n,k = p −1 k 2 n , k+1 2 n , k = 0, . . . , n2 n − 1 and F n = p −1 ([n, ∞)). Since E p dµ < ∞ for any E ∈ M, we have E n,k p dµ = 0 whenever µE n,k = 0, for k = 0, . . . n2 n − 1. Similarly Fn p dµ = 0 whenever µF n = 0. Now we show that lim n→∞ f n = p, point-wise. First assume that p(x) < ∞. Then ∃ n ∈ Z + ∋ p(x) ≤ n. Also ∃ k ∈ Z + , 0 ≤ k ≤ n2 n −1 ∋ k 2 n ≤ p(x) < k+1 2 n and k 2 n ≤ f n (x) < k+1 2 n . This implies 0 ≤ |p−f n | < 1 2 n as required. If p(x) = ∞, for some x ∈ X, then x ∈ F n for all n, and therefore f n (x) ≥ n for all n; hence lim n→∞ f n (x) = ∞ = p(x). Finally we have X f n dµ = n(m) k=1 1 µ(E n,k ) E n,k p dµ µ(E n,k ) = n(m) k=1 E n,k p dµ = X p dµ = 1 The above construction of a sequence of simple functions which approximate a measurable function is similar to the approximation theorem [21,pp.6,Theorem 1.8(b)] in the theory of integration. But, approximation in Lemma 6.1 can be seen as a mean-value approximation where as in the later case it is the lower approximation. Further, unlike in the case of lower approximation, the sequence of simple functions which approximate p in Lemma 6.1 are neither monotone nor satisfy f n ≤ p. Now one can define a sequence of pmfs {p n } corresponding to the sequence of simple functions constructed in Lemma 6.1, denoted byp n = (p n,1 , . . . ,p n,m(n) ), as p n,k = µ(E n,k )f n χ E n,k = E n,k p dµ , k = 1, . . . m(n),(51) for any n. We have m(n) k=1p n,k = m(n) k=1 E n,k p dµ = X p dµ = 1 ,(52) and hencep n is indeed a pmf. We call {p n } as the approximating sequence of pmfs of pdf p. Now we present our main theorem, where we assume that p and r are bounded. The assumption of boundedness of p and r simplifies the proof. However, the result can be extended to an unbounded case. See [29] analysis of Shannon entropy and relative entropy on R. Theorem 6.2. Let p and r be pdf, which are bounded, defined on a measure space (X, M, µ). Letp n andr n be the approximating sequence of pmfs of p and r respectively. Let I α denotes the Rényi relative-entropy as in (20) and I q denote the Tsallis relativeentropy as in (25) then lim n→∞ I α (p n r n ) = I α (p r)(53) and lim n→∞ I q (p n r n ) = I q (p r) Proof. It is enough to prove the result for either Tsallis or Rényi since each are monotone and continuous functions of each other. Hence we write down the proof for the case of Rényi and we use the entropic index α in the proof. Corresponding to pdf p, let {f n } be the approximating sequence of simple functions such that lim n→∞ f n = p as in Lemma 6.1. Let {g n be the approximating sequence of simple functions for r such that lim n→∞ g n = r. Corresponding to simple functions f n and g n there exists a common measurable partition * {E n,1 , . . . E n,m(n) } such that f n and g n can be written as f n (x) = m(n) k=1 (a n,k )χ E n,k (x) , a n,k ∈ R + , ∀k = 1, . . . m(n) , We have h α n f n → f (x) α g(x) α−1 a.e. By Fatou's Lemma [30, pp.23] we obtain that, g n (x) = m(n) k=1 (b n,k )χ E n,k (x) , b n,k ∈ R + , ∀k = 1, . . . m(n) ,(55)lim n→∞ inf X h n (x) α g n (x) dµ(x) ≥ X p(x) α r(x) α−1 dµ(x) .(64) From the construction of f n and g n (Lemma 6.1) we have h n (x)f n (x) = 1 µ(E n,i ) E n,i p(x) r(x) r(x) dµ , ∀x ∈ E n,i .(65) By Jensen's inequality we get h n (x) α f n (x) ≤ 1 µ(E n,i ) E n,i p(x) α r(x) α−1 dµ , ∀x ∈ E n,i .(66) By (55) and (56) we can write (66) as a α n,i b α−1 n,i µ(E n,i ) ≤ E n,i p(x) α r(x) α−1 dµ , ∀i = 1, . . . m(n) .(67) By taking summations both sides of (67) we get m(n) i=1 a α n,i b α−1 n,i µ(E n,i ) ≤ m(n) i=1 E n,i p(x) α r(x) α−1 dµ , ∀i = 1, . . . m(n) .(68) The above equation (68) nothing but X h α n (x)f n (x) µ(x) ≤ X p(x) α r(x) α−1 dµ , ∀n , and hence sup i>n X h α i (x)f i (x) µ(x) ≤ X p(x) α r(x) α−1 dµ , ∀n . Finally we have lim n→∞ sup X h α n (x)f n (x) µ(x) ≤ X p(x) α r(x) α−1 dµ .(69) From (64) and (69) we have lim n→∞ X f n (x) α g n (x) α−1 µ(x) = X p(x) α r(x) α−1 dµ ,(70) and hence (54). On ME of Measure-Theoretic definition of Tsallis entropy With the shortcomings of Shannon entropy that it cannot be naturally extended to the non-discrete case, we have observed that Shannon entropy in its general case on measure space can be used consistently for the ME-prescriptions. One can easily see that generalized information measures of Rényi and Tsallis too cannot be extended naturally to measure-theoretic case, i.e., measure-theoretic definitions are not equivalent to the discrete case in the sense that they can not be defined as a limit of sequence of finite discrete entropies corresponding to pmfs defined on measurable partitions which approximates the pdf. One can use the same counter example we discussed in § 2.1. We have already given the ME-prescriptions of Tsallis entropy in the measure-theoretic case. In this section, we show that the ME-prescriptions in the measure-theoretic case are consistent with the discrete case. Proceeding as in the case of measure-theoretic entropy in § 2.3, measure-theoretic Tsallis entropy S q (P ) (24) in the discrete case can be written as S q (P ) = n k=1 P k ln q µ k P k . By (??) we get S q (P ) = n k=1 P q k [ln q µ k − ln q P k ] = S n q (P ) + n k=1 P q k ln q µ k , where S n q (P ) is the Tsallis entropy in discrete case. When µ is a uniform distribution i.e., µ k = 1 n ∀n = 1, . . . n we get S q (P ) = S n q (P ) − n q−1 ln q n n k=1 P q k . Now we show that the quantity n k=1 P q k is constant in maximization of S q (P ) with respect to the set of constraints (36). The claim is that p(x) q dµ(x) = (Z q ) 1−q ,(74) which holds for Tsallis maximum entropy distribution (40) in general. This can be shown as follows. From the maximum entropy distribution (40), we have p(x) 1−q = 1 − (1 − q) X p(x) q dµ(x) −1 M m=1 β m u m (x) − u m q (Z q ) 1−q , which can be rearranged as (Z q ) 1−q p(x) =   1 − (1 − q) M m=1 β m u m (x) − u m q p(x) q dµ(x)   p(x) q . By integrating both sides in the above equation, and by using (36) we get (74). Now, (74) can be written in its discrete form as n k=1 P q k µ q−1 k = (Z q ) 1−q .(75) When µ is uniform distribution we get n k=1 P q k = n 1−q (Z q ) 1−q(76) which is a constant. Hence by (73) and (76), on can conclude that with respect to a particular instance of ME, measure-theoretic Tsallis entropy S(P ) defined for a probability measure P on the measure space (X, M, µ), is equal to discrete Tsallis entropy up to an additive constant, when the reference measure µ is chosen as a uniform probability distribution. There by, one can further conclude that with respect to a particular instance of ME of measure-theoretic Tsallis entropy is consistent with its discrete definition. Conclusions In this paper we presented measure-theoretic definitions of generalized information measures. We proved that the measure-theoretic definitions of generalized relativeentropies, Rényi and Tsallis, are natural extensions of their respective discrete cases. We also showed that, ME prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case. 2.3) the relation between Shannon entropy and Relative entropy in discrete case we can write Boltzmann H-function in terms of Relative entropy as S [a,b] (p) = ln(b − a) − I(P µ) . Definition 3. 1 . 1Rényi entropy of a pdf p : X → R + on a measure space (X, M, µ) is defined as The Lagrange parameters β m , m = 1, . . . M, are calculated by searching the unique solution (if it exists) of the following system of nonlinear equations: ∂ ∂β m ln Z(β) = − u m , m = 1, . . . M . 34) and (34) are referred to as the thermodynamic equations. Lemma 6 . 1 . 61Let p be a pdf defined on measure space (X, M, µ). Then there exists a sequence of simple functions {f n } (we refer to them as approximating sequence of simple functions of p) such that lim n→∞ f n = p and each f n can be written as * Let ϕ and φ are two simple functions defined on (X, M). Let {E 1 , . . . E n } and {F 1 , . . . , F m } be the measurable partitions corresponding to ϕ and φ respectively. Then partition defined as {E i ∩ E j |i = 1, . . . n, j = 1, . . . m} is a common measurable partition for both ϕ and φ. + This follows from the chain rule for Radon-Nikodym derivative:dP dR a.e = dP dµ dR dµ −1 . where χ E n,k is the characteristic function of E n,k , for k = 1, . . . m(n). By (55) and (56) the approximating sequences of pmfs {p n = (p n,1 , . . . ,p n,m(n) )} and {r n = (r n,1 , . . . ,r n,m(n) )} can be written as (see (51)) p n,k = a n,k µ(E n,k ) k = 1, . . . , m(n) ,Now Rényi relative entropy forp n andr n can be written asTo prove lim n→∞ S α (p n r n ) = S α (p r) it is enough to prove thatsince we have♯Further it is enough to prove thatwhere h n is defined as h n (x) = fn(x) gn(x) . Case 1: 0 < α < 1In this case the Lebesgue dominated convergence theorem[30, pp.26]gives that,and hence (54) Case 2: α > 1 ♯ Since simple functions (f n ) α and (g n ) α−1 can be written as. The Mathematical Theory of Communication. C E Shannon, W Weawer, University of Illinois PressUrbana, IllinoisC. E. Shannon and W. Weawer. The Mathematical Theory of Communication. University of Illinois Press, Urbana, Illinois, 1949. Information Theory. R B Ash, InterscienceNew YorkR. B. Ash. Information Theory. Interscience, New York, 1965. Entropy Optimization Principles with Applications. J N Kapur, H K Kesavan, Academic PressJ. N. Kapur and H. K. Kesavan. Entropy Optimization Principles with Applications. Academic Press, 1997. On the general definition of the amount of information. I M Gelfand, N A Kolmogorov, A M Yaglom, Dokl. Akad. Nauk USSR. 1114In RussianI. M. Gelfand, N. A. Kolmogorov, and A. M. Yaglom. On the general definition of the amount of information. Dokl. Akad. Nauk USSR, 111(4):745-748, 1956. (In Russian). . N Wiener, Cybernetics, Wiley, New YorkN. Wiener. Cybernetics. Wiley, New York, 1948. On the amount of information contained in a σ-field. G Kallianpur, Essays in Honor of Harold Hotelling. I. Olkin and S. G. GhuryeStanfordStanford Univ. PressG. Kallianpur. On the amount of information contained in a σ-field. In I. Olkin and S. G. Ghurye, editors, Essays in Honor of Harold Hotelling, pages 265-273. Stanford Univ. Press, Stanford, 1960. Information and Information Stability of Random Variables and Process. M S Pinsker, Amiel FeinsteinHolden-DaySan Francisco, CAEnglish ed.M. S. Pinsker. Information and Information Stability of Random Variables and Process. Holden- Day, San Francisco, CA, 1960. (English ed., 1964, translated and edited by Amiel Feinstein). On information and sufficiency. S Kullback, R A Leibler, Ann. Math. Stat. 22S. Kullback and R. A. Leibler. On information and sufficiency. Ann. Math. Stat., 22:79-86, 1951. Basic properties of the generalized Boltzmann-Gibbs-Shannon entropy. W Ochs, Reports on Mathematical Physics. 9W. Ochs. Basic properties of the generalized Boltzmann-Gibbs-Shannon entropy. Reports on Mathematical Physics, 9:135-155, 1976. The measure-theoretic aspects of entropy, Part 1. P R Masani, Journal of Computational and Applied Mathematics. 40P. R. Masani. The measure-theoretic aspects of entropy, Part 1. Journal of Computational and Applied Mathematics, 40:215-232, 1992. The measure-theoretic aspects of entropy, Part 2. P R Masani, Journal of Computational and Applied Mathematics. 44P. R. Masani. The measure-theoretic aspects of entropy, Part 2. Journal of Computational and Applied Mathematics, 44:245-260, 1992. Information Theory with Applications. Silviu Guiaşu, McGraw-Hill, Great BritainSilviu Guiaşu. Information Theory with Applications. McGraw-Hill, Great Britain, 1977. Entropy and Information Theory. Robert M Gray, Springer-VerlagNew YorkRobert M. Gray. Entropy and Information Theory. Springer-Verlag, New York, 1990. Some observations on the concepts of information theoretic entropy and randomness. D H Jonathan, Smith, Entropy. 3Jonathan D. H. Smith. Some observations on the concepts of information theoretic entropy and randomness. Entropy, 3:1-11, 2001. A mathematical theory of communication. C E Shannon, Bell System Technical Journal. 27379C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379, 1948. E T Jaynes, sec- 4Prior probabilities. IEEE Transactions on Systems Science and Cybernetics. E. T. Jaynes. Prior probabilities. IEEE Transactions on Systems Science and Cybernetics, sec- 4(3):227-241, 1968. Calculation of maximum entropy distributions and approximation of marginalposterior distributions. Arnold Zellner, Richard A Highfield, Journal of Econometrics. 37Arnold Zellner and Richard A. Highfield. Calculation of maximum entropy distributions and approximation of marginalposterior distributions. Journal of Econometrics, 37:195-209, 1988. On the entropy of continuous probability distributions. C G Aida, Pushpa N Verdugo Lazo, Rathie, IEEE Transactions on Information Theory, IT. 241Aida C. G. Verdugo Lazo and Pushpa N. Rathie. On the entropy of continuous probability distributions. IEEE Transactions on Information Theory, IT-24(1):120-122, 1978. Maximum entropy estimation of density and regression functions. Hang K Ryu, Journal of Econometrics. 56Hang K. Ryu. Maximum entropy estimation of density and regression functions. Journal of Econometrics, 56:397-440, 1993. K B Athreya, Entropy maximization. IMA Preprint Series 1231, Institute for Mathematics and its Applications. MinneapolisUniversity of MinnesotaK. B. Athreya. Entropy maximization. IMA Preprint Series 1231, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, 1994. Introduction to Modern Analysis. Shmuel Kantorovitz, Oxford, New YorkShmuel Kantorovitz. Introduction to Modern Analysis. Oxford, New York, 2003. On generalized entropy. Imre Csiszár, Studia Sci. Math. Hungar. 4Imre Csiszár. On generalized entropy. Studia Sci. Math. Hungar., 4:401-419, 1969. The concept of entropy in probability theory and its applications in the theory of information transmission through communication channels. M Rosenblatt-Roth, Theory Probab. Appl. 92M. Rosenblatt-Roth. The concept of entropy in probability theory and its applications in the theory of information transmission through communication channels. Theory Probab. Appl., 9(2):212-235, 1964. The role of constraints in tsallis' nonextensive treatmentrevisited. G L Ferri, S Martínez, A Plastino, Physica A. 347G. L. Ferri, S. Martínez, and A. Plastino. The role of constraints in tsallis' nonextensive treatmentrevisited. Physica A, 347:205-220, 2005. Necessity of q-expectation value in nonextensive statistical mechanics. Sumiyoshi Abe, G B Bagci, Physical Review E. 7116139Sumiyoshi Abe and G. B. Bagci. Necessity of q-expectation value in nonextensive statistical mechanics. Physical Review E, 71:016139, 2005. Connections between Tsallis' formalism employing the standard linear average energy and ones employing the normalized q-average enery. T Wada, A M Scarfone, Physics Letters A. 335T. Wada and A. M. Scarfone. Connections between Tsallis' formalism employing the standard linear average energy and ones employing the normalized q-average enery. Physics Letters A, 335:351-362, 2005. The role of constraints within generalized nonextensive statistics. Constantino Tsallis, Renio S Mendes, A R Plastino, Physica A. 261Constantino Tsallis, Renio S. Mendes, and A. R. Plastino. The role of constraints within generalized nonextensive statistics. Physica A, 261:534-554, 1998. Some fundamental questions of information theory. Alfred Rényi, MTA III. Oszt. Közl. 10reprinted in [31]Alfred Rényi. Some fundamental questions of information theory. MTA III. Oszt. Közl., 10:251- 282, 1960. (reprinted in [31], pp. 526-552). On the dimension and entropy of probability distributions. Alfred Rényi, Acta Math. Acad. Sci. Hung. 10reprinted in [31]Alfred Rényi. On the dimension and entropy of probability distributions. Acta Math. Acad. Sci. Hung., 10:193-215, 1959. (reprinted in [31], pp. 320-342). Real and Complex Analysis. Walter Rudin, McGraw-HillInternational editionWalter Rudin. Real and Complex Analysis. McGraw-Hill, 1964. (International edition, 1987). Selected Papers of Alfréd Rényi. Akademia Kiado. Pál TuránPál Turán, editor. Selected Papers of Alfréd Rényi. Akademia Kiado, Budapest, 1976.

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{'abstract': 'Though Shannon entropy of a probability measure P , defined as − X dP dµ ln dP dµ dµ on a measure space (X, M, µ), does not qualify itself as an information measure (it is not a natural extension of the discrete case), maximum entropy (ME) prescriptions in the measure-theoretic case are consistent with that of discrete case. In this paper, we study the measure-theoretic definitions of generalized information measures and discuss the ME prescriptions. We present two results in this regard: (i) we prove that, as in the case of classical relative-entropy, the measuretheoretic definitions of generalized relative-entropies, Rényi and Tsallis, are natural extensions of their respective discrete cases, (ii) we show that, ME prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case.PACS numbers: ‡ Corresponding author § Counting or cardinality measure µ on a measurable space (X, M), when is X is a finite set and M = 2 X , is defined as µ(E) = #E, ∀E ∈ M.', 'arxivid': 'cs/0601080', 'author': ['Ambedkar Dukkipati [email protected] \nDepartment of Computer Science and Automation\nIndian Institute of Science\nBangalore-560012India\n', 'Narasimha Murty \nDepartment of Computer Science and Automation\nIndian Institute of Science\nBangalore-560012India\n', 'Shalabh Bhatnagar [email protected] \nDepartment of Computer Science and Automation\nIndian Institute of Science\nBangalore-560012India\n'], 'authoraffiliation': ['Department of Computer Science and Automation\nIndian Institute of Science\nBangalore-560012India', 'Department of Computer Science and Automation\nIndian Institute of Science\nBangalore-560012India', 'Department of Computer Science and Automation\nIndian Institute of Science\nBangalore-560012India'], 'corpusid': 1860, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14577, 'n_tokens_neox': 13029, 'n_words': 7741, 'pdfsha': '751bccc7a3de6b1100c2c435d4e18102bd113523', 'pdfurls': ['https://arxiv.org/pdf/cs/0601080v1.pdf'], 'title': ['On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions', 'On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions'], 'venue': []}

arxiv

Thermodynamically consistent non-isothermal phase-field modelling of elastocaloric effect: indirect vs direct method Wei Tang Devices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering State Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials Nanjing University of Aeronautics and Astronautics (NUAA) 210016NanjingChina Qihua Gong Devices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering State Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials Nanjing University of Aeronautics and Astronautics (NUAA) 210016NanjingChina College of Physics MIIT Key Lab of Aerospace Information Materials and Physics Nanjing University of Aeronautics and Astronautics (NUAA) 211106NanjingChina Min Yi Devices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering State Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials Nanjing University of Aeronautics and Astronautics (NUAA) 210016NanjingChina Bai-Xiang Xu Mechanics of Functional Materials Division Institute of Materials Science Technische Universität Darmstadt 64287DarmstadtGermany Long-Qing Chen Department of Materials Science and Engineering and Materials Research Institute The Pennsylvania State University 16803PAUSA Thermodynamically consistent non-isothermal phase-field modelling of elastocaloric effect: indirect vs direct method Elastocaloric effectPhase-field modelShape memory alloysMartensitic transformationAdiabatic temperature change Modelling elastocaloric effect (eCE) is crucial for the design of environmentally friendly and energy-efficient eCE based solid-state cooling devices. Here, a thermodynamically consistent non-isothermal phase-field model (PFM) coupling martensitic transformation with mechanics and heat transfer is developed and applied for simulating eCE. The model is derived from a thermodynamic framework which invokes the microforce theory and Coleman-Noll procedure. To avoid the numerical issue related to the non-differentiable energy barrier function across the transition point, the austenite-martensite transition energy barrier in PFM is constructed as a smooth function of temperature. Both the indirect method using isothermal PFM with Maxwell relations and the direct method using non-isothermal PFM are applied to calculate the elastocaloric properties. The former is capable of calculating both isothermal entropy change and adiabatic temperature change (∆T ad ), but induces high computation cost. The latter is computationally efficient, but only yields ∆T ad . In a model Mn-22Cu alloy, the maximum ∆T ad (∆T max ad ) under a compressive stress of 100 MPa is calculated as 9.5 and 8.5 K in single crystal (3.5 and 3.8 K in polycrystal) from the indirect and direct method, respectively. It is found that the discrepancy of ∆T max ad by indirect and direct method is within 10% at stress less than 150 MPa, confirming the feasibility of both methods in evaluating eCE at low stress. However, at higher stress, ∆T max ad obtained from the indirect method is notably larger than that from the direct one. This is mainly attributed to that in the non-isothermal PFM simulations, the relatively large temperature increase at high stress could in turn hamper the austenite-martensite transition and thus finally yield a lower ∆T ad . The results demonstrate the developed PFM herein, combined with both indirect and direct method for eCE calculations, as a practicable toolkit for the computational design of elastocaloric devices. temperature window[10], and excellent coefficient of performance[13,14]. The elastocaloric cooling is realized by means of elastocaloric effect (eCE), which originates from the latent heat associated with martensitic transformation (MT) in shape memory alloys (SMAs). Upon loading, the exothermic austenite-martensite transformation (also called the conventional MT) would cause a temperature increase in the adiabatic process. Upon unloading, the endothermic martensite-austenite transformation (also called the inverse MT) occurs, and a rapid drop in temperature arises[8]. The eCE can be quantified by the adiabatic temperature change (∆T ad ) or the isothermal entropy change (∆S iso ).Modelling eCE plays an important role in the computational design of elastocaloric devices and can be an essential complement to experiments. In general, phenomenological constitutive models and phase-field model[15,16](PFM) are utilized to calculate the eCE in SMAs. The phenomenological Tanaka-type model[11,[17][18][19][20][21]has advantages in the description of temperature change and stress-strain behavior. Very good agreement between the measured and calculated values of ∆T ad is obtained. Analogously, the phase transformation kinetic model[22][23][24]and crystal plasticity-based constitutive model[25,26]are also proposed to predict the eCE. These phenomenological constitutive models can be used to obtain the macroscopic elastocaloric properties, but have difficulties in simulating the spatial and temporal evolution of microstructure details during the MT and thus are hardly applicable to the optimization of eCE by microstructure engineering. eCE in SMAs is intrinsically ascribed to MT, which is a first-order phase transformation and can be induced by external stress/strain field or by temperature field. The PFM developed from Landau's theory of phase transformation is widely used to simulate the stress-and temperature-induced MT[27]. Wang et al.[28] built a three-dimensional PFM of MT, taking into account the transformation-induced strain, which comprehensively described a generic cubic-totetragonal MT. In the PFM, each of the martensitic variants is described by an order parameter η I (I = 1, 2, ..., n with n as the total number of various crystallographically equivalent martensitic variants), and the evolution of each of variants is governed by the Ginzburg-Landau equations. Several studies also considered hexagonal-to-orthorhombic [29], cubic-to-tetragonal [30], and tetragonal-to-monoclinic [31] transitions. Over the last decades, the PFM for MT has been constantly improved to include/study more physical phenomena in SMAs. For example, the martensitic reorientation, the temperature-induced transformation, and the stress-induced MT have been of high interests. constructed a thermodynamically consistent PFM for transformations between austenite and martensitic variants and martensitic reorientation. Since then the phase-field (PF) theory is extended for the cases of surface stresses[35,36], large strain[37,38], and eCE[39][40][41][42]. The research that explores eCE by PFM has recently emerged, owing to the rise of efficient solid-state cooling technology by eCE as well as the PFM's advantage in simulating the evolution of microstructure and temperature during the MT process.In 2015, Levitas et al.[43]proposed a multiphase phase-field theory for temperature-and stress-induced MT, which allows for a presence of the third phase at the interface between the two other phases. Then, Cui et al. [44] employed a non-isothermal PFM to study the stress-and temperature-induced MT, as well as the corresponding latent heat in SMAs. However, the discontinuous piecewise energy barrier between austenite and martensite is adverse to the calculation of ∆S iso or ∆T ad , which possibly makes the Maxwell relations based eCE analysis problematic around the transition point. In real materials, the austenite-martensite transition energy barrier varies continuously and increases with temperature in the whole temperature range. Furthermore, Sun et al.[45,46]modified this energy barrier as a continuous piecewise function of temperature in PFM and discussed the effect of grain size, crystal orientation, and loading rate on ∆T ad . Cissé et al.[40]introduced an exponential function to express the energy barrier and computed qualitatively the eCE in CuAlBe, including ∆T ad calculated by the direct method, coefficient of performance and cyclic deformation. Their energy barrier equations are continuous but not differentiable at the transition point (seeFig. 1for more details). In addition, Xu et al.[47,48]constructed a similar PFM by introducing an extra grain boundary energy to investigate the grain size dependent super-elasticity and eCE using the direct method in nanocrystalline NiTi SMAs. A mesoscale model [49] based on the Müller-Achenbach-Seelecke theory without PF order parameters was also developed to describe the evolution of local temperature and strain in SMA films during elastocaloric cycling. Meanwhile, the latent heat effect[44,46], size effect[45,47,50], plasticity[40,41], and microstructure design[42,48]are considered to regulate eCE. Nevertheless, all the above mentioned works are Introduction The solid-state cooling technology features itself as an environmentally friendly and efficient alternative to the traditional vapor compression refrigeration, thus helping move towards carbon neutrality. Typical solid-state cooling by magnetocaloric [1][2][3], electrocaloric [4,5], barocaloric [6,7] and elastocaloric [8][9][10][11] effects has been widely investigated, among which elastocaloric cooling stands out with the large temperature change [9,12], large working focused on the application of PF simulations to directly evaluate ∆T ad during loading and unloading by solving the heat-transfer equation. In general, the indirect and direct methods are two basic approaches to calculate the key parameter ∆T ad [51]. Directly measuring the (adiabatic) temperature change distribution by thermography without thermodynamic calculations is called the direct method. The difficulty of ensuring adiabatic conditions usually leads to the inevitable heat loss and thus the minor ∆T ad calculated by the direct method. Hence, researchers also adopt the indirect method [52][53][54] to calculate ∆S iso and ∆T ad by means of the Maxwell relations (Eq. 1) or Clausius-Clapeyron equation. ∆S iso and ∆T ad can be calculated from the average stress-strain responses at each temperature or average strain-temperature responses at each stress. The Maxwell relations for eCE evaluation read as ρ ∂S ∂σ T = ∂ε ∂T σ or ρ ∂S ∂ε T = − ∂σ ∂T ε .(1) Using Eq. 1, ∆S iso and ∆T ad between the initial and final stress are expressed as [8] ∆S iso (0 → σ) = σ 0 1 ρ ∂ε ∂T σ dσ,(2)∆T ad (0 → σ) = − σ 0 T ρc ∂ε ∂T σ dσ (3) where S is entropy, T temperature, ε strain, σ stress, ρ material's density, and c specific heat. The indirect method is initially extensively utilized to calculate the magnetocaloric effect [2,3]. Bonnot et al. [8] first introduced the Maxwell relations (Eq. 1) to evaluate ∆S iso associated with the MT in Cu-Zn-Al single crystal and found that the indirectly calculated ∆S iso agrees well with the directly measured value. This work has brought about the widespread use of the indirect method for evaluating eCE experimentally. Chen et al. [53] found the directly measured ∆T ad in nanocrystalline Ti-Ni-Cu SMA is consistent with ∆T ad from the indirect method. However, Pataky et al. [55] demonstrated that the directly measured ∆T ad is much lower than the temperature change calculated by the Maxwell or Clausius-Clapeyron relations. Similarly, Qian et al. [56] measured the eCE in CuAlZn and CuAlMn SMAs under compression, and showed that ∆T ad estimated by the indirect method is nearly three times as large as the directly measured one. Experimental results indicate the discrepancy in ∆T ad from the indirect and direct method, resulting in some misunderstandings and disputes about the eCE in the same material, similar observations have been reported for electrocaloric effect of ferroelectric materials [57]. It implies that if the stress-strain curves across the transition point is not measured with a sufficiently small incremental step of temperature, the indirect method may lead to unrealistic ∆S iso and ∆T ad owing to the possibly incorrect estimation of ∂ε/∂T [20,51]. In short, the literature review indicates the calculation method and its effectiveness as a key issue for evaluating eCE by PF simulations. The indirect method has the merit of calculating both ∆S iso and ∆T ad , whereas the direct method only attains ∆T ad . However, the indirect method based on the Maxwell relations, which is widely adopted in experimental studies of eCE, is relatively unexplored in terms of PF simulations. Whether the eCE evaluated by the indirect and direct method using PFM is consistent with each other remains yet unknown. In this work, we aim to develop a thermodynamically consistent non-isothermal PFM coupling MT with mechanics and heat transfer to evaluate eCE using indirect and direct method. The model is derived from a thermodynamic framework which invokes the microforce theory and the Coleman-Noll procedure. In order to account for the possibly non-sharp transition in real materials and avoid the problematic calculation due to the non-differentiable energy barrier function across the transformation temperature, the austenite-martensite transition energy barrier in PFM is introduced as a smooth function of temperature. The PFM details and their derivation accomplished by a systematic use of thermodynamic principles are presented in Sect. 2 for both improper and proper MT. Sect. 3 gives an application of non-isothermal PFM to Mn-22Cu alloy. In Sect. 4, the PFM for both single crystal and polycrystal is numerically implemented by the finite element method. In Sect. 5, benchmark simulations for reproducing the ferroelastic behavior and elastocaloric cycle are performed to verify the PFM. In addition, the elastocaloric properties in Mn-22Cu SMAs are obtained through the indirect and direct method, and the comparison of ∆T ad calculated from these two methods are discussed. Sect. 6 gives the conclusive summary. Thermodynamically consistent phase-field model The phase-field method emerges as a powerful tool for modelling microstructure evolution and predicting eCE in elastocaloric materials. It describes a microstructure using a set of conserved or non-conserved order parameters (OPs) that are continuous across the smooth interfacial regions [58], which could implicitly track the positions of interfaces. For deriving the partial differential equations for the non-isothermal PFM, we consider a closed system of volume Ω in which a pure material undergoes a first-order phase transformation between austenite and martensite. Noticeably, the MT could be divided into improper and proper MT. The improper MT is characterized by the displacements of atoms within a unit cell and is described by soft optical displacement modes (e.g., ceramic materials), where strain is generated as a secondary effect. The proper MT is a homogeneous stress-free (eigen) strain that characterizes a change in crystal lattice parameters [59,60], e.g., fcc to bcc MT in Fe alloys. Therefore, the proper and improper MT in SMAs should be distinguished and modelled by PFM using different OPs. In 1995, Kartha et al [61] proposed a phase field theory for the proper MT by using the strain tensor, which is based on a straightforward approach that uses the Helmholtz free energy defined by the OPs of eigen strain ε 0 i j . In addition, the non-conserved OPs η I are also chosen to distinguish various phases in SMAs regardless of the improper or proper MT [44]. In this work, non-conserved OPs χ is chosen to describe various phases during MT in SMAs. For the improper MT, χ is η I which is chosen to distinguish various phases: η I = 0 represents austenite and η I = 1 represents the I th martensitic variant. For the proper MT, the OPs χ is the eigen strain ε 0 i j , which can directly represent the martensitic variants and austenite. The following subsections present the complete thermodynamic consistent derivation that involves balance law, Coleman-Noll analysis, constitutive relations and evolution equations, and Helmholtz free energy. Balance law • Balance of linear momentum For the body Ω with a boundary ∂Ω, the quasi-static mechanical equilibrium equation is described by σ ji, j + b i = 0 in Ω ,(4)u i =û i on ∂Ω u , σ ji n j =t i on ∂Ω σ(5) where σ ji is the Cauchy stress tensor and b i is the body force. Here, we assume b i = 0. The Latin indices (i, j) run over the range of 1-3.û i is the displacement prescribed on the boundary ∂Ω u , n j is the outward surface unit vector, andt i is the surface traction on the boundary ∂Ω σ [62]. • Balance of angular momentum The stress tensor is symmetric, i.e., σ i j = σ ji .(6) • Balance of microforce associated with OPs χ Based on the Gurtin's microforce theory [63], we assume that there exist a set of forces that accounts for the phase transition dynamics. These forces are called microforces because they are involved with the local transformation of the material, rather than the macroscopic movements. In this work, the microforce is associated with the non-conserved OPs χ, describing the phase transformation between austenite and martensite. The relevant microforce definitions are as follows. ξ χ : the microstress, ξ χ n i : the surface microforce with n i as the unit outer normal to ∂Ω, ζ χ : the internal microforce, ζ χex : the external microforce. The balance of microforce associated with the phase transformation is written as ∂Ω ξ χ n i dA + Ω ζ χ dV + Ω ζ χex dV = 0.(7) By means of the Gauss law, we can obtain the equivalent local microforce balance as ξ χ ,i + ζ χ + ζ χex = 0.(8) Specifically, for the improper MT (χ = η I ) and proper MT (χ = ε 0 k j ), the microforces can be written as [64]              ξ χ = ξ η I i ζ χ = ζ η I for χ = η I ζ χex = ζ η I ex ,                ξ χ = ξ i jk ζ χ = ζ jk for χ = ε 0 k j ζ χex = ζ ex jk .(9) • Balance of energy The principle of the balance of energy can be stated as: the rate of change specific internal energy density e is equal to the sum of power due to external forces and the heat input to the system. In solid with negligible inertia, the thermal heat flux j h i and the heat source per unit volume q h are considered, and the internal microforce does not contribute to the energy change. The first law of thermodynamics can be presented in the form of energy balance equation as Ωė dV = ∂Ω (σ jiu j + ξ χχ − j h i )n i dA + Ω (b iui + ζ χexχ + q h )dV,(10) where u j is the displacement, and n i is the unit outer normal to ∂Ω. During loading or unloading, the elastocaloric material will release or absorb the latent heat associated with the stress-induced MT. Herein, q h is the release/absorb rate of internal heat source from MT, which is only assumed to be related to the evolution rate (χ) of phase transformation [41,65]. For example, the q h for both η I and ε 0 k j in cubic to tetragonal MT can be written as q h =        Q Iη I for χ = η I Q e (ε 0 11 −ε 0 22 ) for χ = ε 0 k j ,(11) where Q and Q e are the latent heat and play a critical role in the temperature change during MT. By Gauss law which converts the surface integration into volume one, Eq. 10 can be rewritten as Ωė dV = Ω (σ jiεi j + σ ji,iu j + ξ χ ,iχ + ξ χχ ,i − j h i,i + b iui + ζ χexχ + q h )dV,(12) whereε i j is the total strain rate withε i j = 1 2 (u i, j +u j,i ). Then, combining Eqs. 4, 6 and 8, as well as considering that the equation holds for any arbitrary volume, the energy balance equation in the local form readṡ e = σ i jεi j + ξ χχ ,i − ζ χχ − j h i,i + q h .(13) Constitutive relations and evolution equations • Second law of thermodynamics For the non-isothermal system, the second law of thermodynamics or entropy inequality, combining the global entropy balance with the Clausius-Duhem inequality [66] for the volume Ω, is expressed as Ωṡ dV + ∂Ω j h i T n i dA − Ω q h T dV ≥ 0(14) where s is the specific entropy per unit volume and T is the temperature. Converting the surface integration in Eq. 14 into volume integration and considering its validity in any volume, we can obtaiṅ s + j h i T ,i − q h T ≥ 0.(15) • Free energy imbalance Herein, the Helmholtz free energy is chosen as a proper thermodynamic potential in the non-isothermal system. The Helmholtz free energy density f per unit volume is defined by f = e − T s(16) Taking time derivatives at both sides, we get the relation 1 T ∂ f ∂t = 1 T ∂e ∂t − s T ∂T ∂t − ∂s ∂t .(17) Substituting the internal energy balance equation Eq. 13 and the second law of thermodynamics Eq. 15 into the above relation Eq. 17, we can get an inequality 1 T ∂ f ∂t ≤ 1 T σ i jεi j + 1 T ξ χχ ,i − ζ χχ − s T ∂T ∂t + j h i 1 T ,i(18) the inequality in Eq. 18 is referred as the free energy imbalance. It plays an analogous role to Eq. 15 in placing restrictions on the constitutive relations. • Coleman-Noll type analysis In order to close the model, the constitutive relations for the Cauchy stress, the internal energy density, the entropy density, the heat flux, and the microforces should be provided. In this part, we derive the explicit form of the constitutive relations and the kinetic equations in terms of a thermodynamic potential. In this derivation, the Coleman-Noll argument [67] is applied so that the resulting constitutive relations will be thermodynamically consistent. Invoking Truesdell's principle of equipresence [68], it is reasonable to assume that f , s, e, σ i j , j h i , ξ χ and ζ χ depend on ε i j , χ,χ, χ ,i , T , T ,i . Specifically, the Helmholtz free energy density f can be written as f = f (ε i j , χ,χ, χ ,i , T, T ,i )(19) The time derivative of f and the chain rule lead to ∂ f ∂t = ∂ f ∂ε i j ∂ε i j ∂t + ∂ f ∂χ ∂χ ∂t + ∂ f ∂χ ∂χ ∂t + ∂ f ∂χ ,i ∂χ ,i ∂t + ∂ f ∂T ∂T ∂t + ∂ f ∂T ,i ∂T ,i ∂t .(20) Now substituting Eq. 20 into the free energy imbalance Eq. 18, and grouping terms together, we can get 1 T ∂ f ∂ε i j − σ i j ε i j + 1 T ∂ f ∂χ + ζ χ χ + 1 T ∂ f ∂χ ∂χ ∂t + 1 T ∂ f ∂χ ,i − ξ χ χ ,i + 1 T ∂ f ∂T + s Ṫ + 1 T ∂ f ∂T ,iṪ ,i − j h i 1 T ,i ≤ 0 (21) Here we provide an analysis of Eq. 21 by invoking the arguments made by Coleman and Noll [67], and notice that Eq. 21 is linear with respect toε i j , ∂χ/∂t,χ ,i ,Ṫ ,Ṫ ,i . Hence, to satisfy Eq. 21 in any admissible thermodynamics process, the coefficients of linear terms must vanish and thus these constitutive relations can be obtained σ i j = ∂ f ∂ε i j , ξ χ = ∂ f ∂χ ,i , ∂ f ∂χ = 0, s = − ∂ f ∂T , ∂ f ∂T ,i = 0.(22) With the relations in Eq. 22, f is independent ofχ and T ,i , and so Eq. 19 can be rewritten as f = f (ε i j , χ, χ ,i , T ). Then, the left nonlinear terms in Eq. 21 are reduced to 1 T ∂ f ∂χ + ζ χ χ − j h i 1 T ,i ≤ 0.(23) • Evolution equations The inequality in Eq. 23 can be satisfied by assuming the following relationṡ χ = −L ∂ f ∂χ + ζ χ (24) j h i = κ i j 1 T , j(25) where positive L and positive semi-definite κ i j are coefficients with respect to temperature. Further, combining Eqs. 8, 22 with 24, the evolution equations for non-conserved OPs χ can be written aṡ χ = −L       ∂ f ∂χ − ∂ f ∂χ ,i ,i − ζ χex       .(26) The Eq. 26 coincides with the general Allen-Cahn equation [69]. In addition, the heat flux equation reads as j h i = −K i j T , j(27) where K i j = κ i j /T 2 is a tensor representing the thermal conductivity. Helmholtz free energy In the non-isothermal PFM, Helmholtz free energy density f of the system with microstructure evolution consists of the chemical free energy density f chem1 + f chem2 , the gradient free energy density f grad , and the elastic energy density f ela , i.e., f (ε i j , χ, χ ,i , T ) = f chem1 + f chem2 + f grad + f ela .(28) Improper MT The chemical free energy or Landau free energy f chem1 of a closed system containing austenite and martensite is determined by the distribution of OPs η I . The minimum energy states at η I = 0 and η = 1 represent the austenite phase and the corresponding martensite variant I, respectively. The associated chemical energy can be constructed as a Landau polynomial [70], i.e., f chem1 = A(T ) I η 2 I − B(T ) I η 3 I + C(T ) I η 2 I 2(29) where A(T ), B(T ), and C(T ) are temperature-dependent coefficients. 7 The gradient energy is often constructed to account for the interface between different phases, which can be expressed in terms of OPs' gradient, i.e., f grad = 1 2 β η (T ) I η I,i η I,i(30) where β η (T ) is the gradient energy coefficient depending on temperature. The total strain tensor ε i j is given as ε i j = ε ela i j + ε tr i j + ε th i j(31) where ε ela i j is the elastic strain, ε tr i j is the transformation strain generated by structural transformation, and ε th i j is the thermal strain caused by temperature change. Therefore, the elastic energy can be expressed as f ela = 1 2 C i jkl ε ela i j ε ela kl = 1 2 C i jkl (ε i j − ε tr i j − ε th i j )(ε kl − ε tr kl − ε th kl ) (32) in which C i jkl = I Φ(η I )C η I i jkl + 1 − I Φ(η I ) C A i jkl with the interpolation function Φ(η I ) = η 3 I (10 − 15η I + 6η 2 I ). C η I i jkl and C A i jkl are the elastic tensors of the I th martensitic variant and austenite, respectively. The chemical energy f chem2 represents the main concave term of the free energy and is related to the heat conduction (e.g., the internal energy density from temperature change). The complete and general form of the f chem2 can be formulated as f chem2 = −c 1 T ln(T + 1) − 1 2 c 2 T 2 − 1 6 c 3 T 3 + · · ·(33) where c i = I Φ(η I )c η I i + (1 − I Φ(η I ))c A i with c η I i and c A i as the i th coefficients for the specific heat (Eq. 34) of I th martensitic variant and austenite, respectively. Meanwhile, the specific heat per unit volume is expressed as a polynomial function of temperature, i.e., c v = c 1 + c 2 T + c 3 T 2 + · · · .(34) In addition, in order to derive the general kinetic equation for temperature T , combining Eqs. 8, 13, 22 and 27, the evolution equation of e reads asė = σ i jεi j + ∂ f ∂η I,iη I ,i + ζ η I exη I + K i j T , j ,i + q h .(35) Neglecting the first three terms on the right side of Eq. 35, one yields the common governing equation for temperature as c vṪ = K i j T , j ,i + q h .(36) Proper MT As an example here, the OPs of eigen strain ε 0 i j is considered to model the proper MT. The corresponding chemical free energy density f chem1 describing the proper MT can be represented as [71] f chem1 = Q 1 e 2 1 + Q 2 (e 2 2 + e 2 3 ) − Q 3 (e 3 3 − 3e 3 e 2 2 ) + Q 4 (e 2 2 + e 2 3 ) 2 + Q 5 (e 2 4 + e 2 5 + e 2 6 ),(37) where Q 1 and Q 5 are bulk and shear modulus, respectively. The coefficients Q 2 , Q 3 , and Q 4 are Landau constants determining the transition temperature T 0 and the transformation strains in the product phase. e i are the symmetryadapted strain defined in term of the transformation strains as e 1 = (ε 0 11 + ε 0 22 + ε 0 33 )/ √ 3, e 4 = ε 0 12 , e 2 = (ε 0 11 − ε 0 22 )/ √ 2, e 5 = ε 0 23 , e 3 = (2ε 0 33 − ε 0 22 − ε 0 11 )/ √ 6, e 6 = ε 0 13 .(38) 8 In addition, the f chem2 is similar to the case of the improper MT in Eq. 33. The austenite and martensite can be described by the eigen strain ε 0 i j . For instance, ε 0 23 = ε 0 13 = ε 0 12 = 0 is set for a cubic to tetragonal MT. Further, the austenite (cubic) is represented as (ε 0 11 = 0, ε 0 22 = 0, ε 0 33 = 0), the three tetragonal martensitic variants are described by tet 1 = (−ε 0 , 1/2ε 0 , 1/2ε 0 ), tet 2 = (1/2ε 0 , − ε 0 , 1/2ε 0 ), and tet 3 = (1/2ε 0 , 1/2ε 0 , − ε 0 ), where ε 0 is the magnitude of the spontaneous strain at a given temperature. Similarly, the gradient energy density can be written as f grad = 1 2 β e (T ) 3 i=1 3 j=1 ε 0 ii, j ε 0 ii, j ,(39) where β e (T ) is the strain gradient coefficient. The elastic strain energy density can be written as f ela = 1 2 C i jkl (ε i j − ε 0 i j − ε th i j )(ε kl − ε 0 kl − ε th kl ).(40) Application of non-isothermal PFM to Mn-22Cu alloy In this section, we apply the above non-isothermal phase-field framework to a model Mn-22Cu alloy and present the detailed formulations for the total free energy, constitutive relations, and governing or evolution equations. In Mn-22Cu SMA, there exist a face-centered cubic high-symmetry austenitic phase at high temperature and three variants of a face-center tetragonal low-symmetry martensitic phase at low temperature [72,73]. Improper MT The three martensitic variants are energetically equivalent. Herein we choose non-conserved OPs η I (I = 1, 2, 3) to represent the improper MT in PFM, and the value of η I varies from 0 to 1. Chemical free energy density The chemical free energy f chem1 represents the chemical driving force of the MT in a stress-free Mn-22Cu alloy, which can be expressed as a Landau 2-3-4 polynomial f chem1 = A(T )(η 2 1 + η 2 2 + η 2 3 ) − B(T )(η 3 1 + η 3 2 + η 3 3 ) + C(T )(η 2 1 + η 2 2 + η 2 3 ) 2(41) where A(T ), B(T ) and C(T ) are positive temperature-dependent coefficients, expressed as A(T ) = 16∆G * , B(T ) = A(T ) − 4∆G m and C(T ) = 0.5A(T ) − 3∆G m . ∆G * is the temperature-dependent energy barrier between austenite and martensite. ∆G m is the driving force of MT. The temperature-dependent profile of f chem1 in non-isothermal system and the different models for austenite-martensite energy barriers are shown in Fig. 1a and b, respectively. η I = 0 or 1 represents the system stable or metastable from the principle of minimization of the free energy. More explicitly, OPs η I = 1 indicates the I th martensitic variant and η I = 0 (I = 1, 2, 3) represents austenite. Since the MT is a first-order diffusionless structural transformation and is not sharp in real materials [74], there are several ways in the literature [40,44,46] to formulate the energy barrier function ∆G * . For instance, Cui et al. [44] proposed that ∆G * =        0.3Q/32 T ≤ T 0 [0.8 + 0.06(T − T 0 )]Q/32 T > T 0 ,(42) Sun et al. [46] proposed that First derivative (a.u.) [40,44,46] compared with what we propose in this work. and Cissé et al. [40] proposed that ∆G * =        Q/ 32 k 1 − k 2 (T − T 0 ) T ≤ T 0 k 3 Q(T − T 0 )/T 0 + Q/32k 1 T > T 0 ,(43)Temperature (K) �G * (a.u.) (a.u.)∆G * =        Qexp a 1 (T − T 0 )/T 0 T ≤ T 0 Qexp a 2 (T − T 0 )/T 0 T > T 0(44) where Q is the specific latent heat and T 0 is the chemical equilibrium temperature. It can be seen from Fig. 1b that the first derivative of all these ∆G * functions [40,44,46] are discontinuous at T 0 , leading to the problematic calculation of driving force and eCE at T 0 . Herein, we formulate the energy barrier as a smooth function of temperature. Specifically, we take advantage of the hyperbolic tangent function to modify the piecewise ∆G * function, i.e., ∆G * = 0.3Q 64 1 − tanh T − T 0 δT + 0.8 + 0.06(T − T 0 ) Q 64 1 + tanh T − T 0 δT .(45) ∆G m is also a continuous function of temperature, i.e., ∆G m = Q(T − T 0 ) T 0 .(46) It should be noted that δT is a new parameter associating with the energy barrier, which could be adjusted according to the transformation temperature window from experimental results. In this work, we assume a moderately sharp transition and set δT = 2 K. As shown in Fig. 1b, compared to the functions proposed in literature [40,44,46], our modified energy barrier function is continuous and differentiate at T 0 . This modification could resolve the difficulty of calculating eCE by indirect and direct method at the transition point. In addition, the internal energy density from temperature change f chem2 reads f chem2 = c v T ln(T + 1)(47) where c v = 3 I=1 Φ(η I )c η I v + (1 − 3 I=1 Φ(η I ))c A v is the specific heat per unit volume. c η I v and c A v are the specific heat per unit volume of the I th martensitic variant and austenite, respectively. The difference between c η I v and c A v is insignificant for Mn-22Cu alloy so that they approximately take the same values. Gradient energy density The gradient energy density can be expressed as a function of the OPs' gradient [60], i.e., f grad = 1 2 β η (T ) 3 I=1 η I,i η I,i(48) where β η (T ) is the gradient energy coefficient related to the interfacial energy and interface thickness [75]. Elastic strain energy density The elastic strain energy density in Mn-22Cu alloys can be given as f ela = 1 2 C i jkl (ε i j − ε tr i j − ε th i j )(ε kl − ε tr kl − ε th kl )(49) where C i jkl is the component of the fourth-order elastic tensor. In this work, c i j in Table 1 denotes the Voigt notation of C i jkl , i.e., c 11 = C 1111 , c 12 = C 1122 , c 44 = C 1212 . We incorporate ε tr i j and ε th i j into the PFM. ε tr i j can be expressed as the stress-free eigen strain, which is in general defined as ε tr i j = ε 00 i j (1)η 1 + ε 00 i j (2)η 2 + ε 00 i j (3)η 3 .(50) ε 00 (i) (i = 1, 2, 3) is determined by the orientation relationship and lattice distortion between martensite and austenite with regard to the FCC-FCT MT. It can be simplified to [60] ε 00 i j (1) =            ε 3 0 0 0 ε 1 0 0 0 ε 1            , ε 00 i j (2) =            ε 1 0 0 0 ε 3 0 0 0 ε 1            , ε 00 i j (3) =            ε 1 0 0 0 ε 1 0 0 0 ε 3           (51) where ε 1 = (a − a c )/a c and ε 3 = (c − a c )/a c . We choose ε 1 = 0.01 and ε 3 = −0.02 [44] in this work. In addition, the thermal strain is computed as ε th i j = α i j (T − T ref )(52) where α i j = 3 I=1 Φ(η I )α η I δ i j + 1 − 3 I=1 Φ(η I ) α A δ i j is a tensor representing thermal expansion, in which α η I and α A are thermal expansion coefficients for the I th martensitic variant and austenite, respectively. T ref is the reference temperature at which there is zero thermal strain. In the PF simulations, T ref is the initial temperature. Constitutive relations for PFM of Mn-22Cu alloy Using Eq. 22, the constitutive relations for PFM of Mn-22Cu alloy is given by σ i j = ∂ f ∂ε i j = C i jkl (ε kl − ε tr kl − ε th kl ) ζ η I i = ∂ f ∂η I,i = β η (T )η I,i e = f + T s = f − T ∂ f ∂T j h i = −K i j T , j(53) The internal energy density e for Mn-22Cu alloy can be simplified as e = c v T 2 T + 1 + A(T ) − T ∂A(T ) ∂T 3 I=1 η 2 I − B(T ) − T ∂B(T ) ∂T 3 I=1 η 3 I + C(T ) − T ∂C(T ) ∂T 3 I=1 η 2 I 2 + 1 2 C i jkl        ε ela i j − T ∂ε ela i j ∂T        ε ela kl + 1 2 β η (T ) − T ∂β η (T ) ∂T 3 I=1 η I,i η I,i .(54) Governing equations for PFM of Mn-22Cu alloy We adopt Eq. 26 to govern the spatial and temporal evolution of η I , which is analogous to the time-dependent Ginzburg-Landau (TDGL) kinetic equation [76,77]. Further, using Eqs. 35 and 54, and assuming quasi-static mechanics, the temperature evolution equation in non-isothermal PFM can be obtained. Finally, The governing equations are deduced and summarized as σ i j, j = 0 η I = −L η ∂ f chem ∂η I + ∂ f ela ∂η I − β η η I,ii − ζ η I ex ė =         β η 3 I=1 η I,iηI         ,i + ζ η I exη I + K i j,i T , j + K i j T ,i j + Q 3 I=1η I .(55) where L η is the kinetic coefficient characterizing the interfacial migration. Proper MT For the proper MT, the OPs ε 0 ii are used to directly describe the austenite and martensite. The corresponding chemical free energy or Landau free energy density for 2D domain in cubic-tetragonal MT of Mn-22Cu can be represented as [78] f chem1 = 1 2 Q 1 e 2 1 + 1 2 Q 2 (T )e 2 2 + 1 2 Q 3 e 2 3 − 1 4 Q 4 (T )e 4 2 + 1 6 Q 5 (T )e 6 2 ,(56) where e 1 = (ε 0 11 + ε 0 22 )/ √ 2, e 2 = (ε 0 11 − ε 0 22 )/ √ 2, and e 3 = (ε 0 12 + ε 0 21 )/2. These coefficients (Q 1 to Q 5 ) in Landau free energy could be obtained by fitting the experimental results. For cubic to tetragonal MT, the eigen strains for martensitic variants are ε 00 = −ε 0 0 0 ε 0 for V1, or ε 0 0 0 −ε 0 for V2,(57) Similar to the case of the improper MT, the internal energy density, gradient energy density, and elastic strain energy density can be written as f chem2 = c v T ln(T + 1), f grad = 1 2 β e ε 0 11,1 2 + ε 0 11,2 2 + ε 0 22,1 2 + ε 0 22,2 2 , f ela = 1 2 C i jkl (ε i j − ε 0 i j − ε th i j )(ε kl − ε 0 kl − ε th kl ),(59) respectively. The constitutive relations of proper MT PFM are analogous to Eq. 53, and the governing equations can be written as σ i j, j = 0 ε 0 ii = −L e       ∂ f chem ∂ε 0 ii + ∂ f ela ∂ε 0 ii − β e ε ii, j j − ζ ex ii       e =         β e 2 i=1 ε ii, jε 0 ii         , j + ζ ex iiε 0 ii + K i j,i T , j + K i j T ,i j + Q e (ε 0 11 −ε 0 22 ).(60) Finite-element implementation Herein, we use finite element method to solve the governing equations in Eqs. 55 and 60, and convert the strong forms into weak forms by introducing a test function. Note that the degrees of freedom η I and ε 0 ii are replaced by the generalized degree of freedom χ in the finite-element implementation. Therefore, the degrees of freedom are set as u 1 , u 2 , u 3 , χ, T . Assuming K i j is a constant K in Eqs. 55 and 60, the weak forms are formulated as 0 = Ω σ i j φ i, j dv − ∂Ω σ i j n j φ i ds 0 = Ω ψ χ L + ∂ f chem ∂χ + ∂ f ela ∂χ − ζ χex + ψ ,i βχ ,i dv − ∂Ω ψβχ ,i n i ds 0 = Ω ϑ ė − q h − ζ χexχ + ϑ ,i β χ ,iχ + KT ,i dv − ∂Ω ϑ β χ ,iχ n i + KT ,i n i ds,(61) where φ i , ψ and ϑ are the test function for u i , χ and T , respectively. Note that the surface terms (σ i j n j and T ,i n i ) in Eq. 61 represent the surface traction and heat flux boundary conditions. n i is the normal vector of the boundary ∂Ω. By introducing the shape functions for independent variables and test functions, the discretized equations can be written as u i = N L u L i χ = N L χ Lχ = N LχL I T = N L T L T = N LṪ L φ i = N L φ L i ψ = N L ψ L ϑ = N L ϑ L(62) where L denotes the node number. N L is the shape function. Here we assume quasi-static mechanics, and neglect ζ χex , and do not consider the dynamic PFM. After the insertion of Eq. 62 into Eq. 61, the following elemental residuals can be obtained R L u i = Ω σ i j N L , j dv − ∂Ω N L σ i j n j ds R L χ = Ω N L 1 Lχ + ∂ f ela ∂χ + ∂ f chem ∂χ + βχ ,i N L ,i dv − ∂Ω N L βχ ,i n i ds R L T = Ω N L ė − q h + N L ,i β χ ,iχ + KT ,i dv − ∂Ω N L β χ ,iχ n i + KT ,i n i ds.(63) With regard to the time dependence of the residuals, we use the implicit backward Euler method to realize the time discretization [80]. The residual equation for the current time step t n+1 is R L n+1 = R L       d J n+1 , d J n+1 − d J n ∆t       ,(64) where (d J n+1 − d J n )/∆t =ḋ J n+1 and ∆t is time step. d J n+1 should be solved in this equation. For solving these non-linear equations, the Newton iteration scheme is performed at each time step. The corresponding iteration matrix is S LJ = K LJ + 1 ∆t D LJ ,(65) where K LJ is the stiffness matrix and D LJ is the damping matrix. We adopt the open source Multiphysics Object Oriented Simulation Environment (MOOSE) [81] to implement this PFM. In order to evaluate the eCE of polycrystalline material, we make use of the rotation matrix to establish the polycrystalline PFM. The rotation matrices spinning around the x-axis, y-axis and z-axis are given by where α, β, γ are the Euler angles. The three-dimensional rotation matrix Q is given by Q x =            1 0 0 0 cosα sinα 0 −sinα cosα            , Q y =            cosβ 0 −sinβ 0 1 0 sinβ 0 cosβ            , Q z =            cosγ sinγ 0 −sinγ cosγ 0 0 0 1           (66)Q = Q z Q y Q x(67) The relationship between the global and the local variables are u = Q u σ = K σ σ ε = K ε ε(68) in which u, ε and σ are the values in the global coordinate, while u , ε and σ are values in the grain local coordinate. K σ and K ε are the stress rotation matrix and strain rotation matrix, respectively. The eigen strain (ε tr i j ) and elastic tensor (C i jkl ) should be rotated by K σ and K ε . It is not difficult to deduce K ε and C from Eq. 68 that K ε = F K σ F −1 C = K σ C K −1 ε(69) where F is a diagonal matrix with diagonal element values (1, 1, 1, 2, 2, 2). The stress rotation matrix (K σ ) is written in Appendix A. Simulation results and discussions In this work, we utilize a 2D domain to calculate the eCE for reducing computation cost. The Mn-22Cu material parameters used for PF simulations are taken from literature [44] and summarized in Table 1. The material parameters of austenite and martensite in Mn-22Cu are regarded to be the same owing to their negligible differences. The finite element mesh size should be smaller than the minimum value of interface thickness (δ = β/2∆G * ≈ 14.8 nm) between austenite and martensite or between different austenite variants, thus the mesh size is chosen as ∆l = 10 nm. In the calculation of eCE by indirect and direct method, the uniaxial compressive stress is applied in the y-direction (favor the martensitic variant 2 (V2) formation), the left and right boundaries are set mechanically free (inserts in Fig. 6a and 7a), and adiabatic boundary conditions are specified by assuming zero heat flux. Comparison of two models for improper and proper MT The MT behavior and eCE in Mn-22Cu are modelled through two models, the "OP-η" model using OPs η I for improper MT and "OP-ε" model using OPs ε 0 ii for proper MT. As shown in Fig. 2, under a 500 MPa uniaxial compressive stress, the stress-strain behavior and the temperature change during MT between these two models show good agreement. The consistency obviously prove the validity and generality of PF simulation in MT and eCE between the two models. The distinction in the temperature change during the beginning of loading is caused by the difference in PF kinetic coefficient L. In addition, the results of stress-strain behavior and microstructure evolution are consistent between these two models. In fact, Sun et al. [45,46] have utilized the above two PFMs to simulate the cubic-tetragonal MT in SMA. Besides, the OPs η I are utilized in PFM built by Khachaturyan et al. [59,60,82] to model the proper and improper MT. This shows the correctness and validity of using OPs η to model the proper or improper MT at least in term of phenomenological results. The main discrepancy between the two models is the choice of OPs. The eigen strain ε 0 i j for the two models is ε 00 i j (I)η I and ε 0 ii , respectively. Therefore, the transformation strains in the two models are all 0.02 (η I = 1 for the former, and ε 0 ii = −0.02 for the latter), when the austenite to martensite transformation completely occurs. For the cubic-tetragonal MT in Mn-22Cu, the two models are equivalent in PF simulation results. Note that the following results of MT behavior and eCE in PF simulation are based on the "OP-η" model. Benchmark simulation of ferroelastic behavior Phase transformation, mainly including stress-and temperature-induced MT, would occur in Mn-22Cu SMAs under external fields, and leads to extraordinary macroscopic behaviors, e.g., the shape memory effect and superelasticity. In order to validate the model, benchmark simulations including that stress-and temperature-induced MT in isothermal (see Fig. 3) and non-isothermal PFM (see Fig. 4), and thermodynamic cycle (see Fig. 5) are carried out. For the single grain case in Fig. 3, austenite (A) with a small portion of V2 martensite embryo is set as the initial condition (inset of Fig. 3c). Figs. 3a and b show the stress-and temperature-induced MT in isothermal PF simulation, respectively. In Fig. 3a, a large enough compressive stress, along the easy axis of V2, would transform the initial A into V2. The mean value of η 2 (η 2 ) increases as the stress is imposed and decreases as the stress is released, indicating the conventional MT during loading and the inverse MT during unloading. In Fig. 3b, upon cooling, high-temperature austenite would turn into low-temperature martensite, or vice versa. The superelasticity effect during stress-induced MT is also clearly presented in Fig. 3c. Upon stress loading, the small martensitic embryo (red stripes) grows to be a large martensitic domain owing to the stress-induced MT. Upon unloading, the austenitic embryo (blue stripes) grows and the original state restores. Mean value of order parameter, Simulating, the superelasticity effect also can be used as a benchmark for validating the non-isothermal PFM for polycrystalline, as shown in Fig. 4. The polycrystalline model size is 1000 nm × 1000 nm and has nine grains with random orientations. The maximum compressive stress (along the easy axis of V2) is 300 MPa and the initial temperature is 245 K. During the loading, the strain increases linearly with the stress at first, corresponding to the elastic response of the austenite. In this regime, there is no MT and temperature and η 2 change a little. As the stress increases to 230 MPa, the stress-induced MT occurs, V2 bands start to grow, and temperatures apparently changes (inset of Fig. 4a). As shown in Fig. 4b, the temperature and η 2 rapidly rise and then recover to the initial values upon unloading. The temperature is increased from 245 to 255 K, resulting in ∆T ad = 10 K. The temperature distribution is almost homogeneous because of the high thermal conductivity and no thermal barrier in the domain wall. The temperature is found to be approximately proportional to order parameters, which is consistent with the results reported in [44]. The refrigeration cycle of eCE is schematically shown in the insert of Fig. 5. When a SMA in the austenitic (cubic) phase is axially stressed or strained, an exothermic austenitic-martensitic transformation occurs, which under adiabatic conditions makes the material heated up (x in Fig. 5). This heated material then releases heat to the surroundings and cools down to the ambient temperature (y in Fig. 5). When the stress is removed, the crystal structure transforms back to the austenitic phase (z in Fig. 5). Finally, the material cools down and is now able to absorb heat from the surroundings ({ in Fig. 5). The simulated temperature vs time profile under loading and unloading is also shown in Fig. 5, showing good agreement with the above process from x to {. These simulation results of MT and thermodynamic cycle indicate that eCE can be soundly handled by the isothermal and non-isothermal PFM. Indirect method The indirect method is usually based on the data of superelastic response, such as the stress-strain curves at various temperatures or strain-temperature curves at various stresses. According to the Eqs. 2 and 3, indirect method is used to calculate the isothermal entropy change ∆S iso and adiabatic temperature change ∆T ad by means of the isothermal PFM. We investigate the eCE under 50, 100, 150, 200, 250, and 300 MPa compressive stress, which is lower than the maximum stress that Mn-22Cu alloy can endure. eCE of single crystal The single crystal model size is 300 nm × 500 nm. The typical stress-strain curves at different temperatures from 215 to 275 K are shown in Fig. 6a, which clearly indicates three stages involving elastic stage in austenite, transformation stage, and elastic stage in martensite. A minimum critical transformation stress is found at T = 245 K, because the energy barrier between austenite and martensite vanishes there. Fig. 6b shows the strain-temperature curves at different compressive stresses from 50 to 300 MPa. During the loading of 100 MPa stress, the strain firstly notably increases and then significantly decreases, indicating that both the inverse and conventional MT occur. By means of the Eqs. 2 and 3, ∆S iso and ∆T ad can be calculated, as summarized in Figs. 6c and d, respectively. Large ∆S iso and ∆T ad observably appear near the transformation temperature (245 K). This is caused by the first-order phase transformation which releases or absorbs lots of heat at the transformation temperature. The values of ∆S max (J/kg/K) and ∆T max (K) at varied stresses are: ∆S max = −3.3 and ∆T max = 2.3 for σ = 50 MPa, ∆S max = −13.5 and ∆T max = 9.5 for σ = 100 MPa, ∆S max = −27.8 and ∆T max = 19.5 for σ = 150 MPa. The maximum ∆T ad 9.5 K at 100 MPa is consistent with the experimentally measured value of 11.6 K [56]. It is found that ∆S max and ∆T max do not increase further for the stress beyond 150 MPa, because of the already complete MT under 150 MPa stress. However, the operating temperature window is effectively broadened if the stress increases. In addition, there exists an inverse/negative eCE, i.e., positive ∆S iso or negative ∆T ad under an external stress. This is a consequence of the inverse MT occurring in the case of co-existed austenite and martensite when the ambient temperature is below T 0 [8,83,84]. The further explanation in terms of microstructure for the negative eCE will also be discussed in Fig. 7e. Fig. 7 shows the simulation results for eCE of polycrystal. ∆T ad = 3.5 K for σ = 100 MPa is close to the measured 3.9 K under 4.0% strain [56]. In general, ∆S iso and ∆T ad in polycrystalline are lower than that in single crystal. The internal interaction caused by the grain boundary, grains with adverse orientations, and the large negative eCE together contribute to the low ∆S iso and ∆T ad in polycrystal. Besides, the large negative ∆T ad of polycrystal (-13 K for 200 MPa) in Fig. 7d results from the inverse MT and the local large compressive stress due to the grain boundary, as shown the microstructure evolution in Figs. 7e and f. Below 245 K, the initial phase is martensite, and the inverse MT would occur under a large enough compressive stress. In the case of 150 MPa and 241 K (Fig. 7e), V1 is changed into A at first, and then into V2. This inverse MT absorbs an amount of heat, resulting in the large inverse eCE. At 248 K (Fig. 7f), the initial austenite phase is transformed into V2 accompanied with a small portion of V1. eCE of polycrystal Direct method In experiment or numerical simulation, the straightforward way to assess eCE is the direct method, which directly measures the ∆T ad during loading and unloading. Cissé et al. [40] and Sun et al. [46] used the non-isothermal PFM to directly calculate ∆T ad in CuAl 11 Be 2 and FePd SMAs, respectively. Here we use the modified non-isothermal PFM to evaluate eCE in Mn-22Cu alloys based the direct method. The applied compressive stress is 100 MPa, the loading rate is 0.2 MPa/ns, and other conditions are consistent with the indirect method for a comparison. eCE of single crystal For single crystal, the strain-stress response and the adiabatic temperature change calculated by the direct method are shown in Fig. 8. It can be found in Fig. 8a that the sample almost stays at the linear elastic regime at 250 K greater than the equilibrium temperature. As the initial temperature decreases to 245 K, the critical transformation stress decreases and the area with MT increases, leading to in a larger ∆T ad (Fig. 8b). The transformation strain is 0.8% at 245 K under 100 MPa compressive stress and the corresponding ∆T ad is around 8.5 K. ∆T ad in Fig. 8b shows a strong temperature dependence. In other words, controlling the ambient temperature to obtain high transformation strain is a feasible neat idea to improve the eCE of SMAs. eCE of polycrystal The settings of polycrystalline model are consistent with those in the indirect method. Fig. 9a and b show the stress-strain response and ∆T ad at different temperatures, respectively. Under the same stress, the maximum transformation strain and ∆T ad occurs at T = 245 K and the corresponding ∆T ad is around 3.8 K. Polycrystal is more closer to the microstructure of experimental samples and thus its maximum ∆T ad of 3.8 K by the direct method and 3.5 K by the indirect method is more closer to the experimental data (3.9 K) [56]. Compared to the maximum ∆T ad of 8.5 K in single crystal, the smaller ∆T ad in polycrystal is probably a consequence of domain wall and adverse orientation grain in the polycrystalline model. The grain misorientation may hinder the martensitic nucleation and growth, as shown in Fig. 4a. In addition, comparing Fig. 8a with Fig. 9a, it is obvious that a smaller hysteresis exists in polycrystal. Indirect vs direct method In order to compare the indirect and direct method which are based on PFM, in Fig. 10 we present the ∆T ad -T curves calculated by both methods under 100 MPa compressive stress. It is clear that the overall trend of ∆T ad varying with T is consistent for both methods. For single crystal in Fig. 10a, ∆T ad calculated by the two methods at different ambient temperatures is close. Under 100 MPa, the maximum ∆T ad (∆T max ad ) calculated by indirect and direct method is 9.5 and 8.5 K (3.5 and 3.8 K) for single crystal (polycrystal), respectively. This indicates the consistency between indirect and direct method and the reliability of calculating eCE by PF simulations. However, the peak temperature corresponding to ∆T max ad calculated by the two methods is slightly different. For the direct method, the 20 peak temperature is around 245 K (the chemical equilibrium temperature), which is lower than the indirect method (247 K for single crystal and 248 K for polycrystal). The insets in Fig. 10 present the ∆T max ad as a function of the applied compressive stress. It can be seen that eCE can be significantly tuned by the uniaxial compressive stress. Results from the direct method is essentially in agreement with those from the indirect method when the stress is less than 150 MPa, with a discrepancy of ∆T max ad within 10%. At higher compressive stress (above 150 MPa), ∆T max ad calculated by the indirect method is apparently larger than that form the direct one. This could be attributed to the temperature-induced MT in the non-isothermal PFM simulation, i.e., the relatively large temperature increase at high stress could in turn remarkably hinder the austenite-martensite transition and thus lower the temperature [85]. Cheng et al. [86] also report the similar phenomena in electrocaloric effect. In addition, ∆T ad /σ max is usually utilized as a parameter to evaluate the field normalized caloric effect, also known as the specific adiabatic temperature [87]. It is found that ∆T ad /σ max is 0.131 and 0.091 K/MPa for single crystal (0.054 and 0.037 K/MPa for polycrystal) by the indirect and direct method, respectively. A large ∆T ad /σ max would be beneficial to enhance the overall performance and efficiency of the elastocaloric refrigerator. The agreement between the calculated and experimental results is satisfactory, implying that the direct method based on the non-isothermal PFM and the direct method based on isothermal PFM are reliable for the calculation and evaluation of eCE in SMAs. Nevertheless, the computational cost and the available information on eCE are also critical factors to be considered. The great advantage of the indirect method is that all the ∆S iso , ∆T ad , and refrigerating capacity can be readily obtained. But the indirect method depends on the temperature dependent strain-stress curves with small temperature intervals for an accurate integration, leading to heavy computation. If one is only interested in ∆T ad , the direct method is the best choice since it doe not require the overall stress-strain-temperature data and thus is computationally efficient. Conclusion In summary, we have developed a thermodynamically consistent non-isothermal PFM for the simulation of eCE. The PFM couples MT with mechanics and heat transfer to evaluate eCE by using the indirect and direct method. The model considering both improper and proper MT is derived from a thermodynamic framework which invokes the microforce theory accommodating non-local effects, thermodynamic laws, and the Coleman-Noll procedure. In order to avoid the problematic calculation due to the non-differentiable energy barrier function across the transformation temperature and consider the possibly non-sharp transition in real materials, the austenite-martensite transition energy barrier in PFM is introduced as a smooth function of temperature by using the hyperbolic tangent function. On one hand, all constitutive relations are represented in terms of a thermodynamic potential. Therefore, the PF modelling work is reduced to the design of a proper form of the thermodynamic potential. On the the hand, the framework automatically satisfies the first, second, and third laws of thermodynamics. After being numerically implemented by the finite element method, the developed PFM is demonstrated to be capable of recapturing the microstructure response and calculating eCE via both indirect and direct method in a model material (i.e., Mn-22Cu SMA) under an external loading. Under a compressive stress of 100 MPa, ∆T max ad calculated by the indirect and direct method is 9.5 and 8.5 K for single crystal (3.5 and 3.8 K for polycrystal), respectively. A large ∆T ad exists in single crystal, but the working temperature window is narrow and can be improved by increasing the compressive stress. Besides, negative eCE caused by the inverse MT is found, especially for polycrystal, which reduces ∆T max ad . Overall, the direct method based on the non-isothermal PFM and the direct method based on isothermal PFM are reliable for the calculation and evaluation of eCE in SMAs. But there are still some differences between these two methods. ∆T ad calculated by the indirect and direct method shows tiny discrepancy (within 10%) under a low stress (≤150 MPa). At higher stress, ∆T max ad from the indirect method is apparently larger than that from the direct one, mainly owing to the fact that the relatively large temperature increase at high stress in the non-isothermal PFM could in turn remarkably hinder the austenite-martensite transition and thus lower the temperature. The indirect method is computational expensive, but can yield all the ∆S iso , ∆T ad , and refrigerating capacity. The direct method is computationally efficient, but only yields ∆T ad . The developed non-isothermal PFM along with the indirect and direct method could be a computational toolkit to unveil strategies for designing high-performance elastocaloric devices. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Rotation matrix K σ =                             Q                            (A.1) Fig. 1 . 1(a) Chemical free energy density as a function of order parameter. (b) Energy barrier functions from literature where ε 0 0= 0.02 is the transformation strain in Mn-22Cu. Here, (e 1 = 0, e 2 = 0, e 3 = 0) represents for austenite, and (e 1 = 0, e 2 = 0.04, e 3 = 0) represents for martensitic variant 1 (V1), and (e 1 = 0, e 2 = −0.04, e 3 = 0) represents for martensitic variant 2 (V2). For a first-order transition, the coefficients of Q 4 and Q 5 is positive and is needed for stability[79]. Here, we assume that the value of Q 2 (T ), Q 4 (T ),Q 5 (T ) are similar to A(T ), B(T ), C(T ) in this work owing to the lack of experimental value for Mn-22Cu. Thus, Q 2 (T ), Q 4 (T ), Q 5 (T ) are assumed as Q 2 (T ) = A(T )/(2ε 0 ) 2 Q 4 (T ) = B(T )/(2ε 0 ) 4 Q 5 (T ) = C(T )/(2ε 0 ) 6 . Fig. 2 . 2(a) Stress-strain behaviors and (b) temperature change evolution under different temperatures by using the two models. "OP-η" represents PFM with OPs η (improper MT) and "OP-ε" PFM with OPs ε 0 ii (proper MT). Fig. 3 . 3Temporal evolution of order parameters under (a) stress-induced MT and (b) temperature-induced MT. (c) The stress-strain curve of single crystal upon stress loading and unloading at T = 275 K (above T 0 ). Blue and red are austenite (A) and V2, respectively. Fig. 4 . 4(a) The stress-strain response of polycrystal under T = 245 K and σ = 300 MPa. Blue and red colors represent A and V2, respectively. (b) Temporal evolution of temperature and η I . Fig. 5 . 5Temperature vs time profile and schematics during the thermodynamic cycle. Fig. 6 . 6(a) Stress-strain responses of single crystal under 500 MPa compressive stress at various temperatures. (b) Strain-temperature curves under compressive stresses of 50, 100, 150, 200, 250, and 300 MPa. (c) ∆S iso and (d) ∆T ad calculated by Eq. 2 and 3 at various temperatures. Fig. 7 . 7(a) Stress-strain responses of polycrystal under 500 MPa compressive stress at various temperatures. (b) Strain-temperature curves, (c) ∆S iso , and (d) ∆T ad under various compressive stresses. Phase evolution under 150 MPa at (e) 241 K (x in (c)) and (f) 248 K (y in (c)). V1, V2, and A are coloured in yellow, red and blue, respectively. Fig. 8 .Fig. 9 . 89(a) Stress-strain curves and (b) ∆T ad of single crystal under 100 MPa compressive stress at different temperatures. (a) Stress-strain curves and (b) ∆T ad of polycrystal under 100 MPa compressive stress at different temperatures. Fig. 10 . 10∆T ad -T curves of (a) single crystal and (b) polycrystal calculated by the indirect and direct method under a compressive stress of 100 MPa. The insets show the ∆T max ad as a function of compressive stress. Formal analysis, Investigation, Visualization, Data curation, Writing -original draft. Qihua Gong: Conceptualization, Resources, Supervision, Investigation, Data curation, Writing -original draft & review & editing. Min Yi: Conceptualization, Resources, Supervision, Project administration, Funding acquisition, Writing -original draft & review & editing. Bai-Xiang Xu: Conceptualization, Supervision, Writing -original draft & review. Long-Qing Chen: Conceptualization, Supervision, Writing -original draft & review. Table 1 . 1Material parameters and simulation parameters of Mn-22Cu for improper and proper MT PFMparameter name value c 11 elastic constant 76.588 GPa c 12 elastic constant 14.588 GPa c 44 elastic constant 31 GPa T 0 chemical equilibrium temperature 245 K Q latent heat for improper MT 4.84 × 10 7 J/m 3 L η kinetic coefficient for improper MT 50 m 3 /s/J L e kinetic coefficient for proper MT 1 m 3 /s/J β η gradient energy coefficient for improper MT 2.5×10 −9 J/m β e gradient energy coefficient for proper MT 1×10 −9 J/m K thermal conductivity 40 J/m/s/K ρ density 7500 kg/m 3 c v specific heat per unit volume 2.64×10 6 J/m 3 /K α thermal expansion coefficient 10 −5 K −1 11 Q 21 Q 12 Q 22 Q 13 Q 23 Q 11 Q 22 + Q 12 Q 21 Q 12 Q 23 + Q 13 Q 22 Q 11 Q 23 + Q 13 Q 21 Q 21 Q 31 Q 22 Q 32 Q 23 Q 33 Q 21 Q 32 + Q 22 Q 31 Q 22 Q 33 + Q 23 Q 32 Q 21 Q 33 + Q 23 Q 31 Q 31 Q 11 Q 12 Q 32 Q 13 Q 33 Q 31 Q 12 + Q 32 Q 11 Q 32 Q 13 + Q 33 Q 12 Q 31 Q 13 + Q 11 Q 332 11 Q 2 12 Q 2 13 2Q 11 Q 12 2Q 12 Q 13 2Q 11 Q 13 Q 2 21 Q 2 22 Q 2 23 2Q 21 Q 22 2Q 22 Q 23 2Q 21 Q 23 Q 2 31 Q 2 32 Q 2 33 2Q 31 Q 32 2Q 33 Q 32 2Q 31 Q 33 Q AcknowledgementsThe authors acknowledge the support from National Natural Science Giant magnetocaloric effect of MnAs 1−x Sb x. 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{'abstract': "Modelling elastocaloric effect (eCE) is crucial for the design of environmentally friendly and energy-efficient eCE based solid-state cooling devices. Here, a thermodynamically consistent non-isothermal phase-field model (PFM) coupling martensitic transformation with mechanics and heat transfer is developed and applied for simulating eCE. The model is derived from a thermodynamic framework which invokes the microforce theory and Coleman-Noll procedure. To avoid the numerical issue related to the non-differentiable energy barrier function across the transition point, the austenite-martensite transition energy barrier in PFM is constructed as a smooth function of temperature. Both the indirect method using isothermal PFM with Maxwell relations and the direct method using non-isothermal PFM are applied to calculate the elastocaloric properties. The former is capable of calculating both isothermal entropy change and adiabatic temperature change (∆T ad ), but induces high computation cost. The latter is computationally efficient, but only yields ∆T ad . In a model Mn-22Cu alloy, the maximum ∆T ad (∆T max ad ) under a compressive stress of 100 MPa is calculated as 9.5 and 8.5 K in single crystal (3.5 and 3.8 K in polycrystal) from the indirect and direct method, respectively. It is found that the discrepancy of ∆T max ad by indirect and direct method is within 10% at stress less than 150 MPa, confirming the feasibility of both methods in evaluating eCE at low stress. However, at higher stress, ∆T max ad obtained from the indirect method is notably larger than that from the direct one. This is mainly attributed to that in the non-isothermal PFM simulations, the relatively large temperature increase at high stress could in turn hamper the austenite-martensite transition and thus finally yield a lower ∆T ad . The results demonstrate the developed PFM herein, combined with both indirect and direct method for eCE calculations, as a practicable toolkit for the computational design of elastocaloric devices. temperature window[10], and excellent coefficient of performance[13,14]. The elastocaloric cooling is realized by means of elastocaloric effect (eCE), which originates from the latent heat associated with martensitic transformation (MT) in shape memory alloys (SMAs). Upon loading, the exothermic austenite-martensite transformation (also called the conventional MT) would cause a temperature increase in the adiabatic process. Upon unloading, the endothermic martensite-austenite transformation (also called the inverse MT) occurs, and a rapid drop in temperature arises[8]. The eCE can be quantified by the adiabatic temperature change (∆T ad ) or the isothermal entropy change (∆S iso ).Modelling eCE plays an important role in the computational design of elastocaloric devices and can be an essential complement to experiments. In general, phenomenological constitutive models and phase-field model[15,16](PFM) are utilized to calculate the eCE in SMAs. The phenomenological Tanaka-type model[11,[17][18][19][20][21]has advantages in the description of temperature change and stress-strain behavior. Very good agreement between the measured and calculated values of ∆T ad is obtained. Analogously, the phase transformation kinetic model[22][23][24]and crystal plasticity-based constitutive model[25,26]are also proposed to predict the eCE. These phenomenological constitutive models can be used to obtain the macroscopic elastocaloric properties, but have difficulties in simulating the spatial and temporal evolution of microstructure details during the MT and thus are hardly applicable to the optimization of eCE by microstructure engineering. eCE in SMAs is intrinsically ascribed to MT, which is a first-order phase transformation and can be induced by external stress/strain field or by temperature field. The PFM developed from Landau's theory of phase transformation is widely used to simulate the stress-and temperature-induced MT[27]. Wang et al.[28] built a three-dimensional PFM of MT, taking into account the transformation-induced strain, which comprehensively described a generic cubic-totetragonal MT. In the PFM, each of the martensitic variants is described by an order parameter η I (I = 1, 2, ..., n with n as the total number of various crystallographically equivalent martensitic variants), and the evolution of each of variants is governed by the Ginzburg-Landau equations. Several studies also considered hexagonal-to-orthorhombic [29], cubic-to-tetragonal [30], and tetragonal-to-monoclinic [31] transitions. Over the last decades, the PFM for MT has been constantly improved to include/study more physical phenomena in SMAs. For example, the martensitic reorientation, the temperature-induced transformation, and the stress-induced MT have been of high interests. constructed a thermodynamically consistent PFM for transformations between austenite and martensitic variants and martensitic reorientation. Since then the phase-field (PF) theory is extended for the cases of surface stresses[35,36], large strain[37,38], and eCE[39][40][41][42]. The research that explores eCE by PFM has recently emerged, owing to the rise of efficient solid-state cooling technology by eCE as well as the PFM's advantage in simulating the evolution of microstructure and temperature during the MT process.In 2015, Levitas et al.[43]proposed a multiphase phase-field theory for temperature-and stress-induced MT, which allows for a presence of the third phase at the interface between the two other phases. Then, Cui et al. [44] employed a non-isothermal PFM to study the stress-and temperature-induced MT, as well as the corresponding latent heat in SMAs. However, the discontinuous piecewise energy barrier between austenite and martensite is adverse to the calculation of ∆S iso or ∆T ad , which possibly makes the Maxwell relations based eCE analysis problematic around the transition point. In real materials, the austenite-martensite transition energy barrier varies continuously and increases with temperature in the whole temperature range. Furthermore, Sun et al.[45,46]modified this energy barrier as a continuous piecewise function of temperature in PFM and discussed the effect of grain size, crystal orientation, and loading rate on ∆T ad . Cissé et al.[40]introduced an exponential function to express the energy barrier and computed qualitatively the eCE in CuAlBe, including ∆T ad calculated by the direct method, coefficient of performance and cyclic deformation. Their energy barrier equations are continuous but not differentiable at the transition point (seeFig. 1for more details). In addition, Xu et al.[47,48]constructed a similar PFM by introducing an extra grain boundary energy to investigate the grain size dependent super-elasticity and eCE using the direct method in nanocrystalline NiTi SMAs. A mesoscale model [49] based on the Müller-Achenbach-Seelecke theory without PF order parameters was also developed to describe the evolution of local temperature and strain in SMA films during elastocaloric cycling. Meanwhile, the latent heat effect[44,46], size effect[45,47,50], plasticity[40,41], and microstructure design[42,48]are considered to regulate eCE. Nevertheless, all the above mentioned works are", 'arxivid': '2303.00309', 'author': ['Wei Tang \nDevices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering\nState Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials\nNanjing University of Aeronautics and Astronautics (NUAA)\n210016NanjingChina\n', 'Qihua Gong \nDevices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering\nState Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials\nNanjing University of Aeronautics and Astronautics (NUAA)\n210016NanjingChina\n\nCollege of Physics\nMIIT Key Lab of Aerospace Information Materials and Physics\nNanjing University of Aeronautics and Astronautics (NUAA)\n211106NanjingChina\n', 'Min Yi \nDevices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering\nState Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials\nNanjing University of Aeronautics and Astronautics (NUAA)\n210016NanjingChina\n', 'Bai-Xiang Xu \nMechanics of Functional Materials Division\nInstitute of Materials Science\nTechnische Universität Darmstadt\n64287DarmstadtGermany\n', 'Long-Qing Chen \nDepartment of Materials Science and Engineering and Materials Research Institute\nThe Pennsylvania State University\n16803PAUSA\n'], 'authoraffiliation': ['Devices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering\nState Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials\nNanjing University of Aeronautics and Astronautics (NUAA)\n210016NanjingChina', 'Devices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering\nState Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials\nNanjing University of Aeronautics and Astronautics (NUAA)\n210016NanjingChina', 'College of Physics\nMIIT Key Lab of Aerospace Information Materials and Physics\nNanjing University of Aeronautics and Astronautics (NUAA)\n211106NanjingChina', 'Devices of Ministry of Education & Institute for Frontier Science & College of Aerospace Engineering\nState Key Laboratory of Mechanics and Control for Aerospace Structures & Key Laboratory for Intelligent Nano Materials\nNanjing University of Aeronautics and Astronautics (NUAA)\n210016NanjingChina', 'Mechanics of Functional Materials Division\nInstitute of Materials Science\nTechnische Universität Darmstadt\n64287DarmstadtGermany', 'Department of Materials Science and Engineering and Materials Research Institute\nThe Pennsylvania State University\n16803PAUSA'], 'corpusid': 257255581, 'doi': '10.2139/ssrn.4399202', 'github_urls': [], 'n_tokens_mistral': 36457, 'n_tokens_neox': 29294, 'n_words': 15960, 'pdfsha': 'aa0884f901b1fdc363b61efd9071ce8a4b81e604', 'pdfurls': ['https://export.arxiv.org/pdf/2303.00309v4.pdf'], 'title': ['Thermodynamically consistent non-isothermal phase-field modelling of elastocaloric effect: indirect vs direct method', 'Thermodynamically consistent non-isothermal phase-field modelling of elastocaloric effect: indirect vs direct method'], 'venue': []}

arxiv

ON THE COMPACTNESS OF WEAK SOLUTIONS TO THE NAVIER-STOKES-KORTEWEG EQUATIONS FOR CAPILLARY FLUIDS 10 Aug 2018 Paolo Antonelli Stefano Spirito ON THE COMPACTNESS OF WEAK SOLUTIONS TO THE NAVIER-STOKES-KORTEWEG EQUATIONS FOR CAPILLARY FLUIDS 10 Aug 2018 In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. In contrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy. Introduction This paper is concerned about the following Navier-Stokes-Korteweg system ∂ t ρ + div(ρu) = 0, ρ ≥ 0, (1.1) ∂ t (ρu) + div(ρu ⊗ u) + ∇ρ γ − 2ν div(ρDu) − 2k 2 ρ∇∆ρ = 0, (1.2) in a three dimensional periodic domain, so that (t, x) ∈ (0, T ) × T 3 . We endow system (1.1)-(1.2) with initial data ρ(0, x) = ρ 0 (x), (ρu)(0, x) = ρ 0 (x)u 0 (x). (1. 3) The positive scalar function ρ represents the density of the fluid and the three dimensional vector field u is the velocity. The positive constants ν and κ, respectively, are the viscosity and the capillarity constants. The aim of this paper is to prove the compactness of solutions to (1.1)-(1.2). More precisely, given a sequence of solutions to (1.1)-(1.2), we show there exists a subsequence converging to a weak solution of the same system. This is one of the key steps in studying the existence of solutions for fluid dynamical systems like (1.1)-(1.2), the other one being the construction of a suitable sequence of approximate solutions. The system (1.1)-(1.2) falls in the class of Navier-Stokes-Korteweg equations, which in their general form read ∂ t ρ + div(ρu) = 0 ∂ t (ρu) + div(ρu ⊗ u) + ∇p = 2ν div S + 2κ 2 div K, (1.4) where S is the viscosity stress tensor given by S = h(ρ) Du + g(ρ) div uI, (1.5) the coefficients h and g satisfying h ≥ 0, h + 3g ≥ 0, and the capillarity term K satisfies div K = ∇ ρ div(k(ρ)∇ρ) − 1 2 (ρk ′ (ρ) − k(ρ))|∇ρ| 2 − div(k(ρ)∇ρ ⊗ ∇ρ). (1.6) The system (1.1)-(1.2) is then obtained from (1.4)-(1.6) by choosing k(ρ) = 1, h(ρ) = ρ and g(ρ) = 0. Systems of Korteweg type arise in modeling several physical phenomena, e.g. capillarity phenomena in fluids with diffuse interface, where the density experiences steep but still smooth change of value. K is called the Korteweg tensor and is derived rigorously from thermodynamic considerations by Dunn and Serrin in [17]. Local existence of smooth solutions and global existence with small data for the system (1.1)-(1.2) have been proved in [22,23]. Regarding the theory of weak solutions few result are available. By exploiting some novel a priori estimates yielded by the so-called Bresch-Desjardins (BD) entropy, [9], in [10] the authors prove the global existence of weak solutions for the system (1.1)-(1.2), by considering test functions of the type ρφ, with φ smooth and compactly supported. This particular notion of weak solutions has the advantage to avoid some mathematical difficulties which arise in the definition of the velocity field in the vacuum region. The result was later extended in [21] to the case of Quantum-Navier-Stokes, namely when we choose k(ρ) = 1/ρ in (1.6). When system (1.1)-(1.2) is augmented by a damping term in the equation for the momentum density, then it is possible to prove the existence of global solutions by using the standard notion of weak solutions [9]. Indeed the presence of the damping term allows to define the velocity field everywhere in the domain. However when dealing with general finite energy weak solutions to (1.1)-(1.2), a major mathematical difficulty arises in defining the velocity field in the vacuum region, due to the degeneracy of the viscosity coefficient h(ρ) = ρ. The momentum density is always well defined, but unfortunately the standard a priori estimates given by the physical energy (and by the BD entropy) do not avoid a possible concentration which would prevent the convergence of the convective term in the compactness argument. Furthermore due to the presence of the capillarity term, a Mellet-Vasseur type estimate [27] does not seem to be available for the system (1.1)-(1.2). This problem was overcome for the quantum case when the viscosity coefficients are chosen to be h(ρ) = ρ and g(ρ) = 0. In [4], by defining a suitable velocity it is possible to consider an alternative formulation of the system where the third order term vanishes, thus allowing the derivation of a Mellet and Vasseur type estimate for the new velocity. Alternatively, in [24] the authors replace the Mellet-Vasseur argument a by truncation method, so that they can recover the necessary compactness. In both the results in [4] and [24] it is crucial that the viscosity and capillarity coefficients satisfy k(ρ) = h ′ (ρ) 2 ρ . (1.7) Note that this relation (1.7) plays a crucial role in the theory, see for example [12] where the authors study the vanishing viscosity limit for the quantum Navier-Stokes equations, or [5] where (1.7) is extensively exploited to construct the approximating system and [8] where numerical methods are performed. We stress that in (1.1)-(1.2) the viscosity and capillarity coefficients do not satisfy the relation (1.7) and hence in this paper we cannot rely on a similar analysis. In order to prove our compactness result, we also exploit a truncation argument. Contrarily to [24], here it is not sufficient to truncate only the velocity field because of the lack of control on the third order term. To overcome this issue we also perform an additional truncation of the density. Unfortunately, this approach is not as straightforward as it would appear at a first glance. Indeed when truncating for example the convective term, some remainders cannot be simply controlled from the a priori estimates. Thus we need to introduce several scales of truncations, in order to control all the error terms. As already remarked, inferring compactness properties for solutions to fluid dynamical systems like (1.1)-(1.2) are only the first step towards an existence result for global in time finite energy weak solutions. Usually this is combined with the construction of a suitable sequence of smooth approximate solutions. Potentially, this latter step could be achieved by considering the following approximating system ∂ t ρ ε + div(ρ ε u ε ) = 0, ∂ t (ρ ε u ε ) + div(ρ ε u ε ⊗ u ε ) − 2ν div(ρ ε Du ε ) + ∇ρ γ 2 ε + ερ ε |u ε | 2 u ε + εu ε = 2κ 2 ρ ε ∇∆ρ ε + 2ε 2 ρ ε ∆ √ ρ ε √ ρ ε . and by adapting, probably in a non trivial way, the regularisation procedure in [24] in order to rigorously derive the truncated formulation of the momentum equations. On the other hand, providing a smooth approximating system as in [5] seems to be very challenging due to the the very rigid structure of the approximation procedure. We plan to attack this problem in future works. We conclude this introduction by describing the state of art of the analysis of the Cauchy problem for the general system (1.4)-(1.6). In the case κ = 0 (1.4) reduces to the system of compressible Navier-Stokes equations. When the viscosity coefficient h(ρ) is chosen degenerating on the vacuum region {ρ = 0} the Lions-Feireisl theory, [26], [18], and the recent approach in [11] cannot be used because it is not possible to define the velocity in the vacuum regions. The global existence of weak solutions has been proved independently in [28] and [25] in the case h(ρ) = ρ and g(ρ) = 0. In both cases, non trivial approximation procedures are required to prove the BD entropy and the Mellet and Vasseur inequality. When the viscosity ν = 0, the system (1.4) is called Euler-Korteweg and it has been also extensively studied. In [7] local well-posedness has been proved, while in [6] the global existence of smooth solutions with small data has been proved. Moreover, when k(ρ) = 1/ρ the system (1.4) is called Quantum Hydrodynamic system (QHD) and arises for example in the description of quantum fluids. The global existence of finite energy weak solutions for the QHD system has been proved in [2,3] without restrictions on the regularity or the size of the initial data. Non uniqueness results by using convex integration methods has been proved in [14]. Moreover, relative entropy methods to study singular limits for the equations (1.4)-(1.6) have been exploited in [12,14,20,16], in particular we mention the incompressible limit in [1] in the quantum case, the quasineutral limit [15] for the constant capillarity case and the vanishing viscosity limit in [12]. Finally, the analysis of the long time behaviour for the isothermal Quantum-Navier-Stokes equations has been performed in [13]. Organization of the paper. The paper is organized as follows. In Section 2 we fix the notations and give the precise definition of weak solutions of (1.1)-(1.2). In Section 3 we recall the formal a priori estimates for solutions of the system (1.1)-(1.2), namely the energy estimate and the BD entropy. Finally, in the Section 4 we prove Theorem 2.2. Preliminaries Notations. Given Ω ⊂ R 3 , the space of compactly supported smooth functions with value in R d will be D((0, T ) × Ω; R d ). We will denote with L p (Ω) the standard Lebesgue spaces and with · L p their norm. The Sobolev space of L p functions with k distributional derivatives in L p is W k,p (Ω) and in the case p = 2 we will write H k (Ω). The spaces W −k,p (Ω) and H −k (Ω) denote the dual spaces of W k,p ′ (Ω) and H k (Ω) where p ′ is the Hölder conjugate of p. Given a Banach space X we use the the classical Bochner space for time dependent functions with value in X, namely L p (0, T ; X), W k,p (0, T ; X) and W −k,p (0, T ; X) and when X = L p (Ω), the norm of the space L q (0, T ; L p (Ω)) is denoted by · L q t (L p x ) . We denote by Du = (∇u + (∇u) T )/2 the symmetric part of the gradient and by Au = (∇u − (∇u) T )/2 the antisymmetric one. Finally, given a matrix M ∈ R 3×3 we denote by Symm M , the symmetric part of M and by Asymm M the antisymmetric one. (1) Integrability conditions. ρ ∈ L ∞ (0, T ; H 1 (T 3 )) ∩ L 2 (0, T ; H 2 (T 3 )), √ ρ u ∈ L ∞ (0, T ; L 2 (T 3 )), ρ γ 2 ∈ L ∞ (0, T ; L 2 (T 3 )) ∩ L 2 (0, T ; H 1 (T 3 )), ∇ √ ρ ∈ L ∞ (0, T ; L 2 (T 3 )). (2) Equations. For any φ ∈ C ∞ c ([0, T ); C ∞ (T 3 ); R). ρ 0 φ(0) dx + ρφ t + √ ρ √ ρu∇φ dxdt = 0. For any fixed l = 1, 2, 3 and ψ ∈ C ∞ c ([0, T ); C ∞ (T 3 ); R) ρ 0 u 0,l ψ(0) dx + √ ρ( √ ρu l )ψ t dxdt + √ ρu √ ρu l : ∇ψ dxdt + ν √ ρ √ ρu l ∆ψ dxdt + ν √ ρ √ ρu∇∇ l ψ dxdt + 2ν ∇ l √ ρ √ ρ u∇ψ dxdt 2ν √ ρu l ∇ √ ρ∇ψ dxdt − 2 ∇ρ γ 2 ρ γ 2 · ψ dxdt − 2κ 2 ∇ l ρ∆ρψ dxdt − 2κ 2 ρ∆ρ∇ l ψ dxdt = 0. (3) Energy Inequality. There exist S ∈ L 2 ((0, T )×T 3 ) such that √ ρS = Symm(∇(ρu))−2∇ √ ρ⊗ √ ρu) in D ′ and Λ such that ρ u = √ ρΛ satisfying the following energy inequality sup t∈(0,T ) T 3 |Λ(t, x)| 2 2 + ρ(t, x) γ γ − 1 + κ 2 |∇ρ(t, x)| 2 dx + |S(s, x)| 2 dxds ≤ T 3 ρ 0 (x)|u 0 (x)| 2 + ρ 0 (x) γ γ − 1 + κ 2 |∇ρ 0 (x)| 2 dx. (4) BD Entropy. There exists A ∈ L 2 ((0, T ) × T 3 ) such that √ ρA = Asymm(∇(ρu)) − 2∇ √ ρ ⊗ √ ρu) in D ′ such that sup t∈(0,T ) T 3 |Λ(t, x) + 2ν∇ √ ρ(t, x)| 2 2 + ρ(t, x) γ γ − 1 + κ 2 |∇ρ(t, x)| 2 dx + |A(s, x)| 2 dxds + 8ν γ |∇ρ γ 2 (s, x)| 2 dxds + 4κ 2 ν |∆ρ(s, x)| 2 dxds ≤ T 3 | ρ 0 (x)u 0 (x) + 2ν∇ ρ 0 (x)| 2 2 + ρ 0 (x) γ γ − 1 + κ 2 |∇ρ 0 (x)| 2 dx. In order to state our main result, we first specify the assumptions on the initial data. We consider {ρ 0 n } n being a sequence of smooth and strictly positive functions and ρ 0 be a strictly positive function such that ρ 0 n > 0, ρ 0 n → ρ 0 strongly in L 1 (T 3 ), {ρ 0 n } n is uniformly in bounded in L 1 ∩ L γ (T 3 ), {∇ ρ 0 n } n is uniformly bounded in L 2 (T 3 ),(2.1) Regarding the initial velocity, let {u n 0 } be a sequence of smooth vector fields and u 0 be a smooth vector field such that { ρ 0 n u 0 n } is uniformly bounded in L 2 (T 3 ), ρ 0 n u 0 n → ρ 0 u 0 in L p (R 3 ) with p < 2. (2. 2) The main theorem of this paper is the following. Remark 2.4. We stress that (2.3) do not implies the convergence of the convective term, which comes from the truncation arguments. Remark 2.5. The notion of weak solution in Definition 2.1 is weaker compared with the one in the quantum case in [4,5]. Indeed, in [4,5] it can be proved that Λ = √ ρ u because √ ρ n u n → √ ρ u strongly in L 2 ((0, T ) × T 3 ). As a consequence the energy inequality and the entire weak formulation can be written only in terms of ρ, Λ, S and A. On the contrary, in the proof of Theorem 2.2, we are not able to prove that Λ = √ ρ u, but only that √ ρΛ = ρ u. Indeed, it is not clear whether √ ρ n u n ⇀ √ ρ u weakly in L 2 ((0, T ) × T 3 ), since we do not know that Λ = 0 on {ρ = 0}. A priori estimates In this section we recall the two formal a priori estimates available for solutions of (1.1)-(1.2). The first lemma is the basic energy estimate for the system (1.1)-(1.2). ρ n |u n | 2 2 + ρ γ n γ − 1 + κ 2 |∇ρ n | 2 dx + 2ν ρ n |Du n | 2 dxdt = ρ 0 n |u 0 n | 2 2 + ρ 0 n γ γ − 1 + κ 2 |∇ρ 0 n | 2 dx. (3.1) The second main a priori estimates is the so-called BD entropy. Although this estimate is well-known, see [9], we give a sketch of the proof for completeness. ρ n |w n | 2 2 + ρ γ n γ − 1 + κ 2 |∇ρ n | 2 dx + 8ν γ |∇ρ γ 2 n | 2 dx + 2ν ρ n |Au n | 2 dx + 4κ 2 ν |∆ρ n | 2 dx = ρ 0 n |w 0 n | 2 2 + ρ 0 n γ γ − 1 + κ 2 |∇ρ 0 n | 2 dx. (3.2) Proof. We first perform the effective velocity transformation. Let c ∈ R to be chosen later. Let us consider w n = u n + c∇ log ρ n . Then, ∂ t ρ n + div(ρ n w n ) = ∂ t ρ n + div(ρ n (u n + c∇ log ρ n )) = c∆ρ n . We recall the following elementary identities, c(ρ n ∇ log ρ n ) t = −c∇ div(ρ n u n ), c div(ρ n u n ⊗ ∇ log ρ n + ρ n ∇ log ρ n ⊗ u n ) = c∆(ρ n u n ) − 2c div(ρ n Du n ) + c∇ div(ρ n u n ), c 2 div(ρ n ∇ log ρ n ⊗ ∇ log ρ n ) = c 2 ∆(ρ n ∇ log ρ n ) − c 2 div(ρ n ∇ 2 log ρ n ). By using these identities it is easy to prove that ∂ t (ρ n w n ) + div(ρ n w n ⊗ w n ) + ∇ρ γ n − c∆(ρ n w n ) = 2(ν −c) div(ρ n Dw n ) − (c 2 +2(ν −c)c) div(ρ n ∇ 2 log ρ n ) + 2κ 2 ρ n ∇∆ρ n . Then, by choosing c = 2ν we obtain the following system ∂ t ρ n + div(ρ n w n ) = 2ν∆ρ n , 3) ∂ t (ρ n w n ) + div(ρ n w n ⊗ w n ) + ∇ρ γ n − 2ν∆(ρ n w n ) + 2ν div(ρ n Dw n ) = 2κ 2 ρ n ∇∆ρ n . (3.4) The BD Entropy (3.2) is nothing else than the energy estimate associated with the system (3.3)-(3.4). By multiplying (3.4) by w n , by integrating in space and by using (3.3) we get d dt ρ n |w n | 2 2 dx + ∇ρ γ n w n dx + 2ν ρ n |Au n | 2 dx − 2κ 2 ρ n ∇∆ρ n w n . Finally, by multiplying (3.3) by −2κ 2 ∆ρ n we have d dt κ 2 |∇ρ n | 2 dx + 4νκ 2 |∆ρ n | 2 dx − 2κ 2 div(ρ n w n )∆ρ n = 0. (3.7) By summing up (3.5), (3.6) and (3.7) and integrating by parts we get (3.2). Compactness In this Section we are going to prove the main result of our paper. Bounds independent on n. First of all we collect the a priori bounds we can deduce from the Proposition 3.1 and Proposition 3.2. By the energy estimates in Proposition 3.1 and the assumptions (2.1), (2.2) we have the following uniform bounds. √ ρ n u n L ∞ t L 2 x ≤ C, ∇ρ n L ∞ t L 2 x ≤ C ρ n L ∞ t (L 1 x ∩L γ x ) ≤ C, √ ρ n Du n L 2 t,x ≤ C (4.1) The uniform bounds obtained by the BD Entropy, Proposition 3.2, are the following √ ρ n w n L ∞ t L 2 x ≤ C, √ ρ n Au n L 2 t,x ≤ C ∇ρ γ/2 n L 2 t,x ≤ C, ∆ρ n L 2 t,x ≤ C. Combining some of the bounds in (4.1) and in (4.2) we obtain the following bounds ∇ √ ρ n L ∞ t L 2 x ≤ C, √ ρ n ∇u n L 2 t,x ≤ C. (4.3) Of course, additional bounds can be easily obtained by interpolation and Sobolev embeddings. Here we list only the ones will be used in the sequel. By Sobolev embeddings and interpolation inequalities we get ρ n L 2 t L ∞ x ≤ C, ∇ρ n L 10 3 t,x ≤ C, ρ γ 2 n L 10 3 t,x ≤ C. (4.4) By using (4.1), (4.2), (4.4) and Hölder inequality we have ρ n u n L 2 t,x ≤ C, ∇(ρ n u n ) L 2 t (L 1 x ) ≤ C. (4.5) Finally, by using the continuity equation (1.1) we have that ∂ t ρ n L 2 t L 1 x ≤ C. (4.6) 4.2. Convergence Lemma. By using the above uniform bounds we are now able to prove the following convergences. (1) Up to subsequences there exist, ρ, m, S, A and Λ such that ρ n → ρ strongly in L 2 (0, T ; H 1 (T 3 )), (4.7) ρ n u n → m strongly in L p (0, T ; L p (T 3 )) with p ∈ [1, 2), (4.8) √ ρ n D(u n ) ⇀ S weakly in L 2 ((0, T ) × T 3 ), (4.9) √ ρ n A(u n ) ⇀ A weakly in L 2 ((0, T ) × T 3 ), (4.10) √ ρ n u n * ⇀ Λ weakly* in L ∞ (0, T ; L 2 (T 3 )). (4.11) Moreover, Λ is such that √ ρΛ = m. (2) The following additional convergences hold for the density Then, since {ρ n } n is uniformly bounded in L 2 (0, T ; H 2 (T 3 )), by using Aubin-Lions Lemma we get (4.7). Next, by using the momentum equations and the bounds (4.1)-(4.4), it is easy to prove that {∂ t (ρ n u n )} n is uniformly bounded in L 2 (0, T ; W −2, 3 2 (T 3 )). Then, by using (4.4), (4.5) and Aubin-Lions Lemma, (4.8) follows. The convergences (4.9), (4.10) and (4.11) follow by standard weak compactness theorems and the equality √ ρΛ = m follows easily from (4.7) and (4.11). Next, the convergences (4.12), (4.13) follow from the the uniform bounds (4.1)-(4.3) and standard weak compactness arguments. Finally, The convergence (4.14) is easily obtained by using (4.7) and the bound (4.3), the convergence (4.15) follows by (4.2) and (4.7). ∇ √ ρ n ⇀ ∇ √ ρ weakly in L 2 ((0, T ) × T 3 ), Lemma 4.2. Let f ∈ C ∩ L ∞ (R 3 ; R) and (ρ n , u n ) be a solution of (1.1)-(1.2) and let u be defined as follows: u = m(t,x) ρ(t,x) = Λ(t,x) √ ρ(t,x) (t, x) ∈ {ρ > 0}, 0 (t, x) ∈ {ρ = 0}. (4.16) Then, the following convergences hold. ρ n f (u n ) → ρ f (u) strongly in L p ((0, T ) × T 3 ) for any p < 6, (4.17) ∇ρ n f (u n ) → ∇ρ f (u) strongly in L p ((0, T ) × T 3 ) for any p < 10 3 , (4.18) ρ n u n f (u n ) → ρu f (u) strongly in L p ((0, T ) × T 3 ) for any p < 2, (4.19) ρ γ 2 n f (u n ) → ρ γ 2 f (u) strongly in L p ((0, T ) × T 3 ) for any p < 10 3 . (4.20) Proof. We first first note that, up to a subsequence non relabelled, (4.7) and (4.8) imply that ρ n → ρ a.e. in (0, T ) × T 3 , ρ n u n → m a.e. in (0, T ) × T 3 , ∇ρ n → ∇ρ a.e. in (0, T ) × T 3 . suppβ ⊂ (−2, 2) and 0 ≤β ≤ 1. Givenβ, we defineβ : R → R as follows: β(z) = z 0β (s) ds. For y ∈ R 3 we define for any δ > 0 the functions β 1 δ (y) := 1 δβ (δ y 1 )β(δ y 2 )β(δ y 3 ), β 2 δ (y) := 1 δβ (δ y 1 )β(δ y 2 )β(δ y 3 ), β 2 δ (y) := 1 δβ (δ y 1 )β(δ y 2 )β(δ y 3 ). Note that for fixed l = 1, 2, 3 the function β l δ : R 3 → R is a truncation of the function f (y) = y l . Finally, for any δ > 0 we defineβ δ : R 3 → R aŝ β δ (y) :=β δ (δ y 1 )β δ (δ y 2 )β δ (δ y 3 ), and for any λ > 0 we defineβ λ : R → R as β λ (s) =β(λ s). In the next Lemma we collect some of the main properties of β l δ ,β δ andβ λ . Those properties are elementary and can be deduced directly from the definitions. (1) For any δ > 0 and l = 1, 2, 3 β l δ L ∞ ≤ C δ , ∇β l δ L ∞ ≤ C, ∇ 2 β l δ L ∞ ≤ C δ,(4.23) (2) For any λ > 0 β λ L ∞ ≤ 1, β ′ λ L ∞ ≤ C λ, |s|β λ (s) ≤ C √ λ . (4.24) (3) For any δ > 0 β δ L ∞ ≤ 1, ∇β δ L ∞ ≤ Cδ, |y||β δ (y)| ≤ C δ , (4.25)(4) The following convergences hold for l = 1, 2, 3, pointwise on R 3 , as δ → 0 β l δ (y) → y l , (∇ y β l δ )(y) → ∇ y l y,β δ (y) → 1. By using (4.7), (4.8) and (2.1) is straightforward to prove that ρ 0 n φ(0, x) + ρ n φ t dxdt + ρ n u n ∇φ dxdt converges to ρ 0 φ(0, x) + ρφ t dxdt + ρ u∇φ dxdt, for any φ ∈ C ∞ c ([0, T ) × T 3 ) . Let us consider the momentum equations. Let l ∈ {1, 2, 3} fixed. By multiplying (1.2) by ∇ y β l δ (u n ) and by using the continuity equation (1.1) we have that ∂ t (ρ n β l δ (u n )) + div(ρ n u n β l δ (u n )) − 2ν div(ρ n D(u n ))∇ y β l δ (u n ) + ∇ρ γ n ∇ y β l δ (u n ) − 2κ 2 ρ n ∇∆ρ n ∇ y β l δ (u n ) = 0. (4.28) Let ψ ∈ C ∞ c ([0, T ) × T 3 ; R) , by multiplying (4.28) byβ λ (ρ n )ψ and integrating by parts we get ρ 0 n β l δ (u 0 n )β λ (ρ 0 n )ψ(0, x) dx + ρ n β l δ (u n )β λ (ρ n )∂ t ψ − ρ n u n β l δ (u n )β λ (ρ n ) · ∇ψ dxdt − 2ν √ ρ n Du n : √ ρ n ∇ y β l δ (u n )β λ (ρ n ) ⊗ ∇ψ dxdt − 2 ρ γ 2 n ∇ρ γ 2 n · ∇ y β l δ (u n )β λ (ρ n )ψ dxdt − 2κ 2 ∇ρ n ∆ρ n ∇ y β l δ (u n )β λ (ρ n )ψ dxdt − 2κ 2 ρ n ∆ρ n ∇ y β l δ (u n )β λ (ρ n )∇ψ dxdt + R δ,λ n ψ dxdt = 0. (4.29) where the remainder is R δ,λ n = 6 i=1 R δ,λ n,i = ρ n β l δ (u n )β ′ λ (ρ n )∂ t ρ n + ρ n u n β l δ (u n )β ′ λ (ρ n )∇ρ n − 2ν √ ρ n Du n : √ ρ n ∇ y β l δ (u n ) ⊗ ∇ρ nβ ′ λ (ρ n ) + 2κ 2 ρ n ∆ρ n ∇ 2 y β l δ (u n ) : ∇u nβλ (ρ n ) + 2κ 2 ρ n ∆ρ n ∇ y β l δ (u n )β ′ λ (ρ n )∇ρ n − 2νρ n Du n ∇ 2 y β l δ (u n )∇u nβλ (ρ n ). (4.30) We first perform the limit as n goes to ∞ for δ and λ fixed. Notice that, sinceβ λ ∈ L ∞ (R), and {ρ n } n converges almost everywhere, we have that β λ (ρ n ) →β λ (ρ) strongly in L q ((0, T ) × T 3 ) for any q < ∞. By using (4.17) with p = 2 and choosing q = 2 in (4.31) we have that ρ n β l δ (u n )β λ (ρ n )∂ t ψ dxdt → ρβ l δ (u)β λ (ρ)∂ t ψ dxdt. Next, by (4.19) with p = 3/2 and choosing q = 3 in (4.31) we get ρ n u n β l δ (u n )β λ (ρ n ) · ∇ψ dxdt → ρ uβ l δ (u)β λ (ρ) · ∇ψ dxdt. By using (4.9), (4.17) with p = 4 and (4.31) with q = 4 it follows √ ρ n Du n : √ ρ n ∇ y β l δ (u n )β λ (ρ n ) ⊗ ∇ψ dxdt → ρ S : ∇ y β l δ (u)β λ (ρ) ⊗ ∇ψ dxdt. By using (4.15), (4.20) with p = 3 and (4.31) with q = 6 it follows ρ γ 2 n ∇ρ γ 2 n · ∇ y β l δ (u n )β λ (ρ n )ψ dxdt → ρ γ 2 ∇ρ γ 2 · ∇ y β l δ (u)β λ (ρ)ψ dxdt. By using (4.13), (4.18) with p = 3 and (4.31) with q = 6 it follows ∇ρ n ∆ρ n ∇ y β l δ (u n )β λ (ρ n )ψ dxdt → ∇ρ∆ρ∇ y β l δ (u)β λ (ρ)ψ dxdt. Next, by using (4.13), (4.17) with p = 3 and (4.31) with q = 6 it follows ρ n ∆ρ n ∇ y β l δ (u n )β λ (ρ n )∇ψ dxdt → ρ∆ρ∇ y β l δ (u)β λ (ρ)∇ψ dxdt. Finally, by using (2.1) the convergence of the term involving the initial data can be easily proved. It remains to study the remainder R δ,λ n . We claim that there exists a C > 0 independent on n, δ and λ such that R δ,λ n L 1 t,x ≤ C δ √ λ + λ δ + λ + δ . (4.32) In order to prove (4.32) we estimate all the terms in (4.30) separately. By using (4.4), (4.6), (4.23) and (4.24) we have R δ,λ n,1 L 1 t,x ≤ ρ n L 2 (L ∞ ) ∂ t ρ n L 2 (L 1 ) β l δ (u n ) L ∞ t,x β ′ λ (ρ n ) L ∞ t,x ≤ C λ δ . By using (4.1), (4.4), (4.23) and (4.24) it holds R δ,λ n,2 L 1 t,x ≤ ρ n u n L 2 t,x ∇ρ n L 2 t,x β l δ (u n ) L ∞ t,x β ′ λ (ρ n ) L ∞ t,x ≤ C λ δ . By using (4.1), (4.4), (4.23) and (4.24) we get R δ,λ n,3 L 1 t,x ≤ ρ n L 2 (L ∞ ) √ ρ n Du n L 2 t,x ∇ρ n L ∞ (L 2 ) ∇ y β l δ (u n ) L ∞ t,x β ′ λ (ρ n ) L ∞ t,x ≤ Cλ. By using (4.1), (4.2), (4.23) and (4.24) we have that R δ,λ n,4 L 1 t,x ≤ ∆ρ n L 2 t,x √ ρ n Du n L t,x ∇ 2 y β l δ (u n ) L ∞ t,x √ ρ nβλ (ρ n ) L ∞ t,x ≤ C δ √ λ . By using R δ,λ n,5 L 1 t,x ≤ ρ n L 2 (L ∞ ) ∆ρ n L 2 t,x ∇ρ n L ∞ (L 2 ) ∇ y β l δ (u n ) L ∞ t,x β ′ λ (ρ n ) L ∞ t,x ≤ Cλ. Finally, by using (4.1), (4.23) and (4.24) we have R δ,λ n,6 L 1 t,x ≤ ρ n ∇u n 2 L 2 t,x ∇ 2 y β l δ (u n ) L ∞ t,x β λ (ρ n ) L ∞ t.x ≤ Cδ. where the remainder is R δ n,j = ρ n u l n ∇ y kβ δ (u n )∇ j u k n + ρ n u j n ∇ y kβ δ (u n )∇ l u k n . (4.36) For fixed δ, by using the convergence (4.9) and (4.17) with p = 4, we have that 2 √ ρ nβδ (u n ) √ ρ n (D(u n )) l,j ∂ j φ dxdt → 2 √ ρS l,jβδ (u)∂ j φ dxdt Next, we have that ρ n u l nβδ (u n )∆φ dxdt → ρ u lβ δ (u)∆φ dxdt ρ n u j nβδ (u n )∇ 2 j,l φ dxdt → ρ u jβ δ (u)∇ 2 j,l φ dxdt because of (4.19) with p = 1. By using (4.25), (4.17) with p = 2 and the weak convergence of ∇ √ ρ n in L 2 t,x we get ∇ l √ ρ n √ ρ n u nβδ (u n )∇φ dxdt → ∇ l √ ρ √ ρ uβ δ (u)∇φ dxdt ∇ √ ρ n √ ρ n u l nβδ (u n )∇φ dxdt → ∇ √ ρ √ ρ u lβ δ (u))∇φ dxdt Finally, by using (4.1), (4.2) and (4.25) we have that R δ n L 1 t,x ≤ C √ ρ n L ∞ (L 2 t,x √ ρ n D(u n ) L 2 t,x ∇ yβδ (u n ) L ∞ t, x ≤ Cδ, and then there exists a measureμ δ such that R δ n · ∇φ dxdt → μ δ , ∇ψ , (4.37) and its total variation satisfies |μ δ |(T 3 ) ≤ Cδ. Collecting the previous convergences, we have 2 √ ρS l,jβδ (u)∇ j φ dxdt = − ρ u lβ δ (u)∆φ dxdt − ρ u jβ δ (u)∇ 2 j,l φ dxdt − 2 ∇ l √ ρ √ ρ uβ δ (u)∇φ dxdt − 2 ∇ √ ρ √ ρ u lβ δ (u))∇φ dxdt − μ δ , ∇ψ . Finally, by using (4.26), Dominated Convergence Theorem and (4.37) we get that 2 √ ρS l,j ∇ j φ dxdt = − ρ u l ∆φ dxdt − ρ u j ∇ 2 j,l φ dxdt − 2 ∇ l √ ρ √ ρ u∇φ dxdt − 2 ∇ √ ρ √ ρ u l ∇φ dxdt. By the very same arguments we identify also the tensor A. Finally, the energy inequality and the BD Entropy follow from the lower semicontinuity of the norms. 2. 2 . 2Definition of weak solutions and statement of the main result. The definition of weak solution for the system (1.1)-(1.2) is the following Definition 2.1. A pair (ρ, u) with ρ ≥ 0 is said to be a weak solution of the Cauchy problem (1.1)-(1.2)-(1.3) if the following conditions are satisfied. Theorem 2. 2 . 2Assume {ρ 0 n } n and {ρ 0 n u 0 n } n are sequences of initial data for (1.1)-(1.2) satisfying (2.1) and (2.2). Let {(ρ n , u n )} n with ρ n > 0 be a sequence of smooth solutions of (1.1)-(1.2) with initial data {ρ 0 n } n and {ρ 0 n u 0 n } n , then, up to subsequences not relabelled, there exist (ρ, u) such that ρ n → ρ strongly in L 2 ((0, T ); H 1 (T 3 )), ρ n u n → ρ u strongly in L p ((0, T ) × T 3 ) for any p < 2, (2.3) and (ρ, u) is a weak solutions of (1.1)-(1.2)-(1.3) in the sense of Definition 2.1. Remark 2.3. We stress that the velocity field is not uniquely defined on the vacuum region {ρ = 0}. Proposition 3. 1 . 1Let (ρ n , u n ) be a smooth solution of (1.1)-(1.2), then sup t∈(0,T ) Proposition 3. 2 . 2Let (ρ n , u n ) be a smooth solution of (1.1)-(1.2). Then, w n = u n + 2ν∇ρ n and ρ n satisfy sup t∈(0,T ) Lemma 4. 1 . 1Let {(ρ n , u n )} n be a sequence of solutions of (1.1)-(1.2). n ⇀ ∆ρ weakly in L 2 ((0, T ) × T 3 ), (4.13) ρ γ n → ρ γ strongly in L 1 ((0, T ) × T 3 in L 2 ((0, T ) × T 3 ). (4.15)Proof. By using (1.1) and (4.4), we have that {∂ t ρ n } n is uniformly bounded in L 2 (0, T ; H −1 (T 3 )). that m = 0 on {ρ = 0} and √ ρ u ∈ L ∞ (0, T ; L 2 (T 3 )).Moreover, m = ρ u = √ ρΛ. Let us prove (4.17). On {ρ > 0} by using (4.21) we have thatρ n f (u n ) → ρ f (u) a.e. in {ρ > 0}.On the other hand, since f ∈ L ∞ (R 3 ; R) we have|ρ n f (u n )| ≤ |ρ n | f ∞ → 0 a.e. in {ρ = 0}.Then, ρ n f (u n ) → ρ f (u) a.e. in (0, T ) × T 3 and the convergence in (4.17) follows by the uniform bound ρ n L 6 t,x ≤ C. Regarding(4.18), from Lemma 4.1 we have that ρ is a Sobolev function, then ∇ρ = 0 a.e. in {ρ = 0}. From (4.21) we have that ∇ρ n f (u n ) → ∇ρ f (u) a.e. in {ρ > 0} |∇ρ n f (u n )| ≤ |∇ρ n | f ∞ → 0 a.e. in {ρ = 0}. Then, ∇ρ n f (u n ) → ∇ρ f (u) a.e. in (0, T ) × T 3 and (4.18) follows from the uniform bound (4.4). Concerning (4.19), again (4.21) implies the following convergences ρ n u n f (u n ) → m f (u) a.e. in {ρ > 0}, |ρ n u n f (u n )| ≤ |ρ n u n | f ∞ → 0 a.e. in {ρ = 0}, which, together with (4.4), imply (4.19). Finally, (4.20) follows by the same arguments used to prove (4.17) and the uniform bounds on the pressure in (4.1) and (4.2). 4.3. The Truncations. Letβ : R → R be an even positive compactly Lemma 4 . 3 . 43Let λ, δ > 0 and K := β W 2,∞ . Then, there exists C = C(K) such that the following bounds hold. ( 5 ). 5The following convergence holds pointwise on R as λ → 0 β λ (s) Proof of the main Theorem. We are now ready to prove Theorem 2.2.Proof of Theorem 2.2. Let (ρ n , u n ) be a solution of (1.1)-(1.2). By Lemma 4.1 there exist ρ, m, Λ such that the convergences (4.7), (4.8) and (4.11) hold. Moreover, by defining the velocity u as in Lemma 4.2 we have that √ ρ u ∈ L ∞ (0, T ; L 2 (T 3 ), m = √ ρΛ = ρ u. Then, (4.32) is proved and, when n goes to infinity, we have that (ρ, u) satisfies the following integral equalitywhere µ δ,λ is a measure such that R δ,λ n → µ δ,λ in M(T 3 ; R) and its total variations satisfiesand by (4.26), (4.27) and the Lebesgue Dominated Convergence Theorem we have that (4.33) converge toNext we need to identify the tensor S. Let φ ∈ C ∞ c ([0, T ) × T 3 ; R) and l = 1, 2, 3 fixed. 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SpiritoSasso Science Institute, Viale Francesco Crispi 7, 67100, L'Aquila, Italy E-mail addressGSSI -Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100, L'Aquila, Italy E-mail address: [email protected] (S. Spirito) address: [email protected] di Ingegneria e Science dell'Informazione e Matematica. Via Vetoio, 67100, L'Aquila, Italy E-mailDISIM-Dipartimento di Ingegneria e Science dell'Informazione e Matematica, Via Vetoio, 67100, L'Aquila, Italy E-mail address: [email protected]

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{'abstract': 'In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. In contrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy.', 'arxivid': '1808.03495', 'author': ['Paolo Antonelli ', 'Stefano Spirito '], 'authoraffiliation': [], 'corpusid': 52576884, 'doi': '10.1016/j.na.2019.03.020', 'github_urls': [], 'n_tokens_mistral': 15895, 'n_tokens_neox': 13849, 'n_words': 7694, 'pdfsha': '09c1b37a0a812e7e08415a98851fc27a626de4d0', 'pdfurls': ['https://arxiv.org/pdf/1808.03495v1.pdf'], 'title': ['ON THE COMPACTNESS OF WEAK SOLUTIONS TO THE NAVIER-STOKES-KORTEWEG EQUATIONS FOR CAPILLARY FLUIDS', 'ON THE COMPACTNESS OF WEAK SOLUTIONS TO THE NAVIER-STOKES-KORTEWEG EQUATIONS FOR CAPILLARY FLUIDS'], 'venue': []}

arxiv

Memory Based Online Learning of Deep Representations from Video Streams Federico Pernici MICC -Media Integration Communcation Center University Of Florence Federico Bartoli MICC -Media Integration Communcation Center University Of Florence Matteo Bruni MICC -Media Integration Communcation Center University Of Florence Alberto Del Bimbo MICC -Media Integration Communcation Center University Of Florence Memory Based Online Learning of Deep Representations from Video Streams We present a novel online unsupervised method for face identity learning from video streams. The method exploits deep face descriptors together with a memory based learning mechanism that takes advantage of the temporal coherence of visual data. Specifically, we introduce a discriminative feature matching solution based on Reverse Nearest Neighbour and a feature forgetting strategy that detect redundant features and discard them appropriately while time progresses. It is shown that the proposed learning procedure is asymptotically stable and can be effectively used in relevant applications like multiple face identification and tracking from unconstrained video streams. Experimental results show that the proposed method achieves comparable results in the task of multiple face tracking and better performance in face identification with offline approaches exploiting future information. Code will be publicly available. Introduction Visual data is massive and is growing faster than our ability to store and index it, nurtured by the diffusion and widespread use of social platforms. Their fundamental role in advancing object representation, object recognition and scene classification research have been undoubtedly assessed by the achievements of Deep Learning [1]. However, the cost of supervision remains the most critical factor for the applicability of such learning methods as linear improvements in performance require an exponential number of labelled examples [2]. Efforts to collect large quantities of annotated images, such as ImageNet [3] and Microsoft coco [4], while having an important role in advancing object recognition, don't have the necessary scalability and are hard to be extended, replicated or improved. They may also impose a ceiling on the performance of systems trained in this manner. Semi or unsupervised Deep Learning from image data still remains hard to achieve. An attracting alternative would be to learn the object appearance from video streams with no supervision, both exploiting the large quantity of video available in the Internet and the fact that adjacent video frames contain seman- tically similar information. This provides a variety of conditions in which an object can be framed, and therefore a comprehensive representation of its appearance can be obtained. Accordingly, tracking a subject in the video could, at least in principle, support a sort of unsupervised incremental learning of its appearance. This would avoid or reduce the cost of annotation as time itself would provide a form of self-supervision. However, this solution is not exempt of problems [5]. On the one hand, parameter re-learning of Deep Networks, to adequately incorporate the new information without catastrophic interference, is still an open challenge [6,7], especially when re-learning should be done in real time while tracking, without the availability of labels and with data coming from a stream which is often nonstationary. On the other hand, classic object tracking [8] has substantially divergent goals from continuous incremental learning. While in tracking the object appearance is learned only for detecting the object in the next frame (the past information is gradually forgotten), continuous incremental learning would require that all the past visual information of the object observed so far is collected in a comprehensive and cumulative representation. This requires that tracking does not drift in the presence of occlusions or appearance changes and that incremental learning should be asymptotically stable in order to converge to an univocal representation. In this paper, we present a novel online unsupervised method for face identity learning from unconstrained video streams. The method exploits CNN based face detectors and descriptors together with a novel incremental memory based learning mechanism that collects descriptors and distills them based on their redundancy with respect to the current representation. This allows building a sufficiently compact and complete appearance representation of the individual identities as time advances (Fig. 1). While we obtained comparable results with offline approaches exploiting future information in the task of multiple face tracking, our model is able to achieve better performance in face identification from unconstrained video. In addition to this, it is shown that the proposed learning procedure is asymptotically stable and the experimental evaluation confirms the theoretical result. In the following, in Section 2, we cite a few works that have been of inspiration for our work. In Section 3 we highlight our contributions, in Section 4 we expounded the approach in detail and finally, in Section 5, experimental results are given. Related Work Memory Based Learning: Inclusion of a memory mechanism in learning [9] is a key feature of our approach. On domains that have temporal coherence like Reinforcement Learning (RL) memory is used to store the past experience with some priority and to sample mini-batches to perform incremental learning [10] [11]. This makes it possible to break the temporal correlations by mixing more and less recent experiences. More recently, Neural Turing Machine architectures have been proposed in [12,13] and [14] that implement an augmented memory to quickly encode and retrieve new information. These architectures have the ability to rapidly bind never-before-seen information after a single presentation via an external memory module. However, in these cases, training data are still provided supervisedly and the methods don't scale with massive video streams. Open Set: In addition to the incremental learning procedure, the system needs to have the capability to discriminate between already known and unknown classes (open set) [15]. The open set classification is a problem of balancing known space (specialization) and unknown open space (generalization) according to the class rejection option. Formalization for open space risk is considered as the relative measure of open space compared to the overall measure space [15,16,17,18]. The underlying assumption in these approaches is that data is I.I.D. which allows sampling the overall space uniformly. However, in a continuously data stream context, as in this paper, data is no longer independent and identically distributed, therefore balancing the known space vs the unknown space is more difficult since space with absence of data may be misinterpreted for open space. Storing data in a memory module can limit these effects [19,20]. Open World: The other fundamental problem is incorporating the identified novel classes into the learning system (open world) [21]. This requirement favors non-parametric methods, since they can quickly learn never seen before information by simply storing examples. The Nearest Class Mean (NCM) classifier proposed in [22], has been shown to work well and be more robust than standard parametric classifiers in an incremental learning setting [22] [23] [24]. NCM's main shortcomings are: it is not a discriminative classifiers and nonlinear data representation and/or non I.I.D. data streams limit the effectiveness of using the mean. We adopt from NCM the idea of prototype-based classification. However, the prototypes we use are not the average features vectors but we keep a representative non redundant discriminative subset. Multiple Object Tracking: All the methods we described so far make use of ground truth labels and typically address the categorization problem in which data is manually cropped around the object boundary. An alternative approach that in principle accomplishes the classincremental learning criteria expounded above (i.e. open set and open world) but with the addition of unknown labels and with data coming from the output of a detector (i.e. no manual cropped data) is Multiple Object Tracking (MOT) [25,26]. Recent Multiple Object Tracking algorithms typically adopt appearance and motion cues into an affinity model to estimate and link detections to form tracklets which are afterwards combined into final trajectories [27,28,29,30,31,32]. Most existing MOT methods are applied to pedestrian tracking and either use simple color histogram features [28,33,34,35,36] or hand-crafted features [37,38,39,40] as the appearance representation of objects and have simple appearance update mechanisms. Few exceptions can operate online and use deep features [41,42,43,44] but they still assume continuous motion and do not update the appearance. MOT methods are not suited to abrupt changes across different shots or scenes since the assumptions of continuous motion no longer hold. Abrupt changes across different shots are typically handled offline by exploiting tracklets into predetermined non-overlapping shots as in clustering face descriptors [45] [46] [47] [48]. Long Term Object Tracking: Finally, another relevant research subject to our learning setting is long-term object tracking [49]. The aim of long-term object tracking is to track a specific object over time and re-detect it when the target leaves and re-enters the scene. Only a few works on tracking have reported drift-free results on on very long video sequences ( [50,51,52,53,54] among the few), and only few of them have provided convincing evidence on Figure 2. Block diagram presenting the major work flow and functional components in the proposed method. The gray shaded region highlights the components discussed in this paper. The memory module and the matching strategy run on the GPU. the possibility of incremental appearance learning strategies that are asymptotically stable [50] [52]. However, all of these works only address tracking and perform incremental learning to detect the target in the next frame. Contributions 1. We firstly combine in a principled manner Multiple Object Tracking in an online Open World learning system in which the learning strategy is shown to be asymptotically stable. 2. The proposed method performs very well with respect to offline clustering methods which exploits future information. 3. Different from several existing approaches, our proposed method operates online and and hence have a wider range of applications particularly face recognition with auto-enrollment of unrecognized subjects. The proposed approach In our system, deep face descriptors are computed on face regions detected by a face detector and stored in a memory module as: M(t) = {(x i , Id i , e i , a i )} N (t) i=1 (1) where x i is the deep descriptor, Id i is the object identity (an incremental number), e i is the eligibility factor (discussed in the following), a i tracks the age of items stored in memory and N (t) is the number of descriptors at time t in the memory module. The block diagram of the proposed system is shown in Fig. 2. As video frames are observed, new faces are detected and their descriptors are matched with those already in the memory. Each newly observed descriptor will be assigned with the object identity of its closest neighbour according to a discriminative strategy based on reverse nearest neighbor described in the next section. Unmatched descriptors of the faces in the incoming frame are stored in the memory module with a new Id. They ideally represent hypothesys of new identities that have not been observed yet and will eventually appear in the following frames. In order to learn a cumulative and comprehensive identity representation of each observed subject, two distincts problems are addressed. They are concerned with matching in consecutive frames and control of the memory module. These are separately addressed in the following subsections respectively. Reverse Nearest Neighbour Matching While tracking in consecutive frames, it is likely that the face of the same individual will have little differences from one frame to the following. In this case, highly similar descriptors will be stored in the memory and quickly a new face descriptor of the same individual will have comparable distances to the nearest and the second nearest descriptor already in the memory. In this case, a discriminative classifier like the Nearest Neighbor (NN) based on the distanceratio criterion [55] does not work properly and matching cannot be assessed. We solved this problem by performing descriptor matching according to Reverse Nearest Neighbour (ReNN) [56]: M = (x i , Id i , e i , a i ) ∈ M(t) | ||xi−1NNI t (xi)|| ||xi−2NNI t (xi)|| <ρ,(2) whereρ is the distance ratio threshold for accepting a match, x i is a deep face descriptor in the memory module and 1NN It (x i ) and 2NN It (x i ) are respectively its nearest and second nearest neighbor deep face descriptor in the incoming frame I t . Fig. 3 shows the effects of this change of perspective: here two new observations are detected (two distinct faces, respectively marked as o 1 and o 2 ). They both have distance ratio close to 1 to the nearest x i s in the memory (the dots inside the grey region S). Therefore both their matchings are undecidable. Differently from NN, ReNN is able to correctly determine the nearest descriptor for each new 1 2 1 Figure 3. Reverse Nearest Neighbor for a repeated temporal visual structure (S) with the distance ratio criterion. All elements xi match with o1, for clarity only one of them is highlighted to show the distances (thick black lines). descriptor in the incoming frame. In fact, with ReNN, the roles of x i and o i are exchanged and the distance ratio is computed between each x i and the o i as shown in figure for one of the x i s (the yellow dot is associated to the newly observed red dot). Due to the fact that with ReNN a large number of descriptors (those accumulated in the memory module) is matched against a relatively small set of descriptors (those observed in the current image), calculation of the ratio between distances could be computationally expensive if sorting is applied to the entire set. However, minimum distances can be efficiently obtained by performing twice a brute force search, with parallel implementation on GPU [57]. This technique not only leverages the very efficient CUDA matrix multiplication kernel for computing the squared distance matrix but it also exploits the GPU parallelism since each query is independent. GPU limited bandwidth is not an issue being the memory incrementally populated. The other important advantage of using ReNN is that all the descriptors x i of the shown repeated structure S of Fig. 3 match with the descriptor o 1 resulting in a one to many correspondence: {o 1 } ↔ {x i }. This capability provides a simple and sound method in the selection of those redundant descriptors that need to be condensed into a more compact representation. The feature o 1 will be used, as described in the next section, to influence the other matched (redundant) features x i regarding the fact that they belong to the same repeated structure. Therefore not only ReNN restores the discriminative matching capability under the distance ratio criterion but it also creates the foundation for the development of memory control strategies to correctly forget the redundant feature information. Memory Control Descriptors that have been matched according to ReNN ideally represent different appearances of a same subject face. However, collecting these descriptors indefinitely could quickly determine memory overload. To detect redundant descriptors and discard them appropriately, we defined a dimensionless quantity e i referred to as eligibility. This is set to e i = 1 as a descriptor is entered in the memory module and hence decreased at each match with a newly observed descriptor, proportionally to the distance ratio: e i (t + 1) = η i e i (t).(3) When doing this, we also re-set the age: a i = 0. Eligibility allows to take into account both discriminative spatial redundancy at a rate proportional to the success of matching in consecutive frames. In fact, as the eligibility e i of a face descriptor x i in the memory drops below a given threshold e (that happens after a number of matches), that descriptor with its associated identity, age and relative eligibility is removed from the memory module: if (e i <ē) then M(t + 1) = M(t) \ {(x i , Id i , e i , a i )}. (4) The value η i is computed according to: η i = 1 ρ d 1 i d 2 i α ,(5) where d 1 i and d 2 i are respectively the distances between x i and its first and second nearest neighbour o i , the valueρ is the distance-ratio threshold of Eq. 2 here used to normalize η i in the unit interval. The value of α emphasizes the effect of the distance-ratio. With every memory update we also increment the age a i of all non-matched elements by 1. Eq. 5 defines a density that weights more the eligibility around the matched features and less the eligibility far apart from their second nearest neighbor. This definition is similar to discriminative distance metric learning in which the features belonging to two different classes are expected to be separated as much as possible in the feature space. The density defined by Eq. 5 can be visualized in Fig. 4 for some values of the distance ratio below the matching thresholdρ. Each 2D circle in the figure visually represents the density weighting the eligibility of the matching descriptors. The geometric shape of the density is a generalization to multiple dimensions of the Apollonious circle 1 . In particular, the asymmetric shape of the density induced by the distance ratio encourages learning feature diversity in the open space. Therefore not only the matching is discriminative and indicated for rejecting hypotheses (Open Set) but also well suited for learning in an Open World. Temporal Coherence in Image Space The model previously described exploits video temporal coherence in the deep descriptor space, further spatiotemporal coherence is exploited in the image space introducing the following constraints: 1. Id novelty: Potential novel identities in the current frame are included in the memory only if at least one known identity is recognized in the current frame. This allows introducing novel identity information which is known to be reasonably different from the recognized ones. 2. Id temporal coherence: An identity is assigned and included in the memory only if has been assigned in two consecutive frames. After the assignment (i.e. memory inclusion) it must match at least once in the following 3 frames, otherwise it is discarded. 3. Id uniqueness: Duplicated Ids in the current frame are not considered. 4. Id ambiguity: A subject may match with multiple identities. This ambiguity is resolved by assigning all the matched descriptors with the Id having the largest number of matched descriptors as shown in Fig. 5. Bounding box overlap, typically used in multiple object tracking, is not exploited since not effective in unconstrained video with abrupt motions. Video temporal coherence in the image space is explicitly enforced by the 2nd constraint. Memory Overflow Control Our method, operating online, does not require any prior information about how many identity classes will occur, and can run for an unlimited amount of time. However, since the memory requirement is dominated by the size of the stored exemplars, if the number of identities increases indefinitely the exemplar removal based on eq. 4 may not be sufficient in handling redundancy and the system may overflow its bounded memory. In this condition the system is forced to remove some stored exemplars by the memory limita-tions. To overcome this issue we follow a strategy similar to [14,58] that involves the use of a policy based on removing from the memory the Least Recently Used Access (LRUA) exemplars. This is achieved by finding memory items with maximum age a i in the memory, and write to one of those. Therefore the system preserves recently encoded information according to the Eligibility strategy, or writes to the last used location according to the LRUA strategy. The latter can function as an update of the memory with newer, possibly more relevant information by avoiding the deletion of rare but useful descriptors. A benefit of the LRUA strategy is that of handling those features collected in the memory that will never obtain matches. This effect is largely due to scene occluders or with descriptors extracted from bounding boxes computed from false positives of the face detector. In the long run such features may waste critical space in the memory buffer. Asymptotic stability Under the assumption that descriptors are sufficiently distinctive (as in the case of deep face descriptors), the incremental learning procedure described above stabilizes asymptotically around the probability density function of the descriptors of each individual subject face. This can be proved by studying the discrete dynamic system of Eq. 3 relating e(t + 1) to e(t) by the map T : X → X as e(t + 1) = T (e(t)). A fixed point of T corresponds to an equilibrium state of the discrete dynamical system. In particular if T is a contraction there is a unique equilibrium state and the system approaches this state as time goes to infinity starting from any initial state. In this case the fixed point is globally asymptotically stable. More formally: Theorem (Contraction Mapping) 1 Let (X, d) be a complete metric space and T : X → X be the map of Eq. 3 such that d(T (e), T (e )) ≤ c · d(e, e ) for some 0 < c ≤ 1 and all e and e ∈ X. Then T has a unique fixed point in X. Moreover, for any e(0) ∈ X the sequence e(n) defined as e(n + 1) = T (e(n)), converges to the fixed point of T . The key element that guarantees such theoretical asymptotic stability is that the ReNN distance ratio is always below 1. In fact, it is easy to demonstrate that the updating rule of Eq. 3 is a contraction and converges to its unique fixed point 0 according to the Contraction Mapping theorem (Banach fixed-point theorem). The asymptotic stability of the method is illustrated in Fig. 6 with a simple one-dimensional case. Two patterns of synthetic descriptors, respectively modeling the case of a distinctive identity (red curve) and a non distinctive identity (black curve) are generated by two distinct 1D Gaussian distributions. The learning method was ran for 1000 iterations for three different configurations of the two distributions. The configurations reflect the limit case in which the dis- tinctiveness assumption of the deep descriptors no longer holds. Mismatches might therefore corrupt the distinctive identity. The blue points represent the eligibility of the distinctive identity. The histogram in yellow represents the distribution of the distinctive identity as incrementally learned by the system. The three figures represent distinct cases in which the non distinctive identity is progressively overlapping the distinctive one. The ReNN matching mechanism and the memory control mechanism still keep the learned distinctive identity close to its ground truth pdf. Quantitative Experiments We focus on tracking/identifying multiple faces according to their unknown identities in unconstrained videos consisting of many shots typically taken from different cameras. We used the Music-dataset in [48] which includes 8 music videos downloaded from YouTube with annotations of 3,845 face tracks and 117,598 face detections. We also add the first 6 episodes from Season 1 of the Big Bang Theory TV Sitcom (referred as BBT01-06) [36]. Each video is about more than 20 minutes long with 5-13 people and is taken mostly indoors. The main difficulty lies in identifying faces of the same subject from a long video footage. The two algorithm parameters in Eq. 5 are set empirically to:ρ = 1.6 and α = 0.01. Deep face descriptor are extracted according to [62]. We firstly show the capability of the proposed method to perform online learning without drifting using the long sequences of the BBT dataset. This consists on monitoring the performance degradation of the system as time advances. A decrease in performance may eventually hinder learning being the system in a condition from which is not possible to recover. In order to build a pic- ture of the performance over time we evaluate the method with the metric set commonly used in multiple object tracking [63]. In particular we report the MOTA: The Multiple Object Tracking Accuracy that takes into account false positives, wrongly rejected identities and identity switches as: MOTA = 1 − t (FNt+FPt+IDSt) t GTt where GT t , FN t , FP t and IDS t are respectively the number of ground truth objects, the number of false negatives, the number of false positives and the number of identity switches at time t. The identity switches are defined as the total number of times that a tracked trajectory changes its matched GT identity. Fig. 7 shows the MOTA curves as time progresses for each video sequence of the BBT dataset for about 30000 frames. Each individual frame is used to test the model before it is used for training by the incremental learning procedure [64]. As can be seen from the figure the curves reveal the stability of the learning mechanism confirming the theoretical result of Sec. 4.5. The initial fluctuations typically vary from sequence to sequence and reflect the approximate invariance of the original representation. That is, the few descriptors entering in the memory at the beginning of each sequence do not provide substantial improvement with respect to the original representation. However, as time advances, the reduction of fluctuations reveal that the proposed method is able to learn by collecting all the non-redundant descriptors it can from the video stream until no more improvement is possible. We further compare the proposed algorithm with other state-of-the-art MOT trackers, including modified versions of TLD [65], ADMM [60], IHTLS [61]. We specifically compare with two multi-face tracking methods using the TLD method implemented as described in [48]. The first method, called mTLD, runs multiple TLD trackers for all targets, and each TLD tracker is initialized with the ground truth bounding box in the first frame. The second method, referred as mTLD2, is used to generate shot-level trajectories within each shot initializing TLD trackers with untracked detections, and link the detections in the following frames according to their overlap scores with TLD outputs. The methods indicated as Pre-trained, SymTriplet, Triplet and Siamese refers to the four alternatives methods proposed in [48]. In these methods including ADMM, mTLD, mTLD2 and IHTLS, shot changes are detected and the video is divided into non-overlapping shots. Within each shot, a face detector is applied and adjacent detections are linked into tracklets. The recovered collection of tracklets are used as face pairs (Siamese) or face triplets (Triplet and SymTriplet) to fine-tune a CNN initial face feature descriptor based on the AlexNet architecture trained on the CASIA-WebFace (Pre-trained). Then, appearance of each detection is represented with the fine-tuned feature descriptors and tracklets within each shot are linked into shotlevel tracklets according to a global multiple object tracking [34,66]. Finally tracklets across shots are subsequently merged across multiple shots into final trajectories according to the Hierarchical Agglomerative Clustering. We reported two alternative versions using the (Deformable Part Model) DPM [67] and the Tiny [68] face detectors. They are indicated as MuFTiR-DPM and MuFTiR-TINY respectively. For such comparisons we also include the multiple target metric MOTP: The Multiple Object Tracking Precision. MOTP is the average dissimilarity between all true positives and their corresponding ground truth targets. MOTP is a measure of localization precision. Given the quite different nature between offline and online this comparison is to be considered a proof-of-concept. However, given the good performance of the offline methods we compare to, it is certainly non-trivial for our online method to do any better. Table 1 shows that our online tracking algorithm does reasonably well with respect to offline algorithms, although there are some exceptions. In HEL-LOBUBBLE, BRUNOMARS, DARLING, TARA and WEST-LIFE our best performing method has the MOTA score similar to the ADMM and IHTLS methods with little less identity switches. Despite the on par performance, our method achieves the results without exploiting future information. Performance are still good in APINK, the identity switches are still comparable despite a decrease in MOTA. Excluding Siamese, Triplet and SymTriplet that use a refined descriptor specifically tailored to the clustered identities extracted with the multiple passes over the sequence, our method is on par with the other offline methods. Our main observation is that with modern CNN based face detector and descriptor, the state-of-the-art offline trackers do not have expected advantages over the simpler online ones. Advantages further thin when processing long video sequences that do not fit into memory. Results are confirmed in the BBT dataset as shown in Table 2. As in the previous comparison on the Music dataset, except for the Siamese, Triplet and SymTriplet the overall performance are very good. In the Episode four we achieved better results. Considering that CNN descriptor fine-tuning takes around 1 hour per sequence on a modern GPU, our method perform favorably in those applications operating on real time streaming data. Currently our approach runs at 10 frame per second on 800x600 video frame resolution on a Nvidia GeForce GTX TITAN X (Maxwell). MOTA 2 , while largely used to evaluate performance in multiple object tracking, it is not fully appropriate to evaluate the performance of identification in a open world scenario. In fact, it does not explicitly handle target reidentification. Different identities assigned to the same individual in two distinct scenes are not accounted as an identity switch. This effect has particular impact with videos obtained from multiple cameras or with many shots. In order to take into account this case, for each sequence we also report the weighted cluster purity, defined as: W = 1 M c m c p c , where c is the identity cluster, m c the number of assigned identities, p c the ratio between the most frequently occurred identity and m c , and M denotes the total number of identity detections in the video. Table 3 Finally, Fig. 8 shows four frames of the of the BRUNO MARS sequence with the learned identities superimposed. Faces appear sensibly diverse (see f.e. individual number 1), nonetheless it can be observed that the learning mechanism is capable to extend the original representation to preserve identities under large pose variations including face profiles not included in the original representation. Conclusion In this paper we exploited deep CNN based face detection and descriptors coupled with a novel memory based learning mechanism that learns face identities from video sequences unsupervisedly, exploiting the temporal coherence of video frames. Particularly, all the past observed information is learned in a comprehensive representation. We demonstrate the effectiveness of the proposed method with respect multiple face tracking on the Music and BBT datasets. The proposed method is simple, theoretically sound, asymptotically stable and follows the cumulative and convergent nature of human learning. It can be applied in principle to any other context for which a detectordescriptor combination is available (i.e. car, person, boat, traffic sign). Figure 1 . 1Memory based appearance representation. Left: Each element in the memory consists of a descriptor with an associated identity (indicated by box color) and an associated scalar value reflecting the degree of redundancy (indicated by grey box area) with respect to the current representation. Right: The shaded regions represent the original appearance representation (i.e. VGGface). The descriptors outside those regions are learned from the video and extend the original appearance representation. Figure 4 . 4The shape of the density (here in 2D) down-weighting the eligibility associated to each matched descriptor in the memory. Features xi in proximity of the observed descriptor o1 have their eligibility decreased to encourage their redundancy. The asymmetric shape of the density encourages more diversity in the open space far from the identity o2 rather than close. Figure 5 . 5Matching with multiple identities. The identity o1 matches with two identities (yellow and green). The ambiguity is resolved by assigning o1 with the Id having the largest number of matched descriptors (i.e. the yellow identity). Figure 6 . 6Asymptotic stability of incremental learning of a face identity in a sample sequence . Figure 7 . 7MOTA for each video sequence in the BBT dataset. Figure 8 . 8Four frames from the BRUNOMARS video sequence with the superimposed estimated identities are shown.4 show the quantitative results of the comparison with the Music and the BBT datasets. HOG, AlexNet and VGGface indicate the method[48] using alternative descriptors. HOG uses a conventional hand-crafted feature with 4356 dimensions, AlexNet uses a generic feature representation with 4096 dimensions. Our proposed approach achieves the best performance in six out of eight videos in the Music dataset and it achieves state of the art in all the BBT video sequences. Table 1 . 1Quantitative comparison with other state-of-the-art multi-object tracking methods on the Music video datasetAPINK Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD [59] Offline 31 -2.2 71.2 ADMM [60] Offline 179 72.4 76.1 IHTLS [61] Offline 173 74.9 76.1 Pre-Trained [48] Offline 100 54.0 75.5 mTLD2 [59] Offline 173 77.4 76.3 Siamese [48] Offline 124 79.0 76.3 Triplet [48] Offline 140 78.9 76.3 SymTriplet [48] Offline 78 80.0 76.3 MuFTiR-dpm Online 121 21.8 61 MuFTiR-tiny Online 191 55.1 65.4 BRUNOMARS Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 35 -8.7 65.3 ADMM Offline 428 50.6 85.7 IHTLS Offline 375 52.7 85.8 Pre-Trained Offline 151 48.3 88.0 mTLD2 Offline 278 52.6 87.9 Siamese Offline 126 56.7 87.8 Triplet Offline 126 56.6 87.8 SymTriplet Offline 105 56.8 87.8 MuFTiR-dpm Online 78 4.5 61 MuFTiR-tiny Online 420 48.8 65.5 DARLING Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 24 -22.0 69.9 ADMM Offline 412 53.0 88.4 IHTLS Offline 381 62.7 88.4 Pre-Trained Offline 115 42.7 88.5 mTLD2 Offline 278 59.8 89.3 Siamese Offline 214 69.5 88.9 Triplet Offline 187 69.2 88.9 SymTriplet Offline 169 70.5 88.9 MuFTiR-dpm Online 64 2.2 63.7 MuFTiR-tiny Online 449 62.1 66.0 GIRLSALOUD Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 9 -1.1 71.0 ADMM Offline 487 46.6 87.1 IHTLS Offline 396 51.8 87.2 Pre-Trained Offline 138 42.7 87.7 mTLD2 Offline 322 46.7 88.2 Siamese Offline 112 51.6 87.8 Triplet Offline 80 51.7 87.8 SymTriplet Offline 64 51.6 87.8 MuFTiR-dpm Online 51 -2.7 61 MuFTiR-tiny Online 339 49.3 66.1 HELLOBUBBLE Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 7 -3.5 66.5 ADMM Offline 115 47.6 69.9 IHTLS Offline 109 52.0 69.9 Pre-Trained Offline 71 36.6 68.5 mTLD2 Offline 139 52.6 70.5 Siamese Offline 105 56.3 70.6 Triplet Offline 82 56.2 70.5 SymTriplet Offline 69 56.5 70.5 MuFTiR-dpm Online 170 4.0 59.0 MuFTiR-tiny Online 88 51.4 69.9 PUSSYCATDOLLS Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 24 3.1 71.3 ADMM Offline 287 63.2 63.5 IHTLS Offline 248 70.3 63.5 Pre-Trained Offline 128 65.1 64.9 mTLD2 Offline 296 68.3 64.9 Siamese Offline 107 70.3 64.9 Triplet Offline 99 69.9 64.9 SymTriplet Offline 82 70.2 64.9 MuFTiR-dpm Online 55 -13.5 61.1 MuFTiR-tiny Online 83 30.7 62.7 TARA Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 130 1.4 67.9 ADMM Offline 251 29.4 63.8 IHTLS Offline 218 35.3 63.8 Pre-Trained Offline 143 57.3 72.4 mTLD2 Offline 251 56.0 72.6 Siamese Offline 106 58.4 72.5 Triplet Offline 94 59.0 72.5 SymTriplet Offline 75 59.2 72.4 MuFTiR-dpm Online 124 15 68 MuFTiR-tiny Online 270 39.5 76.4 WESTLIFE Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 20 -34.7 56.9 ADMM Offline 223 62.4 87.5 IHTLS Offline 113 60.9 87.5 Pre-Trained Offline 85 57.0 88.2 mTLD2 Offline 177 58.1 88.1 Siamese Offline 74 64.1 88.0 Triplet Offline 89 64.5 88.0 SymTriplet Offline 57 68.6 88.1 MuFTiR-dpm Online 47 -0.2 61.5 MuFTiR-tiny Online 76 58.9 66.1 Table 2 . 2Quantitative comparison with other state-of-the-art multi-object tracking methods on the BBT dataset.BBT_S01E01 Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD [59] Offline 1 -16.3 74.8 ADMM [60] Offline 323 42.5 64.0 IHTLS [61] Offline 312 45.7 64.0 Pre-Trained [48] Offline 171 41.9 73.3 mTLD2 [59] Offline 223 58.4 73.8 Siamese [48] Offline 144 69.0 73.7 Triplet [48] Offline 164 69.3 73.6 SymTriplet [48] Offline 156 72.2 73.7 MuFTiR-tiny Online 24 59.9 70.3 BBT_S01E02 Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 1 -7.6 82.8 ADMM Offline 395 41.3 71.3 IHTLS Offline 394 42.4 71.4 Pre-Trained Offline 130 27.4 74.5 mTLD2 Offline 174 43.6 75.9 Siamese Offline 116 60.4 75.8 Triplet Offline 143 60.2 75.7 SymTriplet Offline 102 61.6 75.7 MuFTiR-tiny Online 57 45.1 68.8 BBT_S01E03 Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 5 -2.1 69.4 ADMM Offline 370 30.8 68.1 IHTLS Offline 376 33.5 68.0 Pre-Trained Offline 110 17.8 67.5 mTLD2 Offline 142 38.0 67.9 Siamese Offline 109 52.6 67.9 Triplet Offline 121 50.7 67.8 SymTriplet Offline 126 51.9 67.8 MuFTiR-tiny Online 14 43.6 68.4 BBT_S01E04 Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 0 -15.9 76.8 ADMM Offline 298 9.7 65.8 IHTLS Offline 295 13.3 65.8 Pre-Trained Offline 46 0.1 66.3 mTLD2 Offline 103 11.6 66.3 Siamese Offline 85 23.0 66.4 Triplet Offline 103 18.0 66.4 SymTriplet Offline 77 19.5 66.4 MuFTiR-tiny Online 84 53.2 69.6 BBT_S01E05 Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 1 -15.5 76.9 ADMM Offline 380 37.4 68.2 IHTLS Offline 360 33.8 68.2 Pre-Trained Offline 98 32.3 75.0 mTLD2 Offline 169 46.4 74.9 Siamese Offline 128 60.7 75.0 Triplet Offline 118 60.5 74.9 SymTriplet Offline 90 60.9 74.9 MuFTiR-tiny Online 36 44.5 69.3 BBT_S01E06 Method Mode IDS ↓ MOTA ↑ MOTP ↑ mTLD Offline 0 -3.9 89.3 ADMM Offline 527 47.5 97.6 IHTLS Offline 515 43.2 97.7 Pre-Trained Offline 191 27.8 98.2 mTLD2 Offline 192 37.7 97.8 Siamese Offline 156 46.2 97.9 Triplet Offline 185 45.4 98.0 SymTriplet Offline 196 47.6 98.0 MuFTiR-tiny Online 222 42.9 69.2 and 2 andProvided by www.motchallenge.org Table 3. Clustering results on Music Dataset. Weighted purity of each video is measured on ideal number of clusters. MUSIC DATASET Videos Apink B. Mars Darling Girls A. Hello B. P. Dolls T-ara WestlifeHOG 0.20 0.36 0.19 0.29 0.35 0.28 0.22 0.27 AlexNet 0.22 0.36 0.18 0.30 0.31 0.31 0.25 0.37 Pre-trained 0.29 0.50 0.24 0.33 0.34 0.31 0.31 0.37 VGG-Face 0.24 0.44 0.20 0.31 0.29 0.46 0.23 0.27 Siamese 0.48 0.88 0.46 0.67 0.54 0.77 0.69 0.54 Triplet 0.60 0.83 0.49 0.67 0.60 0.77 0.68 0.52 SymTriplet 0.72 0.90 0.70 0.69 0.64 0.78 0.69 0.56 MuFTiR-tiny 0.51 0.96 0.73 0.89 0.59 0.97 0.72 0.98 Table 4 . 4Clustering results on Big Bang Theory Dataset. Weighted purity of each video is measured on ideal number of clusters.BIG BANG THEORY Episodes BBT01 BBT02 BBT03 BBT04 BBT05 BBT06 HOG 0.37 0.32 0.38 0.35 0.29 0.26 AlexNet 0.47 0.32 0.45 0.35 0.29 0.26 Pre-trained 0.62 0.72 0.73 0.57 0.52 0.52 VGG-Face 0.91 0.85 0.83 0.54 0.65 0.46 Siamese 0.94 0.95 0.87 0.74 0.70 0.70 Triplet 0.94 0.95 0.92 0.74 0.68 0.70 SymTriplet 0.94 0.95 0.92 0.78 0.85 0.75 MuFTiR-tiny 0.98 0.98 0.98 0.85 0.98 0.94 Apollonius of Perga (c. 262 BC -c. 190 BC) showed that a circle may also be defined as the set of points in a plane having a constant ratio of distances to two fixed foci. AcknowledgmentThis research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA contract number 2014-14071600011. 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Shun Zhang, Yihong Gong, Jia-Bin Huang, Jongwoo Lim, Jinjun Wang, Narendra Ahuja, Ming-Hsuan Yang, European Conference on Computer Vision. Springer7Shun Zhang, Yihong Gong, Jia-Bin Huang, Jongwoo Lim, Jinjun Wang, Narendra Ahuja, and Ming-Hsuan Yang. Tracking persons-of-interest via adaptive discriminative fea- tures. In European Conference on Computer Vision, pages 415-433. Springer, 2016. 2, 6, 7, 8 Long term detection and tracking workshop. Conjunction With The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops (LTDT2014). Long term detection and tracking workshop. In Conjunction With The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops (LTDT2014), June 2014. 2 P-n learning: Bootstrapping binary classifiers by structural constraints. Z Kalal, J Matas, K Mikolajczyk, CVPR. 23Z. Kalal, J. Matas, and K. Mikolajczyk. P-n learning: Boot- strapping binary classifiers by structural constraints. 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Zdenek Kalal, Krystian Mikolajczyk, Jiri Matas, IEEE transactions on pattern analysis and machine intelligence. 34Zdenek Kalal, Krystian Mikolajczyk, and Jiri Matas. Tracking-learning-detection. IEEE transactions on pattern analysis and machine intelligence, 34(7):1409-1422, 2012. 6 Multiobject tracking through occlusions by local tracklets filtering and global tracklets association with detection responses. Junliang Xing, Ai Haizhou, Shihong Lao, Computer Vision and Pattern Recognition. IEEECVPR 2009. IEEE Conference onJunliang Xing, Haizhou Ai, and Shihong Lao. Multi- object tracking through occlusions by local tracklets filter- ing and global tracklets association with detection responses. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 1200-1207. IEEE, 2009. 7 Face detection without bells and whistles. Markus Mathias, Rodrigo Benenson, Marco Pedersoli, Luc Van Gool, European Conference on Computer Vision. SpringerMarkus Mathias, Rodrigo Benenson, Marco Pedersoli, and Luc Van Gool. Face detection without bells and whistles. In European Conference on Computer Vision, pages 720-735. Springer, 2014. 7 Finding tiny faces. Peiyun Hu, Deva Ramanan, The IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Peiyun Hu and Deva Ramanan. Finding tiny faces. In The IEEE Conference on Computer Vision and Pattern Recogni- tion (CVPR), July 2017. 7

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{'abstract': 'We present a novel online unsupervised method for face identity learning from video streams. The method exploits deep face descriptors together with a memory based learning mechanism that takes advantage of the temporal coherence of visual data. Specifically, we introduce a discriminative feature matching solution based on Reverse Nearest Neighbour and a feature forgetting strategy that detect redundant features and discard them appropriately while time progresses. It is shown that the proposed learning procedure is asymptotically stable and can be effectively used in relevant applications like multiple face identification and tracking from unconstrained video streams. Experimental results show that the proposed method achieves comparable results in the task of multiple face tracking and better performance in face identification with offline approaches exploiting future information. Code will be publicly available.', 'arxivid': '1711.07368', 'author': ['Federico Pernici \nMICC -Media Integration\nCommuncation Center University Of Florence\n\n', 'Federico Bartoli \nMICC -Media Integration\nCommuncation Center University Of Florence\n\n', 'Matteo Bruni \nMICC -Media Integration\nCommuncation Center University Of Florence\n\n', 'Alberto Del Bimbo \nMICC -Media Integration\nCommuncation Center University Of Florence\n\n'], 'authoraffiliation': ['MICC -Media Integration\nCommuncation Center University Of Florence\n', 'MICC -Media Integration\nCommuncation Center University Of Florence\n', 'MICC -Media Integration\nCommuncation Center University Of Florence\n', 'MICC -Media Integration\nCommuncation Center University Of Florence\n'], 'corpusid': 30312999, 'doi': '10.1109/cvpr.2018.00247', 'github_urls': [], 'n_tokens_mistral': 21006, 'n_tokens_neox': 18159, 'n_words': 10053, 'pdfsha': 'b85795a47dfcd6351510eaa2fb36e6df83b1c5a2', 'pdfurls': ['https://arxiv.org/pdf/1711.07368v1.pdf'], 'title': ['Memory Based Online Learning of Deep Representations from Video Streams', 'Memory Based Online Learning of Deep Representations from Video Streams'], 'venue': []}

arxiv

ON THE THREE-CIRCLE THEOREM AND ITS APPLICATIONS IN 30 Jan 2018 Sasakian Manifolds Shu-Cheng Chang ANDYingbo Han Chien Lin ON THE THREE-CIRCLE THEOREM AND ITS APPLICATIONS IN 30 Jan 2018 This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yau's uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.1991 Mathematics Subject Classification. Primary 32V05, 32V20; Secondary 53C56. Introduction In 1896, J. Hadamard ([Ha]) published the so-called classical three-circle theorem which says that, on the annulus A with inner radius r 1 and outer radius r 2 , the logarithm for the modulus of a holomorphic function on the closure A of the annulus is convex with respect to log r for r lying between r 1 and r 2 . Recently, G. Liu ([Liu1]) generalized the three-circle theorem to complete Kähler manifolds and characterized a necessary and sufficient condition for the three-circle theorem. Here a complete Kähler manifold M is said to satisfy the threecircle theorem if, for any point p ∈ M, R > 0, any holomorphic function f on the geodesic ball B (p, R), log M f (r) is convex with respect to log r, namely, for 0 < r 1 ≤ r 2 ≤ r 3 < R, (1.1) log r 3 r 1 log M f (r 2 ) ≤ log r 3 r 2 log M f (r 1 ) + log r 2 r 1 log M f (r 3 ) where M f (r) = sup x∈B(p,r) |f (x)|. More precisely, he showed that a complete Kähler manifold satisfies the three-circle theorem if and only if its holomorphic sectional curvature is nonnegative. The proof employed the Hessian comparison and the maximum principle. There are many substantial applications pertaining to the three-circle theorem, especially to the uniformization-type problems proposed by Yau ([ScY]) on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. It could be summarized as follows. The first Yau's uniformization conjecture is that if M is a complete noncompact n-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature, then dim C (O d (M n )) ≤ dim C (O d (C n )) . The equality holds if and only if M is isometrically biholomorphic to C n . Here O d (M n ) denotes the family of all holomorphic functions on a complete n-dimensional Kähler manifold M of polynomial growth of degree at most d. In [N], Ni established the validity of this conjecture by deriving the monotonicity formula for the heat equation under the assumption that M has maximal volume growth lim r→+∞ V ol (B p (r)) r 2n ≥ c for a fixed point p and a positive constant c. Later, in [CFYZ], the authors improved Ni's result without the assumption of maximal volume growth. In recent years, G. Liu ([Liu1]) generalized the sharp dimension estimate by only assuming that M admits nonnegative holomorphic sectional curvature. Note that there are noncompact complex manifolds admitting complete Kähler metrics with positive holomorphic sectional curvature but not admitting complete Kähler metrics with nonnegative Ricci curvature (see [Hi]). The second Yau's uniformization conjecture is that if M is a complete noncompact ndimensional Kähler manifold with nonnegative holomorphic bisectional curvature, then the ring O P (M) of all holomorphic functions of polynomial growth is finitely generated. This one was solved completely by G. Liu ([Liu2]) quite recently. He mainly deployed four techniques to attack this conjecture via Cheeger-Colding-Tian's theory ( [ChCo1], [ChCo2], [CCT]), methods of heat flow developed by Ni and Tam ([N], [NT1], [NT4]), Hörmander L 2 -estimate of ∂ ( [De]) and three-circle theorem ( [Liu1]) as well. The third Yau's uniformization conjecture is that if M is a complete noncompact ndimensional Kähler manifold with positive holomorphic bisectional curvature, then M is biholomorphic to the standard n-dimensional complex Euclidean space C n . The first giant progress relating to the third conjecture could be attributed to Mok, Siu and Yau. In their papers ( [MSY] and [M1]), they showed that, under the assumptions of the maximal volume growth and the scalar curvature R (x) decays as 0 ≤ R (x) ≤ C (1 + d (x, x 0 )) 2+ǫ for some positive constant C and any arbitrarily small positive number ǫ, a complete noncompact n-dimensional Kähler manifold M with nonnegative holomorphic bisectional curvature is isometrically biholomorphic to C n . A Riemannian version was solved in [GW2] shortly afterwards. Since then there are several further works aiming to prove the optimal result and reader is referred to [M2], [CTZ], [CZ], [N2], [NT1], [NT2] and [NST]. For example, A. Chau and L. F. Tam ( [CT]) proved that a complete noncompact Kähler manifold with bounded nonnegative holomorphic bisectional curvature and maximal volume growth is biholomorphic to C n . Recently, G. Liu ([Liu3]) confirmed Yau's uniformization conjecture when M has maximal volume growth. Later, M.-C. Lee and L.-F. Tam ([LT]) also confirmed Yau's uniformization conjecture with the maximal volume growth condition. For the corresponding first uniformization conjectures in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry (see the next section for its definition and some properties), it was settled that the CR sharp dimension estimate for CR holomorphic functions of polynomial growth with nonnegative pseudohermitian bisectional curvature in [CHL] of which proof is inspired primarily from [N] and [CFYZ]. So it's natural to concern whether the second and third CR Yau's uniformization conjectures hold as well. However, as inspired by recent works of G. Liu ([Liu2], [Liu3]), it is crucial to have the CR analogue of the three-circle theorem which is a step towards establishing the validity of such CR Yau's uniformization conjectures. In this paper, we mainly focus on the CR three-circle theorem in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature which is weak than nonnegative pseudohermitian bisectional curvature. A smooth complex-valued function on a pseudohermitian (2n + 1)-manifold (M, J, θ) is called CR-holomorphic if ∂ b f = 0. We recall O CR (M) the family of all CR-holomorphic functions f with T f (x) = f 0 (x) = 0 ([CHL]) O CR (M) = {f (x) ∈ C ∞ C (M) |∂ b f (x) = 0 and f 0 (x) = 0 }, where the extra condition T f (x) = 0 is included, the interested readers could refer to [CHL] or [FOW]. Next we give the definition of the CR three-circle theorem generalizing the classical Hadamard's three-circle theorem to CR manifolds : Definition 1.1. Let (M, J, θ) be a complete pseudohermitian (2n + 1)-manifold. (M, J, θ) is said to satisfy the CR three-circle theorem if, for any point p ∈ M, any positive number R > 0, and any function f ∈ O CR (B cc (p, R)) on the ball B cc (p, R), log M f (r) is convex with respect to log r for 0 < r < R. Here M f (r) = sup x∈Bcc(p,r) |f (x)| and B cc (p, R) is the Carnot-Carathéodory ball centered at p with radius R. Now we state our main theorem in this paper as follows: Theorem 1.1. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold with vanishing torsion, then the CR three-circle theorem holds on M if the pseudohermitian sectional curvature is nonnegative; moreover, we have that, for any f ∈ O CR (M), (1.2) M f (kr) M f (r) is increasing with respect to r for any positive number k ≥ 1. Remark 1.1. 1. In the course of the proof of the CR three-circle theorem, we derive the following CR sub-Laplacian comparison ∆ b r ≤ (2n − 1) r if (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature. It is sharp in case of n = 1. See Corollary 3.1 for details. 2. As a consequence of the sub-Laplacian comparison, Liouville theorem holds for positive pseduoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature ( [CKLT]). As an application of the preceding theorem, we have the enhanced version of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity which is served a generalization of authors previous results ( [CHL] Besides, the CR three-circle theorem could be extended to the case when M admits the pseudohermitian sectional curvature bounded below: Theorem 1.3. Let (M, J, θ) be a complete noncompact pseudohermitian (2n + 1)-manifold, r (x) = d cc (p, x) and Z 1 = 1 √ 2 (∇ b r − iJ∇ b r) for some fixed point p ∈ M. If the pseudohermitian sectional curvature R 1111 (x) has the inequality (1.3) R 1111 (x) ≥ g (r (x)) for g ∈ C 0 ([0, +∞)) and the pseudohermitian torsion vanishes, a function u (r) ∈ C 1 (R + ) satisfies (1.4) 2u 2 + u ′ + g 2 ≥ 0 with u (r) ∼ 1 2r as r → 0 + and a function h (r) ∈ C 1 (R + ) satisfies (1.5) h ′ (r) > 0 and (1.6) 1 2 h ′′ (r) + h ′ (r) u (r) ≤ 0 with h (r) ∼ log r as r → 0 + , then log M f (r) is convex with respect to the function h (r) for f ∈ O CR (M); moreover, if the vanishing order ord p (f ) of f ∈ O CR (M) at p is equal to d, then M f (r) exp(dh(r)) is increasing with respect to r. As precedes, there is also a sharp dimension estimate when the pseudohermitian sectional curvature is asymptotically nonnegative. Theorem 1.4. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold, r (x) = d cc (p, x) for some fixed point p ∈ M, and there are two positive constants ǫ, A such that R jjjj (x) ≥ − A (1 + r (x)) 2+ǫ for any Z j ∈ T 1,0 x M with |Z j | = 1, then there is a constant C (ǫ, A) > 0 such that, for any d ∈ N, dim C O CR d (M) ≤ C (ǫ, A) d n . Furthermore, if d ≤ e − 3A ǫ , then dim C O CR d (M) = 1. Finally, if A ǫ ≤ 1 4d for d ∈ N, then we have dim C O CR d (M) ≤ dim C O CR d (H n ) . The method we adopt here is inspired from [Liu1]. This paper is organized as follows. In Section 2, we introduce some basic notions about pseudohermitian manifolds and the necessary results for this paper. In Section 3, we show that the CR analogue of the three-circle theorem and some of its applications; specially, we confirm the first CR Yau's uniformization conjecture on the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity. As a by-product, we obtain the CR sub-Laplacian comparison theorem. In Section 4, we generalize the CR three-circle theorem to the case when the pseudohermitian sectional curvature is bounded below. It enables us to derive the dimension estimate when the pseudohermitian sectional curvature is asymptotically nonnegative. Preliminaries We introduce some basic materials about a pseudohermitian manifold (see [L] and [DT] for more details). Let (M, ξ) = (M, J, θ) be a (2n + 1)-dimensional, orientable, contact manifold with the contact structure ξ. A CR structure compatible with ξ is an endomorphism J : ξ → ξ such that J 2 = −Id. We also assume that J satisfies the integrability condition: If X and Y are in ξ, then so are [JX, Y ]+[X, JY ] and J([JX, Y ]+[X, JY ]) = [JX, JY ]−[X, Y ]. A contact manifold (M, ξ) = (M, J, θ) with a CR structure J compatible with ξ together with a contact form θ is called a pseudohermitian manifold or a strictly pseudoconvex CR manifold as well. Such a choice induces a unique vector field T ∈ Γ (T M) transverse to the contact structure ξ, which is called the Reeb vector field or the characteristic vector field of θ such that ι T θ = 0 = ι T dθ. A CR structure J could be extended to the complexified space ξ C = C ⊗ ξ of the contact structure ξand decompose it into the direct sum of T 1,0 M and T 0,1 M which are eigenspaces of J corresponding to the eigenvalues 1 and −1, respectively. Let {T, Z α , Zᾱ} α∈In be a frame of T M ⊗C, where Z α is any local frame of T 1,0 M, Zᾱ = Z α ∈ T 0,1 M, and T is the Reeb vector field (or the characteristic direction) and I n = {1, 2, ..., n}. Then {θ α , θᾱ, θ}, the coframe dual to {Z α , Zᾱ, T }, satisfies dθ = ih αβ θ α ∧ θ β for some positive definite matrix of functions (h αβ ). Usually, we assume such matrix (h αβ ) is the identity matrix A pseudohermitian manifold (M, J, θ) is called a Sasakian manifold if The Tanaka-Webster connection of (M, J, θ) is the connection ∇ on T M ⊗C (and extended to tensors) given, in terms of a local frame Z α ∈ T 1,0 M, by the pseudohermitian torsion τ = ι T T D = 0 The Levi form , L θ is the Hermitian form on T 1,0 M defined by Z, W L θ = −i dθ, Z ∧ W . We can extend , L θ to T 0,1 M by defining Z, W L θ = Z, W L θ for all Z, W ∈ T 1,∇Z α = ω α β ⊗ Z β , ∇Zᾱ = ωᾱβ ⊗ Zβ, ∇T = 0, where ω α β are the 1-forms uniquely determined by the following equations:          dθ α + ω α β ∧ θ β = θ ∧ τ α τ α ∧ θ α = 0 ω α β + ω β α = 0 . We can write (by Cartan lemma) τ α = A αγ θ γ with A αγ = A γα . The curvature of Tanaka- Webster connection ∇, expressed in terms of the coframe {θ = θ 0 , θ α , θᾱ}, is    Π α β = Π α β = dω α β + ω α γ ∧ ω γ β Π α 0 = Π 0 α = Π α 0 = Π 0 α = Π 0 0 = 0 . Webster showed that Π β α can be written Ω α β = Π α β + iτ α ∧ θ β − iθ α ∧ τ β = R α β γδ θ γ ∧ θ δ + W α βγ θ γ ∧ θ − W α βγ θ γ ∧ θ where the coefficients satisfy R βᾱρσ = R αβσρ = Rᾱ βσρ = R ρᾱβσ ; W α βγ = A βγ , α ; W α βγ = A γ α , β . Besides R αβγδ , the other part of the curvature of Tanaka-Webster connection are clear: (2.1)                  R αβγµ = −2i (A αµ δ βγ − A αγ δ βµ ) R αβγµ = −2i A βµ δ αγ − A βγ δ αµ R αβ0γ = A γα,β R αβ0γ = −A βγ,α . Here R α β γδ is the pseudohermitian curvature tensor field, R αβ = R γ γ αβ is the pseudohermitian Ricci curvature tensor field and A αβ is the pseudohermitian torsion. R = h αβ R αβ denotes the pseudohermitian scalar curvature. Moreover, we define the pseudohermitian bisectional curvature tensor field R αᾱββ (X, Y ) = R αᾱββ X α X α Y β Yβ, the bitorsion tensor field T αβ (X, Y ) = 1 i (A αγ X γ Y β − A βγ X γ Y α ), and the torsion tensor field T or (X, Y ) = tr T αβ = 1 i (A αβ X β Y α − A αβ X β Y α ), where X = X α Z α , Y = Y α Z α in T 1,0 M. We will denote the components of the covariant derivatives with indices preceded by comma. The indices {0, α,ᾱ} indicate the covariant derivatives with respect to {T, Z α , Zᾱ}. For the covariant derivatives of a real-valued function, we will often omit the comma, for instance, u α = Z α u, u αβ = ZβZ α u − ω α γ (Zβ)Z γ u. The subgradient ∇ b ϕ of a smooth real- valued function ϕ is defined by ∇ b ϕ, Z L θ = Zϕ for Z ∈ Γ (ξ) where Γ (ξ) denotes the family of all smooth vector fields tangent to the conact plane ξ. We could locally write the subgradient ∇ b ϕ as ∇ b u = u α Z α + u α Zᾱ. Accordingly, we could define the subhessian Hess b as the complex linear map Hess b : T 1,0 M ⊕ T 0,1 M −→ T 1,0 M ⊕ T 0,1 M by (Hess b ϕ) Z = ∇ Z ∇ b ϕ for Z ∈ Γ (ξ) and a smooth real-valued function ϕ. Also, the sub-Laplacian is defined by ∆ b u = tr (Hess b u) = u α α + u α α . Now we recall the following commutation relations (see [L]). Let ϕ be a smooth real-valued function, σ = σ α θ α be a (1, 0)-form and ϕ 0 = T ϕ, then we have (2.2)                                            ϕ αβ = ϕ βα , ϕ αβ − ϕβ α = ih αβ ϕ 0 ϕ 0α − ϕ α0 = A αβ ϕ β σ α,βγ − σ α,γβ = i (A αγ σ β − A αβ σ γ ) σ α,βγ − σ α,γβ = −i(h αγ A δ β σ δ − h αβ A δ γ σ δ ) σ α,βγ − σ α,γβ = ih βγ σ α,0 + R δ α βγ σ δ σ α,0β − σ α,β0 = A γ β σ α,γ − A αβ,γ σ γ σ α,0β − σ α,β0 = A γβ σ α,γ + Aγβ ,α σ γ . Subsequently, we introduce the notion about the Carnot-Carathéodory distance. Definition 2.1. A piecewise smooth curve γ : [0, 1] → (M, ξ) is said to be horizontal if γ ′ (t) ∈ ξ whenever γ ′ (t) exists. The length of γ is then defined by L(γ) = 1 0 γ ′ (t) , γ ′ (t) 1 2 L θ dt. The Carnot-Carathéodory distance between two points p, q ∈ M is d cc (p, q) = inf {L(γ)| γ ∈ C p,q } , where C p,q is the set of all horizontal curves joining p and q. We say (M, ξ) is complete if it's complete as a metric space. By Chow's connectivity theorem, there always exists a horizontal curve joining p and q, so the distance is finite. The diameter d c is defined by d c (M) = sup {d cc (p, q)| p, q ∈ M} . Note that there is a minimizing geodesic joining p and q so that its length is equal to the distance d cc (p, q). (2.3) deg (f ) = inf    d ≥ 0 |f (y)| ≤ C (1 + d cc (x, y)) d ∀ y ∈ M, for some d ≥ 0 and C = C (x, d, f )    . With these notions, we could define the family O CR d (M) of all CR-holomorphic functions f of polynomial growth of degree at most d with T f (x) = f 0 (x) = 0 : (2.4) O CR d (M) = {f ∈ O CR (M) | deg (f ) ≤ d }. Finally, we denote by ord p (f ) = max {m ∈ N | D α f (p) = 0, ∀ |α| < m } the vanishing order of CR-holomorphic function f at p where D α = j∈In Z α j j with α = (α 1 , α 2, ..., α n ). CR Three-Circle Theorem In this section, we will derive the CR analogue of three-circle theorem on a complete noncompact pseudohermitian (2n + 1)-manifold. Before that, we need a lemma which is essential in the course of the proof of the CR three-circle theorem as follows: 1 = 1 √ 2 (∇ b r − iJ∇ b r), then (3.1) r 11 ≤ 1 2r . In particular, we have (log r) 11 ≤ 0. Proof. Let e j , e j , T j∈In be an orthonormal frame at q where e j = Je j and e 1 = ∇ b r. By Corollary 2.3 in [DZ] and vanishing pseudohermitian torsion, we could parallel transport such frame at q to obtain the orthonormal frame along the radial ∇-geodesic γ from p to q. Hence we have an orthonormal frame {Z j , Z j , T } j∈In along γ where Z j = 1 √ 2 e j − ie j and Z j = Z j . By the fact that γ is the ∇-geodesic, we have r 11 = − 1 2 (ie 2 e 1 + e 2 e 2 ) r − (∇ Z 1 Z 1 ) r = − 1 2 (ie 2 e 1 + e 2 e 2 ) r + 1 2 i∇ (J∇ b r) ∇ b r + J ∇ (J∇ b r) ∇ b r and r 11 = 1 2 (ie 2 e 1 + e 2 e 2 ) r − ∇ Z 1 Z 1 r = 1 2 (ie 2 e 1 + e 2 e 2 ) r − 1 2 i∇ (J∇ b r) ∇ b r + J ∇ (J∇ b r) ∇ b r . Therefore along γ (3.2) r 11 = −r 11 . Moreover, by computing r 1 = Z 1 r = 1 √ 2 (∇ b r − iJ∇ b r) r = 1 √ 2 |∇ b r| 2 − i ∇ b r, J∇ b r = 1 √ 2 and r 11 = Z 1 Z 1 r − Γ 1 11 r 1 = Z 1 Z 1 r − g θ ([Z 1 , Z 1 ] , Z 1 ) r 1 = Z 1 Z 1 r − 1 √ 2 g θ ([Z 1 , Z 1 ] , Z 1 ) , we derive that r 11 is real by the commutation formula. Therefore, we have = α |r α1 | 2 + |r α1 | 2 + r α11 r α + r α11 r α ≥ |r 11 | 2 + |r 11 | 2 + r 111 r 1 + r 111 r 1 = 2r 2 11 + r 111 + ir 10 + R 1 111 r 1 r 1 + (r 11 − ir 0 ) 1 r 1 = 2r 2 11 + ∇ b r 11 , ∇ b r L θ + 1 2 R 1111 ≥ 2r 2 11 + (∇ b r) r 11 = 2r 2 11 + (∇r) r 11 = 2r 2 11 + ∂r 11 ∂r . Here we use the facts that r 1 = 1 √ 2 and r 11 is real, the equality (3.2), (3.3), and the commutation formulas (2.2). Together with the initial condition of r 11 as r goes to zero, we have r 11 ≤ 1 2r . In particular, (3.1) indicates that (log r) 11 = r 11 r − |r 1 | 2 r 2 ≤ 0. This completes the proof. Actually, the similar deductions enable us to derive the substantial CR sub-Laplacian comparisons below. Corollary 3.1. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curvature, then we have the CR sub-Laplacian comparison, for n ≥ 2, (3.5) ∆ b r ≤ 2 n r ; furthermore, if the pseudohermitian bisectional curvature is nonnegative, then (3.6) ∆ b r ≤ 2n − 1 r . Finally, the equality holds in (3.6) only if, for any j ∈ I n , (3.7) R 11jj = 0. Proof. By the similar computation as precedes, for any j = 1, (3.8) 0 = 1 2 |∇ b r| 2 jj = α |r αj | 2 + r αj 2 + r αjj r α + r αjj r α ≥ r jj 2 + r 1jj r 1 + r 1jj r 1 = r 2 jj + r 1jj + ir 10 + R 1 1jj r 1 r 1 + r 1jj r 1 ≥ r 2 jj + ∇ b r jj , ∇ b r L θ = r 2 jj + (∇ b r) r jj = r 2 jj + ∂ ∂r r jj , and the inequality (3.1), it's easy to derive ∆ b r ≤ 2n − 1 r .) = {x ∈ M | r 1 < r (x) = d (p, x) < r 3 } for 0 < r 1 < r 3 , we define F (x) = (log r 3 − log r (x)) log M f (r 1 ) + (log r (x) − log r 1 ) log M f (r 3 ) and G (x) = (log r 3 − log r 1 ) log |f (x)| . May assume that M f (r 1 ) < M f (r 3 ). Let f be a CR-holomorphic function on M with (3.10) f 0 = 0. It suffices to claim G ≤ F on A (p; r 1 , r 3 ). It's clear that G ≤ F on the boundary ∂A (p; r 1 , r 3 ) of the annulus A (p; r 1 , r 3 ) . Suppose that G (x) > F (x) for some interior point x in A (p; r 1 , r 3 ), then we could choose a point q ∈ A (p; r 1 , r 3 ) such that the function (G − F ) attains the maximum value at q. If q / ∈ Cut (p), then i∂ b ∂ b (G − F ) (q) ≤ 0 by observing that the inequality i∂ b ∂ b (G − F ) (q) > 0, which says that (G − F ) αβ is positive definite, implies the positivity of the sub-Lplacian ∆ b (G − F ) (q) > 0 (this contradicts that q is a maximum point of (G − F )). In particular, (3.11) (G − F ) 11 (q) ≤ 0 where Z 1 = 1 √ 2 (∇ b r − iJ∇ b r) . Note that (G − F ) 0 (q) = 0 due to (3.10) and (3.3). On the other hand, it follows from [FOW] or [CHL] that there is a transverse Kähler structure at the point q and we denoted such local coordinates in some open neighborhood U of the point q by {z α , x} α∈In with T = ∂ ∂x and however, it's not enough to obtain the contradiction from (3.11). So we take a modified function F ǫ for replacing the original function F so that it enables us to get the contradiction. Z 1 (q) = ∂ ∂z 1 − θ ∂ ∂z 1 T | q . Set F ǫ (x) = a ǫ log (r (x) − ǫ) + b ǫ for any sufficiently small positive number ǫ. Here the two constants a ǫ and b ǫ are determined by the following equations: F ǫ (r j ) = F ǫ (∂B (p, r j )) = log r 3 r 1 log M f (r j ) for j = 1, 3. It's apparent to see F ǫ → F on the annulus A (p; r 1 , r 3 ) as ǫ → 0 + and a ǫ > 0. Let q ǫ be a maximum point in A (p; r 1 , r 3 ) of the function (G − F ǫ ). With the same deduction, we have (3.12) (G − F ǫ ) 11 (q ǫ ) ≤ 0 and G 11 (q ǫ ) = 0. From Lemma 3.1 again, we obtain (log (r − ǫ)) 11 (q ǫ ) = r 11 (r − ǫ) − 1 2 (r − ǫ) 2 < 0. Accordingly, we have (G − F ǫ ) 11 (q ǫ ) > 0. It contradicts with the inequality (3.12). This indicates that (G − F ǫ ) ≤ 0 in the annulus A (p; r 1 , r 3 ). If q ǫ ∈ Cut (p), then we adopt the trick of Calabi as follows. Choose a number ǫ 1 ∈ (0, ǫ) and the point p 1 lying on the minimal D-geodesic from p to q ǫ with d (p, p 1 ) = ǫ 1 . Set r (x) = d (p 1 , x) and consider the slight modification of the function F ǫ (x) F ǫ,ǫ 1 (x) = a ǫ log ( r + ǫ 1 − ǫ) + b ǫ . It's not hard to observe that F ǫ (q ǫ ) = F ǫ,ǫ 1 (q ǫ ) and F ǫ ≤ F ǫ,ǫ 1 ; hence, we know (G − F ǫ,ǫ 1 ) also attains the maximum value at q ǫ . Then, applying the similar argument as precedes, we still have G − F ǫ,ǫ 1 ≤ 0 in the annulus A (p; r 1 , r 3 ). Letting ǫ 1 → 0 + , then ǫ → 0 + , the validity of the CR three circle theorem is settled. As for the monotonicity of (1.2), it's easily derived by taking the 3-tuples (r 1 , r 2 , kr 2 ) and (r 1 , kr 1 , kr 2 ) into the convexity of the CR three circle theorem for 0 < r 1 ≤ r 2 < +∞. It's clear that we have the following sharp monotonicity: Proposition 3.1. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold on which CR three-circle theorem holds and ord p f ≥ k ≥ 1 for some f ∈ O CR (M) and p ∈ M, then M f (r) r k is increasing in terms of r. Proof. Let 0 < r 2 ≤ r 3 < +∞. Since the vanishing order of f at p is at least k, for any ǫ > 0, there is a sufficiently small number 0 < r 1 < r 2 , such that (3.13) log M f (r 1 ) ≤ log M f (r 3 ) + (k − ǫ) log r 1 r 3 . Substituting (3.13) into the inequality (1.1), we get M f (r 2 ) r k−ǫ 2 ≤ M f (r 3 ) r k−ǫ 3 . The proposition is accomplished by letting ǫ → 0 + . Therefore, Theorem 1.1 and Proposition 3.1 imply Before giving a variety of applications of the CR three-circle theorem, we need some notations about the CR-holomorphic functions of polynomial growth. Definition 3.1. Let (M, J, θ) be a complete noncompact pseudohermitian (2n + 1)-manifold. We denote the collections Proof. It's straightforward to see that the sufficient part holds. On the other hand, let 0 < r 1 ≤ r 2 < +∞, by the assumption that f ∈ O CR (M) lim sup r→+∞ M f (r) r d < +∞ and f ∈ O CR (M) lim inf r→+∞ M f (r) r d < +∞ by O CR d (M) and O CR d (M), respectively. It's clear that O CR d (M) ⊆ O CR d (M) ∩ O CR d (M).f ∈ O CR d (M), for any positive number ǫ, there's a sequence {λ j } ր +∞ such that log M f (λ j ) ≤ log M f (r 1 ) + (d + ǫ) log λ j . From Theorem 1.1, by taking r 3 = λ j , it follows that log M f (r 2 ) ≤ log M f (r 1 ) + (d + ǫ) log r 2 r 1 . Let ǫ go to zero, the necessary part follows. From the last proposition, it's easy to deduce the conclusion below. Corollary 3.3. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature, then O CR d (M) = O CR d (M). In addition, we have the asymptotic property for the degree of CR-holomorphic functions of polynomial growth as follows: Corollary 3.4. Let (M, J, θ) be a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature. If f ∈ O CR d+ǫ (M) for any number ǫ > 0, then f ∈ O CR d (M). Proof. From Proposition 3.2 and Corollary 3.3, we know M f (r) r d+ǫ is decreasing with respect to r for any positive number ǫ. By acontradiction, it's easy to validate the monotonicity of M f (r) r d . Then, by the last two corollaries, we have f ∈ O CR d (M). From Corollary 3.4, we know As an application, we could recover and generalize the CR sharp dimension estimate in [CHL] under the assumption of nonnegative pseudohermitian sectional curvature instead of nonnegative pseudohermitian bisectional curvature. Proof. (of Theorem 1.2) Suppose on the contrary, i.e. dim C O CR d (M) > dim C O CR d (H n ) for some positive integer d ∈ N. Then for any point p ∈ M, there's a nonzero CR-holomorphic function f of polynomial growth of degree at most d with ord p f ≥ d + 1 Concerning the existence of such function f , one could refer to the proof of Theorem 1.1 in [CHL,(5.17)] which is just a method of the Poincaré-Siegel argument via linear algebra ( [M1]). Therefore, we have lim r→0 + M f (r) r d = 0. However, this contradicts with the monotonicity of the function M f (r) r d as in Proposition 3.2. Hence, the sharp dimension estimate holds. As for the rigidity part, we just claim that if (3.14) dim C O CR d (M) = dim C O CR d (H n ) , then (M, J, θ) is CR-isomorphic to (2n+1)-dimensional Heisenberg group H n . From Proposition 4.1 in [T] (or Theorem 7.15 in [B]), it suffices to show that M has constant J-holomorphic sectional curvature −3. From the equation, for any p ∈ M, Z ∈ (T 1,0 M) p with |Z| = 1, R θ Z, Z, Z, Z = R Z, Z, Z, Z + g θ Z ∧ Z Z, Z + 2dθ Z, Z g θ JZ, Z = R Z, Z, Z, Z − 3, the proof of the rigidity part is completed if we justify that the pseudohermitian sectional curvature vanishes. Adopting the notations as in Theorem 1.1, we just claim that, for simplification, (3.15) R Z 1 , Z 1 , Z 1 , Z 1 (p) = 0 where Z 1 = ∂ ∂z 1 − θ ∂ ∂z 1 T. The equality (3.14) enables us to see that there is a function f ∈ O CR d (M) such that f (z 1 , ..., z n , x) = z d 1 + O r d+1 locally. This impies that ord p f = d. Therefore, from Corollary 3.2 and Proposition 3.2, we obtain M f (r) r d is constant. In the proofs of Theorem 1.1 and Lemma 3.1, it's not difficult to find that G − F attains the maximum value 0 on ∂B (p, r) at the point q (r) for any positive number r and then R (∇ b r, J∇ b r, ∇ b r, J∇ b r) (q (r)) = 0. From the definition of the chosen function f , we could take a subsequence {(∇ b r) (q (r j ))} j∈N such that its limit, as r → 0 + , lies in the tangent space at p spanned by ∂ ∂z 1 | p and T | p . Then the equality (3.15) holds by the formula (2.1). Accordingly, this theorem is accomplished. An extension of CR Three-Circle Theorem Subsequently, we will give the proof of the CR three-circle theorem when the pseudohermitian sectional curvature is bounded from below by a function. Proof. (of Theorem 1.3) Although this proof is similar to the one of the CR three-circle theorem, we give its proof for completeness. Here we adopt the same notations as in the proof of Theorem 1.1. Define F (x) = (h (r 3 ) − h (r (x))) log M f (r 1 ) + (h (r (x)) − h (r 1 )) log M f (r 3 ) and G (x) = (h (r 3 ) − h (r 1 )) log |f (x)| on the annulus A (p; r 1 , r 3 ) for 0 < r 1 < r 3 < +∞. We still assume that (4.1) M f (r 1 ) < M f (r 3 ) . It's clear that G ≤ F on the boundary ∂A (p; r 1 , r 3 ) by (1.5). It suffices to show that G ≤ F on the annulus A (p; r 1 , r 3 ) to reach our first conclusion by the maximum principle. Suppose that G (x) > F (x) for some interior point x in A (p; r 1 , r 3 ), then we could choose a point q ∈ A (p; r 1 , r 3 ) such that the function (G − F ) attains the maximum value at q. If q / ∈ Cut (p), then (4.2) (G − F ) 11 (q) ≤ 0. With the same deduction in Theorem 1.1, we have G 11 (q) = 0 from the Poincaré-Lelong equation and the fact that f (q) = 0 with the help of the transverse Kähler structure. Due to the fact that h (r) ∼ log r as r → 0 + , we could define F ǫ (x) = a ǫ log e h(r) − ǫ + b ǫ for any sufficiently small number ǫ > 0 and the two constants a ǫ and b ǫ are restricted by the following equations F ǫ (r j ) = (h (r 3 ) − h (r 1 )) log M f (r j ) for j = 1, 3. It's obvious that F ǫ → F on the annulus A (p; r 1 , r 3 ) as ǫ → 0 + . Due to the inequality (1.5) and the assumption (4.1), we see that a ǫ > 0. Denote by q ǫ a maximum point in A (p; r 1 , r 3 ) of the function (G − F ǫ ) and modify the point q into the point q ǫ . From (1.4), (3.4), (1.3), and the initial condition u (r) ∼ 1 2r as r → 0 + imply that (4.3) r 11 ≤ u (r) . By the hypotheses (1.5), (1.6) and the inequality (4.3), we get log e h(r) − ǫ 11 (q ǫ ) = −ǫe h (h ′ ) 2 |r 1 | 2 +(e h(r) −ǫ)(e h h ′′ |r 1 | 2 +e h h ′ r 11 ) (e h(r) −ǫ) 2 ≤ − ǫe h (h ′ ) 2 2(e h(r) −ǫ) 2 < 0 for sufficiently small positive number ǫ. It yields that (G − F ǫ ) 11 (q ǫ ) > 0. This contradicts with (4.2). Therefore we obtain G ≤ F ǫ for sufficeintly small number ǫ > 0. If q ∈ Cut (p), then, by the trick of Calabi again, let ǫ 1 ∈ (0, ǫ) and the point p 1 lying on the minimal D-geodesic from p to q ǫ with d (p, p 1 ) = ǫ 1 . Set r (x) = d (p 1 , x) and consider the modified function F ǫ,ǫ 1 (x) of the function F ǫ (x) F ǫ,ǫ 1 (x) = a ǫ log e h( r+ǫ 1 ) − ǫ + b ǫ . Due to the monotonicity of the function h, we see that F ǫ ≤ F ǫ,ǫ 1 . It's clear that F ǫ (q ǫ ) = F ǫ,ǫ 1 (q ǫ ). So the point q ǫ is still a maximum point of (G − F ǫ,ǫ 1 ). Set Z 1 = 1 √ 2 (∇ b r − iJ∇ b r) . By observing the expansion of log e h( r+ǫ 1 ) − ǫ 1 1 (q ǫ ), (1.5), (1.6), and the continuity of the pseudohermitian sectional curvature imply that (F ǫ,ǫ 1 ) 1 1 (q ǫ ) < 0 for sufficiently small ǫ 1 > 0 for fixed ǫ. Here the property that the pseudohermitian sectional curvature is continuous is utilized to obtain the estimate r 1 1 ≤ u + ǫ ′ for small positive error ǫ ′ = ǫ ′ (ǫ 1 ). Then (G − F ǫ,ǫ 1 ) 1 1 (q ǫ ) > 0. However, it contradicts with the fact that (G − F ǫ,ǫ 1 ) attains a maximum point at q ǫ . Accordingly, the inequality G ≤ F ǫ,ǫ 1 holds. Letting ǫ 1 → 0 + , then ǫ → 0 + , we have G ≤ F on the annulus A (p; r 1 , r 3 ). Because ord p (f ) = d and h (r) ∼ log r as r → 0 + , then we have, for any ǫ > 0, log M f (r 1 ) ≤ log M f (r 2 ) + (d − ǫ) (h (r 1 ) − h (r 2 )) for sufficiently small positive number r 1 and r 1 < r 2 . By the convexity of log M f (r) with respect to the function h (r) log M f (r) ≤ h (r 2 ) − h (r) h (r 2 ) − h (r 1 ) log M f (r 1 ) + h (r) − h (r 1 ) h (r 2 ) − h (r 1 ) log M f (r 2 ) for r 1 ≤ r ≤ r 2 , we obtain the monotonicity of (r)) . This completes the proof. M f (r) exp(dh Choosing the functions g (r) = −1, u (r) = (e 2r +1) 2(e 2r −1) , and h (r) = log e r −1 e r +1 in Theorem 1.3, we have the following consequence: Similarly, choosing the functions g (r) = 1, u (r) = 1 2 cot r, and h (r) = log tan r 2 in Theorem 1.3, we obtain Corollary 4.2. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with the pseudohermitian sectional curvature bounded from below by 1, f ∈ O CR (B (p, R)), then log M f (r) is convex with respect to the function log tan r 2 . In particular, M f (r) (tan r 2 ) d is increasing for ord p (f ) = d. With the help of Theorem 1.3, we have the dimension estimate when the pseudohermitian sectional curvature is asymptotically nonnegative. Proof. (of Theorem 1.4) Although the proof is almost the same as in [Liu1], we give its proof for completeness. May assume ǫ < 1 2 . Choose u (r) = 1 2r + A (1 + r) 1+ǫ ; hence, the inequality holds 2u 2 + u ′ − 1 2 A (1 + r) 2+ǫ ≥ 0. Therefore we have the dimension estimate dim C O CR d (M) ≤ dim C O CR [d exp( 2A ǫ )] (H n ) = C (ǫ, A) d n for any d ∈ N. If d ≤ e − 3A ǫ , then dim C O CR d (M) ≤ dim C O d exp( 2A ǫ ) (C n ) ≤ dim C O exp ( − A ǫ ) (C n ) = 1. Last, if A ǫ ≤ 1 4d , then d exp 2A ǫ < d + 1 and dim C O CR d (M) ≤ dim C O CR d (H n ) . This theorem is accomplished. It's not difficult to observe that Theorem 1.4 includes the case when the pseudohermitian sectional curvature is nonnegative outside a compact set as follows: Furthermore, there exists a number δ (λ) > 0 such that dim C O CR δ(λ) (M) = 1. The Levi form induces naturally a Hermitian form on the dual bundle of T 1,0 M, denoted by , L * θ , and hence on all the induced tensor bundles. Integrating the Hermitian form (when acting on sections) over M with respect to the volume form dµ = θ ∧ (dθ) n , we get an inner product on the space of sections of each tensor bundle. For any fixed point x ∈ M, a CR-holomorphic function f is called to be of polynomial growth if there are a nonnegative number d and a positive constant C = C (x, d, f ), depending on x, d and f , such that |f (y)| ≤ C (1 + d cc (x, y)) d for all y ∈ M, where d cc (x, y) denotes the Carnot-Carathéodory distance between x and y. Furthermore, we could define the degree of a CR-holomorphic function f of polynomial growth by Lemma 3. 1 . 1If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature, the Carnot-Carathéodory distance function r (x) = d cc (p, x) from a fixed point p to a point x in M is smooth at q ∈ M and Z Remark 3. 1 . 1When (M, J, θ) is a complete noncompact pseudohermitian 3-manifold of vanishing torsion with nonnegative pseudohermitian Ricci curvature, it's also easy to derive ∆ b r ≤ 1 r from the estimate (3.9). The estimate is sharp in the sense of the equality holds only if M is flat as in the proof of Theorem 1.2 . Restrict to the leaf space D = [x = 0] and write the point y in U as ( y, x). It's clear that q = ( q, 0). Hereafter the quantity with the tilde means such one lies in the slice D. This enables us to transfer the local property of the Kähler manifolds to the CR manifolds. Let G and F denote the restrictions of G and F to the leaf space D. So q is a maximum point of G − F and f = f | D is a holomorphic function on U ∩ D. Because G − F attains the maximum value at q (this implies that f ( q) f is the divisor of f . However, the condition (3.10) implies that G is independent of the characteristic direction T. So we have the equality G 11 (q) = 0. This and Lemma 3.1 indicate that (G − F ) 11 (q) ≥ 0; Corollary 3. 2 . 2If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature, and ord p f ≥ k ≥ 1 for some f ∈ O CR (M) and p ∈ M, then M f (r) r k is increasing in terms of r. Proposition 3. 2 . 2If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature, then f ∈ O CR d (M) if and only if M f (r) r d is decreasing with respect to r for any f ∈ O CR (M). Corollary 4 . 1 . 41If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with the pseudohermitian sectional curvature bounded from below by −1, f ∈ O CR (M), then log M f (r) is convex with respect to the function log e r −1 e r +1 . In particular, M f (r) ( e r −1 e r +1 ) d is increasing for ord p (f ) = d. h (r) ≥ exp − 2A ǫ log r + Cfor any number r ≥ 1. Here C = C (A, ǫ). Theorem 1.3 implies that if the vanishing orderord p (f ) of f ∈ O CR (M) at p is equal to d, then M f (r) exp(dh(r)) is increasing with respect to r. So (4.4) M f (r) ≥ exp (dh (r)CR d (M) −→ C q([d exp( 2A ǫ )]) f −→ (D α f ) |α|≤[d exp( 2A ǫ )]where q (m) = n+m n for any m ∈ N and [a] denotes the greatest integer less than or equal to a. We would claim that Φ is injective;for if 0 = f ∈ O CR d (M) and D α f = 0 for any |α| ≤ d ′ = d exp 2A ǫ , then ord p (f ) ≥ d ′ + 1.Hence, by (4.4), we obtainM f (r) ≥ C 1 r (1+d ′ ) exp(− 2A ǫ ) ;however, this contradicts with the fact f ∈ O CR d (M) . Corollary 4. 3 . 3Let (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion of which the pseudohermitian sectional curvature is nonnegative outside a compact subset K and is bounded from below by −a for some a > 0 on M. If λ = a (d c (K)) 2 where d c (K) denotes the diameter of K, then there is a positive constant C (λ, n) such that dim C O CR d (M) ≤ C (λ, n) d n for any positive integer d. For any d ∈ N, there is a positive number ǫ (d) such that if λ ≤ ǫ (d), then we have dim C O CR d (M) ≤ dim C O CR d (H n ) . ).Theorem 1.2. If (M, J, θ) is a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion with nonnegative pseudohermitian sectional curvature, then dim C O CR d (M) ≤ dim C O CR d (H n ) for any positive integer d ∈ N; morever, if M is simply connected, then the equality holds if only if (M, J, θ) is CR-isomorphic to (2n + 1)-dimensional Heisenberg group H n . Remark 1.2. 1. In the forthcoming paper, one of the most important application for CR three-circle theorem, we expect that there exists a nonconstant CR-holomorphic function of polynomial growth on a complete noncompact pseudohermitian (2n + 1)-manifold (M, J, θ) of vanishing torsion, nonnegative pseudohermitian bisectional curvature and maximal volume growth. This is the first step toward the second CR Yau's uniformization conjecture ([Liu2]). 2. As in the paper of G. Liu ([Liu3]), by applying Cheeger-Colding theory for the Webster metric on CR manifolds and Hörmander L 2 -technique of ∂ b on the space of basic forms and this CR three-circle theorem, we shall work on the third CR Yau's uniformization conjecture as well. * SHU-CHENG CHANG, † YINGBO HAN, AND * * CHIEN LIN D E Blair, Riemannian geometry of contact and symplectic manifolds. Birkhauser203D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, Volume 203, Birkhauser 2002. 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{'abstract': "This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the first step toward second and third CR Yau's uniformization conjecture. Moreover, in the course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian comparison theorem. Then Liouville theorem holds for positive pseudoharmonic functions in a complete noncompact pseudohermitian (2n + 1)-manifold of vanishing torsion and nonnegative pseudohermitian Ricci curvature.1991 Mathematics Subject Classification. Primary 32V05, 32V20; Secondary 53C56.", 'arxivid': '1801.08858', 'author': ['Sasakian Manifolds ', 'Shu-Cheng Chang ', 'ANDYingbo Han ', 'Chien Lin '], 'authoraffiliation': [], 'corpusid': 113398126, 'doi': '10.1007/s00526-019-1538-8', 'github_urls': [], 'n_tokens_mistral': 21026, 'n_tokens_neox': 17767, 'n_words': 10397, 'pdfsha': 'ce29c249d11e7913a8e35c3be52c99caf23381dc', 'pdfurls': ['https://arxiv.org/pdf/1801.08858v2.pdf'], 'title': ['ON THE THREE-CIRCLE THEOREM AND ITS APPLICATIONS IN', 'ON THE THREE-CIRCLE THEOREM AND ITS APPLICATIONS IN'], 'venue': []}

arxiv

Connecting Tikhonov regularization to the maximum entropy method for the analytic continuation of quantum Monte Carlo data Khaldoon Ghanem Leopoldstrasse 18080804Quantinuum, MunichGermany Erik Koch Jülich Supercomputer Centre Forschungszentrum Jülich 52425JülichGermany JARA High-Performance Computing 52425JülichGermany Connecting Tikhonov regularization to the maximum entropy method for the analytic continuation of quantum Monte Carlo data (Dated: December 13, 2022) Analytic continuation is an essential step in extracting information about the dynamical properties of physical systems from quantum Monte Carlo (QMC) simulations. Different methods for analytic continuation have been proposed and are still being developed. This paper explores a regularization method based on the repeated application of Tikhonov regularization under the discrepancy principle. The method can be readily implemented in any linear algebra package and gives results surprisingly close to the maximum entropy method (MaxEnt). We analyze the method in detail and demonstrate its connection to MaxEnt. In addition, we provide a straightforward method for estimating the noise level of QMC data, which is helpful for practical applications of the discrepancy principle when the noise level is not known reliably. Analytic continuation is an essential step in extracting information about the dynamical properties of physical systems from quantum Monte Carlo (QMC) simulations. Different methods for analytic continuation have been proposed and are still being developed. This paper explores a regularization method based on the repeated application of Tikhonov regularization under the discrepancy principle. The method can be readily implemented in any linear algebra package and gives results surprisingly close to the maximum entropy method (MaxEnt). We analyze the method in detail and demonstrate its connection to MaxEnt. In addition, we provide a straightforward method for estimating the noise level of QMC data, which is helpful for practical applications of the discrepancy principle when the noise level is not known reliably. I. ANALYTIC CONTINUATION: AN ILL-POSED PROBLEM From a mathematical perspective, the analytic continuation problem corresponds to solving a Fredholm integral equation of the first kind g(y) = dxK(y, x)f (x) ,(1) where f (x) is the unknown spectrum, a non-negative integrable function. K(y, x) is the kernel of the integral equation and is known analytically, while g(y) is noisy data, typically obtained from QMC simulation at a finite number of points y j . To solve the analytic continuation numerically, the integral is discretized using a grid of n points x i , giving a linear system of equations g = Kf ,(2) where the elements of the matrix K are the kernel values K(y j , x i ), g contains m measured data values g(y j ) and f i is the spectrum integral over the i-th grid interval. The most naive and straightforward way of solving Eq. (2) is, as with any other linear system of equations, using the weighted least squares method f LS = arg min f χ 2 (f ) ,(3) which finds the spectrum minimizing the fit to the data χ 2 (f ) := (g − K f ) T C −1 (g − K f ) .(4) The fit is weighted by the inverse of C, the covariance matrix of the noise on the data. By factorizing the covariance matrix into C −1 = T T T, one can always replace the kernel matrix and data vector by the weighted ones TK and Tg, respectively. Then the covariance matrix of the weighted data becomes the identity matrix, and one can use the ordinary least squares method instead. In the following, we will always assume that such transformation has been applied to the kernel and the data despite using the same notation K and g to denote the weighted ones. Using the least squares solution for solving the analytic continuation problem gives generally bad results plagued by noise, as exemplified in Fig. 1. The reason is that the matrices in analytic continuation problems are highly ill-conditioned such that the inevitable small noise on the data leads to disastrous noise on the least-squares solution [1,2]. This can be seen more explicitly using the singular value decomposition (SVD) of the kernel matrix K = USV T ,(5) where S is a diagonal matrix of size m × n, and U and V are unitary matrices of sizes m×m and n×n, respectively. The columns of the matrix U form an orthonormal basis of the data space and are called the data modes, while the columns of the matrix V, which span the space of spectra, are called the spectral modes. The diagonal elements of S are the singular values, and they are sorted in descending order. Using the SVD, the least squares solution can be written as f LS = min(m,n) i u T i g s i v i .(6) For matrices arising from analytic continuation problems, the singular values decay exponentially to zero (see Fig. 2). Dividing by these vanishing singular values hugely amplifies any small noise present in the data. This is the main problem with the least-squares solution. The other source of ill-posedness is the incompleteness of the data, i.e., we only know the data at a finite number of points m < n, where n is typically chosen large enough to resolve the desired features of the spectrum. Therefore, even for numerically exact data, if no regular- Least squares solution (bottom panel) for the analytic continuation of optical conductivity σ(ω) using noisy data of its correlation function. The exact correlation function is computed analytically from the exact optical conductivity (top panel) on the first m = 60 bosonic Matsubara frequencies with inverse temperature β = 15. The input data includes relative Gaussian noise with standard deviation 10 −2 . This test case is an adaptation of the ones proposed by Ref. [3] and studied further in Refs. [4][5][6]. In the notation of the latter reference, the optical conductivity used here differs in the values of the following parameters: Γe = 20, 1 = 15. We denote this data set as test case 1. ization/additional information is provided, one can only ever hope to recover at most the first m modes of the spectrum. II. NOISE ESTIMATION The SVD of the kernel matrix allows an accurate estimation of the overall scale of noise on QMC data. This can be valuable in practical situations where such an estimate is unavailable, or as an important cross-check of the validity of the noise level estimate. As a start, let us assume, as usual, that an estimate of the covariance matrix C already exists and that the data and kernel have been weighted by T, the square root of its inverse. Consequently, the noise on the different components of the weighted data vector g is uncorrelated and has a unit variance. Since the matrix U is unitary, the noise i present in the expansion coefficients of the data u T i g is also uncorrelated and has a unit variance. These noisy data coefficients are then related to the exact Singular values of the (weighted) kernel of test case 1. The singular values decay exponentially until leveling off at a value determined by the machine epsilon. In the inset, we show some of the spectral modes. The leading spectral modes are smooth and slowly varying functions. As the mode index increases, the number of nodes increases, and the modes become more oscillatory. Once the singular values reach numerical accuracy, the corresponding modes become numerically degenerate so that the SVD routine returns arbitrary linear combinations of the exact modes. spectrum via the relation u T i g = s i v T i f exact + i .(7) Given that the exact spectrum has a finite norm and that the singular values in analytic continuation decay exponentially, there is some index k, after which the exact data coefficients become negligible compared to the noise. For these indices, the measured data coefficients are practically plain noise u i T g ≈ i : k < i ≤ m ,(8) and can be used to estimate the variance of the noise i as σ 2 ( ) ≈ 1 m − k m i=k+1 (u i T g) 2 ,(9) where the formula for estimating population variance with a known mean of value zero has been employed. In practice, the cutoff k can be safely chosen as the numerical rank of K, i.e., the index at which the singular values hit numerical accuracy. When the covariance matrix C is properly scaled, we expect this value to be close to one. This is illustrated in Fig. 3 for test case 1, where the data coefficients decay exponentially till they reach the noise level σ( ) = 1 and fluctuate around it. However, when a covariance matrix with the wrong scaling is used, the aforementioned plateau of data coefficients will be scaled accordingly, and σ( ) will deviate from the expected value of one. Values much larger than one indicate that the noise level has been underestimated, while values much lower than one indicate an overestimation of the noise level. An important practical use case of the above formula is estimating the noise level of uncorrelated relative Gaussian noise. In this case, as an initial Ansatz, one can use a diagonal covariance matrix whose diagonal elements are the squares of the data values. Eq. (9) then provides an estimate of σ 2 , which can be multiplied by the ansatz to obtain a properly-scaled covariance matrix. III. TIKHONOV REGULARIZATION The expansion of the least squares solution using SVD modes [cf. Eq. (6)] already suggests a direct remedy to the ill-posedness; namely, truncating the later modes, which are dominated by noise, while keeping the leading ones that are more stable. This is known as the Truncated SVD solution. Tikhonov regularization [7,8] is a more refined method, where the noisy modes are turned off continuously, with each term in the least squares solution multiplied by a filtering function φ(s; α) := s 2 / s 2 + α that depends on its singular value s and an adjustable parameter α: f Tikhonov (α) = min(m,n) i φ(s i ; α) u T i g s i v i .(10) Terms corresponding to very small singular values s 2 i α are practically removed, while ones corresponding to large singular values s 2 i α are hardly modified [9]. It can be shown that the above Tikhonov solution is the least squares solution of an alternative problem with extended data and an extended kernel f Tikhonov (α) = arg min f K √ α I f − g 0 2 ,(11) where I is the unit matrix in the n-dimensional space of spectra. This formulation has a computational advantage for large-scale problems because it allows getting the Tikhonov solution using any linear solver without explicit computation of the singular value decomposition. Moreover, this least squares problem can be written as the following minimization problem f Tikhonov (α) = arg min f χ 2 (f ) + α f 2 ,(12) that aims to balance the fit to the data with the L 2 -norm of the spectrum vector. The balance is controlled by the regularization parameter α. When α is very small, we approach the least squares solution, which fits the data very well but has a very large L 2 -norm. As α increases, more modes get filtered, and the norm gets smaller while the fit gets worse. The smoothness typically associated with Tikhonov solutions comes from the fact that the leading modes are smoother than later ones for analytic continuation kernels (see, for example, the insets of Fig. 2). While the aforementioned form of Tikhonov regularization is the most basic and widely used one in the inverse problem literature [10], it has two drawbacks for analytic continuation problems. The first is that the discretized L 2 -norm is grid-dependent because the spectral values f i := w i f (x i ) include the full weight of the grid interval at point x i . Using a grid with n points and a grid density ρ(x), these weights are defined as w i := 1/ [N ρ(x i )] and the L 2 -norm of the spectrum reads f 2 = i f 2 i = i [w i f (x i )] 2 ≈ 1 N dx f 2 (x) ρ(x) . (13) This shows that the basic form of Tikhonov has an implicit dependence on the grid density [11]. We suggest replacing this implicit dependence with an explicit one on a default model d(x). Let d i := w i d(x i ) be the integral of the default model over the i-th grid interval, then we replace the usual L 2 -norm i f 2 i with the weighted L 2 -norm i f 2 i /d i . It can be easily verified that the weighted norm is indeed grid-independent. The second drawback is that the solution approaches zero in the limit of large regularization parameter α. In analytic continuation, however, we know that the spectrum must have a finite L 1 -norm, so it would be desirable if the solution would approach some properly normalized spectrum in the limit of large α. We choose to center our regularization term at the default model d instead of zero. In summary, we propose using the following form of Tikhonov regularization in analytic continuation prob- lems f Tikhonov (α, d) = arg max f − 1 2 χ 2 (f ) + αT (f |d) , (14) where the Tikhonov penalty term is defined as T (f |d) = − 1 2 i (f i − d i ) 2 d i .(15) It is worth noting that, like the original form, this new formulation can be solved as an extended least squares problem T . Its solution can be similarly expressed in terms of the SVD of the rescaled kernel K √ D as shown in appendix A. f Tikhonov (α, d) = arg min f K √ α D −1 f − g √ αDe 2 ,(16) IV. DISCREPANCY PRINCIPLE Choosing the value of the regularization parameter α is an essential ingredient of any regularization method. Apart from the obvious criterion that α should be smaller for more accurate data, there is no unique procedure for actually determining its value. Any such procedure should strike a balance between fitting the noise and biasing the solution. A common method in the inverse problem literature is the discrepancy principle [12,13]. According to the discrepancy principle, a good spectrum would produce data such that the residual vector r := g − Kf is dominated by noise. Therefore, we should choose α such that the norm of the residual r 2 = χ 2 (f ) equals the expected norm of the noise vector. Assuming, as usual, that data and kernel have been reweighed with the square root of the noise covariance, the expected norm-squared of the noise vector follows the well-known chi-squared distribution. The mean value of this distribution equals the number of data points m, and its variance equals 2m. To avoid accidental over-fitting of noise, one may apply the discrepancy principle using a value (in terms of the standard deviation) somewhat larger than the mean. In this work, however, we always use the mean value. Interestingly, the Tikhonov solution using the discrepancy principle can be written in a form independent of any regularization parameter α as a maximization of the Tikhonov penalty f Tikhonov (d) = arg max f ∈C T (f |d) ,(17) over the manifold C defined by the discrepancy principle C := f ∈ R n : χ 2 (f ) = m .(18) Starting from some spectrum on the manifold C, the Tikhonov solution can then be found by following the gradient of T (f |d), projected on C: a ⊥ = I − z z T z T z a ,(19) where a := ∇T is the gradient of the Tikhonov penalty with a i = − f i − d i d i ,(20) and z := − 1 2 ∇χ 2 is the gradient of the fit function i.e. the surface normal of C with z i = k T i [g − Kf ] ,(21) where k i is the i-th column of the Kernel matrix K. At the optimal point, the projection vanishes, and the gradient of T must be anti-parallel to the fit gradient α a = −z ,(22) which is nothing but the stationarity condition for (14). The optimal regularization parameter α thus reemerges as the ratio of the two gradients at the optimal point. In practice, this constrained optimization problem is converted, using the method of Lagrange multiplier, into an unconstrained optimization of the objective function f Tikhonov (d) = arg max f ,β T (f |d) − β 2 χ 2 (f ) − m ,(23) where the Lagrange multiplier β corresponds to the inverse of the regularization parameter α. V. SELF-CONSISTENT TIKHONOV Tikhonov regularization provides a simple and fast method to obtain a decent first impression of the analytic continuation solution. Its obvious disadvantage, however, is ignoring the non-negativity of the spectrum (see Fig. 4). One can enforce the non-negativity by explicitly restricting the optimization problem to non-negative spectra. This can be done straightforwardly by using the non-negative least squares method [14] with the extended kernel and data of Eq. (16). Nevertheless, enforcing the non-negativity in this artificial way does not improve the results as desired. As shown in Fig. 4, the non-negative Tikhonov solution looks like a clamped version of the original Tikhonov solution where the negative parts are set to zero, while the positive part stays roughly the same with minor adjustments to account for the truncated negative values. Instead of enforcing the non-negativity constraint directly, one can reduce violations by increasing the regularization parameter α, which encourages the solution to be close to the non-negative default model. Under the discrepancy principle, the regularization parameter is determined implicitly and only has a large value if the default model fits the data well. This transforms the problem of satisfying non-negativity into one of improving the fit of the default model. In the limit, when the default model itself satisfies the discrepancy principle, it is its own Tikhonov solution, and thus non-negativity is guaranteed. A simple way of improving the fit of a default model is by linearly mixing it with its Tikhonov solution under the discrepancy principle d ← [1 − µ] d + µ f Tikhonov (d) .(24) Assuming the fit of the starting default model is worse than m, the new default model is guaranteed to have a better fit due to the convexity of the fit function χ 2 . Additionally, if the starting default model is strictly positive, we can always choose the positive mixing parameter µ small enough such that the new default model is also positive. The values of the mixing parameter that guarantee the positivity of the new default model can be calculated explicitly from the values of the starting default model and its Tikhonov solution as µ < min d i d i − f i : f i < d i .(25) These observations suggest an iterative approach to obtain an improved non-negative Tikhonov solution. In this approach, we keep linearly mixing the default model with its Tikhonov solution to obtain a new, improved default model until the difference between the default model and its Tikhonov solution becomes negligible. We call this method Self-Consistent Tikhonov (SCT). For the mixing parameter µ, we use half the maximum allowed value [cf. Eq. (25)]. Using this value implies that the updated default model has at least half its original value at any point. This mixing strategy works well for most cases, but it can sometimes lead to slow convergence when the exact spectrum has values very close to zero (e.g., at the tail of a Gaussian peak). To accelerate the convergence of such cases, we put a lower limit on the mixing parameter µ. This may lead to a violation of the positivity of the default model, which can be directly reinforced by truncating values lower than some positive threshold. It should be emphasized that these limits are not strictly necessary, but help accelerate convergence in pathological cases. In Fig. 5, we plot a set of default models produced by SCT for test case 1 at different iterations. The default model gradually transforms and fits the data till it converges, with the converged solution satisfying the discrepancy principle. This solution represents a significant improvement over the original Tikhonov solution and its non-negative counterpart (see Fig. 4). Besides providing a smooth non-negative spectrum, the shape and width of the peaks are much better reproduced. In the same plot, we also show the solution of the Max-Ent method using the same starting default model, d (0) , and a regularization parameter that is also determined by the discrepancy principle. Remarkably, the MaxEnt solution is indistinguishably close to SCT solution. By examining different other test cases, we have always found that the solutions of MaxEnt and SCT are quite similar and in many cases virtually identical (see Fig. 6 for another example). The following sections will examine and clarify this surprising connection between MaxEnt and SCT. In this context, it is worth noting that MaxEnt has also been recently connected to a specific variant of the average spectrum method, a stochastic method for analytic continuation [15]. Figure 6. Comparison of MaxEnt and SCT for a variant of test case 1. This case differs by the location of the second peak and the width of the envelope. In the notation of reference [6], the optical conductivity used here differs in the values of the following parameters: Γe = 4, 1 = 3. We denote this data set as test case 2. The default model used here is a scaled Gaussian of width 6. VI. MAXIMUM ENTROPY METHOD Similarly to Tikhonov regularization, the Maximum Entropy Method (MaxEnt) introduces a term that penalizes the mismatch between a spectrum and a default model [16][17][18][19]. The penalty term, known as Shannon entropy, is defined as S(f |d) := N i=1 f i − d i − f i ln f i d i .(26) It represents the expected amount of information in a spectrum f relative to the default model d. This entropy is then optimized in MaxEnt simultaneously alongside the data fit f MaxEnt (α, d) = arg max − 1 2 χ 2 (f ) + αS (f | d) . (27) The fit and entropy trade-off is controlled via the regularization parameter α. When α is infinitesimally small, MaxEnt formally gives the non-negative least-squares solution, but as α increases, the solution gets smoother and closer to the default model. There are different "flavors" of MaxEnt depending on how α is chosen [20]. The most relevant for our purpose is the one known as Historic MaxEnt. In this method, α is chosen such that the fit χ 2 equals the number of the data points m. This choice is equivalent to the discrepancy principle when the data and the kernel are transformed so that the noise on the data becomes uncorrelated and has unit variance. Other commonly-used methods for choosing α are the classical MaxEnt and Brayn's MaxEnt. Both methods derive a probability distribution over α using Bayesian theory and use either the maximum of this distribution (Classical MaxEnt) or its average (Bryan's MaxEnt) as the final solution. In the rest of the paper, we will always assume that the discrepancy principle is applied, and thus, MaxEnt refers to the original way of choosing α, i.e., f MaxEnt (d) = arg max f ∈C S (f |d) ,(28) where C is the manifold defined by the discrepancy principle in Eq. (18). The Shannon entropy is directly related to the Tikhonov regularization term, T (f |d) being the entropy expanded to second order in ∆ i : = f i − d i S(f |d) = i ∆ i − ∆ i + d i ln 1+ ∆ i d i ≈ i ∆ i − ∆ 2 i d i − d i ∆ i d i − ∆ 2 i 2d 2 i = T (f |d) . (29) This means that the Tikhonov method can be considered an approximation to MaxEnt. The quality of this approximation depends on how close the starting default model d is to the hypersurface defined by the discrepancy principle C. When the default model satisfies the discrepancy principle, then MaxEnt and Tikhonov give the same solution -the default model itself. As the fit of the default model deteriorates, it gets further away from that hypersurface, and the maxima of the penalty terms S and T in C start to diverge. A more quantitative analysis of the difference between MaxEnt and Tikhonov solutions is given in Appendix B. VII. MAXENT FAMILY OF EQUIVALENT DEFAULT MODELS Analogously to the discussion in Sec. IV about optimizing the Tikhonov penalty, maximizing the Shannon entropy under the discrepancy constraint can also be achieved by following its gradient, projected on C: b ⊥ = I − z z T z T z b ,(30) where b := ∇S is the gradient of the entropy with b i = − ln f i d i .(31) At the MaxEnt solution f , the gradient of Shannon entropy and the gradient of the fit function must be antiparallel This gives rise to the following self-consistent system of equations satisfied by any MaxEnt solution αb = −z .(32)f i = d i exp z i α ,(33) where the fit gradient of the MaxEnt solution z depends on the solution itself. By rearranging this equation, it becomes clear that the same MaxEnt solution can be obtained using a whole family of other equivalent default models d and their corresponding regularization parameters α. This family can be constructed explicitly using the MaxEnt solution and its fit gradient: d i := f i exp − z i α .(34) Alternatively, given a default model d with regularization parameter α, we can construct an entire family of default models d α that result in the same MaxEnt solution f : d α i = d i exp −z i 1 α − 1 α .(35) Note that lim α →∞ d α = f . In Fig. 7, we show a set of equivalent default models for test case 1. Besides establishing the existence of equivalent default models, Eq. (34) can be used to study the stability of MaxEnt solution with respect to perturbations to these default models. The partial derivatives of the default model with respect to variations in MaxEnt solution f and regularization parameter α read ∂d i ∂f j = d i f i δ i,j + d i α k T i k j ,(36)∂d i ∂α = z i α 2 d i .(37) Therefore, an infinitesimal change in the MaxEnt solution df and an infinitesimal change in the regularization parameter dα induce the following relative change in the default model δ := D −1 dd = L df + dα α 2 z ,(38) where D = diag(d) and L is the scaled Hessian of the MaxEnt objective function − αL := − K T K + αF ,(39) with F := diag(f ). Inverting Eq. (38) gives the changes in the MaxEnt solution in terms of perturbations to its default model. Under the discrepancy principle, the change in the regularization parameter is fixed by the constraint z T df = 0 (ensuring that df has no component perpendicular to C) to the value dα = α 2 z T L −1 δ z T L −1 z ,(40) and the corresponding change in the MaxEnt solution is df = L −1 I − z z T L −1 z T L −1 z δ =: L −1 δ ⊥ ,(41) where δ ⊥ is the part of the vector δ perpendicular to the surface normal z , under the inner product defined by the matrix L −1 . Relative changes in the default model along the direction of z give an equivalent default model and thus have no effect on the MaxEnt solution. To assess the effect of changes in the default model along orthogonal directions, we need to look into the spectral decomposition of the matrix L. The eigenvectors of L match the spectral modes of the rescaled kernel K := K √ F and the eigenvalues of the former λ i are related to the singular values of the later s i as λ i = α + s 2 i α .(42) We now distinguish two limiting cases depending on the direction of the vector δ ⊥ . When s 2 i α, then λ −1 i ≈ α/s 2 i . Therefore, changes along the leading modes have little effect on the MaxEnt solution, and the effect is smaller the further away the default model is from C. On the other hand, when s 2 i α, then λ −1 i ≈ 1. Therefore, changes along the trailing modes are directly reflected in the MaxEnt solution. Assuming that a MaxEnt solution is smooth, the leading modes of K are smooth and slowly varying functions while the trailing ones are highly oscillating. These results then confirm and elucidate the common wisdom that slowly varying details of the default model have little to no effect on MaxEnt solutions, while sharp features tend to introduce strong biases. Finally, note that having more accurate data scales up the singular values s i , and thus MaxEnt solution becomes less sensitive to changes in the default model, as one would intuitively anticipate. VIII. CONNECTING SCT TO MAXENT Let d (t) be the default model at step t of SCT and f (t) and z (t) be the corresponding Tikhonov solution and its fit gradient. By combining Eq. (20) with Eq. (22), we see that the Tikhonov solutions satisfy the following selfconsistent equation (analogous to Eq. (33) of MaxEnt) f (t) i = d (t) i 1 + z (t) α (t) .(43) Using mixing parameters µ (t) , the default models at subsequent iterations are then related by d (t+1) i = 1−µ (t) d (t) i +µ (t) f (t) i = d (t) i 1 + µ (t) α (t) z (t) .(44) Applying this relation recursively and assuming very small µ (t) /α (t) , we get the following exponential form for the default models produced by SCT d (t) i = d (0) i exp t−1 τ =0 µ (τ ) α (τ ) z (τ ) = d (0) i exp z (t) α (t) ,(45) where in the last equation we defined the effective fit gradientsz (t) and the effective regularization parameters α (t) as Indeed, the effective gradients of SCT provide an excellent approximation to the MaxEnt gradient. In Fig. 8, we plot the normalized overlap between the two at different iterations of SCT. The starting effective gradient is nothing but the original Tikhonov gradient, which already has a very good overlap of 0.83. This is to be expected since, as discussed in the previous section, Tikhonov provides an approximation to MaxEnt. As the SCT procedure iterates, the effective gradient not only maintains the good initial overlap, but the overlap improves until it saturates at about 0.99 when the procedure converges. Interestingly, the overlap with the "bare" gradients z (t) , i.e., the gradients of Tikhonov solutions at different iterations, does not necessarily increase. The plot shows that the bare overlap actually drops after a couple of iterations. We observed cases where the bare overlap even drops below its starting value (see Fig. 9). Nevertheless, in all Figure 9. Normalized overlap between the MaxEnt fit gradient z and the fit gradients produced at different SCT iterations (denoted as t) in test case 2. z (t) :=α (t) t−1 τ =0 µ (τ ) α (τ ) z (τ ) , 1 α (t) := t−1 τ =0 µ (τ ) α (τ ) .(46)<z |z (t) > z z (t) <z |z (t) > z z (t)<z |z (t) > z z (t) <z |z (t) > z z (t) cases we investigated, the effective gradients always had a monotonically-increasing overlap with the MaxEnt gradient. An argument for this behavior of the fit gradients is detailed in Appendix C. These results demonstrate that the set of default models produced by SCT provides an approximation to the MaxEnt family of equivalent default models, and thus solving the MaxEnt problem with any one of them gives a solution that is close to the solution of the original MaxEnt problem. At convergence, the default model of SCT satisfies the discrepancy principle, and thus, it is trivially the solution of its own MaxEnt problem and a good approximation of the original MaxEnt solution. In Appendix D, we give an alternative perspective in which SCT can be seen as an approximate and simplified variant of Newton's method for obtaining the MaxEnt solution. IX. SUMMARY In this paper, we used singular value decomposition to derive a generally-applicable method for estimating the noise level on QMC data. Having a reliable error estimate is crucial when using the discrepancy principle/Historic MaxEnt. We then introduced a particular form of the Tikhonov regularization that is more suitable for analytic continuation problems. Besides solving the implicit grid dependence and normalization issues, this form is closely connected to Shannon entropy. A quadratic approximation of the entropy around its default model gives precisely the introduced Tikhonov penalty term. This form allows approximating the MaxEnt solution using the Tikhonov method when the default model already has a good fit to the data (i.e., in the limit of large regularization parameter). In the typical cases where the default model does not fit the data well, we showed that an iterative procedure where the default model is repeatedly mixed with its Tikhonov solution still gives similar results to MaxEnt. We investigated the connection between the two methods, which revealed that the same MaxEnt solution could be produced by a whole family of equivalent default models. This family is approximately traced by the the self-consistent Tikhonov procedure. SCT provides a simple and efficient alternative to MaxEnt that could be easily implemented using any linear algebra library. In particular, we expect SCT to be useful for the analytic continuation of matrix-valued Green functions, where MaxEnt is trickier to implement [21,22]. where H is minus the Hessian of the Tikhonov objective function of Eq. (14) H := α D −1 + K T K , and ∇T f is its gradient at the MaxEnt solution ∇T f i = z i + α a i = z i − α f i − d i d i = z i − α exp z i α − 1 = − 1 2α z i 2 + O(α −2 ) .(B3) Therefore, the gradient scales linearly with the inverse of α. To analyze how the difference ∆ scales, we look at the spectral decomposition of the Hessian matrix H. Its eigenvectors are the same as the spectral modes of the rescaled matrixK = K √ D, and its eigenvalues h i are related to the singular values ofK as following h i = α +s 2 i .(B4) The i-th component of the difference then scales as 1/(α 2 + αs 2 i ), and thus, the difference between Tikhonov and MaxEnt vanishes quadratically in the limit of strong regularization. Note that the components of the gradient along the leading spectral modes (i.e., the smooth components with large singular values ) get suppressed more than the trailing ones (i.e., the oscillating components with small singular values). The vector x := H −1 ∇S d is the solution of Log Spectrum Space K T K + α D −1 x = K T [g − Kd] ⇔ K T K + α D −1 [x + d] = K T g + αe . (D2) Comparing with Eq. (A2), we see that x + d equals the Tikhonov solution, and thus Newton's update formula can be written as d = d + γ f Tikhonov − d ,(D3) which is precisely the mixing formula used in SCT. Note that the entropy has no contribution to the gradient vector ∇S d at the starting default model. However, at later steps there is an additional term −α (t) ln(d (t) i /d(0) i ). SCT ignores this term; thus, SCT is equivalent to Newton's method, where the default model is always reset to its most recent solution. Interestingly, the missing entropy contributions can be expressed in terms of the effective fit gradients − α (t) ln d (t) i d (0) i = − α (t) α (t)z (t) .(D4) Therefore, we can recover the full Newton's method as a variant of the SCT method where the data is modified at each step to take into account the residuals of the previous Tikhonov solutions. Figure 1 . 1Figure 1. Least squares solution (bottom panel) for the analytic continuation of optical conductivity σ(ω) using noisy data of its correlation function. The exact correlation function is computed analytically from the exact optical conductivity (top panel) on the first m = 60 bosonic Matsubara frequencies with inverse temperature β = 15. The input data includes relative Gaussian noise with standard deviation 10 −2 . This test case is an adaptation of the ones proposed by Ref. [3] and studied further in Refs. [4-6]. In the notation of the latter reference, the optical conductivity used here differs in the values of the following parameters: Γe = 20, 1 = 15. We denote this data set as test case 1. Figure 2 . 2Figure 2. Singular values of the (weighted) kernel of test case 1. The singular values decay exponentially until leveling off at a value determined by the machine epsilon. In the inset, we show some of the spectral modes. The leading spectral modes are smooth and slowly varying functions. As the mode index increases, the number of nodes increases, and the modes become more oscillatory. Once the singular values reach numerical accuracy, the corresponding modes become numerically degenerate so that the SVD routine returns arbitrary linear combinations of the exact modes. Figure 3 . 3Absolute values of the exact and noisy data coefficients of test case 1. While the exact coefficients decay to the machine epsilon, the noisy ones decay until they hit the noise level and then fluctuate around it. Here the noise level equals one because the data is weighted by the proper covariance matrix. Notice that large noisy coefficients are close to their exact values and that the deviation becomes significant only when their values drop to near the noise level. with D = diag(d) and e :=(1, 1, . . . , 1) Figure 4 . 4Tikhonov solutions for test case 1 using a Gaussian default model centered at 0 with width 10. The values used for the regularization parameter α are determined by the discrepancy principle. Figure 5 . 5Comparison of MaxEnt and default models produced by SCT at different iterations. The superscript of the default model represents its iteration number with d (0) being the starting default model. For MaxEnt, the starting default model d (0) was used, and the regularization parameter was determined by the discrepancy principle. Figure 7 . 7Different default models equivalent to d for test case 1. The value of α is determined via the discrepancy principle. Figure 8 . 8Normalized overlap between the MaxEnt fit gradient z and the fit gradients produced at different SCT iterations (denoted as t) in test case 1. Both the bare gradients z (t) (gradients of Tikhonov solutions) and the effective gradients z (t) are shown. The overlaps and norms are calculated using the inner product x, y := x T L −1 y, where L is the scaled Hessian of MaxEnt objective function defined in Eq. (39). Figure 10 . 10Schematic diagram illustrating how the default models and their MaxEnt and Tikhonov solutions evolve with the SCT iterations. The diagram is depicted in the logarithmic space of spectra. Note that Tikhonov solutions are assumed here to be strictly positive, although, in general, they may have either sign. d = d + γH −1 ∇S d ,(D1)where γ is a small step size and H is minus the Hessian of the objective function at the default model (which coincides with minus the Tikhonov Hessian in Eq. (B2)) and ∇S d is its gradient, also evaluated at the default model. Appendix A: Tikhonov solution using SVDThe minimization problem of Tikhonov in Eq.(14)can be written as the following least squares problem with an extended kernel matrix and extended data vector(A1) where D = diag(d) and e = (1, 1, . . . ,1)T . The normal equation of this least-squares problem readswhere a rescaled kernel matrixK is defined asK := K √ D. Using SVD of the rescaled matrixK =ŨSṼ T , the normal equation in the mode space readsThe Tikhonov solution can then be expressed in terms of the rescaled modes of the rescaled matrix V := √ DṼ asThe first term is similar to the expansion of the original grid-dependent Tikhonov in Eq. (10), while the second term comes from centering the regularization term around the default model.Unlike the spectral modes v i in Eq. (10), however, the modes v i are not orthonormal under the standard inner product. They are instead orthonormal under the modified inner productMoreover, these vectors can be seen as the modes of the original kernel matrix, with the orthogonality being defined under this modified inner product. This view holds sinceandEq. (A5) can then be seen as a direct expansion of the Tikhonov solution in terms of the spectral modes of the kernel matrixwithNote how each component of the Tikhonov solution is an interpolation between the components of the least squares spectrum and the default model. Different components, however, are mixed differently (each according to its singular value), and thus the overall Tikhonov solution is generally not a simple interpolation of the two spectra.Appendix B: Difference between MaxEnt and TikhonovWe can quantify the difference between the Tikhonov and MaxEnt solutions of the same default model and regularization parameter as followingThen the relevant relative change in the default model is δ ⊥ = µ/α dz . From Eq. (41), we see that the corresponding change in the fit gradient of the MaxEnt solution readsGiven that the matrices L −1 and K T K are positive semidefinite, the overlap between dz and dz is non-positive, i.e., the fit gradient of the MaxEnt solution moves op-posite to the change in the fit gradient that induced it. Since Tikhonov solutions generally follow the MaxEnt solutions, the new Tikhonov gradients would be closer to the original MaxEnt gradient than the previous ones. This explains why the bare fit gradient vectors in SCT initially move closer to the original MaxEnt gradient vector(Figs. 8 and 9). However, the MaxEnt solutions using SCT default models keep drifting away in the same direction, so the Tikhonov solutions and their fit gradients would eventually also start moving away from the original MaxEnt. The effective fit gradient, on the other hand, is an average of these bare gradients and thus can be closer to the original MaxEnt than any of its summands. This happens when the bare gradients circulate around the original MaxEnt gradient, which is the case in SCT.The dynamics described above is depicted schematically inFig. 10. In this diagram, we represent the log spectra as points, so the family of equivalent default models d(1), d(2), . . . all lie on a straight line between the initial default model d (0) and its MaxEnt solution f . This line is specified by the fit gradient vector z . In SCT, z is replaced by z (t) , the bare fit gradients at the Tikhonov solutions f (t) , leading to a set of alternative default models d (t) that approximates the equivalent family d(t). Each approximate default model d (t) has itsown MaxEnt solution f (t) and Tikhonov solution f (t) . In this two-dimensional case, according to Eq. (C2), the Tikhonov solution f (t) and the MaxEnt solution at the next iteration f (t+1) must be on opposite sides of the MaxEnt solution f (t) . Therefore, the fit gradients of Tikhonov z (t) would initially get closer to z before moving away. Also, note how the effective fit gradients (i.e., consecutive weighted averages of z (t) ) get monotonically closer to z . This is the result of z (t) moving from one side of z to the other, and the weights µ (t) /α (t) getting lower for higher iterations.Appendix D: SCT as Reset Newton MethodAnother perspective on SCT is seeing it as a variant of Newton's method for optimization. Assuming that the optimal regularization parameter for satisfying the discrepancy principle is somehow known in advance, solving the MaxEnt problem of Eq. (28) reduces to optimizing the MaxEnt objective function of Eq. (27). Using the default model d as an initial guess, an improved solution can be obtained using Newton's method as Numerical tools for analysis and solution of fredholm integral equations of the first kind, Inverse Prob. P C Hansen, 8849P. C. Hansen, Numerical tools for analysis and solution of fredholm integral equations of the first kind, Inverse Prob. 8, 849 (1992). P C Hansen, 10.1137/1.9780898718836Discrete Inverse Problems. SIAMP. C. Hansen, Discrete Inverse Problems (SIAM, 2010). Analytic continuation of imaginary axis data for optical conductivity. O Gunnarsson, M W Haverkort, G Sangiovanni, 10.1103/PhysRevB.82.165125Phys. Rev. B. 82165125O. Gunnarsson, M. W. Haverkort, and G. Sangiovanni, Analytic continuation of imaginary axis data for optical conductivity, Phys. Rev. B 82, 165125 (2010). Stochastic Analytic Continuation: A Bayesian Approach. K Ghanem, 10.18154/RWTH-2017-06704RWTH Aachen UniversityPh.D. thesisK. Ghanem, Stochastic Analytic Continuation: A Bayesian Approach, Ph.D. thesis, RWTH Aachen Uni- versity (2017). Average spectrum method for analytic continuation: Efficient blocked-mode sampling and dependence on the discretization grid. K Ghanem, E Koch, 10.1103/PhysRevB.101.085111Phys. Rev. B. 10185111K. Ghanem and E. Koch, Average spectrum method for analytic continuation: Efficient blocked-mode sampling and dependence on the discretization grid, Phys. Rev. B 101, 085111 (2020). Extending the average spectrum method: Grid point sampling and density averaging. K Ghanem, E Koch, 10.1103/PhysRevB.102.035114Phys. Rev. B. 10235114K. Ghanem and E. Koch, Extending the average spec- trum method: Grid point sampling and density averag- ing, Phys. Rev. B 102, 035114 (2020). A technique for the numerical solution of certain integral equations of the first kind. D L Phillips, 10.1145/321105.321114J. ACM. 9D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9, 84-97 (1962). Solutions of ill-posed problems. A N Tikhonov, V Y Arsenin, John Wiley & SonsV. H. Winston & Sons, Washington, D.C.; New YorkA. N. Tikhonov and V. Y. Arsenin, Solutions of ill-posed problems (V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977). We choose to deviate from that convention in order to make the correspondence with the regularization parameter of MaxEnt more seamless. In the inverse-problem literature, it is common for Tikhonov regularization parameter α to appear squaredIn the inverse-problem literature, it is common for Tikhonov regularization parameter α to appear squared. We choose to deviate from that convention in order to make the correspondence with the regularization param- eter of MaxEnt more seamless. The most general form of Tikhonov is obtained by replacing the L2-norm with a bilinear function f − f0 2 M , where M is some positive-definite matrix and f0 is an arbitrary vector that acts as a default model. The most general form of Tikhonov is obtained by re- placing the L2-norm with a bilinear function f − f0 2 M , where M is some positive-definite matrix and f0 is an arbitrary vector that acts as a default model. In this case, the discretized L2-norm of f corresponds to the continuous l2-norm of f (x). However, using this form implies a specific choice of the measure on x. One can obtain trivial grid independence by including one square root of the grid weights in the spectrum vector and the other square root in the kernel matrix. that is equivalent to fixing the grid density ρ(x) to be uniformOne can obtain trivial grid independence by including one square root of the grid weights in the spectrum vec- tor and the other square root in the kernel matrix. In this case, the discretized L2-norm of f corresponds to the con- tinuous l2-norm of f (x). However, using this form implies a specific choice of the measure on x that is equivalent to fixing the grid density ρ(x) to be uniform. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind , Chapman & Hall/CRC research notes in mathematics series. C Groetsch, Pitman Advanced Pub. ProgramC. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind , Chapman & Hal- l/CRC research notes in mathematics series (Pitman Ad- vanced Pub. Program, 1984). Criteria for selection of regularization parameter, in Methods for Solving Incorrectly Posed Problems. V A Morozov, 10.1007/978-1-4612-5280-1_2SpringerNew York, New York, NYV. A. Morozov, Criteria for selection of regularization pa- rameter, in Methods for Solving Incorrectly Posed Prob- lems (Springer New York, New York, NY, 1984) pp. 32- 64. C L Lawson, R J Hanson, 10.1137/1.9781611971217Solving Least Squares Problems. SIAM, PhiladelphiaC. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, Philadelphia, 1995). Progress on stochastic analytic continuation of quantum monte carlo data. H Shao, A W Sandvik, 10.48550/ARXIV.2202.09870H. Shao and A. W. Sandvik, Progress on stochastic ana- lytic continuation of quantum monte carlo data (2022). Maximumentropy method for analytic continuation of quantum monte carlo data. R N Silver, D S Sivia, J E Gubernatis, 10.1103/PhysRevB.41.2380Phys. Rev. B. 412380R. N. Silver, D. S. Sivia, and J. E. Gubernatis, Maximum- entropy method for analytic continuation of quantum monte carlo data, Phys. Rev. B 41, 2380 (1990). Bayesian inference and the analytical continuation of imaginary-time quantum. M Jarrell, J E Gubernatis, M. Jarrell and J. E. Gubernatis, Bayesian inference and the analytical continuation of imaginary-time quantum Monte Carlo data. 10.1016/0370-1573(95)00074-7Phys. Rep. 269133Monte Carlo data, Phys. Rep. 269, 133 (1996). Analytical continuation of imaginary axis data using maximum entropy. O Gunnarsson, M W Haverkort, G Sangiovanni, 10.1103/PhysRevB.81.155107Phys. Rev. B. 81155107O. Gunnarsson, M. W. Haverkort, and G. Sangiovanni, Analytical continuation of imaginary axis data using maximum entropy, Phys. Rev. B 81, 155107 (2010). Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation. D Bergeron, A.-M S Tremblay, 10.1103/PhysRevE.94.023303Phys. Rev. E. 9423303D. Bergeron and A.-M. S. Tremblay, Algorithms for opti- mized maximum entropy and diagnostic tools for analytic continuation, Phys. Rev. E 94, 023303 (2016). The maximum entropy method: Analytic continuation of QMC data. M Jarrell, Correlated Electrons: From Models to Materials. E. Pavarini, E. Koch, F. Anders, and M. Jarrell (Forschungszentrum JülichJülichM. Jarrell, The maximum entropy method: Analytic con- tinuation of QMC data, in Correlated Electrons: From Models to Materials, edited by E. Pavarini, E. Koch, F. Anders, and M. Jarrell (Forschungszentrum Jülich, Jülich, 2012). Maximum entropy formalism for the analytic continuation of matrix-valued green's functions. G J Kraberger, R Triebl, M Zingl, M Aichhorn, 10.1103/PhysRevB.96.155128Phys. Rev. B. 96155128G. J. Kraberger, R. Triebl, M. Zingl, and M. Aichhorn, Maximum entropy formalism for the analytic continua- tion of matrix-valued green's functions, Phys. Rev. B 96, 155128 (2017). Analytical continuation of matrix-valued functions: Carathéodory formalism. J Fei, C.-N Yeh, D Zgid, E Gull, 10.1103/PhysRevB.104.165111Phys. Rev. B. 104165111J. Fei, C.-N. Yeh, D. Zgid, and E. Gull, Analytical con- tinuation of matrix-valued functions: Carathéodory for- malism, Phys. Rev. B 104, 165111 (2021).

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{'abstract': 'Analytic continuation is an essential step in extracting information about the dynamical properties of physical systems from quantum Monte Carlo (QMC) simulations. Different methods for analytic continuation have been proposed and are still being developed. This paper explores a regularization method based on the repeated application of Tikhonov regularization under the discrepancy principle. The method can be readily implemented in any linear algebra package and gives results surprisingly close to the maximum entropy method (MaxEnt). We analyze the method in detail and demonstrate its connection to MaxEnt. In addition, we provide a straightforward method for estimating the noise level of QMC data, which is helpful for practical applications of the discrepancy principle when the noise level is not known reliably.', 'arxivid': '2212.05475', 'author': ['Khaldoon Ghanem \nLeopoldstrasse 18080804Quantinuum, MunichGermany\n', 'Erik Koch \nJülich Supercomputer Centre\nForschungszentrum Jülich\n52425JülichGermany\n\nJARA High-Performance Computing\n52425JülichGermany\n'], 'authoraffiliation': ['Leopoldstrasse 18080804Quantinuum, MunichGermany', 'Jülich Supercomputer Centre\nForschungszentrum Jülich\n52425JülichGermany', 'JARA High-Performance Computing\n52425JülichGermany'], 'corpusid': 254564134, 'doi': '10.1103/physrevb.107.085129', 'github_urls': [], 'n_tokens_mistral': 13550, 'n_tokens_neox': 12040, 'n_words': 8396, 'pdfsha': '01ab947d096fb9d62112b62d8294202b3eae595d', 'pdfurls': ['https://export.arxiv.org/pdf/2212.05475v1.pdf'], 'title': ['Connecting Tikhonov regularization to the maximum entropy method for the analytic continuation of quantum Monte Carlo data', 'Connecting Tikhonov regularization to the maximum entropy method for the analytic continuation of quantum Monte Carlo data'], 'venue': []}

arxiv

7 Jul 2010 7 Jul 2010THE GL(2, C) MCKAY CORRESPONDENCE MICHAEL WEMYSS In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2, C), it means that the endomorphism ring of the special CM C[[x, y]] G -modules can be used to build the dual graph of the minimal resolution of C 2 /G, extending McKay's observation for finite subgroups of SL(2, C) to all finite subgroups of GL(2, C). Introduction When working with quotient singularities V /G from the viewpoint of resolution of singularities and derived categories the object one first writes down is the skew group ring C[V ]#G (=G-equivariant sheaves) since this object is the canonical way to encode the representation theory of G into the geometry. This algebra satisfies many nice homological properties, in particular it has finite global dimension and so we often view it as a 'noncommuatitive resolution' over its centre. For quotients with G ≤ SL(V ) there is much evidence which suggests that this is a good idea, but there is always a problem extracting and understanding the geometric information that C[V ]#G encodes. The point in this paper is that for non-Gorenstein surface quotient singularities (i.e. those by finite small groups G ≤ GL(2, C) that are not inside SL(2, C)), in order to be able to see the geometry in the most clear way the skew group ring C[x, y]#G is far too large; one should instead pass to a much smaller algebra, the so-called reconstruction algebra. The benefit of passing to this smaller algebra is twofold -firstly we are able to recover the link with the dual graph of the minimal resolution which is obscured in the world of G-equivariant sheaves (which we do in this paper), and secondly it is much easier to extract the geometry (which we do in other papers) since via the reconstruction algebra understanding resolutions of quotients by large non-abelian groups can turn out to be as easy as understanding those arising from vastly smaller cyclic groups. For non-Gorenstein surface quotient singularities the number of exceptional curves in the minimal resolution of C 2 /G is strictly less than the number of irreducible representations. This problem led Wunram [18] to develop the idea of a special representation (equivalently special CM module) so that after passing to the non-trivial indecomposable special representations the 1-1 correspondence with the exceptional curves is recovered. Note that the definition of special representation is homological since it is defined by the vanishing of cohomology of the dual of a certain vector bundle on the minimal resolution. Even in fairly easy examples, determining which representations are special is not a straightforward task [10]. Wunram's results give the necessary 1-1 correspondence and so here we study the noncommutative algebra given by the endomorphism ring of the sum of the indecomposable special CM modules. In [15] it was discovered algebraically that for finite small cyclic subgroups of GL(2, C) the quiver of this non-commutative algebra determines and is determined by the dual graph of the minimal resolutionX of the singularity C 2 /G, labelled with self-intersection numbers. This correspondence is purely on the level of the underlying quiver; it was further discovered that if we add in the extra information of the relations then in fact one can recover the whole spaceX (not just the dual graph) as a certain GIT quotient, and also that the endomorphism algebra describes the derived category ofX. Furthermore the global dimension of the endomorphism algebra was found to be either 2 or 3, which since we are studying surfaces is a little surprising. In this paper we relate Wunram's work to that of Bridgeland [4] and Van den Bergh [14] to give a non-explicit geometric proof of some of the above results which furthermore works in greater generality. Note that by using Riemann-Roch and Serre duality the proofs are quite routine, giving not only the number of arrows but also the number of relations. Our first main theorem is the following: Theorem 1.1 (=3.3 and 3.6). Let R be any affine complete rational surface singularity and letX be the minimal resolution of Spec R. Then the quiver of End R ( M ), where the sum is taken over all indecomposable special CM R-modules, can be computed combinatorially from the dual graph ofX labelled with self-intersection numbers. We explain the terminology in §2 and state this more precisely in §3 however we note here that the combinatorics are easy and so the computation of the quiver is very quick. We call the endomorphism ring in the above correspondence the reconstruction algebra, since its quiver can be reconstructed from the dual graph. Of course, as is the case for preprojective algebras in the classical McKay Correspondence, to obtain End R ( M ) from the dual graph requires us to add information in the form of an extra vertex, whereas to obtain the dual graph from End R ( M ) one must lose information by killing a specified vertex. Using the computation of the ext groups in the proof of the above theorem we also obtain the following, extending the result [10, 2.15]: Corollary 1.2 (=3.7) . Let R be an affine complete rational surface singularity and set A := End R ( M ) where the sum is taken over all indecomposable special CM R-modules. Then gl.dim A = 2 if R is Gorenstein 3 else. When R is Gorenstein all simple left A-modules and all simple right A-modules have projective dimension 2. When R is not Gorenstein all simple right A-modules have projective dimension 2 except the simple corresponding to ⋆, which has projective dimension 3. As left A-modules, the projective dimension of the simples at ⋆ and all curves corresponding to (−2)-curves have projective dimension 2, whereas all other simples have projective dimension 3. Thus not only does the homologically homogeneous property fail for reconstruction algebras, it fails asymmetrically. Applying Theorem 1.1 to quotients of C 2 by finite subgroups of GL(2, C) we also obtain the result which motivated this work. This says, provided we use Auslander's endomorphism ring perspective [2], that the representation theory of the special CM modules determines the dual graph of the minimal resolution in exactly the same way as in the classical SL(2, C) case. Again we emphasize that the reconstruction algebra is in general much smaller than G-equivariant sheaves, allowing us to extract the geometry much more easily. Although the geometric proof of Theorem 1.1 is quite slick, the main content in reconstruction algebras is found in their relations and other than telling us their number the geometric proof does nothing more. Furthermore End C[[x,y]] G (⊕ ρ special (ρ⊗C[[x, y]]) G ) is an entirely representation-theoretic gadget and so turning Corollary 1.3 around we should be able to deduce the dual graph by using only the representation theory. This motivates us to provide a second proof of Corollary 1.3 in which we show how to compute End C[[x,y]] G (⊕ ρ special (ρ ⊗ C[[x, y]]) G ) purely representation-theoretically, assuming none of the geometry. This method actually turns out to give more information since it immediately tells us the grading of the algebra and it also provides a method to obtain the relations. For brevity in this paper we restrict ourselves to only determining the quiver and so we ignore the relations; we consider this problem in separate papers [15,16,17] by using a combination of the two approaches. The main idea behind the representation-theoretic proof for quotient singularities is that the number of irreducible maps between the special CM modules can be determined by using a simple counting argument on the AR quiver of C[[x, y]] G . The relations on the reconstruction algebra are then induced by the mesh relations on the AR quiver. This method involves case-by-case analysis and relies on the classification of the special CM modules in [10] and so is quite space-intensive, thus we prove only a few examples. The structure of this paper is as follows: we begin in §2 with preliminaries involving intersection theory and perverse sheaves. In §3 we give the geometric proof of the main results above, and in §4 we give the alternative representation-theoretic method for quotient singularities. The remainder of the paper is devoted to the quotient case: in §5 we translate the intersection theory into simple rules and in the remaining sections we draw the quivers for all finite subgroups of GL(2, C). Preliminaries Throughout this paper let X = Spec R be an affine complete rational surface singularity over C, let f : X → Spec R be some resolution and denote the exceptional curves by {E i } i∈I . Resolutions will not be minimal unless specified. We shall always assume that our dual graphs are labelled with the corresponding self-intersection numbers: Definition 2.1. Suppose {E i } i∈I is a collection of P 1 s forming the exceptional locus in a resolution of some affine rational surface singularity. The dual graph is defined as follows: for each curve draw a vertex, and join two vertices if and only if the corresponding curves intersect. Furthermore label every vertex with the self-intersection number of the corresponding curve. Definition 2.2 ([1]). For a given exceptional {E i } i∈I , define the fundamental cycle Z f = i∈I r i E i (with each r i ≥ 1) to be the unique smallest element such that Z f · E i ≤ 0 for all i ∈ I. Our notational convention for writing Z f is as follows: Example 2.3. (i) −2 −2 −2 −2 −5 −2 −3 Z f = 1 2 1 2 1 1 1 (ii) −2 −2 −2 −2 −3 −2 Z f = 1 2 1 2 1 1 As is standard, we denote the canonical cycle by Z K . It is the rational cycle defined by the condition Z K · E i = −K X · E i for all i ∈ I. By adjunction this means that Z K · E i = E 2 i + 2 for all i ∈ I. Note that if the resolution is minimal then Z K · E i ≤ 0 for all i ∈ I. Now perverse sheaves were introduced by Bridgeland [4] to prove the existence of flops of certain 3-folds; here we use this theory for surfaces. The key point from our perspective is the following commutative diagram, proved by Van den Bergh [14, 3.2.8, 3.5.5] D b (coh X) ≈ D b (mod End X (O X ⊕ M I )) ∪ ∪ −1 Per X ≈ mod End X (O X ⊕ M I ) where M I = i∈I M i with each M i a certain vector bundle satisfying det M i · E j = δ ij . Furthermore the simple modules in mod End X (O X ⊕ M I ) are, viewed inside D b (coh X), precisely [14, 3.5.7] O Z f and O Ei (−1)[1] for all i ∈ I. The same set of objects also appears in the work of Ishii [8] for quotient singularities as the homology of the lift of the AR sequences to the minimal resolution. Since we are assuming R is complete, for each exceptional curve E i there is a divisor D i intersecting E i transversally at one point and not intersecting any of the other exceptional curves (see [14, 3.4.4]). Thus defining D := i∈I D i then the simples are O Z f and O Ei (−D) [1] for all i ∈ I. Ext Groups and Corollaries When working with quivers by ab we mean a followed by b; similarly when composing morphisms by f g we mean f followed by g. With these conventions representations of quivers correspond to right modules, and further the functor RHom(O X ⊕ M I , −) takes us to the derived category of left End X (O X ⊕ M I ) modules. In what follows denote the simples in −1 Per X (i.e. left End X (O X ⊕ M I )-modules) by S ⋆ = O Z f , and S i = O Ei (−D)[1] for all i ∈ I. The following is immediate from the definition of Z f : Lemma 3.1. If E i is some exceptional curve then E 2 i ≤ Z f · E i . Furthermore the in- equality is strict if E i intersects some other exceptional curve. Proof. Write Z f = a i E i . If a i = 1 then Z f · E i = E 2 i + i =j a j E j · E i ≥ E 2 i where the inequality is strict provided that E i · E j = 1 for some j. If a i > 1 write Z ′ = Z f − E i = (a i − 1)E i + i =j a j E j then Z ′ has all co-efficients ≥ 1 with Z ′ · E j = (Z f − E i ) · E j ≤ 0 for all j = i. Further Z ′ · E i = Z f · E i − E 2 i , thus Z f · E i > E 2 i else Z f is not minimal. Let e denote the embedding dimension of R, i.e. e − 2 = −Z K · Z f + 1 = −1 − Z f · Z f . In what follows, for every integer a ∈ Z denote a + := a if a ≥ 0 0 if a < 0 and a − = 0 if a ≥ 0 −a if a < 0 . Theorem 3.2. Let X → Spec R be some resolution of an affine complete rational surface singularity. Then the dimension of the Ext groups between the simples in −1 Per X are ext 1 (Si,Sj)=(Ei·Ej )+ ext 2 (Si,Sj)=(−1−Ei·Ej)+ ext 3 (Si,Sj)=0 ext 1 (S⋆,S⋆)=0 ext 2 (S⋆,S⋆)=e−2 ext 3 (S⋆,S⋆)=0 ext 1 (S⋆,Si)=−Ei·Z f ext 2 (S⋆,Si)=0 ext 3 (S⋆,Si)=0 ext 1 (Si,S⋆)= ((ZK − Z f ) · Ei)+ 1 − Z f · Ei ext 2 (Si,S⋆)= ((ZK − Z f ) · Ei)− 1 ext 3 (Si,S⋆)= −E 2 i − 2 0 where in the split for ext t (S i , S ⋆ ) the bottom option corresponds to when E i is a (−1)curve, the top option when E i is a non-(−1)-curve. All higher ext groups are zero. Proof. We start by computing ext t (S i , S j ) = ext t (O Ei , O Ej ). Taking the short exact sequence 0 → O X (−E i ) → O X → O Ei → 0 and applying Hom(−, O Ej ) gives 0 → Hom(O Ei , O Ej ) → H 0 (O Ej ) → H 0 (O Ej (E i · E j )) → Ext 1 (O Ei , O Ej ) → H 1 (O Ej ) = 0 and 0 = H 1 (O Ej ) → H 1 (O Ej (E i · E j )) → Ext 2 (O Ei , O Ej ) → H 2 (O Ej ) = 0 with ext t (O Ei , O Ej ) = 0 for all t ≥ 3. From the second exact sequence ext 2 (O Ei , O Ej ) = h 1 (O Ej (E i · E j )) = 0 i = j −1 − E i · E i i = j. From the first exact sequence, if i = j then H 0 (O Ei (E i ·E j )) = 0 forces ext 1 (O Ei , O Ej ) = 0. If i = j then hom(O Ei , O Ej ) = 0 and so ext 1 (O Ei , O Ej ) = h 0 (O Ej (E i · E j )) − h 0 (O Ej ) = E i · E j . We now compute ext t (S ⋆ , S ⋆ ) = ext t (O Z f , O Z f ). First note that h 1 (O Z f (−Z K )) = 0 follows immediately from taking sections of the sequence 0 → O X (−Z f − Z K ) → ω → O Z f (−Z K ) → 0, since H 1 (ω) = 0 by Grauert-Riemenschneider vanishing. Further we know by [1, 3.4] that h 0 (O Z f ) = 1 and h 1 (O Z f ) = 0. Thus applying Hom(−, O Z f ) to the short exact sequence 0 → O X (−Z f ) → O X → O Z f → 0 gives 0 → Hom(O Z f , O Z f ) → H 0 (O Z f ) → H 0 (O Z f (Z f )) → Ext 1 (O Z f , O Z f ) → H 1 (O Z f ) = 0 and 0 = H 1 (O Z f ) → H 1 (O Z f (Z f )) → Ext 2 (O Z f , O Z f ) → H 2 (O Z f ) = 0 with ext t (O Z f , O Z f ) = 0 for all t ≥ 3. By Serre duality on Z f we know that h 0 (O Z f (Z f )) = h 1 (O Z f (−Z K )) = 0 and so the first exact sequence shows that ext 1 (O Z f , O Z f ) = 0. Now using Serre duality on Z f the second short exact sequence gives ext 2 (O Z f , O Z f ) = h 1 (O Z f (Z f )) = h 0 (O Z f (−Z K )). But h 0 (O Z f (−Z K )) = h 0 (O Z f (−Z K )) − h 1 (O Z f (−Z K )) = χ(O Z f (−Z K )) and by Riemann-Roch on Z f χ(O Z f (−Z K )) = −Z f · Z K + 1 = −1 − Z f · Z f as required. Now consider ext t (S ⋆ , S i ). Due to the shift in the simples, ext t (S ⋆ , S i ) = ext t+1 (O Z f , O Ei (−1)). Simply applying Hom(−, O Ei (−1)) to the short exact sequence 0 → O X (−Z f ) → O X → O Z f → 0 gives 0 = H 1 (O Ei (−1)) → H 1 (O Ei (−1 + E i · Z f )) → Ext 2 (O Z f , O Ei (−1)) → H 2 (O Ei (−1)) = 0 and → H 2 (O Ei (−1 + E i · Z f )) = 0 → Ext 3 (O Z f , O Ei (−1)) → H 3 (O Ei (−1)) = 0. with all higher ext groups vanishing. From the second sequence ext 2 (S ⋆ , S i ) = 0, whereas the first shows that ext 1 (S ⋆ , S i ) = ext 2 (O Z f , O Ei (−1)) = h 1 (O Ei (−1 + E i · Z f )) which equals h 0 (O Ei (−1 − E i · Z f )) = −E i · Z f . Finally we consider ext t (S i , S ⋆ ) = ext t−1 (O Ei (−1), O Z f ). Applying Hom(−, O Z f ) to the exact sequence 0 → O X (−E i − D i ) → O X (−D i ) → O Ei (−1) → 0 gives 0 → Hom(O Ei (−1), O Z f ) → H 0 (O Z f (D i )) → H 0 (O Z f (D i +E i )) → Ext 1 (O Ei (−1), O Z f ) → H 1 (O Z f (D i )) → H 1 (O Z f (D i + E i )) → Ext 2 (O Ei (−1), O Z f ) → 0 with all higher terms zero. By summing dimensions ext 1 (S i , S ⋆ )−ext 2 (S i , S ⋆ )+ext 3 (S i , S ⋆ ) = χ(O Z f (D i ))−χ(O Z f (E i +D i )) = −E i ·Z f .(1) We now split into cases -firstly assume that E i is not a (−1)-curve. By Serre duality ext 1 (S i , S ⋆ ) and ext 2 (S i , S ⋆ ) are ext 2 (O Z f , O Ei (−1 − Z K · E i )) and ext 1 (O Z f , O Ei (−1 − Z K · E i )) respectively. Now applying Hom(−, O Ei (−1 − Z K · E i )) to the short exact sequence 0 → O X (−Z f ) → O X → O Z f → 0 gives 0 → Hom(O Z f , O Ei (−1−Z K ·E i )) → H 0 (O Ei (−1−Z K ·E i )) → H 0 (O Ei (−1+(Z f −Z K )·E i )) → Ext 1 (O Z f , O Ei (−1 − Z K · E i )) → H 1 (O Ei (−1 − Z K · E i )) = 0 and 0 = H 1 (O Ei (−1−Z K ·E i )) → H 1 (O Ei (−1+(Z f −Z K )·E i )) → Ext 2 (O Z f , O Ei (−1−Z K ·E i )) → 0 since −Z K · E i ≥ 0 (E i is not a (−1)-curve) . The second exact sequence shows that ext 2 (O Z f , O Ei (−1 − Z K · E i )) = h 1 (O Ei (−1 + (Z f − Z K ) · E i )) = ((Z K − Z f ) · E i ) + . and by the first exact sequence we have a surjection C ((ZK −Z f )·Ei)− → Ext 1 (O Z f , O Ei (−1 − Z K · E i )) → 0 which shows that ext 2 (S i , S ⋆ ) ≤ ((Z K − Z f ) · E i ) − . Thus if (Z K − Z f ) · E i ≥ 0 then ext 2 (S i , S ⋆ ) = 0, so we may assume that (Z K − Z f ) · E i < 0 in which case (by definition of Z f ) necessarily we must have Z K · E i ≤ −1. This means that the first exact sequence reduces to 0 → Hom(O Z f , O Ei (−1 − Z K · E i )) → C −ZK ·Ei → C ((ZK −Z f )·Ei)− → Ext 1 (O Z f , O Ei (−1 − Z K · E i )) → 0. But composing the surjection O Z f ։ O Ei with a basis of Hom(O Ei , O Ei (−1 − Z K · E i )) (which has dimension −Z K · E i ) we see that hom(O Z f , O Ei (−1 − Z K · E i )) ≥ −Z K · E i . By the above exact sequence equality holds, and so by summing dimensions we conclude that ext 1 (O Z f , O Ei (−1 − Z K · E i )) = ((Z K − Z f ) · E i ) − . By (1) we thus obtain ext 3 (S i , S ⋆ ) = −Z f ·E i +((Z K −Z f )·E i ) − −((Z K −Z f )·E i ) + = −Z K ·E i = −E 2 i −2, finishing the proof when E i is not a (−1)-curve. Finally, consider the case where E i is a (−1)-curve, then the statement is well-known if E i is the only curve in the exceptional locus (in which case Z f · E i = E i · E i = −1), thus we may assume that E i intersects some other exceptional curve. In this case Z f · E i = 0 by Lemma 3.1. Now by Serre duality ext 1 (S i , S ⋆ ) and ext 2 (S i , S ⋆ ) are ext 2 (O Z f , O Ei (−2)) and ext 1 (O Z f , O Ei (−2)) respectively, so applying Hom(−, O Ei (−2)) to the exact sequence 0 → O X (−Z f ) → O X → O Z f → 0 gives H 0 (O Ei (−2 + Z f · E i )) → Ext 1 (O Z f , O Ei (−2)) → H 1 (O Ei (−2)) → → H 1 (O Ei (−2 + Z f · E i )) → Ext 2 (O Z f , O Ei (−2)) → 0, which since Z f · E i = 0 is just 0 → Ext 1 (O Z f , O Ei (−2)) → C → C → Ext 2 (O Z f , O Ei (−2)) → 0. Thus ext 1 (S i , S ⋆ ) = ext 2 (S i , S ⋆ ) ≤ 1. To see that both are precisely one we exhibit a non-zero map from O Ei (−1) to O Z f , then ext 1 (S i , S ⋆ ) = hom(O Ei (−1), O Z f ) = 0. But since E i intersects some other curve, Z f − E i > 0 so there is an exact sequence 0 → O Ei (−(Z f − E i )) → O Z f → O Z f −Ei → 0 which since Z f · E i = 0 is simply 0 → O Ei (−1) → O Z f → O Z f −Ei → 0, providing us with the required non-zero map. Lastly by (1) ext 3 (S i , S ⋆ ) = −Z f · E i + 1 − 1 = −Z f · E i = 0, finishing the proof. Corollary 3.3. Let X → Spec R be some resolution of an affine complete rational surface singularity. Then End X (O X ⊕ M I ) can be written as a quiver with relations as follows: for every exceptional curve E i associate a vertex labelled i, and also associate a vertex ⋆ corresponding to O X . Then the number of arrows and relations between the vertices is given as follows: Number of arrows Number of relations i → j (E i · E j ) + (−1 − E i · E j ) + ⋆ → ⋆ 0 −Z K · Z f + 1 = −1 − Z f · Z f i → ⋆ −E i · Z f 0 ⋆ → i ((Z K − Z f ) · E i ) + 1 − Z f · E i ((Z K − Z f ) · E i ) − 1 where in the split for ⋆ → i, the bottom option corresponds to when E i is a (−1)-curve, the top option when E i is a non-(−1)-curve. Proof. Denote A := End X (O X ⊕ M I ). In the conventions here right modules are the same as representations of quivers, so to write A as a quiver with relations we need to take the simple right A-modules and calculate the dimensions of the ext groups between them. In the above the simples which were denoted by S are left modules (since the functor Hom X (O X ⊕ M I , −) has image in left modules) and so we need to reverse the order of the simples, i.e. number of arrows in A i → j = ext 1 A (S j , S i ) number of relations in A i → j = ext 2 A (S j , S i ) The fact that this gives a presentation of the algebra is mostly well-known. The statement on the ext 1 is always true whereas for the statement on ext 2 we are using the fact that we are in the formal case so our path algebras are complete; see for example [6, 3.4(b)]. Corollary 3.4. Let X → Spec R be some resolution of an affine complete rational surface singularity, then gl.dim End X (O X ⊕ M I ) = 3 if there exists E i with E 2 i < −2 2 else. Proof. By Theorem 3.2 Ext t (X, Y ) = 0 for all t ≥ 3 and all simples X and Y , except possibly the case Ext 3 (S i , S ⋆ ) which is zero unless E 2 i < −2. We now relate the above to the work of Wunram so below we restrict our attention to the minimal resolution π :X → Spec R. Recall the following: is an isomorphism, where the right hand sum is taken over all indecomposable special CM R-modules. Proof. It is well-known that if M is a CM R-module then π * M = M [7, 2.2]. Thus taking global sections gives the natural map, which is an isomorphism away from the unique singular point. We know that End R (⊕M ) is reflexive since it is CM, thus if we prove that EndX (OX ⊕ M I ) is reflexive then it follows that the map is an isomorphism. But now each HomX (M 1 , M 2 ) ∼ = π * (M ∨ 1 ⊗ M 2 ) and trivially M ∨ 1 ⊗ M 2 is locally free. Further We finish this section with the following trivial but convenient lemma which reduces the calculation of the quiver to simply adding arrows to a certain base quiver, as is true for reconstruction algebras of type A [15]. H 1 ((M ∨ 1 ⊗ M 2 ) ∨ ⊗ ω) = H 1 (M ∨ 2 ⊗ M 1 ⊗ ω) equals zero since H 1 (M ∨ 2 ⊗ ω) =2 i ≤ −E 2 i for all i. Then (i) −Z f · E i = −Z f · F i + (−E 2 i + F 2 i ) (ii) (Z E K − Z f ) · E i = (Z F K − Z f ) · F i−Z f · E i = −Z f · F i + r i (−E 2 i + F 2 i ). However the point is that by combinatorics on rational surfaces, the condition −F 2 i < −E 2 i forces r i = 1 (see e.g. [13, 3.9]) and so (i) follows. The remaining part (ii) is now trivial. The Representation Theoretic Method Here we compute End C[[x,y]] G (⊕ ρ special (ρ ⊗ C[[x, y]]) G ) directly, without assuming any of the geometry. As stated in the introduction, combining this with Theorem 3.3 and Lemma 3.6 gives a method to recover the dual graph directly from the representation theory, and also provides us with more information than the non-explicit proof. Denote R = C[[x, y]] G , where G is some small finite subgroup of GL(2, C). Note that the AR quiver of the category of CM R-modules (i.e. all irreducible maps between the CM modules) coincides with the McKay quiver by a result of Auslander [2], and further all such quivers for small finite subgroups of GL(2, C) were classified in [3]. Below let S be the set of indecomposable special CM R-modules. For two indecomposable special CM R-modules M and N we wish to determine the number of arrows from M to N in the quiver of the reconstruction algebra. Because everything is graded, this is just the dimension of the space of morphisms from M to N which don't factor through a special CM module via maps of strictly smaller positive degree. Proceed as follows: V = 0 if V ∈ S λ (1) V else. In the AR quiver, at every first-step vertex M 1 write the corresponding number λ V = 0 if V ∈ S λ (2) V else. At every second-step vertex M 2 write the corresponding number λ (2) M2 and then circle the second-step vertices which belong to S. (iv) Next consider all arrows in the AR quiver out of the second-step vertices. The heads of these arrows are called the third-step vertices. For every CM module V define λ (3) V = max{0, −µ (1) τ (V ) + a:L→V µ (2) L } if V is a third-step vertex 0 else and µ (3) V = 0 if V ∈ S λ (3) V else. At every third-step vertex M 3 write the corresponding number λ Remark 4.1. The proof below is similar to [10, §4] (which itself was inspired by knitting), where there S = {R} was taken to establish the dimension of the vector space Hom R (M, N ). However the proof here is more subtle since now both M and N belong to S and so Hom S (M, N ) = 0, making it harder to use functorial proofs. Proof of algorithm: It is clear that the start of the algorithm is correct and so we only need to verify the general induction step. Thus assume that the result is true for smaller n. Consider the AR quiver Q AR of the category of CM C[[x, y]] G -modules. By [3] this is just Z∆/G where ∆ is some extended Dynkin quiver and G is some group of automorphisms; in fact Z∆ → Q AR is a covering. Fix M to be in degree 0 -it is more convenient to work with Z∆ since there the grading is evident. Consider the mesh category C := k(Z∆); in the language of [9], C is a τ -category. Now define S n to be all those objects of C lying in degrees between 1 and n − 1 (inclusive) which belong to S, and consider the quotient category C/[S n ]; this too is a τ -category. Since the AR quiver records all irreducible maps, it is clear by construction that if V is any CM module then the dimension of the space of maps of degree n between M and V which don't factor through an object of S via maps of strictly smaller positive degree is dim k Hom C/[Sn] (M, V ) where the M sits in degree 0 and the V sits in degree n. By the theory of ladders in τ -categories [9] this can be calculated explicitly using a recursion formula (see e.g. [10, 4.5]). Now the AR quiver of C/[S n ] is just the quiver of C with the vertices in S n deleted, and furthermore by inductive hypothesis (and construction of the smaller C/[S n−1 ]) any term in the recursion at a previous vertex is given by the µ associated to that vertex. Consequently the recursion formula in [10, 4.5] gives dim k Hom C/[Sn] (M, V ) = max{0, −µ (n−2) τ (V ) + a:L→V µ (n−1) L } if V is a step n vertex 0 else which by definition is λ (n) V . Thus indeed λ (n) V records the dimension of the space of maps between M and V of degree n which don't factor through an object of S via maps of strictly smaller positive degree. We now use the above algorithm to re-prove Theorem 3.3 for some examples of groups I m (see §5 for notation). By [3] the AR quiver of C[[x, y]] Im is where • represents the free module and there are precisely m repetitions of the original E 8 shown in dotted lines. The left and right hand sides of the picture are identified, and there is no twist in this AR quiver. The AR translation τ moves each dotted segment one place to the left; τ −1 therefore moves it one place to the right. Example 4.2. Consider the group I 30(b−2)+1 with b ≥ 3. By [10, 9.2] the indecomposable special CM modules are R, A 1 , A 2 , A 3 , A 4 , B 1 , B 2 , C and M where A i := τ −6i R for all 1 ≤ i ≤ 4 B i := τ −10i R for all 1 ≤ i ≤ 2 C := τ −15 R M := τ −30 R To determine the number of irreducible maps from R to the other special CM modules we place a 1 in the position of R (double circled below) and start counting: . . . 1 0 0 1 0 1 0 0 1 0 . . . 1 1 0 1 1 1 1 0 1 1 0 . . . . . 1 1 1 0 1 1 1 0 1 1 2 1 1 0 1 1 1 0 1 1 1 0 Thus there is precisely one map from R to A 1 (of grade 12), one map from R to B 1 (of grade 20) and one map from R to C (of grade 30). To determine the number of irreducible maps out of A 1 , we place a 1 in the position of A 1 and start counting: In the above picture the second special circled (i.e A 2 ) absorbs a 1 and the calculation continues, repeating the segment between the dotted lines. Careful analysis shows that this repetition misses all the specials and keeps repeating until it reaches R again, which absorbs a 1 (on the right hand side of the picture above), forcing the calculation to finish. We deduce that there is precisely one irreducible map from A 1 to A 2 , and one irreducible map from A 1 to R, and these are all the maps out of A 1 . The calculations for A 3 , A 4 , B 1 , B 2 and C are very similar. The only remaining calculation which is non-trivial is determining the number of irreducible maps out of M . For this begin by placing a 1 in the position of M and begin to count. Since there are no special CM modules between M and R this calculation between M and R is just the free expansion starting at M (for terminology, see [10, 7.1]). This continues until it reaches R (a grading of 2 + 60(b − 3) away) and so [10, 9.1] specifies the first two columns below, where the first circle corresponds to R: 2b-6 2b-6 2b-6 2b-5 b-3 2b-6 2b-5 b-3 2b-5 b-2 4b-12 4b-12 4b-11 3b-8 3b-9 4b-11 3b-8 3b-8 3b-7 2b-5 6b-18 3b-9 6b-18 3b-9 6b-17 3b-8 5b-14 2b-6 5b-14 3b-8 5b-14 2b-6 5b-14 3b-8 5b-13 2b-5 4b-10 2b-5 4b-10 2b-5 5b-15 5b-14 4b-12 5b-14 4b-11 4b-11 4b-11 4b-10 3b-8 4b-10 4b-12 4b-11 3b-9 4b-12 4b-11 3b-8 3b-8 3b-8 3b-8 3b-8 3b-8 2b-6 3b-9 3b-9 3b-8 2b-5 2b-5 2b-6 3b-8 2b-5 2b-5 b-3 2b-6 2b-6 2b-6 2b-5 b-2 b-3 2b-6 2b-5 b-3 b-3 b-3 b-3 b-3 b-2 0 b-3 b-3 b-2 b-3 2b-5 b-2 b-2 b-2 b-2 b-2 b-2 b-1 0 3b-8 3b-7 2b-4 2b-4 2b-4 2b-4 2b-4 2b-3 b-1 b-2 4b-10 2b-5 4b-10 2b-5 4b-9 2b-4 3b-6 b-2 3b-6 2b-4 3b-6 b-2 3b-6 2b-4 3b-5 b-1 2b-3 b-2 2b-3 b-1 3b-7 3b-7 3b-7 3b-6 2b-4 3b-6 2b-3 2b-4 2b-3 2b-3 3b-7 2b-4 2b-5 3b-7 2b-4 2b-4 2b-3 b-2 2b-4 2b-3 2b-4 b-2 2b-5 2b-5 2b-4 b-1 b-2 b-2 2b-4 b-1 b-2 b-2 b-2 b-3 2b-5 b-1 0 b-2 b-2 b-2 0 b-2 0 b-3 b-2 1 0 b-2 0 b-2 b-2 b-1 0 b-1 0 0 b-1 0 2b-3 b-1 b-1 b-1 0 b-1 b-1 0 2b-3 b-2 2b-3 b-1 2b-2 b-1 b-1 0 b-1 b-1 b-1 0 b-1 b-1 b-1 0 0 b-1 2b-3 b-1 b-1 b-1 b-1 0 b-1 0 b-1 b-1 b-2 b-1 b-1 b-1 0 0 b-1 0 b-1 0 b-2 b-1 b-1 0 0 0 b-1 0 b-1 0 0 b-2 b-1 0 0 0 0 b-1 0 1 0 0 b-2 1 0 0 0 0 b-1 0 Thus there are precisely b − 3 maps from M to R, one map from M to A 4 , one map from M to B 2 and one map from M to C. This proves that the quiver of the reconstruction algebra is Quotient Singularities We now illustrate the details of the previous sections for all quotient singularities. The finite small subgroups of GL(2, C) and their dual graphs were classified by Brieskorn [5], but here we use the notation from Riemenschneider [12]. The classification of finite small subgroups of GL(2, C) is summarized in the following table: Type Notation Conditions A 1 r (1, a) = εr 0 0 ε a r 1 < a < r, (r, a) = 1 D D n,q = ψ 2q , τ, ϕ 2(n−q) if n − q ≡ 1 mod 2 ψ 2q , τ ϕ 4(n−q) if n − q ≡ 0 mod 2 1 < q < n, (n, q) = 1 T T m = ψ 4 , τ, η, ϕ 2m if m ≡ 1,ψ k = ε k 0 0 ε −1 k τ = 0 ε 4 ε 4 0 ϕ k = ε k 0 0 ε k η = 1 √ 2 ε 8 e 3 8 ε 8 ε 7 8 ω = ε 3 5 0 0 ε 2 5 ι = 1 √ 5 ε 4 5 − ε 5 ε 2 5 − ε 3 5 ε 2 5 − ε 3 5 ε 5 − ε 4 5 where ε t is a primitive t th root of unity. Now to build the quiver of the reconstruction algebra associated to C 2 /G, using Theorem 3.3 and Lemma 3.8 it is easy to transfer the intersection theory combinatorics into the following three rules: (1) If Z f is maximal then connect ⋆ to make the extended Dynkin diagram, double the quiver and for every α i > 2, add an extra α i − 2 arrows from that vertex to ⋆. (2) If Z f is reduced but not maximal then we are not in type A so the Dynkin diagram has a vertex which connects to three others; we call this vertex the middle vertex. To build the quiver, connect ⋆ to the vertices at the end of each 'arm' coming out of the middle vertex, double the resulting quiver and then add extra arrows subject to the following rule: if some α i > 2, add an extra α i − 2 arrows from that vertex to ⋆, except at the middle vertex of the Dynkin diagram where we only add an extra α t − 3 arrows. (3) If Z f is neither maximal nor reduced. Denote by C the vertex with α C ≥ 3 which is closest to the middle vertex. Now inside Z f we may find the largest subvector which is the maximal Z f of some Dynkin subdiagram Q ′ of type D. Considering this subdiagram Q ′ , connect ⋆ to form the extended Dynkin diagram of Q ′ . Then also connect ⋆ to those vertices at the end of each arm of the original Q which do not lie in Q ′ , double the resulting quiver and add extra arrows subject to the following rule: if some α i > 2, add an extra α i − 2 arrows from that vertex to ⋆, except at the vertex C, where we only add α C − 3 extra arrows. Remark 5.3. We remark that there are both geometric and algebraic consequences of the above trichotomy. It turns out that geometrically in (1) the minimal resolution is very similar to the minimal resolutions arising from ADE quotients, where in (2) the minimal resolution is almost identical to those found in the toric case. Case (3) is somewhere in between these two extremes. Case (3) is illustrated in Example 5.4 below -5.4(i) is an example from type D whereas 5.4(ii) is I 13 . We label by dotted lines the largest subvector which is the maximal Z f of some Dynkin subdiagram of type D. Example 5.4. (i) −2 −2 −2 −2 −5 −2 −3 Z f = 1 2 1 2 1 1 1 (ii) −2 −2 −2 −2 −3 −2 Z f = 1 2 1 2 1 1 In the sections below, when stating the dual graph we are referring to Brieskorn [5, 2.11]. On the vertices of the quivers, for the convenience of the reader we write Z f and hence the dimension of the irreducible special representation that corresponds to that vertex. In types T, O and I, labelled arrows with ... means that there are b − 3 arrows from that vertex to ⋆, where if b = 3 there are no arrows. Type A Given G = 1 r (1, a) consider the Jung-Hirzebruch continued fraction expansion r a = [α 1 , . . . , α n ]. It was first proved in [15] that the quiver of the reconstruction algebra can be described as follows: take the labelled dual graph of the minimal resolution · · · −α1 −α2 −αn−1 −αn and associate the double quiver of the extended Dynkin diagram: n≥1 1 1 ··· 1 1 ⋆ ⋆ 1 n=1 Then for every α i > 2, add an extra α i − 2 arrows from vertex i to ⋆. Case (i): Z f is reduced. In this case to the labelled Dynkin diagram above attach ⋆ and double the quiver to produce 1 1 1 1 ··· 1 1 ⋆ If N i=1 (α i − 2) ≥ 2 then add extra arrows as follows: • If α 1 > 3 then add an extra α 1 − 3 arrows from that vertex to ⋆. • If α i > 2 with i ≥ 2 then add an extra α i − 2 arrows from that vertex to ⋆. Example 7.1. Consider D 52,11 then 52 11 = [5,4,3] and so the quiver and dual graph is 1 1 1 1 1 ⋆ −2 −5 −2 −4 −3 Case(ii): Z f not reduced or maximal. Let ν be the number of two-dimensional special representations, then to the above labelled Dynkin diagram attach ⋆ to the ν th and N th vertices and double the quiver: 1 1 2 ··· 2 1 ··· 1 ⋆ If N i=1 (α i − 2) = N i=ν+1 (α i − 2) ≥ 2 then add extra arrows as follows: • If α ν+1 > 3 then add an extra α ν+1 − 3 arrows from that vertex to ⋆ • If some other α i > 2 then add an extra α i − 2 arrows from that vertex to ⋆. The case m ≡ 1. In this subfamily m = 6(b − 2) + 1. The case m ≡ 3. In this subfamily m = 6(b − 2) + 3. The case m ≡ 5. In this subfamily m = 6(b − 2) + 5. The case m ≡ 1. In this subfamily m = 12(b − 2) + 1. The case m ≡ 5. In this subfamily m = 12(b − 2) + 5. The case m ≡ 7. In this subfamily m = 12(b − 2) + 7. The case m ≡ 11. In this subfamily m = 12(b − 2) + 11. The case m ≡ 1. In this subfamily m = 30(b − 2) + 1. The case m ≡ 7. In this subfamily m = 30(b − 2) + 7. The case m ≡ 11. In this subfamily m = 30(b − 2) + 11. The case m ≡ 17. In this subfamily m = 30(b − 2) + 17. The case m ≡ 19. In this subfamily m = 30(b − 2) + 19. The case m ≡ 23. In this subfamily m = 30(b − 2) + 23. The case m ≡ 29. In this subfamily m = 30(b − 2) + 29. Corollary 1.3 (The GL(2, C) McKay Correspondence). For any finite small subgroup G of GL(2, C) letX → C 2 /G be the minimal resolution. Then the special representations of G can be used to build the dual graph ofX by taking the quiver of End C[[x,y]] G (⊕ ρ special (ρ⊗ C[[x, y]]) G ) and deleting the vertex corresponding to the trivial representation. Definition 3. 5 5([18]). For a given CM module M of R define M := π * M/torsion to be the corresponding vector bundle onX.A CM R-module M is called special if H 1 (M ∨ ) = 0.By Van den Bergh's construction it is immediate that OX ⊕ M I is equal to the sum, over all special indecomposable CM R-modules, of the corresponding vector bundles M defined in 3.5. Lemma 3. 6 . 6LetX be the minimal resolution, then the natural map EndX (OX ⊕ M I ) → End R (⊕M ) 0 and also M 1 is generated by global sections. Hence by[7, 2.1] we see that each HomX (M 1 ,M 2 ) ∼ = π * (M ∨ 1 ⊗ M 2 ) is reflexive, thus EndX (OX ⊕ M I ) is reflexive.Consequently there is a version of Corollary 3.3 for End R (⊕M ) and so this proves Theorem 1.1. Since in the quotient case the indecomposable special CM modules are precisely (ρ ⊗ C[[x, y]]) G as ρ runs over the special irreducible representations, this also proves Corollary 1.3.Combining Lemma 3.6 and Corollary 3.4 gives the following improvement of[10, 2.15]. Corollary 3. 7 . 7Let R be an affine complete rational surface singularity and set A := End R (⊕M ) where the sum is taken over all indecomposable special CM R-modules. Thengl.dim A = 2 if R is Gorenstein 3 else.When R is Gorenstein all simple left A-modules and all simple right A-modules have projective dimension 2. When R is not Gorenstein all simple right A-modules have projective dimension 2 except the simple corresponding to ⋆, which has projective dimension 3. As left A-modules, the projective dimension of the simples at ⋆ and all curves corresponding to (−2)-curves have projective dimension 2, whereas all other simples have projective dimension 3.Proof. The statement on the global dimension follows immediately from Corollary 3.4 since End R (⊕M ) ∼ = EndX (OX ⊕ M I ) by Lemma 3.6 and on the minimal resolution the configurations containing only (−2)-curves occur precisely when R is Gorenstein. Now the simples S in Theorem 3.2 are left A = End R (⊕M )-modules with proj.dim A (S i ) = min{t ≥ 0 | Ext t (S i , X) = 0 for all simples X} − 1 and so the statement on left modules follows immediately by inspecting the ext groups in Theorem 3.2. Now denote T i to be the simple right A-module corresponding to M i in the decomposition of M , then proj.dim A (T i ) = min{t ≥ 0 | Ext t (X, S i ) = 0 for all simples X} − 1 and so the result follows from Theorem 3.2. Lemma 3. 8 . 8Suppose E = {E i } forms the exceptional curves on a minimal resolution of some affine complete rational surface singularity. Associate the fundamental cycle Z f = r i E i and canonical cycle Z E K . Suppose F = {F i } forms the exceptional curves on a minimal resolution of another affine complete rational surface singularity with the same dual graph and fundamental cycle, such that −F (i) In the AR quiver, begin by writing a 1 at the position corresponding to M and then define for every CM module ) Consider all arrows out of M in the AR quiver. For convenience we call the heads of these arrows the first-step vertices. For every CM module V define ( 1 ) 1M1 and then circle the first-step vertices which belong to S.(iii) Now consider all arrows out of the first-step vertices. The heads of these arrows are called the second-step vertices. For every CM module V define λ ( 3 ) 3M3 and then circle the third-step vertices which belong to S.(v) Continue in this fashion. For any V and any i, λ (i) V records the dimension of the space of maps of degree i from M to V which don't factor through S via maps of strictly smaller positive degree (for the proof, see below). Consequently the dimension of the space of maps from M to N which don't factor through S via maps of strictly smaller positive degree is simply i≥1 λ Finally the number of irreducible maps out of Z 1 is determined by . Type D Given D n,q , the dual graph of the minimal resolution of C 2 /D n,q is 1 , . . . , α N ]. ⋆ Case (iii): Z f maximal. To the above labelled Dynkin diagram first associate the quiver of the preprojective algebra of the extended If α N > 2 then add an extra α N − 2 arrows from vertex N to ⋆.Example 7.3. For the groups D 7,4 and D 7,5 , the quivers and dual graphs are These are the groups T m where m ≡ 1, 3 or 5 mod 6. These are the groups O m where m ≡ 1, 5, 7 or 11 mod 12. These are the groups I m where m ≡ 1, 7, 11, 13, 17, 19, 23 or 29 mod 30. Consequently the quiver for the curve system E is obtained from the quiver of the curve system F by adding −E 2i + F 2 i extra arrows i → ⋆ for every curve E i . Since both F and E have the same Z f , by definition it is true thatProof. (i) If −F 2 i = −E 2 i then there is nothing to prove, so we can assume that −F 2 i < −E 2 i . Now it is clear that the only map out of R is to X. FurtherB1 B2 C M A4 A3 A2 A1 R ... where there are b − 3 arrows from M to R. Example 4.3. Consider the group I 7 . By [10, 9.3] the indecomposable special CM mod- ules occupy the following positions in the AR quiver: . . . . . . Y 2 . . . . . . . . . . . . . . N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X . . . . Z 2 . R . . Y 1 . . Z 1 R Remark 5.2. Case(3)is somewhat artificial since it does not make sense for an arbitrary labelled Dynkin diagram. However for all the labelled Dynkin diagrams coming from quotient singularities (some in type D, then I 13 and I 23 ) it gives the correct quiver. See Example 5.4 below. The case m ≡ 13. In this subfamily m = 30(b − 2) + 13.Acknowledgement. This paper was originally written whilst the author visited Nagoya University with the Cecil King Travel Scholarship from the London Mathematical Society, and was substantially revised when the author held a JSPS Postdoctoral Fellowship, also at Nagoya University. The author would like to thank the Cecil King Foundation, the LMS and the JSPS for funding this work, and Nagoya University for kind hospitality. The author would also like to thank Osamu Iyama and Akira Ishii for helpful conversations and comments. On isolated rational singularities of surfaces. M Artin, Amer. J. Math. 88M. Artin, On isolated rational singularities of surfaces. Amer. J. Math. 88 (1966) 129-136. Rational singularities and almost split sequences. M Auslander, Trans. Amer. Math. Soc. 2932M. Auslander Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), no. 2, 511-531. McKay quivers and extended Dynkin diagrams. M Auslander, I Reiten, Trans. Amer. Math. Soc. 2931M. Auslander and I. Reiten, McKay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc. 293 (1986), no. 1, 293-301. Flops and derived categories. T Bridgeland, Invent. Math. 1473T. Bridgeland, Flops and derived categories. Invent. Math. 147 (2002), no. 3, 613-632 Rationale singularitäten komplexer flächen. E Brieskorn, Invent. Math. 4E. Brieskorn, Rationale singularitäten komplexer flächen, Invent. Math. 4 (1968), 336-358. A Buan, O Iyama, I Reiten, D Smith, arXiv:0804.3813Mutation of cluster-tilting objects and potentials. A. Buan, O. Iyama, I. Reiten and D. Smith, Mutation of cluster-tilting objects and potentials, arXiv:0804.3813 (2008) Reflexive modules on quotient surface singularities. H Esnault, J. Reine Angew. Math. 362H. Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362 (1985), 63-71. On the McKay correspondence for a finite small subgroup of GL(2, C). A Ishii, J. Reine Angew. Math. 549A. Ishii, On the McKay correspondence for a finite small subgroup of GL(2, C), J. Reine Angew. Math. 549 (2002), 221-233 τ -categories I: Ladders. O Iyama, Algebr. Represent. Theory. 83O. Iyama, τ -categories I: Ladders, Algebr. Represent. Theory 8 (2005), no. 3, 297-321. The classification of special Cohen Macaulay modules. O Iyama, M Wemyss, Math. Z. 2651O. Iyama and M. Wemyss, The classification of special Cohen Macaulay modules, Math. Z. 265 (2010), no.1, 41-83. Graphs, singularities, and finite groups. J Mckay, Proc. Sympos. Pure Math. 37J. McKay, Graphs, singularities, and finite groups, Proc. Sympos. Pure Math. 37 (1980), 183-186. . O Riemenschneider, Invarianten endlicher Untergruppen, Math. Z. 153O. Riemenschneider, Invarianten endlicher Untergruppen, Math. Z. 153 (1977), 37-50. Combinatorics of rational singularities. L D Tráng, M Tosun, Comment. Math. Helv. 793L. D. Tráng and M. Tosun, Combinatorics of rational singularities, Comment. Math. Helv. 79 (2004), no. 3, 582-604. Van den Bergh Three-dimensional flops and noncommutative rings. M , Duke Math. J. 1223M. Van den Bergh Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423-455. . M Wemyss, AMSReconstruction algebras of type A, to appear TransM. Wemyss, Reconstruction algebras of type A, to appear Trans. AMS. M Wemyss, arXiv:0905.1154Reconstruction algebras of type D (I). M. Wemyss, Reconstruction algebras of type D (I), arXiv:0905.1154 (2009). M Wemyss, arXiv:0905.1155Reconstruction algebras of type D (II). M. Wemyss, Reconstruction algebras of type D (II), arXiv:0905.1155 (2009). Reflexive modules on quotient surface singularities. J Wunram, Mathematische Annalen. 2794J. Wunram, Reflexive modules on quotient surface singularities, Mathematische Annalen 279 (1988), no. 4, 583-598. Mathematical Institute. St Giles', Oxford, OX1 3LB, UK. E-mail address: [email protected] Institute, 24-29 St Giles', Oxford, OX1 3LB, UK. E-mail address: [email protected]

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{'abstract': "In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2, C), it means that the endomorphism ring of the special CM C[[x, y]] G -modules can be used to build the dual graph of the minimal resolution of C 2 /G, extending McKay's observation for finite subgroups of SL(2, C) to all finite subgroups of GL(2, C).", 'arxivid': '0809.1973', 'author': [], 'authoraffiliation': [], 'corpusid': 16770446, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16277, 'n_tokens_neox': 14418, 'n_words': 9371, 'pdfsha': '5d55be8813c2d0b99cd09517fc484f83f9f38521', 'pdfurls': ['https://arxiv.org/pdf/0809.1973v3.pdf'], 'title': [], 'venue': []}

arxiv

Lecture at the conference on April 3-4, 1992 Edward Witten Institute for Advanced Study Olden Lane Princeton 08540NJ Lecture at the conference on Topics in Quantum Gravity April 3-4, 1992 This lecture surveys a few loosely related topics, ranging from the scarcity of quantum field theories -and the role that this has played, and still plays, in physics -to paradoxes involving black holes in soluble two dimensional string theory and the question of whether naked singularities might be of even greater interest to string theorists than black holes. Consistent quantum field theories are apparently scarce. As far as we understand, truly consistent four dimensional field theories (without Landau ghosts) require Yang-Mills gauge fields, so as to ensure asymptotic freedom. This assertion is an extrapolation beyond what we firmly know, but is supported in various simple examples by mathematical theorems and computer experiments, mostly about φ 4 theories. The scarcity of quantum field theories is arguably one of the most important things we teach our students, and one of the main assets physicists have had in understanding elementary particles. In fact, it has been very fortunate for the progress of physics that a generic Lagrangian, such as a Fermi-type Lagrangian L = d 4 x[ψi ∂ψ − G F ψγ µ ψ ψγ µ ψ](1) does not (presumably) lead to a consistent quantum theory. The search for a consistent quantum theory that reduces to the Fermi theory at low energies led to the discovery of the now-standard electroweak theory, and with it an appreciation of the role of nonabelian gauge fields in physics. The scarcity of quantum field theories also accelerated the discovery of QCD. Of course, questions of consistency also were crucial in the discovery of Maxwell's equations and of special and then general relativity. General relativity -which is based on a geometric concept somewhat analogous to that of non-abelian gauge theory -is the one real experimental signal we have of physics at energies way beyond accelerator energies. Other experiments sensitive to physics of ultra-high energies -notably proton decay and neutrino mass measurements, and searches for magnetic monopoles and other ultra-massive big bang relics -have so far given null results, albeit important ones. The peculiar properties of general relativity are of course intimately tied up with the fact that Newton's constant has dimensions of (Mass) −2 . This leads among other things to the perturbative unrenormalizability of the theory L = M 2 P ℓ d 4 x √ g R(2) which I will take at face value as an indication that the quantum theory does not exist. I consider this the optimistic as well as most likely interpretation. Indeed, the fact that general relativity is unrenormalizable is the one reason that we have some hope of learning something about new physical principles that prevail at higher-than-accelerator energies. If, conversely, this problem were solved -say in the strongest form of showing that at a nonperturbative level an arbitrary Lagrangian coupling general relativity to matter can be successfully quantized -we would feel truly forsaken by what in the twentieth century has become our best friend, the goddess of consistency. Our hopes of learning what the right theory is, or more modestly of learning about fundamental new structures in physics, would greatly dwindle. Taking then at face value the appearance that general relativity is unrenormalizable, let us ask what new consistent framework could have conventional quantum theory, and Riemannian geometry, as limiting cases. Since geometrical structures of the relevant depth do not exist just by accident, this kind of question cannot be expected to have many possible answers -whether right or wrong for physics. In fact, no insight has ever been gained by frontal assault. Happily, a possible answer -in the form of string theory -emerged as an unintended consequence of work on (unsuccessful models of) strong interactions. What we really understand about string theory are rules for perturbative computations of scattering amplitudes. Curiously enough, these rules are much simpler, in key ways, than standard Feynman rules. It can be argued that the tree level scattering of gauge bosons and gravitons -first worked out by Feynman, De Witt, and others in the 1960's -is most easily calculated by embedding Yang-Mills theory or gravity in a string theory, that is, by calculating appropriate string amplitudes and then taking the appropriate limit to extract the field theory result. The advantage of embedding the conventional calculations in string theory is all the greater at the one loop level. This became clear in the calculations by Green and Schwarz of graviton scattering in the early 1980's, and has recently begun to be exploited systematically in work of Bern, Kosower, and others. Indeed, the simplification obtained by embedding the conventional calculations in string theory is substantial enough that this method may well become standard in computations of QCD processes relevant to the SSC. It may also lead to new insights about QCD. In any event, the simplification that arises this way in the standard calculations in the most interesting geometrical theories is one of the symptoms of the power of the mysterious new geometrical structure that underlies string theory. What is missing in our knowledge of string theory is precisely that this structure is mysterious -that we are far from understanding the conceptual-geometrical, and presumably Lagrangian, framework from which all this is to be derived. Roughly, we know the Feynman rules, but we do not understand the classical theory from which they came. Nor are we close, as far as I know. But the spinoffs that have been discovered, even in purely geometrical problems (in-dex theory on loop spaces, soluble conformal field theory and knot invariants, mirror symmetry, cohomology of moduli space of Riemann surfaces -just to name a few) show the exceptional richness of this structure. In groping toward an answer, we grapple with the following paradigm. The string theory analog of a metric is a two dimensional Lagrangian L(X) = 1 8π Σ d 2 σ √ h h αβ ∂ α X µ ∂ β X ν g µν (X λ ) + . . . . (3) (The X's are a map from a two dimensional surface Σ to a spacetime manifold M of arbitrary dimension.) The string theory analog of a solution of the Einstein equations is then an L(X) for which the corresponding quantum theory is conformally invariant. Then on the space of L(X)'s −i.e., the space of two dimensional quantum field theories -one should find a function (perhaps some version of the c function of Zamolodchikov) that is stationary precisely when L(X) defines a conformal field theory. This paradigm is quite beautiful. It is perhaps vaguely reminiscent of an early idea of Penrose that the metric structure of spacetime should be coded in properties of the space of null geodesics. Here null geodesics -one dimensional objects -are replaced by two dimensional minimal surfaces in spacetime (or more generally, by surfaces that are stationary points of L(X)). In one direction, the paradigm is quite correct and extremely useful. Every conformal field theory (obeying a couple of simple conditions) determines a classical solution of string theory. But the paradigm has trouble in the reverse direction, because it clashes with the fundamental first lesson that I recalled at the outset. There is no problem in writing down a generic L(X), but the generic L(X) does not determine a quantum theory; there is therefore no known way to define the beta function associated with a generic L(X), or to discuss the vanishing of this beta function. The problem is particularly acute if one tries to think about sigma models with time dependence (leading to world-sheet operators of negative dimension) or superficially unrenormalizable operators (corresponding to the characteristic massive modes of the string). This clash between what string theorists appear to need and the rest of what we know in physics is in my opinion the main obstacle to progress in string theory; I cannot emphasize this enough. It is a hint that two dimensional field theory as we know it is not an adequate framework. I have long suspected that we need something cruder, that would contain less information in return for less work. This suspicion was one of my main motivations, incidentally, in developing topological field theory. Today, however, I will be sketching some other developments. Though we do not yet know the end of the story, a main focus of effort by many physicists in the last few years has been to experiment with L(X)'s in a simple situation -that in which space-time, as well as the world-sheet, is two dimensional. The simplicity that arises here, though very unexpected technically, is perhaps crudely analogous to the simplicity that appears in many types of problems in two dimensional mathematical physics. My discussion of these matters will be purely qualitative. For more detail I refer to some of the excellent review articles, or to the generally well-known original papers (including the celebrated papers of Brezin and Kazakov, Douglas and Shenker, and Gross and Migdal on the double scaling limit for D < 2). Granted that a generic L(X) does not lead to a quantum field theory, what are some of the good L(X)'s that we know with a two dimensional target space? The simplest is free field theory, with a linear coupling added to achieve conformal invariance: L(X 0 , X 1 ) = 1 8π d 2 σ − ∂ α X 0 ∂ α X 0 + ∂ α X 1 ∂ α X 1 − 2 √ 2R (2) X 1 . (4) The linear term is a little strange. It means that in this two dimensional world, it is impossible to achieve Poincaré invariance. We will offer an intuitive explanation of this later. I will call this puzzle (a). Several other puzzles will appear presently. A slightly more sophisticated example, still leading to a conformal field theory, is obtained by adding an experimental interaction, the so-called Liouville term: L(X) = L(X) + µ d 2 σ e − √ 2X 1 .(5) For µ > 0, this repels us from the region of X 1 → −∞ where, technically, the string coupling is large. Once one finds a classical solution -whether in general relativity or string theory -one would like to understand the scattering theory around this classical solution. In general this is difficult. The greatest progess in D = 2 string theory has come from the remarkable discovery that scattering theory around the particular classical solution of string theory determined by L(X) is exactly soluble. It can be described in terms of a degenerate gas of free fermions in-teracting with an upside-down harmonic oscillator potential: V (λ) = − 1 2 λ 2 .(6) This was discovered by studying the quantum mechanics of an N × N matrix, on the one hand by diagonalizing the matrix and constructing the Hamiltonian, and on the other hand by expanding in Feynman diagrams. The relation of L(X) to the free fermions leads to several additional puzzles about this theory (beyond puzzle (a) that I have already noted above): (b) The physics of L is really related just to the behavior of the free fermions for λ → ∞. What does the second region at λ → −∞ have to do with it? (c) The free fermions are an (elementary) integrable system. They have infinitely many conserved qualities. If one considers a state with incoming fermions of energies (relative to the fermionic surface) ε 1 , . . . , ε k then Q n = k i=1 ε n i (7) is conserved for each n. Of course, Q 1 is just the Hamiltonian. (d) More generally, the canonical transformations of the free fermion phase space enter in deeper study of the model. By now, (c) and (d) have been pretty well explained from a two dimensional field theory point of view, by studying certain discrete operators of the two dimensional theory. What about (a) and (b)? To explain these points, it is judicious to remember that the interest of string theory derives from its interpretation as a theory of spacetime gravity. Look for a more general Lagrangian L(X 0 , X 1 ) = 1 8π d 2 σ √ h h αβ ∂ α X µ ∂ β X ν g µν (X λ ) + Φ(X λ )R (2) . (8) In a one loop approximation, it is easy enough to solve the equations of conformal invariance. One finds that the target space metric g µν must (up to a coordinate transformation) take the form ds 2 = du dv 1 − uv (9) that is characteristic of a black hole. Indeed (9) is the essence of the Schwarz-schild metric (dropping the angular variables and an inessential conformal factor) in a form that exhibits its maximal analytic continuation. The two asymptotically flat regions of the black hole are the two regions of uv large and negative, the exterior of the light cone in figure (1). This apparently explains our problem (b) about the free fermions of the matrix model. The unexpected region λ → −∞ of the free fermions apparently corresponds to the second asymptotically flat world that is discovered upon making a maximal analytic continuation of the original Schwarzschild solution, which as we recall is ds 2 = −(1 − 2GM r )dt 2 + 1 − 2GM r −1 dr 2 + r 2 dΩ 2 .(10) What about problem (a), the lack of Poincaré invariance in our two dimensional target space? Actually, in writing the metric in equation (9), I have not indicated the dilaton field, and so I have not exhibited the one parameter of the solution, which is the mass M of the black hole, and is buried in the possibility of adding a constant to the dilaton field. (In the brand of two dimensional target space gravity that is determined by this model, the dilaton field enters in the ADM mass formula.) One may ask what happens to the solution as M → 0. A four dimensional Schwarzschild black hole goes over to Minkowski space for M → 0; but a charged black hole in four dimensions cannot have M < Q, and goes over for M → Q to a non-trivial and interesting limit, the extreme Reissner-Nordstrom black hole. I believe that the lack of Poincaré invariance in two dimensional string theory can be understood intuitively by making an analogy with the s-wave sector of a Reissner-Nordstrom black hole in four dimensions. It is as if there is a charge sitting at X 1 = −∞; the linear dilaton field in (4) is the field of this charge, and the simple solution (4) is the analog not of Minkowski space but of an extreme Reissner-Nordstrom black hole. Now actually one can do better and find an exact black hole solution in this two dimensional world, by using SL(2, IR) current algebra. To put it differently, and I think this is a vivid illustration of how rich and physical string theory is, the black hole solution pops out of a simple, universal calculation using current algebra and gauge theory. In the black hole analogy, the Liouville term µ exp(− √ 2X 1 ) in L corresponds to a repulsive non-gravitational interaction that does not have a really good counterpart in four dimensions. So far I have outlined, or at least alluded to, answers to our four questions (a)-(d). But at this point, two more basic questions arise: (1) Is the black hole that arises here stable, and if so how is this compatible with Hawking's discovery that conventional black holes radiate? (2) If standard two dimensional string theory is -as I believe -a theory of matter interacting with the analog of an extreme Reissner-Nordstrom black hole, why is this not more obvious in studies of the free fermion model? I will propose answers. I must warn you, however, that the answer to (2) will be frustrating, though in my opinion entirely consistent and even correct. For (1), I want to first step back and ask, can a black hole be in equilibrium with matter, and can such an equilibrium state be described by a pure quantum mechanical state? My answer to this is not really novel, and involves considerations of Hartle and Hawking from the 1970's -with a twist at the end that depends on the linear dilaton field of the two dimensional string theory. (My comments might also be compared to observations by Seiberg and Shenker in a recent Rutgers preprint.) To begin with, we ask whether one can have pure quantum mechanics of any kind in the field of a black hole. There is no problem here, as long as one considers both asymptotically flat ends. One picks a Cauchy hypersurface S, as sketched in figure (1). In the standard way, one defines a quantum Hilbert space H by quantization on this hypersurface. It is convenient to pick S to be a hypersurface of time reflection symmetry, as sketched in the figure. What would be a natural state vector in H? In general, in a time dependent situation such as the field of the black hole, such a question has no answer; there is a natural Hilbert space, but because of particle creation and annihilation, there is no distinguished "vacuum." The black hole is special, however. Recall the Euclidean black hole solution. In two dimensions it is described by a metric of the form ds 2 = dr 2 + tanh 2 r dφ 2 and looks like a semi-infinite cigar. In four dimensions, there is a similar formula, with additional angular variables that we suppress. The Euclidean black hole has a hypersurface S of time symmetry, given by φ = 0, π. S has the same extrinsic and intrinsic geometry as S, and so we could equally well regard S as the boundary not of half of the extended Lorentzian Schwarzschild space but of a half of its Euclidean analog, as sketched in figure (2). We will use the letter W to denote the relevant half of the Euclidean black hole metric, with boundary S ∼ = S. We can now try to define a distinguished state vector ψ ∈ H by following a recipe similar to that of Hartle and Hawking. We denote the quantum field variables on S as X and those on W as Y . We attempt to define a vector ψ(X) ∈ H by integrating over the Y ′ s: ψ(X) = Y | S =X DY exp (−L(Y )) .(12) We will postpone temporarily the discussion of whether the integral is well-behaved, and consider first the physical properties of ψ if it is. The key point is that although the two asymptotically flat ends, say S L and S R , of S are asymptotically infinitely distant in the Lorentzian black hole, in the Euclidean hole the distance between them is finite. In fact they are separated asymptotically by a distance β/2, with β being the asymptotic circumference of the cigar. Thus if X L and X R are quantum field variables on X L and X R , the wave function ψ is approximately (if one considers only observables supported in the asymptotically flat region) ψ(X L , X R ) = X L | exp(− 1 2 βH)|X R ,(13) with H the Hamiltonian. This is a pure state, but if one wants to consider only observables supported on one end, say S R , then one must integrate out X L to form a density matrix in X R . In the approximation of (13), the density matrix in question is simply ρ(X R ′ , X R ) = DX L ψ(X R ′ , X L ) ψ(X L , X R ) = X R ′ | exp(−βH)|X R .(14) This is a thermal state at the Hawking temperature T = 1/β. Thus, though ψ is a pure state, it looks like a mixed, thermal state to an observer at one end. To such an observer, this state appears to describe a black hole in thermal equilibrium with matter, the outgoing Hawking radiation being in balance with the incoming flux of thermal radiation. Is such equilibrium possible? As we will see, the answer is no in the conventional four dimensional world, but yes in two dimensional string theory. The key point is simply to ask whether the integral (12) converges. In any dimension, the one loop approximation to this integral is well-defined. In that approximation, one finds a thermal energy density at temperature T = 1/β. In infinite volume, the total energy due to this thermal energy density is infinite. In the two loop approximation, we will begin to see the gravitational back-reaction of the infinite thermal energy. In four dimensions, this will produce gravitational collapse on a larger scale, showing that thermal equilibrium between a black hole and matter is impossible in four dimensions. (The same conclusion is sometimes reached by considering the negative specific heat of a black hole in four dimensions.) In two dimensional string theory, there is a very elementary but crucial difference. This comes from the linear dilaton term in (4). As a result of this term, the gravitational coupling vanishes exponentially for X 1 → ∞. As a result of this, the gravitational effects of the infinite thermal gas are finite! So the perturbative contributions to the path integral (12) will all be convergent, and there is no difficulty presumably in defining the state ψ. Thus, in two dimensional string theory, it is possible to have a pure quantum state which to an observer at one end appears to describe thermal equilibrium between a black hole and matter. Not only is this possible; it is presumably what one gets by systematically calculating string loop corrections to the SL(2, IR)/U (1) conformal field theory of the black hole. In fact, it is presumably what one would get in any calculation of Lorentzian black hole physics which can be obtained by analytic continuation of a Euclidean black hole calculation. This would be so for the standard conformal field theory, in which the Lorentzian and Euclidean black holes are described by SL(2, IR) cosets that are related by analytic continuation. There may well be an analog of the free fermion description of the black hole interacting with matter. Now as a preliminary to addressing question (2), let me discuss a crude version of that question that is often asked. Can "the" black hole, that is the SL(2, IR) coset model, decay to the standard two dimensional string model, with world-sheet Lagrangian L or L? (I put quotes around the word "the," since I have suggested above that the standard L or L describes "a" black hole-like object.) The answer is clearly no. "The" black hole does not decay, since it is in equilibrium with the thermal gas around it. Here is another naive preliminary to question (2). Can "the" black hole be created in the usual scattering theory around the standard L or L states? The answer again is obviously no. In usual scattering theory, one excites the ground state by a finite energy disturbance, but "the" black hole lies above the ground state (the L or L vacuum) by an infinite energy. Therefore, it cannot be created in conventional scattering theory. This infinite energy may seem worrisome, so let me make a further comment. In two dimensional string theory, the spacetime gravitational and dilaton fields approach their flat space values for X 1 → ∞ exponentially fast. The analog of the ADM mass is the coefficient of exp(−X 1 ). The infinite energy that was crucial above merely means that the coefficient of exp(−X 1 ) grows linearly in X 1 for X 1 → ∞, and thus the ominous-sounding infinite thermal energy just means that the correction to the asymptotic vacuum behavior is proportional to X 1 exp(−X 1 ) instead of exp(−X 1 ). The extra factor of X 1 obviously does not change things drastically. Having disposed of some preliminaries, let us now discuss what I regard as the most incisive version of question (2): why have not black hole effects, such as Hawking radiation, been noticed in scattering theory around the ground state? For I have suggested that this scattering theory involves interaction of matter with a black hole analog. I will give an answer that I consider convincing but frustrating -it will show the price we pay for embedding black hole physics in a soluble model. We noted above that the description by free fermions leads to conservation of infinitely many charges Q n = k i=1 ε n i .(15) These charges are all coupled to gauge fields (as one sees by examining the discrete excitations of L), so the Q n values of a black hole are measurable outside the horizon and so well-defined. Thus there must be a family of black hole solutions depending on infinitely many parameters Q n ; "the" black hole (described by SL(2, IR) current algebra) is just a special case. The more general black holes would be related to more general L(X)'s containing superficially unrenormalizable operators, so they are hard to study. In the theory of Hawking radiation, there is a classical potential conjugate to every conserved charge carried by the black hole. (Electric and magnetic charge are the examples most often considered in four dimensions.) A black hole carrying non-zero values of these other classical potentials does not just radiate thermally. How it radiates depends on the values of the chemical potentials. Now consider scattering theory around the standard D = 2 string vacuum. The initial state is a black hole-like object with some value of the Q n 's, presumably zero. We excite it by sending in some particles from infinity. The values of the Q n 's and hence the chemical potentials of the system so created depend on what we send in. Therefore, in a generic experiment, the radiation that will come back out will not be thermal -it depends on the chemical potentials of the "hole" and hence on what was sent in. Therefore, generically, one will not get thermal radiation as a result of exciting the hole by incoming matter. This answers question (2). Is there, however, any way to excite the hole in such a way that it will emit thermally? The answer is yes, as I will now explainbut in a way that is sure to be frustrating. If one sends in on the hole a thermal distribution of particles, then conservation of the Q n 's ensures that a thermal distribution will come back out. Therefore, it must be that sending in a thermal distribution of particles is a way to excite the vacuum to just the values of the chemical potentials that will lead to thermal radiation. This is an unglamorous way to "see" Hawking radiation, but it seems to be all one can hope for when the black hole is embedded in an integrable system. I wish to add the following remarks. (1) In a scattering experiment with just a few incident particles, instead of a macroscopic number, most probably the resulting chemical potentials have such extreme values that a thermodynamic description is not valid, even allowing for the chemical potentials. (For four dimensional black holes, the analogous issues have been worked out by Preskill, P. Schwarz, Shapere, Trivedi, and Wilczek.) (2) The Liouville interaction, which I have not built into these remarks explicitly, is a repulsive interaction that will scatter many low energy incident particles non-gravitationally. This is very important quantitatively. (3) Ellis, Mavromatos, and Nanopoulos have related the Q n 's to quantum coherence of black holes. Finally, then, I do believe that the standard two dimensional string theory describes the unitary quantum mechanics of matter interacting with an object similar to a black hole. Integrability means that the behavior is rather different from black hole physics as we usually know it. The linear dilaton field means that stable quantum mechanics of the black hole in thermal equilibrium is possible; it is related to SL(2, IR) current algebra, and an analog of the free fermion description may well exist. In this lecture, I have been focussing on black holes to elucidate the surprising role of a black hole look-alike in soluble two dimensional string theory. There are indeed many unsolved conceptual mysteries in black hole physics, and string theory may give a fruitful vantage point for rethinking them. We should be alert, however, to other possibly related problems that might be even more pertinent for string theorists. Here I have in mind the whole question of the role in physics of general relativistic singularities other than black holes. The "cosmic censorship" conjecture of Penrose asserts roughly (in its original form) that black holes are the only type of singularities that can evolve in classical general relativity from good initial data. Attractive though the cosmic censorship hypothesis is, the evidence for it is quite limited. We are pretty well convinced that gravitational collapse that is sufficiently close to being spherically symmetric leads to black holes; but we know very little about highly aspherical collapse. Though a proof of cosmic censorship would extend the scope of classical general relativity in a dramatic fashion, its failure would possibly benefit physics even more, since the breakdown of classical general relativity at a naked singularity might give us the chance to observe effects of quantum gravity -or string theory. Many physicists seem to think that naked singularities would be "objects," in the same sense that black holes are objects. I think that we should be on the lookout for naked singularities as "events," analogous to instantons rather than to solitons, and perhaps looking to a distant observer much like a miniature big bang. In this context, it is very interesting that there are events known to astrophysicists -the cosmic gamma ray bursts, whose extragalactic origin has been pretty well established by the recent BATSE observations -among whose most conservative explanations scenarios involving highly aspherical gravitational collapse (inspiraling neutron star pairs, for instance) are prominent. These events are therefore fairly good candidates as already observed events in which cosmic censorship may have been violated if it is in fact wrong. These events ought to give us a good incentive for thinking about what string theory would have to say if cosmic censorship is false in general relativity.

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{'abstract': 'This lecture surveys a few loosely related topics, ranging from the scarcity of quantum field theories -and the role that this has played, and still plays, in physics -to paradoxes involving black holes in soluble two dimensional string theory and the question of whether naked singularities might be of even greater interest to string theorists than black holes.', 'arxivid': 'hep-th/9206069', 'author': ['Edward Witten \nInstitute for Advanced Study Olden Lane Princeton\n08540NJ\n'], 'authoraffiliation': ['Institute for Advanced Study Olden Lane Princeton\n08540NJ'], 'corpusid': 117590486, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7061, 'n_tokens_neox': 6408, 'n_words': 5013, 'pdfsha': 'f0ce32aa59a9df3038da389d15e1e93d093b3c06', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9206069v1.pdf'], 'title': ['Lecture at the conference on', 'Lecture at the conference on'], 'venue': ['Topics in Quantum Gravity']}

arxiv

2 May 2012 OBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN 2 May 2012 Let G be a finite group and H be a family of subgroups of G which is closed under conjugation and taking subgroups. Let B be a G-CW-complex whose isotropy subgroups are in H and let F = {F H } H∈H be a compatible family of Hspaces (see definition 2.5). A G-fibration over B with the fiber typeIn this paper, we develop an obstruction theory for constructing G-fibrations with the fiber type F over a given G-CW-complex B. Constructing Gfibrations with a prescribed fiber type F is an important step in the construction of free G-actions on finite CW-complexes which are homotopy equivalent to a product of spheres. Introduction In 1925, Hopf stated a problem which was later called the topological spherical space form problem: Classify all finite groups which can act freely on a sphere S n , n > 1. One variant of this problem was solved by Swan [13]. He proved that a finite group acts freely on a finite complex homotopy equivalent to a sphere if and only if it has periodic cohomology. By using Swan's construction and surgery theory, the topological spherical space form problem has been completely solved by Madsen-Thomas-Wall [9]. It turns out that a finite group G acts freely on a sphere if and only if G has periodic cohomology and any element of order 2 in G is central (see [9, Theorem 0.5]). One of the generalizations of this problem is to classify all finite groups which can act freely on a finite CW-complex homotopy equivalent to a product of k-spheres S n 1 × · · · × S n k for some n 1 , . . . , n k . Recently, Adem and Smith [1] gave a homotopy theoretic characterization of cohomological periodicity and as a corollary they obtained a tool to construct free group actions on CW-complexes homotopy equivalent to a product of spheres. More precisely, they have shown that a connected CW-complex X has periodic cohomology if and only if there is a spherical fibration over X with a total space E that has a homotopy type of a finite dimensional CWcomplex. As a consequence they proved that if G is a finite group and X is a finite dimensional G-CW-complex whose isotropy subgroups all have periodic cohomology, 2000 Mathematics Subject Classification. Primary: 57S25; Secondary: 55R91. The author is supported by TÜBİTAK-TBAG/110T712. then there is a finite dimensional CW-complex Y with a free G-action such that Y ≃ S n × X. As remarked in [1], the second result can also be obtained using the techniques given by Connolly and Prassidis in [3]. More recently, Klaus [6] proved that every p-group of rank 3 acts freely on a finite CW-complex homotopy equivalent to a product of three spheres using similar techniques. The method used by Connolly and Prassidis [3] is to construct a spherical fibration inductively over the skeleta by dealing with cells in each dimension separately. This is a standard strategy in obstruction theory. Note that if there is an orientable G-spherical fibration over the n-skeleton of a CW-complex, then its restriction to the boundary of each (n + 1)-cell σ will be an orientable G σ -fibration with the fiber F Gσ where G σ is the isotropy subgroup of σ. Associated to this G σ -fibration over ∂σ, there is a classifying map from ∂σ to the space BAut Gσ F Gσ where Aut Gσ F Gσ is the topological monoid of self G σ -homotopy equivalences of F Gσ . Combining the attaching map of σ with the classifying map gives us an element in the n-th homotopy group of BAut Gσ F Gσ . Therefore we obtain a cellular cochain which assigns a homotopy class in π n (BAut Gσ F Gσ ) to each (n + 1)-cell. This cochain vanishes if and only if the G-fibration over the n-skeleton extends to a G-fibration over the (n + 1)-skeleton. In the situation where Connolly-Prassidis consider, this cochain can be killed by taking fiber joins. Using this idea,Ünlü [17] gives a concrete cell-by-cell construction of G-spherical fibrations in his thesis. In obstruction theory, one often has obstructions as cohomology classes which tells when a construction can be performed on the next skeleton after some modifications. In other words, the cohomological obstruction class vanishes if and only if the restriction of the construction to the (n − 1)-skeleton extends over the (n + 1)-skeleton. Having a cohomological obstruction is better than having a cochain class as an obstruction since a cohomology class is more likely to be zero. Note that if p : E → B is a G-fibration and b ∈ B H then the fiber p −1 (b) is an H-space. When B H is connected for H ≤ G, there is an H-space F H such that for every b ∈ B H , the fiber p −1 (b) is H-homotopy equivalent to F H . Moreover, if B H is connected for every H ≤ G, the family of H-spaces F H forms a compatible family (see 2.5 for a definition). In this case, the G-fibration p : E → B is said to have the fiber type {F H }. In this paper, we notice that the cohomological obstructions for constructing G-fibrations with a given fiber type live in Bredon cohomology of B with coefficients in π n,F (see page 11 for the definition) and we prove the following theorem. Theorem 1.1. Let G be a finite group and H be a family of subgroups of G which is closed under conjugation and taking subgroups. Let B be a G-CW-complex whose isotropy subgroups are in H such that B H is simply connected for every H ∈ Iso(B). Let F = {F H } H∈H be a compatible family of finite H-CW-complexes and p : E n → B n be a G-fibration over the n-skeleton of B with the fiber type {F H } H∈H where n ≥ 2. (1) There is a cocycle α p ∈ C n+1 H (B; π n,F ) which vanishes if and only if p extends to a G-fibration over B n+1 with a total space G-homotopy equivalent to a G-CW-complex. (2) The cohomology class [α p ] ∈ H n+1 G,H (B; π n,F ) vanishes if and only if the Gfibration p| B n−1 : p −1 (B n−1 ) → B n−1 extends to a G-fibration over B n+1 with a total space G-homotopy equivalent to a G-CW-complex. Moreover if B is a finite G-CW-complex then the total space of the obtained fibration has the G-homotopy type of a finite G-CW-complex whenever E n has the G-homotopy type of a finite G-CW-complex. To prove this theorem we first define an obstruction cochain in the chain complex of Bredon cohomology and show that it is a cocycle. We call this cocycle an obstruction cocycle. Then we show that the difference of obstruction coycles of any two extensions of the G-fibration p| B n−1 is the coboundary of a cochain called the difference cochain. If there is an extension of p| B n−1 to a G-fibration over B n+1 , then the obstruction cocycle of the restriction of this extension to B n vanishes. This means that the obstruction cocycle of p is a coboundary and represents a cohomology class which vanishes. This proves the "if" direction of the above theorem. For the "only if" direction it suffices to show that for every cochain d there is a G-fibration q over B n with q| B n−1 = p| B n−1 such that d is the difference cochain of the extensions p and q of p| B n−1 . Here the most technical part is the construction of a G-fibration q with these properties. That is because it is not clear how to glue G-fibration p| B n−1 to G-fibrations over the n-cells corresponding to the cochain d. For quasifibrations it suffices to take the adjunction of the total spaces to glue two quasifibrations over different base. However, in order to obtain a fibration one needs to put some G-tubes between total spaces of these G-fibrations to create enough space to deal with G-homotopies. We use a generalization of a result due to Tulley [15] to produce a G-fibration q with the required properties. The paper is organized as follows: Section 2 contains definitions and preliminary results on equivariant fibrations. In Section 3, we give a method to glue G-fibrations over different base spaces by generalizing a construction due to Tulley [15]. Finally, we prove Theorem 1.1 in Section 4. Equivariant fibrations In this section, we give the basic definitions of the equivariant fibration theory. We refer the reader to [8] and [18] for more details. Definition 2.1. Let G be a finite group. A G-map p : E → B is called a G-fibration if it has the G-homotopy lifting property with respect to every G-space X, that is, given a commutative diagram of G-maps X × {0} E X × I B h H p H there exists a G-map H : X × I → E such that p H = H and H| X×{0} = h. Equivalently, a G-map p : E → B is a G-fibration if there is a G-map λ : Ω p = {(e, ω) ∈ E × B I | p(e) = ω(0)} → E I such that λ(e, ω)(0) = e and pλ(e, ω) = ω. The G-map λ is called a G-lifting function. By using a similar definition, Dold [4] proved that being a fibration is a local property. The same proof applies to the equivariant case. Definition 2.2. A covering U of G-invariant open sets of B is called numerable G- covering if U is locally finite and there is a G-map f U : B → I such that U = f −1 U (0, 1] for every U ∈ U. Theorem 2.3. A G-map p : E → B is a G-fibration if there is a numerable G- covering U of B such that p| U : p −1 (U) → U is a G-fibration for each U ∈ U. The notion of an equivalence between G-fibrations is defined naturally as follows: Let p i : E i → B be a G-fibration for i = 1, 2. A fiber preserving G-map f : E 1 → E 2 is called a G-fiber homotopy equivalence if there is a fiber preserving G-map g : E 2 → E 1 such that the compositions f g and gf are G-homotopy equivalent to identity maps through G-homotopies which are fiber preserving at each time t ∈ I. In this case, we write p 1 ≃ G p 2 . As in the non-equivariant case, a fiber preserving G-homotopy equivalence between G-fibrations is a G-fiber homotopy equivalence (see [10, pg. 50] for the proof of the non-equivariant case). In [12], Stasheff proved a classification theorem for non-equivariant fibrations up to fiber homotopy equivalences. When a G-fibration is over a path-connected space with trivial G-action, the fiber at each point in the base has a natural G-space structure and all fibers are G-homotopy equivalent with respect to this structure. In this case, the theory of G-fibrations is essentially the same as the non-equivariant one and we have the following classification theorem. Theorem 2.4. Let Aut G (F G ) be the monoid of G-equivariant self homotopy equivalences of a finite G-CW-complex F G . If B is a CW-complex with trivial G-action then there is a one-to-one correspondence between the set of G-fiber homotopy equivalence classes of G-fibrations p : X → B with fibers having the G-homotopy type of F G and the set of homotopy classes of maps B → BAut G (F G ). The equivalence is obtained by taking the pullback fibration from the universal fibration over BAut G (F G ). This classification theorem can be proved by using the same techniques and ideas from [12]. Also, Waner constructs a classifying space for a more general set of equivariant fibrations in [19] and the above theorem can be obtained as a special case of his result. The monoid Aut G (F G ) is not connected in general. However, its connected components are homotopy equivalent via the maps (Aut [1] for non-equivariant case). G (F G ), f ) → (Aut G (F G ), g) with φ → gφf −1 where f −1 is the homotopy inverse of f . Furthermore, when the map π 1 (B) → [F G , F G ] G is trivial, BAut I G (F G ) classifies G-fibrations p : E → B with trivial G-actions on the base where Aut I G (F G ) is the connected component of identity in Aut G (F G ) (see For G-fibrations whose G-action on the base is not trivial, we need to consider the collection of equivariant spaces. Note that if p is a G-fibration over B and b ∈ B K , then the space p −1 (b) is closed under K-action and hence a K-space. Moreover, when B K is connected, the spaces p −1 (b) and p −1 (b ′ ) are K-homotopy equivalent for every b, b ′ ∈ B K . On the other hand, when H a ≤ K, we have h(ab) = a(a −1 ha)b = ab for every h ∈ H and b ∈ B K , hence ab ∈ B H . Clearly, the H-space p −1 (b), where the H-action on p −1 (b) is given by conjugation and the H-space p −1 (ab) are Hhomeomorphic. Therefore, when B H is connected for every isotropy subgroup H of B, the spaces p −1 (b) and p −1 (ab) are H-homotopy equivalent for every isotropy subgroups H, K with H a ≤ K. Definition 2.5. Let H be a family of subgroups of G which is closed under conjugation and taking subgroups. A family F = {F H } H∈H of H-spaces is said to be a compatible family if F H is H-homotopy equivalent to F K for every H, K ∈ H with H a ≤ K for some a ∈ G where the H-action on F K is given by h · y = a −1 hay. Let F = {F H } H∈H be a compatible family where H contains the isotropy subgroups of B. We say p : E → B is a G-fibration with the fiber type F = {F H } H∈H if F H ≃ H p −1 (b) for every b ∈ B H and for every isotropy subgroup H of B. When B H is connected for every H ∈ H, every G-fibration over B is a G-fibration with the fiber type F . However, a G-fibration p : E → B does not necessarily have a fiber type. Tulley's theorem for G-fibrations The aim of this section is to prove an equivariant version of a theorem due to Tulley (see [15,Theorem 11]). The proof uses the same ideas and methods from [15] and [16]. We call the G-fibration q : Z → B × I in Theorem 3.1 a G-tube between p 1 and p 2 . Let f : E 1 → E 2 be a fiber preserving G-map between G-fibrations p 1 and p 2 over B. Recall that the mapping cylinder M f of f is the adjunction space E 1 × I ∪ f E 2 where f (e) = (e, 0) for each e ∈ E 1 . We define the G-map p f : M f → B over B by p f (x, s) = p 1 (x) and p f (y) = p 2 (y) for any x ∈ E 1 , y ∈ E 2 , and s ∈ I. Lemma 3.2. Let f : E 1 → E 2 be a fiber preserving G-map between G-fibrations p 1 and p 2 over B. Then the induced G-map p f : M f → B is a G-fibration. Proof. The proof is similar to the proof of Theorem 1 in [16]. Let λ i : Ω p i → E i I be a G-lifting function for p i , i = 1, 2. Since Ω p f = Ω p 1 ×I ∪ f Ω p 2 where f (e, ω) = (f (e), ω), to show that p f is a G-fibration it suffices to construct a G-map λ : , ω), s) = ω, and λ((e, ω), s)(0) = (e, s). Ω p 1 × I → (M f ) I such that λ| Ωp 1 ×{0} = λ 2 • f , p f λ((e Define λ : Ω p 1 × I → M I f by λ(e, ω, s)(t) = (λ 1 (e, ω)(t), s − t), t ≤ s, λ 2 (z, ω s )(t − s), s ≤ t, where z = f • λ 1 (e, ω)(s) and ω s is given by ω s (t) = ω(s + t) when s + t ≤ 1 and ω s (t) = 1, otherwise. Clearly, λ is a continuous G-map which satisfies the relations λ| Ωp 1 ×{0} = λ 2 • f , p f λ((e, ω), s) = ω, and λ((e, ω), s)(0) = (e, s). In order to prove Theorem 3.1, it suffices to construct G-fibrations q 1 : Z 1 → B × I and q 2 : Z 2 → B × I with q 1 | B×{0} = p 1 , q 1 | B×{1} = q 2 | B×{0} = p f and q 2 | B×{1} = p 2 where f is the fiber preserving G-map between p 1 and p 2 . That is because once we have such G-fibrations, we can obtain a G-tube between p 1 and p 2 by gluing q 1 and q 2 as follows. Let Z = Z 1 ∪ i 1 M f × I ∪ i 2 Z 2 where i 1 (x) = (x, 0) and i 2 (x) = (x, 1) for every x ∈ M f . Define the G-map q : Z → B × I by q(z) =    (π 1 (q 1 (z)), 1 3 π 2 (q 1 (z))), z ∈ Z 1 ; (π 1 (q 2 (z)), 2 3 + 1 3 π 2 (q 2 (z))), z ∈ Z 2 ; (p f (x), 1 3 (1 + t)), z = (x, t) ∈ M f × I where π i is the projection map to the i-th coordinate. Since we can extend the lifting functions for q 1 and q 2 to lifting functions for q| B×[0, 6 9 ) and q| B×( 4 9 ,1] , respectively, Gmaps q| B×[0, 6 9 ) and q B×( 4 9 ,1] are G-fibrations. Therefore q : Z → B × I is a G-fibration by Theorem 2.3. Proposition 3.3. Let Z 2 = {(e, s, t) ∈ E 1 ×I ×I| s+t ≤ 1}∪ f E 2 ×I in M f ×I where f : E 1 ×{0}×I is defined by f (e, 0, t) = (f (e), t). Then q 2 = (p f ×id)| Z 2 : Z 2 → B ×I is a G-fibration with q 2 | B×{0} = p f and q 2 | B×{1} = p 2 . Z 2 B × I ∧ s > t q 2 Proof. Let r : M f × I → Z 2 be defined by r| E 2 ×I = id E 2 ×I and r| E 1 ×I×I (x, s, t) = (x, s, t) s + t ≤ 1; (x, t, t), otherwise. Then r is a fiber preserving G-retraction. Let H : X × I → B and h : X → Z 2 be given G-maps with H| X×{0} = p 2 • h. Since p f × id is a G-fibration, there is a G-map H : X × I → M f × I which makes the following diagram commute: X × {0} Z 2 M f × I X × I B × I B × Ī H p 2 h H p f × id H r i Then the G-map H : X × I → Z 2 defined by H = rH satisfies p 2 H = H and H| X×{0} = h. Definition 3.4. Let p i : E i → B be a G-fibration for i = 1, 2 with E 1 ⊆ E 2 and p 2 | E 1 = p 1 . Then p 1 is said to be a G-deformation retract of p 2 if E 1 is a G-deformation retract of E 2 via fiber preserving G-retraction, that is, if there is a G-map H : E 2 × I → E 2 such that H 0 = id E 2 , H(e, 1) ∈ E 1 and p 2 H(e, t) = p 2 (e) for every e ∈ E 2 . If H also satisfies the relation H(e, t) = e for every e ∈ E 1 , we say p 1 is a strong G-deformation retract of p 2 . To show that there is a G-tube q 1 : Z 1 → B × I between p 1 and p f , we need the following proposition. The non-equivariant version of this proposition is proved in [15] and used in a recent paper by Steimle [14]. Proposition 3.5. If p 1 is a strong G-deformation retract of p 2 then the G-map q = ( p 2 × id)| Z : Z → B × I where Z = {(e, t) ∈ E 2 × I| e ∈ E 2 if t > 0, e ∈ E 1 if t = 0} is a G-fibration. Z B × I E 1 q Proof. We are using the same approach that is used in the proof of Theorem 3.1 in [7]. Let H : E 2 × I → E 2 be a strong G-deformation retraction of the G-fibration p 2 onto p 1 . Let λ : Ω p 2 → E I 2 be a G-lifting function for p 2 . Define a G-map λ ′ : Ω q → Z I by π 2 λ ′ ((e, ω 2 (0)), (ω 1 , ω 2 ))(t) = ω 2 (t) and π 1 λ ′ ((e, ω 2 (0)), (ω 1 , ω 2 ))(t) =    H(λ(e, ω 1 )(t), t ω 2 (t) ), ω 2 (t) > 0, ω 2 (t) ≥ t; e, t = ω 2 (t) = 0; H(λ(e, ω 1 )(t), 1), t > 0, t ≥ ω 2 (t). Clearly, qλ ′ ((e, ω 2 (0)), (ω 1 , ω 2 )) = (ω 1 , ω 2 ) and λ ′ ((e, ω 2 (0)), (ω 1 , ω 2 ))(0) = e. Therefore we only need to check the continuity of π 1 λ ′ at t = 0. For this it suffices to show that the adjoint map π 1 λ ′ : Ω q × I → E 2 is continuous at t = 0. Let (e α , ω 1,α , ω 2,α , t α ) be a net converging to (e, ω 1 , ω 2 , 0). Let U be an open neighborhood of e ∈ E 1 . Since H : E 2 × I → E 2 is continuous, V = H −1 (U) is open. Since (e, t) ∈ V for every t ∈ I, there are open neighborhoods A t ∋ e and V t ∋ t such that A t × V t ⊆ V , for all t ∈ I. Since I is compact, there exist t 1 , . . . , t n such that I = ∪ n i=1 V t i . Then A = ∩ n i=0 A t i is an open neighborhood of e with the property that H(A × I) ⊆ U. Since λ is continuous, there is β such that λ(e α , ω 1,α , t α ) ∈ A for every β > α. Therefore π 1 λ ′ (e α , ω 1,α , ω 2,α , t α ) ∈ U for every α > β as desired. It is proved in [11,Corollary 2.4.2] that when f : E 1 → E 2 is a homotopy equivalence then E 1 is a strong deformation retract of M f . The same proof applies to G-fibrations (see [5,Lemma 2.5.2]). Therefore, by Proposition 3.5, there is a G-tube q 1 between p 1 and p f . Note that p 2 is also a strong G-deformation retract of p f and one can also use Proposition 3.5 to construct a G-tube between p 2 and p f . This completes the proof of Theorem 3.1. As an immediate corollary, we have: Z = E 1 ∪ k 1 p −1 1 (B) × [0, 1 3 ] ∪ m 1 Z ∪ m 2 p −1 2 (B) × [ 2 3 , 1] ∪ k 2 E 2 where k 1 : p −1 1 (B) × {0} → E 1 , k 2 : p −1 1 (B) × {1} → E 2 and m 1 : p −1 1 (B) × { 1 3 } → Z, m 2 : p −1 2 (B) × { 2 3 } → Z are the inclusions. Define a G-map q : Z → B 1 ∪ i 1 B × I ∪ i 2 B 2 by q| E i = p i , q| Z×[ 1 3 , 2 3 ] (z, t) = q(z) and by q(e, t) = p j (e) for t ∈ [ 2(j−1) 3 , 2j−1 3 ]), j = 1, 2. B 1 B 2 B × I Z q Clearly the restriction of q to the following subsets are G-fibrations {B 1 ∪ i 1 B × [0, 2 9 ], B × [ 1 9 , 5 9 ], B × [ 4 9 , 8 9 ], B × [ 7 9 , 1] ∪ i 2 E 2 } Therefore, by Theorem 2.3, q is a G-fibration. Theorem 3.7. Let p i : E i → B, i = 1, 2, be G-fiber homotopy equivalent G-fibrations such that E 1 and E 2 have the G-homotopy type of a G-CW-complex. Then there is a G-tube q between p 1 and p 2 whose total space is G-homotopy equivalent to a G-CWcomplex. Proof. Let f : E 1 → E 2 be a G-fiber homotopy equivalence. Recall that the total space of the G-tube q we constructed is Z = Z 1 ∪ i 1 M f × [ 1 3 , 2 3 ] ∪ Z 2 where Z 1 = {(e, t) ∈ M f × [0, 1 3 ]| e ∈ M f if 0 < t ≤ 1 3 , e ∈ E 1 if t = 0} Z 2 = {(x, s, t) ∈ E 1 × I × [ 2 3 , 1]| s + 3t ≤ 3} ∪ f E 2 × [ 2 3 , 1]. Clearly, Z is a strong G-deformation retract of M f . On the other hand, M f is Ghomotopic to E 2 and hence it has a G-homotopy type of a G-CW-complex. Corollary 3.8. A G-fibration p : E → S n−1 over (n − 1)-sphere with trivial G-action on the base extends to a G-fibration over a disk if and only if it is G-fiber homotopy equivalent to a trivial G-fibration. Proof. Since D n is contractible, the "only if" part holds. For the "if" part, let p be G-fiber homotopy equivalent to a trivial fibration. Consider D n as the cone of S n−1 , that is, D n = S n−1 × [0, 2]/ ∼ where (y, 2) ∼ * . Let B 1 = S n−1 × [1, 2]/ ∼, and B 2 = B = S n−1 × {1}. Then ε| B ≃ p where ε : B 1 × F G → B 1 where F G = p −1 (x) for some x ∈ S n−1 . By Corollary 3.6, there is a G-fibration extending p. Remark 3.9. In [3], the statement of Corollary 3.8 appears on page 137 but in there the total spaces were attached directly which results in a G-quasifibration from which one gets a G-fibration by taking the corresponding Hurewicz fibration. Constructing G-fibrations In this section, we introduce an obstruction theory for constructing G-fibrations over G-CW-complexes and we prove Theorem 1.1. An adequate cohomology theory to develop such an obstruction theory is Bredon cohomology. Let us first fix some notation for Bredon cohomology. We refer the reader to [2] and [8] for more detailed information about Bredon cohomology. Let G be a finite group and H be a family of subgroups of G which is closed under conjugation and taking subgroups. The orbit category Or H (G) relative to the family H is defined as the category whose objects are left cosets G/H where H ∈ H and whose morphisms are G-maps from G/H to G/K. Recall that any a ∈ G with H a ≤ K induces a G-mapâ : G/H → G/K defined byâ(H) = aK and conversely, any G-map from G/H to G/K is of this form. Let B be a G-CW-complex whose isotropy subgroups are all in H. We call α p the obstruction cocycle. From now on we assume that the total spaces of G-fibrations that we consider have the homotopy type of a G-CW-complex unless otherwise stated. Moreover if E n has the G-homotopy type of a finite G-CW-complex then the total space of the fibration that we obtain has the G-homotopy type of a finite G-CW-complex. Proof. If the obstruction cocycle is zero, then [φ Gσ • f σ ] = 0 for any (n + 1)-cell σ of B. By Theorem 2.4, the restriction p| ∂σ is G σ -fiber homotopy equivalent to the trivial G σ -fibration ε : ∂σ × F Gσ → ∂σ. Let β σ : ∂σ × F Gσ → p −1 (∂σ) be the G σ -fiber homotopy equivalence between them. By Corollary 3.8, p| ∂σ extends to a G σ -fibratioñ p σ : Z → σ over σ. Let us define E ′ = σ∈I n+1 G × Gσ (Z ∪ i 1 p −1 (∂σ) × I) ∪ i 2 E where I n+1 is the set of representatives of G-orbits of (n + 1)-cells and i j 's are the corresponding inclusions for j = 1, 2. Let q : E ′ → B n+1 be defined by relations q| Z =p σ , q| p −1 (∂σ)×I = p| ∂σ × id and q| E = p. By Theorem 2.3, the G-map q is a G-fibration. Since all the fibers and the base space have the G-homotopy type of a G-CWcomplex, E ′ is G-homotopy equivalent to a G-CW-complex. More precisely, for each orbit representative σ ∈ I n+1 , we can deform G × Gσ Z ∪ i 1 (p −1 (∂σ) × I) ∪ i 2 E to G × Gσ σ × F Gσ ∪ βσ E relative to E via a G-map as shown in Figure 1. result in the standard theory, we have 0 = (δo Ψ )(H)(σ × I) = o Ψ (H)(∂σ × I) + (−1) n+1 o Ψ (H)(σ × {1}) − o Ψ (H)(σ × {0}) = [Ψ q,H • f ∂σ×I ] + (−1) n+1 [Ψ q,H • f σ×{1} ] − [Ψ q,H • f σ×{0} ] = (−1) n+1 (d p 1 ,q,p 2 (H)(∂σ) + α p 2 (H)(σ) − α p 1 (H)(σ)) = (−1) n+1 (δd p 1 ,q,p 2 (H)(σ) + α p 2 (H)(σ) − α p 1 (H)(σ)) for any (n + 1)-cell σ of B H . This implies that for every H ∈ G and for every (n + 1)−cell σ of B H , we have δd p 1 ,q,p 2 (H)(σ) = (α p 1 − α p 2 )(H)(σ) and hence δd p 1 ,q,p 2 = α p 1 − α p 2 . The above proposition immediately implies the "if" direction of the second statement in Theorem 1.1. To see this, note that if the G-fibration p| B n−1 extends to a G-fibrationp over B n+1 then δd p,q,p| Bn = α p − αp | Bn Since d(G τ )(τ ) ∈ π n (BAut Gτ F Gτ ), it is represented by a map Ψ τ : (D n 1 4 , S n−1 1 4 ) → (BAut Gτ F Gτ , * ). Let u ′ Gτ = EAut Gτ F Gτ × Aut Gτ F Gτ F → BAut Gτ F Gτ . Then the restriction of the G τfibration Ψ * τ (u ′ Gτ ) : E 2 τ → X 3 to S n−1 1 4 × {1} is the same as the trivial G τ -fibration with the fiber F . By gluing these fibration with the trivial one over X 2 , we obtain a G τ -fibration E 1 τ ∪ F × X 2 ∪ E 2 τ → D n × {1} over D n × {1}. Let p τ : E τ → τ be the corresponding G τ -fibration over τ . As in the proof of Proposition 4.3, the following G-map E = τ ∈In G × Gτ E τ ∪ i 1 p −1 (∂τ ) × I ∪ i 2 p −1 (B n−1 ) p   B n is a G-fibration over B n . Moreover, when E has the G-homotopy type of a G-CWcomplex, so does E. Let q : E ∪ (p −1 (B n−1 ) × I) ∪ E → (B × I) n be the G-fibration defined by q| E = p, q| E = p, and q| p −1 (B n−1 )×I = p| B n−1 × id. Then d p,q,p (G τ )(τ ) is represented by the classifying map Ψ : D n × {0} ∪ S n−1 × I ∪ X 1 ∪ X 2 ∪ X 3 → BAut Gτ F Gτ where Ψ| D n ×{0}∪ S n−1 ×I = φ p,Gτ π 1fτ , Ψ| X 1 = φ p,Gτ π 1fτ f Ψ| X 2 = c φ p,Gτ (fτ (0)) , Ψ| X 3 = Ψ τ . Here, c φ p,Gτ (fτ (0)) is the constant map at φ p,Gτ (f τ (0)). Since Ψ| D n ×{0}∪S n−1 ×I∪X 1 is homotopic to the constant map c φ p,Gτ (fτ (0)) relative to S n−1 1 2 × {1}, the map Ψ also represents d(G τ )(τ ). Therefore, we have d = d p,q,p . Now we can prove the "only if" of the main theorem as follows. Proof of Theorem 1.1: It only remains to show that if α p is cohomologous to zero then there is a G-fibration over B n+1 which extends p| B n−1 . Let α p = δd for some d ∈ Hom(C n (B), π n,F ). By Proposition 4.5, there is a G-fibration q over B × I such that d = d p,q,p wherep = q| Bn×{1} . Since α p = δd = α p − αp, we have αp = 0 and hencep extends to a G-fibration over B n+1 . Remark 4.6. In Theorem 1.1, one can replace the assumption that B H is simplyconnected for every H ∈ H with the assumption that the map π 1 (B H ) → [F H , F H ] H is trivial for every H ∈ H. In applications, one often has fibers which are homotopy equivalent to spheres and one can take fiber joins to make this map trivial. Theorem 3. 1 . 1Let p 1 : E 1 → B and p 2 : E 2 → B be G-fiber homotopy equivalent G-fibrations. Then there is a G-fibration q over B × I such that q| B×{0} = p 1 and q| B×{1} = p 2 . Corollary 3 . 6 . 36Let B 1 , B 2 , and B be topological spaces such that B ⊆ B 1 ∩ B 2 . If p 1 : E 1 → B 1 and p 2 :E 2 → B 2 are G-fibrations with p 1 | B ≃ p 2 | B then there is a G-fibration over B 1 ∪ i 1 (B × I) ∪ i 2 B 2 extending p 1 and p 2 where i j : B × {j} → B j are inclusions.Proof. By Theorem 3.1, there is a G-tube q : Z → B × I between p 1 | B and p 2 | B . Without loss of generality, we can consider q over B × [ 1 3 , 2 3 ] with q| B×{ 1 3 } = p 1 and q| B×{ 2 3 } = p 2 . Let A coefficient system for the Bredon cohomology is a contravariant functor M : Or H (G) → Ab where Ab is the category of abelian groups. A morphism T : M → N between two coefficient systems is a natural transformation of functors. Note that a coefficient system is a ZOr H (G)-module with the usual definition of modules over a small category. Since the ZOr H (G)-module category is an abelian category, the usual notions for doing homological algebra exist. Given a local coefficient system M : Or H (G) → Ab, one defines the cochain complex C * H (B; M) of B with coefficients in M as follows: Let C n H (B; M) be the submodule of ⊕ H∈H Hom Z (C n (B H ; Z), M(G/H)) formed by elements (f (H)) H∈H which makes the following diagram commute: C n (B K ; Z) H, K ∈ H with H a ≤ K. Hereā : B K → B H is given byā(x) = ax for any x ∈ B K andā * denotes the induced map between the chain complexes. The coboundary map δ : C n H (B; M) → C n+1 H (B; M) is defined by (δf )(H)(τ ) = f (H)(∂τ ) for any H ∈ H and for any (n + 1)-cell τ of B H . Definition 4 . 1 . 41The Bredon cohomology H * G,H (B; M) of G-CW-complex B with coefficients in M is defined as the cohomology of the cochain complex C * H (B; M). Let F = {F H } H∈H be a compatible family. Since F is compatible, we can consider the universal K-fibration u K : E K → BAut I K (F K ) with trivial K-action on the base Proof. For an H ∈ Iso(B), α p (H) ∈ C n+1 (B H , π n (BAut I H F H )) is the obstruction cochain for extending the map φ p,H : B H n → BAut I H F H to the (n + 1)-skeleton of B H . Therefore, by classical obstruction theory, we have (δα p )(H)(τ ) = δ(α p (H))(τ ) = 0 for any (n + 2)-cell τ of B H . On the other hand, for arbitrary H ∈ H, we have (δα p )(H)(τ ) = (Res Gτ H ) * δα p (G τ )(τ ) = 0. So, δα p = 0. Proposition 4.3. A G-fibration p : E n → B n with the fiber type F = {F H } H∈H ,where F H is a finite H-CW-complex, extends to a G-fibration over the (n+1)-skeleton B n+1 if and only if α p = 0. Proposition 4.2. α p is a cocycle. Acknowledgements. This work is part of the author's PhD thesis at the Bilkent University. The author is grateful to her thesis advisor Ergün Yalçın for introducing her to this problem, for valuable discussions and for the careful reading of the first draft. The author thanksÖzgünÜnlü for his crucial comments on this work. We also thank the referee for helpful comments, in particular, for suggesting a simpler map which shortens the proof of Lemma 3.2.as an H-fibration with the fiber F H via conjugation action whenever H a ≤ K. Let a : BAut I K (F K ) → BAut I H (F H ) be the classifying map of this fibration. Define a contravariant functor π n,F : Or H (G) → Ab by letting π n,F (G/H) = π n (BAut I H (F H )) π n,F (â : G/H → G/K) = π * ( a) : π n (BAut I K (F K )) → π n (BAut I H (F H )).From now on we assume that B H is simply connected for every H ∈ Iso(B), where Iso(B) is the set of isotropy subgroups of B.for every H ∈ H and for every (n + 1)-cell σ of B H with an attaching map f σ : S n → B Gσ n . For α p to be a cochain, the following diagram must commute up to homotopyThe first square commutes because B has a G-CW-complex structure. Since a is the classifying map of the G aσ -fibration u Gσ and φ p,Gσ is the classifying map of p Gσ , the pullback of the universal G aσ -fibration u Gaσ by the composition a • φ p,Gσ is G aσ -fiber homotopy equivalent to the fibration p Gσ considered as an G aσfibration. On the other hand, the pullback of the G aσ -fibration p Gaσ byā is G aσ -fiber homotopy equivalent to the fibration p Gσ and hence (φ p,Gaσ •ā) * u Gaσ ≃ Gaσ p Gσ . Therefore the G aσ -fibrations ( a • φ p,Gσ ) * u Gaσ and (φ p,Gaσ •ā) * u Gaσ are G aσ -fiber homotopy equivalent. By Theorem 2.4, the maps a • φ p,Gσ and φ p,Gaσ •ā are homotopic and hence α p is a cochain in C n+1 H (B, π n,F ). The second part of the theorem says that if α p is cohomologous to zero, that is, α p = δd for some cochain d ∈ C n H (B, π n,F ) then the G-fibration p| B n−1 : p −1 (B n−1 ) → B n−1 extends to a Gfibration over B n+1 . In order to prove this, we redefine p over the n-skeleton relative to the (n−1)-skeleton in such a way that the obstruction cocycle of this new G-fibration vanishes.Let p 1 and p 2 be G-fibrations over B n whose restrictions to B n−1 are G-fiber homotopy equivalent. Then by Corollary 3.6, there is a G-fibration q : Z → (B × I) n with q| Bn×{0} = p 1 and q| Bn×{1} = p 2 . Let Ψ q,H : (B H ×I) n → BAut I H F H be the classifying map of the fibration q H for H ∈ Iso(B). Note that the composition Ψ q,H • i j gives a classifying map for the H-fibration p H j where i j 's are the corresponding inclusions. As before, the map Res Gae H • Ψ q,Gae • f ae is homotopic toã • Res Ge K • Ψ q,Ge • f e for every (n + 1)-cell e of B K × I with the attaching map f e : S n−1 → B Gσ n and for every H a ≤ K. Therefore, the map d p 1 ,q,p 2 ∈ H∈H Hom Z (C n (B H ), π n (BAut I H F H )) defined for an n-cell τ of B H by, is a cochain in C n H (B; π n,F ). We call d p 1 ,q,p 2 the difference cochain. As in the standard theory, we have the following. Proof. For an n-cell τ of B, the G τ -mapp τ : p −1 (τ ) ∪ p −1 (∂τ ) × I →τ ∪ ∂τ × I, wherep τ | p −1 (τ ) = p| p −1 (τ ) andp τ | p −1 (∂τ )×I = p| p −1 (∂τ ) × id, is a G τ -fibration and it is classified by the map φ p,Gτ • π 1 where π 1 :τ ∪ ∂τ × I →τ is the projection to the first coordinate. Let E τ be the pullback ofp τ by the mapNote that p 1 | S n−1 1 ×{1} = f * τ (p| ∂τ ) and p 1 | S n−1 1 2 ×{1} is the trivial G τ -fibration with the fiber F = p −1 (f τ (0)), where S n−1 r is the (n − 1)-sphere of radius r. Periodic complexes and group actions. A Adem, J H Smith, Ann. of Math. 154A. Adem and J. H. Smith, Periodic complexes and group actions, Ann. of Math. 154 (2001), 407-435. Equivariant Cohomology Theories. G Bredon, Lecture Notes in Mathematics. 34Spring-VerlagG. Bredon, Equivariant Cohomology Theories, Lecture Notes in Mathematics 34, Spring- Verlag, 1967. F Connolly, S Prassidis, Groups which act freely on R m × S n−1. 28F. Connolly and S. Prassidis, Groups which act freely on R m × S n−1 , Topology 28 (1989), 133-148. Partitions of unity in the theory of fibrations. A Dold, Ann. of Math. 78A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963), 223-255. Obstructions for constructing G-equivariant fibrations. A Güçlükan, PhD thesisA. Güçlükan, Obstructions for constructing G-equivariant fibrations, PhD thesis (2011). M Klaus, arXiv:1011.1274v1Constructing actions of p-groups on product of spheres, preprint. M. Klaus, Constructing actions of p-groups on product of spheres, preprint, arXiv:1011.1274v1. Replacement and extension theorems in the theory of Hurewicz fiber spaces. S Langston, Ph.D. ThesisS. Langston, Replacement and extension theorems in the theory of Hurewicz fiber spaces, Ph.D. Thesis (1968). Transformation Groups and Algebraic K-theory. W Lück, Lecture Notes in Mathematics. 1408Spring-VerlagW. Lück, Transformation Groups and Algebraic K-theory, Lecture Notes in Mathematics 1408, Spring-Verlag, 1989. The topological spherical space form problem II. I Madsen, C B Thomas, C T Wall, Topology. 15I. Madsen, C. B. Thomas, and C .T .Wall, The topological spherical space form problem II, Topology 15 (1978), 375-382. A Concise Course in Algebraic Topology. J P May, Univ. Chicago PressJ. P. May, A Concise Course in Algebraic Topology, Univ. Chicago Press, 1999. R A Piccinini, Lectures on Homotopy Theory. North-HollandR. A. Piccinini, Lectures on Homotopy Theory, North-Holland, 1992. A classification theorem for fiber spaces. J Stasheff, Topology. 2J. Stasheff, A classification theorem for fiber spaces, Topology 2 (1963), 239-246. Periodic resolutions for finite groups. R G Swan, Ann. of Math. 72R. G. Swan, Periodic resolutions for finite groups, Ann. of Math. 72 (1960), 267-291. Higher Whitehead torsion and the geometric assembly map. W Steimle, arXiv:1105.2116v1preprintW. Steimle, Higher Whitehead torsion and the geometric assembly map, preprint, arXiv:1105.2116v1. A strong homotopy equivalence and extensions for Hurewicz fibrations. P , Tulley Mcauley, Duke Math. J. 36P. Tulley McAuley, A strong homotopy equivalence and extensions for Hurewicz fibrations, Duke Math. J. 36 (1969) , 609-619. A note on paired fibrations. P , Tulley Mcauley, Proc. Amer. Math. Soc. 342P. Tulley McAuley, A note on paired fibrations, Proc. Amer. Math. Soc. 34 (2) (1972), 534-540. Constructions of free group actions on products of spheres. Ö Ünlü, PhD. thesisÖ.Ünlü, Constructions of free group actions on products of spheres, PhD. thesis (2004). Equivariant fibrations and transfer. S Waner, Trans. Amer. Math. Soc. 258S. Waner, Equivariant fibrations and transfer, Trans. Amer. Math. Soc. 258 (1980), 369-384. Equivariant classifying spaces and fibrations. S Waner, Trans. Amer. Math. Soc. 258S. Waner, Equivariant classifying spaces and fibrations, Trans. Amer. Math. Soc. 258 (1980), 385-405. E-mail address: guclukan@fen. Asli Güçlükanilhan, Bilkent, Ankara, TurkeyDepartment of Mathematics, Bilkent Universitybilkent.edu.trAsli Güçlükanİlhan, Department of Mathematics, Bilkent University, Bilkent, Ankara, Turkey. E-mail address: [email protected]

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{'abstract': 'Let G be a finite group and H be a family of subgroups of G which is closed under conjugation and taking subgroups. Let B be a G-CW-complex whose isotropy subgroups are in H and let F = {F H } H∈H be a compatible family of Hspaces (see definition 2.5). A G-fibration over B with the fiber typeIn this paper, we develop an obstruction theory for constructing G-fibrations with the fiber type F over a given G-CW-complex B. Constructing Gfibrations with a prescribed fiber type F is an important step in the construction of free G-actions on finite CW-complexes which are homotopy equivalent to a product of spheres.', 'arxivid': '1110.3880', 'author': ['\nOBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN\n\n', '\nOBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN\n\n', '\nOBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN\n\n'], 'authoraffiliation': ['OBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN\n', 'OBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN\n', 'OBSTRUCTIONS FOR CONSTRUCTING EQUIVARIANT FIBRATIONS ASLI GÜÇ LÜKANİLHAN\n'], 'corpusid': 54536701, 'doi': '10.2140/agt.2012.12.1313', 'github_urls': [], 'n_tokens_mistral': 13746, 'n_tokens_neox': 12215, 'n_words': 7630, 'pdfsha': '9dfad007bc1efb400a0a2c234a7f9aaaf1e4324e', 'pdfurls': ['https://export.arxiv.org/pdf/1110.3880v2.pdf'], 'title': [], 'venue': []}

arxiv

World-Wide Web scaling exponent from Simon's 1955 model 31 Aug 2000 Stefan Bornholdt Institut für Theoretische Physik Universität Kiel Leibnizstrasse 15D-24098KielGermany Holger Ebel Institut für Theoretische Physik Universität Kiel Leibnizstrasse 15D-24098KielGermany World-Wide Web scaling exponent from Simon's 1955 model 31 Aug 2000 Recently, statistical properties of the World-Wide Web have attracted considerable attention when self-similar regimes have been observed in the scaling of its link structure. Here we recall a classical model for general scaling phenomena and argue that it offers an explanation for the World-Wide Web's scaling exponent when combined with a recent measurement of internet growth. A quantity important for searching the World-Wide Web [1] is the number k of links that point to a particular web page. Its probability distribution P (k) exhibits power-law scaling [2,3] P (k) ∼ k −γ that is not readily explained by standard random graph theory [4]. An elegant model for scaling in copy and growth processes was proposed by Simon [5] in 1955 which describes scaling behaviour as observed in distributions of word frequencies in texts or population figures of cities [6]. It models the dynamics of a system of elements with associated counters (e.g., words and their frequencies in texts, or nodes in a network and their connectivity k) where the dynamics of the system is based on constant growth via addition of new elements (new instances of words) as well as incrementing the counters (new occurrences of a word) at a rate proportional to their current values. Reformulating this to model network growth consider a network with n nodes with connectivities k i , i = 1 . . . n, forming classes [k] of f (k) nodes with identical connectivity k. Iterate the following steps: (i) With probability α add a new node and attach a link to it from an arbitrarily chosen node. (ii) Else add one link from an arbitrary node to a node j of class [k] chosen with probability P new link to class [k] ∝ kf (k).(1) For this stochastic process, Simon finds a stationary solution exhibiting power-law scaling with exponent γ = 1 + 1 1 − α .(2) The only free parameter of the model α reflects the relative growth of number of nodes versus number of links. In general small values of α, therefore, predict scaling exponents near γ ≈ 2. Let us apply this process to model the evolution of the World-Wide Web, identifying nodes with web pages. Data from two recent comprehensive Altavista crawls [3] provide an estimate for α in the present internet. These two measurements counted 203 million pages and 1466 million links in May 1999, and 271 million pages and 2130 million links in October 1999. The probability for adding a new web page is estimated from the observed increase in counts to α ≃ 0.10. The subsequent prediction of Simon's model for the exponent of the link distribution is γ = 2.1 comparing well to current experimental results γ = 2.1 ± 0.1 [2] and γ = 2.09 [3]. To compare with recently proposed models it may be interesting to note that the model by Barabási and Albert [2] can be mapped to the subclass α = 1/2 of Simon's model, when using the simpler probability for a node being connected to another node i with connectivity k i P new link to i ∝ k i .(3) Note that (3) implies (1) whereas the reverse is not true. Otherwise both models are based on the same two assumptions of growth and preferential linking. From this viewpoint, it is insightful to reconsider a recent discussion of their model. Adamic and Huberman point out that the "rich-get-richer" behaviour of single nodes imposed by (3) correlates age and connectivity of nodes [7]. This, however, is disproven by data they present. They suggest to (and Barabási et al. in response show how to [8]) add individual growth rates to each node. While this solves the correlation problem, the price to pay is a large number of free parameters in the extended model. A simple solution to this problem has already been provided by Simon: Linking is guided by (1) instead of (3), considering not single nodes but classes of nodes with identical connectivities. This allows for different growth rates among class members, leaving just one free parameter. Above we determine this parameter from experimental data, enabling Simon's classical scaling model to estimate the connectivity exponent of the World-Wide Web to γ = 2.1. . D Butler, Nature. 405Butler, D., Nature 405, 112-115 (2000). . A.-L Barabási, R Albert, Science. 286Barabási, A.-L. & Albert, R., Science 286, 509-512 (1999). . A Broder, Computer Networks. 33Broder, A. et al., Computer Networks 33, 309-320 (2000). . P Erdös, A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 517Erdös, P. & Rényi, A., Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960). . H A Simon, Biometrika. 42Simon, H.A., Biometrika 42, 425-440 (1955). Human Behaviour and the Principle of Least Effort. G K Zipf, Addison-WesleyCambridge, MassachusettsZipf, G.K., Human Behaviour and the Principle of Least Effort (Addison-Wesley, Cambridge, Massachusetts, 1949). . L A Adamic, B A Huberman, Science. 2872115Adamic, L.A. & Huberman, B.A., Science 287, 2115a (2000). . A.-L Barabási, R Albert, H Jeong, G Bianconi, Science. 2872115Barabási, A.-L., Albert, R., Jeong, H., & Bianconi, G., Science 287, 2115a (2000).

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{'abstract': "Recently, statistical properties of the World-Wide Web have attracted considerable attention when self-similar regimes have been observed in the scaling of its link structure. Here we recall a classical model for general scaling phenomena and argue that it offers an explanation for the World-Wide Web's scaling exponent when combined with a recent measurement of internet growth.", 'arxivid': 'cond-mat/0008465', 'author': ['Stefan Bornholdt \nInstitut für Theoretische Physik\nUniversität Kiel\nLeibnizstrasse 15D-24098KielGermany\n', 'Holger Ebel \nInstitut für Theoretische Physik\nUniversität Kiel\nLeibnizstrasse 15D-24098KielGermany\n'], 'authoraffiliation': ['Institut für Theoretische Physik\nUniversität Kiel\nLeibnizstrasse 15D-24098KielGermany', 'Institut für Theoretische Physik\nUniversität Kiel\nLeibnizstrasse 15D-24098KielGermany'], 'corpusid': 2582211, 'doi': '10.1103/physreve.64.035104', 'github_urls': [], 'n_tokens_mistral': 1605, 'n_tokens_neox': 1363, 'n_words': 864, 'pdfsha': '4483d9cda376f2e59b38f2a298038e5f822024f8', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0008465v1.pdf'], 'title': ["World-Wide Web scaling exponent from Simon's 1955 model", "World-Wide Web scaling exponent from Simon's 1955 model"], 'venue': []}

arxiv

DOUBLE COVERS OF CURVES ON NIKULIN SURFACES SIMONA D'EVANGELISTA AND MARGHERITA LELLI-CHIESA DOUBLE COVERS OF CURVES ON NIKULIN SURFACES SIMONA D'EVANGELISTA AND MARGHERITA LELLI-CHIESA Dedicated to Peter Newstead on the occasion of his 80 th birthday We survey basic results concerning Prym varieties, the Prym-Brill-Noether theory initiated by Welters, and Brill-Noether theory of general étale double covers of curves of genus g ≥ 2. We then specialize to curves on Nikulin surfaces and show that étale double covers of curves on Nikulin surfaces of standard type do not satisfy Welters' Theorem. On the other hand, by specialization to curves on Nikulin surfaces of non-standard type, we prove that general double covers of curves ramified at b = 2, 4, 6 points are Brill-Noether general; the case b = 2 was already obtained by Bud [Bu] with different techniques. arXiv:2305.06128v1 [math.AG] 10 May 2023 Introduction Double covers of complex curves are a classical and still very hot topic in algebraic geometry. Part of the interest in them stems from the fact that any étale double cover π : C → C of a smooth irreducible curve C of genus g ≥ 2 naturally defines a principally polarized abelian variety (P, Ξ) of dimension g − 1, known as the Prym variety of π. Prym varieties were introduced by Schottky and Jung [SJ] in relation to the Schottky problem and were named after the German mathematician Prym by Mumford ([M1]), who was the first to investigate them from an algebraic point of view. Despite a vast literature on the topic, many questions concerning the geometry of general double covers remain open, both in the étale and in the ramified case. We will here focus on Brill-Noether type questions. We recall that the Brill-Noether theory of a general curve C of genus g is quite well understood since the 1980s. The Brill-Noether Thorem establishes that the Brill-Noether variety W r d (C), parametrizing degree d line bundles on C with a space of global sections of dimension ≥ r + 1, is nonempty if and only if the so-called Brill-Noether number ρ(g, r, d) := g − (r + 1)(g − d + r) is nonnegative. Furthermore, if nonempty, W r d (C) is smooth of dimension ρ(g, r, d) outside of W r+1 d (C) by the Gieseker-Petri Theorem. The existence part of the Brill-Noether Theorem is actually valid for any smooth curve of genus g and is due to Kleiman-Laksov [KL] and Kempf [Ke]. The first proofs of both the non-existence part of the Brill-Noether Theorem and of the Gieseker-Petri Theorem, all based on degeneration techniques, were achieved by Griffiths-Harris [Gr], Eisenbud-Harris [EH1,EH2], and Gieseker [Gi]. A big breakthrough came with Lazarsfeld's alternative proof of the Gieseker-Petri Theorem [La, Pa], that avoided any type of degeneration and proceeded instead by specialization to smooth curves lying on K3 surfaces. If π : C → C is a double cover of a general genus g ≥ 2 curve C, the curve C is obviously nongeneral in moduli and it thus makes sense to investigate its Brill-Noether theory. In the étale case C has genus 2g − 1 and is never Petri general because W 1 2g−2 ( C) turns out to be singular (cf. [We] and Section 3.2 below). Moreover, if g is even, C possesses a pencil of degree g which prevents it from being Brill-Noether general (cf. [AF] and Section 4 later in this paper). Hence, the Brill-Noether behaviour of C seems quite wild. The picture is much nicer if one considers instead the so-called Prym-Brill-Noether theory of C. This is the study of the geometry of the Prym-Brill-Noether varieties V r , that were introduced by Welters [We] and are related to the geometry of the Prym variety P ; indeed, they are obtained by intersecting the Brill-Noether varieties W r 2g−2 ( C) with an appropriate translate of P living in Pic 2g−2 ( C). Using the degeneration techniques developed by Eisenbud and Harris,Welters proved (cf. Thm. 2.3 below) that for a general π the varieties V r are smooth of dimension ρ − (g, r) := g − 1 − (r + 1)r/2. In particular, they are empty if ρ − (g, r) < 0; conversely, V r is nonempty if ρ − (g, r) ≥ 0 for any étale double cover π by a result of Bertram [Ber]. These statements are the analogues in Prym-Brill-Noether theory of the Brill-Noether Theorem and the Gieseker-Petri Theorem. Perhaps discouraged by the intricate picture in the étale case, people ignored the Brill-Noether theory of C in the ramified case until recently, where Bud [Bu] proved that C is both Brill-Noether and Petri general (that is, it satisfies both the the Brill-Noether Theorem and the Gieseker-Petri Theorem) if π : C → C is a double cover of a general genus g curve branched over 2 general points of C. This result is quite amazing as it highlights that ramified double covers behave better then étale ones from a Brill-Noether viewpoint. This paper was born from the desire to make K3 surfaces and Lazarsfeld's techniques enter the picture. This is ensured by specialization to Nikulin surfaces, that is, primitively polarized K3 surfaces (S, H) endowed with a double cover π S : S → S ramified along eight (−2)-curves, which are both pairwise disjoint and disjoint from smooth curves in the linear system |H|. The minimal model of S is again a K3 surface S endowed with an involution having 8 fixed points. The Picard ranks of both S and S are ≥ 9 and, when equality holds, the Picard groups have been described in [VGS, GS]. It turns out that there are two types of Nikulin surfaces, depending on whether the embedding of the rank 9 sublattice generated by H and the (−2)-curves in Pic(S) is primitive or not: according to the terminology introduced in [KLV1], (S, H) is called of standard type in the former case, and of non-standard type in the latter. Note that π induces an étale double cover π| C : C := π −1 (C) → C of any smooth curve C ∈ |H|, and the curve C can be identified with its image in S. In the last decade Nikulin surfaces have been largely exploited in the study of étale double covers of curves (or equivalently, Prym curves) and their moduli space R h . In particular, Farkas and Verra [FV2] proved that for 6 = h ≤ 7, general étale double covers of genus h curves live on Nikulin surfaces of standard type, and used this fact to describe the birational geometry of R h in low genera. Furthermore, in the standard case the curve C ⊂ S has the gonality of a general étale double cover of a genus h curve [AF]. It is thus natural to wonder whether a different proof of Welters' Theorem can be obtained by specialization to étale double covers living on Nikulin surfaces of standard type. Unfortunately, we will answer negatively to this question proving the following theorem. Theorem. Let π : C → C be an étale double cover on a Nikulin surface S of standard type, with C in the primitive linear system |H| of genus h. If h > 7 or h = 6, the curve C does not satisfy Welters' Theorem. Nikulin surfaces of non-standard type have not received as much attention, and their investigation essentially started in [KLV1,KLV2]. As emerged in [KLV1], in the non-standard case the curve C is quite special as it possesses two theta characteristics with many sections cut out by two line bundles R 1 , R 2 ∈ Pic( S). The main result of the paper suggests that Nikulin surfaces of nonstandard type are instead the right environment to investigate double covers of curves ramified at 2, 4, 6 points, which can be realized on a Nikulin surface of non-standard type by restricting π S to the inverse image of smooth curves in the linear systems |R 1 | and |R 2 |. By specialization to ramified double covers of curves on Nikulin surfaces of non-standard type, we obtain the following generalization of Bud's result. Theorem. Let π : C → C be a general double cover of a genus g ≥ 2 curve ramified at 2, 4, or 6 points. Then the curve C is Brill-Noether general. The paper is organized as follows. Section 2 surveys the basic theory of Prym varieties and Prym-Brill-Noether theory, with particular attention to Welters' infinitesimal study of V r in terms of the Prym-Petri map, which is the anti-invariant part µ − 0,L of the Petri map µ 0,L of any line bundle L ∈ V r . The sections ends with a discussion and interpretation of its invariant counterpart µ + 0,L , as well as with a conjectural picture concerning its injectivity. Section 3 recalls some prerequisites on theta characteristics in order to show, following an argument by Beauville [Bea], that for any irreducible étale double cover π : C → C the curve C possesses some invariant vanishing thetanulls: in particular, C is not Petri general. In Section 4 we concentrate on the Brill-Noether theory of C when π is general. More precisely, we recall Schwarz's non-existence result [Sc] concerning linear series whose Brill-Noether number is negative enough, and Aprodu-Farkas' theorem [AF] on the gonality of C. The latter prevents C from being Brill-Noether general if the genus of C is even, while the Brill-Noether generality/speciality in the odd genus case is still unknown in full generality, up to our knowledge. We also recall Bud's result concerning the ramified case. Section 5 is focused on Nikulin surfaces of standard and non-standard type and on the proof of the above theorems. 2. Prym-Brill-Noether theory 2.1. Preliminaries on Prym varieties. Prym varieties are principally polarized abelian varieties arising from étale double covers of curves, and they are useful to link the geometry of curves to that of abelian varieties. Let C be a complex smooth irreducible curve, and let π : C → C be an irreducible étale double cover of C. Denoting by g and g the genus of C and C, respectively, Hurwitz's formula yields g = 2g − 1. The cover π induces the so-called Norm map between the Jacobians J(C) and J( C) of C and C, respectively: Nm : J( C) −→ J(C) O C (D) −→ O C (π(D)). This fits in the following commutative diagram: C π ay 0 / / J( C) Nm C ax 0 / / J(C), where a x 0 and a y 0 are the Abel-Jacobi maps associated with some fixed points x 0 ∈ C and y 0 ∈ C such that π(y 0 ) = x 0 . In particular, the principally polarized Abelian varieties (J(C), Θ) and (J( C), Θ) are related by two maps π * : J(C) → J( C), Nm : J( C) → J(C), such that π * and Nm are dual to each other and Nm • π * : J(C) → J(C) is multiplication by two ( [M1]). Indeed, if N = O C (D) ∈ J(C), then π * N = O C (π −1 (D)) ∈ J( C) and thus Nm(π * N ) = O C (π(π −1 (D))) ∈ J(C); since π is a double cover, we get π(π −1 (D)) = 2D. Denote by ι : C → C the involution that interchanges the sheets of π. For every divisor D on C, the following equality is straightforward: π −1 (π( D)) = D + ι( D). It follows that π * (Nm(M )) = M ⊗ ι * (M ), ∀ M ∈ J( C). Since Nm is surjective, we also get that ι * (π * (N )) = π * N, ∀ N ∈ J(C), that is, the involution ι * acts as the identity on π * (J(C)) ⊂ J( C). On the other hand, ι * acts as −1 on Ker(Nm) ⊂ J( C). In [M2] Mumford decomposed the kernel of Nm into two irreducible components: P 0 := {M ⊗ ι * M ∨ | M ∈ Pic 0 ( C)}, P 1 := {M ⊗ ι * M ∨ | M ∈ Pic 1 ( C)}. More precisely, he proved the following: Lemma 2.1 ([M2] Lem. 1). If L is a line bundle on C such that Nm L O C , then L M ⊗ι * M ∨ for some line bundle M on C. Moreover, M can be chosen of degree 0 or 1. A classical theorem by Wirtinger ([Wi]) yields that the dimension of the space of global sections is constant mod 2 on P 0 and P 1 and it has opposite parity on the two components. The component P := P 0 of Ker(Nm) containing the origin is an abelian subvariety of J( C) of dimension dim P = dim J( C) − dim J(C) = g − 1; furthermore, it turns out that the canonical polarization of J( C) restricts to twice a principal polarization Ξ on P . The principally polarized abelian variety (P, Ξ) is called the Prym variety associated with π (cf. [M1, §3] for more details). By the same argument [M1], the inverse image Nm −1 (ω C ) ⊂ Pic 2g−2 ( C) of the canonical line bundle ω C ∈ Pic 2g−2 (C) breaks up in two components P + := {L ∈ Pic 2g−2 ( C) | Nm(L) ω C , h 0 (C, L) ≡ 0 (mod 2)}, P − := {L ∈ Pic 2g−2 ( C) | Nm(L) ω C , h 0 (C, L) ≡ 1 (mod 2)}, which are both translates of P . We recall that the cover π : C → C defines a class of order two η ∈ J(C)[2] such that π * O C = O C ⊕ η; viceversa, any non-trivial η ∈ J(C)[2] determines an étale double cover of C by setting C := Spec(O C ⊕ η). Therefore, the datum of π is equivalent to that of the pair (C, η), which is called a Prym curve of genus g. The moduli space R g := {(C, η) | C smooth curve of genus g, η ∈ J(C)[2] , η O C } , has attracted much attention in the last decades; we refer to [F3] for a very nice survey on its geometry. Prym-Brill-Noether varieties and Welters' Theorem. In [We] Welters initiated a Brill-Noether study of the varieties P − and P + and introduced the following closed subsets of Nm −1 (ω C ), defined for any integer r ≥ −1: V r (C, η) := {L ∈ Pic 2g−2 ( C) | Nm(L) ω C , h 0 (C, L) ≥ r + 1, h 0 (C, L) ≡ r + 1 (mod 2)}. Scheme-theoretically, these can be realized as the intersections Brill-Noether variety; this justifies the name Prym-Brill-Noether varieties used for the loci V r (C, η). The following result proved in [M2] and [Ha] concerns their expected dimension. V r (C, η) = W r 2g−2 ( C) ∩ P + if r is odd, (2.1) V r (C, η) = W r 2g−2 ( C) ∩ P − if r is even, where W r 2g−2 ( C) = L ∈ Pic 2g−2 ( C) | h 0 (C, L) ≥ r + 1 is the classicalProposition 2.2. Let L ∈ Nm −1 (ω C ) and set h 0 (L) = r + 1. Then, the dimension of V r (C, η) at L satisfies dim L (V r (C, η)) ≥ g − 1 − r + 1 2 . We will refer to the number ρ − (g, r) := g − 1 − r + 1 2 , as the Prym-Brill-Noether number. A priori one may hope that classical results of Brill-Noether theory carry over to étale double covers of smooth curves and to the varieties V r (C, η), but unfortunately this is not the case. Indeed, many statements of this theory, such as the Brill-Noether Theorem and the Gieseker-Petri Theorem, hold only for a general curve C of genus g and they fail in the case where C is the étale double cover of a genus g curve. Nevertheless, an analogue of the Gieseker-Petri Theorem for Prym varieties was obtained by Welters [We] by a degeneration argument similar to the one used by Eisenbud and Harris [EH2] in their proof of the classical statement. Since ω C = π * ω C , the push-pull formula yields the decomposition (2.2) H 0 ( C, ω C ) = H 0 (C, ω C ) ⊕ H 0 (C, ω C (η)). into invariant and anti-invariant forms under the action of ι. Fix a line bundle L ∈ Nm −1 (ω C ) and set h 0 (L) = r + 1. Since the composition Nm • π * is the multiplication by 2, the differential of Nm at L is 2( t π * ), where t π * : (H 0 ( C, ω C )) ∨ → (H 0 (C, ω C )) ∨ is the transpose of the pull-back of differential forms. By taking kernels, we obtain the identification (2.3) T L (Nm −1 (ω C )) = (H 0 (C, ω C (η))) ∨ → (H 0 ( C, ω C )) ∨ , where the inclusion is the transpose of the projection map p : H 0 ( C, ω C ) −→ H 0 (C, ω C (η)) λ −→ 1 2 (λ − ι * λ) . By classical Brill-Noether theory (cf., e.g., [ACGH]), we know that (2.4) T L (W r 2g−2 ( C)) = (Imµ 0,L ) ⊥ ⊂ (H 0 ( C, ω C )) ∨ , where the so-called Petri map µ 0,L : H 0 ( C, L) ⊗ H 0 ( C, ω C ⊗ L ∨ ) → H 0 ( C, ω C ) is multiplication of global sections. In particular, the dimension of the tangent space at L to W r 2g−2 ( C) is: (2.5) dim T L (W r 2g−2 ( C)) = ρ(2g − 1, r, 2g − 2) + dim(Kerµ 0,L ) , where the number ρ(2g − 1, r, 2g − 2) := 2g − 1 − (r + 1) 2 equals the difference between the dimensions of the codomain and the domain of µ 0,L . Recalling (2.1) and (2.3), we obtain: (2.6) T L (V r (C, η)) = (Imµ 0,L ) ⊥ ∩ (H 0 (C, ω C (η)) ∨ = (Im(p • µ 0,L )) ⊥ . By applying π * to the isomorphism Nm L ω C , one gets L⊗ι * L ω C , or equivalently, ω C ⊗L ∨ ι * L. Consider the following composition of maps: H 0 ( C, L) ⊗ H 0 ( C, L) 1⊗ι * / / H 0 ( C, L) ⊗ H 0 ( C, ω C ⊗ L ∨ ) µ 0,L / / H 0 (C, ω C ) p / / H 0 (C, ω C (η)) s ⊗ t / / s ⊗ ι * t / / s · ι * t / / 1 2 (s · (ι * t) − (ι * s) · t), which is clearly skew-symmetric. By restriction to ∧ 2 H 0 ( C, L), we thus obtain the map µ − 0,L : ∧ 2 H 0 ( C, L) → H 0 (C, ω C (η)) s ∧ t → 1 2 (s · (ι * t) − (ι * s) · t), which is called the Prym-Petri map of L. We can rewrite (2.6) in terms of µ − 0,L as T L (V r (C, η)) = (Imµ − 0,L ) ⊥ . Since ρ − (g, r) equals the difference between the dimensions of the codomain and the domain of µ − 0,L , we conclude that (2.7) dim(T L (V r (C, η))) = ρ − (g, r) + dim(Kerµ − 0,L ), and V r (C, η) is smooth of dimension ρ − (g, r) at L if and only if the Prym-Petri map is injective. Welters proved that this is the case for all L ∈ Nm −1 (ω C ) provided that the cover π is general. Theorem 2.3 ( [We] Thm. 1.11). Let (C, η) ∈ R g be general and let π : C → C be the étale double cover defined by η. Then the Prym-Petri map µ − 0,L is injective for all L ∈ Nm −1 (ω C ). The result is analogous to the Gieseker-Petri Theorem, as it yields the smoothness of the Prym-Brill-Noether varieties V r (C, η) for a general (C, η) and their emptiness in the cases where ρ − (g, r) < 0. The analogue of the existence part of the Brill-Noether Theorem for any Prym curve was instead established by Bertram: Theorem 2.4 ( [Ber] Thm. 1.4). Let (C, η) be a Prym curve of genus g and let π : C → C be the étale double cover defined by η. If ρ − (g, r) ≥ 0, the Prym-Brill-Noether variety V r (C, η) is nonempty. Welters' proof is by degeneration to a reduced nodal curve C 0 consisting of a string of rational curves and g elliptic curves E 1 , ..., E g , such that, if x i and y i are the two intersection points of any E i with two adjacent components, the line bundle O E i (x i − y i ) is not a torsion point in Pic 0 (E i ). Consider a line bundle η 0 on C 0 which is trivial on every component except on E g , where it restricts to a non-zero point of Pic 0 (E g )[2]. The étale double cover C 0 of C 0 defined by η 0 then looks as follows: The points P 1 and P 2 on E g are not Z-independent; indeed, O Eg (P 1 −P 2 ) ∈ Pic 0 ( E g ) is a 2-torsion point. This is why Eisenbud and Harris' proof [EH2] of the Gieseker-Petri Theorem fails for C 0 . However, using the theory of limit linear series and the Brill-Noether theory of a (non general) 2-pointed elliptic curve, Welters wrote down the explicit form of any element ρ in the kernel of the relevant Petri maps and, thanks to the skew-symmetry of 1 ⊗ ι * , concluded that such a ρ never lies in the kernel of the corresponding Prym-Petri map. 2.3. The map µ + 0,L . The decomposition (2.2) provides a splitting of the Petri map µ 0,L of any line bundle L ∈ Nm −1 (ω C ) into a ι-anti-invariant part, namely, the Prym-Petri map µ − 0,L , and a ι-invariant part µ + 0,L : Sym 2 H 0 ( C, L) → H 0 (C, ω C ) s ⊗ t + t ⊗ s → s · (ι * t) + (ι * s) · t. As already remarked, the Prym-Petri map µ − 0,L governs the smoothness of the varieties V r (C, η) and has thus been extensively investigated. By contrast, the map µ + 0,L has been almost ignored so far. Up to our knowledge, it only appeared in the proof of the uniruledness of R 8 due to Farkas and Verra [FV2]. We provide an interpretation of the map µ + 0,L by considering the inclusion V r (C, η) ⊂ W r 2g−2 ( C) and the induced short exact sequence of linear maps 0 → T L (V r (C, η)) → T L W r 2g−2 ( C) → N V r (C,η)/W r 2g−2 ( C),L → 0, where N V r (C,η)/W r 2g−2 ( C),L is the normal space at the point L of V r (C, η) in W r 2g−2 ( C). By (2.5) and (2.7), we get: dim C (N V r (C,η)/W r 2g−2 ( C),L ) = = ρ(2g − 1, r, 2g − 2) + dim(Kerµ 0,L ) − ρ − (g, r) − dim(Kerµ − 0,L ) = = g − (r + 1)(r + 2) 2 + dim(Kerµ + 0,L ), where we have used that Kerµ 0,L = Kerµ + 0,L ⊕ Kerµ − 0,L by construction. We set ρ + (g, r) := ρ(2g − 1, r, 2g − 2) − ρ − (g, r) = g − (r + 1)(r + 2) 2 and notice that this number equals the difference between the dimensions of the codomain and the domain of µ + 0,L . We conclude that the dimension of the normal space N V r ( X)/W r 2g−2 ( X),L equals the dimension of the cokernel of µ + 0,L . More precisely, (2.4) and (2.6) imply that N V r ( X)/W r 2g−2 ( X),L = (Imµ 0,L ) ⊥ ∩ (H 0 (ω C )) ∨ = (Im(q • µ 0,L )) ⊥ , where q : H 0 ( C, ω C ) → H 0 (C, ω C ) is the projection mapping a form λ to 1 2 (λ + ι * λ). The composition q • µ 0,L • (1 ⊗ ι * ) : H 0 ( C, L) ⊗ H 0 ( C, L) −→ H 0 (C, ω C ) maps a decomposable tensor s ⊗ t to the invariant form s(ι * t) + (ι * s)t and is thus symmetric. Its restriction to Sym 2 H 0 ( C, L) coincides with the map µ + 0,L and this provides the identification N V r ( X)/W r 2g−2 ( X),L = (Imµ + 0,L ) ⊥ . When (C, η) ∈ R g is general, it is natural to wonder whether the map µ + 0,L is injective for all L ∈ Nm −1 (ω C ). Unfortunately, this trivially fails for any line bundle L ∈ V r (C, η) where r is any fixed integer r such that ρ − (g, r) ≥ 0 but ρ(2g − 1, r, 2g − 2) < 0; indeed, the Prym-Petri map µ − 0,L is injective by Theorem 2.3, but our assumptions prevent µ + 0,L from being injective because dim Kerµ + 0,L ≥ −ρ + (g, r) > 0. More generally, the map µ + 0,L fails to be injective for any line bundle L ∈ V r (C, η) as soon as (2.8) ρ − (g, r) > max{−1, ρ(2g − 1, r, 2g − 2)}, which implies ρ + (g, r) < 0. The inequalities (2.8) can be rewritten only in terms of ρ as (2.9) − r ≤ ρ(2g − 1, r, 2g − 2) < r. In this range, the equality dim Kerµ + 0,L = −ρ + (g, r) is the best one may hope for; since V r (C, η) is smooth at L by Theorem 2.3, this hope is equivalent to the expectation that W r 2g−2 ( C) and V r (C, η) coincide in a neighborhood of L. Expectation 2.5. Let (C, η) ∈ R g be general and let π : C → C be the étale double cover defined by η. Then, the following hold: (i) If −r ≤ ρ(2g − 1, r, 2g − 2) < r, then W r 2g−2 ( C) = V r (C, η). In particular, for all L ∈ V r (C, η) one has dim(Kerµ 0,L ) = dim(Kerµ + 0,L ) = −ρ + (g, r). (ii) If ρ(2g − 1, r, 2g − 2) ≥ r, then both µ 0,L and µ + 0,L are injective for all L ∈ V r (C, η). The situation in the remaining cases ρ(2g − 1, r, 2g − 2) < −r is already clear, as W r 2g−2 ( C) turns out to be empty by a more general result of Schwarz [Sc] that we will recall in Section 4. 3. Theta-characteristics and vanishing thetanulls 3.1. Preliminaries on theta-characteristics. A theta-characteristic on a smooth irreducible curve C is a line bundle θ ∈ Pic g−1 (C) such that θ ⊗2 = ω C . Theta-characteristics are of two types, called odd and even according to the parity of the dimension of the space of their global sections. The set of theta-characteristics on C, denoted by Th(C), is a principal homogeneous space for the space J(C)[2] of two-torsion points in the Jacobian of C, that is, J(C)[2] acts freely and transitively on Th(C). In particular, we have |Th(C)| = |J(C)[2]| = 2 2g . The F 2 -vector space J(C)[2] is endowed with a nondegenerate symplectic form ·, · : J(C)[2] × J(C)[2] → F 2 , which is called Weil pairing and is defined as follows ( [M2]). For any pair of points η, ∈ J(C)[2], one can write η = O C (D) and = O C (E) for two divisors D and E on C with disjoint support, and choose rational functions f and g on C such that div(f ) = 2D and div(g) = 2E. The condition on the supports of D and E ensures the validity of the so-called Weil Reciprocity Law (see [Ha]): f ((g)) = g((f )), where the evaluation of a rational function h at a divisor Z whose support is disjoint from the set of zeros and poles of h is defined as h(Z) := p∈C h(p) multpZ . Hence, we get f (2E) g(2D) = f (E) g(D) 2 = 1, and f (E) g(D) = ±1. The value of the Weil pairing at the pair (η, ) is given by the following formula: (−1) ,η = f (E) g(D) , and one can check that this definition is independent of both the divisors D, E and the rational functions f , g. Given a symplectic vector space (V, ·, · ) over F 2 , we denote by Q(V ) the set of quadratic forms on V with fixed polarity equal to the symplectic form ·, · , that is, all functions q : V → F 2 that satisfy the identity q(x + y) = q(x) + q(y) + x, y , ∀ x, y ∈ V. Having fixed a symplectic basis (e 1 , ..., e g , f 1 , ..., f g ) of V , the so-called Arf invariant of a quadratic form q ∈ Q(V ) is defined as arf(q) := g i=1 q(e i ) · q(f i ) ∈ F 2 , this definition being independent of the basis. A quadratic form q ∈ Q(V ) is called even if arf(q) = 0, and odd if arf(q) = 1. The space of even (respectively, odd) quadratic forms is denoted by Q(V ) + (resp., Q(V ) − ). An easy computation gives |Q(V ) + | = 2 g−1 (2 g + 1) and |Q(V ) − | = 2 g−1 (2 g − 1). Coming back to theta-characteristics, to any θ ∈ Th(C) one associates the so-called theta-form q θ : J(C)[2] → F 2 , by setting q θ (η) := h 0 (C, η ⊗ θ) + h 0 (C, θ) mod 2. For any , η ∈ J(C)[2], the Riemann-Mumford relation ( [M2]) yields: h 0 (C, θ ⊗ ⊗ η) + h 0 (C, θ ⊗ ) + h 0 (C, θ ⊗ η) + h 0 (C, θ) ≡ η, mod 2, or equivalently, q θ ∈ Q(J(C)[2]). We thus identify the two spaces (3.1) Th(C) = Q(J(C)[2]). Since arf(q θ ) = h 0 (C, θ) mod 2 for any θ ∈ Th(C), under this identification even (respectively, odd) theta-characteristics correspond to forms in Q(J(C)[2]) + (resp., Q(J(C)[2]) − ). Hence, on a genus g curve C there are precisely 2 g−1 (2 g + 1) even theta-characteristics and 2 g−1 (2 g − 1) odd ones. Following Atiyah [At], a spin curve of genus g is a pair (C, θ), where C is a smooth irreducible curve of genus g and θ ∈ Th(C). Mumford ([M2]) and Atiyah ([At]) proved that the parity of a spin curve, that is, the value of its Arf invariant, is locally constant in families. As a consequence, the moduli space S g parametrizing spin curves of genus g splits into two connected components S + g and S − g : a pair (C, θ) lies in S + g if θ is an even theta-characteristic, and in S − g otherwise. For each g, r ≥ 0 one can define the locus S r g := {(C, θ) ∈ S g | h 0 (C, θ) ≥ r + 1 and h 0 (C, θ) ≡ r + 1 mod 2}. Harris ( [Ha]) proved that the dimension of every component of S r g is bounded below by the number 3g − 3 − r+1 2 , to which we refer as the expected dimension of S r g . It may well be the case that S r g is nonempty even when its expected dimension is negative. For instance, S [ g−1 2 ] g = ∅ contains hyperelliptic curves and is thus nonempty although in this case 3g − 3 − r+1 2 is very negative. Existence of components of S r g having the expected dimension has been established by Farkas [F2] in the range 1 ≤ r ≤ 11, r = 10 for all g ≥ g(r) where g(r) is an explicit integer. In the case where r = 1, the locus S 1 g parametrizes curves having a so-called vanishing thetanull, that is, an effective even theta characteristic, and is a divisor in S + g . Indeed, every theta characteristic θ on a general curve C of genus g has at most one section. This directly follows from the base-point-free pencil trick, which implies that, as soon as H 0 (C, θ) contains a pencil V , the kernel of the Petri map µ 0,V : V ⊗ H 0 (C, θ) → H 0 (C, ω C ) is nonzero; indeed, one has Kerµ 0,V H 0 (C, O C (B)) = 0, where B is the base locus of V . The Gieseker-Petri Theorem thus excludes the existence of such a pencil if the curve C is general. 3.2. Invariant vanishing thetanulls on étale double covers. Let π : C → C be the irreducible étale double cover associated with a general Prym curve (C, η) of genus g. As explained at the end of Section 2, the Brill-Noether varieties W r 2g−2 ( C) governing the singularities of the theta divisor of J( C) are nonempty as soon as ρ(2g − 1, r, 2g − 2) ≥ −r; in particular, for values of g for which there exists an r yielding −r ≤ ρ(2g − 1, r, 2g − 2) < 0, the curve C is Brill-Noether special. We now fix g, r such that ρ(2g − 1, r, 2g − 2) ≥ 0; under this condition, one may still hope that W r 2g−2 ( C) is smooth and of the expected dimension. However, this expectation fails even for r = 1, as observed by Welters himself [We,Rmk. 1.12]; in particular, C never satisfies the Gieseker-Petri Theorem. By the base-point-free pencil trick, W 1 2g−2 ( C) \ W 2 2g−2 ( C) is singular at any point defining a vanishing thetanull. On the other hand, a vanishing thetanull on C is easily obtained as pullback π * M of a theta-characteristics M on C such that both M and M ⊗ η are odd. The following nice argument taken from [Bea,Prop. 4] counts the number of such M s and thus in particular proves their existence. For any M ∈ Th(C), the push-pull formula yields h 0 ( C, π * M ) = h 0 (C, π * π * M ) = h 0 (C, M ) + h 0 (C, M ⊗ η) = q M (η) mod 2. We are looking for those M ∈ Th(C) such that q M (η) = 0 and arf(q M ) = 1. Pick ∈ J(C) [2] such that the Weil pairing η, = 1 and denote by Σ ⊂ J(C)[2] the plane spanned by η and . Since J(C)[2] = Σ ⊕ Σ ⊥ , any M ∈ Th(C) is completely determined by the restrictions q M | Σ and q M | Σ ⊥ . The condition q M (η) = 0 yields arf(q M | Σ ) = q M (η)q M ( ) = 0. Therefore, M is determined by the value q M ( ) ∈ F 2 and by the restriction q M | Σ ⊥ , which is a quadratic form on Σ ⊥ of arf invariant 1 (because arf(q M ) = arf(q M | Σ ) + arf(q M | Σ ⊥ )). Since dim F 2 Σ ⊥ = 2(g − 1) , we have 2 g−2 (2 g−1 − 1) choices for q M | Σ ⊥ and, having the possibility of choosing q M ( ), we obtain 2 g−1 (2 g−1 − 1) theta-characteristics M on C as above. Since π * (M ⊗ η) π * M , this construction provides precisely 2 g−2 (2 g−1 −1) vanishing thetanulls on C, which are invariant under the covering involution. Brill-Noether theory for double covers Given the étale double cover π : C → C of a general Prym curve (C, η) ∈ R g , it is natural to investigate not only the Prym-Brill-Noether varieties V r (C, η) but any Brill-Noether variety W r d ( C). However, not much is known about the Brill-Noether theory of C. The following nonexistence result follows from [Sc], where Schwarz proved a more general statement concerning étale cyclic covers of arbitrary degree. Theorem 4.1 ( [Sc]). Let π :C → C be the étale double cover associated with a general Prym curve (C, η) ∈ R g . Then the Brill-Noether variety W r d ( C) is empty if ρ(2g − 1, r, d) < −r. The above result can be easily proved using Welters' degeneration and applying [F1,Prop. 4.1] on the Brill-Noether theory of 2-pointed elliptic curves. For r = 1 Theorem 4.1 is known to be optimal and this implies that C is not Brill-Noether general if g is even. Indeed, the existence of a pencil of degree g on C in this case follows from the surjectivity (cf. [ACGH]) of the difference map φ g 2 : C g 2 × C g 2 −→ J(C) (D, E) −→ O C (D − E), yielding η = O C (D − E) for some effective divisors D, E both of degree g/2 on C. The pullback π * (O C (E)) ∈ Pic g ( C) then satisfies h 0 ( C, π * O C (E)) = h 0 (C, π * π * O C (E)) = h 0 (C, O C (E)) + h 0 (C, O C (D)) ≥ 2, where we have again used the push-pull formula. On the other hand, in the odd genus case the gonality of C is maximal, as first proved by Aprodu and Farkas. Theorem 4.2 ([AF] Thm. 0.4). Let π :C → C be the étale double cover associated with a general Prym curve (C, η) ∈ R g . Then the following hold: (i) if g ≡ 1 mod 2, the curveC has maximal gonality, that is, gon(C) = g + 1; (ii) if g ≡ 0 mod 2, then gon(C) = g. In both cases the Clifford index of C equals gon(C) − 2. Two different proofs are provided in [AF], one by degeneration and one by specialization to curves on Nikulin surfaces; the latter are particular K3 surfaces on which we will focus in the next section. Up to our knowledge, the following natural question remains open: Question 1. Let g be an odd positive integer such that the inequalities −r ≤ ρ(2g−1, r, 2g−2) < 0 admit no integral solution r ≥ 1. If (C, η) ∈ R g is general, is the coverC Brill-Noether general? We stress that, quite surprisingly, ramified double covers of curves seem to behave better than étale ones from a Brill-Noether viewpoint. Following [Bu], let R g,2n be the moduli space parametrizing irreducible double covers of smooth genus g curves ramified at 2n points. The space R g,2n can be alternatively defined as follows: R g,2n := (C, x 1 , . . . , x 2n , η) | [C] ∈ M g , x i ∈ C ∀ i, η ∈ Pic −n (C), η 2 = O C (−x 1 − . . . − x 2n ) . While studying its birational geometry, Bud proved the following astonishing result, which is in contrast to the étale case. Let π : C → C be the double cover associated with a general (C, x + y, η) ∈ R g,2 . Then C is both Brill-Noether and Petri general. Bud's proof relies on the study of divisors on a compactification R g,2 of the moduli space. In the next section we will provide a simpler proof of the Brill-Noether generality of C by specialization to ramified double covers of curves on Nikulin surfaces of non-standard type, and we will extend the result to covers ramified at 4 and 6 points. Double covers of curves on Nikulin surfaces 5.1. Nikulin surfaces and their Picard group. Nikulin surfaces represent a rather special class of K3 surfaces, which has been studied in relation to various topics, including the theory of automorphisms [VGS], the study of Prym curves [FK] and the birational geometry of their moduli spaces [FV1,FV2,KLV1,KLV2]. We recall their definition and some basic properties. Definition 1. A polarized Nikulin surface of genus h ≥ 2 is a triple (S, M, H) consisting of a smooth K3 surface S and two line bundles O S (M ), H ∈ Pic S that satisfy the following conditions: • S contains 8 disjoint smooth rational curves N 1 , . . . , N 8 such that N 1 + · · · + N 8 ∼ 2M. • H is nef, H 2 = 2(h − 1) and H · M = 0. We say that (S, M, H) is primitively polarized if the class of H is primitive in Pic S. The line bundle M defines a non-trivial double cover π : S → S branched along 8 i=1 N i . Denoting by σ : S → S the blow-down of the eight (−1)-curves E i := π −1 (N i ) ⊂ S, the surface S is a minimal K3 surface endowed with a so-called Nikulin involution ι ∈ Aut( S) having eight fixed points corresponding to the images σ(E i ) of the exceptional divisors. The quotientS := S/ι has eight ordinary double points and the quotient mapπ : S →S fits in the following commutative diagram: Since the Picard group of a polarized Nikulin surface (S, M, H) contains both the Nikulin lattice and the polarization H, its rank is at least 9. Consider the rank 9 lattice Λ h = Λ(S, M, H) := Z[H] ⊕ ⊥ N ⊂ Pic S; the surface S is said to be a Nikulin surface of standard type if the embedding Λ h ⊂ Pic S is primitive, and a Nikulin surface of non-standard type otherwise. Garbagnati and Sarti [GS] proved that in the latter case h is forced to be odd. The following result describes Pic(S) in the case of minimal Picard number. Proposition 5.1 ( [GS], Prop. 2.1). Let (S, M, H) be a genus h primitively polarized Nikulin surface of Picard number 9. Then either Pic S = Λ h (standard case), or h is odd and Λ h ⊂ Pic S has index two (non-standard case). In the latter situation, possibly after renumbering the curves N i , one falls in one of these cases: • h ≡ 3 mod 4 and there are R 1 , R 2 ∈ Pic S such that R 1 ∼ H − N 1 − N 2 2 , R 2 ∼ H − N 3 − · · · − N 8 2 ; in particular, one has g(R 1 ) = (h + 1)/4, g(R 2 ) = (h − 3)/4. • h ≡ 1 mod 4 and there are R 1 , R 2 ∈ Pic S such that R 1 ∼ H − N 1 − N 2 − N 3 − N 4 2 , R 2 ∼ H − N 5 − N 6 − N 7 − N 8 2 ; in particular, one has g(R 1 ) = g(R 2 ) = (h − 1)/4. The growing interest in Nikulin surfaces is motivated from the fact that some Prym curves live on them. The condition H · M = 0 in Definition 1 ensures that the double cover π restricts to an étale double cover π| C : C := π −1 (C) → C of any smooth curve C ∈ |H|; in other words, the pair (C, M | C ) is a Prym curve of genus h. Since C is disjoint from the (−1)-curves E i , the curve C can be identified with its image in S and we set H := O S ( C). The Picard group of the K3 surface S was described by Van Geemen and Sarti [VGS] by investigating the action of the Nikulin involution ι on the cohomology group H 2 ( S, Z). It turns out that Pic( S) always contains the orthogonal complement (H 2 ( S, Z) ι ) ⊥ , which is isomorphic to the rank 8 lattice E 8 (−2). Since H ∈ E 8 (−2) ⊥ ⊂ Pic( S), the Picard number of S is ≥ 9 and equality holds if and only if the same holds for S. The following proposition summarizes results by Van Geemen and Sarti [VGS,Prop. 2.2 and Prop. 2.7] and Aprodu and Farkas [AF,Prop. 4.3] in the standard case. Proposition 5.2. Let (S, M, H) be a genus h polarized Nikulin surface of standard type and Picard number 9, and let S be the K3 surface with a Nikulin involution obtained from S. Then, the lattice Λ h := Z H ⊕ E 8 (−2) has index 2 in Pic( S). The latter is generated by Λ h and a class H+v 2 , where v is an element of E 8 (−2) satisfying v 2 = −4 if h is even and v 2 = −8 if h is odd. We now turn to the non-standard case. By [GS,Prop. 3.5 (2)], for i = 1, 2 the linear system |R i | contains a smooth irreducible curve as soon as R i has nonnegative self-intersection. We stress that R 1 · M = 2 and R 2 · M = 6 if h ≡ 3 mod 4, while R 1 · M = R 2 · M = 4 for h ≡ 1 mod 4. Therefore, for all smooth curves D 1 ∈ |R 1 | and D 2 ∈ |R 2 |, the cover π induces double covers π D 1 : π −1 (D 1 ) → D 1 and π D 2 : π −1 (D 2 ) → D 2 that are ramified at 2 and 6 points, respectively, if h ≡ 3 mod 4; on the other hand, both π D 1 and π D 2 are ramified at 4 points if h ≡ 1 mod 4. In any case, the curves π −1 (D 1 ) and π −1 (D 2 ) meet each exceptional curve E i in at most 1 point and are thus isomorphic to their images in S, that we denote by D 1 and D 2 respectively. It is trivial to check that σ * D 1 − E 1 − E 2 ∼ π * D 1 , σ * D 2 − E 3 − · · · − E 8 ∼ π * D 2 if h ≡ 3 mod 4, (5.1) σ * D 1 − E 1 − E 2 − E 3 − E 4 ∼ π * D 1 , σ * D 2 − E 5 − E 6 − E 7 − E 8 ∼ π * D 2 if h ≡ 1 mod 4. Note that Hurwitz's formula always yields g( D 1 ) = g( D 2 ) = (h + 1)/2. The following result directly follows from [VGS,Prop. 2.2 and Prop. 2.7] in the non-standard case. Proposition 5.3. Let (S, M, H) be a genus h polarized Nikulin surface of non-standard type and Picard number 9, and let S be the K3 surface with a Nikulin involution obtained from S. Then Pic( S) = Z R ⊕ E 8 (−2), where R is a polarization of genus (h + 1)/2 such that the curves D 1 , D 2 as in (5.1) lie in | R|. 5.2. Standard Nikulin surfaces. As highlighted in [KLV1,Prop. 2.3], a general curve C ∈ |H| on a very general genus h primitively polarized Nikulin surface (S, M, H) of standard type is Brill-Noether general . Furthermore, it was proved by Aprodu and Farkas [AF,Thm. 1.5] that the étale double cover C ⊂ S of C has the gonality of a general curve of genus 2h − 1 that covers a genus h curve, namely, h + 1 if h is odd and h otherwise. It is thus natural to ask whether C is general from a Prym-Brill-Noether viewpoint, that is, if it satisfies Welters' Theorem. A positive answer would provide an alternative proof of Welters' result avoiding degeneration, in analogy with Lazarsfeld's proof of the Gieseker-Petri Theorem by specialization to curves on K3 surfaces. Unfortunately, the answer turns out to be negative as soon h > 7 and h = 6. This bound on the genus agrees with the following theorem by Farkas and Verra. We thus prove the following result. Theorem 5.5. Let (S, M, H) be a genus h primitively polarized Nikulin surface of standard type with either h > 7, or h = 6. For any smooth curve C ∈ |H|, the double cover C ∈ | H| of C defined by M | C does not satisfy Welter's Theorem. Proof. Set A := H+v 2 ∈ Pic( S) with v ∈ E 8 (−2) as in Proposition 5.2. Since C · v = 0, the restriction A| C is a line bundle on C of degree 2h − 2 such that: r := h 0 ( C, A| C ) − 1 = h 0 ( S, A) − 1 ≥ χ(A) − 1 = 1 + 1 2 H + v 2 2 = h 2 ; here, the first equality follows from the strong version of Bertini's Theorem due to Saint-Donat [SD] yielding h 1 ( S, A(− C)) = 0, while in the last equality we have used that v 2 = −4 when h is even and v 2 = −8 otherwise. Since E 8 (−2) = (H 2 ( S, Z) ι ) ⊥ ⊂ Pic( S), one has ι * H = H and ι * v = −v. This implies ι * A| C ω C ⊗ A| ∨ C , or equivalently, Nm(A| C ) = ω C . Hence, A| C defines an element of the Prym-Brill-Noether variety V r (C, M | C ). If h is odd, an easy computation gives ρ − (h, r) = − (h − 1)(h − 7) 8 , which is negative when h > 7. Analogously, for even h one computes the Prym-Brill-Noether number ρ − (h, r) = − (h − 2)(h − 4) 8 , which is negative for h ≥ 6. Hence, the Prym-Petri map of A| C cannot be injective if either h > 7 or h = 6, and this concludes the proof. 5.3. Non-standard Nikulin surfaces. Let (S, M, H) be a very general genus h primitively polarized Nikulin surface of non-standard type. As remarked in [KLV1,Prop. 2.3 and Rmk. 2.4], in this case the line bundles R 1 , R 2 ∈ Pic(S) prevent curves in |H| from being Brill-Noether general; indeed, their restrictions to any smooth curve C ∈ |H| define two theta-characteristics on C with negative Brill-Noether number. Proposition 3.5 (2) in [GS] yields the existence of a smooth irreducible curve in the linear systems |R i | for i = 1, 2 as soon as c 1 (R i ) 2 ≥ 0. Let us pick a smooth curve D lying in either |R 1 |, |R 2 |, and denote by g ≥ 1 its genus. If h ≡ 1 mod 4, then g = (h − 1)/4 and (D, M | C ) ∈ R g,4 . If instead h ≡ 3 mod 4, then either g = (h + 1)/4 and (D, M | C ) ∈ R g,2 , or g = (h − 3)/4 and (D, M | C ) ∈ R g,6 , depending on whether D lies in |R 1 | or |R 2 |. In any case the double cover D ⊂ S of D defined by M | C has genus (h + 1)/2. Proposition 5.6. Let (S, M, H) be a general non-standard Nikulin surface of odd genus h and let D be a smooth curve in either |R 1 | or |R 2 |. Then the following hold: (i) D is always Brill-Noether general, and it is Petri general if it is general in its linear system; (ii) the ramified double cover D of D defined by M | C is Brill-Noether general. Proof. It is easy to verify that neither R 1 nor R 2 can be decomposed as the tensor product of two line bundles on S both satisfying h 0 ≥ 2, and thus (i) follows proceeding as in Lazarsfeld's proof of Petri's Theorem [La, Pa]. Proposition 5.3 yields D ∈ | R|, where R is a generator of Pic( S). Again one can easily exclude the existence of L 1 , L 2 ∈ Pic( S) with R L 1 ⊗ L 2 and h 0 ( S, L i ) ≥ 2. Hence, Lazarsfelds's Theorem implies that all smooth curves in the linear system | R| (thus, in particular, D) are Brill-Noether general. A general curve D as in the above statement is highly expected to be Petri general, as well. However, this does not follow directly from Lazarsfeld's Theorem. Indeed, the latter only implies Petri generality for general curves in | R|; however, a double cover D of a curve D in |R 1 | or |R 2 | is never general in the linear system | R|, as it lies in either the ι-invariant or the ι-anti-invariant part of it. By varying h, one obtains all possible values for the genus g of D. Recalling that the moduli space R g,2n is irreducible for every g ≥ 2 and n ≥ 0 (cf. [Bu,Prop. 2.5]), Proposition 5.6 then implies the following generalization of Bud's result. Theorem 5.7. Fix integers g ≥ 2 and n = 1, 2, 3. Let (D, x 1 , . . . , x 2n , η) ∈ R g,2n be general and let D the double cover of D defined by η. Then the curve D is Brill-Noether general. the contraction of the curves N i to the eight nodes ofS. Definition 2. Let (S, M, H) be a polarized Nikulin surface of genus h. The rank 8 sublattice of Pic S generated by N 1 , . . . , N 8 and M is called Nikulin lattice and its denoted by N = N(S, M ). Theorem 5.4 ([FV2], Thm. 0.2). A general Prym curve (C, η) ∈ R h lies on a Nikulin surface if and only if h ≤ 7 and h = 6. Green's conjecture for general covers. M Aprodu, G Farkas, Contemp. Math. 564M. Aprodu, G. Farkas, Green's conjecture for general covers, Contemp. Math. 564 (2012), 211-226. Geometry of algebraic curves. E Arbarello, M Cornalba, P A Griffiths, J Harris, Grundl. Math. Wiss. 267Springer VerlagE. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves, Grundl. Math. Wiss. 267, Springer Verlag, 1985. Riemann surfaces and spin structures. M Atiyah, Ann. Sci. Éc. Norm. Supér. 4M. Atiyah, Riemann surfaces and spin structures, Ann. Sci. Éc. Norm. Supér. 4 (1971), 47-62. Vanishing thetanulls on curves with involutions. A Beauville, Rend. Circ. Mat. Palermo. 62A. Beauville, Vanishing thetanulls on curves with involutions, Rend. Circ. Mat. Palermo 62 (2013), 61-66. An existence theorem for Prym special divisors. A Bertram, Invent. Math. 90A. Bertram, An existence theorem for Prym special divisors, Invent. Math. 90 (1987), 669-671. The birational geometry of Rg,2 and Prym-canonical divisorial strata. A Bud, math.AG/01718.v2A. Bud, The birational geometry of Rg,2 and Prym-canonical divisorial strata, math.AG/01718.v2. Divisors on general curves and cuspidal rational curves. D Eisenbud, J Harris, Invent. Math. 74D. Eisenbud, J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), 371-418. A simpler proof of the Gieseker-Petri Theorem on special divisors. D Eisenbud, J Harris, Invent. Math. 74D. Eisenbud, J. Harris, A simpler proof of the Gieseker-Petri Theorem on special divisors, Invent. Math. 74 (1983), 269-280. The geometry of the moduli space of curves of genus 23. G Farkas, Math. Ann. 318G. Farkas, The geometry of the moduli space of curves of genus 23, Math. Ann. 318 (2000), 43-65. Gaussian maps, Gieseker-Petri loci and large theta-characteristics. G Farkas, J. Reine Angew. Math. 581G. Farkas, Gaussian maps, Gieseker-Petri loci and large theta-characteristics, J. Reine Angew. Math. 581 (2005), 151-173. Prym varieties and their moduli, in Contributions to Algebraic Geometry. G Farkas, EMS Ser. Congr. Rep. Eur. Math. SocG. Farkas, Prym varieties and their moduli, in Contributions to Algebraic Geometry, EMS Ser. Congr. Rep. (Eur. Math. Soc., Zürich) (2012), 215-255. The generic Green-Lazarsfeld Secant Conjecture. G Farkas, M Kemeny, Invent. Math. 203G. Farkas, M. Kemeny, The generic Green-Lazarsfeld Secant Conjecture, Invent. Math. 203 (2016), 265- 301. Moduli of theta characteristics via Nikulin surfaces. G Farkas, A Verra, Math. Ann. 354G. Farkas, A. Verra, Moduli of theta characteristics via Nikulin surfaces , Math. Ann. 354 (2012), 465- 496. Prym varieties and moduli of polarized Nikulin surfaces. G Farkas, A Verra, Adv. Mat. 290G. Farkas, A. Verra, Prym varieties and moduli of polarized Nikulin surfaces, Adv. Mat. 290 (2016), 314-328. Projective models of K3 surfaces with an even set. A Garbagnati, A Sarti, Adv. Geom. 8A. Garbagnati, A. Sarti, Projective models of K3 surfaces with an even set, Adv. Geom. 8 (2008), 413-440. Stable curves and special divisors: Petri's conjecture. D Gieseker, Invent. Math. 66D. Gieseker, Stable curves and special divisors: Petri's conjecture, Invent. Math. 66 (1982), 251-275. On the variety of special linear systems on an algebraic curve. P A Griffiths, J Harris, Duke Math. J. 47P. A. Griffiths, J. Harris, On the variety of special linear systems on an algebraic curve, Duke Math. J. 47 (1980), 233-272. Theta-characteristics on algebraic curves. J Harris, Trans. Amer. Math. Soc. 271J. Harris, Theta-characteristics on algebraic curves, Trans. Amer. Math. Soc. 271 (1982), 611-638. Schubert methods with an application to algebraic curves. G Kempf, Publ. Math. Centrum. University of AmsterdamG. Kempf, Schubert methods with an application to algebraic curves, Publ. Math. Centrum, University of Amsterdam, 1971. On the existence of special divisors. S Kleiman, D Laksov, Am. J. Math. 94S. Kleiman, D. Laksov, On the existence of special divisors, Am. J. Math. 94 (1972), 431-436. Half Nikulin surfaces and moduli of Prym curves. A L Knutsen, M Lelli-Chiesa, A Verra, J. Inst. Math. Jussieu. 20A. L. Knutsen, M. Lelli-Chiesa, A. Verra, Half Nikulin surfaces and moduli of Prym curves, J. Inst. Math. Jussieu 20 (2021), 1547-1584. Moduli of non-standard Nikulin surfaces in low genus. A L Knutsen, M Lelli-Chiesa, A Verra, Ann. Sc. Norm. Super. Pisa Cl. Sci. 2020A. L. Knutsen, M. Lelli-Chiesa, A. Verra, Moduli of non-standard Nikulin surfaces in low genus, Ann. Sc. Norm. Super. Pisa Cl. Sci. XXI (2020), 361-384. Brill-Noether-Petri without degenerations. R Lazarsfeld, J. Diff. Geom. 23R. Lazarsfeld, Brill-Noether-Petri without degenerations., J. Diff. Geom. 23 (1986), 299-307. Prym varieties I, Contributions to Analysis. D Mumford, Academic PressNew YorkD. Mumford, Prym varieties I, Contributions to Analysis, Academic Press, New York (1974), 325-350. Theta Characteristics of an Algebraic Curve. D Mumford, Ann. scient. Éc. Norm. Sup. 4D. Mumford, Theta Characteristics of an Algebraic Curve, Ann. scient. Éc. Norm. Sup. 4 (1971), 181-192. A proof of Lazarsfeld's Theorem on curves on K3 surfaces. G Pareschi, J. Alg. Geom. 4G. Pareschi, A proof of Lazarsfeld's Theorem on curves on K3 surfaces, J. Alg. Geom. 4 (1995), 195-200. Projective models of K3 surfaces. B Saint-Donat, Amer. J. Math. 96B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602-639. F Schottky, H Jung, Neue Sätze über Symmetralfunktionen und die Abelschen Funktionen der Riemannschen Theorie. S.-B. Preuss. Akad. Wiss. Berlin1F. Schottky, H. Jung, Neue Sätze über Symmetralfunktionen und die Abelschen Funktionen der Rie- mannschen Theorie, S.-B. Preuss. Akad. Wiss. Berlin, Phys. Math. Kl. 1 (1909), 282-287. Noether theory for cyclic covers. I Schwarz Brill, J. Pure Appl. Algebra. 221I. Schwarz Brill-Noether theory for cyclic covers, J. Pure Appl. Algebra 221 (2017), 2420-2430. Nikulin involutions on K3 surfaces. B Van, A Geemen, Sarti, Math. Zeit. 255B. Van. Geemen, A. Sarti, Nikulin involutions on K3 surfaces, Math. Zeit. 255 (2007), 731-753. A theorem of Gieseker-Petri type for Prym varieties. G E Welters, Ann. Sci. Éc. Norm. Supér. 4G. E. Welters, A theorem of Gieseker-Petri type for Prym varieties Ann. Sci. Éc. Norm. Supér, 4 (1985), 671-683. W Wirtinger, Untersuchungen fiber thetafunktionen. BerlinTeubner1895W. Wirtinger, Untersuchungen fiber thetafunktionen, Teubner, Berlin, 1895. Università degli Studi dell'Aquila Via Vetoio I. : Simona D&apos;evangelista, Disim, address: [email protected]. Coppito, 67100 L'Aquila, Italy EmailSimona D'Evangelista: DISIM, Università degli Studi dell'Aquila Via Vetoio I, Loc. Coppito, 67100 L'Aquila, Italy Email address: [email protected] Margherita Lelli-Chiesa, Dipartimento di Matematica e Fisica. Università Roma Tre Largo San Leonardo Murialdo 1, 00146 RomaItaly Email address: [email protected] Lelli-Chiesa: Dipartimento di Matematica e Fisica, Università Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy Email address: [email protected]

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{'abstract': "We survey basic results concerning Prym varieties, the Prym-Brill-Noether theory initiated by Welters, and Brill-Noether theory of general étale double covers of curves of genus g ≥ 2. We then specialize to curves on Nikulin surfaces and show that étale double covers of curves on Nikulin surfaces of standard type do not satisfy Welters' Theorem. On the other hand, by specialization to curves on Nikulin surfaces of non-standard type, we prove that general double covers of curves ramified at b = 2, 4, 6 points are Brill-Noether general; the case b = 2 was already obtained by Bud [Bu] with different techniques. arXiv:2305.06128v1 [math.AG] 10 May 2023", 'arxivid': '2305.06128', 'author': [], 'authoraffiliation': [], 'corpusid': 258587944, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18432, 'n_tokens_neox': 16336, 'n_words': 9797, 'pdfsha': 'bfa19beec23f3a5bce1cacd146d780f09d6311d3', 'pdfurls': ['https://export.arxiv.org/pdf/2305.06128v1.pdf'], 'title': ["DOUBLE COVERS OF CURVES ON NIKULIN SURFACES SIMONA D'EVANGELISTA AND MARGHERITA LELLI-CHIESA", "DOUBLE COVERS OF CURVES ON NIKULIN SURFACES SIMONA D'EVANGELISTA AND MARGHERITA LELLI-CHIESA"], 'venue': []}

arxiv

Exact solutions for one of the extensive chaos model 30 Oct 2003 Nikolai A Kudryashov Department of Applied Mathematics Moscow Engineering and Physics Institute (State university 31 Kashirskoe Shosse115409MoscowRussian Federation Exact solutions for one of the extensive chaos model 30 Oct 2003exact solutionextensive chaos modelnonlinear evolution equa- tion PACS: 0230Hq -Ordinary differential equations; 0545Yv -Solitons Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are given. Nonlinear evolution equation for the considered extensive chaos model is shown to have solitary and periodical waves. Introduction In recent years one can observe a systematical study of a novel type of chaos that is called by "soft-mode turbulence" [1,2,3]. These chaos types are characterized by a smooth interplay of different spatial scales. Properties of these types are qualitatively different from the well known models that are described by the complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. The simplest model exhibiting the soft-mode turbulence can be described by the higher order nonlinear evolution equation with the simplest nonlinearity. This equation was introduced by N.A. Kudryashov [4] and V.N. Nikolaevskiy [5] to describe longitudinal seismic waves in viscoelastic media. The simplest case of this equation takes the form u t + uu x + βu xx + δu xxxx + εu xxxxxx = 0 (1.1) It is known that the Ginzburg-Landau and the Kuramoto-Sivashinskiy equations are not integrable equations because these ones do not pass the Painleve test [6,7]. However these equations have some list of special solutions [6,7,8,9,10,11,12]. Eq. (1.1) can be normalized. Assuming ε = 0, δ = 0 and setting u = δ 2 ε δ ε 1 2 u ′ , x = ε δ 1 2 x ′ , t = ε 2 δ 3 t ′ , σ = βε δ 2 (1.2) Then Eq.(1.1) takes the form u t + uu x + σu xx + u xxxx + u xxxxxx = 0 (1.3) (the primes of the variables are omitted). Equation (1.3) is invariant under transformations u → −u, x → −x (1.4) which allows us to study this equation for x ≥ 0. Eq.(1.1) does not pass the Painlevé test and this is not integrable equation but one can expect that Eq.(1.1) has some special solutions. The aim of this letter is to present some exact solutions of Eq.(1.1). The outline of this letter is as follows. The sampling equation method to look for exact solutions of nonlinear differential equations is discussed in Section 2. Application of this approach to search exact solitary solutions of Eq.(1.1) is considered in Section 3. Exact periodic solutions are presented in Section 4. Sampling equation method It is well known that all nonlinear differential equations can be connectionally divided into three types: exactly solvable, partially solvable and those that have no exact solution. At the present we have a lot of different approaches to look for exact solutions of nonlinear differential equations (see, for a example, refs. [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]). Usually investigators use some sampling functions that are hyperbolic and elliptic functions. However one can note that as a rule partially solvable nonlinear equations have exact solutions that are general solutions of solvable equation of lesser order. In this connection we apply later the sampling equation method to look for exact solutions of Eq.(1.1). Our approach takes into consideration the following simple idea. Let us assume we have two differential equations E[y] = 0 (2.1) and D[u] = 0 (2.2) and let us also assume that Eq. E[y] =ÂD[u] (2.3) where is a operator and y is a transformation that is determined by the formula y = F (u) (2.4) This raises the question as to whether finding transformation (2.4) and exactly solvable equation (2.2) as the sampling equation. One of the impressive method to look for the transformation like (2.4) is the singular manifold method by J.Weiss, M.Tabor and G.Carnevalle [21] that is used to study both integrable and nonintegrable differential equations. Success of this approach for nonintegrable differential equations is explained by so-called truncated expansions that are transformations similar to formula (2.4). In this case for the polynomial class of nonintegrable equations (2.1) one can suggest corresponding exactly solvable equation (2.2) as the Riccati equation, the elliptic equation or other solvable ordinary differential equation [6,7,10,22]. As a example let us consider the ordinary differential equation in the form E[y] = y xxxx + yy xxx − 6yy xx − 6y 2 x − 6y 2 y x − βy = 0 (2.5) This equation is not integrable equation but this one has some exact solutions. Taking into consideration leading members of Eq.(2.5) one can find that solution of Eq.(2.5) have the second degree singularity. In this connection we can find solution of Eq.(2.5) using the truncated expansion y(z) = A 0 + A 1 Y + A 2 Y 2 (2.6) where A 0 , A 1 and A 2 are unknown parameters and Y (z) satisfies to the Riccati equation D[Y ] = Y z + Y 2 − α = 0 (2.7) Here α is a parameter that will be found too. Substituting transformation (2.6) info Eq.(2.5) and taking into account Eq.(2.7) and its consequences One can use another transformation Y zz = 2Y 3 − 2αY (2.8) Y zzz = −6Y 4 + 8αY 2 − 2α 2 (2.9) Y zzzz = 24Y 5 − 40αY 3 + 16α 2 Y(y(z) = B 0 + B 1 R (2.11) Where B 0 and B 1 are constant that are found. As this takes place we take into consideration that R has second degree singularity and R = R(z) is a solution of the elliptic function equation R 2 z = −2R 3 + aR 2 + 2bR + d(y(z) = A 0 + A 1 Y + A 2 Y 2 + A 3 Y 3 + A 4 Y 4 + A 5 Y 5 (3.4) Where Y (z) satisfies to the Riccati equation D[Y ] = Y z + Y 2 − α = 0 (3.5) Constants A 0 , A 1 , A 2 , A 3 , A 4 , A 5 and α are found after substitution of the truncated expansion (3.4) into Eq.(2.1). We need also to take into account the following formulas Y zz = 2Y 3 − 2αY Y zzz = −6Y 4 + 8αY 2 − 2α 2 Y zzzz = 24Y 5 − 40αY 3 + 16α 2 Y Y zzzzz = −120Y 6 + 240αY 4 − 136α 2 Y 2 + 16α 3 (3.6) As a result of calculations we have A 5 = 30240 ε, A 4 = 0, A 3 = 2520δ 11 − 50400εα, A 2 = 0, A 1 = − 2520 11 δ α + 20160 ε α 2 + 1260 251 β − 12600 30371 δ 2 ε , A 0 = C 0 (3.7) Where β is determined by the formula β = − 213811840 ε 3 α 3 − 10204656 δ ε 2 α 2 − 2045 δ 3 − 92400 δ 2 ε α 121ε (9240 ε α + 79 δ) (3.8) Denoting α = δw ε (3.9) we obtain for w the following six values w 1 = − 1 220 , w 2 = − 5 176 , w 3 = −1δ 2 ε Y + C 0 (3.14) where Y = Y (z) is a solution of Eq.(3.5) Y (z) = √ α tanh √ αz + ϕ 0 (3.15) Constant C 1 is determined by formula C 1 = 4112640 11 δ 5 w 4 ε 3 − 9999360 δ 5 w 5 ε 3 − 5080320 251 δ 3 w 3 β ε 2 − 55460160 30371 δ 5 w 3 ε 3 + + 1 2 C 0 2 + 660240 2761 δ 3 w 2 β ε 2 − 25200 30371 δ 5 w 2 ε 3 − 1260 251 β 2 δ w ε + 12600 30371 β δ 3 w ε 2 (3.16) Substituting solution (3.15) into (3.14) and taking into account that α = α i = w i δ/ε (i = 1, ... We can see that solutions of Eq(1.1) have fivth degree singularity and one can also look for exact solution of Eq.(1.1) in the form y (z) = B 1 + B 2 R(z) + B 3 R z + B 4 R 2 + B 5 RR z (4.1) where B k (k = 1, ..., 5) are constants and R = R(z) is a second degree singularity solution of the elliptic function equation R 2 z = −2R 3 + aR 2 + 2bR + d (4.2) From Eq.(4.2) we get that R(z) satisfies also to equations R zz = −3R 2 + aR + b R zzz = −6 RR z + aR z R zzzz = 30 R 3 − 15 aR 2 − 18 bR − 6 d + a 2 R + ab R zzzzz = 90 R 2 R z − 30 aRR z − 18 bR z + a 2 R z R zzzzzz = −630 R 4 + 420 aR 3 + 504 bR 2 + 180 Rd− −63 a 2 R 2 − 108 abR − 30 ab − 18 b 2 + a 3 R + a 2 bδ 5 ε 3 + 1 2 C 0 2 ∓ 2484 161051 √ 21δ 5 ε 3 (4.8) Using (4.4) and (4.5) we obtain as resultant expression for the solution y(z) in the form of periodic waves. y(z) = C 0 + 630 εa + δ 11 − 6εR R z (4.9) where R = R(z) is a solution of the following equations Assuming that R 1 , R 2 and R 3 with R 1 ≥ R 2 ≥ R 3 real roots of equations R 2 z = −2 R 3 + aR 2 −2R 3 − aR 2 + 1 6 a 2 − δ 2 726ε 2 ∓ δ 2 √ 21 2541ε 2 R − 1 108 a 3 − − 13 359379 δ 3 ε 3 ∓ δ 3 √ 21 119790ε 3 + aδ 2 4356ε 2 ± aδ 2 √ 21 15246ε 2 = 0 (4.11) We have solutions of Eq.(4.10) in the form R(z) = R 2 + (R 1 − R 2 )cn 2 (z R 1 − R 2 , S), S 2 = R 1 − R 2 R 1 − R 3 (4.12) Thus Eq.(1.1) have a few exact solutions at different values of equation parameters. These solution are solitary and periodic waves and they are determined by the formulas (3.14) and (4.9). We hope these solutions can be useful for test of the numerical simulations of soft-mode turbulence. This work was supported by the International Science and Technology Center under the project 1379-2. 3 Exact solitary solutions of Eq.(1.1). Let us look for exact solutions of Eq.(1.1) in the form of travelling waves using variablesu(x, t) = y(z), z = x − C 0 t (3.1)Eq.(2.1) takes the form after integration over z C 1 − C 0 y + 1 2 y 2 + βy z + δy zzz + εy zzzzz = into leading members of Eq.(3.2) we have a 0 = 30240ε and p = −5. 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J Weiss, M Tabor, G Carnevalle, J. Math. Phys. 243J. Weiss, M. Tabor and G. Carnevalle, J. Math. Phys. 24, 3 (1983) 522-526 Analytical theary of nonlinear diffeential equations. N A Kudryashov, 360pMoscow-Igevsk, IKIN.A. Kudryashov, Analytical theary of nonlinear diffeential equations, Moscow-Igevsk, IKI (2003) 360 p

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{'abstract': 'Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are given. Nonlinear evolution equation for the considered extensive chaos model is shown to have solitary and periodical waves.', 'arxivid': 'nlin/0310046', 'author': ['Nikolai A Kudryashov \nDepartment of Applied Mathematics Moscow Engineering and Physics Institute (State university\n31 Kashirskoe Shosse115409MoscowRussian Federation\n'], 'authoraffiliation': ['Department of Applied Mathematics Moscow Engineering and Physics Institute (State university\n31 Kashirskoe Shosse115409MoscowRussian Federation'], 'corpusid': 15102962, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5330, 'n_tokens_neox': 4235, 'n_words': 2339, 'pdfsha': 'a5f55656a9fd0af44fde618dd52fe3551d172dc8', 'pdfurls': ['https://export.arxiv.org/pdf/nlin/0310046v1.pdf'], 'title': ['Exact solutions for one of the extensive chaos model', 'Exact solutions for one of the extensive chaos model'], 'venue': []}

arxiv

CONVERGENCE RATES FOR IDENTIFICATION OF ROBIN COEFFICIENT FROM TERMINAL OBSERVATIONS 3 Apr 2023 Subhankar Mondal CONVERGENCE RATES FOR IDENTIFICATION OF ROBIN COEFFICIENT FROM TERMINAL OBSERVATIONS 3 Apr 2023parameter identificationRobin coefficientill-posedregularizationsource condition MSC 2010: 35R2535R3046N10 This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou(2000)and obtain a convergence rate of O(δ 1 2 ), the main goal of this paper, where δ is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that we are getting the above mentioned convergence rate without the need for characterizing the range space of modelling operator, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators, where one obtain the same order of convergence by characterizing the range of the adjoint of the Fréchet derivative of modelling operator, a challenging task for many problems. Introduction Let d ∈ {2, 3} and Ω ⊂ R d be a bounded domain with Lipschitz boundary ∂Ω. Let τ > 0 be fixed and we denote the sets Ω × [0, τ ] and ∂Ω × [0, τ ] by Ω τ and ∂Ω τ , respectively. We consider the PDE (1.1)      u t − ∆u = f in Ω τ , ∂u ∂ν + γ(x)u = g on ∂Ω τ , u(·, 0) = u 0 in Ω, where f ∈ L 2 (0, τ ; L 2 (Ω)), g ∈ L 2 (0, τ ; L 2 (∂Ω)), γ ∈ L ∞ (∂Ω) and u 0 ∈ L 2 (Ω). Here, for a Banach space Y , we used the notation L 2 (0, τ ; Y ) for the space of all Y -valued measurable functions φ on [0, τ ] such that τ 0 φ(t) 2 Y dt < ∞. The Robin coefficient γ is the impedance coefficient that represents the heat exchange on the boundary ∂Ω, ν denotes the outward unit normal of the boundary ∂Ω. The direct or forward problem for (1.1) is to find u(x, t) satisfying (1.1)(possibly in weak sense) for the known impedance coefficient γ and the input data f, g and u 0 , and the existence of solution of the forward problem is well known. The system (1.1) models heat conduction phenomenon where the impedance coefficient characterizes the thermal properties of the conductive material on the interface and certain physical processes, e.g. corrosion, on the boundary [2,20]. Thus, the value of the impedance coefficient γ is of significant interest in thermal imaging such as safety analysis of nuclear reactor and thermal protection of space shuttles [3]. In practice, the impedance coefficient γ cannot be specified from direct measurements since the domain Ω may be embedded in an unknown region [30]. Therefore, one has to deal with the inverse problem of identifying the impedance coefficient from some available observations, for example, a final time observation on the whole spatial domain [30], partial Dirichlet boundary observation for the full time period [2,19], time integral observation on the full spatial boundary [14,23], final or an intermediate time observation on the full spatial boundary [14]. Considering the amount of work that has been devoted for this type of parameter identification problem and various type of observations over the years, it is impossible to list all of them, however, the interested reader may refer to [9,18,20,21,22,31,32]. In the applications of heat transport in the high temperature, it is not possible to measure the temperature distribution in the whole time interval [0, τ ]. Therefore, following [12,5] (see also the recent work [6]), in this paper, we assume that the terminal status observations of the temperature distribution is known and with this knowledge we consider the inverse problem of identifying the spatially dependent impedance coefficient γ. More precisely, we assume that (1.2) u(x, t) = φ(x, t) in Ω × [τ − σ, τ ], is known at hand, where σ > 0 is small so that τ − σ > 0, and then we consider the problem of identifying γ from the knowledge of φ or its noisy approximations φ δ satisfying (1.3) τ τ −σ φ − φ δ 2 L 2 (Ω) dt ≤ δ 2 for some noise level δ > 0. It can be observed that our inverse problem is non linear. Furthermore, as can be seen from next section, the inverse problem is also ill-posed, that is, a small perturbation in the observed data may lead to a large deviation in the corresponding solutions. Thus, some regularization scheme has to be employed in order to obtain stable approximations. We employ the standard Tikhonov regularization for obtaining the approximations for γ. It is well known that the convergence of the stable approximations obtained by regularization can be arbitrary slow [26] unless some apriori conditions, the so-called source conditions (cf. [10,24]) is assumed on the unknown that has to be identified. In Tikhonov regularization theory for non linear operators in Hilbert spaces, generally the source condition involves the adjoint of the Fréchet derivative of the non linear operator involved [10]. More precisely, if X and Y are Hilbert spaces, F : X → Y is a (non linear) operator which is Fréchet differentiable, consider the problem of solving an ill-posed operator equation F (x) = y in the sense that small perturbation in y may lead to a large deviation in the solution of the operator equation. Let y be the exact data and x † be the unique solution (i.e. F (x † ) = y) to be identified from the knowledge of y obs ∈ Y, the observed data satisfying y − y obs Y ≤ δ. The stable approximations are the minimizers of the functional min x∈X F (x) − y obs 2 Y + α x − x * 2 X , for a fixed α > 0, the regularization parameter, and x * is an initial guess for x † that incorporates some apriori smoothness assumptions. Let F ′ denotes the Fréchet derivative of F . Moreover, we assume that the Fréchet derivative of F is Lipschitz continuous with the Lipschitz constant C Lip . Then it is known that(cf. [10,11,25]) if x † − x * = F ′ (x † ) * ϕ for some ϕ satisfying the smallness condition (1.4) C Lip ϕ ≤ 1 then the rate of convergence of the regularized solution is O(δ 1 2 ), provided the regularization parameter α is chosen as α ∼ δ. In many problems it is extremely difficult to characterize these range spaces whereas in many problems these range spaces turns out to be certain Sobolev spaces with higher smoothness, see for e.g. [15]. In addition to these difficulties, verification of the smallness condition (1.4) is another challenging task. In order to overcome these challenges, a new type of source condition was proposed by Engl and Zou in [12] for a parameter identification problem in heat conduction, which is simple and verifiable (atleast for some reasonable regularity assumptions on f, g and u 0 ), does not require any smallness condition and also does not require much higher smoothness assumption on the unknowns. Motivated by the work in [12] and also the recent work in [6], we consider a similar weak source condition and obtain the convergence rates, which is the main goal of this paper. Also, we show explicitly that our source condition is verifiable under certain regularity assumption(see Theorem 4.4). We now discuss the advantages and shortcomings associated with our considered observation (1.2) in comparison to some of the recent works where the observations are different from (1.2). In [23] the authors have considered the problem of reconstruction of γ(x) associated with a system similar to (1.1) with f = 0, u 0 = 0 from the non-local measurement of the form (1.5) τ 0 w(t)u(x, t) dt = h(x) on ∂Ω, for some weight function w. Since the governing PDE is a homogeneous heat equation, by using the fundamental solution of the heat equation, the authors could make use of their boundary observation (1.5) in the analysis of the inverse problem of reconstructing γ. In fact, the inverse problem of reconstructing γ is transformed into a problem of solving a system of ill-posed non linear integral equations, where at first one has to solve for a certain potential q(x, t) that arises from the fundamental solution (see [23, pg.4]) and then solve for the impedance coefficient. Moreover, for the stable reconstruction the authors have considered a semi-Tikhonov regularization scheme in which the penalty term involves the potential q (an auxiliary unknown) in contrast to our approach of traditional Tikhonov regularization functional where the penalty term is comprised of γ, the actual unknown to be identified. Although the observation (1.5) seems to be more realistic (as it deals with measurement only in the spatial boundary) than (1.2), but their analysis requires f = 0 and u 0 = 0, which is not the case in this paper. Moreover, the work in [23] does not provide any error estimate. In [30] the authors have considered the the problem of simultaneous identification of γ(x) and the initial temperature u 0 (x) associated with a system similar to (1.1) from the final time observation on the full spatial domain, that is, the data used for the inversion is of the form (1. 6) u(x, τ ) = h(x) in Ω, and in this work the authors could overcome the restriction of u 0 = 0 that they have considered in their earlier work [23]. Because of the ill-posedness of the inverse problem, the authors in [30] have obtained a stable approximations for γ by a regularization scheme that involves the mollification of inversion input data h, and that is achieved by using higher regularity on h, namely h ∈ W 3,p (Ω) for some p > 2. Since h is an inversion input data, assuming such higher regularity of h is not that much realistic from application point of view. Moreover, for a noisy observation h δ of h the authors have obtained a Hölder rate of convergence O(δ ν ), for some ν ≤ 1 5 (see [30, pg.603]) under some source condition on γ and u 0 which may be not feasible in applications, because it is assumed that the unknowns γ and u 0 are sufficiently regular so that h ∈ W 3,p (Ω). In contrast to these, as mentioned earlier, in this paper we will obtain stable approximations for γ using Tikhonov regularization and our regularity assumption on the inversion input data φ(see (1.2)) is only that φ ∈ L 2 (Ω × [τ − σ, τ ]). Moreover, under a verifiable source condition that only requires γ ∈ H 1 2 (∂Ω) we obtain a better rate of convergence, namely, O(δ 1 2 ). Thus, again it is to be noted that although the observation (1.6) apparently seems more realistic than (1.2), but in terms of source condition and regularity assumption our work is more realistic than [30] with a better convergence rate. This paper is organised as follows: In Section 2 we collect all the existing results related to existence and uniqueness of solutions for forward problem, precisely state the inverse problem that we consider and analyze the existence, uniqueness and ill-posedness of the inverse problem. In Section 3 we do the convergence analysis of the regularized approximations, propose the source condition and proof the convergence rate result, the main result of this paper. In Section 4 we discuss about the source condition, its compatibility, regularity and then show that it is indeed verifiable by a simple construction. The Inverse Problem In this section we recall all the definitions, results related to the PDE (1.1) that will be used later and in addition we formulate the inverse problem more precisely and discuss about its uniqueness and ill-posedness. Throughout the paper whenever we come across a function defined on the boundary ∂Ω, it is to be understood in the sense of trace [1,13]. Definition 2.1. (Weak solution) An element u ∈ L 2 (0, τ ; H 1 (Ω)) ∩ L ∞ (0, τ ; L 2 (Ω)) is said to be a weak solution of (1.1) if τ 0 Ω [−u ∂η ∂t + ∇u · ∇η] dx dt + τ 0 ∂Ω γuη dx dt = τ 0 Ω f η dx dt + τ 0 ∂Ω gη dx dt + Ω u 0 η(·, 0) dx for all η ∈ H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0. ♦ We now state a result about existence and uniqueness of the forward problem. Let A = {γ ∈ L ∞ (∂Ω) : 0 < γ ≤ γ ≤ γ} be the set of admissible parameters, for some constants γ and γ. Theorem 2.2. (cf. [29]) Let γ ∈ A, f ∈ L 2 (0, τ ; L 2 (Ω)), g ∈ L 2 (0, τ ; L 2 (∂Ω)) and u 0 ∈ L 2 (Ω). Then there exists a unique weak solution u of (1.1) satisfying the estimate (2.1) max [0,τ ] u(t) L 2 (Ω) + u L 2 (0,τ ;H 1 (Ω)) ≤ C( f L 2 (0,τ ;L 2 (Ω)) + g L 2 (0,τ ;L 2 (∂Ω)) + u 0 L 2 (Ω) ); where C is a constant depending only on γ, Ω and τ. Before proceeding further, let us first precisely state the inverse problem that is considered. (IP) Identify γ ∈ A from the observation φ ∈ L 2 (τ − σ, τ ; L 2 (Ω)) such that the unique weak solution u of (1.1) satisfies (1.2). We now discuss about the existence and uniqueness of the solution of the inverse problem (IP). Let g ∈ L 2 (τ − σ, τ ; L ∞ (∂Ω)), φ ∈ L 2 (τ − σ, τ ; H 1 (Ω)) be such that φ, ∂φ ∂ν ∈ L 2 (τ − σ, τ ; L ∞ (∂Ω)) and τ τ −σ φ dt = 0 a.e. on ∂Ω. Then from the Robin boundary condition in (1.1) and using (1.2), we have (2.2) γ = τ τ −σ g dt − τ τ −σ ∂φ ∂ν dt τ τ −σ φ dt . Thus, the inverse problem to identify γ from the exact observation φ satisfying (1.2) has a unique solution γ ∈ L ∞ (∂Ω) given in (2.2). Next, we observe that the inverse problem (IP) is non linear since the temperature distribution u(x, t) depends on the impedance coefficient γ. Also, it is ill-posed in the sense that small perturbation in the observation data φ in (1.2) may lead to large deviation in the corresponding solution of the inverse problem. This is the case because the expression (2.2) for γ contains a derivative of the observation φ. Thus, in order to obtain some stable approximations for γ some regularization method has to be employed. We will consider the Tikhonov regularization for obtaining stable approximations in the next section. Convergence rates with weak source condition Let u(γ) denotes the unique weak solution of (1.1) for a fixed γ ∈ A. For δ > 0, let φ δ be the noisy data corresponding to the exact data φ satisfying (1.3). As discussed in the previous section, the inverse problem is ill-posed, and thus we shall use the Tikhonov-regularization in order to obtain stable approximations. Throughout we shall denote by γ † ∈ L 2 (∂Ω) the exact impedance coefficient to be identified for the corresponding exact data φ. For a fixed α > 0, consider the output-least square Tikhonov functional (3.1) J(γ) := τ τ −σ Ω |u(γ) − φ δ | 2 dx dt + α γ − γ * 2 L 2 (∂Ω) , where γ * ∈ L 2 (∂Ω) is an initial guess for γ † that incorporates some apriori smoothness assumption on γ † . By γ α δ we denote a minimizer of the optimization problem (3.2) min γ∈A J(γ). These minimizers are the Tikhonov-regularized solutions. Note that the initial guess γ * may not belong to the admissible set A. Our next two results are about the existence and stability of such minimizers, which in turn ensures that the minimizers are indeed regularized solutions. It is to be noted that by now the arguments for the proof of existence and stability is well established in the literature (cf. [12]), but since the context of this paper is different, in order to keep the paper self contained we include the proof also. Proof. It is clear that A is a convex set. Let {γ n } be a minimizing sequence in A. Clearly, {γ n } is a bounded sequence in L 2 (∂Ω). Thus, there exists a subsequence {γ m } and a γ α ∈ L 2 (∂Ω) such that γ m ⇀ γ α in L 2 (∂Ω). Now the closedness and convexity of A implies that A is weakly closed, thus, γ α ∈ A. Since u(γ m ) is the weak solution of (1.1) for γ = γ m , by the estimate (2.1), it follows that {u(γ m )} is a bounded sequence in L 2 (0, τ ; H 1 (Ω)). Thus, there exists a subsequence, still denoted by {u(γ m )}, and u * ∈ L 2 (0, τ ; H 1 (Ω)) such that u(γ m ) ⇀ u * in L 2 (0, τ ; H 1 (Ω)) as m → ∞. Now, for any η ∈ L 2 (0, τ ; H 1 (Ω)), writing τ 0 ∂Ω γ m u(γ m )η dx dt = τ 0 ∂Ω γ α u(γ m )η dx dt + τ 0 ∂Ω (γ m − γ α ) u(γ m )η dx dt, using the weak convergence of γ m and the boundedness of u(γ m ), it follows that τ 0 ∂Ω γ m u(γ m )η dx dt → τ 0 ∂Ω γ α u * η dx dt as m → ∞. Therefore, using the fact that u(γ m ) is a weak solution of (1.1) for γ = γ m , and the weak convergence of {u(γ m )}, it follows that τ 0 Ω [−u * ∂η ∂t + ∇u * · ∇η] dx dt + τ 0 ∂Ω γ α u * η dx dt = τ 0 Ω f η dx dt + τ 0 ∂Ω gη dx dt + Ω u 0 η(·, 0) dx for all η ∈ H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0. Since u(γ α ) is the unique weak solution of (1.1) for γ = γ α , we have u * = u(γ α ). We now consider the identity lim inf m τ τ −σ Ω |u(γ m ) − φ δ | 2 dx dt = lim inf m τ τ −σ Ω |u(γ m ) − u(γ α )| 2 + |u(γ α ) − φ δ | 2 + 2 u(γ m ) − u(γ α ) u(γ α ) − φ δ dx dt. Since u(γ m ) and u(γ α ) are the weak solutions of (1.1) for γ = γ m and γ α , respectively, using the estimate (2.1), we have lim m→∞ τ τ −σ Ω |u(γ m ) − u(γ α )| 2 dx dt = 0. Therefore, lim inf m τ τ −σ Ω |u(γ m ) − φ δ | 2 dx dt = τ τ −σ Ω |u(γ α ) − φ δ | 2 dx dt. Thus, using the weak lower semi-continuity of the L 2 -norm and using the fact that {γ n } is a minimizing sequence for the minimization problem (3.2), we have τ τ −σ Ω |u(γ α ) − φ δ | 2 dx dt + α γ α − γ * 2 L 2 (∂Ω) ≤ lim inf m τ τ −σ Ω |u(γ m ) − φ δ | 2 dx dt + α γ m − γ * 2 L 2 (∂Ω) = min γ∈A J(γ). This shows that γ α is a minimizer of (3.2). We now prove the stability of the minimization problem (3.2) with respect to the observation data φ δ . That is, we establish that the minimizers of (3.2) are indeed regularized solutions. Theorem 3.2. Let {φ n } be a sequence such that φ n converges to φ δ in L 2 (τ − σ, τ ; L 2 (Ω)). For a fixed α > 0, let γ α n be the minimizer of (3.3) min γ∈A τ τ −σ Ω |u(γ) − φ n | 2 dx dt + α γ − γ * 2 L 2 (∂Ω) . Then there exists a subsequence {γ α n } that converges to a minimizer γ α δ . Proof. Since γ α n is a minimizer of (3.3), we have τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt + α γ α n − γ * 2 L 2 (∂Ω) ≤ τ τ −σ Ω |u(γ) − φ n | 2 dx dt + α γ − γ * 2 L 2 (∂Ω) for any γ ∈ A. Thus, {γ α n } is a bounded sequence in L 2 (∂Ω). Therefore, there exists a subsequence, with abuse of notation, denoted by {γ α n } and a γ α ∈ L 2 (∂Ω) such that γ α n converges weakly to γ α in L 2 (∂Ω) as n → ∞. Now, the closedness and convexity of A implies that γ α ∈ A. Since u(γ α n ) is the unique weak solution of (1.1), by (2.1) it follows that {u(γ α n )} is bounded in L 2 (0, τ ; H 1 (Ω)). Thus, there exist a subsequence, still denoted by u(γ α n ) and a u * ∈ L 2 (0, τ ; H 1 (Ω)) such that u(γ α n ) converges weakly to u * in L 2 (0, τ ; H 1 (Ω)) as n → ∞. Now, for any η ∈ L 2 (0, τ ; H 1 (Ω)), writing τ 0 ∂Ω γ α n u(γ α n )η dx dt = τ 0 ∂Ω γ α u(γ α n )η dx dt + τ 0 ∂Ω (γ α n − γ α ) u(γ α n )η dx dt, using the weak convergence of γ α n and the boundedness of u(γ α n ), it follows that τ 0 ∂Ω γ α n u(γ α n )η dx dt → τ 0 ∂Ω γ α u * η dx dt as n → ∞. Therefore, using the fact that u(γ α n ) is a weak solution, and the weak convergence of {u(γ α n )}, it follows that τ 0 Ω [−u * ∂η ∂t + ∇u * · ∇η] dx dt + τ 0 ∂Ω γ α u * η dx dt = τ 0 Ω f η dx dt + τ 0 ∂Ω gη dx dt + Ω u 0 η(·, 0) dx for all η ∈ H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0. Since u(γ α ) is the unique weak solution of (1.1), we have u * = u(γ α ). Now the weak lower semi continuity of L 2 -norm implies γ α − γ * 2 L 2 (∂Ω) ≤ lim inf n γ α n − γ * 2 L 2 (∂Ω) and τ τ −σ Ω |u(γ α ) − φ δ | 2 dx dt ≤ lim inf n τ τ −σ Ω |u(γ α n ) − φ δ | 2 dx dt. Since φ n → φ δ in L 2 (τ − σ, τ ; L 2 (Ω)), it follows that lim inf n τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt = lim inf n τ τ −σ Ω |u(γ α n ) − φ δ | 2 + |φ n − φ δ | 2 + 2 u(γ α n ) − φ δ (φ δ − φ n ) dx dt = lim inf n τ τ −σ Ω |u(γ α n ) − φ δ | 2 dx dt = lim inf n τ τ −σ Ω |u(γ α n ) − u(γ α )| 2 + |u(γ α ) − φ δ | 2 + 2 u(γ α n ) − u(γ α ) u(γ α ) − φ δ dx dt. Since u(γ α n ) and u(γ α ) are the weak solutions of (1.1) for γ = γ α n and γ α , respectively, using the estimate (2.1), we have lim n→∞ τ τ −σ Ω |u(γ α n ) − u(γ α )| 2 dx dt = 0. Therefore, lim inf n τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt = τ τ −σ Ω |u(γ α ) − φ δ | 2 dx dt. Thus, τ τ −σ Ω |u(γ α ) − φ δ | 2 dx dt + α γ α − γ * 2 L 2 (∂Ω) ≤ lim inf n τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt + α γ α n − γ * 2 L 2 (∂Ω) ≤ lim sup n τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt + α γ α n − γ * 2 L 2 (∂Ω) ≤ lim sup n τ τ −σ Ω |u(γ) − φ n | 2 dx dt + α γ − γ * 2 L 2 (∂Ω) = τ τ −σ Ω |u(γ) − φ δ | 2 dx dt + α γ − γ * 2 L 2 (∂Ω) for any γ ∈ A. This shows that γ α is a minimizer, and we denote this by γ α δ . Also, taking γ = γ α δ in the last equality, we have (3.4) lim n τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt + α γ α n − γ * 2 L 2 (∂Ω) = τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt + α γ α δ − γ * 2 L 2 (∂Ω) We now establish the convergence of γ α n to γ α δ in L 2 (∂Ω) as n → ∞. The proof is by contradiction. Suppose that γ α n does not converge to γ α δ in L 2 (∂Ω). Then, clearly γ α δ − γ * 2 L 2 (∂Ω) < lim sup n γ α n − γ * 2 L 2 (∂Ω) =: ǫ. Thus, there exists a subsequence of {γ α n }, say {γ α m } such that γ α m converges weakly to γ α δ in L 2 (∂Ω) and γ α m − γ * 2 L 2 (∂Ω) → ǫ as m → ∞. Now from (3.4), we have lim m τ τ −σ Ω |u(γ α m ) − φ m | 2 dx dt = τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt + α γ α δ − γ * 2 L 2 (∂Ω) − lim m γ α m − γ * 2 L 2 (∂Ω) = τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt + α γ α δ − γ * 2 L 2 (∂Ω) − ǫ <0 . Hence, lim m τ τ −σ Ω |u(γ α m ) − φ m | 2 dx dt < τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt. Therefore, we have τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt = lim inf n τ τ −σ Ω |u(γ α n ) − φ n | 2 dx dt ≤ lim m τ τ −σ Ω |u(γ α m ) − φ m | 2 dx dt < τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt, a contradiction. Let f ∈ L 2 (0, τ ; L 2 (Ω)), g ∈ L 2 (0, τ ; L 2 (∂Ω)) and u 0 ∈ L 2 (Ω). Then by Theorem 2.2, we know that for each γ ∈ A, (1.1) has a unique weak solution in L 2 (0, τ ; H 1 (Ω)) ∩ L ∞ (0, τ ; L 2 (Ω)) and we have used the notation u(γ) to denote this weak solution. For notational simplicity we denote this by a map F , that is, F : A → L 2 (0, τ ; H 1 (Ω)) ∩ L ∞ (0, τ ; L 2 (Ω)) is defined as γ → F (γ) := u(γ). In order to avoid notational complexity, throughout C will denote a generic constant that may depend only on Ω, τ, σ, γ, γ, u 0 , f, g, the operator norm of the trace operator (and also on some function ψ considered later in Theorem 3.4). LEMMA 3.3. The mapping F : A ⊂ L ∞ (∂Ω) → L 2 (0, τ ; H 1 (Ω)) ∩ L ∞ (0, τ ; L 2 (Ω)) is Fréchet differentiable. Proof. Since u(γ) denotes the unique weak solution of (1.1), we have (3.5) τ 0 Ω [−u(γ) ∂η ∂t + ∇u(γ) · ∇η] dx dt + τ 0 ∂Ω γu(γ)η dx dt = τ 0 Ω f η dx dt + τ 0 ∂Ω gη dx dt + Ω u 0 η(·, 0) dx for all η ∈ H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0. Let h ∈ L ∞ (∂Ω) be such that γ + h ∈ A. Then, we have H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0. Thus, from (3.5) and (3.6), we have (3.6) τ 0 Ω [−u(γ + h) ∂η ∂t + ∇u(γ + h) · ∇η] dx dt + τ 0 ∂Ω γu(γ + h)η dx dt = τ 0 Ω f η dx dt + τ 0 ∂Ω gη dx dt + Ω u 0 η(·, 0) dx for all η ∈(3.7) τ 0 Ω −[u(γ + h) − u(γ)] ∂η ∂t dx dt + τ 0 Ω [∇u(γ + h) − ∇u(γ)] · ∇η dx dt + τ 0 ∂Ω (γ + h)[u(γ + h) − u(γ)]η dx dt + τ 0 ∂Ω hu(γ)η dx dt = 0 for all η ∈ H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0. Now consider the PDE (3.8)      ∂v ∂t = ∆v in Ω τ , ∂v ∂ν + γv = −hu(γ) on ∂Ω τ , v(·, 0) = 0 in Ω. By Theorem 2.2 there exist a unique weak solution v of (3.8) satisfying (3.9) τ 0 Ω v ∂η ∂t dx dt − τ 0 Ω ∇v · ∇η dx dt − τ 0 ∂Ω γvη dx dt = τ 0 ∂Ω hu(γ)η dx dt for all η ∈ h 1 (0; τ, H 1 (Ω)) with η(·, τ ) = 0 in Ω. Moreover, from (3.8) and the estimate (2.1) it follows that max t∈[0,τ ] v(t) L 2 (Ω) + v L 2 (0,τ ;H 1 (Ω)) ≤ C hu(γ) L 2 (0,τ ;L 2 (∂Ω)) ≤ C h L ∞ (∂Ω) u(γ) L 2 (0,τ ;L 2 (∂Ω)) ≤ C h L ∞ (∂Ω) u(γ) L 2 (0,τ ;H 1 (Ω)) . Therefore, for a fixed γ ∈ A, the map h → v, where v is the unique weak solution of (3.8), is a bounded linear operator w(t) L 2 (Ω) + w L 2 (0,τ ;H 1 (Ω)) ≤ C hv L 2 (0,τ ;L 2 (∂Ω)) . Now using the fact that u(γ) is a weak solution of (1.1), from Theorem 2.2 it follows that u(γ) L 2 (0,τ ;H 1 (Ω)) ≤ C h L ∞ (∂Ω) ( f L 2 (0,τ ;L 2 (Ω)) + g L 2 (0,τ ;L 2 (∂Ω)) + u 0 L 2 (Ω) ). from L ∞ (∂Ω) → L ∞ (0, τ ; L 2 (Ω)) ∩ L 2 (0, τ ; H 1 (Ω)). Let w := u(γ + h) − u(γ) − v, Therefore, from (3.11) we have max t∈[0,τ ] w(t) L 2 (Ω) + w L 2 (0,τ ;H 1 (Ω)) ≤ C h 2 L ∞ (∂Ω) ( f L 2 (0,τ ;L 2 (Ω)) + g L 2 (0,τ ;L 2 (∂Ω)) + u 0 L 2 (Ω) ). Thus, it follows that F is Fréchet differentiable, and F ′ (γ)h = v, where v is the unique weak solution of (3.8) for a given γ ∈ A. We are now in a position to state and proof our main result about the convergence rate under a weak source condition. Let us recall that γ † ∈ A is the exact Robin coefficient to be identified and γ * ∈ L 2 (∂Ω) is an apriori initial guess for γ † , and u(γ † ) denotes the unique weak solution of (1.1) for γ = γ † . Theorem 3.4. (Convergence rate) Let ψ ∈ H 1 0 (τ − σ, τ ; L 2 (Ω)) ∩ L 2 (τ − σ, τ ; H 2 (Ω)) be such that (3.12) τ τ −σ u(γ † )ψ dt = γ † − γ * on ∂Ω. For fixed α, δ > 0, let γ α δ be the minimizer of (3.1). Then γ † − γ α δ L 2 (∂Ω) ≤ C δ 2 α + α 1 2 , for some constant C > 0 independent of α, δ, γ † and γ α δ . Proof. Since γ α δ is a minimizer of (3.1) for a fixed α, using (1.3), it follows that τ τ −σ Ω |u(γ α δ ) − φ δ | 2 dx dt + α γ α δ − γ * L 2 (∂Ω) ≤ τ τ −σ Ω |u(γ † ) − φ δ | 2 dx dt + α γ † − γ * L 2 (∂Ω) ≤ δ 2 + α γ † − γ * 2 L 2 (Ω) . Thus, τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) + α γ † − γ α δ 2 L 2 (∂Ω) ≤ δ 2 + α γ † − γ α δ 2 l 2 (∂Ω) + γ † − γ * 2 L 2 (∂Ω) − γ α δ − γ * 2 L 2 (∂Ω) = δ 2 + 2α γ † − γ α δ , γ † − γ * L 2 (∂Ω) . The source condition implies α γ † − γ α δ , γ † − γ * L 2 (∂Ω) = α τ τ −σ u(γ † )ψ dt, γ † − γ α δ L 2 (∂Ω) = α τ τ −σ ∂Ω u(γ † )ψ (γ † − γ α δ ) dx dt. Since F is Fréchet differentiable, taking h = γ α δ − γ † , η = ψ, u(γ) = u(γ † ) and using the identity (3.9), we get τ τ −σ ∂Ω u(γ † )ψ (γ α δ − γ † ) dx dt = τ τ −σ Ω [F ′ (γ † )(γ α δ − γ † ) ∂ψ ∂t − ∇F ′ (γ † )(γ α δ − γ † ) · ∇ψ] dx dt − τ τ −σ ∂Ω γ † F ′ (γ † )(γ α δ − γ † )ψ dx dt. Thus, α γ † − γ α δ , γ † − γ * L 2 (∂Ω) = α τ τ −σ Ω − F ′ (γ † )(γ α δ − γ † ) ∂ψ ∂t + ∇F ′ (γ † )(γ α δ − γ † ) · ∇ψ dx dt + τ τ −σ ∂Ω γ † F ′ (γ † )(γ α δ − γ † )ψ dx dt . Let w α δ := u(γ α δ ) − u(γ † ) − F ′ (γ † )(γ α δ − γ † ). Then, we have (3.13)            α γ † − γ α δ , γ † − γ * L 2 (∂Ω) = α τ τ −σ Ω − [u(γ α δ ) − u(γ † )] ∂ψ ∂t + ∇[u(γ α δ ) − u(γ † )] · ∇ψ dx dt + τ τ −σ ∂Ω γ † [u(γ α δ ) − u(γ † )]ψ dx dt + τ τ −σ Ω w α δ ∂ψ ∂t − ∇w α δ · ∇ψ dx dt − τ τ −σ ∂Ω γ † w α δ ψ dx dt Taking w = w α δ , from (3.10), we have τ τ −σ Ω [−w α δ ∂ψ ∂t + ∇w α δ · ∇ψ] dx dt + τ τ −σ ∂Ω γ α δ w α δ ψ dx dt + τ τ −σ ∂Ω (γ α δ − γ † )F ′ (γ † )(γ α δ − γ † )ψ dx dt = 0. Thus, τ τ −σ Ω [w α δ ∂ψ ∂t − ∇w α δ · ∇ψ] dx dt − τ τ −σ ∂Ω γ † w α δ ψ dx dt = τ τ −σ ∂Ω (γ α δ − γ † )(u(γ α δ ) − u(γ † ))ψ dx dt. Therefore, from (3.13), we have α γ † − γ α δ , γ † − γ * L 2 (∂Ω) = α τ τ −σ Ω − [u(γ α δ ) − u(γ † )] ∂ψ ∂t + ∇[u(γ α δ ) − u(γ † )] · ∇ψ dx dt + τ τ −σ ∂Ω γ α δ (u(γ α δ ) − u(γ † ))ψ dx dt = α τ τ −σ Ω − [u(γ α δ ) − u(γ † )] ∂ψ ∂t + [u(γ † ) − u(γ α δ )]∆ψ dx dt + τ τ −σ ∂Ω [u(γ α δ ) − u(γ † )] ∂ψ ∂ν dx dt + τ τ −σ ∂Ω γ α δ [u(γ α δ ) − u(γ † )]ψ dx dt =: I 1 + I 2 + I 3 + I 4 . We now estimate the integrals I 1 , I 2 , I 3 and I 4 . We will be using the Cauchy-Schwarz inequality and Young's inequality appropriately which will involve an arbitrary ε > 0. |I 1 | = |α τ τ −σ Ω −[u(γ α δ ) − u(γ † )] ∂ψ ∂t dx dt| ≤ α τ τ −σ u(γ α δ ) − u(γ † ) L 2 (Ω) ∂ψ ∂t L 2 (Ω) dt ≤ α τ τ −σ u(γ α δ ) − φ δ L 2 (Ω) ∂ψ ∂t L 2 (Ω) dt + α τ τ −σ φ δ − u(γ † ) L 2 (Ω) ∂ψ ∂t L 2 (Ω) dt ≤ ε τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt + α 2 4ε τ τ −σ ∂ψ ∂t 2 L 2 (Ω) dt +ε τ τ −σ φ δ − u(γ † ) 2 L 2 (Ω) + α 2 4ε τ τ −σ ∂ψ ∂t 2 L 2 (Ω) dt ≤ ε τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt + δ 2 ε + α 2 2ε τ τ −σ ∂ψ ∂t 2 L 2 (Ω) dt. Similarly, we obtain |I 2 | = |α τ τ −σ [u(γ † ) − u(γ α δ )]∆ψ dx dt| ≤ ε τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt + δ 2 ε + α 2 2ε τ τ −σ ∆ψ 2 L 2 (Ω) dt. In order to estimate I 3 and I 4 , additionally we will make use of the continuity of the trace map from H 1 (Ω) to L 2 (∂Ω). |I 3 | = |α τ τ −σ ∂Ω [u(γ α δ ) − u(γ † )] ∂ψ ∂ν dx dt| ≤ α τ τ −σ u(γ α δ ) − u(γ † ) L 2 (∂Ω) ∂ψ ∂ν L 2 (∂Ω) dt ≤ Cα τ τ −σ u(γ α δ ) − u(γ † ) L 2 (Ω) ∂ψ ∂ν L 2 (∂Ω) dt ≤ Cα τ τ −σ u(γ α δ ) − φ δ L 2 (Ω) ∂ψ ∂ν L 2 (∂Ω) dt + Cα τ τ −σ φ δ − u(γ † ) L 2 (Ω) ∂ψ ∂ν L 2 (∂Ω) dt ≤ ε τ τ −σ u(γ α δ ) − u(γ † ) 2 L 2 (Ω) dt + δ 2 ε + C 2 α 2 2ε τ τ −σ ∂ψ ∂ν 2 L 2 (∂Ω) dt. Similarly, we have |I 4 | = |α τ τ −σ ∂Ω γ α δ [u(γ α δ ) − u(γ † )]ψ dx dt| ≤ γ Cα τ τ −σ u(γ α δ ) − u(γ † ) L 2 (Ω) ψ L 2 (∂Ω) dx dt ≤ ε τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt + δ 2 ε + γ 2 C 2 α 2 2ε τ τ −σ ψ 2 L 2 (∂Ω) dt. Therefore, we obtain τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) + α γ † − γ α δ 2 L 2 (∂Ω) ≤ δ 2 + 8ε τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt + 8δ 2 ε + α 2 ε τ τ −σ ∂ψ ∂t 2 L 2 (Ω) + ∆ψ 2 L 2 (Ω) + C 2 ∂ψ ∂ν 2 L 2 (∂Ω) + γ 2 C 2 ψ 2 L 2 (∂Ω) dt . Thus, taking ε = 1 16 , we have τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt + 2α γ † − γ α δ 2 L 2 (∂Ω) ≤ 3δ 2 + 32α 2 τ τ −σ ∂ψ ∂t 2 L 2 (Ω) + ∆ψ 2 L 2 (Ω) + C 2 ∂ψ ∂ν 2 L 2 (∂Ω) + γ 2 C 2 ψ 2 L 2 (∂Ω) dt ≤ C(δ 2 + α 2 ). Therefore, τ τ −σ u(γ α δ ) − φ δ 2 L 2 (Ω) dt ≤ C(δ 2 + α 2 ) and γ † − γ α δ L 2 (∂Ω) ≤ C δ 2 α + α 1 2 . Remark 3.5. From the estimate obtained in the above theorem for γ † − γ α δ L 2 (∂Ω) under the source condition (3.12) it follows that if we choose the regularization parameter α as α ∼ δ, then we have γ † − γ α δ L 2 (∂Ω) = O(δ 1 2 ). ♦ Remark 3.6. As observed, our analysis could produce the rate O(δ 1 2 ). It would be interesting to analyse the convergence rates by proceeding along the recent line of research based on variational source conditions (see for e.g., [7,16,17,28]) which to the best of our knowledge still remains to be explored in the context of impedance coefficient identification. In fact we are aware of only a very recent paper [8] that deals with Robin coefficient identification using variational source condition for elliptic PDE only. ♦ Discussion about source condition We now discuss advantages of the source condition (3.12). From Theorem 3.4 it is clear that our source condition does not require any smallness condition to be verified unlike the smallness condition of the form (1.4), that is required in standard convergence theory for Tikhonov-regularization for non linear operators (cf. [10,11]). Moreover, in order to obtain a convergence rate of O(δ 1 2 ) we do not need to assume that γ † belongs to the range of the adjoint of the Fréchet derivative of the parameter-to-solution operator. This is another big advantage since most often the range of such operators is nothing but some Sobolev spaces with higher smoothness (see for e.g. [15]), and thus our source condition being free from such apriori range condition means that we can obtain the said convergence rate without imposing higher smoothness assumption on γ † , which is obviously not known to us. We now discuss about the apriori regularity of γ † that is embedded in the source condition (3.12). Recall that for defining trace operator, the minimal assumption that we need is that τ τ −σ u(γ † ) ψ dt ∈ H 1 (Ω), and in that case it follows that we must have γ † − γ * ∈ H 1 2 (∂Ω). Thus, in order that the source condition (3.12) makes sense, the regularity assumption that we require is that γ † − γ * ∈ H 1 2 (∂Ω). We shall show that in our setting we do always have τ τ −σ u(γ † )ψ dt ∈ H 1 (Ω), thanks to the following very recent result on multiplication of elements in Sobolev spaces [4]. U V H s (D) ≤ C U H s 1 (D) V H s 2 (D) , ∀ U ∈ H s1 (D), V ∈ H s2 (D). Since ψ ∈ L 2 (τ −σ, τ ; H 2 (Ω)) and u(γ † ) ∈ L 2 (τ −σ, τ ; H 1 (Ω)), by Theorem 4.1 it follows that τ τ −σ u(γ † )ψ dt ∈ H 1 (Ω). Remark 4.2. It is to be noted that one can also derive a sufficient condition for τ τ −σ u(γ † )ψ dt ∈ H 1 (Ω) based on the results related to Banach algebra properties of Sobolev space, but with this argument we need to assume H 2 -spatial regularity of u(γ † ), which of course is a drawback. Indeed, recall that for d ∈ {2, 3}, Ω ⊂ R d , H s (Ω), s ≥ 2, is a Banach algebra. Thus, if we assume that u(γ † ) ∈ L 2 (τ − σ, τ ; H 2 (Ω))(such a regularity indeed holds if the initial profile u 0 and the source function f in (1.1) are assumed to be of appropriate spatial regularity) then τ τ −σ u(γ † ) ψ dt ∈ H 2 (Ω) ⊂ H 1 (Ω) and thus by the property of trace operator it follows that we must have γ † − γ * ∈ H We now look into the compatibility issue related to the source condition. From the expression of the source condition (3.12), one may think that on some portion of the boundary ∂Ω near the terminal time status, it may happen that u(γ † ) vanishes(in the sense of trace) but γ † − γ * does not, and this implies that on such portions one has to know γ † . But this is compatible with the fact that for terminal time status, if u(γ † ) vanishes (in the sense of trace) on some portion of the boundary ∂Ω then it is impossible to recover γ † on such portions, as can be observed from (2.2). Next, we give a procedure to verify the source condition explicitly, motivated from the construction given in [12] for the case of 1-dimensional diffusion coefficient identification problem. However, our construction is valid for the dimension d ∈ {2, 3} as considered in this paper, but provided we have u(γ † ) ∈ L 2 (τ − σ, τ ; H 2 (Ω)). Indeed, we construct a ψ ∈ H 1 0 (τ − σ, τ ; L 2 (Ω)) ∩ L 2 (τ − σ, τ ; H 2 (Ω)) that satisfies the source condition (3.12). Before proceeding further, we recall the following interesting result from [27]. Our next result is about verification of source condition under certain smoothness assumption. Theorem 4.4. Let γ † , γ * ∈ L ∞ (∂Ω) ∩ H 3 2 (∂Ω) and u(γ † ) ∈ L 2 (τ − σ, τ ; H 2 (Ω)). Let ψ 1 ∈ H 1 (τ − σ, τ ) be an arbitrary function. Define u 1 (x) := τ τ −σ (τ − t)(τ − σ − t)ψ 1 u(γ † ) dt. Assume that (4.1) u 1 = 0 a.e. in Ω and 1 u 1 ∈ L ∞ (Ω). Then there exists a ψ ∈ H 1 0 (τ − σ, τ ; L 2 (Ω)) ∩ L 2 (τ − σ, τ ; H 2 (Ω)) satisfying the source condition (3.12). Proof. Let ψ 2 ∈ H 1 (τ − σ, τ ) and ϕ ∈ H m (Ω), m ≥ 2, be two arbitrary functions. We will write u to denote u(γ † ). Define u 2 (x) := τ τ −σ (τ − t)(τ − σ − t)ψ 2 ϕ u dt. Let Υ † , Υ * ∈ H 2 (Ω) be such that γ † and γ * are their traces, respectively. We define ψ := (τ − t)(τ − σ − t) ψ 2 ϕ + ψ 1 Υ † − Υ * − u 2 u 1 . We claim that this ψ satisfies the source condition. Observe that from the L ∞ -assumption in (4.1) and from Theorem 4.3, it follows that 1 u1 ∈ H 2 (Ω) and hence Υ † −Υ * −u2 u1 ∈ L 2 (τ − σ, τ ; H 2 (Ω)). Also, from the definition of ψ it clearly follows that ψ(·, τ − σ) = 0 = ψ(·, τ ) in Ω. Thus, ψ ∈ H 1 0 (τ − σ, τ ; L 2 (Ω)) ∩ L 2 (τ − σ, τ ; H 2 (Ω)). Moreover, τ τ −σ u(γ † )ψ dt = τ τ −σ (τ − t)(τ − σ − t) ψ 2 ϕ u + ψ 1 u γ † − γ * − u 2 u 1 on ∂Ω = γ † − γ * on ∂Ω. This completes the proof. Remark 4.5. Note that the non-zero assumption in (4.1) does make sense, because otherwise it would mean u(γ † ) vanishes on the boundary near the terminal time τ (i.e., [τ − σ, τ ]), but then from (2.2) it follows that the impedance coefficient γ † is impossible to recover. ♦ Conclusion We have considered an inverse problem of identifying a spatially dependent impedance coefficient in a parabolic PDE from a short-time observation of the temperature distribution. We have proposed a weak-type source condition that allowed to obtain a convergence rate of O(δ 1 2 ) provided we choose the regularization parameter α ∼ δ, where δ is the deterministic noise level. As compared to the standard convergence theory of Tikhonov regularization for non linear operators in Hilbert spaces, the above rate of convergence is obtained under a simple source condition which does not require any range condition of the adjoint of Fréchet derivative, no smallness condition to be verified and requires a regularity assumption that γ † ∈ H 1 2 (∂Ω). Moreover, if it is known that u(γ † ) vanishes(in the sense of trace) on some portion of the boundary ∂Ω, then one has to know γ † apriori over those portions, as it is impossible to recover the impedance coefficient on those portions. Theorem 3. 1 . 1The minimization problem (3.2) has a solution. all η ∈ H 1 (0, τ ; H 1 (Ω)) with η(·, τ ) = 0 in Ω. Therefore from (3.10), the definition of weak solution and the estimate (2.1), it follows that there exists a constant C depending only on γ, Ω and τ such that(3.11) max t∈[0,τ ] Theorem 4. 1 . 1(cf. [4, Theorem 7.4]) Let D ⊂ R d , d ∈ {1, 2, 3}, be a bounded domain with Lipschitz continuous boundary. Let s i (i = 1, 2), s be real numbers such that s i ≥ s ≥ 0 and s 1 + s 2 − s > d 2 . Then there exists a constant C depending only on s 1 , s 2 , s, d and Ω such that Let D ⊂ R d be a bounded domain with Lipschitz boundary. Let s > d 2 . If U ∈ H s (D) and U ≥ c > 0 for some constant c, then 1 U ∈ H s (D). Acknowledgments. The author is grateful to Prof. Thorsten Hohage (University of Göttingen) for several insightful comments and for bringing the result in[27]to our notice, which has helped to improve the content. This work was initiated when the author was a postdoctoral researcher in the Institute for Numerical and Applied Mathematics, University of Göttingen. 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Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Problems 17 (2001) 1181-1202. On the simultaneous reconstruction of boundary Robin coefficient and internal source in a slow diffusion system. M Zhang, J Liu, Inverse Problems. 377ppM. Zhang and J. Liu, On the simultaneous reconstruction of boundary Robin coefficient and internal source in a slow diffusion system. Inverse Problems 37 (2021), no. 7, Paper No. 075008, 33 pp. . TIFR Centre for Applicable Mathematics. Bangalore-560065India Email address: [email protected]; [email protected] Centre for Applicable Mathematics, Bangalore-560065, India Email address: [email protected]; [email protected]

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{'abstract': 'This paper deals with the problem of identification of a Robin coefficient (also known as impedance coefficient) in a parabolic PDE from terminal observations of the temperature distributions. The problem is ill-posed in the sense that small perturbation in the observation may lead to a large deviation in the solution. Thus, in order to obtain stable approximations, we employ the Tikhonov-regularization. We propose a weak source condition motivated by the work of Engl and Zou(2000)and obtain a convergence rate of O(δ 1 2 ), the main goal of this paper, where δ is the noise level of the observed data. The obtained rate is better than some of the previous known rates. Moreover, the advantage of the proposed source condition is that we are getting the above mentioned convergence rate without the need for characterizing the range space of modelling operator, which is in contrast to the general convergence theory of Tikhonov-regularization for non linear operators, where one obtain the same order of convergence by characterizing the range of the adjoint of the Fréchet derivative of modelling operator, a challenging task for many problems.', 'arxivid': '2304.00726', 'author': ['Subhankar Mondal '], 'authoraffiliation': [], 'corpusid': 257913137, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19131, 'n_tokens_neox': 15836, 'n_words': 9748, 'pdfsha': 'ed8f333d7db0ec602fed6519601ad69431e0ef75', 'pdfurls': ['https://export.arxiv.org/pdf/2304.00726v1.pdf'], 'title': ['CONVERGENCE RATES FOR IDENTIFICATION OF ROBIN COEFFICIENT FROM TERMINAL OBSERVATIONS', 'CONVERGENCE RATES FOR IDENTIFICATION OF ROBIN COEFFICIENT FROM TERMINAL OBSERVATIONS'], 'venue': []}

arxiv

ON CONVERGENCE TO A FOOTBALL 27 Jan 2015 Hao Fang Mijia Lai ON CONVERGENCE TO A FOOTBALL 27 Jan 2015 We show that spheres of positive constant curvature with n (n ≥ 3) conic points converge to a sphere of positive constant curvature with two conic points, or an (American) football in Gromov-Hausdorff topology when the conic angles of the sequence pass from the subcritical case in the sense of Troyanov to the critical case in the limit.We prove this convergence in two different ways. Geometrically, the convergence follows from Luo-Tian's explicit description of conic spheres as boundaries of convex polytopes in S 3 . Analytically, considering the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.H.F.'s work is partially supported by Simons Foundation and NSF DMS-100829. Introduction We study convergence properties of metrics on Riemann surface with conical singularities. It has been an old topic and been first studied as early as in 1905 by Picard [P], when he was considering the uniformization problem for Riemann surfaces with branched points. In 1990s, it was systematically studied by several authors [Tr,LT,CL1,CL2]. Recent progress on the study of canonical metrics on Fano manifolds has brought up renewed interests in this field [D, JMR]. We first start with some definitions. For a compact closed Riemann surface S, a metric g is said to have a conic singularity of order β ∈ (−1, ∞) at p, if under a local holomorphic coordinate centered at p, g = e f (z) |z| 2β |dz| 2 , where f (z) is continuous and C 2 away from p. The singularity is modeled on the Euclidean cone: C with the metric |z| 2β |dz| 2 is isometric to a Euclidean cone of cone angle 2π(β + 1). In general, we use the triple (S, g, β), where β = n i=1 β i p i , to denote a Riemann surface S endowed with a metric g with conic singularities at each p i of order β i . Let K = K(g) be the Gaussian curvature of g where it is smooth. The Gauss-Bonnet formula for the conic surface (S, g, β) becomes S Kds 2 = 2πχ(S, β), where χ(S, β) := χ + n i=1 β i is the Euler characteristic for the conic surface (S, β). Troyanov [Tr] systematically studied the prescribing curvature problem on conic surfaces. The problem is divided into three main cases according to the sign of the Euler characteristic; while the positive Euler characteristic case is divided further into three sub-cases. Namely, we have 1. negative case: χ(S, β) < 0; 2. zero case: χ(S, β) = 0; 3. positive case: χ(S, β) > 0; 3.a subcritical case: χ(S, β) < min{2, 2 + 2 min i β i }; 3.b critical case: χ(S, β) = min{2, 2 + 2 min i β i }; 3.c supercritical case : χ(S, β) > min{2, 2 + 2 min i β i }. It turns out that cases 1, 2, and 3.a are parallel to the corresponding cases in the prescribing curvature problem on a smooth surface as the corresponding functionals are coercive; while cases 3.b and 3.c are more delicate. we refer readers to, e.g. [BDM,CL1,E,LZ] for details. For the Yamabe problem on surfaces with the conic singularity, where we prescribe the constant curvature, the answer is more complete. Without loss of generality, we assume that S is orientable. If χ(S, β) ≤ 0, it has been shown [Tr] that there always admits a conic metric with constant curvature. While for χ(S, β) > 0, S would necessarily be S 2 , if in addition β i ∈ (−1, 0), then S admits a conic metric of positive constant curvature if and only if : • n = 2, β 1 = β 2 ; • n ≥ 3, χ(S, β) < min{2, 2 + 2 min i β i }. Note that surfaces in the first class are often called (American) footballs, and they belong to the critical case. In [CL2], it has been shown a conic metric of positive constant curvature on S 2 in the critical case is necessarily a football. For the second class, the sufficiency is proved by Troyanov [Tr], the necessity and uniqueness argument is due to Luo-Tian [LT]. Recently, there is renewed interest on metrics with conic singularities in the study of Kähler geometry, namely the Kähler-Einstein metrics with cone singularities along a divisor [D, JMR]. Conic metrics of constant curvature on Riemann surfaces are just one-dimensional examples of Kähler-Einstein metrics with cone singularities along a divisor. In higher dimension, for the existence of Kähler-Einstein metrics with cone singularities, Troyanov's condition can be generalized to the coercivity of twisted Mabuchi K-energy functional [JMR], which can be reinterpreted as the pair (S, β) being logarithmically K-stable [RT]. The smooth version of this connection between algebraic stability and existence of Kahler-Einstein metrics on Fano manifolds is the essence of the recently solved Yau-Tian-Donaldson conjecture [CDS1,CDS2,CDS3,Ti]. From this point of view, Troyanov's classification can be understood as the stability conditions: the subcritical pairs and critical pairs can be viewed as being stable and semi-stable respectively; while the supercritical pairs are considered being unstable. While there exist metrics with constant curvature on stable and semi-stable pairs; the canonical metric on unstable pairs should be conical gradient shrinking Ricci solitons instead. Ricci flow method has been considered for conical surfaces [Y1, Y2]. Yin has proved long time existence as well as the convergence of the flow when χ(S, β) ≤ 0. Another approach was given by Mazzeo, Rubinstein and Sesum [MRS] using extensive machinery of polyhomogeneous expansions. They have also studied flows which have cone angles varying in a prescribed way. In a recent preprint [PSSW], the Ricci flow on conic spheres is shown to converge in stable, semi-stable and unstable cases. In particular, it was shown in [PSSW] that conic metrics on sphere with n (n ≥ 3) conic points in the semi-stable case converges in Gromov-Hausdorff topology to a football along the Ricci flow. In this note, we investigate from another perspective. We consider the moduli space of all conic spheres with constant curvature. According to numbers of the conic points on the sphere, the moduli has complicated topology with components with varying dimensions. Algebraically, points in the moduli can be separated as stable and semistable points. We show that any sequence of spheres of positive constant curvature with n (n ≥ 3) conic points passing from stable case to semi-stable case converges to a football in Gromov-Hausdorff topology. Geometrically, all but one conic points will merge into a single conic point of the limit football. Following method of Luo-Tian [LT], we have the following Theorem 1.1. Let (S 2 , g l , β (l) ) be a sequence of Riemann spheres with conic metric of positive constant curvature 1. Suppose the order of conic points β (l) = (β (l) 1 , · · · β (l) n ) ∈ R n (n ≥ 3) converges to β (∞) = (β (∞) 1 , · · · , β (∞) n ) which is in the critical case. Then (S 2 , ds 2 (l) ) converges in the Gromov-Hausdorff topology to (S 2 , g, β), the unique football of constant curvature 1 with β = min i β (∞) i p + min i β (∞) i q. Moreover, suppose β (l) 1 ≤ · · · ≤ β (l) n , then the corresponding conic points converge in the following fashion: lim l→∞ p (l) 1 = p and lim l→∞ p (l) i = q, for i ≥ 2. To understand this convergence phenomenon in an analytical way, we investigate the problem in the conformal geometrical setting. Let g 0 be the standard Euclidean metric. Under the stereographic projection, a conformal metric g = e 2u g 0 is of constant curvature 1 and represents (S 2 , g, β = i β i p i ) if and only if u satisfies the equation ∆u = −e 2u , z ∈ C \ {z 1 , · · · , z n }. (1.1) The asymptotic behavior of u near z i is: • u ∼ β i ln |z − z i | as z → z i ; • u ∼ −2 ln |z| as |z| → ∞. Notice that u is uniquely associated to a conic metric only up to a conformal normalization. Our main result is the following Theorem 1.2. Let u l be functions on C * representing g (l) given in Theorem 1.1. Under proper normalization, there exist two distinct points p, q ∈ C * such that a subsequence of u l converges to u ∞ on any compact set K ⊂ C * \ {p, q}. Furthermore, e 2u∞ g 0 on C * − {p, q} represents a football. The standard stereographic projection gives a unique correspondence of metrics on sphere and a conformal factor function on C * up to a Möbius transformation. This large conformal transformation group poses analytical difficulty, namely a proper conformal gauge must be chosen with care so that the resulting limit for the conformal factors exists and is non-trivial. This requires precise choices of a shifting factor and a scaling factor for each configuration that we consider. Our main approach is to explore the rotational symmetry of the football solution. This was examined in earlier works (e.g. [CL2]) when a single manifold with conic singularies is considered. Especially, level sets of the conformal factor are considered and isoperimetric inequalities play crucial roles. For our set-up, we follow a similar but more delicate approach. We would like to consider the level set of each conformal factor u l and analyze the isoperimetric defect more carefully. Consequently, our choice of normalization is also connected to the level sets. See Section 3 for more details. While the method applied in this article heavily relies on the rotational symmetry of football solutions, it is a perfect example highlighting the equivalence of different convergence concepts on the moduli space of constant curvature metrics with conic singularities. Several set-up for higher dimensions can be considered from both Kähler geometry and conformal geometry points of views. From a more analytical prospect, it would also be interesting to understand the convergence described in Theorem 1.2 in more details. By Theorem 1.2, all but one singular points will merge into one end of the football solution. It would be interesting to understand the blow-up limit of the sequence near this end. Flat metrics with conical singularities are to be expected while exact formation is unknown. Similar problems can also be considered for the Ricci flow limit. This paper is organized as the following: In Section 2, we provide a proof of Theorem 1.1 following [LT]. In Section 3, we prove Theorem 1.2. Acknowledgements: Both authors would like to thank Jian Song and Lihe Wang for discussion. Part of the work was done when both authors were visiting Beijing International Center for Mathematical Research. We are thankful for its hospitality. Proof of Theorem 1.1 We adopt the geometric setting of [LT] and notations therein. By a theorem of Alexandrov, each spherical conic metric of constant curvature 1 is isometric to the boundary of a convex polytope in S 3 . There are two degenerate cases: one is the metric doubling of a "lens", which is a degenerate spherical triangle with length of three sides being π, π, 0; the other one is the metric doubling of a usual spherical triangle. Clearly the former one corresponds to a football (2-conic points) and the latter one corresponds to a sphere with three conic points. For a convex polytope of n vertices, we denote its angles at vertices by (α 1 , · · · α n ) with each α i ∈ (0, 2π). Let P n be the space of all boundaries of labeled n-vertex convex polytopes in S 3 modulo isometry, with the topology induced by the Hausdorff metric. For each convex polytope P , construct a totally geodesic triangulation, then there are exactly 3(n−2) edges and 2(n−2) triangles. Variation of the length of each edge gives rise to distinct convex polytopes (up to isometry). Therefore, the dimension of P n is 3(n−2). Meanwhile, denote the conformal structure of n-labeled Riemannian sphere by M n . Since Möbius transformations are 3-transitive, it follows that dim M n = 2(n − 3). In [LT], Luo-Tian show that there admits a conic metric on S 2 of positive constant curvature representing β = k i=1 β i p i , (k ≥ 3), if and only if the corresponding cone angles α i = 2π(1 + β i ) satisfy n i=1 α i > 2(n − 2)π, n i=1 α i < 2(n − 2)π + 2 min i α i . (2.1) This condition is exactly same as the subcritical condition of Troyanov. It defines a convex open set in R n , which we denote by A. The critical case corresponds to the equality n i=1 α i = 2π(n − 2) + 2 min i α i . In addition, Luo-Tian [LT] have proved the following Theorem 2.1 (Luo-Tian). The map Π : P n → M n × A, (2.2) P → (conformal structure of P , angles of P at vertices) is a homeomorphism. We are now ready to give a proof of Theorem 1.1. Proof of Theorem 1.1. As mentioned above, each (S 2 , g l , β (l) ) is isometric to the boundary of a convex polytope P (l) in S 3 , and it satisfies the subcritical condition (2.1). By compactness of compact sets in S 3 with respect to the Hausdorff metric, we may assume a subsequence, still denoted by P (l) , converges to a convex polytope P (∞) . P (∞) represents a conic sphere of positive constant curvature, which is either in the subcritical case or the critical case. For the latter, P (∞) is necessarily the so-called lens or lune, which is defined as a region on the 2-sphere bounded between two geodesics lines connecting two antipodal points. By Theorem 2.1, if P (∞) does not degenerate to a lens, then β (∞) = lim l→∞ β (l) = lim l→∞ Π 2 (P (l) ) = Π 2 (P (∞) ) must be in the subcritical case as well, a contradiction. Thus, we conclude that P (∞) is a lens. Thus to prove our result, we are left to show that the conic angle of the corresponding football is 2π(min i β (∞) i + 1). Let α (l) i = 2π(β (l) i + 1) be angles of P (l) , with corresponding vertices denoted by V (l) i , i = 1, 2, · · · , n. It was shown [LT] under this situation that there exists a k such that lim l→∞ d(V (l) k , V (l) i ) > 0, for i = k, and lim l→∞ d(V (l) i , V (l) j ) = 0, for i, j = k. Now consider the triangulation of P (l) which consists of 2(n − 2) triangles {∆ i,(l) } 2(n−2) i=1 . For each ∆ i,(l) we denote its inner angles as α i,(l) 1 , α i,(l) 2 , α i,(l) 3 . For simplicity, if ∆ i,(l) is incident to V (l) k , assume that α i,(l) 1 is the angle at the point V (l) k . We denote, for k = 1, 2, 3 and i = 1, 2, · · · , 2(n − 2), α i,(∞) k = lim l→∞ α i,(l) k . When l is converging to ∞, we have the following two cases. If ∆ i,(l) is incident to V (l) k , we conclude that ∆ i,(l) converges to a lens whose angles satisfy α i,(∞) 1 = α i,(∞) 2 + α i,(∞) 3 − π. (2.3) If ∆ j,(l) is not incident to V (l) k , all its three vertices merge in the limit. Hence its inner angles has the following limit behavior α j,(∞) 1 + α j,(∞) 2 + α j,(∞) 3 = π (2.4) Summing relations (2.3) and (2.4) for all 2(n − 2) triangles, we have α (∞) k + 2(n − 2)π = i =k α (∞) i . Comparing with the critical condition, it follows that α (∞) k = min i α (∞) i . We have thus finished the proof. Proof of Theorem 1.2 In this section, we give a proof of Theorem 1.2. Under the stereographic projection, the conformal factors of positive constant curvature metrics are solutions to a semi-linear elliptic equation in the complex plane C. We shall study the limit behavior of these solutions. More precisely, given an (S 2 , g, β), under the stereographic projection we assume z i 's are the corresponding projection of p i in the complex plane. Let g 0 be the standard Euclidean metric, then 4 (1+|z| 2 ) 2 g 0 is the standard metric on S 2 . A conic metric g = e 2u g 0 is of constant curvature 1 representing β = n i=1 β i p i if and only if u satisfies the equation ∆u = −e 2u , z ∈ C \ {z 1 , · · · , z n }. (3.1) The asymptotic behavior of u near z i is: • u ∼ β i ln |z − z i | as z → z i ; • u ∼ −2 ln |z| as |z| → ∞. Let v = u − n i=1 β i ln |z − z i |, then v is bounded near each singular point z i and satisfies ∆v = −e 2v Π n i=1 |z − z i | 2β i . (3.2) Remark 3.1. One can equally work out this receipt in the sense of complex geometry. For a given (S 2 , g, β) with β = n i=1 β i p i , choose the background metric to be the Kähler metric ω = (1 + n i=1 β i 2 )ω F S ∈ 2π(1 + n i=1 β i 2 )c 1 (S 2 ), where ω F S is the standard Fubini-Study metric on S 2 . Let s i be the defining section of the line bundle [2p i ] = −K S 2 , let h be the hermitian metric on −K S 2 such that its curvature form Θ h = ω F S . A conic metric ω ϕ on S 2 with constant curvature 1 represents the divisor β if and only if ϕ satisfies (ω + ∂∂ϕ) = e −ϕ Π n i=1 |s i | β i h ω,(3.3) or equivalently Ric(ω ϕ ) = ω ϕ − 2π n i=1 β i [p i ],(3.4) where [p i ] is current of integration or Dirac-measure at p i . Note that (3.2) is equivalent to (3.3). Indeed, the defining section s i on the complex plane is (z − z i ) 2 , and the conformal change of metric amounts to scale the hermitian metric h on −K S 2 by the same conformal factor. Remark 3.2. For 0 < α < 1, notice that z → z α maps C * to a football with conic angles 2πα at 0 and ∞. By the standard steorgraphic projection, this constant scalar curvature metric on this football is represented as e 2uα g 0 , where e 2uα = 4α 2 |z| 2α−2 (1 + |z| 2α ) 2 . (3.5) Recall we are given a sequence of conic metrics of constant curvature 1 on S 2 representing β (l) = (β For each l, by the conformal description above, and assuming that we fix z (l) 1 at ∞, we have g l = e 2u l g 0 where u l is the solution of ∆u l = −e 2u l in C \ {z (l) 2 , · · · , z (l) n } ,(3.6) subject to the asymptotic behavior • u l ∼ β (l) i ln |z − z (l) i | as z → z (l) i for i = 2, · · · , n; • u l ∼ −(2 + β (l) 1 ) ln |z| as |z| → ∞. The difficulty of the conformal geometry on sphere lies in the fact there exists a large conformal transformation group. In our set-up, this indicates u l in (3.6) is not unique. In particular, for scaling u λ,0 l (z) := u l (λz) + ln λ; ( †) translation u 0,κ (z) := u(z − κ),( ‡) e 2u λ g 0 and e 2u κ g 0 all represent the same conic metric on the punctured sphere as e 2u g 0 . To clearly state the normalization we shall choose, we first present the main tools of the proof: to study the level set of u l and apply the isoperimetric inequality. While these ideas have been explored before (cf. [CL2]), our problem required more delicate analysis. We would examine the defect of isoperimetric inequality carefully under the limit procedure. With the help of the proper normalizations we prove the Hausdorff convergence of level sets. This convergence leads to a uniform bound of u l on compact sets, which allows to extract a limit function u ∞ . Then the isoperimetric inequality is applied again to prove that u ∞ must be radially symmetric about some point z 0 . For each u l , let Ω u l (t) := {u l > t}, A u l (t) := Ω l (t) e 2u l and B u l (t) = Ω l (t) 1 = |Ω l (t)|. Thus A u l is monotone decreasing and the Gauss-Bonnet formula gives the following R 2 e 2u l = 2π(2 + β (l) 1 + · · · + β (l) n ). (3.7) It follows that lim t→−∞ A u l (t) = 2π(2 + β (l) 1 + · · · + β (l) n ) and lim t→∞ A u l (t) = 0. Under the scaling and translation, we have A u λ,κ l (t) = A u l l (t − ln λ). We can now state our normalization for all u l . For each l, we pick a suitable λ l and κ l such that A u λ l ,κ l (ln(1 + β (∞) 1 )) = π(2 + β (l) 1 + · · · + β (l) n ) = 1 2 A u λ l ,κ l (−∞). (3.8) The centroid of Ω u λ l ,κ l (ln(1 + β (∞) 1 )) is at 0. (3.9) From now on, without confusion we write u l for u λ l ,κ l . For simplicity, we denote Ω l (t) = Ω u l (t), A l (t) = A u l (t), B l (t) = B u l (t). We can thus restate Theorem 1.2 into the following Theorem 3.3. For a sequence of functions {u l } satisfying (3.6), assume that A l (ln(1 + β (∞) 1 )) = π(2 + β (l) 1 + · · · + β (l) n ) = 1 2 A l (−∞), (3.10) The centroid of Ω l (t * ) is at 0, (3.11) where t * ∈ R is a fixed generic point, then u l sub-converges to u ∞ in C ∞ loc (C \ {0}), where u ∞ is given by e 2u∞(z) = 4(1 + β (∞) 1 ) 2 |z| 2β (∞) 1 (1 + |z| 2+2β (∞) 1 ) 2 . (3.12) Moreover, lim l→∞ z (l) i = 0, for i ≥ 2. In view of (3.5), Theorem 1.1 is an immediate consequence of Theorem 3.3. In the rest of this section, we present a proof of Theorem 3.3. Proof. we take a careful look of Ω l (t). In view of asymptotic behavior of u l , Ω l (t) is a bounded region for each t, and A l (t) = Ω l (t) e 2u l = Ω l (t) −∆u l = ∂Ω l (t) |∇u l | + 2π(β (l) 2 + · · · + β (l) n ). (3.13) In general, there are multiple connected components for Ω l (t). For a regular value t of u l , Ω l (t) consists of finitely many disjoint regions bounded by Jordan curves. Each component is simply connected due to the maximum principle. We now present the following estimate relating the size of level sets and the upper bound of the function. l = max Γ l (t 0 ) {u l }. For t ∈ [t 0 , H l ], let Γ l (t) = Ω l (t) ∩ Γ l (t 0 ), a l (t) = Γ l (t) e 2u l and b l (t) = |Γ l (t)|, then we have a l (t) ≥ 4π(1 − e t−H l ). (3.14) Furthermore, for a l (t) ≤ 2π, we have b l (t) ≥ 4πe −H l (e −t − e −H l ). (3.15) Proof. Since Γ l (t 0 ) does not contain any singularity, a similar computation of (3.13) shows a l (t) = Γ l (t) e 2u l = Γ l (t) −∆u l = ∂Γ l (t) |∇u l |, t ∈ [t 0 , H l ], (3.16) and a l (H l ) = 0. By the co-area formula, we have a ′ l (t) = −e 2t ∂Γ l (t) 1 |∇u| , (3.17) and b ′ l (t) = − ∂Γ l (t) 1 |∇u| . (3.18) Hence (a l (t) 2 ) ′ = 2a l (t)a ′ l (t) = −2e 2t ∂Γ l (t) |∇u l | ∂Γ l (t) 1 |∇u l | (3.19) ≤ −2e 2t ( ∂Γ l (t) 1) 2 ≤ −8πe 2t |Γ l (t)| = −8πe 2t b l (t). Here we have used the Hölder's inequality and isoperimetric inequality for Γ l (t): ∂Γ l (t) |∇u l | ∂Γ l (t) 1 |∇u l | ≥ |∂Γ l (t)| 2 ≥ 4π|Γ l (t)|. (3.20) By Fubini's theorem, we also have a l (t) = Γ l (t) e 2u l = ∞ 0 |e 2u l > λ|dλ (3.21) = H −∞ |u > t|2e 2t dt = e 2t b l (t) + H t 2e 2t b l (t)dt. Integrating (3.19) from t to H l and using (3.21), we obtain − a l (t) 2 ≤ −4πa l (t) + 4πe 2t b l (t). (3.14) then follows from (3.23) and the fact that a l (H) = 0. When a l ≤ 2π, (3.15) is thus a consequence of (3.14) and (3.22). Lemma 3.5. Let A ∞ (t) := 4π(1 + β (∞) 1 ) ρ 2+2β (∞) 1 1 + ρ 2+2β (∞) 1 , (3.24) where ρ is determined by e 2t = 4(1 + β (∞) 1 ) 2 ρ 2β (∞) 1 (1+ρ 2+2β (∞) 1 ) 2 , then under the normalization (3.8), we have Proof. We now run a similar argument for level sets including singular points. Let f l (t) := A 2 l (t) − (4π + 4π(β (l) 2 + · · · + β (l) n ))A l (t) + 4πe 2t B l (t). (3.25) Then f ′ l (t) = − 2e 2t Ω l (t) e 2u l ∂Ω l (t) 1 |∇u l | + (4π + 4π(β (l) 2 + · · · + β (l) n ))e 2t ∂Ω l (t) 1 |∇u l | + 8πe 2t B l − 4πe 2t ∂Ω l (t) 1 |∇u l | = − 2e 2t ( ∂Ω l (t) |∇u l | + 2π(β (l) 2 + · · · + β (l) n )) ∂Ω l (t) 1 |∇u l | + (4π + 4π(β (l) 2 + · · · + β (l) n ))e 2t ∂Ω l (t) 1 |∇u l | + 8πe 2t B l − 4πe 2t ∂Ω l (t) 1 |∇u l | =2e 2t (4πB t − ∂Ω l (t) |∇u l | ∂Ω l (t) 1 |∇u l | ) ≤ 0. (3.26) Since R 2 e 2u l dx = ∞ −∞ 2e 2t B l (t)dt < ∞, it follows e 2t B l (t) → 0, as t → ±∞. Let C l = 2π(2 + n i=1 β (l) i ) = lim t→−∞ A l (t), we obtain 0 ≤ f l (t) ≤ C 2 l − (4π + 4π(β (l) 2 + · · · + β (l) n ))C l . lim l→∞ β (l) = β (∞) implies lim l→∞ C 2 l − (4π + 4π(β (l) 2 + · · · + β (l) n ))C l = 0, thus f l converges to 0 uniformly. Moreover f ′ l is integrable with ||f ′ l || L 1 = f l (−∞) − f l (∞) and lim l→∞ ||f ′ l || L 1 = 0. Combining (3.25) and (3.26) we find that A l satisfy A l A ′ l − (2π n i=2 β (l) i )A ′ l − (A 2 l − (4π + 4π n i=2 β (l) i )A l ) = 1 2 (f ′ l − 2f l ). (3.27) For simplicity, we denote n i=2 β (l) i by β (l) . Thus (3.27) can be rearranged as ( a A l + b 4π(1 + β (l) ) − A l )A ′ l = 1 + 1 2 f ′ l − f l A 2 l − 4π(1 + β (l) )A l , (3.28) where a = β (l) 2 β (l) +2 and b = − β (l) +2 2 β (l) +2 . Let t l = sup t {t| lim inf l 4π(1+ β (l) )−A l (t) = 0} and t r = inf t {t| lim inf l A l (t) = 0}. Due to (3.10), we have t l < ln(1 + β (∞) 1 ) < t r . It follows from the definition of t l and t r that 1 A 2 l −4π(1+ β (l) )A l is uniformly bounded on any finite closed interval [r, s] ⊂ (t l , t r ). Recalling that lim l→∞ ||f ′ l || L 1 = 0, thus f ′ l converges to 0 in L 1 and we also have f l converges uniformly to 0. Hence integrating (3.28) and taking the limit, we obtain lim l→∞ ln(A a l (4π(1 + β (l) ) − A l ) −b )| s r = lim l→∞ s r 1 + 1 2 f ′ l − f l A 2 l − 4π(1 + β (l) )A l dt = s − r (3.29) Notice that lim l→∞ A l (ln(1 + β (∞) 1 )) = 2π(1 + β (∞) 1 ), from this single point convergence and (3.29) we conclude that A l has a pointwise limit A ∞ on (t l , t r ), which satisfies ( a A ∞ + b 4π(1 + β (l) ) − A ∞ )A ′ ∞ = 1 (3.30) with A ∞ (ln(1 + β (∞) 1 )) = 2π(1 + β (∞) 1 ). By separation of variables, we have the solution of (3.30) A ∞ (t) = 4π(1 + β (∞) 1 ) ρ 2+2β (∞) 1 1 + ρ 2+2β (∞) 1 , (3.31) where ρ is such that e 2t = 4(1 + β (∞) 1 ) 2 ρ 2β (∞) 1 (1+ρ 2+2β (∞) 1 ) 2 . It is easy to see Combining (3.28) and (3.32), it is easy to see that (t l , t r ) = (−∞, ∞). A simple computation which we will skip here gives the corresponding result for B l (t). We now study the isoperimetric defect. Define, for any region Ω ⊂ R 2 with boundary a Jordan curve ∂Ω, the isoperimetric defect is D(Ω) := |∂Ω| 2 − 4π|Ω|. (3.33) It is easy to show that D(Ω) is super additive. This means, if Ω 1 and Ω 2 are two disjoint sets in R 2 , Ω = Ω 1 ∪ Ω 2 , we have D(Ω) ≥ D(Ω 1 ) + D(Ω 2 ). (3.34) Furthermore, we have the following Lemma 3.6 (Bonnesen's inequality). For a bounded region Ω ⊂ R 2 , let r and R be the radii of incircle and circumcircle of Ω, then D(Ω) ≥ π 2 (R − r) 2 ; The equality holds if and only if Ω is a round disk. In view of Bonnesen's inequality, we can prove Lemma 3.7. Let D l (t) := D(Ω l (t)) be the isoperimetric defect of the level set Ω l (t). Then there exists a subset V ⊂ R such that |R \ V | = 0 and after passing to a subsequence, lim l→∞ D l (t) = 0, ∀t ∈ V. (3.35) Proof. By (3.26) and (3.20), we have 2e 2t D l (t) ≤ 2e 2t ( ∂Ω l (t) |∇u l | ∂Ω l (t) 1 |∇u l | − 4π|Ω l (t)|) ≤ −f ′ l (t). (3.36) For each fixed t 0 , we then conclude 2e 2t 0 ∞ t 0 D l (t)dt ≤ ∞ t 0 2e 2t B l (t)dt ≤ ∞ t 0 −f ′ l (t)dt = f l (t 0 ). Since f l (t 0 ) → 0 as l → ∞, we have D l (t) converges to 0 in L 1 (t 0 ,∞) norm, thus after passing to a subsequence D l (t) converges to 0 almost everywhere for t ≥ t 0 . Repeating the same argument for a sequence of t i → −∞ and using a diagonal argument, it follows D l (t) converges to 0 almost everywhere on R. Thanks to Sard's theorem, after disregarding critical values for all of u l , we still get the convergence (3.35) almost everywhere. lim l→∞ D l (t 0 ) = 0 that a subsequence of Σ l (t 0 ) converges in Hausdorff distance to B r 0 (0) with r 0 = B∞(t 0 ) π . By (3.39), Σ l (t i ) ⊃ Σ l (t 0 ) ⊃ Σ l (t j ), for t i < t 0 < t j . It thus follows that for each fixed t i , the centroid of Σ l (t) is contained in a bounded set. Hence the conclusion follows. Proof of Theorem 3.3. By passing to a subsequence, we shall assume that lim l→∞ z l i = z ∞ l , i ≥ 2, where z ∞ l is possibly at ∞. Also let z 0 = ∩ i∈Z B r i (p i ) be the point given by Lemma 3.9. We will show that u l is uniformly bounded on any compact subset K ⊂ C \ {z ∞ 2 , · · · , z ∞ n , z 0 }. For any given ǫ > 0, we have thus a uniform L 1 ∈ N such that for any l > L 1 , d(z l i , K) < ǫ. Clearly, for such a compact set K, there exist r i > r j such that K ⊂ B r i (p i ) \ B r j (p j ). Hence for a δ > 0 small enough, we have K ⊂ N δ (B r i (p i )) \ N δ (B r j (p j )), where N δ (·) stands for the δ-neighborhood. By Lemma 3.9, there exists a L 2 > 0 such that for l > L 2 , K ⊂ Σ l (t j ) \ Σ l (t i ). It follows that u l (x) ≥ t j , for x ∈ K and l > L 2 . It remains to show a uniform upper bound for u l . By Lemma 3.8, for the chosen ǫ, there exists a L 3 ∈ N such that for all l > L 3 , any connected component Ω of Ω l (t i ) \ Σ l (t i ) satisfies |Ω| ≤ ǫ/2, |∂Ω| ≤ ǫ/2. Now for l > max{L 1 , L 2 , L 3 } large enough, any component of Ω l (t i ) containing any singular point will not intersect K. Thus, if Σ ′ is a connected component of Ω l (t i ) such that K ∩ Σ ′ = ∅, it contains no singular point. By Lemma 3.4, we conclude that max Σ ′ u l ≤ − ln(1 − a l (t i ) 4π ) + t i . (3.40) Since a l (t i ) ≤ C l and lim l→∞ C l = 4π(1 + β (∞) 1 ) < 4π, there exists an L 4 ∈ N such that for all l > L 4 , C l ≤ 4π + πβ (∞) 1 < 4π. (3.41) Combining (3.40) and (3.41), we thus get a uniform upper bound for u l (z) for all z ∈ K and l > max{L 1 , L 2 , L 3 , L 4 }. In summary, a subsequence of {u l } is uniformly bounded in any compact subset K ⊂ C \ {z ∞ 1 , · · · , z ∞ n , z 0 }. In particular, by the standard L p estimates of the Poisson equation (3.6), we have ||u l || W 2,p (K) ≤ C, ∀p > 0. For p > n, we apply Sobolev's embedding and classical Schauder's estimate to get ||u l || C ∞ (K) ≤ C. Let u ∞ := lim l→∞ u l , then u ∞ satisfies ∆u ∞ (z) = −e 2u∞(z) , for z ∈ C \ {z ∞ 2 , · · · , z ∞ n , z 0 }. It follows the corresponding A and B for u ∞ are just A ∞ and B ∞ . In particular, any level set of u ∞ has vanishing isoperimetric deficit, which means each level set must be a circle. In addition, (3.19) being identity shows that |∇u ∞ | are constants on the round circles {x; u ∞ (x) = c}. Hence u ∞ has to be radially symmetric with center z 0 , which in turn has to be the origin by our normalization (3.11). Since u ∞ is unbounded in view of B ∞ , z 0 = 0 has to be a singular point of u ∞ . Henceforth, u ∞ satisfies ∆u ∞ (z) = −e 2u∞(z) , for z ∈ C \ 0. (3.42) All solutions to (3.42) are classified in [CL2]. By direct computation, having A ∞ and B ∞ match with u ∞ 's, u ∞ is necessarily given by (3.12). Finally, we show that lim l→∞ z (l) i = z 0 , for i ≥ 2. (3.43) Suppose on the contrary, there are some singular points going to ∞, then there exists L and T , such that for all l ≥ L, t ≥ T , Σ l (t) contains only parts of singular points, say z (l) i 1 , · · · , z (l) i k . Let β (l) = β (l) i 1 + · · · + β (l) i k . Applying the analysis of Lemma 3.5 only for the quantity a l (t) := Σ l (t) e 2u l , we would have a l a ′ l − (2πβ (l) )a ′ l − (a 2 l − (4π + 4πβ (l) )a l ) ≤ 0 (3.44) hold for t ≥ T and l ≥ L. Sinceβ (l) >β (l) , we deduce, by direct computation, that lim sup l→∞ a l (t) < A ∞ (t), t ≥ T. While for any other component Σ ′ of Ω l (t), since |Σ ′ | → 0 and e 2u l is uniformly integrable, we get lim l→∞ Σ ′ e 2u l = 0. Hence the total contribution to A l (t) does not converge to A ∞ (t), a contradiction. , and lim l→∞ β (l) = β (∞) , where β (∞) is in the critical case. Without loss of generality, we may assume min i β Lemma 3 . 4 . 34For a fixed t 0 ∈ R, let Γ l (t 0 ) be a connected component of Ω l (t 0 ) does not contain any singular point of u l and H lim l→∞ A l (t) = A ∞ (t), ∀t ∈ R,and lim l→∞ B l (t) = B ∞ (t) = πρ 2 . A ∞ (t) = 4π(1 + β (∞) 1) and lim t→∞ A ∞ (t) = 0.(3.32) Lemma 3.8. For each t ∈ V , where V is given as in Lemma 3.7, let Σ l (t) be the connected component of Ω l (t) with largest area. ThenProof. We prove by contradiction. If for Ω 1 l (t) = Σ l (t) and Ω 2Hence we conclude that there is exactly one component whose area tends to B ∞ (t), which we denote by Σ l (t). Moreover, both the area and boundary length of the remaining components must go to zero. Now take a monotone sequence {t i } i∈Z ⊂ V , where V is obtained in Lemma 3.7. By Lemma 3.5 and Lemma 3.8, for any 0 < λ << 1/2, there exists a positive integer L i such that for all l > L i ,Such Σ l (t) is thus unique. By a diagonal argument, we may pick a subsequence of u l (which we still call u l ) and assume that (3.38) holds for all l. Notice that for t i > t j , we haveWe now explain our choice of t * in (3.11). Without loss of generality, let t * = t 0 ∈ V . We have thus the following Lemma 3.9. There exists a sequence of descending ballswith lim i→−∞ r i = ∞ and lim i→∞ r i = 0, such that Σ l (t i ) converges in Hausdorff distance to B r i (p i ).Proof. Without loss of generality, assume t * = t 0 . By our choice of normalization, the centroid of Ω l (t 0 ) is the origin. It thus follows from Supercritical conformal metrics on surfaces with conical singularities. D Bartolucci, F De Marchis, A Malchiodi, Int. Math. Res. Not. IMRN. 24D.Bartolucci, F. De Marchis and A. Malchiodi, Supercritical conformal met- rics on surfaces with conical singularities, Int. Math. Res. Not. IMRN 2011, no. 24, 5625-5643. Kḧler-Einstein metrics on Fano manifolds I: Approximation of metrics with cone singularities. X Chen, S Donaldson, S Song, J. Amer. Math. Soc. 281X. Chen, S. Donaldson and S. Song, Kḧler-Einstein metrics on Fano mani- folds I: Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015), no. 1, 183C197. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. X Chen, S Donaldson, S Song, J. Amer. Math. Soc. 281X. Chen, S. Donaldson and S. Song, Kähler-Einstein metrics on Fano man- ifolds. II: Limits with cone angle less than 2π, J. Amer. Math. Soc. 28 (2015), no. 1, 199C234. III: Limits as cone angle approaches 2π and completion of the main proof. Kähler-Einstein metrics on Fano manifolds. 28Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle ap- proaches 2π and completion of the main proof, J. Amer. Math. Soc. 28 (2015), no. 1, 235C278. Prescribing Gaussian curvatures on surfaces with conical singularities. W Chen, C Li, J. Geom. Anal. 14W. Chen, and C. 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Rubinstein, Kähler-Einstein metrics with edge singularities with an appendix by C. Li and Y. Rubinstein, arXiv:1105.5216. Explicit construction of extremal Hermitian metrics with with finite conical singularities on S 2. C S Lin, X Zhu, Communications in Analysis and Geometry. 101C.S. Lin, and X. Zhu, Explicit construction of extremal Hermitian metrics with with finite conical singularities on S 2 , Communications in Analysis and Geometry 10, no. 1 (2002), 177-216. Lioville equation and spherical convex polytopes. F Luo, G Tian, Proc. Amer. Math. Soc. 1164F. Luo and G. Tian, Lioville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116(1992), no.4, 1119-1129. R Mazzeo, Y Rubinstein, N Sesum, arXiv:1306:6688Ricci flow on surfaces with conic singularities. R. Mazzeo, Y. Rubinstein and N. Sesum, Ricci flow on surfaces with conic singularities, arXiv:1306:6688. De l'intégration de l'équation ∆u = e u sur une surface de Riemann fermée. E Picard, Crell's J. 130E. 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{'abstract': "We show that spheres of positive constant curvature with n (n ≥ 3) conic points converge to a sphere of positive constant curvature with two conic points, or an (American) football in Gromov-Hausdorff topology when the conic angles of the sequence pass from the subcritical case in the sense of Troyanov to the critical case in the limit.We prove this convergence in two different ways. Geometrically, the convergence follows from Luo-Tian's explicit description of conic spheres as boundaries of convex polytopes in S 3 . Analytically, considering the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.H.F.'s work is partially supported by Simons Foundation and NSF DMS-100829.", 'arxivid': '1501.06881', 'author': ['Hao Fang ', 'Mijia Lai '], 'authoraffiliation': [], 'corpusid': 119333534, 'doi': '10.1007/s00208-015-1303-z', 'github_urls': [], 'n_tokens_mistral': 14076, 'n_tokens_neox': 12362, 'n_words': 7400, 'pdfsha': 'cf65427c106b48b376445e75294c2658c770d635', 'pdfurls': ['https://arxiv.org/pdf/1501.06881v2.pdf'], 'title': ['ON CONVERGENCE TO A FOOTBALL', 'ON CONVERGENCE TO A FOOTBALL'], 'venue': []}

arxiv

Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications 24 Dec 2017 Yunxiao Chen Department of Psychology School of Statistics Shanghai Center for Mathematical Sciences Emory University University of Minnesota Fudan University Xiaoou Li Department of Psychology School of Statistics Shanghai Center for Mathematical Sciences Emory University University of Minnesota Fudan University Siliang Zhang Department of Psychology School of Statistics Shanghai Center for Mathematical Sciences Emory University University of Minnesota Fudan University Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications 24 Dec 2017arXiv:1712.08966v1 [stat.ME] Latent factor models are widely used to measure unobserved latent traits in social and behavioral sciences, including psychology, education, and marketing. When used in a confirmatory manner, design information is incorporated, yielding structured (confirmatory) latent factor models. Motivated by the applications of latent factor models to large-scale measurements which consist of many manifest variables (e.g. test items) and a large sample size, we study the properties of structured latent factor models under an asymptotic setting where both the number of manifest variables and the sample size grow to infinity. Specifically, under such an asymptotic regime, we provide a definition of the structural identifiability of the latent factors and establish necessary and sufficient conditions on the measurement design that ensure the structural identifiability under a general family of structured latent factor models. In addition, we propose an estimator that can consistently recover the latent factors under mild conditions. This estimator can be efficiently computed through parallel computing. Our results shed lights on the design of large-scale measurement and have important 1 implications on measurement validity. The properties of the proposed estimator are verified through simulation studies. KEY WORDS: High-dimensional latent factor model, confirmatory factor analysis, identifiability of latent factors, structured low-rank matrix, large-scale psychological measurement J rW,T s . Thus, pΘ˚A J˚Θ J q r1:N,W s " Θr T s A Jr W,T s´Θ rT s J rW,T s . That is, Θr T s A Jr W,T s´Θ rT s J rW,T s is a submatrix of Θ˚A J˚´ΘÂJ . Thus, we have the following bound }Θr T s A Jr W,T s´Θ rT s J rW,T s } 2 ď }Θ˚A J˚´ΘÂJ } 2 ďwhich implies that both Θr Ss and Ar s,rs have full rank, and RpΘr Ss q " RpU S q. By Weyl's perturbation theorem (see, e.g. Stewart and Sun (1990)), we have |σ |S| pΘ rSs J rR Q pSq,Ss q´σ |S| pΘr Ss A˚J rR Q pSq,Ss q| ď }Θ rSs J rR Q pSq,Ss´ΘrSs A˚J rR Q pSq,Ss } 2 .46Combining the above display with(39)and(40), we have σ |S| pΘ rSs J rR Q pSq,Ss q ě ? NJσ 2´}Θ rSs J rR Q pSq,Ss´ΘrSs A˚J rR Q pSq,Ss } 2 ą 0.Thus, we also haveΘ rSs and rs,rs have full rank. We writeΘ rSs J rs,rs "Û SΣSV J S , the reduced singular value decomposition. Then, RpΘ rSs q " RpÛ S q. By the modified Davis- Introduction Multivariate data analysis (Anderson, 2003) has been one of the most important topics in modern statistics that is widely encountered in different disciplines including psychology, education, marketing, economics, medicine, engineering, geography, and biology. Latent factor models, originated from the seminal work of Spearman (1904) on human intelligence, play a key role in multivariate data analysis (Anderson, 2003;Skrondal and Rabe-Hesketh, 2004; Bartholomew et al., 2008Bartholomew et al., , 2011. Latent factor models capture and interpret the common dependence among multiple manifest variables through the use of low-dimensional latent factors. A key application of latent factor models is in psychological measurement, where the latent factors are interpreted as the psychological traits (e.g., cognitive abilities, personality traits, and psychopathic traits) that are not directly observable. By modeling the relationship between the latent factors and the manifest variables, the ultimate goal of latent factor analysis in psychological measurement is to make statistical inference on the individual specific latent traits. Specifically, after fitting a latent factor model, each individual will be assigned latent factor scores, as an estimate of his/her levels on the corresponding psychological traits. Decision will be made based on the factor scores, such as deciding whether a student has mastered a certain skill, ranking the students according to their proficiency levels on a skill, describing the personality profile of an individual, and diagnosing whether a patient suffers from a certain mental health disorder. Due to the confirmatory nature of psychological measurement, item design information, i.e. how each item is associated with latent traits, is often available a priori and used in fitting a latent factor model. Such information is incorporated into the model through constraints on the model parameters, resulting in a structured latent factor model. A good item design is key to measurement validity, guaranteeing that the model-based measurement reflects what it is supposed to measure (Wainer and Braun, 2013). This paper develops statistical theory and methods for the design and the scoring in psychological measurement, which are the central problems to the psychological measurement theory (AERA et al., 2014). Motivated by large-scale assessments, we adopt an asymptotic regime in which the sample size N and the number of items J grow to infinity. Under this regime, we provide insights into the design of measurement based on our theoretical development on the structural identifiability of latent factors, a concept proposed in this paper that is central to large-scale measurement. This notion of identifiability is different from the classic definition of identifiability and may shed lights on the identifiability of infinitedimensional models for non-i.i.d. data that are widely encountered in modern statistics, including in network analysis and spatial-temporo statistics. Moreover, necessary and sufficient conditions are established for the structural identifiability of a latent factor, which formalizes the intuition a latent factor can be identified when it is measured by sufficiently many items that distinguish it from the other factors. This result explains the reason why the "simple structure" design is popular in psychological measurement (Cattell, 2012). Our asymptotic results also provide theoretical guarantee to the use of estimated factor scores in making decisions (e.g. classification and ranking) when the corresponding latent factors are structurally identifiable. The rest of the paper is organized as follows. In Section 2, we introduce a generalized latent factor modeling framework, within which our research questions are formulated. In Section 3, we discuss the structural identifiability for latent factors, the relationship between structural identifiability and estimability, and provide an estimator that consistently estimates the structurally identifiable latent factors. Further implications of our theoretical results on large-scale measurement are provided in Section 4 and extensions of our results to more complex settings are discussed in Section 5. A new perturbation bound on linear subspaces is presented in Section 6 that is useful to statistical analysis of low-rank matrix estimation. Our theoretical results are verified by simulation studies in Section 7. Finally, concluding remarks are provided in Section 8. The proofs of all the technical results are provided in the supplement. Structured Latent Factor Analysis Generalized Latent Factor Model Consider that there are N individuals and J manifest variables (e.g. J test items). Let Y ij be a random variable denoting the ith individual's value on the jth manifest variable and let y ij be its realization. For example, in educational tests, Y ij s could be binary responses from the examinees, indicating whether the answers are correct or not. We further assumes that each individual i is associated with a K-dimensional latent vector, denoted as θ i " pθ i1 , ..., θ iK q J and each manifest variable j is associated with K parameters a j " pa j1 , ..., a jK q J . We give two concrete contexts. Consider an educational test of mathematics, with K " 3 dimensions of "algebra", "geometry", and "calculus". Then θ i1 , θ i2 , and θ i3 represent individual i's proficiency levels on algebra, geometry, and calculus, respectively. In the measurement of Big Five personality factors (Goldberg, 1993), K " 5 personality factors are considered, including "openness to experience", "conscientiousness", "extraversion", "agreeableness", and "neuroticism". Then θ i1 , ..., θ i5 represent individual i's levels on the continuums of the five personality traits. The manifest parameter a j s can be understood as the regression coefficients when regressing Y ij s on θ i s, i " 1, ..., N. The manifest parameter a j s are also known as the factor loadings in the factor analysis literature (e.g. Bartholomew et al., 2008) and the discrimination parameters in the item response theory literature (e.g. Embretson and Reise, 2000). In many applications of latent factor models, especially in psychology and education, the estimations of θ i s and a j s are both of interest (e.g. Bartholomew et al., 2008). Our development is under a generalized latent factor model framework (Skrondal and Rabe-Hesketh, 2004), which extends the generalized linear model framework (McCullagh and Nelder, 1989) to latent factor analysis. This general modeling framework allows for different types of manifest variables. Specifically, we assume that the distribution of Y ij given θ i and a j is a member of the exponential family with natural parameter m ij " a J j θ i " a j1 θ i1`¨¨¨ajK θ iK ,(1) and possibly a scale (i.e. dispersion) parameter φ. More precisely, the density/probability mass function takes the form: f py|a j , θ i , φq " expˆy m ij´b pm ij q φ`c py, φq˙, where bp¨q and cp¨q are pre-specified functions that depend on the member of the exponential family. Given θ i and a j , i " 1, ..., N and j " 1, ..., J, we assume all Y ij s are independent. Consequently, the likelihood function, in which θ i s and a j s are treated as fixed effects, can be written as Lpθ 1 , ..., θ N , a 1 , ..., a J , φq " N ź i"1 J ź j"1 expˆy ij m ij´b pm ij q φ`c py ij , φq˙.(2) This likelihood function is known as the joint likelihood function in the literature of latent variable models (Skrondal and Rabe-Hesketh, 2004). Since the likelihood function depends on a j and θ i only through m ij s, it has rotational and scaling indeterminacy. That is, the likelihood remains unchanged when we replace θ i and a j by Dθ i and pD´1q J a j , for all i and j, where D can be any invertible KˆK matrix. We remark that in the existing literature of latent factor models, there is often an intercept term indexed by j in the specification of (1), which can be easily realized under our formulation by constraining θ i1 " 1, for all i " 1, 2, ..., N. In that case, a j1 serves as the intercept term. This family of models contains special cases, such as the linear factor analysis model (e.g. Anderson, 2003;Bartholomew et al., 2008), multidimensional item response theory model for binary responses that plays a key role in educational assessment (e.g. Reckase, 2009), and the Poisson model that is widely used to analyze multivariate count (e.g. Moustaki and Knott, 2000). We list their forms below. 1. Linear Factor Analysis: Y ij |θ i , a j " Npa J j θ i , σ 2 q, where the scale parameter φ " σ 2 . 2. Multidimensional Item Response Theory (MIRT): Y ij |θ i , a j " Bernoulli´e xppa J j θ i q 1`exppa J j θ i q¯, where the scale parameter φ " 1. Poisson Factor Model: Y ij |θ i , a j " P oisson`exppa J j θ i q˘, where the scale parameter φ " 1. Usually, the likelihood function (2) is not used for maximum likelihood analysis. This is possibly because, under the conventional asymptotic setting that J is fixed and N grows to infinity, the number of parameters in (2) diverges due to the growing number of person parameters, resulting in inconsistent maximum likelihood estimation. This phenomenon is first pointed out in Neyman and Scott (1948) and is further investigated in subsequent developments, including Andersen (1970), Haberman (1977), Fischer (1981), andGhosh (1995). Consequently, in generalized latent factor analysis, the person parameters θ i s are typically assumed to be random effects (i.e., independent and identically distributed samples from a distribution) and are integrated out from the joint likelihood function (2), while the manifest parameter a j s are still regarded as fixed effects. The resulting likelihood function is known as the marginal likelihood. The analysis of this paper focuses on the joint likelihood function (2) where both θ i s and a j s are treated as fixed effects, under an asymptotic setting that both N and J grow to infinity. For ease of exposition, we assume the scale parameter is known in the rest of the paper, while pointing out that it is straightforward to extend all the results to the case where it is unknown. This asymptotic regime is motivated by large-scale assessments in psychology and education, where both the sample size and the number of manifest variables can be very large. Moreover, quantifying the estimation accuracy of θ i s, which is a main focus of latent factor analysis in psychological and educational measurement, is more straightforward and thus easier to interpret under the fix effect point of view. We point out that a similar asymptotic setting has been adopted in Haberman (1977) for the analysis of the Rasch model (Rasch, 1960), a simple unidimensional latent factor model that is widely used in educational measurement. Confirmatory Structure In this paper, we consider a confirmatory setting where domain knowledge is available for the manifest variables. For example, in a personality assessment in psychology, what personality factor each item measures is pre-specified. For instance, the item "I am the life of the party" measures trait "extraversion" and the item "I get stressed out easily" measures trait "neuroticism" in a Big Five personality test. In an educational assessment, the item design is also pre-specified in the test blueprint. For example, one item may measure both algebra and geometry and another may measure calculus solely. Such information is typically reflected by constraints on the manifest parameters a jk s. Specifically, for each item j, there is a prespecified vector q j " pq j1 , ..., q jK q, where q jk " 1 means that latent factor k is measured by manifest variable j and thus no constraint is imposed on a jk , and q jk " 0 implies that latent factor k is independent with manifest variable j and thus a jk is set to 0. Intuitively, a good design leads to superior measurement results. This intuition is formalized in this paper through asymptotic analysis which establishes a relationship between the design information given by q j s and the structural identifiability of the latent factors that is defined in Section 3. Research Questions This paper focuses on the identifiability of the latent factors. To define the identifiability, consider a population of people where N " 8 and a population of manifest variables where J " 8. A latent factor k is a hypothetical construct, defined by the person population. More precisely, it is determined by the individual latent factor scores of the entire person population, denoted by pθ1 k , θ2 k ...q P R Z`, where θi k denotes the true latent factor score of person i on latent factor k and R Z`d enotes the set of vectors with countably infinite real number components. The identifiability of the kth latent factor then is equivalent to the identifiability of a vector in R Z`u nder the distribution of an infinite dimensional random matrix, tY ij : i " 1, 2, ..., j " 1, 2, ...u. The above setting is natural in the context of large-scale measurement, but is a nonstan- In other words, we want to know whether and to what extend the scores of an identifiable latent factor can be recovered from data. The identifiability of latent factor models is an important problem in statistics. Research on this topic dates back to Anderson and Rubin (1956) and has received much attention by statisticians under both low-and high-dimensional settings (e.g. Anderson, 1984;Bai et al., 2012). To the best of our knowledge, this is the first work characterizing the relationship between measurement design information (reflected by constraints) and the identifiability of latent factor models under a high-dimensional setting. In addition, our developments apply to a general model class. Both the incorporation of design information and the general model form make our problem technically challenging, involving the asymptotic analysis of a non-convex optimization problem. As will be shown in the rest of the paper, we tackle these challenges by proving useful probabilistic error bounds and by developing perturbation bounds on the intersection of linear subspaces. Preliminaries In this section, we fix some notations used throughout this paper. Notations. a. Z`: the set of all positive integers. sin =pu, vq " sgnpuv J q d 1´p uv J q 2 }u} 2 }v} 2 , where u, v P R m , u, v ‰ 0, and the function sgnpxq takes value 1, 0, and´1 when x is positive, zero, and negative, respectively. p. }W } F : the Frobenius norm of a matrix W " pw ij q mˆn , }W } F fi b ř m i"1 ř n j"1 w 2 ij . q. }W } 2 : the spectral norm of matrix W , i.e., the largest singular value of matrix. r. σ 1 pW q ě σ 2 pW q ě ... ě σ n pW q: the singular values of a matrix W P R mˆn , in a descending order. s. γpW q: a function mapping from R Z`ˆn to R, defined as γpW q fi lim inf mÑ8 σ n pW r1:m,1:ns q ? m . (3) t. |S|: the cardinality of a set S. u. Prox C pvq: the projection of a vector v P R m onto a subset C Ă R m . Structural Identifiability and Theoretical Results Structural Identifiability We first formalize the definition of structural identifiability. For two vectors with countably infinite components w " pw 1 , w 2 , ...q J , z " pz 1 , z 2 , ..q J P R Z`, we define sin`=pw, zq " lim sup nÑ8 | sin =pw r1:ns , z r1:ns q|, which quantifies the angle between two vectors w and z in R Z`. In particular, we say the angle between w and z is zero when sin`=pw, zq is zero. Definition 1 (Structural identifiability of a latent factor). Consider the kth latent factor, where k P t1, ..., Ku, and a nonempty parameter space S Ă R Z`ˆt1,...,KuˆRZ`ˆt1,...,Ku for pΘ, Aq. We say the k-th latent factor is structurally identifiable in the parameter space S if for any pΘ, Aq, pΘ 1 , A 1 q P S, P Θ,A " P Θ 1 ,A 1 implies sin`=pΘ rks , Θ 1 rks q " 0. We point out that the parameter space S is essentially determined by the design information q jk s. As will be shown shortly in this section, a good design imposes suitable constraints on the parameter space, which further ensures the structure identifiability of the latent factors. This definition of identifiability avoids the consideration of the scale of the latent factor, which is not uniquely determined as the distribution of data only depends on tθ J i a j : i, j P Z`u. Moreover, the sine measure is a canonical way to quantify the distance between two linear spaces that has been used in, for example, the well-known sine theorems for matrix perturbation (Davis, 1963;Wedin, 1972). As will be shown in the sequel, this definition of structural identifiability naturally leads to a relationship between identifiability and estimability and has important implications on psychological measurement. We now characterize the structural identifiability under suitable regularity conditions. We consider a design matrix Q for the manifest variable population, where Q P t0, 1u Z`ˆt1,...,Ku . Throughout the paper, we consider design matrices satisfying the following stability assumption. A1 The limit p Q pSq " lim JÑ8 |tj : q jk " 1, if k P S and q jk " 0, if k R S, 1 ď j ď Ju| J exists for any subset S Ă t1, ..., Ku. In addition, p Q pHq " 0. Note that p Q pSq is the proportion of manifest variables that are associated with and only with latent factors in S. In addition, p Q pHq " 0 implies that there are few irrelevant manifest variables. We also make the following assumption on the generalized latent factor model, that is satisfied under most of the widely used models, including the linear factor model, MIRT model, and the Poisson factor model listed above. A2 The natural parameter space tν : |bpνq| ă 8u " R. Under the above assumptions, Theorem 1 provides a necessary and sufficient condition on the design matrix Q for the structural identifiability of the kth latent factor. This result is established within the parameter space S Q Ă R Z`ˆt1,...,KuˆRZ`ˆt1,...,Ku , S Q " ! pΘ, Aq P R Z`ˆt1,...,KuˆRZ`ˆt1,...,Ku : }θ i } ď C, }a j } ď C, γpΘq ą 0 A rR Q pSq,S c s " 0 for all S Ă t1, ..., Ku, γpA rR Q pSq,Ss q ą 0 for all S, s.t. p Q pSq ą 0 ) , where C is a positive constant, the γ function is defined in (3) and R Q pSq " tj : q jk " 1, if k P S and q jk " 0, if k R Su where we define Ş kPS,p Q pSqą0 S " H if p Q pSq " 0 for all S that contains k. The following proposition guarantees that the parameter space is nontrivial. Proposition 1. For all Q satisfying A1, S Q ‰ H. We further remark on the parameter space S Q . First, S Q requires some regularities on each θ i and a j (}θ i } ď C, }a j } ď C) and the A-matrix satisfying the constraints imposed by Q (A rR Q pSq,S c s " 0) for all S. It further requires that there is enough variation among people, quantified by γpΘq ą 0, where the γ function is defined in (3). Note that this requirement is mild, in the sense that if θ i s are i.i.d. with a strictly positive definite covariance matrix, then γpΘq ą 0 a.s., according to the strong law of large numbers. Furthermore, γpA rR Q pSq,Ss q ą 0 for S satisfying p Q pSq ą 0 requires that each group of items (categorized by S) contains sufficient information if appearing frequently (p Q pSq ą 0). Similar to the justification for Θ, γpA rR Q pSq,Ss q ą 0 can also be justified by considering that a j s are i.i.d. following a certain distribution for j P R Q pSq. We provide two examples to assist with understanding Theorem 1. If K " 2 and p Q pt1uq " p Q pt1, 2uq " 1{2, then the second latent factor is not structurally identifiable, even if it is associated with infinitely many manifest variables. In addition, having many manifest variables with a simple structure ensures the structural identifiability of a latent factor. That is, if p Q ptkuq ą 0, then the kth factor is structurally identifiable. Identifiability and estimability It is well known that for a fixed dimensional parametric model with i.i.d. samples, the identifiability of the model parameter is necessary for the existence of a consistent estimator. We extend this result to the infinite-dimensional parameter space under the current setting. We start with a generalized definition for the consistency of estimating a latent factor. An estimator given N individuals and J items is denoted by pΘ pN,Jq , pN,Jq q, which only depends on Y r1:N,1:Js for all N, J P Z`. Definition 2 (Consistency for estimating latent factor k). The sequence of estimators The next proposition establishes the necessity of the structural identifiability of a latent factor on its estimability. Proposition 2. If latent factor k is not structurally identifiable in S Q , then there does not exist a consistent estimator for latent factor k. Estimation and Its Consistency We further show that the structural identifiability and estimability are equivalent under our setting. For ease of exposition, let Q be the true design matrix in t0, 1u Z`ˆt1,...,Ku satisfying assumption A1. In addition, let pΘ˚, A˚q P S Q be the true parameters for the person and the manifest variable populations. We provide an estimator pΘ pN,Jq , pN,Jq q such that sin =pΘ pN,Jq rks , Θr 1:N,ks q P Θ˚,AÑ 0, N, J, Ñ 8, when Q satisfies (4), which leads to the structural identifiability of latent factor k under Theorem 1. Specifically, we consider the following estimator pΘ pN,Jq , pN,Jq q P arg min´l pθ 1 , ..., θ N , a 1 , ..., a J q , s.t. }θ i } ď C 1 , }a j } ď C 1 , a j P D j , i " 1, ..., N, j " 1, ..., J,(5) where lpθ 1 , ..., θ N , a 1 , ..., a J q " ř N i"1 ř J j"1 y ij pθ J i a j q´bpθ J i a j q, C 1 is any constant greater than C in the definition of S Q , and D j " ta P R K : a jk " 0 if q jk " 0u imposes the constraint on a j . Note that maximizing l pθ 1 , ..., θ N , a 1 , ..., a J q is equivalent to maximizing the joint likelihood (2), due to the natural exponential family form. The next theorem provides an error bound on pΘ pN,Jq , pN,Jq q. Theorem 2. Under assumptions A1-A2 and pΘ˚, A˚q P S Q , there exists N 0 , J 0 , and κ 0 ą 0 such that for all N ě N 0 and J ě J 0 , with probability 1´2{pN`Jq. 1 NJ }Θ pN,Jq p pN,Jq q J´Θr 1:N,1:Ks A˚J r1:J,1: Ks } 2 F ď κ 0 c N`J NJ logpNJq. (6) Moreover, if Q satisfies (4) and thus latent factor k is structurally identifiable, then | sin =pΘr 1:N,ks ,Θ pN,Jq rks q| ď κ 1ˆN`J NJ logpNJq˙1 4 .(7) with probability 1´2{pN`Jq, where κ 1 is a constant independent with N and J. Proposition 2, and Theorems 1 and 2 together imply that the structural identifiability and estimability over S Q are equivalent, which is summarized in the following corollary. Remark 1. The error bound (6) holds even when one or more latent factors are not structurally identifiable. In particular, (6) holds when removing the constraint a j P D j from (5), which corresponds to the exploratory factor analysis setting where no design matrix Q is pre-specified (or in other words, q jk " 1 for all j and k; see e.g. Chen et al., 2017). Remark 2. The proposed estimator (5) and its error bound are related to low-rank matrix completion (e.g. Candès and Plan, 2010;Davenport et al., 2014), where a bound similar to (6) can typically be derived. The key differences are (a) the research on matrix completion is only interested in the estimation of Θ˚A˚J, while the current paper focuses on the estimation of Θ˚that is a fundamental problem of psychological measurement and (b) our results are derived under a generalized latent factor model that covers many models. We end this section by providing an alternating minimization algorithm (Algorithm 1) for solving the optimization program (5), which is computationally efficient through our parallel computing implementation using Open Multi-Processing (OpenMP; Dagum and Menon, 1998). Specifically, we adopt a projected gradient descent update (e.g. Parikh and Boyd, 2014) to handle the constraints, where the projections have closed-form solutions. Similar algorithms have been considered in other works, such as Udell et al. (2016) and Zhu et al. (2016), for solving optimization problems with respect to low-rank matrices. Convergence properties of this type of algorithms have also been studied (e.g. Zhao et al., 2015). Algorithm 1: Alternating minimization algorithm 1. Input: Data py ij : 1 ď i ď N, 1 ď j ď Jq, dimension K, constraint parameter C 1 , initial iteration number l " 1, and initial value θ p0q i and a p0q j P D j , i " 1, ..., N, j " 1, ..., J. 2. Alternating minimization: for l " 1, 2, ... do for i " 1, 2, ..., N do θ plq i " Prox txPR K :}x}ďC 1 u pθ pl´1q i´η g plq i q, where g plq i "´B l i pθq Bθˇθ "θ pl´1q and l i pθq " ř J j"1 y ij pθ J a pl´1q j q´bpθ J a pl´1q j q. η ą 0 is a step size chosen by line search. end for j " 1, 2, ..., J do a plq j " Prox txPR K :}x}ďC 1 u pProx D j pa pl´1q j´ηg plq j qq, whereg plq j "´Bl j paq Baˇa "a pl´1q j and l j paq " ř N i"1 y ij ppθ plq i q J aq´bppθ plq i q J aq. η ą 0 is a step size chosen by line search. end end 3. Output: Iteratively perform Step 2 until convergence. Outputθ i " θ pLq i , a j " a pLq j , i " 1, ..., N, j " 1, ..., J, where L is the last iteration number. Further Implications In this section, we discuss the implications of the above results on large-scale measurement. 1 2 3 4 5 6¨¨1 1 0 1 1 0¨¨Q J 1 0 1 1 0 1¨¨0 1 1 0 1 1¨¨T able 1: An example of the design matrix Q, which has infinite rows and K " 3 columns. The rows of Q are given by repeating the first 3ˆ3 submatrix infinite times. On the design of tests. According to Theorems 1 and 2, the key to the structural identifiability and consistent estimation of factor k is tku " č kPS,p Q pSqą0 S,(8) which provides insights on the measurement design. First, it implies that the "simple structure" design that is advocated in psychological measurement is a safe design. Under the simple structure design, each manifest variable is associated with one and only one factor. If each latent factor k is associated with many manifest variables that only measure factor k, or more precisely p Q ptkuq ą 0, (8) is satisfied. Second, our result implies that a simple structure is not necessary for a good measurement design. A latent factor can still be identified even when it is always measured together with some other factors. For example, consider the Q-matrix in Table 1. Under this design, all three factors satisfy (8) even when there is no item measuring a single latent factor. Third, (8) is not satisfied when there exists a k 1 ‰ k and k 1 P Ş kPS,p Q pSqą0 S. That is, almost all manifest variables that are associated with factor k are also associated with factor k 1 , in the asymptotic sense. Consequently, one cannot distinguish factor k from factor k 1 , making factor k structurally unidentifiable. We point out that in this case, factor k 1 may still be structurally identifiable; for example, when p Q ptk 1 uq ą 0. Finally, (8) is also not satisfied when Ş kPS,p Q pSqą0 S " H. It implies that the factor k is not structurally identifiable when the factor is not measured by a sufficient number of manifest variables. Properties of Estimated Factor Scores A useful result. Let pΘ˚, A˚q P S Q be the true parameters for the person and the manifest variable populations. The following corollary is derived from Theorem 2 that establishes a relationship between the true person parameters and their estimates. This result is the key to the rest of the results in this section. Corollary 2. Under Assumption A1-A2 and (4) is satisfied for some k, then there exists a sequence of random variables c N,J P t´1, 1u, such that › › › › › Θr 1:N,ks }Θr 1:N,ks }´c N,JΘ pN,Jq rks }Θ pN,Jq rks } › › › › › P pΘ˚,A˚q Ñ 0, N, J, Ñ 8. Remark 3. Corollary 2 follows directly from (7). It provides an alternative view on hoŵ On the distribution of person population. In psychological measurement, the distribution of true factor scores is typically of interest, which may provide an overview of the population on the constructs being measured. Corollary 2 implies the following proposition on the empirical distribution of the factor scores. Proposition 3. Suppose assumptions A1-A2 are satisfied and furthermore (4) is satisfied for factor k. We normalize θi k andθ pN,Jq ik by v i " ? Nθi k }Θr 1:N,ks } andv i " c N,J ? Nθ pN,Jq ik }Θ pN,Jq rks } for i " 1, ..., N,(9) where c N,J is defined and discussed in Corollary 2. Let F N andF N,J be the empirical measures of v 1 , ..., v N andv 1 , ...,v N , respectively. Then, W asspF N ,F N,J q P pΘ˚,A˚q Ñ 0, N, J, Ñ 8, where W assp¨,¨q denotes the Wasserstein distance between two probability measures W asspµ, νq " sup h is 1-Lipschitz | ż hdµ´ż hdν|. We point out that the normalization in (9) ř N i"1 pv i´vi q 2 {N " 0, implying that ř N i"1 1 tpv i´vi q 2 ěǫu {N " 0, for all ǫ ą 0. That is, most of thev i s will fall into a small neighborhood of the corresponding v i s. On ranking consistency. The estimated factor scores may also be used to rank individuals along a certain construct. In particular, in educational testing, the ranking provides an ordering of the students' proficiency in a certain ability (e.g., calculus, algebra, etc.). Our results also imply the validity of the ranking along a latent factor when it is structurally identifiable and N and J are sufficiently large. More precisely, we have the following proposition. Proposition 4. Suppose assumptions A1-A2 are satisfied and furthermore (4) is satisfied for factor k. Consider v i andv i , the normalized versions of θi k andθ pN,Jq ik as defined in (9). In addition, assume that there exists a constant κ R such that for any sufficiently small ǫ ą 0 and sufficiently large N, ř i‰i 1 It|v i´vi 1 | ď ǫu NpN´1q{2 ď κ R ǫ.(10) Then, τ pv,vq NpN´1q{2 P pΘ˚,A˚q Ñ 0, N, J, Ñ 8,(11) where τ pv,vq " ř i‰i 1 Ipv i ą v i 1 ,v i ăv i 1 q`Ipv i ă v i 1 ,v i ąv i 1 q is the number of inconsistent pairs according to the ranks of v " pv 1 , ..., v N q andv " pv 1 , ...,v N q. We point out that (10) is a mild regularity condition on the empirical distribution F N . It requires that the probability mass under F N does not concentrate in any small ǫ-neighborhood, which further implies that the pairs of individuals who are difficult to distinguish along factor k, i.e., pi, i 1 qs that v i and v i 1 are close, take only a small proportion among all the pN´1qN{2 pairs. In fact, it can be shown that (10) is true with probability tending to 1 as N grows to infinity, when θi k s are i.i.d. samples from a distribution with a bounded density function. Proposition 4 then implies that if we rank the individuals usinĝ v i (assuming c N,J can be consistently estimated based on other information), the proportion of incorrectly ranked pairs converges to 0. Note that τ pv,vq is known as the Kendall's tau distance (Kendall and Gibbons, 1990), a widely used measure for ranking consistency. On classification consistency. Another common practice of utilizing estimated factor scores is to classify individuals into two or more groups along a certain construct. For example, in an educational mastery test, it is of interest to classify examinees into "mas-tery" and "nonmastery" groups according their proficiency in a certain ability (Lord, 1980;Bartroff et al., 2008). In measuring psychopathology, it is common to classify respondents into "diseased" and "non-diseased" groups based on a mental health disorder. We justify the validity of making classification based on the estimated factor score. Proposition 5. Suppose assumptions A1-A2 are satisfied and furthermore (4) is satisfied for factor k. Consider v i andv i , the normalized versions of θi k andθ pN,Jq ik as defined in (9). Let τ´ă τ`be the classification thresholds, then ř N i"1 Itv i ě τ`, v i ď τ´u`Itv i ď τ´, v i ě τ`u N P pΘ˚,A˚q Ñ 0, N, J, Ñ 8, .(12) Considering two pre-specified thresholds τ´and τ`is the well-known indifference zone formulation of educational mastery test (e.g. Bartroff et al., 2008). In that context, examinees with v i ě τ`are classified into the "mastery" group and those with v i ď τ´are classified into the "nonmastery" group. The interval pτ´, τ`q is known as the indifference zone, within which no decision is made. Proposition 5 then implies that when factor k is structurally identifiable, the classification error tends to 0 as both N and J grow to infinity. Extensions Generalized latent factor models with intercepts As mentioned in Section 2.1, intercepts can be easily incorporated in the generalized latent factor model by restricting θ i1 " 1. Then, a j1 s are the intercept parameters and q j1 " 1 for all j. Consequently, for any S satisfying p Q pSq ą 0, 1 P S and thus the latent factors 2-K are not structurally identifiable according to Theorem 1. Interestingly, these factors are still structurally identifiable if we restrict to the following parameter space S Q,´" ! pΘ, Aq P S Q : lim N Ñ8 1 N 1 J N Θ r1 :N,ms " 0 for m ě 2, and θ i1 " 1 for i P Z`), which requires that Θ rks and Θ r1s are asymptotically orthogonal, for all k ě 2. Proposition 6. Under Assumptions A1-A2, and assuming that q j1 " 1 for all j P Z`and K ě 2, then the kth latent factor is structurally identifiable in S Q,´i f and only if t1, ku " č kPS,p Q pSqą0 S,(13) for k ě 2. The next proposition guarantees that S Q,´i s also non-empty. Proposition 7. For all Q satisfying A1 and q j1 " 1 for all j P Z`, and in addition C ą 1, then S Q,´‰ H. Remark 4. When having intercepts in the model, similar consistency results can be established for the estimator pΘ pN,Jq , pN,Jq q P arg min´l pθ 1 , ..., θ N , a 1 , ..., a J q , s.t. }θ i } ď C 1 , }a j } ď C 1 , a j P D j , θ i1 " 1, N ÿ i 1 "1 θ i 1 k " 0, i " 1, ..., N, j " 1, ..., J, k " 2, ..., K. Extension to Missing Values Our estimator can also handle missing data which are often encountered in practice. Let Ω " pω ij q NˆJ be the indicator matrix of nonmissing values, where ω ij " 1 if Y ij is observed and ω ij " 0 if Y ij is missing. When data are completely missing at random, the joint likelihood function becomes L Ω pθ 1 , ..., θ N , a 1 , ..., a J , φq " ź i,j:ω ij "1 expˆy ij m ij´b pm ij q φ`c py ij , φq2 and our estimator becomes pΘ pN,Jq , pN,Jq q P arg min´l Ω pθ 1 , ..., θ N , a 1 , ..., a J q , s.t. }θ i } ď C 1 , }a j } ď C 1 , a j P D j , i " 1, ..., N, j " 1, ..., J, where l Ω pΘA J q " ř i,j:ω ij "1 y ij m ij´b pm ij q. Moreover, results similar to Theorem 2 can be established even when having missing data. Specifically, we assume A3 ω ij s in Ω are independent and identically distributed Bernoulli random variables with P pω ij " 1q " n NJ . This assumption implies that data are completely missing at random and only about n entries of pY ij q NˆJ are observed. We have the following result. Proposition 8. Under assumptions A1-A3 and pΘ˚, A˚q P S Q , there exists N 0 , J 0 , and κ 0 ą 0 such that for N ě N 0 and J ě J 0 , and n ě pN`Jq logpJNq, then there exists a constant κ 0 ą 0 (independent with N and J) such that with probability 1´2{pN`Jq, 1 NJ }Θ pN,Jq p pN,Jq q J´Θr 1:N,1:Ks pAr 1:N,1: Ks q J } 2 F ď κ 0 c pN`Jq logpNJq n .(14) Moreover, if Q satisfies (4) and thus latent factor k is structurally identifiable, then there exists a constant κ 1 (independent with N and J) such that with probability 1´2{pN`Jq, | sin =pΘr 1:N,ks ,Θ pN,Jq rks q| ď κ 1ˆp N`Jq logpNJq n˙1 4 .(15) Remark 5. Results similar to (14) have also been derived in the literature of matrix completion (e.g. Candès and Plan, 2010;Davenport et al., 2014) under specific statistical models with an underlying low rank structure. Proposition 8 extends existing results on matrix com-pletion to a generalized latent factor model. A Useful Perturbation Bound on Linear Subspace The standard approach (see, e.g., Davenport et al. (2014)) for bounding the error of the maximum likelihood estimator is by making use of the strong/weak convexity of the log-likelihood function. However, in the generalized latent factor model, the log-likelihood function is not convex in pΘ r1:N,1:Ks , A r1:J,1:Ks q. Thus, the standard approach is not applicable for proving (7) in Theorem 2. For this reason, we develop new technical tools to handle this problem. In particular, we where we define θ min,`p L, Mq as the smallest positive principal angle between L and M (defined as 0 if all the principal angles are 0), P M denotes the orthogonal projection onto a linear space M, and αpθq " 2p1`cos θq p1´cos θq 3 . Here, the norm }¨} could be any unitary invariant, uniformly generated and normalized matrix norm. In particular, if we take }¨} to be the spectral norm }¨} 2 , then we have sin =pL 1 X M 1 , L X Mq ď8 maxtαpθ min,`p L, Mqq, αpθ min,`p L 1 , M 1 qqupsin =pL, L 1 q`sin =pM, M 1 qq. We refer the readers to Stewart and Sun (1990) for more details on principal angles between two linear spaces and on matrix norms. In particular, the spectral norm is a unitary invariant, uniformly generated and normalized matrix norm. The result in Proposition 9 holds for all linear subspaces. The right-hand side in (16) is finite if and only if θ max,`p L, Mq ‰ 0 and θ max,`p L 1 , M 1 q ‰ 0. In our problem, L " RpΘ r1:N,S 1 s q and M " RpΘ r1:N,S 2 s q for S 1 , S 2 Ă t1, ..., Ku and S 1 ‰ S 2 . The next lemma further bounds αpθ min,`p L, Mqq when L and M are column spaces of a matrix, which is a key step in proving (7). Lemma 1. Let W P R NˆK for some positive integer N and K and W is not a zero matrix, and S 1 , S 2 Ă t1, ..., Ku be such that S 1 zS 2 ‰ H and S 2 zS 1 ‰ H, then cospθ min,`p RpW rS 1 s q, RpW rS 2 s qqq ď 1´σ Simulation Study Study I. We first verify Theorem 2 and its implications when all latent factors are structurally identifiable. Specifically, we consider K " 5 under the three models discussed in Section 2.1, including the linear, the MIRT and the Poisson models. Two design structures are considered, including (1) a simple structure, where p Q ptkuq " 1{5, k " 1, ..., 5 and (2) a mixed structure, where p Q pSq " 1{5, S " t1, 2, 3u, t2, 3, 4u, t3, 4, 5u, t4, 5, 1u, and t5, 1, 2u. The true person parameters θi s and the true manifest parameters aj s are generated i.i.d. from distributions over the ball tx P R K : }x} ď 2.5u, respectively, (i.e., C " 2.5 in S Q ). Under these settings, all the latent factors are structurally identifiable. Specifically, τ´and τ`are chosen as 0.14 and 0.43 which are the 55% and 65% quantiles of the v i s. From these plots, both the ranking and the classification errors tend to zero, as J and N grow large. Study II. We then provide an example, in which a latent factor is not identifiable. Specifically, we consider K " 2 and the same latent factor models as in Study I. The design structure is given by p Q pt1uq " 1{2 and p Q pt1, 2uq " 1{2. The true person parameters θi s and the true manifest parameters aj s are generated i.i.d. from distributions over the ball tx P R K : }x} ď 3u, respectively, (i.e., C " 3 in S Q ). Under these settings, the first latent factor is structurally identifiable and the second factor is not. For each model, we consider J " 100, 200, ..., 1000. The rest of the simulation setting is the same as Study I. Results are shown in Figures 6 and 7. First, Figure 6 presents the patten that 1 N J }Θ pN,Jq p pN,Jq q J´Θr 1:N,1:Ks A˚J r1:J,1:Ks } 2 F decays to 0 when J increases, even when a latent factor is not structurally identifiable. This is consistent with the first part of Theorem 2. Second, Figure 7 shows the trend of | sin =pΘr 1:N,ks ,Θ which is structurally unidentifiable, while it still decays towards 0 for the identifiable one. Concluding Remarks In this paper, we study a central problem in psychometrics that is the identifiability of latent factors in structured latent factor models. Motivated by large-scale psychological measurement that is becoming more and more popular these days, we adopt an asymptotic setting in which both the numbers of individuals and manifest variables grow to infinity. Under this asymptotic regime, the notion of identifiability of latent factors is formalized through the definition of structural identifiability. Under a generalized latent factor model that covers most of the popular latent factor models, necessary and sufficient conditions are established for the structural identifiability of a latent factor. Moreover, an estimator is proposed that can consistently recover all the structurally identifiable latent factors. This estimator can be efficiently computed through an alternating minimization algorithm which is substantially boosted by parallel computing. Our results have significant implications, including the design of test, inference on the distribution of latent factors, and the validity of making ranking and classification decisions based on the estimated factor scores, which provide theoretical guidance to large-scale psychological measurement. There are many future directions along the current work. First, it is of interest to develop methods, such as information criteria, for model comparison under the current asymptotic regime. These methods can be used to select the design matrix Q that best describes the data structure when there are multiple Q-matrices available, or to determine the underlying latent dimensions. Second, the current results may be further generalized by considering more general latent factor models beyond the exponential family. For example, we may establish similar identifiability and estimability results when the distribution of Y ij is a more complicated function or even an unknown function of θ i and a j . Acknowledgement We would like to thank Prof. Zhiliang Ying for his valuable comments. A Technical Proofs In this supplement, we provide proofs of theoretical results in the main manuscript. A.1 Some facts about matrices In the proof, we will use the following facts about matrices often. • For any matrix V P R mˆn , and a subset S Ă t1, .., nu, σ |S| pV rSs q ě σ n pV q. This is a direct result of Fact 3 in (Hogben, 2006, Chapter 17-7). • For any two matrices V P R mˆn , W P R nˆl , σ n pV q}W } 2 ě σ n pV W q ě σ n pV qσ n pW q. This is a direct result of Fact 7(b) in (Hogben, 2006, Chapter 17-8). A.2 Proof of results in Section 3.1 In this section, we first prove Proposition 1, and then prove the necessary part of Theorem 1. The sufficiency part of Theorem 1 is implied by Proposition 2 together with Theorem 2, whose proof is provided in Section A.4. Proof of Proposition 1. In this proof, we construct pΘ, Aq P S Q . We first construct A. For S with p Q pSq " 0, we construct A rR Q pSq,1:Ks " 0. For each S such that p Q pSq ą 0, we construct A rR Q pSq,Ss as A rR Q pSq,Ss " C » - - - - - - - I |S| I |S| . . . fi ffi ffi ffi ffi ffi ffi fl , where I m denotes the mˆm identity matrix. We also let A rR Q pSq,S c s " 0 so that the Q-matrix requirement is satisfied. It is not hard to calculate that γpA rR Q pSq,Ss q " C|S|´1 {2 ą 0. Now we construct Θ as follows, Θ " C » - - - - - - - I K I K . . . fi ffi ffi ffi ffi ffi ffi fl . We have γpΘq " C{K 1{2 ą 0. Furthermore, from the construction, }a j } ď C and }θ i } " C for all i P Z`. Thus, pΘ, Aq P S Q . Proof of Theorem 1, necessary part. We prove by contradiction. If (4) is not satisfied, then there are two cases: 1) tk, k 1 u Ă Ş kPS,p Q pSqą0 S for some k 1 ‰ k, and 2) H " Ş kPS,p Q pSqą0 S. For these two cases, we will construct pΘ,Ãq, pΘ 1 , A 1 q P S Q with P pΘ,Ãq " P pΘ 1 ,A 1 q and sin`=pΘ rks , Θ 1 rks q ą 0. Case 1: tk, k 1 u Ă Ş kPS,p Q pSqą0 S. Without loss of generality, we assume k " 1 and k 1 " 2. Let pΘ, Aq be constructed in the same way as we did in the proof of Proposition 1 on page 32. LetΘ " 1 2 Θ andà " 1 2 A, then pΘ,Ãq P S Q . Now we construct pΘ 1 , A 1 q as follows. θ 1 im " $ ' ' & ' ' % θ im {2 if m ‰ k pθ ik´θik 1 q{2 if m " k and a 1 jm " $ ' ' & ' ' % a jm {2 if m ‰ k 1 pa jk`ajk 1 q{2 if m " k 1 , for all i, j P Z`. That is, we subtract the k 1 th column from the kth column in the matrixΘ and obtain Θ 1 . We add the kth column to the k 1 th column in the matrixà to construct A 1 . By construction, ř K m"1 θ 1 im a 1 jm " ř K m"1θ imãjm for all i, j P Z`. Thus, P pΘ,Ãq " P pΘ 1 ,A 1 q . Now we verify that pΘ 1 , A 1 q P S Q . First, note that Θ 1 is obtained by an invertible column transformation on Θ and A, so γpΘ 1 q ą 0 given γpΘq ą 0. Second, we verify that A 1 satisfies the Q-matrix requirement. That is, a 1 jm " 0 if q jm " 0 for all j P Z`and all m P t1, .., Ku. According to the construction, it is automatically true if m ‰ k 1 or q jk 1 " 1. For j such that q jk 1 " 0, there are two cases: p Q pS j q " 0 or p Q pS j q ą 0, where we define S j " tm : q jm " 1u for all j P Z`. That is, S j indicates the set of dimensions measured by item j. For the former case, according to the construction of A in the proof of Proposition 1, a jk " a jk 1 " 0. So a 1 jk 1 " 0 as well. For the latter case, q jk 1 " 0, p Q pS j q ą 0, and k 1 P X S:kPS,p Q pSqą0 S implies k R S j . As a result, a jk " a jk 1 " 0, and a 1 jk 1 " 0. Summarizing these two cases, we can see that a 1 jm " 0 for all j P Z`, m P t1, ..., Ku such that q jm " 0. That is, A 1 satisfies the Q-matrix restriction. Third, we verify that γpA 1 rR Q pSq,Ss q ą 0 for all S such that p Q pSq ą 0. This is true because A 1 rR Q pSq,Ss is either an invertible column transformation or equal to A rR Q pSq,Ss , and γpA rR Q pSq,Ss q ą 0. Lastly, it is easy to verify that }θ i }, }θ 1 i }, }ã j }, }a 1 j } ď C for all i, j P Z`. Therefore, we have verified that pΘ 1 , A 1 q P S Q . Now we consider the angle between Θ 1 rks ,Θ rks . To see it clearer, we write down the k and k 1 th columns of Θ 1 andΘ. By construction, we havẽ , and Θ 1 rtk,k 1 us " pC{2q Consequently, sin`=pΘ rks , Θ 1 rks q " 1{ ? 2 ą 0. This contradicts the definition of structural identifiability. We complete the proof for Case 1. Θ rtk,k 1 us " pC{2q » - - - - - - - - - - - - - - - - - - - - - - - 1 0 0 1 0 K´2 0 K´2 1 0 0 1 0 K´2 0 K´2 .» - - - - - - - - - - - - - - - - - - - - - - - 1 0 1 1 0 K´2 0 K´2 Case 2: We first note that if K " 1, then p Q pHq ą 0. This contradicts Assumption A1. In the rest of the proof, we assume K ě 2. Without loss of generality, we assume k " 1. Let pΘ, Aq be constructed the same as the one in the proof of Proposition 1. We letΘ " 1 2 Θ andà " 1 2 A. Obviously, pΘ,Ãq P S Q . We further let A 1 " 1 2 A "Ã. We construct Θ 1 as follows. Θ 1 rms " 1 2 Θ rms for m ě 2, and θ 1 i1 " C{2 for all i P Z`. Since A rR Q pSq,1:Ks " 0 for S with p Q pSq " 0 and p Q pSq " 0 for all k P S in this case, we have a j1 " 0 for all j P Z`. Combining this finding with the fact thatΘ and Θ 1 are different only at the first column, we have a 1 J j θ 1 i "ã J jθ i for all i, j P Z`. Thus, PΘ ,à " P Θ 1 ,A 1 . Note that A 1 " A{2, and Θ 1 is an invertible column transformation of Θ. Thus, from pΘ, Aq P S Q , we have γpΘ 1 q ą 0, and pΘ 1 , A 1 q P S Q . We proceed to verify that sin`=pΘ 1 r1s ,Θ r1s q ą 0. Note that lim N Ñ8 N´1|xΘ 1 r1:N,1s ,Θ r1:N,1s y| " C 2 {p4Kq, lim N Ñ8 N´1 {2 }Θ 1 r1:N,1s } " C{2, and lim N Ñ8 N´1 {2 }Θ r1:N,1s } " C{p2 ? Kq. Therefore, we have lim N Ñ8 cos =pΘ r1:N,1s , Θ 1 r1:N,1s q " K´1 {2 . Thus, sin`=pΘ r1s , Θ 1 r1s q " b K´1 K ą 0. By contradiction, we complete the proof. A.3 Proof of results in Section 3.2 In this section, we provide the proof of Proposition 2. Proof of Proposition 2. We prove the proposition by contradiction. If the k-th dimension is not structurally identifiable, then there exists pΘ, Aq, pΘ 1 , A 1 q P S Q , with P Θ,A " P Θ 1 ,A 1 and sin`=pΘ rks , Θ 1 rks q ą 0. On the other hand, by contradiction there is a consistent estimator tpΘ pN,Jq , pN,Jq q, N, J P Z`u for the k-th dimension, which means This implies lim N,JÑ8 sin =pΘ 1 r1:N,ks , Θ r1:N,ks q " lim N,JÑ8 sin =pΘ pN,Jq rks , Θ 1 r1:N,ks q " 0. By definition, this gives sin`=pΘ rks , Θ 1 rks q " 0, which contradicts our assumption sin`=pΘ rks , Θ 1 rks q ą 0. A.4 Proofs of Results in Section 3.3 In this section, we provide the proof of Theorem 2, which consists of two parts: the proof of (6) and the proof of (7). Throughout this section, all the matrices considered are finite dimensional. For the ease of presentation, we drop the subscripts N and J. We abuse the notation a little bit and write Θ˚:" Θr 1:N,1:Ks , A˚:" Ar 1:J,1:Ks , Θ :" Θ r1:N,1:Ks , A :" A r1:J,1:Ks , and pΘ,Âq " pΘ pN,Jq , pN,Jq q within this section (Section A.4). Furthermore, all the probability considered in the current section are taken under the true value Θ˚and A˚. We will first present the main proof of the theorem and then present the proof of its supporting lemmas in Section A.4.1. We start with the proof of (6). Proof of (6) in Theorem 2. LetM "Θ J and M˚" Θ˚A˚J,then lpM q´lpM˚q ě 0. On the other hand, lpM q´lpM˚q " N ÿ i"1 J ÿ j"1 Y ij pm ij´mi j q´pbpm ij q´bpmi j qq " N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqpm ij´mi j q´N ÿ i"1 J ÿ j"1 bpm ij q´bpmi j q´b 1 pmi j qpm ij´mi j q " N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqpm ij´mi j q´N ÿ i"1 J ÿ j"1 1 2 b 2 pm ij qpm ij´mi j q 2 ď N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqpm ij´mi j q´1 2 min |ν|ďC 12 b 2 pνq}M´M˚} 2 F ,(19) for somem ij " ηmi j`p 1´ηqm ij and η P p0, 1q in the third equation. Combining (18) and (19), we have }M´M˚} 2 F ď 2 min |ν|ďC 12 b 2 pνq N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqpm ij´mi j q.(20) Let C " tM : M " ΘA J , }a j } ď C 1 , }θ i } ď C 1 , a j P D j , for all 1 ď i ď N and 1 ď j ď Ju. We further bound (20) by }M´M˚} 2 F ď 2 min |ν|ďC 12 b 2 pνq 2 sup MPC | N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqm ij | " 4 min |ν|ďC 12 b 2 pνq sup M PC |xX,My|,(21) where we let X ij " Y ij´b 1 pmi j q, and define xX,My " ř N i"1 ř J j"1 X ijmij . In what follows, we proceed to an upper bound of P psupM PC |xX,My| ą tq for some positive t to be chosen later. Let S " tpΘ, Aq : }θ i } ď C 1 , }a j } ď C 1 , a j P D j u. For each θ P R K , we define the ball Bpθ, δq " tθ 1 : }θ 1´θ } ď δu. Now, for an NˆK matrix Θ, we define BpΘ, δq " tΘ 1 : θ 1 i P Bpθ i , δq for all 1 ď i ď Nu. Similarly, for a JˆK matrix A, we define BpA, δq " tA 1 : a 1 j P Bpa j , δq for all 1 ď j ď Ju. Furthermore, for a pair pΘ, Aq, we define BppΘ, Aq, δq " BpΘ, δqˆBpA, δq. For each δ, we consider a covering tBppΘ plq , A plq q, δq, i " 1, ..., Npδqu of the set S, where pΘ plq , A plq q P S. That is, S Ă Ť N pδq l"1 BppΘ plq , A plq q, δq, where Npδq denotes number of balls covering S. In particular, we let Npδq be the smallest covering number and pΘ piq , A piq q be the centers corresponding to the covering with the smallest covering number. Now, the probability P psupM PC |xX,My| ą tq can be controlled using the covering. We have for any γ ą 0, P psup MPC |xX,My| ą tq ďP psup MPC |xX,My| ą t, }X} F ď γq`P p}X} F ą γq ď N pδq ÿ l"1 P p sup pΘ 1 ,A 1 qPSXBppΘ plq ,A plq q,δq |xX, Θ 1 A 1J y| ě t, }X} F ď γq`P p}X} F ą γq.(22) Note that for all pΘ 1 , A 1 q P S X BppΘ plq , A plq q, δq, |xX, Θ 1 A 1J y| ď|xX, Θ plq A plqJ y|`}X} F }Θ 1 A 1 J´Θplq A plqJ } F ď|xX, Θ plq A plqJ y|`}X} F p}A 1´Aplq } F }Θ plq } F`} Θ 1´Θplq } F }A 1 } F q. ď|xX, Θ plq A plqJ y|`}X} F p}A 1´Aplq } F ? N C 1`} Θ 1´Θplq } F ? JC 1 q.(23) The last inequality is due to the constraint on the set S. That is, }θ plq j } ď C 1 implies that }Θ plq } ď ? NC 1 . Similarly, }a 1 j } ď C 1 for 1 ď j ď J implies }A 1 } ď ? JC 1 . Note that for A 1 P BpA plq , δq, }A 1´Aplq } 2 F " J ÿ j"1 }a 1 j´a plq j } 2 ď Jδ 2 . Thus, }A 1´Aplq } F ď ? Jδ. Similarly, }Θ 1´Θplq } F ď ? Nδ. Combine this with (23), we have |xX, Θ 1 A 1J y| ď |xX, Θ plq A plqJ y|`2}X} F ? NJδC 1 . Combine the above inequality with (22), we have for each γ ą 0, P psup MPC |xX,M y| ą tq ď N pδq ÿ l"1 P´|xX, Θ plq A plqJ y|`2}X} F ? NJ δC 1 ě t, }X} F ď γ¯`P p}X} F ą γq ď N pδq ÿ l"1 P´|xX, Θ plq A plqJ y| ě t´2γ ? NJδC 1 , }X} F ď γ¯`P p}X} F ą γq ďNpδq sup pΘ 1 ,A 1 qPS P´|xX, Θ 1 A 1J y| ě t´2γ ? NJδC 1¯`P p}X} F ą γq.(24) We proceed to bound Npδq, sup pΘ 1 ,A 1 qPS P´|xX, Θ 1 A 1J y| ě t´2γ ? NJ δC 1¯, and P p}X} F ě γq separately. We start with Npδq. It is not hard to see that an upper bound of Npδq is Npδq ď p C 1 K δ q KpN`Jq .(25) Next, we find an upper bound of sup pΘ 1 ,A 1 qPS P´|xX, Θ 1 A 1J y| ě t´2γ ? NJ δC 1¯. We use the following lemma on the marginal probability tail bound. Lemma 2. There exist constants δ 0 , ǫ 0 (depending only on the function b and the constant C 1 , φ), such that for 0 ă t ă δ 0 NJ , andM P C, P p|xX,My| ą tq ď 2 expp´ε 0 t 2 NJ q. According to Lemma 2, for 0 ď t´2γ ? NJ δC 1 ď δ 0 NJ, P´|xX, Θ 1 A 1J y| ě t´2γ ? NJδC 1¯ď expt´ε 0 pt´2γ ? NJ δC 1 q 2 NJ u.(26) We proceed to an upper bound for P p}X} F ą γq. By Chebyshev's inequality, we have P p}X} F ą γq " P p}X} 2 F ą γ 2 q ď γ´2Ep}X} 2 F q " γ´2φ N ÿ i"1 J ÿ j"1 b 2 pmi j q ď γ´2NJφ sup |ν|ďC 12 b 2 pνq.(27) Combining (24), (25), (26), and (27), we arrive at that for 0 ď t´2γ ? NJδC 1 ď δ 0 NJ, P psup MPC |xX,M y| ą tq ďp C 1 K δ q KpN`Jq expt´ε 0 pt´2γ ? NJδC 1 q 2 NJ u`γ´2NJ sup |ν|ďC 12 b 2 pνq " exptKpN`Jqplog C 1`l og K´log δq´ε 0 pt´2γ ? NJδC 1 q 2 NJ u`γ´2φNJ sup |ν|ďC 12 b 2 pνq. The above inequality holds for all δ, γ ą 0. Now we choose t, δ, and γ. We choose γ " b NJpN`Jqφ sup |ν|ďC 12 b 2 pνq, δ " pNJq´1 {2 , and t " 2γ ? NJδC 14 1 ? ε 0 a 2NJKpN`Jqplog C 1`l og K´log δq, then we have P psup MPC |xX,M y| ą tq ď expt´KpN`Jqplog C 1`l og K`1 2 logpNJqqu`1 N`J . From the above display, we can see that for a fixed K, there exists a positive constant κ 0 (depending on C 1 , b, and K) such that t ď κ 0 a NJpN`Jq logpNJq, thus, P psup MPC |xX,My| ě κ 0 a NJpN`Jq logpNJqq ď expt´pN`Jq logpNJqqu`1 N`J ď 2 N`J for sufficiently large N and J. Combining this with (21), we complete the proof. Proof of (7) in Theorem 2. Note that by proving (6), we have already proved that with probability 1´2 N`J 1 NJ }Θ˚A˚J´ΘÂ} 2 F ď κ 0 c N`J NJ logpNJq.(28) Note that the right-hand side of the above display tend to zero as N, J Ñ 8. On the event (28) happens and recall the definition of S Q , we can see that in order to prove (7), it is sufficient to prove the following statement. Let N 0 , J 0 and σ be such that σ K pΘ˚q ě σ ? N , σ |S| pAr R Q pSqXt1,...,Ju,Ss q ě σ ? J , and |tj : S j " S, 1 ď j ď Ju| J ě p Q pSq 2 ,(29) and ? NJσ 2 ą 2}Θ˚A˚J´Θ J } 2 ,(30) for all N ě N 0 , J ě J 0 , and S such that p Q pSq ą 0. As (28) happens with probability at least 1´2 N`J , the above display also happens with at least 1´2 N`J probability. In addition, a j P D j , and (4) holds. Then, then there exists a constant C 1 , independent of N, J, Θ, and A, and possibly depend on K, σ and C 1 , such that | sin =pΘr ks ,Θ rks q| ď C 1 }Θ˚A˚J´Θ J } 2 ? NJ . In what follows, we prove this statement. For each S Ă t1, ..., Ku, denote ρpSq " sin =pRpΘr Ss q, RpΘ rSs qq. The rest of the proof is based on Proposition 9 and Lemma 1 (proved in Section A.7.1). Consider two sets S 1 , S 2 Ă t1, ..., Ku that are are not subset of each other. Let L " RpΘr S 1 s q, M " RpΘr S 2 s q,L " RpΘ rS 1 s q, andM " RpΘ rS 2 s q in Proposition 9. Then, we have ρpS 1 XS 2 q ď 8 maxtαpθ min,`p RpΘr S 1 s q, RpΘr S 2 s qq, αpθ min,`p RpΘr S 1 s q, RpΘ rS 2 s qqupρpS 1 q`ρpS 2 qq. By Lemma 1, we have cos`θ min,`p RpΘr S 1 s q, RpΘr S 2 s qq˘ď 1´σ 2 |S 1 YS 2 | pΘr S 1 YS 2 s q }Θ˚} 2 2 ď 1´σ 2 K pΘ˚q }Θ˚} 2 2 . Recall αpθq " 2p1`cos θq p1´cos θq 3 . Thus, according to (29), and }Θ˚} 2 ď }Θ˚} F ď ? NC 1 , αpθ min,`p RpΘr S 1 s q, RpΘr S 2 s qqq ď 4σ´6 K pΘ˚q}Θ˚} 6 2 ď 4C 1 6 σ´6. In order to get a similar result forΘ, we need the following lemma. Lemma 3. Given (29) and (30), we have σ |Y S:p Q pSqą0 S| pΘ rY S:p Q pSqą0 Ss q ě σ 2 ? N 2C 1 . Using the above lemma and Lemma 1, we have cos θ min,`p RpΘ rS 1 s q, RpΘ rS 2 s qq ď 1´σ 2 |Y S:p Q pSqą0 S| pΘ rY S:p Q pSqą0 Ss q }Θ} 2 2 ď 1´σ 4 4C 1 4 . Thus, αpθ min,`p RpΘ rS 1 s q, RpΘ rS 2 s qqq ď 4p σ 4 4C 1 4 q´3 ď 256C 1 12 σ´1 2 . Combining (32), (33), and (34), we arrive at an iteration inequality, ρpS 1 X S 2 q ď 1024C 1 12 σ´1 2 pρpS 1 q`ρpS 2 qq(35) Now we can work on all the sets S such that k P S and p Q pSq ą 0. Without loss of generality, we can arrange them as S 1 , ..., S m , where m ď 2 K´1 . We use (35) repeatedly, then we arrive at ρptkuq " ρpX kPS,SĂt1,...,Ku Sq ď 1024 m C 1 12m σ´1 2m m ÿ i"1 ρpS i q.(36) We bound each ρpS i q on the right-hand side of the above display by the following lemma. Lemma 4. Given (29) and (30), we have sin =pRpΘr Ss q, RpΘ rSs qq ď 2}Θ˚A˚J´Θ J } 2 σ |S| pΘr Ss A˚J rR Q pSq,Ss q , for S such that p Q pSq ą 0. Combining (36) and (37), and notice that σ |S| pΘr Ss A˚J rR Q pSq,Ss q ě σ 2 ? NJ we complete the proof of (31). A.4.1 Proof of Lemmas 2 -4 Proof of Lemma 2. We note that under assumption A2, Epe λX ijmij q " exptpbpmi j`λ φm ij qb pmi j qq{φu. Thus, there exist constants λ 0 and κ 1 (depending on b) uniformly for all i, j such that for all |λ| ď λ 0 and all |m ij | ď C 12 , Epe λX ijmij q ď κ 1 . That is,m ij X ij is sub-exponential. Therefore, based on the Bernstein inequality, there exist constants δ 0 and ε 0 (depending on λ 0 , C 1 , and sup |λ|ďλ 0 ,|m ij |ďC 12 Epe λX ijmij q), P p|xX,My| ą tq ď 2e´ε 0 t 2 {pN Jq , for 0 ă t{pNJq ď δ 0 . Proof of Lemma 3. Let T " Y S:p Q pSqą0 S and W " Y S:p Q pSqą0 R Q pSq X r1, Js. Note that for j P W a jm " 0 for m R T , so we have Θ˚A Jr j,1:Ks " Θr T s A Jr j,T s . Thus, pΘ˚A J˚q r1:N,W s " Θr T s A Jr W,T s . Similarly, pΘ J q r1:N,W s "Θ rT s 1 2 σ 2 ? NJ .(38) On the other hand, we have σ |T | pAr W,T s q " min xPR K ,}x rT s }"1 x J rT s A˚J rW,T s Ar W,T s x rT s " min xPR K ,}x rT s }"1 ÿ jPW x J rT s aj ,rT s a˚J j,rT s x rT s " min xPR K ,}x rT s }"1 ÿ S:p Q pSqą0 ÿ j:jPR Q pSq x J rSs aj ,rSs a˚J j,rSs x rSs " min xPR K ,}x rT s }"1 ÿ S:p Q pSqą0 x J rSs A˚J rR Q pSq,Ss Ar R Q pSq,Ss x rSs ě min xPR K ,}x rT s }"1 ÿ S:p Q pSqą0 σ |S| pAr R Q pSq,Ss q}x rSs } 2 Here, the third equation holds because W " Y S:p Q pSqą0 R Q pSq X r1, Js and R Q pSq are disjoint for different S, and a jm " 0 if j P R Q pSq and m R S. Given (29) and the above display, we further have σ |T | pAr W,T s q ě min xPR K ,}x rT s }"1 ÿ S:p Q pSqą0 ? Jσ}x rSs } 2 ě ? Jσ. Also, according to (29), σ |T | pΘr T s q ě σ K pΘ˚q ě ? Nσ. Combining the above two inequalities, we have σ |T | pΘr T s A Jr W,T s q ě σ |T | pΘr T s qσ |T | pAr W,T s qq ě ? NJ σ 2 . Combining the above inequality with (38), and Weyl's perturbation theorem (see, e.g. Stewart and Sun (1990)), we arrive at σ |T | pΘ rT s J rW,T s q ě ? NJσ 2 2 . Thus, we arrive at σ |T | pΘ rT s q ě σ |T | pΘ rT s J rW,T s q }Â} 2 ě ? NJ σ 2 2 a |W |C 1 ě ? N σ 2 2C 1 . Proof of Lemma 4. Similar to (38), we have Θr Ss A˚J rR Q pSq,Ss´Θ rSs J rR Q pSq,Ss is a submatrix of Θ˚A˚J´Θ J , and thus, }Θr Ss A˚J rR Q pSq,Ss´Θ rSs J rR Q pSq,Ss } 2 ď }Θ˚A˚J´Θ J } 2 ď σ 2 2 ? NJ .(39) The rest of the proof is similar to that of Theorem 2 in Chen et al. (2017). We only state the main steps. We write the reduced singular value decomposition (SVD) of Θr Ss A˚J rR Q pSq,Ss as Θr Ss A˚J rR Q pSq,Ss " U S Σ S V J S , where U S is am NˆM orthonormal matrix, Σ S " diagpσ 1 , ..., σ M q is an MˆM matrix, V S is a p Q pSqJˆM orthonormal matrix, and M " |S|, the cardinality of the set S. By (29), we have σ |S| pΘr Ss A˚J rR Q pSq,Ss q ě σ |S| pΘr Ss qσ |S| pAr R Q pSq,Ss q ě ? NJ σ 2 ,(40) Kahan-Wedin sine theorem (O'Rourke et al., 2013, Theorem 19) and (40), we have sin =pRpU rSs q, RpÛ rSs qq ď 2 }Θ rSs J rR Q pSq,Ss´ΘrSs A˚J rR Q pSq,Ss } 2 σ |S| pΘr Ss A˚J rR Q pSq,Ss q ď 2 }Θ J´Θ˚A˚J } 2 σ |S| pΘr Ss A˚J rR Q pSq,Ss q . This further implies sin =pRpΘr Ss q, RpΘ rSs qq ď 2 }Θ J´Θ˚A˚J } 2 σ |S| pΘr Ss A˚J rR Q pSq,Ss q . A.5 Proof of Results in Section 4 In this section, we provide proofs of Corollary 2 and Proposition 3-5. Proof of Corollary 2. Take }Θ r1:N,ks } } 2 " 2´2c N,J cos =pΘ r1:N,ks ,Θ r1:N,ks q " 2´2| cos =pΘ r1:N,ks ,Θ r1:N,ks q|. c N,J " $ ' ' & ' ' % 1 if Θ J r1:N, The above display converges to 0 in probability because sin =pΘ r1:N,ks ,Θ r1:N,ks q Ñ 0 in probability. This completes the proof. Proof of Proposition 3. From Corollary 2, we know that 1 N ř N i"1 pv i´vi q 2 Ñ 0. Let h be a 1-Lipschitz function, we have | ż hpxqF N,J pxq´ż hpxqF N,J pxq| " | 1 N N ÿ i"1 hpv i q´hpv i q| ď 1 N N ÿ i"1 |v i´vi |. By Cauchy inequality, we further have 1 N N ÿ i"1 |v i´vi | ď d ř N i"1 pv i´vi q 2 N . Combining the above two displays, we have W asspF N,J ,F N,J q " sup h is 1-Lipschitz | ż hpxqF N,J pxq´ż hpxqF N,J pxq| ď d ř N i"1 pv i´vi q 2 N . By Corollary 2, the right-hand side of the above display tend to 0 as N, J Ñ 8 in probability. Proof of Proposition 4. For each ǫ ą 0, we have ÿ i‰j Ipv i ă v j ,v i ąv j q ď ÿ i‰j Ipv i ă v j ,v i ąv j , |v i´vj | ą ǫq`ÿ i‰j Ip0 ă v i´vj ă ǫq ď ÿ i‰j Ipv j´vi ą ǫ,v i ąv j q`ÿ i‰j κ R ǫ ď ÿ i‰j Ipv j´vi´pvj´vi q ą ǫq`NpN´1qκ R ǫ ď ÿ i‰j |v j´vi´pvj´vi q| ǫ`N pN´1qκ R ǫ ď ÿ i‰j |v j´vj |`|v i´vi | ǫ`N pN´1qκ R ǫ "2pN´1q{ǫ N ÿ i"1 |v i´vi |`NpN´1qκ R ǫ ďǫ´12pN´1q g f f e N N ÿ i"1 pv i´vi q 2`N pN´1qκ R ǫ. Thus, ř i‰j Ipv i ă v j ,v i ąv j q NpN´1q ď 2ǫ´1 d ř N i"1 pv i´vi q 2 N`κ R ǫ. Take ǫ " p b ř N i"1 pv i´vi q 2 N {κq 1{2 in the above inequality, then we have ř i‰j Ipv i ă v j ,v i ąv j q NpN´1q ď 3pκ R ř N i"1 pv i´vi q 2 N q 1{2 . The right-hand side converge to 0 as N, J Ñ 8 in probability, so as the left-hand side. This completes the proof. Proof of Proposition 5. We have N´1 N ÿ i"1 Itv i ě τ`, v i ď τ´u`Itv i ď τ´, v i ě τ`u ďN´1 N ÿ i"1 |v i´vi | τ`´τď pτ`´τ´q´1 g f f e N´1 N ÿ i"1 pv i´vi q 2 . We complete the proof by noting the right-hand side of the above display converge to 0 in probability. A.6 Proofs of Results in Section 5 Proof of Proposition 7. We will construct pΘ, Aq P S Q,´. First, we note that S Q,´d oes not impose additional requirement on A beyond those in S Q . So, we construct A in the same way as we did in the proof of Proposition 1. We proceed to construct Θ. We already have θ i1 " 1 for all i P Z`, so we only need to consider Θ r2:Ks . Let w i " c i p1, 1, .., 1,´i, 0 J K´i q J P R K`1 , where c i " pi`i 2 q´1 {2 is the normalizing constants making the w i to be a unit vector (i " 1, ..., K). We can see w i and w j are orthogonal for i ‰ j. We consider a matrix E m " rw 1 , ..., w m s P R pK`1qˆm . Now we construct Θ r2:Ks as Θ r2:Ks " p ? C 2´1 {2q » - - - - - - - E K´1 E K´1 . . . fi ffi ffi ffi ffi ffi ffi fl . Then, Θ " » - - - - - - - 1 K`1 p ? C 2´1 {2qE K´1 1 K`1 p ? C 2´1 {2qEN " 1{pK`1q » - - - 1 0 J K´1 0 K´1 pC 2´1 q{4I K´1 fi ffi ffi fl . Thus, γpΘq " pK`1q´1 2 minp ? C 2´1 {2, 1q ą 0. Furthermore, it is easy to verify that }θ i } ď C for all i P Z`. Thus, pΘ, Aq P S Q,´. Proof of Proposition 6. Necessary part: We prove by contradiction. If (13) does not hold, then there are two cases: 1) t1, k, k 1 u P Ş kPS,p Q pSqą0 S for some k 1 ‰ k, and 2) for all k P S, p Q pSq " 0. For these two cases, we will construct pΘ,Ãq and pΘ 1 , A 1 q such that PΘ ,A " P Θ 1 ,A 1 but sin`=pΘ rks , Θ 1 rks q ą 0. Case 1:t1, k, k 1 u P Ş kPS,p Q pSqą0 S for some k 1 ‰ k. First, we considerΘ " Θ{2 and A " A{2 where pΘ, Aq P S Q,´i s constructed in the same way as we did in the proof of Proposition 7. Now we construct pΘ 1 , A 1 q P S Q,´s uch that PΘ ,à " P Θ 1 ,A 1 . We use the same strategy as we did in the proof of Theorem 1, θ 1 im " $ ' ' & ' ' % θ im {2 if m ‰ k pθ ik´θik 1 q{2 if m " k and a 1 jm " $ ' ' & ' ' % a jm {2 if m ‰ k 1 pa jk`ajk 1 q{2 if m " k 1 . Then, similar as the proof of Theorem 1, we know pΘ 1 , A 1 q P S Q,´. Now we consider the angle betweenΘ rks and Θ 1 rks . We havẽ N´1xΘ r1:N,ks , Θ 1 r1:N,ks y " Θ rks " pC 2´1 q 1{2 {4 » - - - - - - - w k´1 w k´1 . . . fi ffi ffi ffi ffi ffi ffi fl , and Θ 1 rks " pC 2´1 q 1{2 {4 » - - - - - - - w k´1´wk 1´1 w k´1´wk 1´1 . . .C 2´1 4pK`1q xw k´1 , w k´1´wk 1´1y " pC 2´1 q{p4pK`1qq, lim N Ñ8 N´1}Θ r1:N,ks } 2 " pC 2´1 q{p4pK`1qq, and lim N Ñ8 N´1}Θ 1 r1:N,ks } 2 " pC 2´1 q{p2pK`1qq. Thus, lim N Ñ8 cos =pΘ rks , Θ 1 rks q " 1{ ? 2, and sin`=pΘ rks , Θ 1 rks q " 1{ ? 2 ą 0. This contradicts the identifiability assumption. Case 2: for all k P S, p Q pSq " 0. Similar to Case 1, we letΘ " Θ{2 andà " A{2. We proceed to construct pΘ 1 , A 1 q. We let A 1 "à and Θ 1 rms "Θ rms for m ‰ k. For Θ 1 rks , we let Θ 1 rks " ? C 2´1 4 » - - - - - - - w K w K . . . fi ffi ffi ffi ffi ffi ffi fl . Similar to the proof of Proposition 7, it is not hard to show that pΘ 1 , A 1 q P S Q,´. On the other hand, xΘ 1 rks ,Θ rks y " 0. Thus, sin`=pΘ 1 rks ,Θ rks q " 1 ą 0. Again, this contradicts our assumption. Sufficient part: Let pΘ 1 , A 1 q, pΘ, Aq P S Q,´s uch that P Θ 1 ,A 1 " P Θ,A . According to the definition of S Q,´, there exists N 0 , J 0 , 0 ă σ ă C, and 0 ă ε ă σ{8, such that for all N ě N 0 and J ě J 0 , we have σ |S| pA rR Q pSqXr1,Js,Ss q ą ? Jσ, σ |S| pA 1 rR Q pSqXr1,Js,Ss q ą ? Jσ for S such that p Q pSq ą 0, (41) σ K pΘ r1:N,1:Ks q ě σN, σ K pΘ 1 r1:N,1:Ks q ě σN, | 1 N 1 N Θ r1:N,ms | ď ε, and | 1 N 1 N Θ 1 r1:N,ms | ď ε for all m P t2, ..., Ku.(42) Since P Θ 1 ,A 1 " P Θ,A and the generalized latent factor model has a strictly convex loglikelihood Intersecting all S such that k P S and p Q pSq ą 0, we have RpΘ r1:N,X S:kPS,p Q pSqą0 Ss q " X S:kPS,p Q pSqą0 RpΘ r1:N,Ss q " X S:kPS,p Q pSqą0 RpΘ 1 r1:N,Ss q "RpΘ 1 r1:N,X S:kPS,p Q pSqą0 Ss q. function in pa J j θ i , i, j P Z`q, we have a J j θ i " a 1J j θ i for all i, j P Z`. Given (13), we have X S:kPS,p Q pSqą0 S " t1, ku. Thus, from the above display, we have RpΘ r1:N,t1,kus q " RpΘ 1 r1:N,t1,kus q. Recall Θ r1:N,1s " Θ 1 r1:N,1s " 1 N . Thus, the above display implies spanp1 N , Θ r1:N,ks´θN 1 N q " spanp1 N , Θ 1 r1:N,ks´θ 1 N 1 N q, where we writeθ N " 1 N 1 J N Θ r1:N,ks andθ 1 N " 1 N 1 J N Θ 1 r1:N,ks . Since 1 N is orthogonal to Θ r1:N,ksθ N 1 N and Θ 1 r1:N,ks´θ 1 N 1 N , from the above equation, we have spanpΘ r1:N,ks´θN 1 N q " spanpΘ 1 r1:N,ks´θ 1 N 1 N q. That is, there exists λ N P R, such that Θ r1:N,ks´θN 1 N " λ N pΘ 1 r1:N,ks´θ 1 N 1 N q.(44) Because }θ i }, }θ 1 i } ď C for all i P Z`, we have |λ N | " }Θ r1:N,ks´θN 1 N } }Θ 1 r1:N,ks´θ 1 N 1 N } ď ? N C`?N |θ N | ? N C´?N |θ 1 N | . According to (43), we further have |λ N | ď C`ε C´ε .(45) The last equation in the above display is due to (44). According to (42), }Θ r1:N,ks } " σ 1 pΘ r1:N,1:Ks q ě σ K pΘ r1:N,1:Ks q ě σ ? N. Combining this with (46) and (45), we have | sin =pΘ r1:N,ks , Θ 1 r1:N,ks q| ď |pθ N´θ 1 N q|pC`εq pC´εqσ . Taking limit in the above display, we arrive at sin`=pΘ rks , Θ 1 rks q " lim sup N Ñ8 |pθ N´θ 1 N q|pC`εq pC´εqσ " 0. This completes our proof. Proof of Proposition 8. The proof of Proposition 8 is similar to that of Theorem 2. We only state the difference. Throughout this proof, we will use Θ˚, A˚,Θ, and to represent Θr 1:N,1:Ks , Ar 1:J,1:Ks ,Θ pN,Jq , and pN,Jq , respectively, to simplify the notation. We define M˚" Θ˚A˚J andM "Θ J . Then, l Ω pMq´l Ω pM˚q ě 0. Based on the above inequality, we obtain the following inequality which is similar to (20). N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 ω ij ď 2 min |ν|ďC 12 b 2 pνq N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqω ij pm ij´mi j q. Comparing the left-hand side of the above display with its conditional expectation given Y ij s, we further obtain the following inequality. n NJ N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 ď 2 min |ν|ďC 12 b 2 pνq N ÿ i"1 J ÿ j"1 pY ij´b 1 pmi j qqω ij pm ij´mi j q´N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q ď 4 min |ν|ďC 12 b 2 pνq sup M PC |xZ,My|`| N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q|,(47) where Z " pZ ij , 1 ď i ď N, 1 ď j ď Jq with Z ij " X ij ω ij " pY ij´b 1 pmi j qqω ij . We establish probability tail bounds for the two terms on the right-hand side of the above display. We start with the second term. Note that E " N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q ı " E ! E " N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q|Y ı) " 0, and V ar " N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q ı " N ÿ i"1 J ÿ j"1 Epm ij´mi j q 4 pω ij´n NJ q 2 ď N ÿ i"1 J ÿ j"1 p2C 1 2 q 4 n NJ ď n16C 1 8 . Thus, by Chebyshev's inequality, for all s ą 0, P p N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q ą sq ď t´2n16C 1 8 .(48) We proceed to the first term on the right-hand side of (47). Using the same arguments as P´|xZ, Θ 1 A 1J y| ě t´2γ ? NJδC 1¯`P p}Z} F ě γq(49) for any positive δ and γ. Npδq is already bounded by (25). We proceed to bound P´|xZ, Θ 1 A 1J y| ě t´2γ ? NJ δC 1¯. We use the following lemma which is similar to Lemma 2. Lemma 5. There exist constant δ 0 , ε 0 (depending on b, C 1 ) such that for allM P C 0 ă t ă δ 0 n, P p|xZ,My| ą tq ď 2 expp´ε 0 t 2 n q. Using the above lemma, we have for 0 ď t´2γ ? NJ C 1 ď δ 0 n P´|xZ, Θ 1 A 1J y| ě t´2γ ? NJ δC 1¯ď 2 expt´p t´2γ ? NJδC 1 q 2 n u.(50) Now we consider P p}Z} F ě γq. By Chebyshev's inequality, we have P p}Z} F ą γq ď γ´2Ep}Z} 2 F q " γ´2φ n NJ N ÿ i"1 J ÿ j"1 b 2 pmi j q ď γ´2nφ sup |ν|ďC 1 b 2 pνq.(51) Combining (49), (50) and (51) ďp CK δ q KpN`Jq expt´ε 0 pt´2γ ? NJ δC 1 q 2 n u`γ´2nφ sup |ν|ďC b 2 pνq " exptKpN`Jqplog C 1`l og K´log δq´ε 0 ppt´2γ ? NJ δC 1 q 2 q n u`γ´2nφ sup |ν|ďC 12 b 2 pνq. In the above display, we choose γ " b 2nφpN`Jq sup |ν|ďC 1 bpνq, δ " pNJq´1 {2 and t " 2γ ? NJ δC 1`1 ? ε 0 a 2nKpN`Jqplog C 1`l og K´log δq, then P psup MPC |xX,My| ą tq ď expt´KpN`Jqplog C 1`l og K`1 2 logpNJqqu`1 2pN`Jq . Now we let s " a 2npN`Jq16C 1 8 in (48) and obtain P p N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 pω ij´n NJ q ą sq ď 1 2pN`Jq Combining the above two inequalities with (47), we arrive at P´n NJ N ÿ i"1 J ÿ j"1 pm ij´mi j q 2 ě s`t¯ď expt´KpN`Jqplog C 1`l og K`1 2 logpNJqqu`1 N`J . Note that there exists constant κ 1 such that s`t ď κ 1 a npN`Jq logpNJq. Thus, P´}M´M˚} 2 2 NJ ě κ 1 c pN`Jq logpNJq nď expt´KpN`Jqplog C 1`l og K`1 2 logpNJqqu`1 N`J ď 2 N`J , for sufficiently large N and J. This completes the proof of (14). To prove (15), we note that (29) and (30) still hold. Thus we could apply the same proof as the proof of (7). We omit the repetitive details. A.6.1 Proof of Lemma 5 Proof of Lemma 5. Consider the moment generating function for |λ| ă 1, Epe λxZ,My q "E´exp " N ÿ i"1 J ÿ j"1 φ´1tbpmi j`φ λm ij ω ij q´bpmi j qu´λb 1 pmi j qω ij ı" E´exp " N ÿ i"1 J ÿ j"1 1 2 b 2 pm : ij qφλ 2m2 ij ω ij ıď E´exp " κ 2 λ 2 N ÿ i"1 J ÿ j"1 ω ij ı¯, where κ 2 " sup |ν|ďφC 14 bpνqφ. Note that ř N i"1 ř J j"1 ω ij " BinomialpNJ, n{pNJqq. Thus, Epexprκ 2 λ 2 N ÿ i"1 J ÿ j"1 ω ij sq " p1´n NJ`n NJ e κ 2 λ 2 q N J ď e npe κ 2 λ 2´1 q . When λ is sufficiently close to 0, the above display can be further bounded from above by e 2κ 2 nλ 2 . Thus, xZ,My is sub-exponential and we could use Bernstein's inequality and arrive at P pxZ,My ą tq ď 2 expp´ε 0 t 2 n q for some ε 0 , δ 0 and t ď δ 0 n . A.7 Proofs of Results in Section 6 Proof of Proposition 9. Our proof is based on the following three results. The first one is proved in Anderson and Duffin (1969). It provides a characterization of the intersection of two spaces through the pseudo inverse of the sum of two projection matrices. Lemma 6 (Theorem 8 of Anderson and Duffin (1969)). P LXM " 2P L pP L`PM q : P M , where P is the projection operator and A : is the pseudo-inverse (or Moore-Penrose generalized inverse) of a matrix A. The second result is from Stewart (1977). It provides perturbation bounds for the pseudoinverse of a matrix. Lemma 7 (Theorem 3.3 of Stewart (1977)). For any two matrices A and B, }B :´A: } ď 3 maxp}A : } 2 2 , }B : } 2 2 q}B´A}. Here, the norm }¨} could be any uniformly generated, normalized, and unitary invariant norm. See Stewart (1977) for the detailed definition. In order to apply the above lemma to our problem, we need an upper bound of }pP LP M q : } 2 and }pP L 1`P M 1 q : } 2 . The next two Lemmas provide an upper bound through the smallest positive principal angle between L and M. Lemma 8. Let L and M be any two linear subspaces of a finite dimensional vector space. For all x P L`M, }P L x`P M x} ě }x}{βpθ min,`p L, Mqq, where we define βpθq " a αpθq. Lemma 9 (Spectral norm of the pseudo-inverse). For any matrix A, }A : } 2 " 1 inf xPRpA J q,}x}"1 }Ax} . Now we apply Lemma 8 and Lemma 9 to the operator P L`PM , and arrive at }pP L`PM q : } 2 ď βpθ min,`p L, Mqq. Similarly, we have }pP L 1`P M 1 q : } 2 ď βpθ min,`p L 1 , M 1 qq. For the norm }¨} discussed here, it has a nice property that }AB} ď }A} 2 }B} and }AB} ď }A}}B} 2 for any matrices A, B. Now we can combine all the results to bound }P L 1 XM 1Ṕ LXM }. We have }P L 1 XM 1´P LXM } ď2}P L´PL 1 }}pP L`PM q : } 2 }P M } 2`2 }pP L`PM q :´p P L 1`P M 1 q : }}P L 1 } 2 }P M 1 } 2 2}P M 1´P M }}P L 1 } 2 }pP L 1`P M 1 q : } 2 ď2}P L´PL 1 }βpθ min,`p L, Mqq`2}pP L`PM q :´p P L 1`P M 1 q : } 2}P M 1´P M }βpθ min,`p L 1 , M 1 qq ď2}P L´PL 1 }βpθ min,`p L, Mqq`2}P M 1´P M }βpθ min,`p L 1 , M 1 qq 6 maxtβpθ min,`p L, Mqq 2 , βpθ min,`p L 1 , M 1 qq 2 u}P L´PL 1`P M´PM 1 } ď " 6 maxtβpθ min,`p L, Mqq 2 , βpθ min,`p L 1 , M 1 qq 2 u`2 maxtβpθ min,`p L, Mqq, βpθ min,`p L 1 , M 1 qqu ‰ p}P L´PL 1 }`}P M´PM 1 }q ď8 maxtβpθ min,`p L, Mqq 2 , βpθ min,`p L 1 , M 1 qq 2 up}P L´PL 1 }`}P M´PM 1 }q ď8 maxtαpθ min,`p L, Mqq, αpθ min,`p L 1 , M 1 qqup}P L´PL 1 }`}P M´PM 1 }q, where the last inequality holds, as βpθ min,`p L, Mqq ě 1 and βpθ min,`p L 1 , M 1 qq ě 1. Proof of Lemma 1. If σ |S 1 YS 2 | pW rS 1 YS 2 s q " 0, then the right-hand side of (17) is 1 and the lemma holds. In the rest of the proof, we assume σ |S 1 YS 2 | pW rS 1 YS 2 s q ą 0. Let X " W rS 1 zS 2 s´PRpW rS 1 XS 2 s q W rS 1 zS 2 s , Y " W rS 2 zS 1 s´PRpW rS 1 XS 2 s q W rS 2 zS 1 s and Z " W rS 1 XS 2 s . Then, RpZq " RpW rS 1 s q X RpW rS 2 s q. Let θ 1 ď θ 2 ď ... ď θ minp|S 1 |,|S 2 |q be the principal angles between RpW rS 1 s q and RpW rS 2 s q. Since dimpRpZqq " |S 1 X S 2 |, we have θ 1 " θ 2 " ... " θ |S 1 XS 2 | " 0, and θ |S 1 XS 2 |`1 " θ min,`p RpW rS 1 s q, RpW rS 2 s qq. We proceed to find an upper bound of cospθ |S 1 XS 2 |`1 q. In what follows, we find the upper bound according to the definition of θ |S 1 XS 2 |`1 . Note that the space RpXq and RpZq are orthogonal and RpXq`RpZq " RpW rS 1 s q. Similarly, RpY q and RpZq are orthogonal and RpY q`RpZq " RpW rS 2 s q. We have 2´2 cospθ |S 1 XS 2 |`1 q "2´2 max uPRpXq,vPRpY q,}u}"1,}v}"1 xu, vy " min uPRpXq,vPRpY q,}u}"1,}v}"1 t2´2xu, vyu " min uPRpXq,vPRpY q,}u}"1,}v}"1 }u´v} 2 . We use the following lemma to proceed. Lemma 10. σ |pS 1 zS 2 qYpS 2 zS 1 q| prX, Y sq ě σ |S 1 YS 2 | pW rS 1 YS 2 s q, and }rX, Y s} 2 ď }W rS 1 YS 2 s } 2 . From the above lemma, we have σ |S 1 zS 2 | pXq ą σ |S 1 zS 2 YS 2 zS 1 | prX, Y sq ą 0. Similarly, σ |S 2 zS 1 | pY q ą 0. Thus, X J X and Y J Y are invertible. For u P RpXq, v P RpY q, let w P R m , x P R m such that u " XpX J Xq´1 {2 w, and v " Y pY J Y q´1 {2 x. Then, it is easy to verify that }u} " }w} and }v} " }x}. Thus, (52) becomes 2´2 cospθ |S 1 XS 2 |`1 q ě min }w}"1,}x}"1 }XpX J Xq´1 {2 w´Y pY J Y q´1 {2 x} 2 ě min }w} 2`} x} 2 "2 }XpX J Xq´1 {2 w´Y pY J Y q´1 {2 x} 2 "2λ min pW 1 q,(53) where W 1 " » --- Then, we have I |S 1 zS 2 | pX J Xq´1 {2 X J Y pY J Y q´1 {2 pY J Y q´1 {2 Y J XpX J Xq´1 {2IW 1 " D´1 {2 W 2 D´1 {2 . Or, equivalently, W 2 " D 1{2 W 1 D 1{2 Note that W 1 , W 2 , D are all symmetric positively definite matrices. Combine this fact with Lemma 10, we have λ min pW 1 q ě λ min pW 2 q }D 1{2 } 2 2 ě λ min prX, Y s J rX, Y sq }rX, Y s} 2 2 ě σ |S 1 YS 2 | pW rS 1 YS 2 s q 2 }W } 2 2 . Combining this with (53), we arrive at cospθ |S 1 XS 2 |`1 q ď 1´σ |S 1 YS 2 | pW rS 1 YS 2 s q 2 }W } 2 2 . This completes our proof. A.7.1 Proofs of Lemmas 8, 9, and 10 Proof of Lemma 8. Our proof for this lemma has two steps: 1. We first prove the statement for L X M " t0u. 2. We generalize it to the case where dimpL X Mq ě 1. Step 1, L X M " t0u Let θ 1 ď θ 2 ď ... ď θ m be the principal angles between L and M with m " minpdimpLq, dimpMqq. Of particular interest is the smallest principal angle θ 1 , which is defined as θ 1 " max uPL,vPM,}u}"1,}v}"1 arccos |xu, vy|, or, equivalently, cospθ 1 q " max }u}"1,uPL,}v}"1,vPM |xu, vy|. We start with a lower bound for }P L x} 2`} P M x} 2 for x P L`M. Since we assume x P L`M and L X M " t0u, there exists a unique pair x P L and y P M such that x " x`y. }P L x} 2`} P M x} 2 "}x`P L y} 2`} y`P M x} 2 ěp}x}´}P L y}q 2`p }y}´}P M x}q 2 . On the other hand, }P L y} " max uPL,}u}"1 |xu, yy| ď max uPL,}u}"1,vPM,}v}"1 |xu, vy|}y} " cospθ 1 q}y}. Similarly, }P M x} ď cospθ 1 q}x}. Combining this with (54), we have }P L x} 2`} P M x} 2 ěp}x}´}y} cospθ 1 qq 2`p }y}´}x} cospθ 1 qq 2 ě 1 2 tp}x}´}y} cospθ 1 qq``p}y}´}x} cospθ 1 qq`u 2 ě 1 2 p}x}`}y}q 2 p1´cospθ 1 qq 2 . On the other hand, }x} 2 " }y} 2`} x} 2`2 xx, yy ď }y} 2`} x} 2`2 cospθ 1 q}x}}y} ď p1`cospθ 1 qqp}x} 2`} y} 2 q. (56) Combining (55) and (56), we arrive at }P L x} 2`} P M x} 2 ě p1´cospθ 1 qq 2 }x} 2 2p1`cospθ 1 qq . On the other hand, since P L x P L and P M x P M, we have }P L x`P M x} 2 "}P L x} 2`} P M x} 2`2 xP L x, P M xy ě}P L x} 2`} P M x} 2´2 cospθ 1 q}P L x}}P M x} "p1´cospθ 1 qqp}P L x} 2`} P M x} 2 q`cospθ 1 qp}P L x}´}P M x}q 2 ěp1´cospθ 1 qqp}P L x} 2`} P M x} 2 q. Now we combine (57) with (58). Then, we arrive at }P L x`P M x} 2 ě p1´cospθqqp}P L x} 2`} P M x} 2 q ě p1´cospθ 1 qq 3 }x} 2 2p1`cospθ 1 qq " }x} 2 pβpθ 1 qq 2 . Thus, for all x P L`M, }P L x`P M x} ě }x}{βpθ 1 q, Step 2. L X M " S and dimpSq ě 1. Let L´and M´be linear subspaces such that L´K S, L´`S " L, M´K S and M´`S " M. It is not hard to show that L´XM´" t0u. Now, for x P L`M " S`L´`M´, there is a unique pair px S , x´q such that x " x S`xá nd x s P S and x´P L´`M´. We note that P L x`P M x " x S`PL x´`P M x´. Also, P L x´" P L´PL x´`P S P L x´" P L´x´`PS x´" P L´x´. Similarly, P M x´" P M´x´. Combining the above three equations, we have P L x`P M x " x S`PL´x´`PM´x´. Note that x S K P L´x´`PM´x´. Thus, }P L x`P M x} 2 " }x S } 2`} P L´x´`PM´x´} 2 . According to Step 1 of the proof, we further have }P L x`P M x} 2 ě }x S } 2`} x´} 2 β 2 pθ 1 pL´, M´qq . Recall that x " x S`x´a nd x S K x´, and βpθq ě 1 for all θ, we further bound the above display by }P L x`P M x} 2 ě }x} 2 β 2 pθ 1 pL´, M´qq . To complete the proof, the only thing we need to prove is θ 1 pL´, M´q " θ min,`p L, Mq, which is obvious according to the iterative definition of principal angles between linear spaces. Proof of Lemma 9. We write the SVD of A, A " UΣV J " U 1 Σ 1 V J 1 , where Σ " diagpσ 1 , ..., σ m , 0, ..., 0q, and U 1 " U r1:ms , Σ 1 " diagpσ 1 , ..., σ m q and V 1 " V r1:ms . Then, the pseudo inverse has the following form (Hogben, 2006, p. 5-13, fact 2), A : " UΣ : V J , with Σ : " diagp1{σ 1 , ..., 1{σ m , 0, ..., 0q. The spectral norm is unitary invariant, so }A : } 2 " }Σ : } 2 " max i"1,..,m 1{σ i " 1 min 1ďiďm σ m .(59) On the other hand, consider x P RpA J q " RpV 1 q and }x} " 1. We write x " V 1 y for y P R m . Then, }y} " }x} " 1. We have }Ax} 2 " x J V 1 Σ 1 U J 1 U 1 Σ 1 V J 1 x " x J V 1 Σ 2 1 V J 1 x " y J Σ 2 1 y. Thus, inf xPRpA J q,}x}"1 }Ax} 2 " inf yPR m ,}y}"1 y J Σ 2 1 y " min i"1,...,m σ 2 i .(60) Combining (59) and (60), we completes the proof. Proof of Lemma 10. From definition, we have rX, Y s " pI n´PRpZq qrW rS 1 zS 2 s , W rS 2 zS 1 s s " pI n´PRpZq qH, where we write H " rW rS 1 zS 2 s , W rS 2 zS 1 s s " W rpS 1 zS 2 qYpS 2 zS 1 qs . Since σ |S 1 XS 2 | pZq " σ |S 1 XS 2 | pW rS 1 XS 2 s q ě σ |S 1 YS 2 | pW rS 1 YS 2 s q ą 0, we know Z J Z is invertible and P RpZq " ZpZ J Zq´1Z J . Thus we have rX, Y s J rX, Y s " H J pI n´Z pZ J Zq´1Z J qH " H J H´H J ZpZ J Zq´1Z J H. }`H J H´H J ZpZ J Z˘´1 Z J Hq´1} 2 ď › › › › › › › › » - - - H J H H J Z Z J H Z J Z fi ffi ffi fl´1 › › › › › › › › 2 " σ |S 1 YS 2 | prH, Zsq´1. Note that rH, Zs " W rS 1 YS 2 s . Thus, from the above display we have }`H J H´H J ZpZ J Z˘´1 Z J Hq´1} 2 ď σ |S 1 YS 2 | pW rS 1 YS 2 s q´1. Note that }pH J H´H J ZpZ J Zq´1Z J Hq´1} 2 " σ l pH J H´H J ZpZ J Zq´1Z J Hq´1 " σ l prX, Y sq´1. Thus, we arrive at σ l prX, Y sq´1 ď σ |S 1 YS 2 | pW rS 1 YS 2 s q´1. Consequently, σ l prX, Y sq ě σ |S 1 YS 2 | pW rS 1 YS 2 s q. This proves the first statement of the lemma. For the second statement of the lemma, we have }rX, Y s} 2 " }pI n´PRpZq qW rS 1 zS 2 YS 2 zS 1 s } 2 ď }I n´PRpZq } 2 }W rS 1 zS 2 YS 2 zS 1 s } 2 ď }W rS 1 YS 2 s } 2 . Here we used the fact that }I n´PRpZq } 2 ď 1. denotes the set of manifest variables that are associated with and only with latent factors in S. Discussions on the parameter space are provided after the statement of Theorem 1. Theorem 1. Under Assumptions A1 and A2, the k-th latent factor is structurally identifiable in S Q if and only if tku " č kPS,p Q pSqą0 S, tpΘ pN,Jq , pN,Jq q, N, J P Z`u is said to consistently estimate the latent factor k if sin =pΘ pN,Jq rks , Θ r1:N,ks q P Θ,A Ñ 0, for all pΘ, Aq P S Q . Corollary 1 . 1Under Assumptions A1 and A2, there exists an estimator pΘ pN,Jq , pN,Jq q such that lim N,JÑ8 sin =pΘ pN,Jq rks , Θ r1:N,ks q " 0 in P Θ,A for all pΘ, Aq P S Q if and only if the design matrix Q satisfies (4). approximates Θr 1:N,ks . Since the likelihood function depends on Θ r1:N,1:Ks and A r1:J,1:Ks only through Θ r1:N,1:Ks A J r1:J,1:Ks , the scale of Θ r1:N,ks is not identifiable even when it is structurally identifiable. This phenomenon is intrinsic to latent variable models (e.g. Skrondal and Rabe-Hesketh, 2004). Corollary 2 states that Θr 1:N,ks andΘ pN,Jq rks are close in Euclidian distance after properly normalized. The normalized vectors Θr 1:N,ks {}Θr 1:N,ks } and c N,JΘ pN,Jq rks {}Θ pN,Jq rks } are both of unit length. The value of c N,J depends on the angle between Θr 1:N,ks andΘ pN,Jq rks . Specifically, c N,J " 1 if cos =pΘr 1:N,ks ,Θ pN,Jq rks q ą 0 and c N,J "´1 otherwise. In practice, especially in psychological measurement, c N,J can typically be determined by additional domain knowledge. establish a new perturbation bound for the intersection of linear spaces, which may be of independent theoretical interest. Let RpW q denote the column space of a matrix W . Under the conditions of Theorem 2, the result of (6) combined with the Davis-Kahan-Wedin sine theorem (see e.g.Stewart and Sun, 1990) allows us to bound | sin =pRpΘr 1:N,Ss q, RpΘpN,Jq rSs qq|, for any S satisfying p Q pSq ą 0, where =pL, Mq denotes the largest principal angle between two linear spaces L and M, i.e., sin =pL, Mq " max uPM,u‰0 min vPL,v‰0 | sin =pu, vq|. Our strategy is to bound | sin =pΘr 1:N,ks ,Θ pN,Jq rks q| " sin =pRpΘr 1:N,ks q, RpΘ pN,Jq rks qq by sin =pRpΘr 1:N,1:Ks q, RpΘ pN,Jq qq under the assumptions of Theorem 2. Note that RpΘr 1:N,ks q " Ş kPS,p Q pSqą0 RpΘr 1:N,Ss q and similarly RpΘ pN,Jq rks q " Ş kPS,p Q pSqą0 RpΘ pN,Jq rSs q. Consequently, it remains to show that if the linear spaces are perturbed slightly, then their intersection does not change much. To this end, we establish a new perturbation bound on the intersection of general linear spaces in the next proposition.Proposition 9 (Perturbation bound for intersection of linear spaces). Let L, M, L 1 , M 1 be linear subspaces of a finite dimensional vector space. Then, }P L 1 XM 1´P LXM } ď 8 maxtαpθ min,`p L, Mqq, αpθ min,`p L 1 , M 1 qqup}P L´PL 1 }`}P M´PM 1 }q, 2 |S 1 Figure 1 : 211YS 2 | pW rS 1 YS 2 The value of }Θ pN,Jq p pN,Jq q J´Θr 1:N,1:Ks A˚J r1:J,1:Ks } 2 F {pNJq versus the number of manifest variables J under different simulation settings (solid line: simple structure; dashed line: mixed structure). The median, 25% quantile and 75% quantile based on the 50 independent replications are shown by the dot, lower bar, and upper bar, respectively. For each model and each design structure, a range of J values are considered and we let N " 25J. Specifically, we consider J " 100, 200, ..., 1000 for the linear and the Poisson models and J " 200, 400, ..., 2000 for the MIRT model. For each combination of a model, a design structure, and a J value, 50 independent datasets are generated. For each dataset, we apply Algorithm 1 to solve (5), where C 1 " 1.2C.Results are shown inFigures 1-5.Figure 1shows the trend of 1 N J }Θ pN,Jq p pN,Jq q J2 Figure 2 : 2The value of | sin =pΘr 1:N,1s ,Θ pN,Jq r1s q| under different simulation settings (solid line: simple structure; dashed line: mixed structure). The median, 25% quantile and 75% quantile based on the 50 independent replications are shown by the dot, lower bar, and upper bar, respectively. Θr 1:N,1:Ks A˚J r1:J,1:Ks } 2 F (y-axis) when J increases (x-axis), where each panel corresponds to a model. This result verifies (6) in Theorem 2. According to these plots, the normalized squared Frobenius norm between Θr 1:N,1:Ks A˚J r1:J,1:Ks and its estimateΘ pN,Jq p pN,Jq q J decays towards zero, as J and N increase. Figures 2-5 present results based on the first latent factor and we point out that the results based on the other latent factors are almost the same. Figure 2 Figure 3 : 23is used to verify (7) in Theorem 2, showing the pattern that | sin =pΘr 1:N,ks ,Θ pN,Jq rks q| decreases as J and N increase. Moreover,Figure 3provides evidence on the result of Proposition 3. Displayed inFigure 3are the histograms of v i s andv i s, respectively, based on a randomly selected dataset when J " 1000 under the Poisson model and the simple structure. According to this figure, little difference is observed between the empirical distribution of v i s and that ofv i s. Similar results are observed for other datasets under all these three models when J and N are large.Finally,Figures 4 and 5show results on the ranking and the classification, whose theoretical results are given in Propositions 4 and 5. The y-axes of the two figures show the normalized Kendall's tau distance in (11) and the classification error in (12), respectively. Comparison between the histogram of v i s and that ofv i s for the first latent factor under the Poisson Model and the simple structure. Figure 4 : 4The Kendall's tau ranking error calculated from Θr 1:N,1s andΘ pN,Jq r1s , under different simulation settings (solid line: simple structure; dashed line: mixed structure). The median, 25% quantile and 75% quantile based on the 50 independent replications are shown by the dot, lower bar, and upper bar, respectively. Figure 5 : 5The classification error calculated from Θr 1:N,1s andΘ pN,Jq r1s with indifference zone p0.13, 0.43q under different simulation settings (solid line: simple structure; dashed line: mixed structure). The median, 25% quantile and 75% quantile based on the 50 independent replications are shown by the dot, lower bar, and upper bar, respectively. Figure 6 : 6q| as J increases. In particular, the value of | sin =pΘr 1:N,ks ,Θ pN,Jq rks q| stays above 0.2 for most of the data sets for the factor The value of }Θ pN,Jq p pN,Jq q J´Θr 1:N,1:Ks A˚J r1:J,1:Ks } 2 F {pNJq versus the number of manifest variables J under different simulation settings (solid line: Linear Model; dashed line: MIRT Model; dotted line: Poisson Model). The median, 25% quantile and 75% quantile based on the 50 independent replications are shown by the dot, lower bar, and upper bar, respectively. Figure 7 : 7The value of | sin =pΘr 1:N,1s ,Θ pN,Jq r1s q| under different simulation settings (solid line: the first latent trait; dashed line: the second latent trait). The median, 25% quantile and 75% quantile based on the 50 independent replications are shown by the dot, lower bar, and upper bar, respectively. , Θ r1:N,ks q " 0 in P Θ,A and lim N,JÑ8 sin =pΘ pN,Jq rks , Θ 1 r1:N,ks q " 0 in P Θ 1 ,A 1 .Note that P Θ 1 ,A 1 " P Θ,A , so the above display further N,ks q " 0 in P Θ,A . |S 2 zS 1 | fi ffi ffi fl and λ min pW q denotes the smallest eigenvalue of W . |S 1 zS 2 |,|S 2 zS 1 | 0 |S 2 zS 1 |,|S 1 zS 2 | Y J Y fi ffi ffi fl . Recall the formula of inverse of a block matrix, we have pH J H´H J ZpZ J Zq´1Z J Hq´1 "write l " |pS 1 zS 2 qYpS 2 zS 1 q|. Because pH J H´H J ZpZ J Zq´1Z J Hq´1 is a dard asymptotic setting in statistics. Under this setting, this paper addresses three research questions that are central to modern measurement theory. First, how should the identifiability of latent factors be suitably formalized? Second, under what design are the latent factors identifiable? Third, what is the relationship between the identifiability and estimability? b. R Z`: the set of vectors with countably infinite real number components.c. R Z`ˆt1,...,Ku : the set of all the real matrices with countably infinite rows and K columns.d. t0, 1u Z`ˆt1,...,Ku : the set of all the binary matrices with countably infinite rows and K columns. e. Θ: the parameter matrix for the person population, Θ P R Z`ˆt1,...,Ku . f. A: the parameter matrix for the manifest variable population, A P R Z`ˆt1,...,Ku . g. Q: the design matrix for the manifest variable population, Q P t0, 1u Z`ˆt1,...,Ku . h. 0: the vector or matrix with all components being 0. i. P Θ,A : the probability distribution of pY ij , i, j P Z`q, given person and item parameters Θ and A. j. v r1:ms : the first m components of a vector v. k. W rS 1 ,S 2 s : the submatrix of a matrix W formed by rows S 1 and columns S 2 , where S 1 , S 2 Ă Z`. l. W r1:m,ks : the first m components of the k-th column of a matrix W . m. W rks : the k-th column of a matrix W . n. }v}: the Euclidian norm of a vector v.o. sin =pu, vq: the sine of the angle between two vectors, is reasonable. Consider a random design setting where θi k s are i.i.d. samples from some distribution with a finite second moment. Then F N converges weakly to the distribution of η{ a Eη 2 , where η is a random variable following the same distribution. Proposition 3 then implies that when factor k is structurally identifiable and both N and J are large, the empirical distribution ofθ mates the empirical distribution of θ1 k , θ2 k , ..., θN k accurately, up to a scaling. Specifically, for any 1-Lipschitz function h, ş hpxqF N,J pdxq is a consistent estimator for ş hpxqF N pdxq according to the definition of Wasserstein distance. Furthermore, Corollary 2 states that under the regularity conditions, lim N Ñ8pN,Jq 1k ,θ pN,Jq 2k , ...,θ pN,Jq N k approxi- We can see that lim N Ñ8 N´1xΘ r1:N,ks , Θ 1 r1:N,ks y " pC 2 {4Kq, lim N Ñ8 N´1 {2 }Θ r1:N,ks } "1 0 1 1 0 K´2 0 K´2 . . . . . .. fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl C{p2 ? Kq, and lim N Ñ8 N´1 {2 }Θ 1 r1:N,ks } " C{ ? 2K. Thus, lim N Ñ8 cos =pΘ r1:N,ks , Θ 1 r1:N,ks q " lim N Ñ8 |xΘ r1:N,ks , Θ 1 r1:N,ks y| }Θ r1:N,ks }}Θ 1 r1:N,ks } " 1 ? 2 . for all N, J P Z`. Now we focus on N ě N 0 and J ě J 0 . For all S such that p Q pSq ą 0, weConsequently, for all N, J, we have Θ r1:N,1:Ks A J r1:J,1:Ks " Θ 1 r1:N,1:Ks A 1J r1:J,1:Ks have pΘ r1:N,1:Ks A J r1:J,1:Ks q rR Q pSqXr1,Js,1:Ks " Θ r1:N,1:Ks A J rR Q pSqXr1,Js,1:Ks " Θ r1:N,Ss A J rR Q pSqXr1:Js,Ss Similarly, pΘ 1 r1:N,1:Ks A 1J r1:J,1:Ks q rR Q pSqXr1,Js,1:Ks " Θ 1 r1:N,Ss A 1J rR Q pSqXr1:Js,Ss . Therefore, Θ r1:N,Ss A J rR Q pSqXr1:Js,Ss " Θ 1 r1:N,Ss A 1J rR Q pSqXr1:Js,Ss . According to (41) and (42), we know Θ r1:N,Ss , Θ 1 r1:N,Ss , A rR Q pSqXr1:Js,Ss , A 1 rR Q pSqXr1:Js,Ss all have full rank (rank |S|). Thus, RpΘ r1:N,Ss q " RpΘ 1 r1:N,Ss q. Now we find an upper bound on | sin =pΘ r1:N,ks , Θ 1 r1:N,ks q|. We have | sin =pΘ r1:N,ks , Θ 1 r1:N,ks q| " inf λPR }Θ r1:N,ks´λ Θ 1 r1:N,ks q} }Θ r1:N,ks } N q1 N } }Θ r1:N,ks } .ď }Θ r1:N,ks´λN Θ 1 r1:N,ks q} }Θ r1:N,ks } " }λ N pθ N´θ 1 |xZ,My| ą t¯ď Npδq sup pΘ 1 ,A 1 qPS(24), we arrive at P´sup M PC , we haveP psup MPC |xX,My| ą tq Standards for educational and psychological testing. Aera, American Educational Research Association. AERA, APA and NCME (2014). Standards for educational and psychological testing, Amer- ican Educational Research Association, Washington, DC. Asymptotic properties of conditional maximum-likelihood estimators. E B Andersen, Journal of the Royal Statistical Society. 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{'abstract': 'Latent factor models are widely used to measure unobserved latent traits in social and behavioral sciences, including psychology, education, and marketing. When used in a confirmatory manner, design information is incorporated, yielding structured (confirmatory) latent factor models. Motivated by the applications of latent factor models to large-scale measurements which consist of many manifest variables (e.g. test items) and a large sample size, we study the properties of structured latent factor models under an asymptotic setting where both the number of manifest variables and the sample size grow to infinity. Specifically, under such an asymptotic regime, we provide a definition of the structural identifiability of the latent factors and establish necessary and sufficient conditions on the measurement design that ensure the structural identifiability under a general family of structured latent factor models. In addition, we propose an estimator that can consistently recover the latent factors under mild conditions. This estimator can be efficiently computed through parallel computing. Our results shed lights on the design of large-scale measurement and have important 1 implications on measurement validity. The properties of the proposed estimator are verified through simulation studies. KEY WORDS: High-dimensional latent factor model, confirmatory factor analysis, identifiability of latent factors, structured low-rank matrix, large-scale psychological measurement J rW,T s . Thus, pΘ˚A J˚Θ J q r1:N,W s " Θr T s A Jr W,T s´Θ rT s J rW,T s . That is, Θr T s A Jr W,T s´Θ rT s J rW,T s is a submatrix of Θ˚A J˚´ΘÂJ . Thus, we have the following bound }Θr T s A Jr W,T s´Θ rT s J rW,T s } 2 ď }Θ˚A J˚´ΘÂJ } 2 ďwhich implies that both Θr Ss and Ar s,rs have full rank, and RpΘr Ss q " RpU S q. By Weyl\'s perturbation theorem (see, e.g. Stewart and Sun (1990)), we have |σ |S| pΘ rSs J rR Q pSq,Ss q´σ |S| pΘr Ss A˚J rR Q pSq,Ss q| ď }Θ rSs J rR Q pSq,Ss´ΘrSs A˚J rR Q pSq,Ss } 2 .46Combining the above display with(39)and(40), we have σ |S| pΘ rSs J rR Q pSq,Ss q ě ? NJσ 2´}Θ rSs J rR Q pSq,Ss´ΘrSs A˚J rR Q pSq,Ss } 2 ą 0.Thus, we also haveΘ rSs and rs,rs have full rank. We writeΘ rSs J rs,rs "Û SΣSV J S , the reduced singular value decomposition. Then, RpΘ rSs q " RpÛ S q. By the modified Davis-', 'arxivid': '1712.08966', 'author': ['Yunxiao Chen \nDepartment of Psychology\nSchool of Statistics\nShanghai Center for Mathematical Sciences\nEmory University\nUniversity of Minnesota\nFudan University\n\n', 'Xiaoou Li \nDepartment of Psychology\nSchool of Statistics\nShanghai Center for Mathematical Sciences\nEmory University\nUniversity of Minnesota\nFudan University\n\n', 'Siliang Zhang \nDepartment of Psychology\nSchool of Statistics\nShanghai Center for Mathematical Sciences\nEmory University\nUniversity of Minnesota\nFudan University\n\n'], 'authoraffiliation': ['Department of Psychology\nSchool of Statistics\nShanghai Center for Mathematical Sciences\nEmory University\nUniversity of Minnesota\nFudan University\n', 'Department of Psychology\nSchool of Statistics\nShanghai Center for Mathematical Sciences\nEmory University\nUniversity of Minnesota\nFudan University\n', 'Department of Psychology\nSchool of Statistics\nShanghai Center for Mathematical Sciences\nEmory University\nUniversity of Minnesota\nFudan University\n'], 'corpusid': 73650241, 'doi': '10.1080/01621459.2019.1635485', 'github_urls': [], 'n_tokens_mistral': 41148, 'n_tokens_neox': 37217, 'n_words': 20586, 'pdfsha': '42be1d0cbe0e1ea0f29ccce29e25aa33686edcc7', 'pdfurls': ['https://arxiv.org/pdf/1712.08966v1.pdf'], 'title': ['Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications', 'Structured Latent Factor Analysis for Large-scale Data: Identifiability, Estimability, and Their Implications'], 'venue': []}

arxiv

Geometrical Field Representation of Solid, Fluid, and Gas as Continuum in Rational Mechanics Xiao Jianhua Measurement Institute Henan Polytechnic University 454000JiaozuoHenanChina Geometrical Field Representation of Solid, Fluid, and Gas as Continuum in Rational Mechanics 1 PACS: Contents:rock burstingeffective parametersphase transitionintrinsic stretchinglocal rotationclassical strainconstitutive equationsdeformation tensormotion transformationgauge fieldrational mechanicsgasliquidsolidmultiphase materials Any materials have three physical states as solid, liquid, and gas. Basically, the materials at some molecular cluster scales are assumed as the same for all the above physical states. So, a basic gauge tensor field can be attached to the basic molecular cluster elements (basic material element). Under this objective (material element) invariance, the concept of continuum in rational mechanics is studied in this paper. Different material phases have different basic gauge fields. So, the matter phases are expressed by the basic gauge transformations.Based on this general understanding, the different phases have different motion transformations, internally. Based on the points-set transformation concept about the motion transformation in continuum, the macro classical strain is expressed by the additive addition of the intrinsic stretching of material element and its intrinsic local rotation.For zero classical strain (no macro deformation observed on its configuration surface, suitable container is required for liquid and gas to make up macro invariant configuration), the results show that: (1) For solid, the local rotation angular is zero. The material element has no intrinsic stretching. (2) For liquid, the local rotation will not change the basic gauge tensor. The material element has intrinsic plane stretching on the rotation plane. (3) For gas state, the intrinsic local rotation will amplify the basic gauge tensor. The material element has intrinsic stretching along the rotation direction. Hence, under the condition of no macro classical strain be observed, the material element has three different physical states: solid (no intrinsic stretching), fluid (plane intrinsic stretching), and gas (directional intrinsic stretching). Furthermore, for the three states, the free conditions are defined by zero intrinsic stretching. Referring to this free condition, the constitutive equations for the materials at multiple states are established. It is shown that the classical constitutive equations are included in this general unified formulation. Introduction In classical physics, the materials as a continuum are classified as solid, liquid, and gas. For each state, some corresponding mechanics theory or interpretations are formulated. The three states are not unified as a continuum in rational sense but it is in the conceptual sense. In this research, a rational unifying is persuaded. In fact, at actual world, many materials are composed of the combination of solid, liquid, and gas states. The minor gas or liquid states material in a working part of machine may cause the failure of the working part. So, for advanced machinery industry, a constitutive equation which taken the three states into consideration are urgently needed. On physical view-point, although the three states are unified as continuum, the rational formulation which can be used to explain the intrinsic difference among them is still missing. Phenomenally, for idea solid state, the deformation of material configuration is assumed as zero without external action. So, it has no configuration change at natural state. Its deformation stress is completely by the classical strain. For liquid, to make it has a fixed configuration (externally observed), a suitable container must be used. The internal motion of material elements is not zero. So, a static compressive pressure parameter is introduced to express such a kind of internal motion. When the internal motion tends to be zero, it becomes to the solid state. So, the boundary between the solid phase and the liquid phase is not so clear as we are taught by the standard textbook. If the internal motion is strong enough, a closed container must be used to keep the macro external configuration of gas. So, the gas has a natural expansion pressure. This is essentially different from the nature of liquid phase. However, the boundary between both is still not clear in rational sense. By physical facts, two or three phases can exist spontaneously for a suitable temperature. As the temperature is the measure the internal motion of materials, the only conclusion which one can make out is that the internal motion is continuous although the different phase states seems to be so different in phenomenon sense. Therefore, a continuous motion representation should be used to unify the different phase states. Only in such a rational formulation, the continuum concept can be viewed as rational concept rather than conceptual ones. This is the main purpose of this paper. Any materials have three physical states as solid, liquid, and gas. Basically, the materials at some molecular cluster scales are the same for all the above physical states. So, a basic gauge tensor field can be attached to the basic molecular cluster elements (basic material element). By this kind of gauge field theory, different materials have different basic gauge fields. Under this objective (material element) invariance principle, the motion concept of continuum in rational mechanics is expressed by the base vectors transformation [1] . Mathematically, the motion in continuum is a point-set transformation with the time as the implied parameter. (Unfortunately, in many researches (including some textbooks), it is treated as a coordinator transformation [2] . Although mathematically it is acceptable with some extent reservation, however, physically, the admissibility of physical motion and the material objective invariance principle are abandoned [3][4][5] .) The geometrical theory based on the points-set transformation concept has been established by Chen Zhida, in 1987 [1] . He finds that the deformation tensor defined on coordinators embedded in the material is the looking-for motion transformation. Mathematically, Chen shows that the deformation tensor can be decomposed into the additive addition of a symmetrical tensor (represents intrinsic stretching of material element under discussion) and an orthogonal local rotation tensor [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] (represents intrinsic local rotation referring to its original orientation determined by neighboring materials). The researches exposes that the macro classical strain is expressed by addition of the intrinsic stretching of material element and its intrinsic local rotation. Using Chen geometrical field theory of motion in continuum [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] , the solid, liquid, and gas with fixed (invariant external) configuration have a common essential feature of zero classical strain (no macro deformation). Hence, the different phase states can be expressed by the different combination of intrinsic stretching and the local rotation. When the Chen decomposition form I and decomposition form II are used to meet the zero classical strain condition, there are three typical internal motion modes. Hence, it is very natural to find that: For zero classical strain (no macro deformation observed on its configuration surface, suitable container is required for liquid and gas to make up macro invariant configuration), the results show that: (1) For solid, the local rotation angular is zero. The material element has no intrinsic stretching. (2) For liquid, the local rotation will not change the basic gauge tensor. The material element has intrinsic plane stretching on the rotation plane. (3) For gas state, the intrinsic local rotation will amplify the basic gauge tensor. The material element has intrinsic stretching along the rotation direction. Hence, for macro static continuum observed externally, under the understanding that no macro classical strain be observed, the material element has three different physical states: solid (no intrinsic stretching), fluid (plane intrinsic stretching), and gas (directional intrinsic stretching). Then, a natural question will be asked: if the container is removed, what are the classical strains to be determined for different phase states. In this research, such a kind of condition is named as the three states. Rationally, the free conditions are defined by zero intrinsic stretching of material elements. Referring to this free condition, the stress concept is rationalized. Once these works have been done, the constitutive equations for the materials at multiple states are established. Based on the conventional standard, the classical constitutive equations must be included in this general unified formulation. This is certain for this research, as it is shown where it is appropriate. The whole paper will be paragraphed as above. The reference papers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] supply the additional documentation for further inquiring. For simplicity, the related results will be used directly. Motion Transformation of Macro Static Continuum For a material cluster element, embedding three coordinators 3 , 2 , 1 , = i x i with three basic vectors 0 i g r , then the material element is defined in continuum. Such a kind of coordinator system is named as commoving dragging coordinator system. Then the motion of the material element within unit time (suitably selected) is defined as: 0 j j i i g F g r r = (1) Where, i g r is the current base vector, i j F is the motion transformation. In mathematical sense, i j F is a point-set transformation. In mechanics, it is named as deformation tensor. Locally, it is constructed by the displacement field i u (It is measured in initial coordinator system. Here, it is the displacement field of unit time; however, the velocity concept will not be used for the acceleration force is no discussed in the research. In fact, the zero acceleration means the static, here). The deformation tensor is defined as: i j j i i j u F δ + = (2) Where, j i u represents the covariant-derivative, i j δ is a unit tensor. Phenomenally, the static state of continuum is defined as no macro deformation is observed externally. That is the classical strain is zero. For different phase of the same material, its physical and mechanics implications are different. Static Solid Solid in natural environment has a fixed configuration. So, its static state is defined as the identical base transformation. Hence, the solid state is defined by the deformation tensor as: i j i j F δ = , for natural static solid(3) As there is no displacement field, the classical strain is zero. Static Liquid For liquid, the material element has non-zero displacement field referring to its initial position. In natural static state, to make it has fixed configuration, a container is used to keep out its flow on horizontal plane. Using Chen decomposition form I, one has: i j i j i j R S F + =(4) Where: k j i k i j j i i j L L u u S ) cos 1 ( ) ( 2 1 Θ − − + = (5) k j i k i j i j i j L L L R ) cos 1 ( sin Θ − + ⋅ Θ + = δ (6) ) ( sin 2 1 i j j i i j u u L − Θ =(7)] ) ( ) ( ) [( 2 1 sin u u u u u u − + − + − = Θ(8) In above expressions, the parameter Θ ( 2 / 2 / π π < Θ < − ) is called local average rotation angel and tensor k j L defines the local average rotation axis direction. i j S is the intrinsic strain tensor, i j R is an unit orthogonal rotation tensor. Not that the intrinsic strain and intrinsic local rotation are defined on material element referring to its initial configuration (the whole displacement and rigid rotation have no contribution to intrinsic deformation, which is eliminated by commoving feature of coordinator system). As the classical strain is defined as: ) ( 2 1 i j j i ij u u + = ε (9) The static liquid will require that: 0 ≡ ij ε (10) So, the material intrinsic stretching will be: ) )( cos 1 ( ) cos 1 ( i j j i k j i k i j L L L L S δ − Θ − − = Θ − − =(11) Here, for simplicity, the following equations are used: , others are zero (13) It can be seen that the liquid has intrinsic isotropic expansion stretching on fluid surface plane direction. This will make the static liquid surface be an idea plane. Theoretically, as the surface normal direction has zero intrinsic stretching, the static liquid can have laminar flow structure, as it is observed in daily life. It represents our phenomenon experience well. The local rotation will form steam line which will not be discussed in this paper. Static Gas For the gas, to keep it has fixed external configuration, a closed container must be used. For this case, the Chen decomposition form II should be used. By this formulation, he unit time deformation (deformation rate) is decomposed as: i j i j i j R S F) (cos 1 − + = θ(14) Where, the related items are: ) )( 1 cos 1 ( ) ( 2 1 i j k j i k i j j i i j L L u u S δ θ + − − + = (15) ) )( 1 cos 1 ( cos siñ ) (cos 1 i j k j i k i j i j i j L L L R δ θ θ θ δ θ + − + + = − (16) ) ( sin 2 cos i j j i i j u u L − = θ θ (17) ] ) ( ) ( ) [( 4 1 1 ) (cosu u u u u u − + − + − + = − θ(18) Here, the i j S is the intrinsic strain tensor, i j R is an unit orthogonal rotation tensor. Differing from the previous mode, the rotation direction tensor is changed into i j L with angular θ ( 2 / 2 / π θ π < < − ). Here, i u (previously, defined as displacement fields within the unit time duration) is the velocity field. Comparing with the liquid, the important point is that: although the deformation tensor is continuous, the intrinsic strain and/or local rotation are not continuous. On mechanic sense, although the macro deformation is continuous, the intrinsic strain is discontinuous and the local rotation jumps from unit orthogonal rotation into orthogonal rotation with volume expansion (say bubbling in fluid environment or cracking in solid environment). When zero classical strain condition is applied, the material intrinsic stretching will be: j i i j k j i k i j L L L L S) 1 cos 1 ( ) )( 1 cos 1 ( − − = + − − = θ δ θ (19) Where, 3 2 2 3 1 1L L L L − = = = , 1 3 3 1 2 2L L L L − = = = , 2 1 1 2 3 3L L L L − = = = . Near the container surface, the rotation direction should be along the surface normal direction. Taking this direction as the 1 3 = L , 0 2 1 = = L L , the intrinsic stretching is: ) 1 cos 1 ( 3 3 − − = θ S , others are zero (20) It shows that the material element is compressed by the container boundary. On the other hand, the expansion local rotation gives out an isotropic expansion ij δ θ ) 1 cos 1 ( − , so the net expansion is parallel to the container surface. The internal materials isotropic expansion is the intrinsic features of gas. In classical statistic theory, the boundary bounce mode is used. This logically has no contract with the boundary compressing here. Summing above results: (1) the solid has no intrinsic deformation; (2) the liquid has intrinsic plane expansion on local rotation plane, the local rotation is an unit orthogonal rotation; (3) the gas has intrinsic compressive on its local rotation direction, the local rotation is combined with isotropic expansion. Generally, the intrinsic stretching will change the shape of material elements. Motion Transformation of Intrinsic Free Continuum For a free material element in continuum, it should has no intrinsic stretching. Its orientation is determined by the continuum as whole. If in statistical sense the material elements are in free state, the continuum is defined as intrinsic free continuum. Based on previous formulation, the intrinsic free continuum is defined as: 0 = i j S , or 0 = i j S(21) This means that the classical strain may be non-zero, hence macro displacement field or deformation can be observed. Free Solid For free solid, the deformation is Equation (3). It is an identical motion transformation. That means the free solid material elements are fixed in continuum. In fact, this picture is overwhelmingly adopted in many textbooks as the definition of continuum in mechanics. By this definition, the classical strain for free solid is: i j i j F δ = , 0 = ij ε , free solid(22) Comparing with previous results, the static solid is free solid. This is the special feature of solid. It means that the solid material has no motion without external force action. Free Liquid For liquid, its free state is defined by the deformation tensor: k j i k i j i j i j i j L L L R F ) cos 1 ( sin Θ − + ⋅ Θ + = = δ(23) It is a pure local rotation. The gauge tensor of the material element is invariant. That is: 0 0 0 kl l j k i l k l j k i j i ij g R R g g R R g g g = = ⋅ = r r r r(24) For spatial isotropic material element, ij ij g g δ 0 0 = , one has: 0 0 ij ij ij g g g = = δ . Hence, the liquid material element is gauge field invariant. The free state is defined by the local rotation (orientation variation). The local rotation is limited by the continuum feature as a whole, as the macro classical strain is not zero. By Equation (5),, one has: k j i k i j j i ij L L u u ) cos 1 ( ) ( 2 1 Θ − = + = ε (25) For the local rotation along (12)), the classical strain components are: 1 3 = L direction (see Equation) cos 1 ( 11 Θ − − = ε , ) cos 1 ( 22 Θ − − = ε , others are zero (26) Phenomenally, the continuum supplies the essential compressing to keep the liquid element has invariant gauge. So, the free liquid will form flow-stream line on the rotational plane. If the macro mechanical feature of liquid is expressed by viscosity parameters ( µ λ, ), the internal initial stress defined by the free state of continuum is: ) cos 1 )( ( 2 22 11 Θ − + − = = µ λ σ σ , ) cos 1 ( 2 33 Θ − − = λ σ , others are zero (27) It is an anisotropy stress field. Generally speaking, it will drive the liquid element drifts along the rotation direction while rotating along the direction. This phenomenon is typical in daily life as the high viscous fluid and low viscous fluid are separated by itself in free continuum. Note that, for liquid, the isotropic stress components can be defined as: ij ij p δ λ δ ) cos 1 ( 2 0 Θ − − = − (28) So, the rational definition of static pressure of fluid is: ) cos 1 ( 2 0 Θ − = λ p (29) It shows that the static pressure of fluid is determined by the fluid viscosity λ and its local rotation angle. Generally speaking, for the same material composition, the bigger is the local rotation angle, the higher pressure is. The surface tension of liquid is rationally defined as: ij ij δ µ σ ) cos 1 ( 2 Θ − − = , 2 , 1 , = j i on surface(30) Note that plane stress on rotational plane has no contribution to the liquid pressure as the rotation has random distribution. It shows that the free liquid material elements have an intrinsic surface contraction force to keep its shape. For given material features, by measuring the surface tension, the Θ parameter can be measured (generally, for most fluid, the μ can be well measured). In fact, the Equations (29) and (30) can be used to determine the parameter λ . This parameter is omitted in most fluid mechanics textbook. This problem is highly criticized (Lodge, A.S, 1974, [23]). In fact, by the classical strain Equation (25), the general form of stress can also be written as: j i ij ij L L ) cos 1 ( 2 ) cos 1 )( ( 2 Θ − + Θ − + − = µ δ µ λ σ (31) For a small liquid drop, the isotropic compressive stress is interpreted as the atmosphere pressure, the second item on the right can be interpreted as the liquid drop expansion on the surface direction. Hence, the liquid drop can have a fixed shape with an appropriate highness. This equation can found its usage in many cases. Free Gas Free gas is defined by the motion transformation: ) )( 1 cos 1 ( cos siñ ) (cos 1 i j k j i k i j i j i j i j L L L R F δ θ θ θ δ θ + − + + = = − (32) If the initial gauge tensor is selected as: . In conventional nomination, this factor is named as expansion rate of gas. It depends on the internal motion of gas as a continuum. This is represented by the local rotation angle θ . Unlike the free liquid (invariant gauge tensor), free gas material will expansion infinitely. So, a closed container is needed to keep them in a configuration. For free gas in continuum mechanics, the classical strain is: j i ij L L1 cos 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = θ ε (33) When the open surface normal direction is taken as the ( 1 3 = L ) local rotation direction, the classical strain is: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 cos 1 33 θ ε (34) It means that the gas will escape from the surface. As a simple example, if a rectangular container is used, when the H x = 3 plane ( 0 3 = x is the opposite boundary surface) is opened suddenly, the escape velocity distribution for the gas material within the container will be approximated as (in statistical sense): 3 3 1 cos 1 x u ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = θ . The classical stress for free gas material element is: j i ij ij L L) 1 cos 1 ( 2 ) 1 cos 1 ( − + − = θ µ δ θ λ σ (35) The first item on the right is an isotropy expansion pressure, it has the opposite sign referring to the liquid. Hence, in rational mechanics, the gas pressure should be defined as: ) 1 cos 1 ( 0 − = θ λ P(36) Note that the second item has no contribution to the gas pressure as the rotation has random distribution. Summering up above results: (1) continuum composed by intrinsic free solid has no classical strain; (2) continuum (fluid) composed by intrinsic free liquid has a static compressive pressure; (3) continuum (gas or vapor) composed by intrinsic free gas has a static expansion pressure. Note that, although at the free state the material has classical strain, its macro statistical value is zero for random orientation distribution. Therefore, for idea continuum, the static state is equivalent with the free state. However, for modern industry, in many cases, the liquid or gas may have coherent orientation distribution. In these cases, the static state and the free state should be identified independently. Strictly speaking, the static state is defined externally as no configuration variation. The free state is defined internally (physically) as no molecular cluster scale deformation (intrinsic stretching). Constitutive Equations for Continuum For continuum, viewing it from physics, the free states (zero intrinsic strain) should be taken as the reference state. Viewing from mechanics deformation theory, the static state should be taken as the reference configuration. Except for solid, the incremental stress reference and the incremental strain reference are different. In Green stress definition (which is usually used in physics), the intrinsic stretching stress is defined as: ) ( 2 ) 3 ( i j i j ij l l i j F F δ µ δ λ σ − + − = (37-1) In Chen rational mechanics, the classical stress is defined as: i j i j l l i j S S µ δ λ σ 2 + = (37-2) Where, the parameters ( µ λ, ) is determined by the physical features of materials. This equation will be used to derive the classical constitutive equations. Chen Zhida argued that the local rotation has no contribution to the stress. He pointed out that the classical strain in the constitutive definition should be replaced by the intrinsic strain (intrinsic stretching). In this paper, the related problems will be fully discussed from physical and statistical points of view. For continuum, as the local rotation direction (attached on the material element) is random for general cases, they contribute zero stress to macro continuum stress observed externally. So, on statistical sense, the macro stress of continuum is determined by the intrinsic stretching tensor (intrinsic strain tensor). That is: i j i j l l i j i j S S d A µ δ λ σ σ 2 1 + = Σ = ∫ Σ Σ (38) The surface integration expression is used to emphasize the statistical sense. It shows that, for stress, the reference state of material should be its free state. For liquid and gas at free states, the initial strain is has zero average. However, the initial stress is not zero. For infinitesimal incremental deformation, the classical strain is defined as: ) ( 2 1 i j j i ij X U X U e ∂ ∂ + ∂ ∂ =(38) Here, the laboratory coordinator system ( In deformation mechanics point of views, the natural continuum which can be taken as a reference should be: zero statistical classical strain reference. Solid Continuum For solid continuum, referring to its free state, the classical constitutive equation is: Liquid Continuum For liquid continuum, its free state is defined as: ) (Θ = i j i j R F(40) For infinitesimal deformation, the incremental deformation is assumed to have no contribution to the intrinsic local rotation of material element. Hence, the stress is: ij ij ll j i ij ij e e L L µ δ λ µ δ µ λ σ2 ) ( ) cos 1 ( 2 ) cos 1 )( ( 2 + + Θ − + Θ − + − =(41) It is clear, if the contribution from incremental local rotation is included, the static pressure should be modified. As the static pressure is temperature dependent, it can be inferred that the local rotation angle Θ plays the similar role as the temperature parameter. Gas Continuum For gas continuum, its free state is defined as: ) ( cos 1 θ θ i j i j R F =(44) For infinitesimal deformation, the incremental deformation is assumed to have no contribution to the intrinsic local rotation of material element. Hence, the stress is: ij ij ll j i ij ij e e L L µ δ λ θ µ δ θ λ σ2 ) ( ) 1 cos 1 ( 2 ) 1 cos 1 ( + + − + − = (45) Or, in conventional form for random orientation distribution: It is clear, if the contribution from incremental local rotation is included, the static pressure should be modified. As the static pressure is temperature dependent, it can be inferred that the local rotation angle θ plays the similar role as the temperature parameter. For many gas continuums, the shear is omitted, and the constitutive equation becomes: ij ll ij ij e k P δ δ σ ) ( 0 + =(47) Here, the bulk compressibility parameter k is used to replace the viscosity parameter λ as their physical implications are different (although they are almost equal in value). In thermal mechanics, the equation is written as: dV k P P0 + =(48) It is clear that, the overwhelming simplification of constitutive equation of gas continuum cuts its link with the deformation mechanics sharply. No doubt, this empirical altitude will damage the fully development of gas dynamics. Summering above results, the unified constitutive equations for continuum are By these equations, the stress is the classical stress. When the continuity of stress concept is used in deformation theory, the stress defined above is continuous, although the deformation may be not continuous. Here, it should be pointed out that the local rotation to another kind of stress which is related with the internal curvature of continuum. It is related with the micro-deformation of continuum. As a first approximation, it is related with thermo mechanics. This topic will be expanded bellow. Phase Transition The above formulation is about the deformation within an unit time. If the unit time is much longer than the characteristic time of liquid (composing fluid) or gas (composing vapor), the formulation must be modified. For engineering practice, the continuum is assumed to be stable within the unit time. In fact, it is the usually case. However, this does not mean the intrinsic motion of the basic material element has long characteristic time. In fact, introducing a characteristic time scale τ will be useful. Firstly, the within the fraction time 1 1 << ≈ N τ ( τ τ < for gas) duration, the local rotation angle will be expressed: Where, the negative sign is dropped as the absolute value will be discussed bellow. Hence, to modify the local rotation angle definition, on statistical sense, the best way is to express the free liquid deformation as: ) ( τ τ Θ = i j i j R F (52) The similar procedure is applied to the free gas. Viewing the local rotation as a stochastic process: Θ + Θ = Θ δ 0 (54-1) δθ θ θ + = 0 (54-2) For simplicity, it is viewed as a Brownian motion with normal distribution. Under this understanding, the local rotation angle is represented as a stochastic process defined by the characteristic function of normal distribution: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⋅ Θ = Θ Θ 2 2 0 2 1 exp ) ( t t i t f σ (55-1) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⋅ = 2 2 0 2 1 exp ) ( t t i t f θ θ σ θ (55-2) Where, 1 − = i is the sign of imaginary number, 0 Θ and 0 θ are the mean value of Θ and θ process respectively, 2 Θ σ and 2 θ σ are their variance. The corresponding probability distribution function is: ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Θ − Θ − ⋅ ⋅ = Θ Θ Θ 2 2 0 2 ) ( exp 2 1 ) ( σ σ π P (56-1) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − ⋅ ⋅ = 2 2 0 2 ) ( exp 2 1 ) ( θ θ σ θ θ σ π θ P (56-2) Based on above formulations, using the statistical mechanics theory, the temperature incremental dT is related with the variance as: ρ σ dT C ⋅ = Θ Θ 2 (57-1) ρ σ θ θ dT C ⋅ = 2 (57-2) Here, the continuum mass density ρ is invariant for phase transition of isochoric process. Θ C and θ C are material thermal parameters. For phase transition discussion, it can be viewed as a constant (in fact, they should be determined by statistical physics for different micro dynamics processes). With above preliminary preparation, the phase transition problem can be discussed. Solid-Gas Transition For solid, its intrinsic characteristic time is very large. For unit time duration, its average intrinsic local rotation is almost zero. For solid continuum, when the local rotation variance Θ σ is bigger than a critical angle S Θ (which is determined by material features), a fraction of solid will be cracked out from its surroundings. The cracked out materials will be free at low pressure environment. So, they are viewed as gas here. For isotropic cracking process with certain possibility (defined by variance level Θ σ ), initially, the solid deformation is: ) ( Θ + ⋅ = σ δ i j i j i j R S F (58) It produces the stress which is: ij j i ij ij S L L δ µ λ σ µ δ σ µ λ σ ⋅ + + − + − + − = Θ Θ ) 2 3 ( ) cos 1 ( 2 ) cos 1 )( ( 2 (59) For homogenous distribution of rotation direction ( ( ) 3 1 = j i L L mean , here after), the isotropic stress is: S ) 2 3 ( ) cos 1 ( 3 2 ) cos 1 )( ( 2 µ λ σ µ σ µ λ σ + + − + − + − = Θ Θ (60) After cracking, for the cracked out materials, the gas deformation is: 13 Similarly, for homogenous distribution of rotation direction ( ) ( cos 1 0 0 θ θ i j i j R F =(61)( ) 3 1 = j i L L mean , hear, after), its corresponding isotropic stress is: ij ij δ θ µ λ σ ) 1 cos 1 ( ) 3 2 ( 0 − ⋅ + = (62) The macro volume continuity gives out the condition equation: S = − 1 cos 1 0 θ (63) For solid-gas transition, the isotropic stress should be continuous. So, the transition equation is: G P = − ⋅ + = − + − − + Θ ) 1 cos 1 ( ) 3 2 ( ) cos 1 )( 3 2 ( 2 ) 1 cos 1 )( 2 3 ( 0 0 θ µ λ σ µ λ θ µ λ (64) For the solid-gas transition, the G P pressure is the measured externally. For low pressure, the 0 θ is small. When the G P is taken as the variable, the 0 θ is determined by the last equality equation. Here, the mass density is defined for solid state. Hence, the dT is determined. The maximum allowable solid-gas transition pressure range is given by the equation: It is completely determined by the solid features and gas features, also. Liquid-Gas Transition For liquid-gas process with certain possibility (defined by variance level Θ σ ), the liquid deformation tensor is: 14 For such a liquid, its initial pressure is defined as: ) ( 0 Θ + Θ = σ i j i j R F(70)) cos 1 )( 3 2 ( 2 0 0 Θ − + = L L p µ λ(71) Here and after, the liquid parameters are labeled with L to distinguish from gas. Based on phase transition curves, the following condition is met: triple L L P ≥ Θ − + ) cos 1 )( 3 2 ( 2 0 µ λ(72) At the liquid-gas transition process, the liquid stress tensor is: ij L L ij δ σ µ λ σ )] cos( 1 )[ 3 2 ( 2 0 Θ + Θ − + − =(73) Hence, using gas Equations (61) and (62), the stress continuity condition is: G L L P = − + = + Θ − + Θ ) 1 cos 1 )( 3 2 ( )] cos( 1 )[ 3 2 ( 2 0 0 θ µ λ σ µ λ(74) It means that the gas expansion stress is balance by the liquid compressive stress. (In solid-gas transition, the gas expansion stress is supplied by the solid volume expansion force. This can be true only for low temperature case.) It is clear that, here, triple G P P > . Using the G P as the variables, the transition temperature incremental is determined by equation: ) 3 2 ( 2 ) cos( 1 0 L L G L P dT C µ λ ρ + = + Θ − Θ (75) The global temperature is determined by the equation (indirectly): µ λ θ3 2 1 ) ( cos 1 0 + = − G P T(76) Generally speaking, as triple G P P > , the temperature is higher than the solid-gas transition temperature. Solid-Liquid Transition For many solid at medium temperature and pressure, the stochastic local rotation is not zero-mean. For example, when some kinds of coherent patterns happened in the solid materials, the shear stress will crack the materials into several pieces. Hence, the mean local rotation is a finite value S Θ . This feature is expressed by the deformation tensor: ) ( Θ + Θ = σ S i j i j R F(77) Here, the mean value of local rotation angle is determined by the yield stress of solid as: yield S σ µ λ = Θ − + ) cos 1 )( 3 2 ( 2(78) For liquid, the deformation tensor is: ) ( L i j i j R F Θ = (79) The stress continuity equation is: L L L L S P = Θ − + = + Θ − + Θ ) cos 1 )( 3 2 ( 2 )] cos( 1 )( 3 2 ( 2 µ λ σ µ λ(80) As solid parameters ( µ λ, ) are much larger than liquid parameters ( L L µ λ, ), the L P will be at the range: L yield triple P P ≤ ≤ σ (81) The global temperature is indirectly determined by equation: L L L L P T = Θ − + )] ( cos 1 )[ 3 2 ( 2 µ λ (82) Generally speaking, although ) are much larger than gas parameters ( µ λ, ), the temperature is lower than the liquid-gas transition temperature. This can be inferred from the related equations. Using the L P as the variables, the transition temperature incremental is determined by equation: L S S P dT C = + Θ − + Θ )] cos( 1 )[ 3 2 ( 2 ρ µ λ (83) Now, it is time to turn to the supercritical fluid region. Super-Fluid Transition For liquid materials, its deformation is: ) (Θ = i j i j R F(84) For gas, the deformation is: ) ( cos 1 θ θ i j i j R F = (85) By the geometrical Equations (6) and (16) , the pressure tends to infinite. That is: , ) 1 ) ( cos 1 )( 3 2 ( ∞ → − + = T P θ µ λ when 2 / π θ →(87) Hence, the super-fluid region is defined by condition equations: ) 3 2 ( 2 L L C P P µ λ + = ≥ , C T T ≥(88) It shows that, the critical pressure is completely determined by the fluid parameters. Where, the critical temperature is determined (indirectly) by equation: It shows that, the critical temperature is determined by the ratio of liquid parameter over gas parameter. Summering above results: the phase transition phenomenon is well formulated by the unique physical requirements stress continuity (pressure continuity) for continuous temperature variation. The typical phase diagram of continuum be obtained by the related equations given in this paper. Application: Rock as Multiphase Continuum There are two different multiphase continuum definitions. The multiphase continuum in physics is a topic for statistical mechanics, where the materials have the same composition. In this case, the internal interaction is temperature dependent, and the dynamics in a scale less than molecular cluster scale is concerned. It is too complicated for deformation mechanics. This topic will not be discussed here further. In mechanical engineering, the multiphase continuum is referring to the continuum composed by solid, liquid, and gas. They usually have different composition. The solid forms the volume frame, while the liquid and gas are filled in the volume. They have contribution to stress, but usually has no direct contribution to volume variation. For simplicity, taking the volume variant deformation as an example, the macro deformation tensor is: ) ( cos 1 ) ( ) 1 ( θ θ β α δ γ i j i j i j i j R R e F + Θ + + = (90) Where, the solid, liquid, and gas contribution coefficients are , ,α γ and β . Their total should be unit one. 1 = + + β α γ . Then, the stress field is: ij L L ij e δ θ µ λ β µ λ α µ λ γ σ )] 1 cos 1 )( 3 2 ( ) cos 1 )( 3 2 ( 2 ) 2 3 ( [ − + + Θ − + − ⋅ + = (91) The static state, which is taken as the reference configuration is defined by the deformation: ) ( cos 1 ) ( 0 0 0 θ θ β α γδ i j i j i j i j R R F + Θ + = (92) The static state stress is: ij L L ij δ θ µ λ β µ λ α σ )] 1 cos 1 )( 3 2 ( ) cos 1 )( 3 2 ( 2 [ 0 0 0 − + + Θ − + − =(93) Based on the geometrical meaning of each items, in Chen rational mechanics, the length variation is: e g = δ(94) The stress variation is: , water driven bursting is produced. ij L L ij e δ θ θ µ λ β µ λ α µ λ γ δσ Hence, by measuring the compressive parameters of rock, the potential rock bursting can be predicted. Conclusions In this research based on Chen Rational Mechanics frame, the motion of mater in continuum sense is described by the base vector transformation: ) ( ) , ( ) , ( 0 x g x t F x t g i i j j r r ⋅ = (the spatial and time parameters will be dropped, here after). Once an arbitral initial coordinator is selected, a material element will be labeled by this coordinator combining with local initial base vector (on this sense, the coordinator and base vectors are embedded into the continuum), no matter how its current configuration is deformed. By this way, omitting the global translation and rotation of the continuum as a whole, the matter motion in continuum is purely the base vector intrinsic stretching and local relative intrinsic rotation referring to its initial configuration (the should be configuration determined by the continuum motion surrounding the material element under discussion). In observation sense, the commoving dragging coordinator system (defined by coordinators and base vectors) variation is the natural results of matter motion as a continuum. Hence, such a geometrical field theory is a natural selection for continuum motion. (If coordinator transformation method is used, there are too many jobs putting on the relation between the i x coordinator system and the laboratory i X coordinator system. Then, the true physical essential variation of gauge tensor is waiting to be recovered, which is not practical for complicated deformation. Furthermore, the material objective invariance may not be met. Although for infinitesimal deformation this is not serious, it do cause serious error for large deformation or deformation with local rotation.) In such a kind of natural geometrical field selection, the static solid is defined by: i j i j F δ = . It is an identical transformation. For static liquid, within unit time, the basic material element motion is defined as: ) (Θ = i j i j R F . It means that, the liquid material has a local relative rotation referring to its "should be" configuration unit-time-before. For continuum composed by liquid, it has a classical expansion on its rotation plane. Hence, to make the continuum has a fixed external configuration, a container should be used (top open boundary under gravity field). As the local rotation has no classical strain on rotation direction, a natural laminar structure (each layer has different local rotation angular) may be formed. This is the basic feature for static continuum. By this sense, the wall of container has contribution to the static pressure of liquid continuum. In fact, the local rotation will produce an isotropic compressive stress (which contributes the main body of static pressure) and surface contraction stress which is called as surface tension). In the theory, these results are obtained naturally, while in conventional mechanics theory, they are introduced externally. So, I have enough reasons to claim that the Chen rational mechanics theory frame is much powerful than others. Furthermore, the conventional static liquid definition fails to define its internal velocity field (which determines the local rotation in Chen theory). Hence, the static liquid definition given in this paper is much better. Distinguishing from conventional static liquid continuum definition (gauge invariant), the static liquid is defined by motion within unit time. Hence, introducing the thermal parameter Θ σ the static liquid under incremental temperature environment is described by Hence, the solid-liquid transition is formulated under the physical requirement of stress continuity. As the local rotation has no contribution to the gauge tensor, the macro external observable classical strain (hence, stress) variation is caused by local rotation. This feature is expressed as: zero intrinsic stretching (intrinsic strain) and non-zero classical strain. In physical sense, for solid-liquid transition, the material element is invariant (on gauge field sense). However, although the physical stress is continuous, the classical strain is not continuous. By the classical strain definition (strain rate, in most textbooks), the velocity field (displacement field in unit-time) is not continuous for solid-liquid transition. For static gas, it is well known that to keep its fixed external configuration, a closed container must be used. Its internal expansive pressure is balanced by the container closed walls. For free state gas, the deformation is: Hence, the static gas is different from static liquid in the local rotation modes: for liquid, the gauge is invariant; for gas, the gauge is amplified. Hence, for the solid-gas transition, the solid must have a volume explosion to broken into material pieces. However, for liquid-gas transition, the flow-ability of liquid may not require the liquid has such an volume explosion as the linkage between material elements of liquid as a continuum is weak (comparing with solid continuum). In liquid-gas transition, the stress field is continuous. However, the classical strain is not continuous. Hence, very complicated flow patterns are possibly be formed. Unfortunately, they are beyond the coverage of this paper as the dynamic process will be concerned. Although the transition dynamic process is omitted in this research, the phase transition condition is well formulated. They give out a rational interpretation about the phase diagram of conventional materials. As a limit case, the super-fluid transition conditions are formulated, also. Therefore, it can conclude that: (1) a geometrical field representation of solid, liquid, and gas as a continuum id established in rational mechanics frame; (2) a geometrical deformation theory of solid, liquid, and gas phases transition are formulated; (3) the conventional constitutive equations of solid, liquid, and gas are unified into a constitutive equation which express the stress by intrinsic strain. As an example, the rock bursting is formulated by the phase formulation in this paper through formulating a suitable multi-phase model. Theoretically, it is hoped that an unified geometrical field theory which represents the solid, liquid, and gas in the same motion concept will help to solve the multiphase continuum mechanics problem. However, this job is extremely difficult. This difficulty is apparent in this research, as the temperature is not given in a rational formulation although it is the representation of internal motion of matter. However, "step by step" is the only way to attend to the idea target. i L is the component of unit vector along the rotation axe direction.For liquid in container, taken the3 x on the free surface normal direction, as the flow direction used. i U is the spatial macro displacement within the unit time under discussion. The classical strain attached on the material element has zero contribution to the average, because their rotation direction and orientation have randomly distribution. , Chen decomposition form II (49-2) interpreted as the local rotation angle within time τ . To meet our formulation, way, the Θ and θ still is defined by unit time duration.For simplicity, in the following discussion, the 1 = τ is taking as the unite time duration. The above discussion is to emphasize the statistical sense for the related formulations. gas local rotation 0 θ is temperature dependent, so the global temperature is determined by this equation, also.For given material parameters, putting Equation (57-1) into Equation (64), the incremental temperature is determined by the equation: , the critical temperature for the triple point of solid-liquid-gas is defined by the pressurecompletely determined by the solid features and gas features. The triple temperature can be determined, indirectly, by the equation: continuum, because there are no continuous displacement field (here, it is velocity) for such cases. Observing equation: are at different physical phase. Theoretically, there is no sharp boundary between both. keep the fixed macro external configuration, the container must supply the compressive stress to produce a deformation (in container scale): For random orientation distribution, it is simplified as:ij ij ll ij ij e e µ δ λ δ µ λ σ2 ) ( ) cos 1 )( ( 2 + + Θ − + − = (42) Or, in conventional form: ij ij ll ij ij e e p µ δ λ δ σ2 ) ( 0 + + − = Hence, if one define the stress by the effective parameters ( µ λ , ), then the stress is expressed as:By comparing the Equations (97) and (98), one has:It says the effective mechanic parameters are the weight sum of solid, liquid, and gas features.δθ are material feature constants. In mining industry, the infinitive effective mechanical parameter means rock bursting. For gas driven bursting in mining industry, the condition can be predicted by)] cos 1 cos 1 )( 3 2 ( ) cos )(cos 3 2 ( 2 ) 2 3 ( [ 0 0 − + + Θ − Θ + − ⋅ + = (95) This equation can be simplified as: ij L L ij e δ δθ θ θ µ λ β δ µ λ α µ λ γ δσ ] ) cos sin )( 3 2 ( ) )(siñ 3 2 ( 2 ) 2 3 ( [ 0 2 0 0 + + Θ Θ + + ⋅ + = (97) e ij ⋅ + = ) 2 3 ( µ λ δσ (98) e e L L δθ θ θ µ λ β δ µ λ α µ λ γ µ λ ⋅ + + Θ Θ + + + ⋅ = + ) cos sin )( 3 2 ( ) )(siñ 3 2 ( 2 ) 2 3 ( 2 3 0 2 0 0 (100) Generally speaking, the e / Θ δ and e / gas θ θ ≥ 0 , where the gas θ can be measured by laboratory experiments. For 2 / 0 π → Θ Chen Zhida, Rational Mechanics, Xuzhuo: China University of Mining & Technology Press. In ChineseChen Zhida, Rational Mechanics, Xuzhuo: China University of Mining & Technology Press., 1987 (In Chinese) The Mechanical Foundation of Elasticity and Fluid Mechanics. C Truesdell, Gordon and Breach Science Pub. IncNew YorkTruesdell, C., The Mechanical Foundation of Elasticity and Fluid Mechanics. New York: Gordon and Breach Science Pub. Inc., 1966 Evolution of Continuum from Elastic Deformation to Flow. Xiao Jianhua, arXiv:physics/0511170physics.class-phXiao Jianhua. Evolution of Continuum from Elastic Deformation to Flow. E-print, arXiv: physics/0511170(physics.class-ph), 2005, 1-25 Xiao Jianhua, Chen Rational Mechanics I: Introduction, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics I: Introduction, Sciencepaper Online, http://www.paper.edu.cn, 2007-01-30 Xiao Jianhua, Chen Rational Mechanics II: Geometrical Equations, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics II: Geometrical Equations, Sciencepaper Online, http://www.paper.edu.cn, 2007-01-31 Chen Rational Mechanics III: Deformation decomposition and strain definition. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics III: Deformation decomposition and strain definition, Sciencepaper Online, http://www.paper.edu.cn, 2007-02-06 Xiao Jianhua, Chen Rational Mechanics IV: Constitutive Equations, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics IV: Constitutive Equations, Sciencepaper Online, http://www.paper.edu.cn, 2007-02-06 Xiao Jianhua, Chen Rational Mechanics V: Motion Equations, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics V: Motion Equations, Sciencepaper Online, http://www.paper.edu.cn, 2007-02-07 Chen Rational Mechanics VI: Micro geometry and Macro Geometry. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics VI: Micro geometry and Macro Geometry, Sciencepaper Online, http://www.paper.edu.cn, 2007-03-09 Xiao Jianhua, Chen Rational Mechanics VII: Viscoelastisity, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics VII: Viscoelastisity, Sciencepaper Online, http://www.paper.edu.cn, 2007-03-09 Xiao Jianhua, Chen Rational Mechanics VIII: Fatigue and Cracking, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics VIII: Fatigue and Cracking, Sciencepaper Online, http://www.paper.edu.cn, ,2007-03-19 Xiao Jianhua, Chen Rational Mechanics IX: Dynamic Instability and Cracking Deformation. Xiao Jianhua. Chen Rational Mechanics IX: Dynamic Instability and Cracking Deformation, Sciencepaper Online, http://www.paper.edu.cn, 2007-03-12 Chen Rational Mechanics X: Description of Path-dependence for one dimension deformation. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics X: Description of Path-dependence for one dimension deformation, Sciencepaper Online, http://www.paper.edu.cn, 2007-03-19 Chen Rational Mechanics XI: Multi-scale and Locality of Periodic Structure of Solid. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics XI: Multi-scale and Locality of Periodic Structure of Solid, Sciencepaper Online, http://www.paper.edu.cn,, 2007-04-04 Chen Rational Mechanics XII: Determine the elastic parameters from stress-strain experiment curves. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua. Chen Rational Mechanics XII: Determine the elastic parameters from stress-strain experiment curves, Sciencepaper Online, http://www.paper.edu.cn, 2007-05-04 Intrinsic Knots Produced by Large Deformation in 3-Space I: Curvatures. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua, Intrinsic Knots Produced by Large Deformation in 3-Space I: Curvatures, Sciencepaper Online, http://www.paper.edu.cn, 2007-10-17 Intrinsic Knots Produced by Large Deformation in 3-Space II: Multi-scale. Xiao Jianhua, Sciencepaper Online. Xiao Jianhua, Intrinsic Knots Produced by Large Deformation in 3-Space II: Multi-scale, Sciencepaper Online, http://www.paper.edu.cn, 2008-1-10 Decomposition of Displacement Gradient and Strain Definition. Xiao Jianhua, J. Central South University of Technology. 14Advances in Rheology. Suppl.1Xiao Jianhua. Decomposition of Displacement Gradient and Strain Definition. Advances in Rheology, J. Central South University of Technology, Vol.14 (Suppl.1), 401-404, 2007 Determining Loading Field based on Required Deformation for Isotropic Hardening Materials. Xiao Jianhua, J. Shanghai Jiaotong Uni. (Science). 126Xiao Jianhua, Determining Loading Field based on Required Deformation for Isotropic Hardening Materials, J. Shanghai Jiaotong Uni. (Science), E12(6):805-812, 2007 Bubble dynamics equations for Newton fluid, ISND2007. Xiao Jianhua, 10.1088/1742-6596/96/1/012134Journal of Physics: Conference Series. 9612134Xiao Jianhua, Bubble dynamics equations for Newton fluid, ISND2007, Journal of Physics: Conference Series 96, 2008, 012134, doi:10.1088/1742-6596/96/1/012134, 2008 Motion Equation of Vorticity for Newton Fluid. Xiao Jianhua, arXiv:physics/0512051physics.flu-dynXiao Jianhua. Motion Equation of Vorticity for Newton Fluid. E-print, arXiv: physics/0512051(physics.flu-dyn), 2005, 1-7 Xiao Jianhua, arXiv:physics/0601006Intrinsic Structure of Turbulent Flow in Newton Fluid. E-print. physics.flu-dynXiao Jianhua. Intrinsic Structure of Turbulent Flow in Newton Fluid. E-print, arXiv: physics/0601006(physics.flu-dyn), 2006, 1-16 A S Lodge, Body tensor fields in continuum mechanics. New YorkAcademic PressLodge, A.S., Body tensor fields in continuum mechanics, Academic Press, New York, 1974

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{'abstract': 'Any materials have three physical states as solid, liquid, and gas. Basically, the materials at some molecular cluster scales are assumed as the same for all the above physical states. So, a basic gauge tensor field can be attached to the basic molecular cluster elements (basic material element). Under this objective (material element) invariance, the concept of continuum in rational mechanics is studied in this paper. Different material phases have different basic gauge fields. So, the matter phases are expressed by the basic gauge transformations.Based on this general understanding, the different phases have different motion transformations, internally. Based on the points-set transformation concept about the motion transformation in continuum, the macro classical strain is expressed by the additive addition of the intrinsic stretching of material element and its intrinsic local rotation.For zero classical strain (no macro deformation observed on its configuration surface, suitable container is required for liquid and gas to make up macro invariant configuration), the results show that: (1) For solid, the local rotation angular is zero. The material element has no intrinsic stretching. (2) For liquid, the local rotation will not change the basic gauge tensor. The material element has intrinsic plane stretching on the rotation plane. (3) For gas state, the intrinsic local rotation will amplify the basic gauge tensor. The material element has intrinsic stretching along the rotation direction. Hence, under the condition of no macro classical strain be observed, the material element has three different physical states: solid (no intrinsic stretching), fluid (plane intrinsic stretching), and gas (directional intrinsic stretching). Furthermore, for the three states, the free conditions are defined by zero intrinsic stretching. Referring to this free condition, the constitutive equations for the materials at multiple states are established. It is shown that the classical constitutive equations are included in this general unified formulation.', 'arxivid': '0911.1397', 'author': ['Xiao Jianhua \nMeasurement Institute\nHenan Polytechnic University\n454000JiaozuoHenanChina\n'], 'authoraffiliation': ['Measurement Institute\nHenan Polytechnic University\n454000JiaozuoHenanChina'], 'corpusid': 118394468, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14984, 'n_tokens_neox': 13140, 'n_words': 8688, 'pdfsha': '8a5b5f1e3a87f4e34c25ed45ae2364fb10b0d42a', 'pdfurls': ['https://export.arxiv.org/pdf/0911.1397v1.pdf'], 'title': ['Geometrical Field Representation of Solid, Fluid, and Gas as Continuum in Rational Mechanics', 'Geometrical Field Representation of Solid, Fluid, and Gas as Continuum in Rational Mechanics'], 'venue': []}

arxiv

P3-82 Ion energy measurements on MAST using a midplane RFEA P3-82 2 S Y Allan EURATOM/CCFE Fusion Association Culham Science Centre OX14 3DBAbingdonUnited Kingdom S Elmore EURATOM/CCFE Fusion Association Culham Science Centre OX14 3DBAbingdonUnited Kingdom The University of Liverpool Brownlow HillL69 3GJLiverpoolUnited Kingdom A Kirk EURATOM/CCFE Fusion Association Culham Science Centre OX14 3DBAbingdonUnited Kingdom M Kočan Max-Planck-Institut für Plasmphysick Boltzmannstrasse, 2D-85748GarchingGermany P Tamain Culham Science Centre CEA IRFM F-13108, OX14 3DB, 1UKSaint-Paul-lez-Durance, AbingdonFrance Scott Allan P3-82 Ion energy measurements on MAST using a midplane RFEA P3-82 2 1 *Corresponding author address: CCFE, Presenting author:5270Ds5255Fa PSI-20 keywords: Edge plasmaMASTProbes Ion energy measurements have been made in the scrape off layer of the Mega Amp Spherical Tokamak (MAST) using a midplane retarding field energy analyser (RFEA) in H-mode plasmas during the inter-edge localised mode (ELM) period and during type I and type III ELMs. During the inter-ELM period at distances of 3 to 8 cm from the last closed flux surface (LCFS), ion temperatures of 20 to 70 eV have been measured giving an ion to electron temperature ratio of 2 to 7 with a mean of 4. During type III ELMs, an ion temperature of 50 eV has been measured 3 to 6 cm from the LCFS which decreases to 30 eV at distances 11 to 16 cm from the LCFS. During type I ELMs, an ion temperature of 40 eV has been measured at a distance of 10 to 15 cm from the LCFS. P3-82 2 Introduction One of the most important parameters in determining the extent of wall damage in tokamaks is the energy of ions impacting on the surface. Ion temperature determines sputtering yields and the quantity of impurities injected back into the plasma. Some of the most significant damage to the first wall occurs during ELMs where bursts of core plasma are released into the scrape off layer (SOL). The measurement of ion energies during both steady state and transient events such as ELMs is important for determining sputtering yields and for the development of particle and energy transport models of the scrape off layer where experimental measurements can provide boundary conditions for computer simulations. One of the most widely used methods of measuring ion temperature is using an electrical probe called a Retarding Field Energy Analyser (RFEA) [1]. RFEAs have been used in a number of tokamaks including Alcator C [2], JET [3,4], ASDEX Upgrade [5], Tore Supra [6] and MAST [7,8]. Measurements in a number of tokamaks [9] have shown that the ratio of the ion temperature (T i ) to electron temperature (T e ) in the SOL of L-mode plasmas can range from 1 to 10. Ion energy measurements during ELMs have been made on ASDEX Upgrade [10] showing energies between 20 and 200 eV, 3.5 to 6 cm from the LCFS and during inter-ELM periods at approximately 5 cm from the LCFS, T i was found to be approximately 12 eV [5]. Measurements on JET during ELMs have measured ions with energies in excess of 400 eV, 5 cm from the LCFS [3]. Preliminary measurements on MAST during ELMs have measured ion energies between 200 and 500 eV at distances of 10 to 20 cm from the LCFS [7] but no measurement of ion temperature was possible. P3-82 3 In this paper, the results of inter-ELM and ELM T i measurements made on MAST [11] using a RFEA mounted on a reciprocating probe at the midplane will be presented. In section 2, a brief introduction to RFEAs is given along with a description of the plasma conditions used. In section 3, the results of inter-ELM and ELM measurements are presented. Method Retarding field energy analysers A standard RFEA consists of a series of metal grids inside a probe body. Plasma enters the RFEA through a slit in the electrically grounded shell. Electrons are repelled by a negatively biased slit plate and ions are discriminated using a swept positive voltage on grid 1. The RFEA is aligned with its grid faces perpendicular to the magnetic field so that ions are discriminated based on the component of their velocity parallel to the magnetic field lines. The ion current is measured at a metal collector at the rear of the RFEA as a function of the grid 1 voltage. Grid 2 is biased negatively to minimise the effects of secondary electron emission on ion current measurements. For a given discriminator voltage, the collector current (I col ) is a measure of the ion velocity distribution function parallel to the magnetic field [1]. Assuming that the energy distribution of the ions along the magnetic field lines is described by a Maxwellian distribution, the effective ion temperature (T i * ) can be determined by fitting a graph of I col versus grid 1 voltage (V gr1 ) to equation (1): ( ) off s gr i i o col I V V T Z I I +       − − = 1 * exp .....(1) where I 0 is the ion saturation current, Z i is the effective ion charge (this work assumes Z i =1), V s is the plasma sheath voltage and I off is an offset current [12]. P3-82 4 MAST RFEAs The measurements presented here were performed in a range of H-mode deuterium plasmas using a RFEA located at the midplane on a reciprocating probe (RP). MAST is also fitted with a RFEA at the divertor target and T i measurements in H-mode from this RFEA are reported elsewhere [13]. The RP RFEA has two identical grid stack modules mounted in an electrically grounded graphite shell allowing bidirectional measurements of T i . More detailed information on the design of the RFEAs and their electronics and the results of T i measurements in L-mode plasmas using both the RP and divertor RFEAs can be found in reference [8]. Results Inter-ELM Measurements Inter-ELM measurements were obtained in a double null H-mode discharge with 3.3 MW of neutral beam heating and a plasma current of 900 kA [14]. The inter-ELM pedestal electron density and temperature measured using Thomson laser scattering was approximately 5x10 19 m -3 and 200 eV respectively. The RFEA was reciprocated into the plasma and ion current measurements were made as a function of radial position as grid 1 of the RFEA was swept up to 600 V at a frequency of 2 kHz. The results from four repeat shots were combined to obtain a more comprehensive data set. Due to damage sustained to the RFEA graphite shell on the side facing the lower divertor (facing into the parallel plasma flow), only data from the side facing toward the upper divertor could be analysed. Fig. 1 shows the ion saturation current (J sat ) measured using the RFEA slit plate as a function of distance from the LCFS. Also shown is a exponential fit to the data with a decay length of 1.2 cm. Moving outwards from the LCFS, the results show a large drop in J sat within the first 2 cm followed by a more gradual fall off over the P3-82 5 next 8 cm. As J sat is dependent on both density and temperature, its fall off with increasing distance from the plasma also shows the fall off in these quantities. An example of inter-ELM data obtained 1.8 cm from the LCFS is shown in fig. 2. The collector measurements from two grid 1 sweeps were combined and binned and fitted using eq. (1) giving a T i of 68 eV. Ion temperature measured during the inter-ELM period is shown in fig. 3 as a function of distance from the LCFS. Also shown are T e measurements made using Thomson scattering (TS) [15] and T i measurements in the plasma core obtained using charge exchange recombination spectroscopy (CXRS) [16]. ELM P3-82 7 The results of ion energy measurements made during ELMs in a type III ELMing discharge at a distance of 3 to 17 cm from the LCFS are shown in fig. 4. Data from two repeat shots were combined to provide a larger data set. Ion saturation current measurements in fig. 4a show a fall off length of approximately 5 cm. Fig. 4b shows the collector current measured during the ELMs as a function of grid 1 voltage. The closed circles correspond to measurements made 3 to 6 cm from the LCFS and are shown along with a fit using equation (1) with a T i =50 eV. The open circles correspond to measurements made 11 to 16 cm from the LCFS and are shown along with a fit using equation (1) with a T i =30 eV. The error associated with fitting the data using equation (1) was approximately ± 10 eV. Collector current measurements made during type I ELMs at a distance of 10 to 15 cm from the LCFS are shown in fig. 5 as a function of grid 1 voltage. For the type I ELMs, data from five repeat shots was accumulated to provide a larger data set. A fit to the measured data using equation (1) is also shown and gives a T i of 40 eV which is greater than measured in the type III ELMing discharge at a similar distance from the LCFS. The error associated with fitting the data using equation (1) was approximately ± 10 eV. The temperature of an ELM leaving the plasma is determined by the pedestal temperature and the radial velocity of the ELM. The radial velocity determines the extent of cooling the ELM will undergo before reaching a specific point. In [3] and ASDEX-Upgrade [10]. However, these other measurements were made closer to the LCFS (approximately 3.5 to 8 cm from the LCFS) where higher ion temperatures would be expected. Conclusion Ion temperature measurements have been made in MAST using a midplane RFEA in H-mode plasmas during the inter-ELM period and during ELMs as a function of distance from the LCFS. Figure Captions Linear (dotted line) and exponential (dashed line with a fall off length of 5 cm) fits to the CXRS measurements in the pedestal are also shown and act as a guide to what may be expected in the SOL. The RFEA data show that in the SOL, T i / T e is approximately 2 to 6 with a mean value of 4. While the T e profile is relatively flat in the SOL, the T i profile is consistent with a fall off over the 2 to 10 cm region from the LCFS. The fall off of T i with distance from the LCFS lies between the linear and exponential extrapolations of the CXRS data. Previous measurements using CXRS [17] in a high collisionality plasma (υ * >1) have shown T i ≈ T e in the pedestal region. Measurements in a plasma of similar collisionality to the one studied in this work (υ * ≈0.5) have measured a flat T i profile in the pedestal region and extending out beyond the LCFS with T i ≈ T e(pedestal) = 300 eV. Comparison of RFEA measurements in this work with those made by CXRS in reference [17] show a lower value of T i in the region beyond the LCFS where the two measurements overlap. . Investigation of the reason for this difference will form the basis for future work. Fig. 1 . 1Ion saturation current (J sat ) as a function of distance from the LCFS during the inter-ELM period of a 900 kA H-mode plasma. The different coloured/shaped points represent data from different repeat shots. Also shown is an exponential fit (fall off length of 1.2 cm) to the data. Fig. 2 . 2A typical collector current versus grid 1 voltage graph for the RFEA obtained during inter-ELM measurements. The results of two grid 1 voltage sweeps have been combined and binned (square points) and fitted using eq. (1).Fig. 3. T i measurements made using the RFEA compared with T e measurements made using Thomson scattering (TS) and T i measurements made using charge exchange recombination spectroscopy (CXRS) as a function of radius during the inter-ELM period. The dotted and dashed lines are linear and exponential (fall off length 5 cm) fits to the CXRS T i data. Fig. 4 . 4Type III ELMing discharge results obtained using the RFEA. (a) Ion saturation current measurements made using the RFEA slit plate during ELMs as a function of distance from the LCFS. (b) Collector current measured during ELMs as a function of grid 1 voltage. The closed circles represent points measured 3 to 6 cm from the LCFS and are shown fitted using equation (1) with a T i =50 eV. The open circles represent points measured 11 to 16 cm from the LCFS and are shown fitted using equation (1) with a T i =30 eV. Fig. 5 . 5Collector current measured during type I ELMs as a function of grid 1 voltage. ELM measurements were made in both a type I and type III ELMing discharge. The type I ELMing discharge was a lower single null plasma with 1.9 MW of neutral beam heating and a plasma current of 650 kA. The pedestal electron density and temperature measured usingThe filaments ejected into the SOL during ELMs which are detected by the RP, occur on a timescale of the order of tens of microseconds[18] which is less than the fastest sweep time of grid 1 (approximately 100 µs) and hence the ion T i of a single ELM cannot be measured. Instead, to determine the average T i in an ELM, slow sweeps of the grid 1 voltage were made (at a frequency of 20 Hz) in a method similar that used for ELM T i measurements on ASDEX Upgrade[10]. Due to the filamentry nature of ELMs, not all ELMs generated by the plasma will strike the RFEA. The plasma Dα emission trace was used to detect when an ELM occurred and the ELM was determined to have struck the RFEA if a spike was seen on the RFEA slit plate current trace during the ELM. When an ELM was detected by the RFEA, the collector current and grid 1 voltage at the time of the ELM were recorded. The accumulation of these measurements from a large number of ELMs allows graphs of ion current versus grid one voltage to be made at different distances from the LCFS. From these graphs, the ion temperature can be determined using equation(1). For both the type I and type III ELMing discharges only data from the side of the RFEA pointing away from the plasma flow is presented.Measurements P3-82 6 Thomson scattering during the H-mode periods was approximately 3x10 19 m -3 and 200 eV respectively. The type III ELMing discharge was a connected double null plasma with 1.9 MW of beam heating and a plasma current of 600 kA. The pedestal electron density and temperature measured using Thomson scattering were approximately 2x10 19 m -3 and 100 eV respectively. During the inter-ELM period, ion temperatures of 10 to 70 eV were measured at distances of 3 to 8 cm from the LCFS. In the inter-ELM period in the SOL, T i /T e was found to be 2 to 6 with a mean of 4. Measurements during type III ELMs gave T i ≈ 50 eV, 3 to 6 cm from the LCFS, decreasing to give T i ≈ 30 eV, 11 to 16 cm from the LCFS. Measurements during type I ELMs give T i ≈ 40 eV at a distance of 10 to 15 cm from the LCFS. The results obtained show that the ions in both type I and type III ELMs can carry significant energy into the far SOL and this qualitatively supports the results obtained in other machines during type I ELMs. AcknowledgementsThis work was part-funded by the RCUK Energy Programme under grant EP/I501045 and the European Communities under the contract of Association between EURATOM and CCFE. The views and opinions expressed herein do not necessarily reflect those of the European Commission.References I Hutchinson, Principles of Plasma Diagnostics. New YorkCambridge University Presssecond ed.I. Hutchinson, Principles of Plasma Diagnostics, second ed., Cambridge University Press, New York, 2002. . A S Wan, Rev. Sci. Intrum. 571542A. S. Wan, et al., Rev. Sci. Intrum. 57 (1986) 1542. . R A Pitts, Nucl. Fusion. 4682R. A. Pitts, et al., Nucl. Fusion 46 (2006) 82. . R A Pitts, Rev. Sci. Intrum. 744644R. A. Pitts, et al., Rev. Sci. Intrum. 74 (2003) 4644. . M Kočan, Plasma Phys. Control. Fusion. 5365002M. Kočan, Plasma Phys. Control. Fusion 53 (2011) 065002. . M Kočan, Contrib. Plasma Phys. 50836M. Kočan, et al., Contrib. Plasma Phys. 50 (2010) 836. . P Tamain, J. Nucl. Mater. 4151139P. Tamain, et al., J. Nucl. Mater. 415 (2011) S1139. . S Elmore, Plasma Phys. Control. Fusion. 5465001S. Elmore, et al., Plasma Phys. Control. Fusion, 54 (2012) 065001. . M Kočan, J. Nucl. Mater. 4151133M. Kočan, et al., J. Nucl. Mater. 415 (2011) S1133. . M Kočan, Nucl. Fusion. 5223016M. Kočan, et al., Nucl. Fusion 52 (2012) 023016. . B Lloyd, Nucl. Fusion. 5194013B. Lloyd, et al., Nucl. Fusion 51 (2011) 094013. . G F Matthews, Plasma Phys. Control. Fusion. 361595G. F. Matthews, Plasma Phys. Control. Fusion 36 (1994) 1595. . S Elmore, J. Nucl. Mater. in printS. Elmore, et al., J. Nucl. Mater., in print. . A Kirk, Plasma Phys. Control. Fusion. 5365011A. Kirk, et al., Plasma Phys. Control. Fusion 53 (2011) 065011. . R Scannell, Rev. Sci. Intrum. 81R. Scannell, et al., Rev. Sci. Intrum. 81 (2010) 10D520. . N J Conway, Rev. Sci. Intrum. 77N. J. Conway, et al., Rev. Sci. Intrum. 77 (2006) 10F131 T W Morgan, Proc. of the 37 th EPS Conference on Plasma Physics. of the 37 th EPS Conference on Plasma PhysicsDublin, IrelandT. W. Morgan, et al., Proc. of the 37 th EPS Conference on Plasma Physics, Dublin, Ireland. . A Kirk, Plasma Phys. Control. Fusion. 47A. Kirk, et al., Plasma Phys. Control. Fusion 47 (2005) 315. P3-82

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{'abstract': 'Ion energy measurements have been made in the scrape off layer of the Mega Amp Spherical Tokamak (MAST) using a midplane retarding field energy analyser (RFEA) in H-mode plasmas during the inter-edge localised mode (ELM) period and during type I and type III ELMs. During the inter-ELM period at distances of 3 to 8 cm from the last closed flux surface (LCFS), ion temperatures of 20 to 70 eV have been measured giving an ion to electron temperature ratio of 2 to 7 with a mean of 4. During type III ELMs, an ion temperature of 50 eV has been measured 3 to 6 cm from the LCFS which decreases to 30 eV at distances 11 to 16 cm from the LCFS. During type I ELMs, an ion temperature of 40 eV has been measured at a distance of 10 to 15 cm from the LCFS.', 'arxivid': '1306.6505', 'author': ['S Y Allan \nEURATOM/CCFE Fusion Association\nCulham Science Centre\nOX14 3DBAbingdonUnited Kingdom\n', 'S Elmore \nEURATOM/CCFE Fusion Association\nCulham Science Centre\nOX14 3DBAbingdonUnited Kingdom\n\nThe University of Liverpool\nBrownlow HillL69 3GJLiverpoolUnited Kingdom\n', 'A Kirk \nEURATOM/CCFE Fusion Association\nCulham Science Centre\nOX14 3DBAbingdonUnited Kingdom\n', 'M Kočan \nMax-Planck-Institut für Plasmphysick\nBoltzmannstrasse, 2D-85748GarchingGermany\n', 'P Tamain \nCulham Science Centre\nCEA\nIRFM\nF-13108, OX14 3DB, 1UKSaint-Paul-lez-Durance, AbingdonFrance\n', 'Scott Allan '], 'authoraffiliation': ['EURATOM/CCFE Fusion Association\nCulham Science Centre\nOX14 3DBAbingdonUnited Kingdom', 'EURATOM/CCFE Fusion Association\nCulham Science Centre\nOX14 3DBAbingdonUnited Kingdom', 'The University of Liverpool\nBrownlow HillL69 3GJLiverpoolUnited Kingdom', 'EURATOM/CCFE Fusion Association\nCulham Science Centre\nOX14 3DBAbingdonUnited Kingdom', 'Max-Planck-Institut für Plasmphysick\nBoltzmannstrasse, 2D-85748GarchingGermany', 'Culham Science Centre\nCEA\nIRFM\nF-13108, OX14 3DB, 1UKSaint-Paul-lez-Durance, AbingdonFrance'], 'corpusid': 55820752, 'doi': '10.1016/j.jnucmat.2013.01.263', 'github_urls': [], 'n_tokens_mistral': 5175, 'n_tokens_neox': 4381, 'n_words': 2974, 'pdfsha': '0af84d60a28e159ba90feb120536a7a1173c7642', 'pdfurls': ['https://arxiv.org/pdf/1306.6505v1.pdf'], 'title': ['P3-82 Ion energy measurements on MAST using a midplane RFEA P3-82 2', 'P3-82 Ion energy measurements on MAST using a midplane RFEA P3-82 2'], 'venue': []}

arxiv

Learning without Forgetting for Vision-Language Models Da-Wei Zhou State Key Laboratory for Novel Software Technology Nanjing University Yuanhan Zhang [email protected] S-Lab Nanyang Technological University Jingyi Ning [email protected] State Key Laboratory for Novel Software Technology Nanjing University Han-Jia Ye State Key Laboratory for Novel Software Technology Nanjing University De-Chuan Zhan State Key Laboratory for Novel Software Technology Nanjing University Ziwei Liu [email protected] S-Lab Nanyang Technological University Learning without Forgetting for Vision-Language Models Class-Incremental Learning (CIL) or continual learning is a desired capability in the real world, which requires a learning system to adapt to new tasks without forgetting former ones. While traditional CIL methods focus on visual information to grasp core features, recent advances in Vision-Language Models (VLM) have shown promising capabilities in learning generalizable representations with the aid of textual information. However, when continually trained with new classes, VLMs often suffer from catastrophic forgetting of former knowledge. Applying VLMs to CIL poses two major challenges: 1) how to adapt the model without forgetting; and 2) how to make full use of the multi-modal information. To this end, we propose PROjectiOn Fusion (PROOF) that enables VLMs to learn without forgetting. To handle the first challenge, we propose training task-specific projections based on the frozen image/text encoders. When facing new tasks, new projections are expanded and former projections are fixed, alleviating the forgetting of old concepts. For the second challenge, we propose the fusion module to better utilize the cross-modality information. By jointly adjusting visual and textual features, the model can capture semantic information with a stronger representation ability. Extensive experiments on nine benchmark datasets validate PROOF achieves state-of-the-art performance. * Han-Jia Ye and Ziwei Liu are corresponding authors.Preprint. Under review. Introduction In our ever-changing world, training data often comes in a stream format with new classes, requiring a learning system to absorb them continually [19,18]. To address the challenge of learning emerging new classes, Class-Incremental Learning (CIL) has been proposed [47]. However, in CIL, the absence of former classes triggers catastrophic forgetting [16], where learning new concepts overwrites the knowledge of old ones and results in decline in performance [33]. Numerous efforts have been made [37,15,79,53,62,77] to combat catastrophic forgetting in the machine learning field. With the rapid development of pre-training techniques [20], recent years have witnessed the transition of CIL research from training from scratch [67,21,78] to utilizing pre-trained models (PTM) [63,64,49]. With the help of PTM, e.g., Vision Transformers [13], incremental models are born with strong transferability to grasp the visual features. Facing the domain gap introduced by the incremental classes, they only need to learn a limited number of additional parameters [26,11,34] as the patches to bridge the gap, which significantly simplifies the challenge of incremental learning. While pre-trained ViT-based CIL methods focus on learning the visual features to recognize new concepts, recent advances in Vision-Language Models (VLM) have demonstrated the potential of textual information in building generalized feature representations. A typical work, i.e., contrastive language-image pre-training [46] (CLIP), maps the visual and textual information in the shared embedding space, enabling robust learning and recognition of concepts from diverse sources. This integration of visual and textual modalities presents a promising avenue for developing continual learning models that can effectively adapt to real-world scenarios. Extending VLMs to CIL faces two significant challenges. First, sequentially tuning the VLM overwrites the innate generalizability and former concepts, leading to forgetting and poor performance on future tasks. Second, relying solely on textual information for classification neglects the valuable cross-modal features present in the multi-modal inputs. To fully utilize this information, it is necessary to explore methods for cross-modal fusion beyond textual features. Correspondingly, we aim to turn a VLM into a continual learner that is both retentive and comprehensive. Retentive refers to the model's ability to maintain its pre-trained capabilities, thereby preserving generalizability and enabling it to perform well on future tasks without forgetting. Comprehensive refers to the model's capacity to integrate and adjust information from multiple modalities. By leveraging these characteristics, we can mitigate catastrophic forgetting and use cross-modal features to build more robust classifiers as data evolves. In this paper, we propose PROjectiOn Fusion (PROOF) to address catastrophic forgetting in VLM. To make the model retentive, we freeze the pre-trained image/text backbones and append liner projections on top of them. The task-specific information is encoded in the corresponding projection layer by mapping the projected features. When facing new tasks, new projections are extended while old ones are frozen, preserving former knowledge. Besides, we aim to fuse the information from different modalities via cross-model fusion, which allows for the query embedding to be adjusted with context information. Consequently, PROOF efficiently incorporates new classes and meanwhile resists forgetting old ones, achieving state-of-the-art performance on nine benchmark datasets. We also investigate the zero-shot performance of VLM with new evaluation protocols and metrics, and find that PROOF maintains its zero-shot performance with a simple modification. Related Work Vision-Language Model (VLM) Tuning: Recent years have witnessed the prosperity of research in VLMs, e.g., CLIP [46], ALIGN [25], CoCa [70], Florence [73], BLIP [31], CLIPPO [54], and Flamingo [1]. These models are pre-trained on vast amounts of images and texts, achieving a unified embedding space across modalities. With great generalizability, they can be applied for downstream tasks in a zero-shot manner. However, a domain gap still exists between the pre-trained and downstream datasets, requiring further tuning for better performance. CoOp and CoCoOp [85,84] apply prompt learning [32] into VLM tuning with learnable prompt tokens. Subsequent works explore VLM tuning via adapter tuning [17], prompt distribution learning [39], task residual learning [72], similarity learning [76], descriptor learning [42], and optimal transport mapping [10]. However, they only focus on adapting VLM to downstream tasks while overlooking the forgetting of former ones. Class-Incremental Learning (CIL): aims to learn from evolutive data and absorb new knowledge without forgetting [81]. Replay-based methods [40,4,8,38,9] save and replay former instances to recover old knowledge when learning new ones. Knowledge distillation-based methods [47,33,14] build the mapping between models as regularization. Parameter regularization-based methods [27,2,74,3] weigh the importance of different parameters as regularization. Model rectification-based methods [50,78,67,71] rectify the inductive bias for unbiased predictions. Dynamic networks [69,58,82,59] show strong performance by expanding the network structure as data evolves. CIL with VLM: Aforementioned CIL methods aim to train an incremental model from scratch, while it would be easier to start with a pre-trained model [30]. The integration of pre-trained Vision Transformer [13] into CIL has attracted the attention of the community, and most methods [63,64,49] employ parameter-efficient tuning techniques to learn without forgetting. S-Prompt [61] explores CLIP in domain-incremental learning, but the application of VLM in CIL remains relatively unexplored. WiSE-FT [66] utilizes weight ensemble for robust finetuning, while it cannot be extended to multiple tasks. This paper aims to address this research gap by presenting a comprehensive solution for tuning vision-language models without suffering from forgetting. From Old Classes to New Classes In this section, we introduce the background information about class-incremental learning and vision language models. We also discuss the naïve solutions for tuning VLM in CIL. Class-Incremental Learning Given a data stream with emerging new classes, class-incremental learning aims to continually incorporate the knowledge and build a unified classifier [81]. We denote the sequence of B training sets without overlapping classes as D 1 , D 2 , · · · , D B , where D b = {(x i , y i )} n b i=1 is the b-th training set with n b instances. A training instance x i ∈ R D belongs to class y i ∈ Y b . Y b is the label space of task b, and Y b ∩ Y b ′ = ∅ for b ̸ = b ′ . Following the typical CIL setting [47,22,67], a fixed number of exemplars from the former classes are selected as the exemplar set E. During the b-th incremental stage, we can only access data from D b and E for model training. The target is to build a unified classifier for all seen classes Y b = Y 1 ∪ · · · Y b continually. In other words, we hope to find a model f (x) : X → Y b that minimizes the expected risk: f * = argmin f ∈H E (x,y)∼D 1 t ∪···D b t I (y ̸ = f (x)) ,(1) where H denotes the hypothesis space and I(·) is the indicator function. D b t denotes the data distribution of task b. Following [63,64,61], we assume that a pre-trained vision-language model is available as the initialization for f (x), which will be introduced in Section 3.2. Vision-Language Model This paper focuses on contrastive language-image pre-training (CLIP) [46] as the VLM. During pretraining, CLIP jointly learns an image encoder g i (·) : R D → R d and a text encoder g t (·) : R Dt → R d in a contrastive manner, where D/Dt are input dimensions of image/text, and d is the embedding dimension. CLIP projects a batch of image-text pairs into a shared embedding space. It maximizes the cosine similarity of paired inputs and minimizes it for unmatched ones. Benefiting from the massive training data, CLIP can synthesize a zero-shot classifier that generalizes to unseen classes. The output of CLIP is formulated as: p(y i | x) = exp (cos (z, w i ) /τ ) |Y b | j=1 exp (cos (z, w j ) /τ ) ,(2) where cos(·, ·) denotes cosine similarity, τ is learnable temperature parameter, z = g i (x) is the image embedding. Correspondingly, w i is the text embedding of class y i obtained by feeding templated texts, e.g., "a photo of a [CLASS]" into the text encoder. We denote the templated text of class i as t i . Eq. 2 aims to find the most similar text t i that maximizes the cosine similarity to the query image. Overcome Forgetting in Class-Incremental Learning CIL, as a long-standing problem, has garnered significant attention from the research community. In this section, we introduce two typical solutions for adapting pre-trained models with new classes. Vision-Based Learning: Traditional CIL methods primarily rely on the image encoder to capture the patterns of new classes. One such method, L2P [64], leverages visual prompt tuning [26] to enable incremental updates of a pre-trained Vision Transformer [13]. By keeping the image encoder frozen, L2P trains a learnable prompt pool Pool and combines it with patch embeddings to obtain instance-specific embeddings. The optimization target can be formulated as: L = ℓ (h (ḡ i (x i , Pool)) , y i ) + L reg ,(3) where h(·) is the classification head,ḡ i is the frozen image encoder, L reg is the regularization loss for prompt selection. By freezing the encoder, Eq. 3 grasps the new pattern with humble forgetting. CLIP Tuning: The issue of tuning VLM without forgetting in CIL remains unaddressed, as previous works have solely focused on transferring CLIP to downstream tasks without considering the performance of former tasks. For instance, CoOp [85] converts text inputs into a learnable prompt, i.e., t i = [V] 1 [V] 2 · · · [V] M [CLASS] i . The posterior probability in Eq. 2 is transformed into: p(y i | x) = exp (cos (z, g t (t i )) /τ ) |Y b | j=1 exp (cos (z, g t (t j )) /τ ) .(4) With the help of the learned prompt, Eq. 4 enables the model to be transferred to the downstream task. However, since the prompt template is shared for all tasks, sequentially tuning CoOp will suffer catastrophic forgetting of former concepts. Discussions: Current methods focus on different aspects of CIL. Vision-based methods (e.g., Eq. 3) address the issue of forgetting but neglect the valuable semantic information conveyed in texts. Conversely, CLIP's pre-trained text encoder captures class-wise relationships that can enhance model learning. Meanwhile, transfer learning methods (e.g., Eq. 4) effectively leverage the cross-modal information, while sequentially tuning them suffers the catastrophic forgetting of former concepts. Is it possible to combine the cross-modal information and meanwhile resist catastrophic forgetting? PROOF: Projection Fusion for VLM Observing the limitations of typical vision-based methods in utilizing textual information and forgetting in CLIP tuning, we aim to leverage cross-modality knowledge in CLIP while effectively mitigating forgetting. To this end, we must make the model retentive and comprehensive. Retentive represents the ability to adapt to downstream tasks without forgetting, and we propose projections to map the pre-trained features in the projected feature space. Our unique training strategy ensures the preservation of former knowledge by freezing old projections and expanding new ones for new tasks. The comprehensive aspect involves co-adapting and utilizing cross-modal information to enhance unified predictions. The query instance's embedding is influenced by both visual and textual information, allowing for instance-specific adaptation and enabling comprehensive predictions. In the following sections, we introduce the learning paradigm and the co-adaptation process. Lastly, we provide detailed guidelines for training and inference. Expandable Feature Projection CLIP is known for its strong zero-shot performance [46], i.e., Eq. 2 obtains competitive results even without explicit training on the specific tasks. However, given the domain gap between pre-trained and downstream tasks, an adaptation process is still necessary to capture the characteristics of the latter. Specifically, we introduce a linear layer (denoted as "projection") which is appended after the frozen image and text embeddings to facilitate the matching of pair-wise projected features. Denoting the projection of image/text as P i (·) : R d → R d and P t (·) : R d → R d , Eq. 2 is transformed into: p(y i | x) = exp (cos (P i (z) , P t (w i )) /τ ) |Y b | j=1 exp (cos (P i (z) , P t (w j )) /τ ) Projected Matching .(5) We denote the classification based on Eq. 5 as f PM (x). By freezing the image and text encoders, it aligns the downstream features in the projected space, allowing the model to encode the relevant downstream information into projection layers. Since the pre-trained model outputs generalizable features, the projection layer learns to recombine features in a data-driven manner. For instance, in a task involving 'birds,' the projection would assign a higher weight to features like 'beaks' and 'wings.' This adaptation enables the projected features to better discern and recognize downstream tasks. Expandable Projections: However, sequentially training a single projection layer still leads to forgetting of former tasks, resulting in confusion when combining old and new concepts. To this end, we expand task-specific projections for each new task. Specifically, we append a newly initialized projection layer P b i , P b t when a new task D b arrives. This results in a set of projections: {P 1 i , P 2 i , · · · P b i , }, {P 1 t , P 2 t , · · · P b t , }, and we adopt the aggregation as the output, i.e., P i (z) = b m=1 P m i (z) , P t (w) = b n=1 P n t (w) .(6) In Eq. 6, projected features from different stages are mapped and aggregated to capture the different emphases of former and latter tasks. For example, former tasks might emphasize 'beak' features Figure 1: Illustration of PROOF. The model learns expandable projections and aggregates them to get the aggregated features. The query instance, prototype features, textual features, and context prompts are fed into the cross-modal fusion. The fusion process utilizes self-attention to co-adapt the input set, which outputs the adapted features. The adapted query embedding is separately matched among the visual prototypes and textual features to get the final prediction. Red parts are trainable while gray ones are frozen. Cross-Modal Fusion for bird recognition, while later tasks may focus on 'beards' features to differentiate cats. The aggregation of different projections produces a comprehensive representation of the query instance. By substituting Eq. 6 into Eq. 5, the model aligns the unified features in the joint space. How to resist forgetting of former projections? To overcome forgetting old concepts, we freeze the projections of former tasks when learning new ones, i.e., {P 1 i ,P 2 i , · · · P b i , } (same for P t ). It allows the newly initialized projection to learn the residual information of new tasks, incorporating new concepts while preserving the knowledge of former ones. During the learning process of task b, we optimize the cross-entropy loss to encode the task-specific information into the current projections. Effect of projections: The illustration of projections are shown in Figure 1 (left). PROOF learns projections based on the pre-trained encoders, which fits new patterns and maintains the generalizability of pre-trained model. The parameter number of each projection layer is d × d, which is neglectable for the pre-trained model. Furthermore, the model learns new projections for new tasks, and task-specific projections fit new concepts easily. Since we only optimize the current projections and freeze old ones, the former knowledge is preserved, and forgetting is alleviated. Contextualizing Projections with Projection Fusion In Eq. 5, the projected visual and textual features are directly matched in the joint space. However, it would be beneficial to further refine these features to capture the contextual relationship between images and texts. For instance, when the query instance is a 'panda,' it is desirable to adjust the visual and textual features in a coherent manner to highlight discriminative attributes such as black eyes and ears. Similarly, when the query instance is a 'cat,' features like beards and tails should be emphasized. This adjustment process involves jointly adapting the query embedding and the context (e.g., textual information) to obtain a contextualized embedding. Correspondingly, we propose a set-to-set function that contextualizes and fuses the query embeddings and contextual information. Specifically, we denote the adaptation function as T (·). It receives the query instance and context information as bags, i.e., [P i (z), Context], and outputs the set of adjusted embeddings while being permutation-invariant: T ([P i (z), Context]) = [P i (z),Context] . T (·) encodes the set information and performs adaptation on each component. In the following, we describe the construction of the context information Context and provide details on the implementation of the set-to-set function. How to define the context? In Eq. 5, the mapping is established between the query instance and the textual information (i.e., classifiers). The classifiers represent the typical textual description of the corresponding class, i.e., the common feature. Hence, a naïve idea is to utilize textual features as the context, i.e., Context = W, W = [P t (w 1 ), P t (w 2 ), · · · , P t (w |Y b | )] ∈ R |Y b |×d is the concatenation of all textual classifiers. However, recent works find an inherent domain gap [35] between the visual and textual embeddings in VLM. The gap leads to visual and textual embeddings residing in two separate clusters in the embedding space, which hinders effective pair-wise mapping. Correspondingly, we leverage visual prototype features [51] as a useful tool for capturing the common characteristics of each class. Define the visual prototype of class k as: p k = 1 N |D b | j=1 I(y j = k)g i (x j ), where N = |D b | j=1 I(y j = k). They are calculated via forward pass at the beginning of each incremental stage and stay fixed in subsequent tasks. Visual prototypes are representative features of the corresponding class, which can serve as the visual context to adjust the embeddings. Hence, we augment the context with projected visual information, i.e., Context = [P, W], where P = [P i (p 1 ), P i (p 2 ), · · · , P i (p |Y b | )] ∈ R |Y b |×d is the concatenation of all visual prototypes. Combining prototypes from multiple modalities help the model adapt and fuse information in a cross-modal manner, which goes beyond simple visual-textual matching. Implementing T with Self-Attention: In our implementation, we use the self-attention (SA) mechanism [55,36] as the cross-modal fusion function T . Being permutation invariant, SA is good at outputting adapted embeddings even with long dependencies, which naturally suits the characteristics of the adaptation function. Specifically, SA keeps the triplets (query Q, key, K, and value V). The inputs are projected into the same space, i.e., K = W ⊤ K [ k k ; ∀k k ∈ K ] ∈ R d×|K| . Similar projections are made for Q and V. The query x q ∈ Q is matched against a list of keys K where each key has a value V . The output is the sum of all the values weighted by the proximity of the key to the query point:P i (z) = P i (z) + k α qk V :,k ,(7) where α qk ∝ exp Pi(z) ⊤ W Q ·K √ d , V :,k is the k-th column of V . The adaptation process is the same for other components in Context. Specifically, we have Q = K = V = [P i (z), Context]. Effect of Cross-Modal Fusion: The illustration of the projection fusion is shown in Figure 1 (right). We utilize the visual and textual information of seen classes as context information to help adjust the instance-specific embeddings. The fusion model is trained incrementally to adjust embeddings to reflect the context information as data evolves. With the contextualized embeddings, we can conduct the visual mapping and textual matching: p(y i | x) = exp cos P i (z),P i (p i ) /τ |Y b | j=1 exp cos P i (z),P i (p j ) /τ Visual Matching + exp cos P i (z),P t (w i ) /τ |Y b | j=1 exp cos P i (z),P t (w j ) /τ Textual Matching .(8) In Eq. 8, the model assigns logits to the query instance by the similarity to the adapted visual and textual prototypes. The incorporation of cross-modal matching enhances the prediction performance. Learning Context Prompts: In addition to visual prototypes and textual classifiers, we also introduce a set of learnable context prompts {c 1 , · · · , c b }, c i ∈ R c×d to be optimized as data evolves. c denotes the length of each prompt. Similar to projection layers, we make the context prompts expandable to catch the new characteristics of new tasks. We initialize a new context prompt while learning a new task and freeze others {c 1 ,c 2 , · · · , c b }. The context prompts serve as adaptable context information, enhancing the co-adaption. Hence, the context information is formulated as Context = [P, W, C], where C is the aggregation of all context prompts. Note that C only encodes the task-specific information into the self-attention process, which does not serve as the matching target in Eq. 8. Summary of PROOF In PROOF, we first enable learning new concepts via projected mapping. Then, to accommodate new concepts without interference from previous ones, we initialize new projections for each new task and freeze the former ones. Besides, we utilize self-attention to adjust the embeddings of the query instance and the context information to promote cross-modal fusion. Figure 1 illustrates three predictions, i.e., projected matching (Eq. 5), visual/textual matching (Eq. 8). We denote these models as f PM (x), f VM (x), f TM (x), respectively. During training, we optimize the cross-entropy loss: min {P b i ,P b t ,T ,c b } ℓ(f PM (x), y) + ℓ(f VM (x), y) + ℓ(f TM (x), y) .(9) In Eq. 9, all pre-trained weights are frozen, and we only optimize these additional parameters. For inference, we aggregate the three logits, i.e., f (x) = f PM (x) + f VM (x) + f TM (x) . We give the pseudo-code of PROOF in the supplementary. Table 1 are not plotted due to their inferior performance. ImageNet-R CUB UCF B0 Inc20 B100 Inc20 B0 Inc20 B100 Inc20 B0 Inc10 B50 Inc10 A A BĀ A BĀ A BĀ A BĀ A BĀ A B Finetune ✗ 1.1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (a) Aircraft Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (b) CIFAR100 Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (c) Cars Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (d) SUN Base150 Inc30 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (e) Food Base50 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (f) ObjectNet Base100 Inc20 Experiment In this section, we compare PROOF in comparison to state-of-the-art methods on benchmark datasets to investigate the capability of overcoming forgetting. We also conduct ablations to analyze the effect of each component in the model. Furthermore, we address a fundamental issue in VLM training known as zero-shot degradation. Finally, we extend PROOF to other VLMs to verify the universality of proposed method. Further details and experimental results can be found in the supplementary. Experimental Setup Dataset: Following the benchmark CIL settings [47,64,63,71,83], we evaluate the performance on CIFAR100 [29], CUB200 [57], ObjectNet [6], and ImageNet-R [12]. We also follow the setting in VLM tuning [85], and formulate FGVCAircraft [41], StanfordCars [28], Food101 [7], SUN397 [68] and UCF101 [52] into CIL setting. Specifically, we sample (a subset of) 100 classes from CIFAR100, Aircraft, Cars, Food, UCF, 200 classes from CUB200, ObjectNet, ImageNet-R, and 300 classes from SUN to ease the data split. Following [47], the class order of training classes is shuffled with random seed 1993. The dataset splits are denoted as Base-x, Inc-y, where x represents the number of classes in the first stage, and y represents the number of new classes in each subsequent task. x = 0 means each task contains y classes. More details are reported in the supplementary. (a) OpenAI weight Comparison methods: We first compare to SOTA CIL methods iCaRL [47], MEMO [82], Sim-pleCIL [83] L2P [64], DualPrompt [63]. Denote the baseline of sequential finetuning as Finetune; we combine it with different tuning techniques, e.g., LiT [75] and CoOp [85]. We also report the zero-shot performance of CLIP as ZS-CLIP by matching the query instance to the template (Eq. 2). 1XPEHURI&ODVVHV $FFXUDF\ =6&/,3 )XVLRQ 3URMHFWLRQ 3URMHFWLRQ )XVLRQ 3URMHFWLRQ )XVLRQ &RQWH[W3URPSW (b) Compositional components 1XPEHURI&RQWH[W3URPSWV $FFXUDF\ /DVW$FFXUDF\ $YHUDJH$FFXUDF\ (c) Context prompt length Implementation details: We deploy all methods with PyTorch [44] and PyCIL [80] on Tesla V100. We use the same network backbone ViT-B/16 for all compared methods for fair comparison. We experiment with two commonly used pre-trained CLIP weights, i.e., OpenAI [46] and OpenCLIP LAION-400M [24]. The model is trained with a batch size of 64 for 5 epochs, and we use SGD with momentum for optimization. The learning rate starts from 0.001 and decays with cosine annealing. Following [47], we use the herding [65] algorithm to select 20 exemplars per class for rehearsal. The context prompt length is set to 3, and the head of self-attention is set to 1. The template for classification is the same as [43]. The source code will be made publicly available upon acceptance. Performance Measure: Denote the Top-1 accuracy after the b-th stage as A b , we follow [47] to use A B (last stage performance) andĀ = 1 B B b=1 A b (average performance) for evaluation. Benchmark Comparison We report the results on nine benchmark datasets using ViT-B/16 (OpenCLIP LAION-400M) in Table 1 and Figure 2. These splits include the scenarios with large and small base classes. Notably, PROOF consistently achieves the best performance among all the methods compared. Sequential finetuning of the model with contrastive loss leads to significant forgetting, irrespective of the tuning techniques employed (e.g., LiT and CoOp). Since SimpleCIL and ZS-CLIP do not finetune the model parameters, they achieve competitive results by transferring the knowledge from the pre-training stage into the downstream tasks. However, most methods achieve better performance than ZS-CLIP, indicating the importance of incremental learning on downstream tasks. Specifically, we can draw three key conclusions from these results. 1) The first stage performance of PROOF surpasses that of the typical prompt learning method, CoOp, thus validating the effectiveness of learning projections for downstream tasks. 2) The performance curve of PROOF consistently ranks at the top across all methods, demonstrating its capability to resist forgetting. 3) Compared to vision-only methods (i.e., L2P and DualPrompt), PROOF exhibits substantial improvement, indicating textual and visual information can be co-adapted to facilitate incremental learning. Ablation Study Different backbone weights: The comparison in Section 5.2 is based on LAION-400M pre-trained CLIP. As another popular pre-trained weight, we also explore the performance of the weights provided by OpenAI. We report the last accuracy A B of four competitive methods on nine benchmarks in Figure 3(a). We report the full results of the incremental performance in the supplementary. As depicted in the figure, PROOF still performs the best on all datasets among all compared methods. Compositional components: We experiment on CIFAR100 B0 Inc10 to investigate the importance of each part in PROOF. Specifically, we compare the performance of PROOF and its sub-modules, i.e., projections and cross-modal fusion. The results, shown in Figure 3( Middle: LAION score during incremental learning. Right: accuracy of seen, unseen, and harmonic mean (HM) at the last incremental stage. PROOF † strikes a balance between adaptivity and the ZS performance. the learning process. Furthermore, when combining them together, we find 'Projection & Fusion' further shows better performance than any of them, verifying that they can work together by fusing the expandable representations. Lastly, when incorporating the context prompts, the model shows the best performance among all variations, verifying the effectiveness of expandable task-specific prompts in incremental learning. Ablations verify the importance of each component in PROOF. Number of context prompts: Figure 3 Exploring Zero-Shot Performance CLIP is known to have the zero-shot (ZS) ability, i.e., even if the model has not been trained for recognizing the image, it can still predict the possibility of an image x belonging to the class y by matching the cosine similarity via Eq. 2. The strong generalizability of CLIP makes it a popular model in computer vision. However, in CIL, the model is continuously updated with the downstream task, which weakens the generalizability and harms the ZS performance [66] on subsequent tasks. In this section, we explore the ZS performance degradation of CLIP and propose a variation of PROOF to maintain the ZS performance. Evaluation protocol for ZS performance: Current CIL methods focus on evaluating 'seen' classes, i.e., evaluating Y b = Y 1 ∪ · · · Y b after learning task b. However, since CLIP exhibits ZS performance, we can also assess the performance on 'unseen' classes Y u = Y b+1 ∪ · · · Y B to investigate the ZS performance. Correspondingly, we can obtain the performance metrics A S (seen classes), A U (unseen classes), and A HM (harmonic mean of A S and A U ) after each task. Additionally, based on the LAION400M [48] pre-trained CLIP, we also utilize a subset of 10,000 image-text pairs from LAION400M, and calculate the matching score of them, i.e., cosine similarity of image-text embeddings. We denote the average matching score as LAION score, which indicates the matching degree of the adapted model on the upstream tasks. Given the relationship between generalizability and the upstream task, the LAION score serves as an effective measure of ZS performance. Results: We compare the aforementioned measures on CIFAR100 B0 Inc10. Apart from compared methods in Section 5.2, we also report a variation of PROOF, namely PROOF † . The only difference lies in the design of the projection, where PROOF † uses a residual format P i (z) = b m=1 (P m i (z) + z) as the output (same for P t ). To investigate the ZS performance as model updates, we show the accuracy on unseen classes A U along incremental stages in Figure 4(a), where ZS-CLIP shows the best performance. Due to the incorporation of pre-trained information into the projected features, PROOF † maintains competitive ZS performance. Conversely, other methods experience a decline in ZS performance as their focus shifts to downstream tasks. We observe a similar trend in Figure 4(b), where PROOF † achieves a LAION score similar to that of ZS-CLIP. Lastly, we report A S , A U , A HM in the last incremental stage in Figure 4(c). We can infer a trade-off between the adaptivity on downstream tasks and the generalizability of ZS performance. Compared to PROOF, PROOF † sacrifices the adaptivity to maintain ZS performance, strikings a balance between seen and unseen classes. Therefore, when ZS performance is essential, using PROOF † is the preferred choice. Task 5: Play -A man plays a purple guitar while sitting next to a man playing the accordion -A man is on a golf course playing golf -People playing hockey on ice Figure 5: The training protocol of five incremental stages in Flickr30K. We split training instances into five tasks, i.e., walk, stand, run, ride, and play. The training/testing sets do not include images that do not fall into these tasks. We use the pre-trained BEiT-3 as the initialization and sequentially learn cross-modal retrieval tasks. At the end of each task, the model is evaluated on all previously learned concepts. Extension to Other Vision Language Models In the main paper, we use CLIP as an exemplar VLM due to its popularity and representativeness. However, the field of vision-language models is rapidly advancing, and various models are available. Therefore, in this section, we extend our PROOF framework to another widely used vision-language model, namely BEiT-3 [60], focusing on the cross-modal retrieval task. BEiT-3 is a popular VLM that demonstrates promising performance across multiple vision-language tasks. When fine-tuning BEiT-3 for cross-modal retrieval, it functions as a dual encoder, similar to CLIP, featuring a dual-branch structure. As the retrieval task differs from classification, we adopt a degradation of PROOF by solely employing the projection expansion strategy without implementing cross-modal fusion. We refer the readers to the BEiT-3 paper [60] for more details about the backbone model. For evaluation, we employ the Flickr30K dataset [45] to assess the performance of incremental cross-modal retrieval. Flickr30K comprises 31,783 images collected from the Flickr image-sharing platform, encompassing diverse themes such as daily life, travel, people, food, and scenes. Each image in the dataset is accompanied by five manually annotated textual descriptions, which provide descriptive information capturing the main content and context of the images. To formulate an incremental data stream, we utilize keyword matching to identify images containing different actions (e.g., walk, stand, run, ride, play). Then, we split the training instances into five subsets based on these specific actions. Figure 5 illustrates the formulation of the stream, while images not associated with these actions are excluded from training. To create a balanced testing set, we maintain a 5:1 training-to-testing ratio for splitting the training and testing pairs. Following the instructions provided by BEiT 2 , we use 'beit3_base_itc_patch16_224 3 ' as the VLM's initialization. (f) TR@10 Figure 6: Incremental performance of each method. IR means the recall of image retrieval, and TR denotes the recall of text retrieval. PROOF consistently outperforms other compared methods with a substantial margin on the incremental cross-modal retrieval task. For evaluation, we employ standard cross-modal retrieval measures, namely R@1, R@5, and R@10. The retrieval is conducted in two directions: image → text and text → image. Similarly to the CIL evaluation, we also report the last recall R B @1 and the average recallR@1 across incremental stages. To provide a comparative analysis, we compare PROOF against typical fine-tuning as the baseline and modify MEMO [82] and DER [69] for comparison. These methods represent state-of-the-art CIL approaches that can be adapted with minor modifications to the current task. However, methods such as L2P and DualPrompt are unsuitable for cross-modal retrieval tasks as they do not focus on cross-modal matching. The experimental results are presented in Table 2, and the incremental performance of each measure is depicted in Figure 6. As evident from these figures, fine-tuning the model with new concepts leads to catastrophic forgetting in cross-modal retrieval tasks. However, equipping the model with incremental learning abilities alleviates forgetting. Among all the compared methods, PROOF consistently achieves the best performance across different retrieval tasks and metrics, thereby verifying its effectiveness in mitigating forgetting in VLMs. Experiments conducted on different VLMs and tasks establish PROOF as a unified and general framework. Future work involves extending PROOF to other VLMs and applications, such as image captioning [56] and VQA [5]. Conclusion Real-world learning systems necessitate the ability to continually acquire new knowledge. In this paper, we aim to equip the popular VLM with the CIL ability. Specifically, we learn the expandable projections so that visual and textual information can be aligned incrementally. This expansion technique allows for the integration of new concepts without compromising previous ones. Additionally, we enforce cross-modality fusion with self-attention mechanism, where visual and textual information are jointly adapted to produce instance-specific embeddings. Extensive experiments validate the effectiveness of our proposed PROOF. Furthermore, we demonstrate that a simple variation of PROOF preserves the model's zero-shot capability during updating. Limitations: Possible limitations include the usage of exemplars, where storage constraints and privacy issues may happen. Future works include extending the model to exemplar-free scenarios. Supplementary Material In the main paper, we present a method to prevent forgetting in vision-language models through projection expansion and fusion. The supplementary material provides additional details on the experimental results mentioned in the main paper, along with extra empirical evaluations and discussions. The organization of the supplementary material is as follows: • Section A presents the pseudo code of PROOF, explaining the training and testing pipeline. • Section B reports comprehensive experimental results from the main paper, including the full results of nine benchmark datasets with two data splits, as well as the results obtained using OpenAI weights. Furthermore, this section includes additional ablations such as variations of projection types, results from multiple runs, and an analysis of the number of parameters. • Sections C and D provide detailed information on the experiments, including dataset and exemplar selection details, an introduction to the compared methods, and a discussion of the broader impacts. A Pseudo Code In this section, we provide a detailed explanation of PROOF by presenting the pseudo-code in Alg 1. In each incremental stage, we are provided with the training dataset D b and the exemplar set E, with the objective of updating the current model f (·). Prior to training, we initially extract visual prototypes for the new classes (Line 1). These prototypes are calculated using the frozen visual embedding g i (·), ensuring their stability throughout model updates. Subsequently, we freeze the former projections and context prompts, while initializing new projections and context prompts specifically for the new incremental task (Line 2 to Line 4). These steps represent the model expansion process, which is followed by the subsequent learning process. During the learning process, we concatenate the training instances from the current dataset and the exemplar set, initiating a for-loop. For each instance-label pair, we calculate the projected visual and textual embeddings (Line 6 to Line 9). Subsequently, we compute the projected matching loss (Line 10) to encode task-specific information into the current projection layers. Based on the projected features, we derive context information and perform cross-modal fusion (Line 11 to Line 13). Consequently, we obtain three logits for model updating and utilize the cross-entropy loss to update these modules (Line 14). The updated model is then returned as the output of the training process. Discussions: Besides the simple addition operation, there exist alternative methods for aggregating information from multiple projections. However, due to the requirement of fixed input dimensionality for cross-modal fusion, we refrain from using concatenation as the aggregation function. Furthermore, it is worth noting that MEMO [82] can be viewed as a specific case where concatenation is employed for aggregation. Nonetheless, its inferior performance (as shown in Table 3) suggests that summation is a more favorable choice. B Additional Experimental Results This section presents further experimental results of PROOF, including comparisons with multiple runs, analysis of parameter numbers, and ablations on projection types. Additionally, we report the results of using OpenAI pre-trained CLIP and provide the full results mentioned in the main paper. B.1 Multiple Runs Following [47], we conduct typical CIL comparisons by randomly splitting the classes with a fixed seed of 1993, and these results are reported in the main paper. In this supplementary section, we perform multiple runs by varying the random seed among {1993, 1994, 1995, 1996, 1997}. We repeat the comparison on CIFAR100 Base50 Inc10 and ImageNet-R Base100 Inc20 five times and present the results in Figure 7. The solid line represents the mean performance, while the shaded area indicates the standard deviation. From these figures, it is evident that PROOF consistently outperforms Calculate the visual embedding z = g i (x); 7: Calculate the projected visual feature P i (z); 8: Calculate the textual embedding w of all seen classes; 9: Calculate the projected textual embeddings of all seen classes P t (w); 10: Calculate the logits for projected matching f PM (x) via Eq. 5; ▷ Projected matching 11: Calculate the projected visual features for all visual prototypes p; 12: Conduct cross-modal fusion via Eq. 7; ▷ Cross-modal fusion 13: Calculate the logits for visual and textual matching via Eq. 8; ▷ Visual & textual matching 14: Calculate the loss via Eq. 9; update the model; return the updated model; other methods by a significant margin across different dataset splits. These results validate the robustness of PROOF. B.2 Parameter Analysis As mentioned in the main paper, the additional parameters in PROOF come from two sources: the projections and the fusion module. The projection layers are implemented with a single linear layer, each containing d × d parameters, where d = 512 is the embedding dimension. Similarly, the cross-modal fusion is implemented with a single-head self-attention mechanism, and the number of parameters is determined by the weight matrices W Q , W K , and W V , each containing d × d parameters. These extra parameters are negligible compared to the large backbone of the pre-trained CLIP model, which has approximately 150 million parameters. To provide a clear comparison of the parameter numbers for each method, we present the details in Figure 8 using CIFAR100 B0 Inc10 as an example. The figure illustrates that PROOF has a similar parameter scale to other finetune-based methods, while achieving significantly stronger performance. SimpleCIL, which only utilizes the vision branch, requires fewer parameters for the textual branch but lacks the zero-shot capability. L2P and DualPrompt also only require the vision branch but need an additional encoder to identify the appropriate prompt, resulting in a higher parameter count compared to PROOF. B.3 Variation of Projection Types Apart from simple linear layers, there are other methods to implement the projection layers, such as layer-wise rescale (SSF) [34] and Adapter [23]. SSF learns a d-dimensional rescale parameter to project the features, while Adapter learns both the down-projection and up-projection for feature mapping. In this section, we explore the performance of these projection methods on CIFAR100 B0 Inc10 and present the results in Figure 9. The figure clearly demonstrates that using a single linear layer as the projection layer achieves the best performance among all methods, indicating its superiority. Furthermore, this result suggests that a simple linear mapping can effectively bridge the gap between visual and textual domains. B.4 Variation of Context Information In the main paper, we discuss the composition of the context information Context, which should include information from visual prototypes, textual classifiers, and context prompts. In this section, we conduct ablations to demonstrate the effectiveness of constructing Context with [P, W, C]. Specifically, we perform experiments on CIFAR100 B0 Inc10 and change the context construction to Context = P (visual prototypes only), Context = W (textual prototypes only), Context = [P, W] (visual and textual prototypes), and Context = [P, W, C] (current choice). We keep the same classification rule for these ablations, i.e., classification via Eq. 9. When visual/textual prototypes are not included in the context, we use the projected features without adaptation as the matching target in Eq. 8. The results are presented in Figure 10. From the results, we observe that using visual prototypes or textual prototypes alone yields similar performance, and the impact of adjustment is marginal. However, when both visual and textual prototypes are jointly utilized as context information, the model can learn from cross-modality and achieve better performance. Lastly, the introduction of context prompts into the context further enhances the performance of PROOF, resulting in the best performance among all variations. B.5 Different Pre-trained Weights In the main paper, we discussed two popular weights for pre-trained CLIP: OpenAI [46] 4 and OpenCLIP [24] 5 . We primarily presented the results of the OpenCLIP pre-trained model in the main paper, while providing the results of the OpenAI weights using a radar chart. In this section, we present the full results of the OpenAI pre-trained CLIP on nine benchmark datasets in Figure 11. The results demonstrate that PROOF consistently achieves the best performance among all methods, regardless of the pre-trained weights used. This highlights the robustness of PROOF in the learning process. B.6 Full Results We provide the complete results of the benchmark comparison in the main paper, which are presented in Table 3 and Figures 12 and 13. These results are obtained using OpenCLIP pre-trained weights on LAION-400M [24]. Table 3 displays the average and last accuracy for the nine benchmark datasets. Figures 12 and 13 (i) ObjectNet Base0 Inc20 Figure 11: Incremental performance of different methods when using OpenAI weights. We report the performance gap after the last incremental stage of PROOF and the runner-up method at the end of the line. PROOF consistently achieves the best performance regardless of the pre-trained weights used. Across all these evaluations, PROOF consistently outperforms the compared methods, demonstrating its superior performance. C Experimental Details This section provides detailed information about the experiments conducted, including the introduction of datasets, exemplar selection, and the methods compared in the paper. C.1 Dataset Introduction In our evaluation, we utilize nine datasets, which are introduced in Table 4 in the main paper. It is worth noting that some of these datasets have a larger number of classes, but we select a subset of classes for ease of data split and evaluation. Exemplar Selection: As mentioned in the main paper, we follow the exemplar selection approach in [47,67,22] to utilize herding algorithm [65]. In addition, there are two typical methods [81] to store these exemplars in memory. 1. Fixed Memory Budget: In this approach, a fixed memory budget of K instances is allocated. Given the number of seen classes denoted as |Y b |, the model selects K |Y b | exemplars per class after each incremental stage. 2. Expandable Exemplar Set: In this method, an expandable exemplar set is maintained as the data evolves. With the number of exemplars per class denoted as k, the model stores |Y b | × k exemplars in total after each incremental stage. We evaluate both protocols using these benchmark datasets in our experiments. Specifically, we employ the first policy for CIFAR100 and Food, keeping a total of 2,000 exemplars. Since these datasets consist of 100 classes, the average number of exemplars per class after the last incremental stage is 20. We adopt the second policy for the other datasets and store 20 exemplars per class. C.2 Compared methods introduction This section provides an overview of the compared methods discussed in the main paper. These methods, listed in the order presented in Table 3, include: • Finetune: This baseline method involves finetuning the pre-trained CLIP model using contrastive loss. No regularization terms are set, and no part of the model is frozen, allowing us to observe the forgetting phenomenon in sequential learning. (i) ObjectNet Base0 Inc20 Figure 12: Incremental performance of different methods. We report the performance gap after the last incremental stage of PROOF and the runner-up method at the end of the line. • Finetune LiT [75]: Following LiT, which freezes the image encoder and only finetunes the text encoder, we design Finetune LiT with CIL. Similar to finetune, we sequentially tune the model with contrastive loss while the image encoder is frozen during optimization. • Finetune CoOp [85]: Following the CoOp method, this approach freezes both the image encoder and text encoder. It optimizes a learnable prompt tensor t (as in Eq.4) using contrastive loss without utilizing any historical data for rehearsal. • SimpleCIL [83]: This method relies on the pre-trained image encoder and does not involve the text encoder. The frozen image encoder extracts class centers (prototypes) for each new class, and a cosine classifier is utilized for classification. Since the model is not updated via backpropagation, it showcases the generalizability of the pre-trained vision encoder on downstream tasks. • ZS-CLIP [46]: This baseline freezes the pre-trained CLIP model and predicts the logits of each incoming class using cosine similarity (Eq. 2). It serves as a reference for the performance of pre-trained CLIP on downstream tasks. • CoOp (with exemplars): This method combines the CoOp approach with exemplar rehearsal. During the learning of new classes, the model utilizes a combination of the current dataset and exemplar set to optimize the learnable prompt. (i) ObjectNet Base100 Inc20 Figure 13: Incremental performance of different methods with large base classes. We report the performance gap after the last incremental stage of PROOF and the runner-up method at the end of the line. • iCaRL [47]: iCaRL is a typical class-incremental learning algorithm that employs knowledge distillation and exemplar replay to mitigate forgetting. It combines contrastive loss with distillation loss to learn new classes while retaining knowledge of old classes. • MEMO [82]: As a state-of-the-art class-incremental learning algorithm based on network expansion, MEMO is modified to be compatible with the CLIP structure. The image and text encoders are expanded for new tasks, and the concatenated features are used for prediction based on cosine similarity. • L2P [64]: L2P is a state-of-the-art class-incremental learning algorithm utilizing pre-trained vision transformers. In this case, the text encoder of CLIP is dropped, and a prompt pool (as in Eq. 3) is learned to adapt to evolving data. Another pre-trained image encoder is required to select the appropriate prompt during inference. • DualPrompt [63]: DualPrompt is an extension of L2P that incorporates two types of prompts: general prompts and expert prompts. It also relies on another pre-trained image encoder for prompt retrieval. It is important to note that these methods are compared fairly, meaning they are initialized with the same pre-trained weights for incremental learning. Since some compared methods are not designed with the CLIP encoder, we modify their backbone into pre-trained CLIP for a fair comparison. We use the same number of exemplars for a fair comparison of the methods with exemplars. D Broader Impacts In this work, we address the class-incremental learning problem with vision-language models, which is a fundamental challenge in machine learning. Our focus is on tackling the forgetting problem that arises when sequentially finetuning a vision-language model. We propose solutions to project and integrate features from multiple modalities for unified classification. Our research provides valuable insights for applications that struggle with managing the forgetting issue in large pre-trained vision-language models. However, there are still ample opportunities for further exploration in this field. Therefore, we aspire to stimulate discussions on class-incremental learning in real-world scenarios and encourage more research to develop practical models for this purpose. We also acknowledge the ethical considerations associated with this technology. It is crucial to recognize that individuals expect learning systems to refrain from storing any personal information for future rehearsal. While there are risks involved in AI research of this nature, we believe that developing and demonstrating such techniques are vital for comprehending both the beneficial and potentially concerning applications of this technology. Our aim is to foster discussions regarding best practices and controls surrounding these methods, promoting responsible and ethical utilization of technology. Figure 2 : 2Incremental performance of different methods. We report the performance gap after the last incremental stage of PROOF and the runner-up method at the end of the line. Finetune-based methods in Figure 3 : 3Ablation study. Left: experiments on 9 benchmarks with OpenAI weights. Middle: ablation study on compositional components in PROOF. Every part improves the performance of CIL. Right: AB andĀ with change of context prompts. The performance is robust to the change of context prompt length. Figure 4 : 4Experiment on zero-shot performance. Left: accuracy on unseen classes during incremental learning. (b) verifies the strong performance of context prompts, and we explore the appropriate length c of the context prompt on CIFAR100 B0 Inc10. By varying the number of c among {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 30, 50, 100}, we report the average performance and last performance of PROOF inFigure 3(c). As shown in the figure, the performance of PROOF is robust with the change of the prompt length, and we set c = 3 as the default length. Training dataset: D b ; Exemplar set: E; Current model: f (·); Output: Updated model; 1: Extract prototypes p for each new class in D b ; 2: Freeze current projections and context prompts; 3: Initialize new projections for the visual and textual branches, P b i , P b t ; ▷ Expand projections 4: Initialize new context prompt c b ; 5: for (x, y) ∈ D b ∪ E do ▷ Incremental learning 6: Figure 7 : 7Results of multiple runs for CIFAR100 and ImageNet-R. The solid line represents the mean performance, while the shaded area indicates the standard deviation. PROOF consistently and robustly outperforms other methods by a substantial margin. Figure 8 :Figure 9 : 89Number of parameters in different methods. The shaded area represents the parameters used during training but dropped during inference. PROOF achieves state-of-the-art performance with a comparable number of parameters to other methods. Variations of projection layers. The choice of using a single linear layer as the projection layer achieves the best performance. Figure 10 : 10Variations of context information. The choice of using visual prototypes, textual prototypes, and context prompts as the context information achieves the best performance. Table 1 : 1Average and last performance of different methods. The first and second columns represent the methods with and without exemplars. The performance of L2P and DualPrompt are reproduced with the source code with exemplars. The best performance is shown in bold. Full results are reported in supplementary.Method Exemplar Task 1 : 1Walk-A basket vendor walking down a busy city street -An old man in a suit is smoking a cigar and walking forward -Young Asian individuals walking in a busy city streetTask 2: Stand -Three women in black outfits hold black umbrellas and signs while a man stands by -Four people in casual clothing are standing outside holding garbage bags -A Muslim girl is standing on a street corner listening to music in a crowded city Task 3: Run -A woman in the blue sweater is running through a brown field -Two black and white dogs running towards each other in the grass -A rugby player running the ball between two downed opponents Task 4: Ride -Two people riding dirt bikes on a bike trail -A young woman riding a bike down a street past a crowd of people -Two men , both wearing green cycling clothes and helmets , are riding bicycles Table 2 : 2Average and last performance of different methods. The best is in bold. The first row stands for the text retrieval task, and the second is the image retrieval task.MethodText → ImageR B @1R@1 R B @5R@5 R B @10R@10Method Image → Text R B @1R@1 R B @5R@5 R B @10R@10 Finetune 48.79 62.89 76.38 85.04 85.68 91.84 DER [69] 78.37 84.48 96.34 98.23 99.06 99.59 MEMO [82] 83.18 87.79 96.57 98.27 99.16 99.66 PROOF 85.68 89.43 97.07 98.68 99.79 99.86 Finetune 37.35 51.33 67.38 77.77 77.95 85.55 DER [69] 66.71 74.18 89.63 93.00 94.84 96.69 MEMO [82] 69.53 76.35 91.89 94.44 96.09 97.32 PROOF 72.10 78.01 93.10 95.27 96.92 97.90 illustrate the incremental performance with varying numbers of base classes.1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (a) Aircraft Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (b) CIFAR100 Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (c) Cars Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (d) ImageNet-R Base0 Inc20 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (e) CUB Base0 Inc20 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (f) UCF Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (g) SUN Base0 Inc30 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) (h) Food Base0 Inc10 1XPEHURI&ODVVHV $FFXUDF\ L&D5/ /3 'XDO3URPSW 6LPSOH&,/ 0(02 &R2S =6&/,3 3522) Table 3 : 3Average and last performance comparison of different methods. The first and second columns represent the methods with and without exemplars. The performance of L2P and DualPrompt are reproduced with the source code with exemplars. The best performance is shown in bold.70.82 72.98 69.18 83.21 74.94 78.06 74.27 89.48 85.41 86.96 84.65 PROOF ✓ 85.34 80.10 82.32 80.30 84.93 79.43 81.67 79.18 92.34 89.92 91.70 89.16 64.84 38.23 20.00 40.62 12.96 29.74 12.05 43.27 17.46 32.85 17.17 Finetune CoOp [85] ✗ 45.93 23.11 39.33 24.89 36.01 14.18 33.13 18.67 21.24 6.29 16.21 6.82Method Exemplar Aircraft CIFAR100 Cars B0 Inc10 B50 Inc10 B0 Inc10 B50 Inc10 B0 Inc10 B50 Inc10 A A BĀ A BĀ A BĀ A BĀ A BĀ A B Finetune ✗ 3.16 0.96 1.72 1.05 7.84 4.44 5.30 2.46 3.14 1.10 1.54 1.13 Finetune LiT [75] ✗ 27.74 14.28 25.10 13.77 44.66 14.69 27.69 7.67 84.12 72.37 83.08 78.23 Finetune CoOp [85] ✗ 14.54 7.14 13.05 7.77 47.00 24.24 41.23 24.12 36.46 21.65 37.40 20.87 SimpleCIL [83] ✗ 59.24 48.09 53.05 48.09 84.15 76.63 80.20 76.63 92.04 86.85 88.96 86.85 ZS-CLIP [46] ✗ 26.66 17.22 21.70 17.22 81.81 71.38 76.49 71.38 82.60 76.37 78.32 76.37 CoOp [85] ✓ 44.26 39.87 41.81 39.18 83.37 73.36 78.34 73.04 89.73 84.91 87.98 86.60 iCaRL [47] ✓ 53.60 43.98 50.40 45.33 79.91 63.94 71.94 63.00 89.38 84.95 86.71 84.19 MEMO [82] ✓ 42.24 25.41 38.16 27.75 84.67 74.98 80.75 75.34 88.23 81.31 84.90 81.83 L2P [64] ✓ 55.06 44.88 47.78 43.37 76.42 66.21 72.67 67.88 83.81 72.44 79.76 73.47 DualPrompt [63] ✓ 55.95 46.53 50.93 46.50 79.07 70.06 74.81 70.75 85.30 74.35 81.32 75.85 PROOF ✓ 61.00 53.59 59.99 58.90 86.70 79.05 82.92 78.87 93.26 89.84 90.53 89.54 Method Exemplar ImageNet-R CUB UCF B0 Inc20 B100 Inc20 B0 Inc20 B100 Inc20 B0 Inc10 B50 Inc10 A A BĀ A BĀ A BĀ A BĀ A BĀ A B Finetune ✗ 1.37 0.43 1.01 0.88 2.06 0.64 0.56 0.47 4.51 1.59 1.21 0.80 Finetune LiT [75] ✗ 64.88 30.42 57.75 29.77 58.15 35.28 51.95 35.96 79.25 64.84 81.79 65.40 Finetune CoOp [85] ✗ 60.73 37.52 54.20 39.77 27.61 8.57 24.03 10.14 47.85 33.46 42.02 24.74 SimpleCIL [83] ✗ 81.06 74.48 76.84 74.48 83.81 77.52 79.75 77.52 90.44 85.68 88.12 85.68 ZS-CLIP [46] ✗ 83.37 77.17 79.57 77.17 74.38 63.06 67.96 63.06 75.50 67.64 71.44 67.64 CoOp [85] ✓ 82.40 76.20 79.76 77.13 77.34 68.70 74.09 67.47 90.13 86.24 88.36 85.71 iCaRL [47] ✓ 72.22 54.38 68.67 60.15 82.04 74.74 78.57 75.07 89.47 84.34 88.51 84.11 MEMO [82] ✓ 80.00 74.07 76.72 73.95 77.32 65.69 72.88 66.41 84.02 74.08 82.58 75.48 L2P [64] ✓ 75.73 67.22 74.15 71.20 79.23 68.54 75.85 71.12 88.71 83.93 86.51 83.22 DualPrompt [63] ✓ 78.47 Method Exemplar SUN Food ObjectNet B0 Inc30 B150 Inc30 B0 Inc10 B50 Inc10 B0 Inc20 B100 Inc20 A A BĀ A BĀ A BĀ A BĀ A BĀ A B Finetune ✗ 4.51 1.59 0.78 0.72 3.49 1.71 2.14 1.52 1.34 0.47 0.69 0.54 Finetune LiT [75] ✗ 79.25 SimpleCIL [83] ✗ 82.13 75.58 78.62 75.58 87.89 81.65 84.73 81.65 52.06 40.13 45.11 40.13 ZS-CLIP [46] ✗ 79.42 72.11 74.95 72.11 87.86 81.92 84.75 81.92 38.43 26.43 31.12 26.43 CoOp [85] ✓ 80.46 73.44 77.68 73.06 85.38 76.15 81.74 76.35 46.16 33.81 40.40 34.47 iCaRL [47] ✓ 78.56 67.30 74.74 69.07 84.12 71.68 78.86 70.64 45.28 26.97 37.22 26.15 MEMO [82] ✓ 81.48 73.45 78.00 73.87 89.18 82.85 86.50 83.08 46.98 33.37 41.62 34.67 L2P [64] ✓ 79.83 72.14 76.16 72.32 84.48 75.22 85.04 80.56 46.18 34.00 43.90 39.57 DualPrompt [63] ✓ 80.14 73.06 77.25 73.82 87.12 81.27 85.37 82.36 53.13 40.59 45.84 40.37 PROOF ✓ 83.57 77.28 80.70 77.49 90.04 84.73 87.52 84.74 55.28 44.36 49.64 43.65 Table 4 : 4Introduction about benchmark datasets.Dataset # training instances # testing instances # Classes LinkCIFAR100 50,000 10,000 100 Link CUB200 9,430 2,358 200 Link ImageNet-R 24,000 6,000 200 Link ObjectNet 26,509 6,628 200 Link Aircraft 6,667 3,333 100 Link Cars 4,135 4,083 100 Link UCF 10,053 2,639 100 Link SUN 72,870 18,179 300 Link Food 79,998 20,012 100 Link https://github.com/microsoft/unilm/blob/master/beit3/README.md 3 https://conversationhub.blob.core.windows.net/beit-share-public/beit3/ pretraining/beit3_base_itc_patch16_224.pth https://github.com/openai/CLIP 5 https://github.com/mlfoundations/open_clip Flamingo: a visual language model for few-shot learning. 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{'abstract': 'Class-Incremental Learning (CIL) or continual learning is a desired capability in the real world, which requires a learning system to adapt to new tasks without forgetting former ones. While traditional CIL methods focus on visual information to grasp core features, recent advances in Vision-Language Models (VLM) have shown promising capabilities in learning generalizable representations with the aid of textual information. However, when continually trained with new classes, VLMs often suffer from catastrophic forgetting of former knowledge. Applying VLMs to CIL poses two major challenges: 1) how to adapt the model without forgetting; and 2) how to make full use of the multi-modal information. To this end, we propose PROjectiOn Fusion (PROOF) that enables VLMs to learn without forgetting. To handle the first challenge, we propose training task-specific projections based on the frozen image/text encoders. When facing new tasks, new projections are expanded and former projections are fixed, alleviating the forgetting of old concepts. For the second challenge, we propose the fusion module to better utilize the cross-modality information. By jointly adjusting visual and textual features, the model can capture semantic information with a stronger representation ability. Extensive experiments on nine benchmark datasets validate PROOF achieves state-of-the-art performance. * Han-Jia Ye and Ziwei Liu are corresponding authors.Preprint. Under review.', 'arxivid': '2305.19270', 'author': ['Da-Wei Zhou \nState Key Laboratory for Novel Software Technology\nNanjing University\n\n', 'Yuanhan Zhang [email protected] \nS-Lab\nNanyang Technological University\n\n', 'Jingyi Ning [email protected] \nState Key Laboratory for Novel Software Technology\nNanjing University\n\n', 'Han-Jia Ye \nState Key Laboratory for Novel Software Technology\nNanjing University\n\n', 'De-Chuan Zhan \nState Key Laboratory for Novel Software Technology\nNanjing University\n\n', 'Ziwei Liu [email protected] \nS-Lab\nNanyang Technological University\n\n'], 'authoraffiliation': ['State Key Laboratory for Novel Software Technology\nNanjing University\n', 'S-Lab\nNanyang Technological University\n', 'State Key Laboratory for Novel Software Technology\nNanjing University\n', 'State Key Laboratory for Novel Software Technology\nNanjing University\n', 'State Key Laboratory for Novel Software Technology\nNanjing University\n', 'S-Lab\nNanyang Technological University\n'], 'corpusid': 258967796, 'doi': '10.48550/arxiv.2305.19270', 'github_urls': ['https://github.com/microsoft/unilm/blob/master/beit3/README.md', 'https://github.com/openai/CLIP', 'https://github.com/mlfoundations/open_clip'], 'n_tokens_mistral': 30872, 'n_tokens_neox': 26201, 'n_words': 13992, 'pdfsha': 'fd928577d67dd01048d13f284a6256164bbcf2f0', 'pdfurls': ['https://export.arxiv.org/pdf/2305.19270v1.pdf'], 'title': ['Learning without Forgetting for Vision-Language Models', 'Learning without Forgetting for Vision-Language Models'], 'venue': []}

arxiv

THE RADIUS IN MATRIX ALGEBRAS-EXAMPLES AND REMARKS 9 Nov 2015 Moshe Goldberg THE RADIUS IN MATRIX ALGEBRAS-EXAMPLES AND REMARKS 9 Nov 2015To my daughter Maya on her 40th birthday The main purpose of this note is to illustrate how the radius in a finite-dimensional power-associative algebra over a field F, either R or C, may change when the multiplication in this algebra is modified. Our point of departure will be F n×n , the familiar algebra of n × n matrices over F with the usual matrix operations, where it is known that the radius is the classical spectral radius. We shall alter the multiplication in F n×n in three different ways and compute, in each case, the radius in the resulting algebra. Introduction: the radius and its basic properties Let A be a finite-dimensional algebra over a field F, either R or C. We shall assume that A is power-associative, i.e., that the subalgebra of A generated by any one element is associative; thus ensuring that powers of each element in A are unambiguously defined. As usual, by a minimal polynomial of an element a in A we mean a monic polynomial of lowest positive degree with coefficients in F that annihilates a. With this familiar definition, we may cite: Theorem 1.1 ([G1, Theorem 1.1]). Let A be a finite-dimensional power-associative algebra over F. Then: (a) Every element a ∈ A possesses a unique minimal polynomial. (b) The minimal polynomial of a divides every other polynomial over F that annihilates a. Denoting the minimal polynomial of an element a ∈ A by p a , we follow [G1] and define the radius of a to be the nonnegative quantity r(a) = max{|λ| : λ ∈ C, λ is a root of p a }. The radius has been computed for elements in several well-known finite-dimensional powerassociative algebras. For instance, it was recently shown in [GL1] that the radius in the Cayley-Dickson algebras is given by the corresponding Euclidean norm. Another example emerged in [G1, page 4060], where it was established that if A is an arbitrary finite-dimensional matrix algebra over F with the usual matrix operations, then the radius of a matrix A ∈ A is given by the classical spectral radius, ρ(A) = max{|λ| : λ ∈ C, λ is an eigenvalue of A}. 2010 Mathematics Subject Classification. Primary 11C08, 16P10, 17A05, 17D05. Key words and phrases. Radius of an element in a finite-dimensional power-associative algebra, matrix algebras, standard matrix multiplication, Hadamard product in matrices, Jordan product in matrices. With this last example in mind, we recall the following theorem which asserts that the radius retains some of the most basic properties of the spectral radius not only in finite-dimensional matrix algebras with the usual matrix operations, but in the general finite-dimensional powerassociative case as well. As a final introductory remark we mention that an analysis of the relevance of the radius to stability of subnorms and to the Gelfand formula can be found in [G1], [G2], [G3], and [GL1]. Examples of radii in matrix algebras Our main purpose in this note is to illustrate how the radius in a finite-dimensional powerassociative algebra may change when the multiplication in this algebra is modified. Selecting a positive integer n, n ≥ 2, our point of departure will be F n×n , the familiar algebra of n × n matrices over F, either R or C, with the usual matrix operations. By what we already know about the radius in arbitrary finite-dimensional matrix algebras over F with the usual operations, we may register the following result which can also be derived directly from the fact that the roots of the minimal polynomial of a matrix A in F n×n are the eigenvalues of A. Theorem 2.1. The radius of a matrix A in F n×n is given by r(A) = ρ(A), where ρ denotes the spectral radius. The multiplication in F n×n can be altered, of course, in a myriad of ways. Often, however, computing the radius in the newly obtained algebra will remain out of reach. In what follows, we shall modify the multiplication in F n×n in three different ways, and calculate the radius in each case. We embark on our plan by replacing the standard multiplication in F n×n by the well-known Hadamard product which, for any two n × n matrices A = (α ij ) and B = (β ij ), is defined entry-wise by A • B = (α ij β ij ). The resulting algebra, denoted by F n×n H , has been extensively studied in the literature (see for example Chapter 5 in [HJ] and the references at the end of that chapter). Obviously, F n×n H is distributive, commutative, and associative; and its unit element is given by E, the n × n matrix all of whose entries are 1. Denoting the k-th power of a matrix A = (α ij ) in F n×n H by A [k] , we see that (2.1) A [k] = (a k ij ), k = 1, 2, 3, . . . . Assisted by this observation, we can now post: Theorem 2.2. The radius of a matrix A = (α ij ) in F n×n H is given by the sup norm of A, i.e., r(A) = max i,j |α ij |. Proof. Select a matrix A = (α ij ) in F n×n H , and let ζ 1 , . . . , ζ s (1 ≤ s ≤ n 2 ) be a list of all the distinct entries of A (so that each α ij equals precisely one of the ζ l 's). Let p A (t) = t m + α m−1 t m−1 + · · · + α 1 t + α 0 be the minimal polynomial of A in F n×n H , hence A [m] + α m−1 A [m−1] + · · · + α 1 A [1] + α 0 E = 0. By (2.1), this can be equivalently written as ζ m l + α m−1 ζ m−1 l + · · · + α 1 ζ l + α 0 = 0, l = 1, . . . , s. It follows that the ζ l are roots of p A ; and since these roots are distinct, we infer that the monic polynomial q(t) = (t − ζ 1 )(t − ζ 2 ) · · · (t − ζ s ) must divide p A . On the other hand, we notice that (A − ζ 1 E) • (A − ζ 2 E) • · · · • (A − ζ s E) = 0; so q annihilates A in F n×n H . Appealing to Theorem 1.1(b), we conclude that p A must divide q; hence p A = q, and the rest of the proof follows without difficulty. Another way of altering the standard multiplication in F n×n is to replace it by the familiar Jordan product A · B = 1 2 (AB + BA), which turns F n×n into the special Jordan algebra F n×n+ (e.g., [J, page 4, Definition 2]). Since F n×n is distributive, so is F n×n+ . Further, both F n×n+ and F n×n share the same unit element, the n × n identity matrix I. We observe, however, that F n×n+ , unlike F n×n , is commutative. Moreover, in contrast with F n×n , the algebra F n×n+ is not associative, nor even alternative. 2 Indeed, consider the matrices A = 0 1 0 0 ⊕ O n−2 , B = 0 0 1 0 ⊕ O n−2 , where O n−2 is the (n − 2) × (n − 2) zero matrix. Then, (A · B) · B = 1 4 (AB 2 + 2BAB + B 2 A) = 1 2 BAB = 1 2 B = 0 = A · (B · B), and alternativity is shattered. Despite the fact that F n×n+ is not alternative, it is power-associative. This is so because powers of matrices in F n×n+ coincide with those in F n×n , and hence are uniquely defined. Turning to compute the radius in F n×n+ , we realize that it is a simple task: Since F n×n+ and F n×n have an identical linear structure, and since raising to powers in F n×n+ and F n×n coincide, the minimal polynomials of a matrix A in F n×n+ and in F n×n are one and the same; thus, the radii of A in F n×n+ and in F n×n come to the same thing, yielding: Theorem 2.3. The radius of a matrix A in F n×n+ is given by r(A) = ρ(A). This result is of particular interest, precisely because it tells us that altering the multiplication in F n×n does not necessarily result in a different radius. In our last example, we shall modify the multiplication in F n×n in a more intricate way, by introducing the product A * B = (A ′ B ′ ) ′ , where A ′ is the matrix obtained from A by replacing α 1n , the (1, n) entry of A, by its negative, and where A ′ B ′ is the usual product of A ′ and B ′ in F n×n . Denoting our new algebra by F n×n * , we remark that it is distributive and associative. Indeed, for all A, B, C ∈ F n×n * we have, A * (B + C) = (A ′ (B + C) ′ ) ′ = (A ′ B ′ + A ′ C ′ ) ′ = (A ′ B ′ ) ′ + (A ′ C ′ ) ′ = A * B + A * C and similarly, (A + B) * C = A * C + B * C; so the distributive laws are in the bag. Furthermore, since (A * B) * C = (A ′ B ′ ) ′ * C = ((A ′ B ′ )C ′ ) ′ = (A ′ (B ′ C ′ )) ′ = A * (B ′ C ′ ) ′ = A * (B * C), associativity holds as well. We also observe that the identity matrix I constitutes the unit element in F n×n * , since for all A ∈ F n×n * , A * I = (A ′ I ′ ) ′ = (A ′ I) ′ = A, 2 As usual, we call an algebra A alternative if the subalgebra generated by any two elements of A is associative. and analogously, I * A = A. Lastly, we note that since F n×n is not commutative, neither is F n×n * . It seems interesting to mention that the algebra F n×n * possesses certain exotic properties which are not shared by either F n×n , F n×n H , or F n×n+ . For instance, F n×n * contains nilpotent matrices which have nonzero eigenvalues. To substantiate this statement, let A = (α ij ) be the n × n matrix all of whose entries are zero except for α 11 , α 1n , α n1 , and α nn which are given by 1, −i, i, and -1, respectively. It is not hard to verify that A * A = 0, so A is a nilpotent matrix of index 2 in F n×n * . At the same time, we have det(tI − A) = t n−2 (t 2 − 2), so √ 2 and − √ 2 are eigenvalues of A. Another property of F n×n * which is not shared by our previous matrix algebras lies in the fact that F n×n * admits positive matrices whose squares are negative. 3 For example, consider the n × n matrix A = (α ij ) where α 1n = α n1 = 1 and the rest of the entries vanish. While A is positive, its squaring in F n×n * provides the negative matrix all of whose entries are zero, except for the first and last entries along its diagonal which equal -1. Turning to compute the radius in F n×n * , we denote the k-th power of a matrix A in this algebra by A k , and offer the following elementary observation. Lemma 2.1. If A ∈ F n×n * , then (2.2) A k = ((A ′ ) k ) ′ , k = 1, 2, 3, . . . , where (A ′ ) k is the usual k-th power of A ′ in F n×n . Proof. For k = 1 the assertion is trivial. So assuming (2.2) for k, we get A k+1 = A * A k = A * ((A ′ ) k ) ′ = (A ′ (A ′ ) k ) ′ = ((A ′ ) k+1 ) ′ , and we are done. With the above lemma in our grip, we may now proceed to record: Theorem 2.4. The minimal polynomial of a matrix A in F n×n * coincides with the minimal polynomial of A ′ in F n×n . Proof. Let p(t) = α m t m + · · · + α 1 t + α 0 be a polynomial over F that annihilates A in F n×n * ; that is, α m A m + · · · + α 1 A 1 + α 0 I = 0. By (2.2), this is equivalent to α m ((A ′ ) m ) ′ + · · · + α 1 (A ′ ) ′ + α 0 I ′ = 0; or in other words, to α m (A ′ ) m + · · · + α 1 A ′ + α 0 I = 0. It follows that p annihilates A in F n×n * if and only if p annihilates A ′ in F n×n ; so aided by Theorem 1.1, the proof follows. An immediate consequence of Theorem 2.4 reads: Corollary 2.1. The radii of A in F n×n * and of A ′ in F n×n coincide. Finally, since the radius in F n×n is the spectral radius, we get: Theorem 2.5. The radius of a matrix A in F n×n * is given by r(A) = ρ(A ′ ). We conclude this note by pointing out that all our findings regarding F n×n * hold verbatim when the product is defined by A * B = (A ′ B ′ ) ′ where now, A ′ is obtained from A by negating α n1 , the (n, 1) entry of A. The author is truly grateful to Thomas Laffey for helpful discussions. Theorem 1.2 ([G1, Theorems 2.1 and 2.4]). Let A be a finite-dimensional power-associative algebra over F. Then:(a) The radius r is a nonnegative function on A.(b) The radius is homogeneous, i.e., for all a ∈ A and α ∈ F, r(αa) = |α|r(a). ( c ) cFor all a ∈ A and all positive integers k, r(a k ) = r(a) k . (d) The radius vanishes only on nilpotent elements of A. (e) The radius is a continuous function on A. 1 Naturally, a real-valued function on a finite-dimensional algebra A is said to be continuous if it is continuous with respect to the (unique) finite-dimensional topology on A. By a positive matrix we mean here a nonzero matrix all of whose entries are nonnegative. Similarly, a negative matrix is a nonzero matrix whose entries are all non-positive. Minimal polynomials and radii of elements in finite-dimensional power-associative algebras. Moshe Goldberg, Trans. Amer. Math. Soc. 3598Moshe Goldberg, Minimal polynomials and radii of elements in finite-dimensional power-associative algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 4055-4072. Radii and subnorms on finite-dimensional power-associative algebras, Linear Multilinear Algebra. Moshe Goldberg, 55Moshe Goldberg, Radii and subnorms on finite-dimensional power-associative algebras, Linear Multilin- ear Algebra, 55 (2007), no. 5, 405-415. Stable subnorms on finite-dimensional power-associative algebras. Moshe Goldberg, Electron. J. Linear Algebra. 17Moshe Goldberg, Stable subnorms on finite-dimensional power-associative algebras, Electron. J. Linear Algebra 17 (2008), 359-375. On the radius in Cayley-Dickson algebras. Moshe Goldberg, Thomas J Laffey, Proc. Amer. Math. Soc. 14311Moshe Goldberg and Thomas J. Laffey, On the radius in Cayley-Dickson algebras, Proc. Amer. Math. Soc. 143, (2015), no. 11, 4733-4744. A Roger, Charles R Horn, Johnson, Topics in Matrix Analysis. CambridgeCambridge Univ. PressRoger A. Horn and Charles R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge 1991. Nathan Jacobson, Structure and representations of Jordan algebras. Providence, R.IAmer. Math. SocXXXIXNathan Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloquium Publications, Vol. XXXIX, Amer. Math. Soc., Providence, R.I. 1968.

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{'abstract': 'The main purpose of this note is to illustrate how the radius in a finite-dimensional power-associative algebra over a field F, either R or C, may change when the multiplication in this algebra is modified. Our point of departure will be F n×n , the familiar algebra of n × n matrices over F with the usual matrix operations, where it is known that the radius is the classical spectral radius. We shall alter the multiplication in F n×n in three different ways and compute, in each case, the radius in the resulting algebra.', 'arxivid': '1511.02732', 'author': ['Moshe Goldberg '], 'authoraffiliation': [], 'corpusid': 119147608, 'doi': '10.1016/j.laa.2015.10.024', 'github_urls': [], 'n_tokens_mistral': 4503, 'n_tokens_neox': 4080, 'n_words': 2558, 'pdfsha': '396d851292e9e3eb801607c46688997cb02fa28d', 'pdfurls': ['https://arxiv.org/pdf/1511.02732v1.pdf'], 'title': ['THE RADIUS IN MATRIX ALGEBRAS-EXAMPLES AND REMARKS', 'THE RADIUS IN MATRIX ALGEBRAS-EXAMPLES AND REMARKS'], 'venue': []}

arxiv

Realization of A Non-Markov Chain in A Single 2D Crystal RRAM Rongjie Zhang Shenzhen Geim Graphene Center Tsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School Tsinghua University 518055ShenzhenP. R. China Wenjun Chen Shenzhen Geim Graphene Center Tsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School Tsinghua University 518055ShenzhenP. R. China Changjiu Teng Shenzhen Geim Graphene Center Tsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School Tsinghua University 518055ShenzhenP. R. China Wugang Liao Institute of Microscale Optoelectronics Shenzhen University 518060ShenzhenP. R. China Bilu Liu [email protected] Shenzhen Geim Graphene Center Tsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School Tsinghua University 518055ShenzhenP. R. China Hui-Ming Cheng [email protected] Shenzhen Geim Graphene Center Tsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School Tsinghua University 518055ShenzhenP. R. China Institute of Metal Research Shenyang National Laboratory for Materials Science Chinese Academy of Sciences 110016ShenyangP. R. China Realization of A Non-Markov Chain in A Single 2D Crystal RRAM 1 # These authors contributed equally to this work.2D materialsmicaion transportRRAMnon-Markov chain 3 The non-Markov processes widely exist in thermodymanic processes, while it usually requires packing of many transistors and memories with great system complexity in traditional device architecture to minic such functions. Two-dimensional (2D) material-based resistive random access memory (RRAM) devices show potential for next-generation computing systems with much-reduced complexity. Here, we achieve the non-Markov chain in an individual RRAM device based on 2D mica with a vertical metal/mica/metal structure. We find that the internal potassium ions (K + ) in 2D mica gradually move along the direction of the applied electric field, making the initially insulating mica conductive. The accumulation of K + is tuned by electrical field, and the 2D-mica RRAM possesses both unipolar and bipolar memory windows, high on/off ratio, decent stability and repeatability. 2 Importantly, the non-Markov chain algorithm is established for the first time in a single RRAM, in which the movement of K + is dependent on the stimulated voltage as well as their past states. This work not only uncovers the inner ionic conductivity of 2D mica, but also opens the door for such novel RRAM devices with numerous functions and applications. Introduction Information technology plays increasingly important roles in human society while conventional von-Neumann architecture is approaching its limit when processing the exponentially growing amount of data. 1,2 Combining memory/sensor cells and algorithms into cutting edge hardware and devices 3 is important in next-generation computing. 4,5 To this end, resistive random access memory (RRAM) or memristors 6,7 have been put forward to complment the traditional non-volatile memory devices owing to their high operation speed, high integration density, as well as applications in sophisticated condition. [8][9][10][11] Recently, two-dimensional (2D) materials have emerged as a versatile platform for RRAM applications. [12][13][14][15][16][17] Compared with bulk materials, the RRAM devices based on 2D materials possess outstanding performance. For example, 2D material-based RRAM devices have demonstrated ultra-fast switching of < 10 ns, 18 the thinnest nonvolatile resistive switching (RS), 19 sub-pA switching current, 20 ultralow operation voltage of 100 mV, 21 high-frequency operation at GHz level, 22 and excellent stability at high temperature of up to 340 °C. 23 The high integration of RRAMs is advantageous to accomplish specific functional algorithms in a machine learning system. Although deep learning algorithms can be realized by software, numerous transistors are usually needed, 24 which may degrade the operation stability and robustness of the system. Several algorithms, such as sparse coding, convolution neural network and reinforcement learning, are accomplished by using RRAM arrays. [25][26][27][28][29][30] The application of RRAM arrays requires software and hardware assistance. To overcome these limitations, the Markov chain, which is usually realized by software, was achieved in a single RRAM based on 2D SnSe. 31 The Markov chain simplifies and reduces the calculation in the system, which shows great potential in machine learning and automatic speech recognition. However, various physical phenomena are related to the non- 4 Markov 32 property, where the probability of the next state is related to both current and past states. For instance, the Markov chain is not applicable in the systems which are related with their historical states, such as time-dependence human neural memory brain, environmental-related quantum entangled state, 33 thermodynamic processes, and cumulative effect, etc. However, the achievement of a hardware non-Markov chain is elusive as it refers to complicated process, which hindrads the application of non-Markov process in historical related physical processes. In addition, the vacancybased or external ion based RS mechanism may cause the variation of RRAM performance, such as random telegraph noise 34 and set voltage variability, 35 which is challenging to be addressed. Therefore, it is urgent to explore new RS mechanism to achieve non-Markov process with high performance. Here, we report the realization of a non-Markov chain in a single 2D crystal RRAM with a novel internal ionic conduction mechanism. We find that the migration of the internal K + inside 2D mica leads to the formation of conductive filament (CF) paths, which is comfirmed by electrical measurements and the depth anylysis of K + . The 2D mica RRAM exhibits both unipolar and bipolar conductive behaviors with a high on/off current ratio of 10 3 , high stability and repeatability, verified by conductive atomic force microscopy (CAFM) measurements. The mica-based RRAM is capable to remember the polarity of input voltages, resulting in the different response behaviors of K + in mica under positive and negative electrical fields. Accordingly, it is the first time to realize the non-Markov process in an individual RRAM device. The internal ion transport demonstrated in this paper provides additional dimensions in memory devices, opening a new avenue in the design and fabrication of intelligent devices. 5 Figure 1a shows the structure and measurement schematics of the RRAM based on 2D mica, in which the AFM tip is an electrode to apply voltage and the bottom Au electrode is another one. The application of electric field leads to the accumulation of K + along the direction of the electrical field, which forms the CF in the vertical direction of mica ( Figure 1b). Here, trilayer mica with a thickness of 3 nm is selected to reveal the RS behaviors. Figure 1c shows the optical image of the mica flake, and its thickness is determined by AFM (inset of Figure 1d). The current-voltage (I-V) curves ( Figure 1e) displays that the mica-based device is initially insulating then conductive under the application of a cyclic triangular wave voltage with the maximum value of 2 V. The typical unipolar RRAM window is observed after 30 swept cycles, representing the formation of CF in the device. The mica is initially insulating (with a resistance >10 GΩ), and gradually switches to a low resistance state (< 0.1 GΩ) at a positive voltage of 1.5 V. The formation of CF results in stable RRAM features in the following sweeps. Results and discussion When the amplitude of the ramped voltage is increased to 4 V, the mica exhibits stable bipolar RRAM windows after 10 swept cycles (Figure 1f), indicating the easier formation of CF in mica under a higher electrical field. On the contrary to the traditional RRAM with either unipolar or bipolar windows, the mica-based device uniquely combines unipolar and bipolar performance, the transformation between which can be simply tuned by controlling intensity of the applied electric field. The accumulation and migration of K + driven by an electric field are proposed to reveal the 7 conductive behaviors of mica. To verify the electrodialysis ability of K + in mica, a device based on aqueous potassium iodide (KI)/mica/Au with free K + was fabricated ( Figure 2a). In detail, a layer of photoresist (PR) was spin-coated and dried to cover the whole substrate. Then a hole was made in the PR layer by laser ablation that serves as a window to allow the external K + offered by KI/DI water solution to move through the mica (see details in methods). The I-V curves of the device with one probe attached to the bottom Au electrode (ground) and another immersed into KI solution droplet, were measured in air. As shown in Figure 2b, the KI/mica/Au device is initially at a high-resistance state (>1 GΩ), and transformed into a low-resistance state (<200 kΩ) after the application of positive voltage with a maximum value of 20 V. The reason is that the K + in KI solution would gather to the top of mica and then gradually migrate through the Al2(AlSi3O10)(OH)2layer in mica, which causes the conduction of mica. To further confirm the migration mechanism of K + , a pulse voltage (±2.5 V) was used to stimulate the KI/mica/Au device after the formation of CF. A clear current vibration at the voltage of 1 V is exhibited in 150 pulse cycles ( Figure 2c). The conductivity of the KI/mica/Au device is decreased under the negative pulse voltage, since the K + is forced to move back to the top layer of mica. On the contrary, the device becomes more conductive under the positive voltage because of the continuous accumulation of K + . The results reveal that the external K + can transport and migrate in the out-of-plane direction of mica under the electrical field. In addition to the external K + , the accumulation and rearrangement of the internal ones in mica also contribute to the RS behaviors. To confirm the migration of internal K + in mica, a field effect transistor (FET) with mica as the dielectric, MoS2 as the channel, and graphite as the electrodes, was fabricated. The migration of K + in mica plays a role in gating effect due to their positive charge ( Figure S1). As a result, the MoS2 FET shows synaptic responses at the gate pulse voltage, which indicates the 8 accumulation and rearrangement of the internal K + in 2D mica ( Figure S2). Moreover, another device based on a vertical graphite/mica/graphite structure ( Figure S3a) was fabricated to rule out the possibility that the formation of CF from the electrodes. Figure S3b displays a similar RRAM window of the graphite/mica/graphite device. It is known that the carbon atoms in graphite are immovable under the electrical field because of the strong covalent bonds. From all the above results, we conclude that the vertical conduction of mica is ascribed to the movement of its internal ions. Next, we analysis the conduction mechanism of the device by examing the concentrations of K + at different stages. We use time-of-flight secondary-ion mass spectrometry (TOF-SIMS), a common technique for depth profiling of materials 36 , to explore the migration of K + in mica. Here, we study the graphite/mica/graphite structure (Figure 2d) with TOF-SIMS. Figure in the RRAM window. Noteworthily, the accumulated K + are regressed to the initial position immediately once the electrical field is changed to negative, which turns the mica to the insulation state. Consequently, the device shows unipolar RRAM windows under the ramped voltage sweep from -2 V to 2 V. As a contrast, in the case of higher electrical field (cyclic triangular wave voltage sweep with the amplitude of 5V), the mica-based RRAM shows bipolar behavior, the mechanism of which is schemed in Figure 3d. The higher negative voltage turns over the arrangement of K + compared with the positive electrical filed. The accumulation of K + connects the two electrodes as well, contributing to the conduction of mica. As a result, the mica-based RRAM exhibits low resistance state in 0 V to 5 V sweep. The accumulation of K + would gradually disperse under the positive electrical field, which provides enough energy to reinstate mica to its original insulation state. That makes the device exhibit a clockwise RRAM window in the positive voltage. From 0 V to -5 V sweep, the K + slightly reaccumulate under higher input electrical field. Further application of negative voltage produces higher current in mica than before with the increasing accumulation of K + , leading to the RRAM window in negative voltage. In brief, the K + in mica migrate through the Al2(AlSi3O10)(OH)2layer under electrical field, forming the CF in the mica-based RRAM. The different accumulation directions of K + under positive or negative electrical field contribute to unipolar or bipolar RRAM windows in the mica-based RRAM. Besides RRAM windows, repeatability and on/off ratio are two important criteria to evaluate the performance of RRAM. As indicated in Figure S4a and b, both the unipolar and bipolar I-V curves 11 keep stable during ten cycles of triangular wave voltage, illustrating the repeatability of the RRAM behaviors. In addition, a high on/off ratio of 10 3 is obtained, which is sufficient to realize the functions of a RRAM (> 10). 37 Apart from 3-nm-thick mica, similar phenomena ( Figure S5a and b) were also observed for the devices based on mica with the thickness of 4.1, 6.3 and 7.9 nm. We find that the thicker the mica is, the higher the sweep voltage is needed to form the CF. To avoid electrical performance variations among samples, more than three mica flakes in each thickness were measured and they exhiti reproducible behaviors. Figure According to the current differences, the patterns of "T", "B", "S", and "I" are achieved in the array ( Figure 3g). These results show that the mica-based RRAM displays high stability, repeatability, reproducibility and multi-state storage. 16 envison that the scalable production of 2D mica and film assemblywould facilitate fabrication of largescale device arrays for applications. Experimental Section Device Fabrication: For the mica-based RRAM, a Cr/Au (5 nm/50 nm) film was first deposited on a 300 nm-thick-SiO2/Si substrate by an e-beam evaporation system (TSV-1500, Tianxingda Vacuum Coating Equipment Co., Ltd., China). Subsequently, mica flakes were mechanically exfoliated using scotch tape from the commercial bulk material (HQ graphene, Nederland) and directly transferred onto the Cr/Au-coated substrate. Fabrication of probe/KI solution/mica/Au devices. First, PR (AZ 5214) was spin-coated (2000 rpm) to cover the whole substrate, followed by baking the sample on a hot plate at 125 ℃ for 1 min. Second, the shape of the bottom Au electrode was defined by a laser writer system (Dall, Aresis), followed by Au deposition using an e-beam evaporation system. Third, the mica was mechanically exfoliated by using scotch tape and transferred onto the bottom Au electrode with the assistance of PDMS using a homemade alignment station. Forth, another layer of PR was spin-coated and baked in the substrate. Fifth, laser writing process was performed to ablate a hole in the PR layer as a window to expose the mica flake to air. For device testing, a droplet of KI/DI water solution was dipped into the window. 6 Figure 1 . 61Device structure and electrical characteristics of 2D-mica-based RRAM. Schematic of the distribution of K + in the device (a) before and (b) after the application of a positive electric field. (c) Optical microscopy image of 2D mica on Au/SiO2/Si substrate. (d) AFM image of the thin-layer mica in (c). Inset is the height analysis of the mica, showing its thickness of ~3 nm. (e) The 1 st , 10 th , and 30 th RS characteristics of the micabased device at a cyclic triangular wave voltage with the amplitude of 2V. (f) The 1 st , 5 th , and 10 th RS characteristics of the mica-based device at a cyclic triangular wave with the amplitude of 4V. 9 Figure 2 . 922e shows the profiles of C, Si, K, and Al elements as a function of the depth in the device. The two separated regions of C signal are corresponding to the top and bottom graphite, respectively, between which the signals of Si, K, and Al belong to the mica layer sandwiched by two graphite electrodes. Before the application of voltage, the intensity of K keeps nearly constant, suggesting the homogeneous distribution of K + in mica. After the application of a positive vertical voltage (Figure 2f), the increasing gradient of K profile reflects the movement of K + from the top of mica toward the bottom, as schemed in Figure 2a. Therefore, the TOF-SIMS analyses reveal that the conduction of mica in out-of-plane direction relies on the migration of K + under electrical field. K + transport along the vertical direction in 2D mica. (a) Schematic of the probe/KI solution/mica/Au device for the measurements of K + transport. RS characteristics (b) of the device in (a). (c) The current in a read voltage of 1 V of the device in (a) after the application of voltage pulses (±2.5 V). (d) Schematic of the graphite/mica/graphite device. The TOF-SIMS depth profiling of C, Si, K and Al elements in graphite/mica/graphite (e) without and (f) after the application of a positive voltage. The unique RRAM behaviors in 2D-mica-based devices can be understood by the migration of K + under electrical field. Figure 3a exhibits the counterclockwise unipolar RRAM window of the device aroused by a cyclic triangular wave voltage with an amplitude of 2 V. Interestingly, the unipolar window is transformed to a clockwise bipolar one when the amplitude of the periodic voltage is increased to 5 V (Figure 3b). The formation of two distinguished RRAM windows is mainly attributed to the scanning of voltage at different ranges. Figure 3c reveals the dynamics of K + after the device was stimulated by a triangular wave voltage with the amplitude of 2 V. The K + from the top layer of mica go through the Al2(AlSi3O10)(OH)2layer under the 0 V to 2 V voltage sweep. The local accumulation of K + enables the conduction of mica, which causes the increasing current in the vertical direction of mica. The subsequent positive electric field (2 V to 0 V sweep) continuously motivates the gathering of the positive ions. Thus, the mica is more conductive at this stage than the former, resulting 3e exhibits the storage time of the mica-based RRAM after set and reset process by pulse voltage. The obvious resistance difference is observed after measuring the device for 800 s, with a current on/off ratio around 10 2 , indicating that the migration of K + can retain the accumulated state persistently. Taking advantage of the high on/off ratio, three memory states are demonstrated by use of the pulse voltage on the mica-based RRAM. Figure 3f shows the current of the device at a read voltage of 1 V after each pulse. The device shows distinct current states in 200 cycles. The low resistance (<0.1 GΩ), high resistance (0.1-1 GΩ) and insulating state (>1 GΩ) were achieved through the application of +10 V pulse, +5 V pulse and -10 V pulse, respectively. Furthermore, a device arry with 8 × 8 positions in the 2D mica flake are selected (Figure S6a). The current of all points in the array are below 10 pA under the read voltage of 1 V, indicating the high uniformity in resistance at the initial state (Firuge S6b). In comparison, the current increases to the compliance value (10 nA) after the application of a 10 V voltage to set some positions of the array. Figure 3 . 3Electrical performance of the 2D-mica-based RRAM. (a) Unipolar RRAM behavior of the device at the stimulation of periodic triangular wave voltage with the amplitude of -2 to 2 V. (b) Bipolar RRAM behavior of the device at the stimulation of periodic triangular wave voltage with the amplitude of -5 to 5 V. Schematic of the accumulation of K + in mica under the periodic triangular wave voltage sweep with the amplitude of (c) 2 V and (d) 5 V. (e) The current evolution of the set and reset states in 13 the device after 800s. (f) The current distribution of more than 200 switching cycles of three memory states. The current in (e) and (f) were read at a voltage of 1 V. (g) Pattern of a device array with 8 × 8 positions in 2D mica measured under 1 V, in which the composed areas of "T", "B", "S", and "I" patterns are pre-set to be on with a pulse voltage of 10 V. Interestingly, the state maintenance of K + in mica enables the related device to exhibit the property of a non-Markov chain, which can be considered for the relevant applications based on the non-Markov process. The typical non-Markov process can be described asprobability of the states, −1 , −2 , …, 0 are the past states, is the current state, and +1 represents the next state of the non-Markov chain. Figure 4a shows the transfer matrix of a typical non-Markov chain, in which the transfer probability in each state is a function of both current and past states. In our design, three non-Markov states are defined, the insulating (State Ⅰ), high resistance (HR) (State Ⅱ) and low resistance (LR) (State Ⅲ), according to the resistance of the micabased RRAM. The cartoons in Figure 4b exhibit parts of the transfer paths of the three states in the device. The positive and negative voltages are applied to imitate different transformations of states in the non-Markov chain. Two steps of input voltages are applied to inspire the device, in which the first acts as the past transition path while the second serves as the evolution from current state to the next state, for example, Path 1 (State Ⅰ→State Ⅱa→State IIIa) and Path 2 (State Ⅰ→State Ⅱb→State Ⅰ), which are schemed inFigure 4b. The State Ⅱ is considered as the current state for both paths. Specifically, even at the same state, K + in mica have different memory effects due to their different accumulation pathways induced by the past states. Thus, for the second step, the same positive voltage can switch14 the device from State Ⅱa to State IIIa or State Ⅱb to State Ⅰ. This feature suggests that the next state of the system not only relates to the current state, but also to the transport trail of the past states, which is defined as a typical non-Markov process. The experimental results inFigure 4cshow the changing process of the resistance of the two examples.Table 1summarizes all the possible non-Markov paths that can be achieved by the mica-based RRAM. Altogether, by using mica as the RRAM active layer, the non-Markov chain can be realized in a single device. Figure 4 . 4Realization of a non-Markov chain in a single RRAM. Schematics of (a) a typical non-Markov process and (b) a non-Markov chain achieved by a mica-based device. The three resistance states of the device are defined as State I, State II and State III, which correspond to insulating, HR and LR states, respectively. The applied positive and negative pulse voltages are indicated by solid and15 dotted arrows, respectively. The first and second step of the state transformation are represented by green and orange arrows, respectively. Examples of the non-Markov process by following the paths of (c) State Ⅰ→State Ⅱa→State IIIa and State Ⅰ→State Ⅱb→State Ⅰ.Table 1. Realization of all possible non-Markov paths in a single mica-based RRAMConclusionWe have achieved a non-Markov chain in a single 2D crystal RRAM device. The conductive mechanism of the device is related to the vertical migration of internal K + in the 2D mica under an electric field, which has been verified by both electrical transport characterization and depth profile with TOF-SIMS analysis. The accumulated K + show distinguished behaviors under electrical fields in opposite directions, resulting in the combination of unipolar and bipolar features of the 2D mica RRAM. Simultaneously, the device exhibits a high on/off ratio of 10 3 , good repeatability, and decent stability. The realization of a non-Markov chain in a single device opens a new horizon in highlyefficient and integrated systems toward future in-memory computing and intelligent devices. WeComparison of two non- Fabrication of graphite/mica/graphite and mica/MoS2/graphite devices. First, the graphite (HQ graphene), muscovite mica (HQ graphene, Nederland), fluorophlogopite mica (Tiancheng Fluor phlogopite Mica Co., Ltd., China) and MoS2 (HQ graphene, Nederland) flakes were exfoliated by using scotch tape from bulk crystal onto PDMS and identified by an optical microscope. Second, the target 2D flake was staked on a SiO2 (300 nm)/Si substrate in a homemade alignment station with the 17 assistance of PDMS. Third, PR (AZ 5214) was spin-coated (2000 rpm) to cover the whole substrate and baked on a hot plate (125 ℃, 1 min). Forth, the electrodes were fabricated by the laser writer system and e-beam evaporation system. Electrical Measurements: AFM characterization and CAFM measurements were performed by AFM (Cypher ES, Oxford Instruments, USA). For all the CAFM measurements, a fixed compliant current of 10 nA was applied. 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{'abstract': 'The non-Markov processes widely exist in thermodymanic processes, while it usually requires packing of many transistors and memories with great system complexity in traditional device architecture to minic such functions. Two-dimensional (2D) material-based resistive random access memory (RRAM) devices show potential for next-generation computing systems with much-reduced complexity. Here, we achieve the non-Markov chain in an individual RRAM device based on 2D mica with a vertical metal/mica/metal structure. We find that the internal potassium ions (K + ) in 2D mica gradually move along the direction of the applied electric field, making the initially insulating mica conductive. The accumulation of K + is tuned by electrical field, and the 2D-mica RRAM possesses both unipolar and bipolar memory windows, high on/off ratio, decent stability and repeatability. 2 Importantly, the non-Markov chain algorithm is established for the first time in a single RRAM, in which the movement of K + is dependent on the stimulated voltage as well as their past states. This work not only uncovers the inner ionic conductivity of 2D mica, but also opens the door for such novel RRAM devices with numerous functions and applications.', 'arxivid': '2108.13244', 'author': ['Rongjie Zhang \nShenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China\n', 'Wenjun Chen \nShenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China\n', 'Changjiu Teng \nShenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China\n', 'Wugang Liao \nInstitute of Microscale Optoelectronics\nShenzhen University\n518060ShenzhenP. R. China\n', 'Bilu Liu [email protected] \nShenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China\n', 'Hui-Ming Cheng [email protected] \nShenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China\n\nInstitute of Metal Research\nShenyang National Laboratory for Materials Science\nChinese Academy of Sciences\n110016ShenyangP. R. China\n'], 'authoraffiliation': ['Shenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China', 'Shenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China', 'Shenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China', 'Institute of Microscale Optoelectronics\nShenzhen University\n518060ShenzhenP. R. China', 'Shenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China', 'Shenzhen Geim Graphene Center\nTsinghua-Berkeley Shenzhen Institute and Tsinghua Shenzhen International Graduate School\nTsinghua University\n518055ShenzhenP. R. China', 'Institute of Metal Research\nShenyang National Laboratory for Materials Science\nChinese Academy of Sciences\n110016ShenyangP. R. China'], 'corpusid': 234848233, 'doi': '10.1016/j.scib.2021.04.025', 'github_urls': [], 'n_tokens_mistral': 13994, 'n_tokens_neox': 11452, 'n_words': 6244, 'pdfsha': '9948659b11e491439b54839aaac172ed492d1b5b', 'pdfurls': ['https://arxiv.org/pdf/2108.13244v1.pdf'], 'title': ['Realization of A Non-Markov Chain in A Single 2D Crystal RRAM', 'Realization of A Non-Markov Chain in A Single 2D Crystal RRAM'], 'venue': []}

arxiv

An application-oriented scheduler J.-C Sibel [email protected] Mitsubishi Electric R&D Centre Europe Rennes France N Gresset [email protected] Mitsubishi Electric R&D Centre Europe Rennes France V Corlay [email protected] Mitsubishi Electric R&D Centre Europe Rennes France An application-oriented scheduler Index Terms-Schedulingapplication-oriented systemscross- layer We consider a multi-agent system where agents compete for the access to the radio resource. By combining some application-level parameters, such as the resilience, with a knowledge of the radio environment, we propose a new way of modeling the scheduling problem as an optimization problem. We design accordingly a low-complexity solver. The performance are compared with state-of-the-art schedulers via simulations. The numerical results show that this application-oriented scheduler performs better than standard schedulers. As a result, it offers more space for the selection of the application-level parameters to reach any arbitrary performance. I. INTRODUCTION In modern wireless communications, latency and reliability are as important as throughput. Indeed, new applications, such as the industrial internet of things, involve new use cases with very high latency and reliability requirements. For instance, [1] describes latency requirements of 0.5 ms and reliability of 99.999999% for motion control. Similar figures are provided for mobile automation. These requirements should be met in the context of systems with many agents sharing the same communication resources. Moreover, for agents with different missions, their application requirements and the quality of their channel may vary. In this scope, we believe that all available degrees of freedom should be used in the design of the communication system. More specifically, the application requirements should be taken into account in the optimization to maximize the performance. Having this paradigm in mind, we shall focus on the critical MAC layer in this paper, namely the scheduling. Of course, there exist many studies in the literature that propose design guidelines with respect to these new requirements. A relevant emerging field is the Age of Information (AoI). AoI measures the time elapsed between the generation of a message u(t) and its delivery time t, i.e., the AoI is t − u(t). This metric enables to assess existing queuing and scheduling strategies and can serve as a design guideline for new algorithms. AoI is a finer metric than the quality of service (QoS), used for resource allocation, e.g. in the LTE [2], as the latter is transmission centric while the former focuses on the quality of the information effectively received 1 . AoI is also different from resource allocation mechanisms considering communication metrics, such as the data rate or the channel capacity, for their scheduling decision [3]. However, the standard AoI metric does not take into account application requirements. It is possible to weight the AoI of each agent (in the case where a sum AoI is optimized) but this does not enable to directly optimize with respect to these requirements. Moreover, according to many definitions a realtime constraint means meeting a deadline. This is slightly different than the metric optimized with AoI. Related work -In [4], the average time status update with the queue algorithm first-come-first-served is investigated. In [5], the same authors study AoI in the context of a vehicular network. It is shown that the source rate should not be too high to maximize the AoI metric. Moreover, several scheduling algorithms have been proposed to optimize AoI: the AoI in the case of multiple agents, each having multiple sources, and sharing a common channel is considered in [6]. The authors prove that the problem is NP-hard and proposed suboptimal algorithms. In [7], the case of a base-station delivering information to several agents is studied. The context is similar to the one in this paper, but the optimization is done with respect to a different metric: they investigate a transmission scheduling policy that minimizes the expected weighted sum of the AoI of each agent. Main contributions -In this work, we define a new multi-agent scheduling problem. It takes several application parameters into account, including the resilience, i.e., how often an application needs to receive fresh information to work properly. This enables to assess the quality of a scheduler with respect to the application failure probability. As a result, similarly to what is done with semantic communications [8], the communication system is optimized directly with respect to its final real-time requirements. We show with numerical evaluations a strong confidence in the proposed scheduler design as the performance gain is significant compared with state-of-art schedulers. II. CONTEXT A. Description of the system We consider a discrete-time system, divided in time slots whose length is denoted by dt, e.g., dt = 1 msec. Let N be the number of agents in the system. Any k th agent A k is represented by a data stream characterized by three parameters: • The period T : the time duration between two consecutive packet arrivals from the application in the agent buffer for transmission. • The packet lifetime D (with D ≤ T ): the time duration for which the packet is alive. • The resilience R: the maximum time duration for which the agent can survive without a successful transmission of a packet. We define the event resilience violation for any agent (E1) as "no packet successfully transmitted during the last R time slots". For an agent A k at (discrete) time t, we accordingly introduce r k (t) as the remaining time before resilience violation. This means that 0 ≤ r k (t) ≤ R and r k (t) is a decreasing function of t. The complementary event of (E1), called success (E0), is: "successful transmission of a packet at time t". With both the events (E0) and (E1) for any agent A k at time t, r k (t) is immediately set to its maximum value R. We define a resilience window as a time window that starts just after one event (E0) or one event (E1), i.e., when r k (t) = R, and that ends when A k meets either the next (E0) or the next (E1). Fig. 2 shows an arbitrary example of the time evolution of r k (t) and the associated resilience windows. Additionally, we define a transmission opportunity for an agent A k as a time slot for which a packet is alive. Accordingly, we define the quantity q k (t) as the remaining number of transmission opportunities before meeting (E1). In case D = T , i.e., if there is no "hole" in the time-line, then q k (t) = r k (t). In case D < T , then q k (t) ≤ r k (t). B. Description of the environment We consider a single available radio resource per time slot. We assume that the resource is suited to the transmission of one packet by any agent. Therefore, at any time, all the agents with an alive packet compete for the access to the single radio resource but only one agent finally obtain the resource. We denote by δ k (t) = 1 the event "A k is allocated at time t" and by δ k (t) = 0 the event "A k is not allocated at time t". When allocated the resource, A k observes an unsuccessful transmission of the packet with probability p k being the channel error probability. Thus, an agent A k meets (E1) either if it is never allocated the resource or if the channel strongly impacts the transmission each time the agent is allocated. C. Example Consider a mobile robot with a trajectory monitoring application: T is the time duration between two consecutive position measurements of the agent, D is the time duration for which the measured position remains relevant, and R represents the capacity to interpolate/extrapolate the trajectory without consecutive measurements. We consider that the agent's mission is to reach a geographical point. Assume that due to numerous clutters in its surrounding environment, some packets are dropped by the channel resulting in event (E1) for the agent. The trajectory cannot be interpolated/extrapolated with a sufficient accuracy to ensure the success of the mission. Therefore, the agent stops its motion to wait for a full restart which generates, among others, a significant delay. Such an event is then detrimental to the system performance regarding the application purpose. In our study, for simplification purpose, the restart delay is not considered. In other words, the event (E1) does not stop the agent. III. OPTIMIZATION PROBLEM In this section, we first formalize the scheduling problem as an optimization problem. Then, we introduce some heuristics to allow for a practical solution. A. Presentation of the problem The problem we propose to solve is the opportunistic centralized scheduling problem: "which agent, at a given time slot, should get the radio ressource given the knowledge of the system and the environment?". For any agent A k we define V k (t) as the accumulated sum of resilience violations until time t. V k (t) is updated based on the events (E0) and (E1) as follows: Meet (E0): V k (t) ← V k (t − dt),(1)Meet (E1): V k (t) ← V k (t − dt) + 1.(2) If none of (E0) or (E1) occurs at time t, then V k (t) is naturally extended as V k (t) ← V k (t − dt) . From these quantities, we define the local long-term experimented probability of resilience violation at any time t as: F k (t) def = V k (t) t .(3) From a system perspective, we define the optimization problem as the search for the scheduling decision δ * (t) = {δ * 1 (t), ..., δ * N (t)} that minimizes the average probability of resilience violation F (t) embodied by the sum of F k 's s.t.: F (t) def = k V k (t) t .(4) B. Proposed alternative objective As a first step to design an efficient scheduler, we build a heuristic J t, δ(t) whose aim is to predict F (t) by predicting actually: S(t) def = k V k (t)(5) under a given scheduling decision δ(t). The rationale behind considering S(t) instead of F (t) is that t is common to all the agents therefore t only scales the problem, i.e., there is no need to insert it within the optimization problem. The solver in the following section then consists in choosing the allocation δ(t) such that the heuristic J t, δ(t) is minimized. To establish the said heuristic, first, we build a functionV t, δ k (t) to estimate V k (t) for any agent A k . Then, the heuristic J t, δ(t) is obtained by summing these local estimates. 1) Local predict function: As V k (t) is an observation metric, we define an associated local long-term predicted accumulated sum of resilience violationsV k t, δ k (t) : V k t, δ k (t) def = V k (t − dt) +f k t, δ k (t) ,(6) where V k (t − dt) is the observed value from the previous time instant andf k t, δ k (t) is the local short-term predicted probability of resilience violation within the current resilience window for time t given an allocation decision δ k (t). The functionf k (·, ·) acts as a correction term especially when t is small. For example, when all V k 's are zero at the beginning, they are not well-representing the near future. We constructf k (·, ·) to fairly represent the current application status as well as the radio conditions: •f k (·, ·) increases as p k increases: a worse radio conditions makes greater the probability of resilience violation •f k (·, ·) increases as r k (t) decreases: getting closer to the resilience violation makes greater the probability of resilience violation We propose the heuristic f r (·, ·) that fulfills these requirements f r (t, k) def = p r k (t) k .(7) In the case of holes in the time-line, i.e., when D < T , two agents A k1 , A k2 with the same values r k1 (t) = r k2 (t) can observe two different time-lines q k1 (t) < q k2 (t). We propose the following enhancement of the heuristic to distinguish between A k1 and A k2 : f q (t, k) def = p q k (t) k .(8) Now, we build the functionf k (·, ·) to integrate any of these heuristics. The function should distinguish the case the allocation is not granted from the case the allocation is granted. In the latter case, indeed,f k (·, ·) must depend on p k because a channel transmission is assumed. As a result, we propose the following definition: f k t, δ k (t) = f (t, k) if δ k (t) = 0, p k f (t, k) if δ k (t) = 1,(9) where f could be either f r or f q . This can be simplified as: f k t, δ k (t) = 1 − δ k (t)(1 − p k ) f (t, k).(10) Now thatV k is fully constructed, the local heuristic is nearly completed. We use the utility-based formalism to define the local predict function: j k t, δ k (t) def = U α V k t, δ k (t) ,(11) where U α (·) is a non-linear monotonic function identical for all the agents. In this paper, we consider the α-fair utility function [9]: U α (x) def = x 1−α 1−α α = 1, log x α = 1 , U α (x) = x −α .(12) The α-fair utility framework allows us for considering a family of schedulers with a good performance/fairness tradeoff [10]. The rationale behind the introduction of U α is to stick with such a well-known family of schedulers. Several values of α are considered in this paper to observe if the fairness concern indeed exerts any influence on the scheduling performance. 2) Global predict function: We define the global predict function of S(t) as the sum of the local predict functions: J t, δ(t) def = k j k t, δ k (t) .(13) The goal of the solver is then to find the scheduling decision δ * (t) that minimizes J: δ * (t) = arg min δ(t) J t, δ(t) .(14) IV. SOLVER In this section, we present two manners for solving (14): the exact solver and the approximated solver. These two solvers are important regarding the computational complexity because a great number of agents leads the exact solver to overload any computation resource. A. Exact solver for J: On-line sum Let us first compute j k (t, 1) and j k (t, 0) for any agent A k having an alive packet at time t. This corresponds to the events "A k is allocated" and "A k is not allocated", respectively. If t is a time slot out of the lifetime packet of A k , i.e., the packet of A k is dead, A k is not considered for the scheduling decision. j k (t, 1) and j k (t, 0) are set to infinite values to explicitly exclude A k . This provides us Table I. Table I, we sum all the rows to obtain a list of N cost values. Thirdly, we extract the column index k * whose cost value is lower than any other cost value. This leads to consider agent A k * as the agent to allocate. This solver requires at least N times the computations of j k (t, 1), j k (t, 0), then N sums of N terms each (so N 2 operations at least), then a comparison between N values. Consequently, it might cause computational issues when N increases. k * 0 1 . . . N − 1 j 1 t, δ 1 (t) j 1 (t, 1) j 1 (t, 0) . . . j 1 (t, 0) j 2 t, δ 2 (t) j 2 (t, 0) j 2 (t, 1) . . . j 2 (t, 0) . . . . . . . . . . . . . . . j N t, δ N (t) j N (t, 0) j N (t, 0) . . . j N (t, 1) B. Approximated solver for J: On-line Taylor For great values of N , let us use the Taylor expansion of U α (·) around the predictionV k t, δ k (t) considering the experimented V k (t − dt): U α V k t, δ k (t) ≈ U α V k (t − dt) (15) + V k t, δ k (t) − V k (t − dt) × U α V k (t − dt) . We only keep the terms that depend on the current scheduling decision at t. In addition, as only one agent is provided the resource at a time, this leads (14) to become: k * = arg min k kV k t, δ k (t) U α V k (t − dt) .(16) First, we replace the differential from (12), then we replace the prediction from (6) and finally we only keep what depends on the scheduling decision δ(t). We get: k * = arg min k kf k t, δ k (t) V k (t − dt) −α .(17) Using (10) and keeping again only the terms that depend on the current scheduling decision δ(t), the previous equation becomes: k * = arg max k k δ k (t)(1 − p k )f (t, k)V k (t − dt) −α . (18) One agent A k * is allocated at a time t, therefore, δ k * (t) = 1 whereas δ k =k * (t) = 0. Accordingly, the previous equations amounts to: k * = arg max k {M k (t)} k ,(19)M k (t) def = (1 − p k )f (t, k)V k (t − dt) −α . The on-line Taylor solver requires a comparison between N metrics M 1 (t), . . . , M N (t), each one requiring few computations, which is dramatically less than what requires the on-line sum. For a large quantity of agents, therefore, provided that the Taylor expansion holds, i.e., |V k t, δ k (t) −V k (t−dt)| < with 1, it is highly recommended to consider this solver. V. NUMERICAL OBSERVATIONS This section presents the evaluation results of the proposed solution in comparison with other schedulers from the stateof-the-art. A. Challengers We perform the On-Line Sum (OLS) scheduler from IV-A and the On-Line Taylor (OLT) scheduler from IV-B considering either f = f r (the schedulers are then called OLS-R and OLT-R, respectively) or f = f q (the schedulers are then called OLS-Q and OLT-Q, respectively) when D = T and when D < T . We set the α parameter of the utility function U α to α ∈ {−5, −2, −1, 0}, see [10] for details on the fairness impact. We confront OLS and OLT to the Round-Robin scheduler [11] used for network scheduling. It consists in allocating the radio resource to the agents one by one following a buffer. The said buffer is a random permutation of [0, . . . , N − 1]. In case an agent A k does not have an alive packet at time t, i.e., the packet is dead, the scheduler scans the next buffer indexes to extract the first agent with an alive packet. In addition, when the Round-Robin has finished a round in its buffer, i.e., after N allocation steps, the Round-Robin replaces its buffer with a new random permutation of [0, . . . , N −1]. This randomization prevents an agent from being always out at each period of the Round-Robin. We also confront the on-line scheduler to a proportional-fair like scheduler (PF-like) whose allocation rule is based on the channel capacity: k * = arg max k log 2 1 + γ k (t) t−dt t =0 ω k (t ) log 2 1 + γ k (t ) . (20) with γ k (t) the instantaneous signal-to-noise ratio at time t (dual value of p k ) and ω k (t ) = 1 if A k had an alive packet at time t and ω k (t ) = 0 otherwise. The denominator comprises the accumulated quantity of resources allocated in the past to the agent in terms of the channel capacity and the numerator indicates the instantaneous capacity the agent can reach at time t. Therefore, with two agents with the same past, the scheduler allocates the agent with the greatest capacity. With two agents with the same instantaneous capacity, the scheduler allocates the agent whose accumulated capacity is the lowest. Therefore, PF-like balances between good performance and fairness. B. Environment We perform 1000 iterations of 10000 time slots duration each for each of the challengers. At each iteration, each probability p k is drawn randomly around a mean valuep k . This actually creates a non-static environment for the agents. The mean values are linearly selected in [10 −3 , . . . , 10 −1 ] such that each agent A k is providedp k =p k =k . We consider a drop of N = 100 agents with a packet period T = 100, with two lifetimes D = 90, 100 to observe the behavior difference by selecting f = f q and f = f r . We consider that the time-line of an agent does not necessarily starts at the same time slot as another agent. Therefore, at each iteration, the time start of each agent is randomly drawn between zero and T . C. Performance metric We consider F (t) defined in (4) as the performance metric. First, let us observe the time evolution of S(t) from (5) for some challengers, see Fig. 3. The curves reach a linear steady state after around 5000 time slots whatever the scheduler. F (t) being the slope of S(t), we conclude that F (t) reaches a constant value F cst for a sufficiently large amount of time t. Practically speaking, we consider the performance metric to be F cst computed as the slope of the curve of S(t) over the last 5000 time slots of the simulation. To ensure an even more reliable F cst value, we average over the previously mentioned 1000 iterations for each scheduler, for each value of R. D. Results We show the results for D = 100 in Fig. 4 and for D = 90 in Fig. 5. In Fig. 4, as the time-lines of the agents are full, OLS-Q behaves exactly as OLS-R and OLT-Q behaves exactly as OLT-R therefore we only display OLS-R and OLT-R. First of all, the performance of the Taylor approximation OLT-R are exhibited in comparison with the performance of the exact solver OLS-R in Fig. 4 for α = 0. We observe that both are very similar, e.g., for R = 105, OLS-R and OLT-R both perform nearly with F cst = 1.2 · 10 −3 . This confirms that the Taylor approximation is relevant enough to continue only with OLT-R for the other values of α and for D = 100, 90. Secondly, we observe that the various values of α for OLT does not bring much different performance. The curves are all very close one to each other therefore we conclude that the value selection for α is sufficiently free, the fairness does not exert a strong constraint on the scheduling performance. Thirdly, on both figures, we observe that Round-Robin and PF-like similarly perform with poor results, e.g., F cst does not go below 10 −1 even with great values of R. As a matter of fact, Round-Robin does not perform well because it does not take into account the time-line of the agents and it does not take into account the channel probabilities. In addition, the allocation rule of PF-like does not integrate any form of prediction from the time-line, it only focuses on a past situation of the agents to provide a scheduling decision. Consequently, it integrates somehow the channel error probabilities but it does not take into account the time-lines too which explains the associated bad performance. Fourthly, we observe a significant gap between the values of F cst for OLT (R or Q) and PF-like or Round-Robin whatever D. This said gap can be exploited in the two following ways: • We search for the best performance given an application parameter. • We search for the least application constraint that is able to reach a given performance. Regarding the first way, for the same application constraint, say R = 100 and D = 100, OLT-R exhibits F cst = 3 · 10 −2 whereas PF-like exhibits F cst = 3 · 10 −1 , i.e., OLT-R offers a ten times better performance result than PF-like. Regarding the second way, to reach the same F cst performance, e.g., F cst = 2 · 10 −1 , OLT-R needs a resilience value of R = 90 whereas PF-like requires R = 105, i.e., PF-like enlarges by 17% the constraint on the application parameters. Furthermore, we observe that PF-like -and Round-Robin too -cannot satisfy a performance of 10 −1 or lower whatever the application parameter R. Increasing R to even greater values than 115, the maximum value we considered in the simulations, may indeed not lead to significant better performance for PF-like and Round-Robin. This is not the case of OLT-(R or Q) as the slope of F cst still increasing (in absolute values) when values of R approach R = 115. OLT-(R or Q) reaches a F cst value more than a thousand times less than PF-like at this extreme R value. In other words, playing with R can bring significant benefits for OLT-(R or Q), contrary to PFlike and Round-Robin. This means that the proposed scheduler better integrates the application requirements than the other schedulers. Comparing now OLT-R with OLT-Q when D = 90, i.e., when the time-lines get some holes, we observe a performance gap when R ≥ 105. More precisely, the slope of F cst for OLT-Q changes with an increase in R whereas the slope of F cst for OLT-R remains constant. From the application point of view, increasing R means having more application computation capacity, e.g., some extrapolation algorithms in the case of the mobile robot, see II-C. This is a constraint regarding the cost of a deployment, therefore, we think it is more beneficial to lower R as much as possible. However, considering the holes in the time-lines requires a slightly more complex scheduler because the computation of q k (t) is not as easy as the computation of r k (t). As a matter of fact, we don't need to take into account the time-line to compute r k (t), only the resilience windows are enough. When computing q k (t), though, there is a need to couple the knowledge of the resilience windows with the knowledge of the time-lines. In case of some jitter in the application traffic, one needs strong robustness to obtain the exact values of q k (t). Consequently, selecting either OLT-R or OLT-Q is a question of computation capacity at the application level. VI. CONCLUSION We designed an application-oriented multi-agent system to better formalize the problem of competition for the access to the radio resource. We jointly considered applicationlevel parameters, characterized mainly by the resilience, and radio environment variables embodied by the channel error probability. We then proposed an on-line scheduler to solve the optimization problem either exactly or approximately with a low-complexity approach. We also introduced a novel way of observing the schedulers' behavior by focusing on an application-level metric instead of focusing on other usual radio-level metrics. From the performance evaluations, we observed that the proposed on-line scheduler design significantly outperform the other schedulers whatever the value of the resilience. Moreover, we highlighted that the proposed on-line scheduler allows for more degrees of freedom in the selection of the application parameters to reach an arbitrary performance. To conclude, the design of an application-oriented scheduler proved to be a promising method to efficiently integrate application requirements as well as radio parameters. Fig. 1 1shows a representation, called time-line, of T and D. Fig. 1 . 1Example of a time-line with T = 4 and D = 3 (in number of dt's). Fig. 2 . 2Example of a time evolution of r k (t) with R = 10 (top) and the associated resilience windows (bottom). At time instants t = 7, 11, 31, 33, A k meets (E0) whereas at time instants t = 22, 44, A k meets (E1). Fig. 3 . 3Accumulated sum of violations S(t) over the time. Fig. 4 . 4Steady state Fcst of F (t) with D = 100. 5 Fig. 5 . 55OLT-R, α = 0 OLT-Q, α = 0 OLT-R, α = −1 OLT-Q, α = −1 OLT-R, α = −2 OLT-Q, α = −2 OLT-R, α = −5 OLT-Q, α = −Steady state Fcst of F (t) with D = 90. TABLE I TABLE IOF THE ON-LINE SUM SOLVER Secondly, for each scheduler decision, i.e., for each column in S Baek, D Kim, M Tesanovic, A , 3gpp new radio release 16: Evolution of 5g for industrial internet of things. 59S. Baek, D. Kim, M. Tesanovic, and A. Agiwal, "3gpp new radio release 16: Evolution of 5g for industrial internet of things," IEEE Communications Magazine, vol. 59, no. 1, pp. 41-47, 2021. Quality of service based resource determination and allocation apparatus and procedure in high speed packet access evolution and long term evolution systems. S E T Chandra, Jin Wang, Patent. S. E. T. Arty Chandra, Jin Wang, "Quality of service based resource determination and allocation apparatus and procedure in high speed packet access evolution and long term evolution systems," Patent, 2009, wO2007092245A3. [Online]. Available: https: //patents.google.com/patent/WO2007092245A3/en Fair preemption for joint delay constrained and best effort traffic scheduling in wireless networks. N Gresset, H Bonneville, N. Gresset and H. Bonneville, "Fair preemption for joint delay con- strained and best effort traffic scheduling in wireless networks," 05 2015. Real-time status: How often should one update. S Kaul, R Yates, M Gruteser, 2012 Proceedings IEEE INFOCOM. S. Kaul, R. Yates, and M. Gruteser, "Real-time status: How often should one update?" in 2012 Proceedings IEEE INFOCOM, 2012, pp. 2731- 2735. Minimizing age of information in vehicular networks. S Kaul, M Gruteser, V Rai, J Kenney, 2011 8th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks. S. Kaul, M. Gruteser, V. Rai, and J. Kenney, "Minimizing age of information in vehicular networks," in 2011 8th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks, 2011, pp. 350-358. Optimal link scheduling for age minimization in wireless systems. Q He, D Yuan, A Ephremides, IEEE Transactions on Information Theory. 647Q. He, D. Yuan, and A. Ephremides, "Optimal link scheduling for age minimization in wireless systems," IEEE Transactions on Information Theory, vol. 64, no. 7, pp. 5381-5394, 2018. Scheduling policies for minimizing age of information in broadcast wireless networks. I Kadota, A Sinha, E Uysal-Biyikoglu, R Singh, E Modiano, IEEE/ACM Transactions on Networking. I. Kadota, A. Sinha, E. Uysal-Biyikoglu, R. Singh, and E. Modiano, "Scheduling policies for minimizing age of information in broadcast wireless networks," IEEE/ACM Transactions on Networking, vol. PP, 01 2018. Semantic Communication Systems for Speech Transmission. Z Weng, Z Qin, Z. Weng and Z. Qin, "Semantic Communication Systems for Speech Transmission," 2021. [Online]. Available: https://arxiv.org/abs/2102. 12605 An axiomatic theory of fairness in network resource allocation. T Lan, D Kao, M Chiang, A Sabharwal, 2010 Proceedings IEEE INFOCOM. T. Lan, D. Kao, M. Chiang, and A. Sabharwal, "An axiomatic theory of fairness in network resource allocation," in 2010 Proceedings IEEE INFOCOM, 2010, pp. 1-9. Throughput maximizing multiuser scheduling with adjustable fairness. S Schwarz, C Mehlfuhrer, M Rupp, 2011 IEEE International Conference on Communications (ICC). S. Schwarz, C. Mehlfuhrer, and M. Rupp, "Throughput maximizing mul- tiuser scheduling with adjustable fairness," in 2011 IEEE International Conference on Communications (ICC), 2011, pp. 1-5. R H Arpaci-Dusseau, A C Arpaci-Dusseau, Operating Systems: Three Easy Pieces. 1st ed. Arpaci-Dusseau BooksR. H. Arpaci-Dusseau and A. C. Arpaci-Dusseau, Operating Systems: Three Easy Pieces, 1st ed. Arpaci-Dusseau Books, August 2018.

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{'abstract': 'We consider a multi-agent system where agents compete for the access to the radio resource. By combining some application-level parameters, such as the resilience, with a knowledge of the radio environment, we propose a new way of modeling the scheduling problem as an optimization problem. We design accordingly a low-complexity solver. The performance are compared with state-of-the-art schedulers via simulations. The numerical results show that this application-oriented scheduler performs better than standard schedulers. As a result, it offers more space for the selection of the application-level parameters to reach any arbitrary performance.', 'arxivid': '2302.09926', 'author': ['J.-C Sibel [email protected] \nMitsubishi Electric R&D Centre Europe Rennes\nFrance\n', 'N Gresset [email protected] \nMitsubishi Electric R&D Centre Europe Rennes\nFrance\n', 'V Corlay [email protected] \nMitsubishi Electric R&D Centre Europe Rennes\nFrance\n'], 'authoraffiliation': ['Mitsubishi Electric R&D Centre Europe Rennes\nFrance', 'Mitsubishi Electric R&D Centre Europe Rennes\nFrance', 'Mitsubishi Electric R&D Centre Europe Rennes\nFrance'], 'corpusid': 257038835, 'doi': '10.1109/wcnc55385.2023.10118729', 'github_urls': [], 'n_tokens_mistral': 8516, 'n_tokens_neox': 7748, 'n_words': 5209, 'pdfsha': 'a9f142d10420e0e057e74e5bcb9d36b69df5ab29', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09926v1.pdf'], 'title': ['An application-oriented scheduler', 'An application-oriented scheduler'], 'venue': []}

arxiv

Realization of a Hopf insulator in circuit systems 3 Feb 2023 Zhu Wang Wuhan Institute of Quantum Technology 430206WuhanChina School of Physics and Technology Wuhan University 430072WuhanChina Xu-Tao Zeng School of Physics and Technology Wuhan University 430072WuhanChina School of Physics Beihang University 100191BeijingChina Yuanchuan Biao Wuhan Institute of Quantum Technology 430206WuhanChina School of Physics and Technology Wuhan University 430072WuhanChina Zhongbo Yan School of Physics Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices Sun Yat-sen University 510275GuangzhouChina Rui Yu Wuhan Institute of Quantum Technology 430206WuhanChina School of Physics and Technology Wuhan University 430072WuhanChina Realization of a Hopf insulator in circuit systems 3 Feb 2023(Dated: February 6, 2023) Three-dimensional (3D) two-band Hopf insulators are a paradigmatic example of topological phases beyond the topological classifications based on powerful methods like K-theory and symmetry indicators. Since this class of topological insulating phases was theoretically proposed in 2008, they have attracted significant interest owing to their conceptual novelty, connection to knot theory, and many fascinating physical properties. However, because their realization requires special forms of long-range spin-orbit coupling (SOC), they have not been achieved in any 3D system yet. Here we report the first experimental realization of the long-sought-after Hopf insulator in a 3D circuit system. To implement the Hopf insulator, we construct basic pseudo-spin modules and connection modules that can realize 2 × 2-matrix elements and then design the circuit network according to a tight-binding Hopf insulator Hamiltonian constructed by the Hopf map. By simulating the band structure of the designed circuit network and calculating the Hopf invariant, we find that the circuit realizes a Hopf insulator with Hopf invariant equaling 4. Experimentally, we measure the band structure of a printed circuit board and find the observed properties of the bulk bands and topological surface states (TSS) are in good agreement with the theoretical predictions, verifying the bulk-boundary correspondence of the Hopf insulator. Our scheme brings the experimental study of Hopf insulators to reality and opens the door to the implementation of more unexplored topological phases beyond the known topological classifications. In 2008, the pioneering tenfold-way classification based on non-spatial symmetries provided the first systematic understanding of non-interacting topological phases of matter 1,2 , and founded the basis for the later discovery of a long list of symmetry-protected topological phases based on powerful methods such as symmetry indicators [3][4][5][6][7][8] . Despite its systematicity and fundamental significance, the existence of topological phases beyond the tenfold way classification was soon noticed. Just in the same year, Moore, Ran, and Wen theoretically showed that a class of 3D two-band magnetic topological insulators 9 , later dubbed Hopf insulators as characterized by an integer-valued Hopf invariant 10,11 , exist outside the tenfold-way periodic table 1,2 . Besides the prominent conceptual significance, the two-band Hopf insulators have attracted considerable interest both in theory and experiment due to their many fascinating properties 12,13 . The bulk-boundary correspondence, a central property of topological phases, is also unique in Hopf insulators. The uniqueness is manifested through the dependence of TSS on the surface's orientation and the support of gapless surface Dirac cones, even though the time-reversal symmetry is broken. Besides enriching topological phases, the study of Hopf insulators also substantially advances the understanding of 2D out-ofequilibrium topological phases. The Hopf invariant is found to play an important role in the topological characterization of quenched Chern insulators 14,15 , quenched Euler insulators 16,17 , and Floquet Chern insulators 18 . Although Hopf insulators have been proposed for more than one decade and the great importance of their physical realization is well appreciated 9,19,20 , to date they have only been simulated in a single-qubit quantum simulator 21 and have not been implemented in any 3D system yet. The main challenges for implementing Hopf insulators are the demand of having exactly two bands and a peculiar pattern of long-range SOC. These requirements rule out the implementation in many quantum material systems as well as many artificial systems. In this paper, we report the first bulk realization of the long-sought-after Hopf insulators in a 3D circuit. Because of the extremely high level of connection freedom, circuit networks have been used to realize many novel states of matter, such as 2D topological insulators 22,23 , 3D topological semimetals [24][25][26] , and even 4D topological phases [27][28][29][30] . To carry out the experiment, we use basic building blocks, which in principle admit the implementation of any arbitrary two-band model, Hermitian or non-Hermitian, to design a 3D periodic circuit according to a Hopf insulator model constructed by the Hopf map. By numerically simulating the band structure and calculating the Hopf invariant N h , we find a Hopf insulator phase with N h = 4 exists in a sizable region of the parameter space. By experimentally measuring the bulk and boundary energy spectra of a printed circuit board sample, we find the experimental results agree well with the theoretical predictions and verify the defining bulkboundary correspondence of the Hopf insulator. Model Hamiltonian.-We start with the theoretical model for two-band Hopf insulators. It is known that any two-band model can be expressed via the Pauli matrices σ=(σ 1 ,σ 2 ,σ 3 ) as where σ 0 is the 2 × 2 identity matrix and k = (k x , k y , k z ). Focusing on band topology, all essential information is encoded in the three-component d-vector. The first term on the right-hand side is irrelevant and can be neglected. Moore, Ran, and Wen showed that theoretical models for two-band Hopf insulators can be systematically constructed when the d vector is descended from a complex spinor via the Hopf map 9 , i.e., d(k) = z(k) † σz(k), where z(k) = (z 1 (k), z 2 (k)) T , z 1 = η 1 (k) + iη 2 (k), z 2 = η 3 (k) + iη 4 (k), with η 1,2,3,4 (k) being real functions of momentum. The map is characterized by the Hopf invariant defined as 9 H(k) = d 0 (k)σ 0 + d(k) · σ,(1)N h = − 1 4π 2 d 3 k µνρ A µ ∂ ρ A ν ,(2) where A µ = −i u(k)|∂ µ |u(k) with µ, ν, ρ = {k x , k y , k z } and |u being the negative-energy eigenfunction of H(k) is the Berry connection. It is worth noting that N h does not have any gauge ambiguity even though the Berry connection A µ is gauge dependent. Such a property allows us to numerically calculate the Hopf invariant through discretization of the Brillouin zone 31 . When N h is nonzero, the resulting d(k) · σ model realizes a two-band Hopf insulator. There are infinite choices for η 1,2,3,4 (k) to achieve a nonzero N h . For the convenience of experimental implementation, in this work we consider η 1 (k) = t 1 sin k x , η 2 (k) = t 2 cos(k x + k y + k z ), η 3 (k) = t 3 sin k y , and η 4 (k) = t 4 sin k z . Accordingly, we find N h = 4 and d 1 (k) = 2t 1 t 3 sin k x sin k y + 2t 2 t 4 cos k d sin k z , d 2 (k) = 2t 1 t 4 sin k x sin k z − 2t 2 t 3 cos k d sin k y , d 3 (k) = t 2 1 sin 2 k x + t 2 2 cos 2 k d − t 2 3 sin 2 k y − t 2 4 sin 2 k z .(3) Here we have introduced k d ≡ k x + k y + k z to shorten the notation. Apparently, all three components of the d vector involve long-range hopping processes in real space. What raises a particular challenge is that the hopping parameters involving different length scales need to be comparable in magnitude and satisfy a stringent phase pattern. Hopf insulator circuit.-In this paper, we overcome the challenge and implement the Hopf insulator Hamiltonian (3) in circuit networks as follows. We first create the pseudo-spin space via the module depicted in Fig. 1(a), where three identical inductors form a triangle with C 3 rotational symmetry. Accordingly, the pseudo-spin space in Eq.(3) is provided by the twofold degenerate eigenstates characterized by the 2D representation of the C 3 group. Connecting pseudo-spin modules with connection modules, in which the components form a braided network, allows electrical signals to flip the pseudo-spin, resulting in a SOC-like effect when signals are transmitted between the pseudo-spin modules. Based on this idea, we design connection modules to generate couplings of the form ±(i)σ n (n = 1, 2, 3) as shown in Figs. 1(b-d) 32,33 . As the parameters of the capacitors, inductors, and resistors are positive real numbers, we incorporate the nega- tive sign and the imaginary unit i into the network structure of the connection modules in order to obtain the hopping matrix with complex coefficients in Eq. (4) below. With these modules, the challenging long-range SOC in Eq. (3) can be achieved since components in electronic circuits can be connected between nodes at arbitrary distances by wires, which is in sharp contrast to condensed solid materials and many artificial materials. Performing Fourier transformation on Eq. (3), the realspace tight-binding Hamiltonian reads H T B = l 10 m=0 (c † l+δm,lÛ m c l + h.c.),(4) where l indicate lattice sites, δ m are hopping vectors,Û m are SOC operators as given in Table (I). The operatorŝ U m can be implemented with the connection modules illustrated in Figs. 1(b-d). For example,Û 1 can be built by m −σ1 module andÛ 5 can be constructed by connecting the m −σ3 and m iσ3 modules in parallel. To reduce the number of operational amplifiers used in the experiment, we have exchanged the expressions for d 1 and d 3 . This operation does not change the band topology since it is equivalent to a redefinition of the spin basis. According to Eq. (4), we construct the 3D Hopf insulator circuit network using pseudo-spin modules and connection modules. The unit cell of the Hopf insulator circuit is shown in Figs. 2(a-c). It is worth noting that the resistors in the modules m ±iσ2 and m ±σ3 induce energy loss, which makes the Hamiltonian non-Hermitian. To address this issue, we use the H-module (Fig. 2(b)) to compensate for the energy loss and restore the hermiticity 33 . m 0 1 2 δm 0 2x 2ŷ Um 1 2 (t 2 1 +t 2 2 −t 2 3 −t 2 4 )σ1 − 1 4 t 2 1 σ1 1 4 t 2 3 σ1 m 3 4 5 δm 2ẑx −ŷx +ŷ Um 1 4 t 2 4 σ1 1 2 t1t3σ3 1 2 (it2t4 − t1t3)σ3 m 6 7 8 δmx −ẑx +ẑx + 2ŷ +ẑ Um 1 2 t1t4σ2 − 1 2 (it2t3 + t1t4)σ2 i 2 t2t3σ2 m 9 10 δmx +ŷ + 2ẑ 2x + 2ŷ + 2ẑ Um − i 2 t2t4σ3 1 4 t 2 2 σ1 Kirchhoff's equations for the Hopf insulator circuit can be written as (h 1 (k) ⊕ H circuit h (k))ṽ = ω −2 (0 ⊕ I 2 )ṽ(5) where ⊕ stands for a direct sum of the constant representation space and the pseudo-spin space of the C 3 symmetry group,ṽ = U † v, v = (v 1 , v 2 , v 3 ) T are the node 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 Fig. 2(a). voltages in the unit cell, and U is defined in Supplemental Material 33 . h 1 (k) is the Hamiltonian in the constant representation space. H circuit h (k) = 3 i=0 f i (k)σ i is the Hamiltonian in the pseudo-spin space, where f 0 (k) = L 3 4C 1 + 2C 2 + 2C 3 + 2C 4 + 12(C 5 + C 6 ) , f 1 (k) = − 4L 3 C 1 sin 2 k x + C 2 cos 2 k d − C 3 sin 2 k y −C 4 sin 2 k z , f 2 (k) = − 4 √ 3L 3 − 1 R 1 ω cos k d sin k y + C 5 sin k x sin k z , f 3 (k) = − 4 √ 3L 3 C 6 cos k d sin k z + 1 R 2 ω sin k x sin k y ,(6) and R α (α = 1, 2), C β (β = 1 to 6), L are parameters of the components in the circuit. By choosing appropriate parameters, one can separate the eigenfrequencies of H circuit h (k) well from that of h 1 (k). Therefore, we focus on H circuit h (k) and examine its topological properties below. For simplicity of discussion, we set the inductance L=2.7 µH, R α =R (α = 1, 2), and C β =C (β = 1 to 6). The phase diagram of the frequency band gap as a function of R and C is shown in Fig. 3(a), where the band gaps are finite in the yellow and green regions and tend to zero in the dark blue region as shown in Fig. 3(b). In the gapped region, we find N h = 4 as expected, agreeing with the considered Hopf map. In the regions with vanishingly small gaps, the Hopf invariant does not converge as it is not well-defined in the presence of band degeneracy. One defining characteristic of the Hopf insulator is the unique correspondence between the number of topologically protected TSS and the Hopf invariant. Measuring this bulk-surface correspondence can faithfully identify the topological nature of the system. In Figs. 3(c-d), we show the numerically-calculated band structures for a 16-layers thickness slab structure terminated in x-and z-direction, respectively. There are four TSS on the xdirection surface of the system in Fig. 3(c), where the red (blue) color refers to TSS located on the x=1 (x=16) layer, and the green color refers to bulk states. On the z-direction termination, the TSS localized on z=1 and z=16 layers are degenerate in frequency, as shown in Fig. 3(d). Remarkably, on each z-direction surface, the crossings of the surface frequency spectra labeled by the same color at time-reversal invariant momenta reveal the existence of surface Dirac cones even though the Hamiltonian does not have time-reversal symmetry as already mentioned. Experimentally, these exotic surface Dirac cones provide a unique signature to identify the Hopf insulator. In the following, we experimentally verify the topological nature of the Hopf insulator designed above. According to the circuit diagram in Figs. 2(a-c), we prepare a printed circuit board with the number of unit cells being 3×20×6 in x-, y-, and z-directions and set peri- To extract the frequency spectrum of the circuit lattice, we perform frequency-domain measurements to obtain the voltage vectors v(r, f )=(v 1 (r, f ), v 2 (r, f ), v 3 (r, f )), where r=(x, y, z) labels the unit cell, f is the frequency, and the subscripts indicate the nodes in each unit cell. By performing Fourier transformation in the xand y-directions, the voltage v(k , z, f ) can be obtained in the momentum space, where k =(k x , k y ). The frequency dispersions shown in Figs. 4(c-e) are obtained by plotting ρ(k , z, f )=|v(k , z, f )| 2 , where the peaks of ρ indicate the resonance frequency of the circuit system. The frequency dispersions for the excitation signal applied to the z=1 and z=6 layers are depicted in Fig. 4(c) and (e), where the TSS appear in the band gap and locate around the time-reversal invariant momenta, which is in good agreement with the results calculated from the model Hamiltonian (color dotted lines). The TSS disappear for the excitation signal applied to the z=3 layer because the signal on the middle layer cannot excite the TSS located on the surface layers, as shown in Fig. 4(d). These experimental results confirm the predicted properties of the TSS. In Figs. 4(c-e), we have shifted the experimental data upward by 2.9×10 4 Hz to compare with the theoretical results. Details of the distribution of the TSS in the z-direction are provided in the Supplemental Material 33 . As a final remark, the TSS in the circuit lattice can also be detected by other methods, like impedance measurements 34,35 . Conclusions and discussions.-In this work, we present a general scheme for the implementation of longrange SOC in electric circuits, where the spatial dependence of the SOC can be modulated with a high degree of freedom. Using this property, we have successfully implemented the long-sought-after Hopf insulator in the circuit and observed the TSS enforced by bulk-boundary correspondence. Our general scheme brings the experimental study of Hopf insulators to reality and paves the way for exploring other exotic topological phases associated with peculiar SOC. With our established platforms, the experimental exploration of links and knots with very rich topological structures becomes accessible [36][37][38] . Moreover, the idea behind our scheme can be applied to the future design of systems with more but fixed bands to implement the novel topological phases beyond the known topological classifications, such as Hopf insula-tors in three-band systems 39 and models constructed by higher-dimensional generalizations of the Hopf map 40 . FIG. 1 . 1(a) A connection module connects pseudo-spin modules p1 and p2. The three ports on the connection module's left (right) side are connected to the three nodes of the left (right) pseudo-spin module. (b-d) The list of designed connection modules m ±(i)σ 1,2,3 that gives ±(i)σ1,2,3 types of tunneling matrices between the pseudo-spin space, where solid lines indicate capacitors, dashed lines indicate resistors. In each module, the red solid and dashed lines indicate that their impedances are half those of the black solid and dashed lines. FIG. 2 . 2(a) The unit cell of the Hopf insulator circuit. The connection modules are used for 3D connections. The ports of the connection modules marked with δm (m = 1 to 10) indicate that they are connected to the pseudo-spin module in cell l + δm, while unmarked ones indicate that they are connected to the pseudo-spin module in cell l. (b) The H-module consists of voltage followers, resistors, and adder-subtractor operational amplifiers. (c) Detailed circuit diagram of the red ports in FIG. 3 . 3(a) The calculated band gap as a function of the parameters of C and R. The color map represents the ratio of the band gap to the total frequency band width. (b) The frequency spectrum of H circuit h (k) at points A, B, C, and D in the phase diagram, where Γ=(0,0,0), Y=(0,1,0), M=(1,1,0), R=(1,1,1) are the high symmetry points in the Brillouin zone in units of π with the unit cell lattice parameters set to 1. The parameters (R, C) are equal to (0.03 kΩ, 0.03 nF) at A point, (0.2 kΩ, 0.56 nF) at B point, (1.2 kΩ, 1.2 nF) at C point, and (2.3 kΩ, 1.8 nF) at D point. The inductance is fixed as L=2.7 µH in all calculations. (c) The calculated frequency spectrum along ky and kz directions of a 16-layers slab with x-direction surfaces. The green color refers to bulk states. The red (blue) color refers to states localized on the x=1 (x=16) layer. (d) The frequency spectrum of a 16-layers slab with z-direction surfaces. The TSS on the z=1 (red) and z=16 (blue) layers degenerate and overlap in frequency. The parameters at point B are used to calculate the TSS in Figs. 3(c-d). odic boundary conditions in the x-and y-directions and open boundary conditions in the z-direction. The circuit structure of the unit cell and the H-module are shown in Figs. 4(a-b). A global view of the printed circuit board is shown in Supplemental Material 33 . FIG. 4 . 4(a) The unit cell of the printed circuit board (in the red box) is fabricated according to Fig. 2(a). (b) The unit cell of the printed circuit board of the H-module (in the yellow box). (c), (d) and (e) show the experimentally measured band structure along ky direction with the signal source connected to a cell at z=1 (bottom surface), z=3 (bulk), and z=6 (top surface), respectively. 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{'abstract': 'Three-dimensional (3D) two-band Hopf insulators are a paradigmatic example of topological phases beyond the topological classifications based on powerful methods like K-theory and symmetry indicators. Since this class of topological insulating phases was theoretically proposed in 2008, they have attracted significant interest owing to their conceptual novelty, connection to knot theory, and many fascinating physical properties. However, because their realization requires special forms of long-range spin-orbit coupling (SOC), they have not been achieved in any 3D system yet. Here we report the first experimental realization of the long-sought-after Hopf insulator in a 3D circuit system. To implement the Hopf insulator, we construct basic pseudo-spin modules and connection modules that can realize 2 × 2-matrix elements and then design the circuit network according to a tight-binding Hopf insulator Hamiltonian constructed by the Hopf map. By simulating the band structure of the designed circuit network and calculating the Hopf invariant, we find that the circuit realizes a Hopf insulator with Hopf invariant equaling 4. Experimentally, we measure the band structure of a printed circuit board and find the observed properties of the bulk bands and topological surface states (TSS) are in good agreement with the theoretical predictions, verifying the bulk-boundary correspondence of the Hopf insulator. Our scheme brings the experimental study of Hopf insulators to reality and opens the door to the implementation of more unexplored topological phases beyond the known topological classifications.', 'arxivid': '2302.01591', 'author': ['Zhu Wang \nWuhan Institute of Quantum Technology\n430206WuhanChina\n\nSchool of Physics and Technology\nWuhan University\n430072WuhanChina\n', 'Xu-Tao Zeng \nSchool of Physics and Technology\nWuhan University\n430072WuhanChina\n\nSchool of Physics\nBeihang University\n100191BeijingChina\n', 'Yuanchuan Biao \nWuhan Institute of Quantum Technology\n430206WuhanChina\n\nSchool of Physics and Technology\nWuhan University\n430072WuhanChina\n', 'Zhongbo Yan \nSchool of Physics\nGuangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices\nSun Yat-sen University\n510275GuangzhouChina\n', 'Rui Yu \nWuhan Institute of Quantum Technology\n430206WuhanChina\n\nSchool of Physics and Technology\nWuhan University\n430072WuhanChina\n'], 'authoraffiliation': ['Wuhan Institute of Quantum Technology\n430206WuhanChina', 'School of Physics and Technology\nWuhan University\n430072WuhanChina', 'School of Physics and Technology\nWuhan University\n430072WuhanChina', 'School of Physics\nBeihang University\n100191BeijingChina', 'Wuhan Institute of Quantum Technology\n430206WuhanChina', 'School of Physics and Technology\nWuhan University\n430072WuhanChina', 'School of Physics\nGuangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices\nSun Yat-sen University\n510275GuangzhouChina', 'Wuhan Institute of Quantum Technology\n430206WuhanChina', 'School of Physics and Technology\nWuhan University\n430072WuhanChina'], 'corpusid': 256561322, 'doi': '10.1103/physrevlett.130.057201', 'github_urls': [], 'n_tokens_mistral': 10530, 'n_tokens_neox': 8651, 'n_words': 4734, 'pdfsha': 'fc1553d14707eeb7e1f99a521dbf6887ad4284c0', 'pdfurls': ['https://export.arxiv.org/pdf/2302.01591v1.pdf'], 'title': ['Realization of a Hopf insulator in circuit systems', 'Realization of a Hopf insulator in circuit systems'], 'venue': []}

arxiv

ANTIGRAVITING BUBBLES WITH THE NON-MINKOWSKIAN ASYMPTOTICS arXiv:hep-ph/9610548v1 31 Oct 1996 Andro Barnaveli Institute of Physics Georgian Academy of Sciences Tamarashvili str. 6380077TbilisiRepublic of Georgia Merab Gogberashvili Institute of Physics Georgian Academy of Sciences Tamarashvili str. 6380077TbilisiRepublic of Georgia ANTIGRAVITING BUBBLES WITH THE NON-MINKOWSKIAN ASYMPTOTICS arXiv:hep-ph/9610548v1 31 Oct 1996 The conventional approach describes the spherical domain walls by the same state equation as the flat ones. In such case they also must be gravitationally repulsive, what is seemingly in contradiction with Birkhoff's theorem. However this theorem is not valid for the solutions which do not display Minkowski geometry in the infinity.In this paper the solution of Einstein equations describing the stable gravitationally repulsive spherical domain wall is considered within the thin-wall formalism for the case of the non-Minkowskian asymptotics. For the last two decades great attention has been paid to the investigation of gravitational properties of topological structures such as domain walls, strings and monopoles. It was obtained, that cosmic strings do not produce any gravitational force on the surrounding matter locally while global monopoles, global strings and planar domain walls exhibited repulsive nature [1,2,3]. In this paper we shall consider some problems which arose at studying the gravitational properties of spherical domain walls and show the existence of the solution of Einstein's equations corresponding to a stable gravitationally repulsive spherical domain wall. It is assumed, that the flat domain walls are described by the state equation [1]: σ = −p = const,(1) where σ is the surface density and p is the strong tension in two spatial directions. This state equation corresponds to de Sitter's expansion in the wall-plane and the borders of the wall running away with the horizon. One can speak about the gravitational field of the wall only in the normal direction to the wall. If, for such objects, it is possible to use Newtonian approximation with the mass described by Tolman's formula M = (T 0 0 − T 1 1 − T 2 2 − T 3 3 ) · √ −gdV = (σ + 2p) · √ −gdV = − σ · √ −gdV ,(2) then the tension p acts as a repulsive source of gravity and the planar domain wall has a negative gravitational mass exhibiting repulsive gravitational field [1]. It is natural to think that the same behavior (gravitational repulsion) must occur for the spherical domain walls (bubbles), since usually it is assumed that they are described by the same state equations (1) (e.g. see [4,5]). On the other hand, according to Birkhoff's theorem, the empty space, surrounding any spherical body (including bubbles), is described by Schwarzschild metric. This metric contains the parameter m (corresponding to the mass of gravitating body) m = T 0 0 · √ −gdV,(3) which independently of the state equation is positive. While for planar domain walls (stretching the horizon) the negative gravitational mass (2) can be admissible, for bubbles the negativeness of mass (3) from the first glance looks surprising, since T 0 0 is positively defined everywhere. The above-mentioned problem emerged also when investigating bubble dynamics within the thin-wall formalism [6]. It was obtained that active gravitational mass of the spherical domain wall is positive, i.e. its gravitational field is attractive [4,5,7]. The disagreements in gravitational properties of planar and spherical domain walls were explained by instability of the latter [5], or by existence of a positive energy source stabilizing the bubble [7]. However there still remain various paradoxes (appearing in the models with large pressure [5,8,9]) which can be solved only if bubbles with the state equation (1) are repulsive. The negative mass problem can be solved by the assumption that domain walls are not described by the state equation (1). One must take into account the flux out from the volume of integration, or some external forces stabilizing the domain wall. As a result a state equation can have a principally different form and both the spherical and planar walls can be gravitationally attractive. The other possible solution of discrepancy may be the assumption that the planar domain walls are described by the state equation (1) while the bubbles are not. Recently we have investigated the bubble dynamics within the thin-wall formalism when the state equation for spherical domain walls nevertheless has the form (1). We have found a solution describing repulsive spherical domain walls with outer the Schwarzschild geometry [11], but only in the case when the time coordinate changes its direction on the wall-surface. In this paper we consider the different case, when the domain walls are described by the state equation (1), the time-flow has the same direction in whole space, however the metric far from the spherical domain wall is not Minkowskian. We show that in such case there also exists a solution of the Einstein equations which corresponds to a gravitationally repulsive stable bubble. The assumption about non-Minkowskian asymptotics is reasonable, since in the case of spherical domain walls it is impossible to surround the full source by any boundary inside the horizon (just as it is for planar domain walls). The domain wall is only the "part" of the scalar field solution which fills the whole Universe up to horizon and which has a nonzero vacuum expectation value even in the infinity. The result is that the quantity T µν · dS ν is not a 4-vector of energy-momentum and one can not define the energy simply as T 00 · dxdydz. For example, the energy density of an expanding spherical domain wall remains constant (see (1)) despite increasing of its surface, i.e., this object "takes" the energy from vacuum. In pure Einstein's theory it has been proved that the total energy carried by an isolated system, generating an asymptotic Minkowski geometry, is positive [10]. Due to the essential role played by the asymptotic condition this theorem can not be applied to solutions of Einstein's theory which do not display a Minkowskian asymptotic structure. In order to demonstrate that the sign of the gravitational potential depends on the asymptotical geometry let us consider the zero-zero component of the metric tensor for the isolated source in Newton's approximation g 00 = g ∞ 00 + Φ, where g ∞ 00 is the asymptotic value of metric tensor and Φ = g 00 − g ∞ 00(4) is Newton's potential. When far from the source we have Minkowskian geometry, then g ∞ 00 reaches the maximal value, 1, and, since g 00 ≤ 1, Φ is always negative, i.e., we have gravitational attraction. For non-Minkowskian asymptotics, when g ∞ 00 < 1, Newton's potential (4) can be positive or zero depending on the state equation of the source. The examples of sources with non-Minkowskian asymptotics and with unusual gravitational behavior, as it was mentioned above, are topological objects [1,2,3]. Since the exact solution of the coupling Einstein-Higgs equations for the spherically domain wall is unknown we shall work within the thin-wall formalism. Then Einstein's equations describing motion of spherical domain walls in the case when the time-flow has the same direction in whole space have the form [4,6]: f + +Ṙ 2 − f − +Ṙ 2 = −κGR,(5) where κ = 4πσ and f ± are the zero-zero components of the metric tensor in the outer and inner regions of the bubble; G is the gravitational constant, R is the bubble radius and the overdot denotes the derivative with respect to proper time τ on the shell. Let us investigate a general case of a spherically symmetrical charged bubble, when the metric outside the bubble is f + = 1 − ∆ − 2Gm r + Ge 2 r 2 − GΛ + r 2 , while inside we have f − = 1 − GΛ − r 2 , where Λ ± ≡ 8π 3 · ρ ± , ρ ± being the vacuum energy density in the outer and inner regions. The parameters m and e are the Schwarzschild mass and the charge of the shell, respectively, and g ∞ 00 = 1 − ∆ is the value of the metric tensor in the infinity. Now the equation of motion(5) takes the form Ṙ2 + 1 − ∆ − Λ + GR 2 − 2Gm R + Ge 2 R − Ṙ2 + 1 − Λ − GR 2 = −κGR. Finding m from this equation we obtain: m = − ∆ 2G · R − a 2 · R 3 + e 2 2R + κR 2 · Ṙ2 + 1 − Λ − GR 2 ,(6) where a ≡ Λ + − Λ − + Gκ 2 . It is easy to understand the meaning of terms in (6). The first term is the asymptotical energy of the Higgs field forming the bubble. The second term represents the volume energy of the bubble (a difference between the old and new vacuum energy densities) and the energy of gravitational self-interaction of the shell (the surfacesurface binding energy). The third term is the electrostatic energy lying in the threespace outside the bubble. The last term contains the kinetic energy of the shell and the surface-volume binding energy. Introducing new dimensionless variables z ≡ Rb 1/6 (−2m) 1/3 , τ ′ ≡ τ b 1/2 2κ ,(7) where b = a 2 + 4κ 2 Λ − G, and dimensionless parameters A ≡ ab −1/2 , E ≡ −4κ 2 (−2m) −2/3 b −2/3 , Q 2 ≡ e 2 (−2m) −4/3 b 1/6 , D ≡ ∆(−2m) −2/3 b 1/3 , we can represent the equation of motion (6) as dz dτ ′ 2 + U(z) = E, which is identical to that of the point-like particle with the energy E, moving in one dimension under the influence of the potential U(z) = − z 2 − 2A z · 1 + Q 2 z + Dz + 1 z 4 · 1 + Q 2 z + Dz 2 ,(8) In the equilibrium statė z |z=z 0 = 0, ∂U(z) ∂z z=z 0 = 0, where z 0 is the equilibrium point, U(z 0 ) = E and one can find the critical mass and the equilibrium radius of the bubble m 0 = − 4κ 3 bU 0 3/2 , R 0 = 2κz 0 b 1/2 U 0 1/2 ,(9) where U 0 = |U(z 0 )| > 0. Note that m 0 is negative for the positive b. For the real trajectories potential (8) must be negative since E < 0. Such a potential, for the case of uncharged shells, Q = 0, and with m > 0, was discussed in [9,12], while for the case of Minkowskian asymptotics, D = 0 and m < 0, in [11]. Investigating potential (8) in [11] we have found that in case when D = 0 it has the single maximum and equilibrium state with (9) is unstable for any values of parameters. However in the case when D = 0 the term Dz in (8) for some values of D causes the appearing of a minimum of the potential and gives the stable configuration. Here we would like to note that sometimes for applications it is more easy to evaluate the critical radius and mass of the bubble directly from the equation (6) imposing the conditions [4] Ṙ = 0, ∂m(R,Ṙ) ∂R |Ṙ =0 = 0.(10) The sign of the last term in equation (6) is principal when we investigate the problem of stability of the spherical shells. For the ordinary matter this sign is negative, thus ∂m(R,Ṙ) ∂(Ṙ 2 ) |Ṙ =0 < 0 and the equilibrium state (10) is stable if the function m(R,Ṙ = 0) takes a maximum value at the point R 0 [4]. For the case of domain walls, due to Tolman's formula (2), sign of the last term in (6) is positive, ∂m(R,Ṙ) ∂(Ṙ 2 ) |Ṙ =0 > 0, and the equilibrium state (10) is stable if the function m(R,Ṙ = 0) takes a minimum at the point R 0 [4]. Now let us discuss some particular cases. The simplest example of the antigraviting stable configuration is the case of the Minkowski metric inside the bubble, f − = 1, and the Schwarzschild metric with the non-Minkowskian asymptotics, f + = 1 − ∆ − 2m/r, outside the bubble. In this case equation (6) has the form: m = − ∆ 2G · R − Gκ 2 2 · R 3 + κR 2 · Ṙ2 + 1.(11) From this equation it is easy to find using (10) the radiuses of the critical configurations: R 0 = 2 ± √ 4 − 3∆ 3Gκ , one of which (with the lower sign) is stable, since ∂ 2 m/∂R 2 |Ṙ =0 > 0 for this value of R. Inserting the value of R 0 into (11) one can find that the mass of such configuration is negative. In more simplified case, if we neglect the second term in equation (11), the critical radius and mass of the configuration are R 0 = ∆ 4Gκ , m 0 = − ∆ 4Gκ 2 . This is a stable configuration, since R 0 is a minimum point of the function m(R,Ṙ = 0). As the other example let us consider the case when the surface density σ in equation (6) can be neglected. However, as it was mentioned above, its sign governs the stability of the system. From relations (10) for this case one can find R 2 0 = ∆ 6G(−a) · 1 + 1 − 12ae 2 ∆ 2 ,(12)m 0 = ∆R G · −2 + 1 − 12ae 2 ∆ 2 .(13) From this relations one can notice that the stable configuration is possible only for the negative a = Λ + − Λ − . The sign of the mass of the critical bubble depends on values of parameters a, e and ∆ and for different models can be positive, negative or zero. The next example of the stable spherical remnant of the false vacuum surrounded by a spherical domain wall and with non-Minkowskian asymptotics is the global monopole. We want to treat the monopole problem within the thin-wall approximation, i.e. we could regard that the whole variation of the scalar field forming the monopole is concentrated near some value of the radius R 0 . In the spherical coordinates the zero-zero component of the energy-momentum tensor of the global monopole configuration reads (see for example [2]): T 0 0 = η 2 ξ 2 r 2 + 1 2 ∂ξ ∂r 2 + λ 4 η 2 (ξ 2 − 1) 2 ,(14) where ξ = 0 r < R 0 , ξ = 1 r > R 0 . In other words, we are modeling the monopole by a pure false vacuum inside the core, and an exactly true vacuum at the exterior. In the outer region (14) does not contains a constant term. Thus for the monopole the outer vacuum energy density ρ + is zero. In the inner region ρ − = λη 4 /4, and we have e = Λ + = 0, Λ − = 8π 3 · λη 4 4 .(15) For the surface density σ of the monopole within the thin-wall approximation we find σ = R 0 +δ R 0 −δ T 0 0 dr = R 0 R 0 −δ λη 4 4 dr + R 0 +δ R 0 η 2 r 2 dr ≈ λη 4 4 + η 2 R 2 0 δ,(16) where δ ≪ R 0 is the width of the wall. Thus κ ≡ 4πσ ≪ Λ and we can neglect it in equation (6) for the monopole. We can find the quantity ∆ in (6), formed from the first term of (14), from the solution of Einstein's equations for the monopole: g 00 = 1 + 8πG r ∞ 0 T 0 0 r 2 dr = 1 − ∆ − 2Gm r ,(17) where ∆ = 8πG r r ∞ T 0 0 r 2 dr = 8πGη 2 = 0. Using expressions (15), (16) and (18) from (12) and (13) for the monopoles radius and mass we find: R 0 ∼ ( ∆ 3GΛ − ) 1/2 = 2 η √ λ , m ∼ − ∆ 3/2 3G 3/2 Λ 1/2 − = − 16πη 3 √ λ < 0. These values are in good agreement with the exact solutions for the global monopole obtained in paper [2]. At the end we would like to notice that in case of t'Hooft-Polyakov's monopole the gauge field energy cancels nonzero energy of scalar field in infinity. Thus ∆ = 0 and we have the ordinary Schwarzschild metric, as it was considered in paper [13]. The research described in this publication was made possible in part by Grant MXL200 of Georgia Government and the International Science Foundation. A Vilenkin, E P S Shellard, Cosmic Strings and Other Topological Defects. CambridgeCambridge Univ. PressA. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge Univ. Press, Cambridge, 1994). . D Harrari, C Lousto, Phys.Rev. 422626D. Harrari and C. Lousto, Phys.Rev. D42 (1990) 2626. . D Harrari, P Sikivie, Phys. Rev. 373438D. Harrari and P. Sikivie, Phys. Rev. D37 (1988) 3438. . V A Berezin, V A Kuzmin, I I Tkachev, Phys.Rev. 362919V.A. Berezin, V.A. Kuzmin and I.I.Tkachev, Phys.Rev. D36 (1987) 2919. . J Ipser, P Sikivie, Phys.Rev. 30712J. Ipser and P. Sikivie, Phys.Rev.D30 (1984) 712. . W Israel, Nuovo Cim. 441W. Israel, Nuovo Cim. B44 (1966) 1. . C Lopez, Phys.Rev. 30313C. Lopez, Phys.Rev.D30 (1984) 313. . O Grøn, Phys.Rev. 312129O. Grøn, Phys.Rev. D31 (1985) 2129. . S K Blau, E I Guendelman, A H Guth, Phys.Rev. 351747S.K. Blau, E.I. Guendelman and A.H.Guth, Phys.Rev. D35 (1987) 1747. . R Schoen, S T Yau, Math. Phys. 79231R. Schoen and S.T. Yau, Math. Phys. 79 (1981) 231. . A Barnaveli, M Gogberashvili, GRG. 261117A. Barnaveli and M. Gogberashvili, GRG 26 (1994) 1117; . hep- ph/9505412New Frontiers in Gravitation. Hadron Pressin "New Frontiers in Gravitation" (Hadron Press,Palm Harbor, 1996); hep- ph/9505412. . A Aurilia, R Kissack, R Mann, E Spallucci, Phys.Rev. 352961A. Aurilia, R. Kissack, R. Mann and E. Spallucci, Phys.Rev. D35 (1987) 2961. . Y Cho, P Freund, Phys.Rev. 121588Y.Cho and P.Freund, Phys.Rev. D12 (1975) 1588.

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{'abstract': "The conventional approach describes the spherical domain walls by the same state equation as the flat ones. In such case they also must be gravitationally repulsive, what is seemingly in contradiction with Birkhoff's theorem. However this theorem is not valid for the solutions which do not display Minkowski geometry in the infinity.In this paper the solution of Einstein equations describing the stable gravitationally repulsive spherical domain wall is considered within the thin-wall formalism for the case of the non-Minkowskian asymptotics.", 'arxivid': 'hep-ph/9610548', 'author': ['Andro Barnaveli \nInstitute of Physics\nGeorgian Academy of Sciences\nTamarashvili str. 6380077TbilisiRepublic of Georgia\n', 'Merab Gogberashvili \nInstitute of Physics\nGeorgian Academy of Sciences\nTamarashvili str. 6380077TbilisiRepublic of Georgia\n'], 'authoraffiliation': ['Institute of Physics\nGeorgian Academy of Sciences\nTamarashvili str. 6380077TbilisiRepublic of Georgia', 'Institute of Physics\nGeorgian Academy of Sciences\nTamarashvili str. 6380077TbilisiRepublic of Georgia'], 'corpusid': 17705145, 'doi': '10.1007/bf02634175', 'github_urls': [], 'n_tokens_mistral': 5276, 'n_tokens_neox': 4568, 'n_words': 2913, 'pdfsha': '52c07f9a6eca894be4bc9a70f4a5c8ddb579fd59', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/9610548v1.pdf'], 'title': ['ANTIGRAVITING BUBBLES WITH THE NON-MINKOWSKIAN ASYMPTOTICS', 'ANTIGRAVITING BUBBLES WITH THE NON-MINKOWSKIAN ASYMPTOTICS'], 'venue': []}

arxiv

An efficient method for quantum impurity problems out of equilibrium Julian Thoenniss Department of Theoretical Physics University of Geneva Quai Ernest-Ansermet 301205GenevaSwitzerland Michael Sonner Department of Theoretical Physics University of Geneva Quai Ernest-Ansermet 301205GenevaSwitzerland Alessio Lerose Department of Theoretical Physics University of Geneva Quai Ernest-Ansermet 301205GenevaSwitzerland Dmitry A Abanin Department of Theoretical Physics University of Geneva Quai Ernest-Ansermet 301205GenevaSwitzerland An efficient method for quantum impurity problems out of equilibrium (Dated: 22nd November 2022) We introduce an efficient method to simulate dynamics of an interacting quantum impurity coupled to non-interacting fermionic reservoirs. Viewing the impurity as an open quantum system, we describe the reservoirs by their Feynman-Vernon influence functionals (IF). The IF are represented as matrix-product states in the temporal domain, which enables an efficient computation of dynamics for arbitrary interactions. We apply our method to study quantum quenches and transport in an Anderson impurity model, including highly non-equilibrium setups, and find favorable performance compared to state-of-the-art methods. The computational resources required for an accurate computation of dynamics scale polynomially with evolution time, indicating that a broad class of out-of-equilibrium quantum impurity problems are efficiently solvable. This approach will provide new insights into dynamical properties of mesoscopic devices and correlated materials. arXiv:2211.10272v2 [cond-mat.str-el] Introduction. Non-equilibrium many-body dynamics is actively investigated in condensed matter and synthetic quantum systems such as ultracold atoms [1]. The aim of the ongoing quest is to find regimes where a nonequilibrium system exhibits desired physical properties, which may be qualitatively different compared to equilibrium. Theoretically, out-of-equilibrium many-body problems are extremely challenging, both for analytical and numerical methods [2,3]. Quantum impurity models (QIM), where a small quantum system such as a quantum dot is coupled to reservoir(s) of itinerant electrons, naturally arise in a variety of systems, including mesoscopic conductors [4] and ultracold atoms [5,6]. Even relatively simple QIM such as the celebrated Anderson impurity model (AIM) [7], exhibit rich many-body physics including the Kondo effect whereby the impurity spin is screened by itinerant electrons [8]. Fermionic QIM, including the Anderson models, also play a central role in state-of-the-art methods for strongly correlated materials such as dynamical mean-field theory (DMFT), where the material properties are expressed via a self-consistent QIM [3,9]. A large number of methods for non-equilibrium QIM, and in particular for the AIM, have been developed in recent years. These include iterative path-integral approximations [10][11][12], non-Markovian [13,14] or auxiliary master equations (AME) [15,16], hierarchical equations of motion (HEOM) [17][18][19], time-dependent numerical renormalization group (NRG) [20][21][22] and density matrix renormalization group (tDMRG) [23][24][25][26][27][28], various variants of Quantum Monte Carlo (QMC) [29][30][31][32][33][34], as well as variational [35,36] techniques. Recent advances including inchworm algorithm [37] and increasingly sophisticated high-order diagrammatic calculations [38,39] ameliorated the sign problem of QMC, thereby giving access to * These authors contributed equally to this work. Top: Illustration of single impurity Anderson model [Eq. (1)] with an impurity (red) tunnel-coupled to two reservoirs (gray). Bottom: Tensor-network representation of a time-dependent observable Ô (t) . The dynamical influence of the environment is encoded in a single IF per orbital degree of freedom (here, two gray tensors for σ =↑, ↓, left) which can be efficiently represented as MPS in the temporal domain (right) and hence contracted with the local impurity evolution (product of red tensors). Foreground [background] layer represents forward [backward] branch of the Keldysh contour. longer evolution times. However, despite recent developments, the current methods cannot provide guarantees of computational efficiency for out-of-equilibrium QIM, which remain a subject of active research. In this Article, we present a conceptually simple and efficient method for fermionic QIM, building on recent developments in describing interacting [40][41][42][43][44][45][46][47][48][49][50] and noninteracting [51][52][53] quantum baths using temporal tensor networks. The starting point of our approach is to treat the impurity as an open quantum system coupled to the "bath" that consists of fermionic leads (Fig. 1). The effect of the leads is then represented by the fermionic extension of the Feynman-Vernon influence functional (IF) [54], which can be obtained in closed form for arbitrary noninteracting reservoirs [3,14,52]. As a key ingredient of our approach, the IF can be efficiently represented as a matrix-product state (MPS) in the temporal domain with controlled bond dimension, thanks to the favorable scaling of temporal entanglement of the IF [43,52]. This enables an efficient computation of time-dependent observables at the impurity location (e.g. charge, spin, currents) via straightforward tensor contraction. We demonstrate the efficiency of our method for paradigmatic non-equilibrium QIM setups, including (i) a quantum quench, where impurity site is connected to equilibrium leads at time t = 0 and (ii) a biased AIM with two imbalanced leads. In all cases, our method is capable of reproducing and going beyond the state-ofthe-art results obtained by inchworm and diagrammatic QMC. Besides conceptual simplicity, the method presented here has a number of advantages. First and foremost, required resources grow polynomially in evolution time. In terms of computational complexity [55,56], this implies that QIM are efficiently solvable even far away from equilibrium. Furthermore, the method is non-perturbative, in contrast e.g. to QMC, which involves perturbative expansions either in the impurity-reservoirs hybridization or in the on-site Coulomb interaction. In addition, from a practical viewpoint, once an efficient MPS representation of the reservoirs' IF is found, dynamics of impurities with an arbitrary choice of time-dependent local Hamiltonian can be subsequently computed with modest effort. Description of the method. We consider the singleimpurity Anderson model, described by the Hamiltonian H = k σ=↑,↓ α=L,R t k d † σ c k,α,σ + h.c. + k c † k,α,σ c k,α,σ + H imp ,(1)with H imp = ( d − U/2) σd † σdσ + Ud † ↑d ↑d † ↓d ↓ . The impurity level described by fermions d σ is coupled to two baths (α = L, R) of free fermions c k,α,σ with identical dispersion k and tunnel couplings t k , initially in thermal equilibrium (see top illustration in Fig. 1). Coulomb interaction U = 0 in H imp gives rise to strong correlations in and out of equilibrium. We are primarily interested in the real-time evolution of an impurity observable Ô (t) starting from a factorized initial state ρ(0) = ρ L ⊗ ρ imp ⊗ ρ R , with ρ L,R equilibrium states at inverse temperatures β L,R and chemical potentials µ L,R . While conventional tensor-network approaches attempt to compactly represent ρ(t) [2], we instead express Ô (t) as a Keldysh path integral over Grassmann trajectories of impurity and baths. Gaussian integration over the bath trajectories gives Ô (t) ∝ σ,τ dη σ,τ dη σ,τ O(η t , η t ) × exp C dτ ση σ,τ ∂ τ η σ,τ − iH imp (η τ , η τ ) ×ρ imp [η 0 , η 0 ] σ=↑,↓ exp C dτ C dτ η σ,τ ∆(τ, τ )η σ,τ .(2) Hereη τ = (η ↑,τ ,η ↓,τ ) and η τ = (η ↑,τ , η ↓,τ ) parametrize the impurity trajectory. The IF is the last exponential in Eq. (2), defined by the hybridization function ∆(τ, τ ) = α ∆ α (τ, τ ), where ∆ α fully encodes the dynamical influence of the bath α, ∆ α (τ, τ ) = dω 2π Γ(ω)g α τ,τ (ω).(3) The latter is determined by the bath's spectral density Γ(ω) = 2π k |t k | 2 δ(ω − k ) and non-interacting Green's function g α τ,τ (ω) = n α F (ω) − Θ C (τ, τ ) e −iω(τ −τ ) , where n α F is the Fermi distribution at inverse temperature β α and chemical potential µ α and Θ C is the Heaviside step function on the Keldysh contour C (see e.g. Ref. [3]). Equation (2) is the starting point of advanced techniques for impurity dynamics such as AME, HEOM or QMC. The difficulty in evaluating the path integral arises from the combination of non-Gaussianity (in H imp ) and time-non-locality (in ∆(τ, τ )). The key idea of our method is to interpret Eq. (2) as a scalar product of fictitious states and operators defined in a fermionic Fock space on a temporal lattice. To that end, we note that the textbook expression in Eq. (2) is defined as the limit M → ∞ of a discrete-time expression, obtained by dividing the full time evolution window [0, T ] into M steps of size δt = T /M ; we fix a sufficiently large M . For our purpose, it is convenient to use a Trotter scheme that further splits the Trotter step into impurity and hybrization, leading to 8M trajectory variables per spin species along the discretized Keldysh contour, see Supplemental Material (SM) for details. We arrange these in two arrays, η σ = (η σ,0 + , η σ,0 − , . . . , η σ,(2M −1) + , η σ,(2M −1) − ) and analogouslyη σ , with degrees of freedom alternating on the forward (+) and backward (−) branch of the Keldysh contour. A series of manipulations with the discrete-time path integral, including partial "particle-hole transformations" η ↔η, allows us to rewrite Eq. (2) in a scalar product form (see SM for details): Ô (t) ∝ σ dη σ dη σ × I[η ↓ ]e −η ↓ η ↓ D O,t [η ↓ , η ↑ ] e −η ↑ η ↑ I[η ↑ ] ≡ I|DÔ ,t |I .(4) Here, the kernel D O,t [η ↓ , η ↑ ], which is non-Gaussian, describes impurity's own dynamics, and has a simple product form due to time locality. This gives rise to a product operatorDÔ ,t =D 1 ⊗ · · · ⊗D M , where eacĥ D m is a 16 × 16 matrix (except the first and last: see superimposed red tensors in Fig. 1) andD m * =t/δt con-tainsÔ. The discrete-time IF has a Gaussian form, I[η σ ] = exp η T σ B η σ , where the antisymmetric matrix B is related to the time-discretization of ∆(τ, τ ) (see SM). The Gaussian many-body wave function |I associated with I (gray tensors in Fig. 1 bottom left) is obtained by replacing Grassmann variables by corresponding creation operators acting on the Fock space vacuum, c † ≡ (c † 0 + , c † 0 − , . . . , c † (2M −1) + , c † (2M −1) − ), |I = exp c † T B c † |∅ .(5) Such a state formally has a Bardeen-Cooper-Schrieffer form, regardless of the fermion-number conservation of the original problem, cf. Eq. (2); this is related to the "particle-hole transformations" performed to arrive at Eq. (4). We note that particle number conservation shows up as a sublattice symmetry in Eq. (5). Next, we aim to represent the state |I as a MPS. Correlations of this state, described by the function ∆(τ, τ ), reflect non-Markovianity of the bath. The possibility of a compact MPS representation is determined by the entanglement properties of a wave function; we previously showed [52] that Gaussian IF wave functions arising in QIM exhibit at most logarithmic scaling of temporal entanglement with evolution time for both equilibrium and certain non-equilibrium initial states of the reservoirs. This suggests that such wave functions can be described by a polynomial-in-T number of parameters. Previous works [57,58] proposed algorithms for representing a fermionic Gaussian wave function as a MPS. Here we apply the Fishman-White (FW) algorithm [57], extended to BCS-like wave functions [52]. We first approximately represent the Gaussian state determined by B [Eq. (5)] as a quantum circuit of nearest-neighbor Gaussian unitary gates applied to the vacuum (a product state in a temporal chain of 4M spins). The approximation is controlled by a threshold parameter of the algorithm [52,57], which determines the maximum number D of gates acting on a given site in this circuit (which we refer to as "local depth" below). Second, we compress the circuit with standard singular-value truncations to produce a MPS approximation of |I with bond dimension χ ≤ 2 D . Once the MPS is obtained (gray tensors in Fig. 1 bottom right), the impurity's reduced density matrix time evolved with an arbitrary (possibly timedependent) impurity Hamiltonian H imp can be efficiently computed by tensor contraction in the time direction. This method is straightforwardly applicable to the computation of multi-time observables, e.g. the impurity Green's function, as well as currents (see below). A quantum quench. As a first application of our method, we study a local quantum quench, where tunneling between impurity and the bath -initially in equilibrium at equal β and µ -is turned on at time t = 0. We monitor the real-time evolution of the impurity level population at t > 0. In the Kondo regime (strong interaction and low temperature), strong correlations develop in real time between the impurity and the bath, corresponding to the formation of a local screening cloud over a nonperturbatively long timescale -a real-time manifestation of the Kondo effect, which was previously investigated with other methods [20,21,24,35,59]. Here we benchmark the state-of-the-art results of inchworm QMC in Ref. [37]: We consider a bath defined by a flat band with smooth edges, Γ(ω) = Γ/ (1 + e ν(ω−ωc) )(1 + e −ν(ω+ωc) ) with ω c = 10Γ and ν = 10/Γ. Moreover, we set β = 50/Γ, µ = 0. We prepare the impurity in a singly occupied state ρ imp = |↑ ↑|, with d = 0 and U = 8Γ, and couple it to the bath at time t = 0. In Fig. 2 we report our results for the evolution of the diagonal components of the impurity's reduced density matrix. Data are converged with respect to all simulation parameters (see caption), demonstrating accuracy beyond the data of Ref. [37]. These results showcase the ability of our method to capture the slow dynamical formation of a spin singlet in the Kondo regime, which will be further investigated elsewhere. Non-equilibrium transport. The system described by Eq. (1) with a temperature or chemical potential bias between L and R reservoirs models paradigmatic nonequilibrium setups with correlated nanodevices. Capturing the full transient charge and spin dynamics after a quench (either of tunnel-couplings or of interactions) toward the non-equilibrium stationary state is a recurrent challenging test for novel advanced numerical techniques [22,38,[60][61][62]. Here we benchmark the state-of-the-art computation of the system's current-voltage characteristics in Ref. [38]. We model the reservoirs as two homogenous tight-binding chains with nearest-neighbor hopping t hop = 1, coupled to the impurity with tunneling amplitude t hop = 0.3162, corresponding to a resonance width Γ( d = 0) = 0.1 (cf. Ref. [24]). We initialize the two reservoirs at zero temperature and chemical potentials ±V /2, and monitor the time-dependent current flowing through the impurity for several values of U , until the stationary state is reached. Unlike the contraction illustrated in Fig. 1 and used above for the quench simulation, computing the current into either reservoir requires one to keep track of the separate influence of reservoirs L and R. A suitable Trotter decomposition (see Ref. [52] and SM) allows us to couple the two reservoirs with the impurity alternatively in discrete time steps δt. The current of spin σ electrons flowing into reservoir α, can then be computed as I α,σ (t) = 1 δt d † σ (t+δt)d σ (t+δt) − d † σ (t)d σ (t) , where the impurity interacts only with reservoir α during the time step from t to t + δt. Keeping track of L and R separately results in a tensor contraction with four IF MPS. This considerably limits the bond dimension we can afford for each IF, as the final impurity evolution entails storing matrices acting on a 16χ 4 -dimensional space (while it was 16χ 2 before). Nonetheless, we found that the value of the current is converged over the full transient to the stationary state for bond dimension as low as χ = 32 (see inset of Fig. 3). Figure 3 shows the results of our computations, as well as the corresponding data from Fig. 15 of Ref. [38]. We find a fairly good agreement throughout the wide explored parameter regime. The unit slope of the dot- ted line represents the universal Landauer linear-response conductance, I = (e 2 /h)V (recall e = = 1 in our units). We note that small discrepancies are to be expected at large biases V Γ due to non-universal effects of finite bath bandwidth (t hop = 10Γ here). We further remark that for small bias and large interaction the nonequilibrium Kondo regime is approached, characterized by slow relaxation. Accordingly, in the computation with smallest bias V = 0.36Γ and largest interaction U = 8Γ in Fig. 3, the time-dependent current has not yet fully reached its stationary value at time T . Computational efficiency. Finally, we report on the computational efficiency of our method. Previous works found that for Gaussian ground states [57] and IFs [52] (including states with algebraic correlations), the FW algorithm produces a quantum circuit of "local depth" D = D(T ) that scales at most logarithmically with evolution time T . We note that the FW control parameter affects the prefactor of log T scaling of D. In turn, D puts an exact upper bound on the bond dimension of the corresponding MPS as χ ≤ 2 D [52,57], indicating that computational complexity of the algorithm scales at most polynomially with evolution time. We found that compression of the FW circuit using conventional singular-value truncation typically leads to a further significant reduction of the required computational resources. For example, for the data shown in Fig. 2, we find a maximum "local depth" D = 28 which sets the hard upper bound χ ≤ 2 28 . However, this circuit could be accurately approximated by a MPS with a much smaller bond dimension χ = 256 = 2 8 . We finally investigated how this MPS compression affects the a posteriori error of observables. To this end, we considered an environment that consists of a single tight-binding chain [63]. Having fixed an extremely low FW threshold (which makes this source of error negligible), we estimated the residual error of time-dependent observables in t ∈ [0, T ] due to the truncated bond dimension, as the trace distance e(t, χ) = ρ 1 between the reduced density matrix computed with a cutoff χ on the IF MPS and the fully converged result (computed using a much higher χ = 512). (χ) imp (t) − ρ (∞) imp (t) The behavior of the error e as a function of t and χ is illustrated in Fig. 4. We observe that the bond dimension χ = χ(t) required to achieve a fixed error e grows approximately linearly with t, indicating the efficiency of the approach. We similarly found in other examples we studied, that representing IF with an MPS with a moderate bond dimension is sufficient to accurately compute impurity observables. Thus, we conclude that our approach indeed has a polynomial complexity [43,52], allowing one to access long-time impurity dynamics using resources available in present-day computers. Summary and outlook. To summarize, we introduced a method for studying dynamics of QIM, based on a tensor-network representation of reservoir's IF. We applied this approach to paradigmatic quantum quenches in AIM, demonstrating that it compares favorably to state-of-the art QMC computations. The approach is non-perturbative and offers several other advantages: in particular, it applies to both equilibrium and highly non-equilibrium QIM setups. Moreover, once a MPS form of the IF is obtained, arbitrary choices of impurity interactions can be analyzed with modest extra effort. We showed that the required computational resources scale polynomially with the evolution time. Combined with previous results on temporal entanglement scaling [52], this demonstrates that a broad range of nonequilibrium QIM problems are efficiently solvable using our approach. While here we focused on quenches of the impurity-reservoir tunnel-coupling in the single-impurity Anderson model, the approach can be extended to a number of other setups, including multi-orbital impurities and initial states where entanglement between impurity and reservoirs is present. Another promising application is to DMFT, which will require imaginary-time extension of the technique introduced here. We expect the computational efficiency of the approach to enable long-time simulations of dynamics in such setups as well, opening the door to analyzing non-equilibrium behavior of mesoscopic devices and quantum materials. We start by recalling the standard derivation of the path integral in Eq. (2). Defining the evolution operator U = exp(i δt H) for a time step δt = T /M (M 1) and the Hamiltonian H from Eq. (1), the expectation value of an impurity observable can be expressed as Ô (t m * ) = Tr imp Tr bath U M −m * Ô U m * ρ imp ⊗ρ bath (U † ) M . (A1) Here, t m * = m * · δt denotes a point on the discrete-time lattice and m * ∈ {0, 1, . . . , M }. This expression is cast into path integral form by inserting Grassmann resolutions of identity 1 τ = ⊗ σ 1 σ,τ , where 1 σ,τ = d(η σ,τ , η σ,τ )d(ξ σ,τ , ξ σ,τ ) e −ησ,τ ησ,τ −ξσ,τ ξσ,τ |η σ,τ , ξ σ,τ η σ,τ ,ξ σ,τ |, (A2) between every multiplication of operators. Here, η σ,τ , η σ,τ are impurity variables with index τ = 0 ± , . . . , M ± on the (discretized) Keldysh contour, whileξ σ,τ = (ξ j=1,σ,τ , . . . ,ξ j=L,σ,τ ) T , ξ σ,τ = (ξ j=1,σ,τ , . . . , ξ j=L,σ,τ ) T are the degrees of freedom of the environment (made of L fermionic modes). In total, we thus insert 2(M + 1) identity resolutions per fermionic mode. In the limit δt → 0, Eq. (2) is retrieved following standard textbook passages. To define a clean prescription for our temporal wave functions overlap, however, it is more convenient to further split the evolution operator U into a local impurity and environment+tunneling evolution operators, i.e. U ≈ U imp · U hyb (with the same error O(δt 2 ) as before). Here, we defined U hyb = exp i δt H hyb with H hyb = H − H imp = = k σ=↑,↓ α=L,R t k d † σ c k,α,σ + h.c. + k c † k,α,σ c k,α,σ ,(A3) and, for later convenience, we choose [While this choice is convenient for the conceptual derivation within the Grassmann formalism, we found that a slightly modified prescription, where we replace 1 imp in Eq. (A4) by the perfectly depolarizing channel, is numerically more efficient, see App. B.] In total, with this modified Trotter decomposition, we insert 4M Grassmann identity resolutions per spatial site and τ = 0 ± , . . . , (2M − 1) ± . In Fig. S1, we illustrate how the resulting 8M Grassmann variables are associated with the legs of the IF and impurity tensors, respectively. In order to arrive at the overlap form of Eq. (4), we manipulate the path integral in a way that results in the following structure: All variables associated with the kernel of the spin-up (down) IF should be conjugate (nonconjugate) and opposite for the impurity kernel. This is achieved by making appropriate variable substitutions in the system-variables of the identity resolution, Eq. (A2). We define these modified identity resolutions as 1 σ,τ with substitutionη σ,τ → η σ,τ , η σ,τ → −η σ,τ , 1 σ,τ with substitutionη σ,τ → −η σ,τ , η σ,τ →η σ,τ . U imp = e i With this, Grassmann identities are inserted between the hybridization-and impurity evolution operators on the arXiv:2211.10272v2 [cond-mat.str-el] 21 Nov 2022 forward branch in the following way: U imp · 1 (2m+1) + · U hyb · 1 (2m) + ,(A5) with 1 (2m+1) + =1 ↑,(2m+1) + ⊗ 1 ↓,(2m+1) + ,(A6)1 (2m) + =1 ↑,(2m) + ⊗ 1 ↓,(2m) + .(A7) On the backward branch, we insert identities as follows: 1 (2m) − · U † hyb · 1 (2m+1) − U † imp ,(A8) with 1 (2m) − =1 ↑,(2m) − ⊗ 1 ↓,(2m) − ,(A9)1 (2m+1) − =1 ↑,(2m+1) − ⊗ 1 ↓,(2m+1) − .(A10) With these insertions, one arrives at Eq. (4). Note that these variable substitutions alter the signs of some components of the impurity kernel, while they amount to a simple renaming of variables for the IF. The resulting discrete-time IF has Gaussian form, I[η σ ] = exp m,m η T σ,m B mm η σ,m ,(A11) with η σ,m = (η σ,(2m) + , η σ,(2m) − , η σ,(2m+1) + , η σ,(2m+1) − ) T . The matrix B that appears here is the exact Gaussian influence action of the trotterized (Floquet) environment [52]. To understand its relation to the continuoustime result, it is convenient to express it in terms of a discrete-time hybridization matrix ∆, I[η σ ] = exp m≥m η T σ,m ∆ mm η σ,m e η σ,0 + η σ,0 − × e M −1 m=0 η σ,(2m+1) + η σ,(2m) + +η σ,(2m) − η σ,(2m+1) − , (A12) where the terms in the second and third exponential in Eq. (A12) stem from the overlap of Grassmann coherent states. For a general discrete-time (Floquet) unitary evolution of the environment, the matrix ∆ has a complicated structure, which we evaluated exactly e.g. for setups with chain-environments [52]. However, the structure greatly simplifies in the Trotter limit δt → 0, where ∆ can be written in the form ∆ mm = (δt) 2 α dω 2π Γ(ω) G α mm (ω) + O δt . (A13) Here, G α mm (ω) is a matrix of non-interacting Green's functions of the environment. For fermion-numberconserving Hamiltonians as considered in this work, this yields (omitting ω-dependence for simplicity) G α m>m =     0 g α,> mm * g α,< mm * 0 −g α,> mm 0 0 −g α,< mm −g α,> mm 0 0 −g α,< mm 0 g α,> mm * g α,< mm * 0     , (A14) G α m=m = 1 2     0 g α,> mm * g α,> mm 0 −g α,> mm 0 0 −g α,> mm * −g α,> mm 0 0 −g α,< mm 0 g α,> mm * g α,< mm * 0     , (A15) with g α,< mm (ω) ≡ −n α F (ω) e −iω(m−m )δt (A16) g α,> mm (ω) ≡ 1 − n α F (ω) e −iω(m−m )δt .(A17) These equations make the connection between ∆ mm and the standard textbook hybridization function ∆(τ, τ ) [Eq. (2) of the main text] manifest. Equation (A13) allows to use our formalism to compute impurity dynamics with an environment defined by an arbitrary spectral density Γ(ω). We emphasize that Eq. (A11) here represents the exact discrete-time IF of a trotterized system, computed from its unitary Floquet dynamics. Thus, in contrast to a brute-force discretization of the textbook expression in Eq. (2) The impurity tensorsD m appearing in the overlap form are obtained directly from the Grassmann kernels corresponding to the impurity evolution: We convert each time-local impurity kernel at time step m to an op-eratorD m that acts between a "↑" two-fermion space (originally corresponding to the tensor product of input and output Hilbert spaces of the ↑ impurity fermion) and a "↓" two-fermion space (originally corresponding to the tensor product of input and output Hilbert spaces of the ↓ impurity fermion), see e.g. superimposed red squares in Fig. S1. Their tensor product,DÔ ,t =D 1 ⊗. . .⊗D M (see main text), defines the product operator which we contract with the IF-MPS as shown in Fig. S1 and described in App. B. 1) and v ↑,β (M − 2), necessary to extract the impurity density matrix from the augmented density matrices, are shaded blue. The whole tensor network corresponds to the vectorized impurity density matrix ρ ef (3) (after the action of IF→impurity→ IF ). Bottom: To compute non-equilibrium observables like the current-here for a two-terminal setup-we time evolve ρimp(m) by contracting it successively with the IF-MPS of the right (light gray) and left (dark gray) environment before applying the impurity gate. In this example, we show the density matrix of the impurity after two full time steps. given by Λ τ τ =   I|cτ c † τ |I I|I I|cτ c τ |I I|I I|c † τ c † τ |I I|I I|c † τ c τ |I I|I   ,(B1) where τ, τ are points on the discretized Keldysh contour. The FW algorithm encodes Λ as a quantum circuit consisting of unitary gates. The accuracy of the circuit representation of |I is set by an external parameter that is chosen beforehand. For → 0, the circuit representation becomes exact. Applying the circuit to the vacuum (product) state of 4M spins yields the MPS representation of |I . The "local depth" D of the circuit scales logarithmically in evolution time [42,49,52], with a prefactor that increases as is decreased. For numerical stability, it is advantageous to fix a very small and continuously reduce the bond dimension χ of the MPS during the circuit contraction. For this, we use conventional singular value decomposition (SVD) where we fix the maximal bond dimension χ max . In practice, we group all four tensor legs at equal time index m into a single, larger physical Hilbert space of dimension d = 2 4 ; the bond dimension χ refers to these enlarged tensors as depicted in Fig. S2 (top). This procedure yields a MPS that is proportional to the IF wave function, up to errors through finite and SVD truncation errors. To get the overall normalization I|I , we are in principle free to insert arbitrary impurity evolution operators U imp , imposing that the Keldysh partition function equals 1. For environments defined by a continuous spectral density Γ(ω), however, different choices of U imp lead to slightly different normalizations, as time discretization slightly violates the CPTP property of the impurity evolution, as remarked in App. A. For the sake of efficiency, we choose to apply perfectly depolarizing channels on the impurity, which has the effect of connecting the forward and backward legs at a given variable index τ [65] (see upper half of top panel in Fig. S2). This allows us to minimize the cost of normalizing each IF separately. Once the properly normalized IF-MPS have been obtained, we proceed as follows: The individual tensors of the IF-MPS {A σ (m)} M m=1 (gray bricks in Fig. S2) can formally be viewed as "superoperators" acting on the impurity with its physical legs and on the (compressed) environment with its virtual legs [66]. To obtain the full time evolution of the density matrix, we apply these "superoperators" alternately with the local superoperator associated with the impurity Hamiltonian (red bricks in Fig. S2). Contracting with a single environment. First, let us consider the evolution of an impurity coupled to a single environment which is encoded by two IF-MPS (one for each spin species). This setup is depicted in Fig. S2 (top) and corresponds to the calculations in Fig. 2 and Fig. 4 of the main text. In the following notation, we use Latin letters to denote combined indices of the forward and backward branch of the Keldysh contour. These indices correspond to physical legs of the IF-MPS. Furthermore, we use Greek letters to label the virtual legs of the MPS. These represent a fictitious state of the compressed environment associated with the MPS virtual bond space as depicted in Fig. S2 (top). Our starting point is the vectorized initial density mat-rix ρ ab of the impurity, where the indices a, b correspond to the spin up-and down-fermion respectively. We then add two additional indices, one for each IF, to obtain an augmented density matrixρ(0) ab,αβ . Each time step consists of i) the combined evolution of the impurity with the environment and ii) the evolution of the impurity only. The former is formally expressed as: ρ ab,αβ (2m + 1) = A αα ↓,aa (m)A ββ ↑,bb (m)ρ a b ,α β (2m), Physically, this is equivalent to completing the Keldysh contour to the final time T by evolving the impurity with the perfectly depolarizing channel for each timestep later then m. Numerically, this is slightly more convenient than using identities as in Eq. (A4), as the matrices to be multiplied in Eq. (B4) are smaller. Computation of the current. To compute the current as in Fig. 3 of the main text, it is necessary to compute the IF of the left and right environment separately, such that we can access the impurity density matrix before and after the individual interaction with the left and with the right environments, as depicted in the bottom panel of Fig. S2. This leads to a modified prescription for time evolution including four independent IF: The augmented density matrix has now four environment indices corresponding to the virtual bond of each of the four IF. For each timestep, we perform the operation in Eq. (B2) twice, once for the left and once for the right environment IF. To extract the impurity density matrix from the augmented density matrix at each substep, all of the additional indices have to be contracted with the corresponding vector v L/R,↑/↓,α . This vector is obtained individually for each IF according to Eq. (B4). The required computer memory of the contraction algorithm scales thus as O(16χ 4 ), where χ is the maximal bond dimension of each IF. While this requirement of memory resources imposes a bound on manageable bond dimensions in practice, we demonstrate in Fig. 3 that even bond dimensions as low as χ = 32 yield converged results that are competitive with state-of-the-art methods for nonequilibrium dynamics. Figure 1 . 1Figure 1. Top: Illustration of single impurity Anderson model [Eq. (1)] with an impurity (red) tunnel-coupled to two reservoirs (gray). Bottom: Tensor-network representation of a time-dependent observable Ô (t) . The dynamical influence of the environment is encoded in a single IF per orbital degree of freedom (here, two gray tensors for σ =↑, ↓, left) which can be efficiently represented as MPS in the temporal domain (right) and hence contracted with the local impurity evolution (product of red tensors). Foreground [background] layer represents forward [backward] branch of the Keldysh contour. Figure 2 . 2Real-time evolution of the impurity density matrix after a quench. The plot reports diagonal entries ραα, with α = ∅, ↑, ↓, ↑↓ as a function of time. The environment is modelled as in Ref.[37] (see main text), with β = 50/Γ and µ = 0. Simulation parameters: Bond dimension χ = 256 per spin species, FW threshold = 5 · 10 −13 , Trotter step δt = 0.02/Γ. Figure 3 . 3Current-voltage characteristics of an AIM. Reservoirs L and R are tight-binding chains as in Refs. [24, 38] (see main text), of L = 600 sites each, at zero temperature and chemical potentials ±V /2. Simulation parameters: Bond dimension χ = 32 per reservoir per spin species, FW threshold = 1 · 10 −12 , Trotter step δt = 0.007/Γ. For all values of V and U we evolve until time T = 4.2/Γ and verify that at this time stationary state is reached. Inset: At fixed V /Γ = 14.8 and U/Γ = 2, we demonstrate convergence in bond dimension for all four components of the transient current, Iα,σ(t) with α = L, R and σ =↑, ↓ . Figure 4 . 4Error e(t, χ) of the time-evolved impurity density matrix as a function of bond dimension and evolution time (see main text for a precise definition), for an impurity starting from ρimp(0) = |↑ ↑| and coupled with tunneling amplitude t hop = 0.3162 to a single tight-binding chain of L = 400 sites with homogeneous nearest-neighbor hopping t hop = 1, initially at zero temperature and half filling (cf. Ref.[24]). The constant-error e = 10 −5 dashed line indicates that the required bond dimension grows slowly with simulation time.Here we fixed T = 4, δt = 0.01, = 10 −12 . Figure S1 . S1δtHimp for evolution up to time t m * 1 imp for evolution in range [t m * , T ]. (A4) After inserting Grassmann resolutions of identity into Eq. (A1), each leg of the IF and impurity tensor gets associated with a Grassmann variable [red (blue) color refers to the forward (backward) Keldysh branch]. Here, we show the density matrix of the impurity after two time steps. It is obtained from an IM containing M = 4 time steps. The impurity gates at time steps later than m * are replaced by the identity operator. of the main text, Eq. (A11) produces a physically meaningful evolution of the impurity [completely positive and trace preserving (CPTP)] -close by O(δt) to the exact continuous-time Hamiltonian dynamics. Conversely, plugging a spectral density Γ(ω) in Eq. (A13) and neglecting the O(δt) corrections generally (slightly) breaks CPTP. Figure S2 . S2Top: Illustration of the IF-MPS contraction procedure for a single environment and two spin species. Pairs of impurity-legs on the forward and backward branch are combined and labelled by Latin letters, while virtual MPS indices are labelled by Greek letters. Each tensor A of the IF-MPS can be viewed as a linear operator connecting two physical and two virtual spaces. The augmented density matrix ρ ab,αβ (1) for the first half time step is highlighted by dashed lines. The vectors v ↑,γ (M − where A αα σ,aa (m) is the IF-MPS tensor at time step m. For step ii), we apply the local impurity evolution represented by the superoperator D cd ab (m) (cf. Eq. (4) in main text): ρ ab,αβ 2m + 2 = D ab a b (m)ρ a b ,αβ (2m + 1). (B3) To obtain the density matrix of the impurity from the augmented density matrices at arbitrary intermediate timesρ(m), we recursively compute a set of vectors v σ,α (m) for each of the two IF (corresponding to σ =↑ and σ =↓, respectively): v σ,α (m − 1) = A αα σ,aa (m)v σ,α σ,α (M ) = 1 and 1 a being vectorized identities, see upper panel of Fig. S2. 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Schwarz, J. von Delft, and E. Arrigoni, Nonequilibrium kondo effect in a magnetic field: auxiliary master equation approach, New Journal of Physics 20, 013030 (2018). The choice of an environment defined by unitary evolution allows us to avoid errors associated with timediscretization of a pre-defined spectral density Γ(ω). The choice of an environment defined by unitary evol- ution allows us to avoid errors associated with time- discretization of a pre-defined spectral density Γ(ω). Real time evolution of anderson impurity models via tensor network influence functionals. N Ng, G Park, A J Millis, G K L Chan, D R Reichman, N. Ng, G. Park, A. J. Millis, G. K.-L. Chan, and D. R. Reichman, Real time evolution of anderson impurity models via tensor network influence functionals (2022). M M Wilde, Quantum information theory. Cambridge University PressM. M. Wilde, Quantum information theory (Cambridge University Press, 2013). In the strict sense, {Aσ(m)} M m=1 are not superoperators as their bond dimension (and thus the equivalent of the environment operator space) is not guaranteed to be a square integer, and CPTP property is not enforced. Furthermore. this property would require to fix the gauge freedom of the MPSIn the strict sense, {Aσ(m)} M m=1 are not superoperators as their bond dimension (and thus the equivalent of the environment operator space) is not guaranteed to be a square integer, and CPTP property is not enforced. Fur- thermore, this property would require to fix the gauge freedom of the MPS.

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{'abstract': 'We introduce an efficient method to simulate dynamics of an interacting quantum impurity coupled to non-interacting fermionic reservoirs. Viewing the impurity as an open quantum system, we describe the reservoirs by their Feynman-Vernon influence functionals (IF). The IF are represented as matrix-product states in the temporal domain, which enables an efficient computation of dynamics for arbitrary interactions. We apply our method to study quantum quenches and transport in an Anderson impurity model, including highly non-equilibrium setups, and find favorable performance compared to state-of-the-art methods. The computational resources required for an accurate computation of dynamics scale polynomially with evolution time, indicating that a broad class of out-of-equilibrium quantum impurity problems are efficiently solvable. This approach will provide new insights into dynamical properties of mesoscopic devices and correlated materials. arXiv:2211.10272v2 [cond-mat.str-el]', 'arxivid': '2211.10272', 'author': ['Julian Thoenniss \nDepartment of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland\n', 'Michael Sonner \nDepartment of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland\n', 'Alessio Lerose \nDepartment of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland\n', 'Dmitry A Abanin \nDepartment of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland\n'], 'authoraffiliation': ['Department of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland', 'Department of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland', 'Department of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland', 'Department of Theoretical Physics\nUniversity of Geneva\nQuai Ernest-Ansermet 301205GenevaSwitzerland'], 'corpusid': 253708179, 'doi': '10.1103/physrevb.107.l201115', 'github_urls': [], 'n_tokens_mistral': 18774, 'n_tokens_neox': 16107, 'n_words': 9449, 'pdfsha': '42501243a5d0e560593f9d6bba7d448369e9043c', 'pdfurls': ['https://export.arxiv.org/pdf/2211.10272v2.pdf'], 'title': ['An efficient method for quantum impurity problems out of equilibrium', 'An efficient method for quantum impurity problems out of equilibrium'], 'venue': []}

arxiv

FINITE GROUPS WITH MAXIMAL NORMALIZERS I 2 Jun 2009 Joseph P Bohanon FINITE GROUPS WITH MAXIMAL NORMALIZERS I 2 Jun 2009 We examine p-groups with the property that every non-normal subgroup has a normalizer which is a maximal subgroup. In particular we show that for such a p-group G, when p = 2, the center of G has index at most 16 and when p is odd the center of G has index at most p 3 . Introduction In this paper, and the one to follow, we will examine groups in which the normalizer of every non-normal subgroup is a maximal subgroup. We call such a group an MN-group. This paper will examine p-groups. The second paper will examine all other MN-groups. The principal result of this paper is the following: Theorem 1.1. Let G be a p-group in which every non-normal subgroup has p conjugates. Then [G : Z(G)] ≤ 16 if p = 2, p 3 if p > 2. If H ≤ G has at most p conjugates, then [G : N G (H)] ≤ p, therefore G is an MNgroup. Also, as Φ(G) = G ′ G p , every p-th power must normalize every subgroup of G. In Section 2 we provide some of the background material and preliminary results. In Section 3, we examine p-groups with element breadth 1. In Section 4, we examine 2-groups with element breadth 2. In Section 5, we examine p-groups with element breadth 2 for odd p. In Section 6, we make a conjecture that would generalize 1.1 to groups with subgroup breadth k. In Section 7, we provide details about our use of GAP to solve this problem. Preliminary Results and Definitions All groups in this paper will be finite. Recall that a group is Hamiltonian if every subgroup is normal. The following theorem of Dedekind is the starting point for this paper: Theorem 2.1. Let G be a Hamiltonian group. Then G is either abelian, or G ∼ = Q 8 × (Z 2 ) n × A where A is an abelian group of odd order. Proof. See [15]. We begin with some definitions: 2000 Mathematics Subject Classification. 20D15. The author would like to thank his advisor, John Shareshian, as well as Yakov Berkovich and Eamonn O'Brien for all of their helpful comments on this paper. Definition 2.2. The element breadth of an element x of a p-group G, ebr(x), is defined to be the integer such that [G : C G (x)] = p ebr(x) . The element breadth of G, ebr(G), is the maximum value that ebr(x) takes over all the elements of G. It should be noted that "element breadth" is a non-standard term (breadth is the standard term). We use the term to distinguish between the previous definition and the following one. The cyclic breadth of G, cbr(G) is the maximum value that sbr(H) takes over all cyclic subgroups of G. As an example, consider an extra-special 2-group G of plus-type of order 2 2m+1 . Clearly every non-normal subgroup must be elementary abelian of order at most 2 m . Let H be a non-normal subgroup of order 2 m generated by non-central involutions x 1 , · · · , x m . Then C G (H) = m i=1 C G (x i ) has index at most 2 m so that sbr(G) ≤ m. As G is of plus-type, we have generators i 1 , j 1 , · · · , i m , j m such that i k , j k ∼ = Q 8 for 1 ≤ k ≤ m, each such Q 8 commutes with every other Q 8 and the −1 from each of the Q 8 's have been identified. In particular, a non-central involution of G is a product of involutions of the form a s b t where s = t, a s is one of i s , j s or k s and b s is one of i t , j t or k t . Consider the subgroup K = i 1 i 2 , j 1 j 2 · · · , i 2m−1 i 2m , j 2m−1 j 2m . This group is elementary abelian and is clearly non-normal. Moreover, if some element g normalizes K it cannot conjugate the element i s i t to j s j t or k s k t therefore N G (K) = C G (K). It is easily seen that the centralizers of the generators of K are mutually distinct, therefore [G : N G (K)] = p m so that sbr(G) ≥ m, therefore sbr(G) = m. It can similarly be shown that an extra-special group of minus-type has subgroup breadth m − 1. Note also that every cyclic subgroup of order more than 4 must contain the center, therefore must be normal, therefore cbr(G) = ebr(G) = 1. Therefore cyclic breadth has no influence on subgroup breadth. We will need use of GAP [6] as well. In particular, we will use the notation [n, m] to be group m of order n in the Small Group Library [3]. (It should be noted that the Small Group Library is not specific to GAP.) We access the library and use the GAP functions ConjugacyClasses and ConjugacyClassesSubgroups to determine the element and subgroup breadths of a p-group. We will repeatedly use the results of section 7 to show that certain groups involved in minimal counterexamples do not have subgroup breadth 1. Finally, it should be noted that some of the groups that we must construct in GAP are constructed as finitely-presented groups, therefore we use the GAP function IsomorphismPcGroup to speed up necessary computations over what can be expected in a finitely presented group. For p = 2, 1.1 was first posed as a conjecture in [10]. In that paper it was shown that condition (TC), that is sbr(G) = 1, is equivalent to each of the following two conditions: (CO) The core H G of every subgroup H of G "requires" all the conjugates of H, in the sense that the intersection of a proper subset of the set of distinct conjugates of H properly contains H G . (NC) The normal closure H G of every subgroup H of G "requires" all the conjugates of H, in the sense that the subgroup generated by a proper subset of the set of distinct conjugates of H is a proper subgroup of H G . The question of whether such a bound on the index of the center exists or not is found in [2] as suggested research problem 830. In [11] it is shown that [G : Z(G)] < p 81sbr(G)(log 2 (p sbr(G) 2 )) , therefore the results in this paper are a great improvement over this result in the case that sbr(G) = 1. The central product of Q 8 and D 8 has subgroup breadth 1 and a center of index 16, so the bound is the best possible when p = 2. For p = 3, the group a, b, c | a p 3 = b p = c p = 1, [a, b] = a p 2 , [a, c] = a p 2 b, [b, c] = 1 shows the bound is sharp. For p > 3 the group a, b, c | a p 2 = b p = c p = 1, [a, b] = a p , [a, c] = a p b, [b, c] = 1 shows the bound is sharp. The following result bounding the element breadth of a p-group with given cyclic breadth [5]. Proposition 2.4. If G is a p-group, then ebr(G) ≤ 2cbr(G) + 1 if p = 2, 2cbr(G) if p > 2. In particular, this says that ebr(G) ≤ 3 when sbr(G) = 1. Proposition 2.6. [14] If G is a p-group, then ebr(G) = 2 if and only if one of the following holds: (1) |G ′ | = p 2 or (2) [G : Z(G)] = p 3 and |G ′ | = p 3 . The following can be found in [4]: Proposition 2.7. Let G be a p-group in which the intersection of all the nonnormal subgroups is non-trivial. Then p = 2 and G is isomorphic to one of the following: (1) C ∼ = Q 8 × Z 4 × (Z 2 ) k . (2) C ∼ = Q 8 × Q 8 × (Z 2 ) k . Here Q 8 × Q 8 does not have subgroup breadth 1. (3) C = g, A with A abelian but not elementary abelian, 1 = g 2 ∈ A and a g = a −1 . The first reduction we make is the following. Theorem 2.8. If sbr(G) = 1 then ebr(G) ≤ 2. We give several results about metacyclic groups and determine which metacyclic groups have subgroup breadth 1. The following may be found in [9]. Lemma 2.9. If G is a metacyclic p-group, then G has a presentation of the form a, b | a p m = 1, b p n = a k , a b = a r where m, n ≥ 0, 0 < r, k < p m , p m | k(r − 1), and p m | r p n − 1. Up to picking different generators, we may assume that k = 0 or p j . Proof. The first statement is proved in [9]. For the second statement, if k = 0, we may obtain the result by replacing a or b with powers of a or b. Proof. By 2.6 we know that G ′ has size either 4 or 8 and we can write G = a, b | a 2 m = 1, b 2 n = a 2 m−ℓ , [a, b] = a m−2 or G = a, b | a 2 m = 1, b 2 n = a 2 m−ℓ , [a, b] = a m−3 . We assume that ℓ ≥ 3. Consider the first case. Note that we have that b is also normal in G, so without loss of generality, we may assume that |a| ≥ |b|. Note that |G| = 2 m+n . Now, a 4 and b 4 are central, therefore if m ≥ 5, b −4 a 2 m−3 ⊳ G. Quotienting out by this subgroup, we get a corresponding metacyclic group with n = 2 and ℓ = 3. When m = 5, consider the subgroup a 3 b and when m > 5 consider the subgroup a −2 n−4 b . These groups both have order 4, and intersect a trivially, therefore by picking different generators, we may assume that G is a split extension Z 2 m ⋊ Z 4 . Letting c be the generator of Z 4 , we still have that [a, c] = a ±2 m−2 . In this group c has four conjugates. Therefore, we may assume that m = 4 (when m = 3, G is clearly a split extension), hence ℓ = 3. It is easily verified, in GAP, that the groups with the given presentations for n = 2, 3 or 4, have element breadth 1. Consider the second case. Again, b is normal, so we assume |a| ≥ |b|. Also, b −8 a 2 m−3 is normal in G if m ≥ 6. When m = 6 consider the subgroup a 3 b and when m > 6 consider the subgroup a −2 m−4 b . As before, these groups intersect a trivially, therefore, we can write G as a semi-direct product Z 2 m ⋊ Z 8 . However, if c is a generator of Z 8 , then c has more than two conjugates. Therefore m < 6. If m = 5, suppose ℓ = 3. When n = 1, G has element breadth 1. When n = 2 we get [64,28] and when n = 3 we get [128,130] neither of which have subgroup breadth 1 by 7.2 and 7.3. It is easily checked that when ℓ = 4 that G has element breadth 1 in all possible cases. When m = 4, similarly, all remaining groups have element breadth 1. When m = 3, G is cyclic, therefore we are done. Now, we prove some structure theorems regarding metacyclic 2-groups with subgroup breadth 1. Lemma 2.11. Let G be a metacyclic p-group with the presentation as in 2.9. Then we have (ba i ) j = b j a i(1+r+r 2 +···+r j−1 ) Proof. By induction. When j = 1 this is obvious. Now suppose we have the statement for j − 1. Then (ba i ) j = (ba i ) j−1 ba i = b j−1 a i(1+r+r 2 +···+r j−2 ) ba i = b j a ri(1+r+r 2 +···+r j−2 ) a i = b j a i(1+r+r 2 +···+r j−1 ) . We prove some results about 2-groups. Theorem 2.12. If a 2-group G is metacyclic, non-abelian and sbr(G) = ebr(G) = 1 then with the notation from 2.9 we have r = 2 m−1 + 1. Proof. We have [a, b] = a r−1 . Since ebr(G) = 1, by 2.5 we must have that a r−1 has order 2. This says that 2r ≡ 2 (mod 2 m ), which says that r = 2 m−1 + 1 (r = 1 since G is non-abelian). Theorem 2.13. If a 2-group G is metacyclic, sbr(G) = 1, and ebr(G) = 2 then with the notation as above, there is some n such that G = a, b | a 8 = b 2 n = 1, a 4 = b 2 n−1 , [a, b] = a ±2 . Proof. Note first that G ′ = [a, b] . This is true because clearly a 2 ⊳ G and G/ a 2 is abelian, therefore G ′ ≤ a 2 and no proper subgroup can have an abelian quotient. Suppose that a ∩ b = 1. If a has order 2, then a and b commute and clearly G is abelian. Consider the subgroup b . We have b a = ba −r+1 and b a 2 = ba −2r+2 . Hence we must have that b = ba −2r+2 . In this case, r = 2 m−1 +1 so G has element breadth 1 by 2.5, because G ′ has order 2. This shows that a and b intersect non-trivially. We showed in 2.10 that we cannot have that |G ′ | = 8. Therefore G ′ must have order 4, and we may assume that r = 2 m−2 + 1 or 3 · 2 m−2 + 1. Suppose first that b = b a = ba −r+1 . Then a 2 m−2 = b ±2 n by 2.10. We must examine these groups for some small values of m and n first. Suppose that m = 4 so that r = 5 or 13 and we have a 4 = b ±2 n . When n = 2, G=[64,28] and when n = 3, G =[128,130], neither of which have subgroup breadth 1 by 7.2 and 7.3. Both of these groups also have a unique normal subgroup of order 2 generated by a 8 , hence no quotient groups satisfy our hypothesis that a has order more than 8. So we may assume that n ≥ 4. Now, since H = H b 2 we have that a 8 = b 2 n−1 is a power of b 2 n−2 a ∓1 which has order 4 (throughout this proof the use of ± and ∓ indicates that one must consistently pick the sign on the top or the sign on the bottom). Since a 8 has order 2, we must have (b 2 n−2 a ∓1 ) 2 = b 2 n−1 a ∓2 = a 8 . This says that a 2 is a power of b. However (a 2 ) b = a 10 which says that a 2 = a 10 and a has order 8, a contradiction. Therefore m = 4. Now, assume that m > 4. We proceed by induction on the size of G. Given the relations for G it is easily verified that a 4 and b 4 are central. Moreover, a 2 m−2 and a 2 m−3 b 2 n−1 are two central subgroups of order 4, therefore Z(G) is not cyclic. Therefore, there is some involution z which is not a commutator. We get that G = G/ z is a metacyclic group with element breadth 2. By induction the order of a in this quotient group must be 8. This says that a has order 8 or 16, contradicting our assumption. Hence we must have that n = 1 or 2. If n = 1, then b 2 commutes with a. However a b 2 = a 2 m−1 +1 . This says that a has order 2 m−1 and we get the result by induction. Now, we assume that n = 2. We claim that this group is actually a split extension isomorphic to Z 2 m ⋊ Z 4 . Let a be the subgroup isomorphic to Z 2 m . Consider the subgroup a 2 m−4 b ∓1 . When m = 5 we again have that G is a quotient of [128,130], so we assume that m ≥ 6. Then a 2 m−4 and b commute, hence (a 2 m−4 b ∓1 ) 4 = a 2 m−2 b ∓4 = 1. This contradicts our earlier statement that the generators of a metacyclic group with element breadth 2 and subgroup breadth 1 generate subgroups with non-trivial intersection. Therefore, we may assume that m = 3. So G has a presentation of the form a, b | a 8 = 1, b 2 n = a 2 k , [a, b] = a ±2 . Note that such a group has a unique central involution, namely a 4 , hence no quotient groups still satisfy the requirement that a has order 8. If k = 3, the subgroup generated by b has four conjugates. If k = 1, note that 1 = [b 2 n , b] = [a 2 , b] = a 4 . So we may assume that k = 2, which gives the lemma. By a result in [5], if G is a metacyclic 2-group, then ebr(G) ≤ sbr(G) + 1, therefore there are no metacyclic groups with element breadth 3 and subgroup breadth 1. We remark that in the element breadth 2 case, the conjugacy class of b is the only one with four elements. Also the subgroup b has two conjugates. Recall that the only non-trivial automorphisms of order 2 of a cyclic group of order 2 n send a generator a to a −1 , a 2 n−1 −1 or a 2 n−1 +1 . We call these automorphisms dihedral, semi-dihedral and modular, respectively. We can use the above results to show the following. Theorem 2.14. If G is a 2-group with ebr(G) = 3 then sbr(G) = 1. Proof. Let a be an element with 8 conjugates. We aim to show that a has order 8. Let A = a and H = N G ( a ). Suppose that a has order 2 n where n > 3. Then Aut(A) ∼ = Z 2 n−2 × Z 2 where the first factor is the automorphism a → a 5 and the second factor is the automorphism a → a −1 and we get a homomorphism φ : H → Aut(A). If φ(H) contains a dihedral automorphism, let t be a preimage. Then consider a, t and the subgroup K = t . Then K = K a 2 = ta −4 . Therefore a 4 ∈ t , so a has order 16 by 2.10. However, in this case [a, t] has order 8, a contradiction. Similarly, if φ(H) contains a semi-dihedral automorphism, we again get that a 4 ∈ t . Therefore φ(H) is cyclic. Let t be a pre-image of a generator. If H = G then we get that a, t must contain all eight conjugates of a, however, this group is metacyclic, a contradiction. Therefore [G : H] = 2. Let C = C G (a). Then |H/C| = 4. If H/C is cyclic, then let t be the pre-image of a generator. Again a, t is metacyclic and since it has element breadth 2, a would have order 8. However, this is impossible since Aut(A) has no elements of order 4. Therefore N/C ∼ = Z 2 × Z 2 . Let b and c be pre-images of the generators of this group. We may assume that b induces a dihedral automorphism and c induces a semi-dihedral automorphism. Now, b 2 and c 4 are central elements. Then a, b has element breadth at least 2, hence a must have order 8 and a 4 = b 2 n and similarly a 4 = c 2 ℓ . Suppose first that b does not have eight conjugates. In a, b , b already has four conjugates, hence b c must be one of these, that is, [b, c] = a 2i where i = 0, 1, 2 or 3. Note that this implies that b 2 and c 4 are central elements. Suppose first that m ≥ 3. We look at the subgroup H = ab 2 m−2 . Conjugating by b and c we get H b = a −1 b 2 m−2 and H c = a 3 b 2 m−2 respectively. If H = H b or H = H c then a 2 is a power of ab 2 m−2 which has order 4. This implies that a is a power of b, a contradiction. If H b = H c then a 4 is a power of ab 2 m−2 . Since a 4 has order 2, this implies that a 4 = a 2 b 2 m−1 which is also impossible by the structure of a metacyclic group with element breadth 2 and subgroup breadth 1. This shows that if b has 4 conjugates, it must have order at most 8. Similarly, c has order at most 16. Note that since b has two conjugates in a, b , we must have that a, b ⊳ a, b, c . Therefore | a, b, c | ≤ 256 and has element breadth 3. By 7.4 there are no such groups. This completes the proof. Next we prove two unpublished results of John Shareshian. Proof. Let G be a 2-group and let s and t be involutions such that [s, t] = 1. Then s, t is a dihedral group. Since D 2 n does not have subgroup breadth 1 for n ≥ 4, we clearly get s, t = D ∼ = D 8 . Since s and t already have two conjugates in D, we must have D ⊳ G. Let C = C G (D). Then G = CD (see, for instance, 4.17 in [16]). Let z generate Z(D). We must have C ∩ D ≤ z . If H < C is not normal in C, then H must contain z (otherwise if H = K and H is conjugate to K, the four groups t, H , t, K , tz, H and tz, K are distinct and conjugate). If C has no non-normal subgroup, then C is Hamiltonian and it is straightforward that the proposition holds. Otherwise, by 2.7, one of the following occurs: (1) C ∼ = Q 8 × Z 4 × (Z 2 ) k . It is straightforward to verify that the proposition holds in this case. (2) C ∼ = Q 8 × Q 8 × (Z 2 ) k . Here Q 8 × Q 8 does not have subgroup breadth 1. (3) C = g, A with A abelian but not elementary abelian, 1 = g 2 ∈ A and a g = a −1 . Since g both centralizes and inverts g 2 , g has order 4 and g 2 = z. Suppose some a ∈ A has order eight. If g 2 = a 4 then the involutions stg, stga 2 , tsg and tsga 2 are distinct and conjugate, a contradiction. Next suppose there is some b ∈ A such that b has order 4 and b 2 = g 2 . Then g b = gb 2 , so stg, stgb 2 , tsg and tsgb 2 are distinct and conjugate, a contradiction. Therefore A is the direct product of a cyclic group of order 4 and an elementary abelian group. Therefore [C : Z(C)] = 4 and it is straightforward to verify the proposition. Proof. Let z be the unique element of order 2 in Z(G) and let t = z have order 2 in G. Let t ′ = t be a conjugate of t. Since [G : C G (t)] = 2 we have C G (t) ⊳ G, so C G (t ′ ) = C G (t). Therefore if g ∈ C G (t), (tt ′ ) g = tt ′ , while if g ∈ G − C G (t), then (tt ′ ) g = t ′ t = tt ′ . Thus tt ′ ∈ Z(G) and since t and t ′ commute, we have |tt ′ | = 2, so t ′ = tz. Suppose there is some involution s in G besides t, t ′ and z. Since s and t commute, |st| = 2 and st / ∈ {z, t, t ′ }. If C G (s) = C G (t), let g ∈ G − C G (t). Then (st) g = sztz = st, so C G (st) = C G (t). Then s, t , s, tz , sz, t and sz, tz are distinct and conjugate, a contradiction. The result of Blackburn (2.7) will be used multiple times. We examine the groups in case 3 in more detail. Since g 2 ∈ A any element of this group not in A has the form ga where a ∈ A. As mentioned above, g must have order 4. Also, g 2 is central. Now, every involution in A must centralize g, therefore so does Ω 1 (A). We must have that [A : Ω 1 (A)] ≤ 4. If A = Ω 1 (A) then A is elementary abelian and C is abelian. Therefore, A is either isomorphic to Z 8 × (Z 2 ) n or Z 4 × Z 4 × (Z 2 ) n . Consider the former case. If g 2 does not lie in Z 8 , then letting t be a generator of Z 8 , g, t is [32,14] which does not have subgroup breadth 1 by 7.1. Otherwise, g, t is isomorphic to Q 16 and therefore contains a subgroup isomorphic to Q 8 . Consider the latter case. If g 2 is not the square of an element of A of order 4, then C contains a subgroup isomorphic to Z 4 × Z 4 . This group contains quotients isomorphic to Q 8 and D 8 . If g 2 is the square of an element of order 4 then C contains a subgroup isomorphic to Q 8 . Suppose that we have some a ∈ C of order 8. If g 2 = a 4 then in g, a we have g , ga 2 , ga 4 and ga 6 are conjugate. If C ≥ Z 8 × Z 8 , one of the generators of the direct factors of Z 8 × Z 8 cannot be equal to g 2 . So we can assume that C ∼ = Z 2 i ×(Z 4 ) m ×(Z 2 ) n or C ∼ = (Z 4 ) m ×(Z 2 ) n . Suppose now that g 2 is the generator of a Z 2 . (Clearly if g 2 is not a square in C, we can always pick a presentation for C such that g 2 is such a generator.) Let a and b be elements such that |a|, |b| > 2 and a ∩ b = 1. Then in g, a, b , consider the subgroup H = g . We have H a = ga 2 and H b = gb 2 . Since g 2 cannot be an element of one of the non-Z 2 factors, these two groups are clearly distinct from H and from each other. This says that we may assume that C ∼ = Z 2 i × (Z 2 ) n . Also, if g 2 is the square of some element of order 4, then B contains a quaternion subgroup. However, only Q 8 and Q 16 have subgroup breadth 1, which says that i = 2 or i = 3. Hence, in this case, B contains a subgroup isomorphic to Q 8 . Otherwise, since g inverts the element of order 2 i we have a subgroup isomorphic to a, b | a 2 i = b 4 = 1, [a, b] = a 2 . If i > 2, the subgroup generated by b has at least four conjugates, hence we may assume that B ∼ = Z 4 ⋊ Z 4 (where the action is irreducible). By quotienting out by b 2 we get a quotient group isomorphic to D 8 and by quotienting out by a 2 b 2 we get a quotient group isomorphic to Q 8 . 3. p-groups with element breadth 1 By 2.5 we know that |G ′ | = p. We first state two results from [1]. Recall that a minimal non-abelian group is a group all of whose proper subgroups are abelian. Theorem 3.1. [1] (1.18a) A minimal non-abelian p-group is isomorphic to one of the following: ( Using these results we will show the following: 1) Q 8 ,(2)P i,j = a, b | a p i = b p j = 1, [a, b] = a p i−1 , i ≥ 2, j ≥ 1,(3)P i,1,k = a, b, c | a p i = b p = c p k , [a, b] = [b, c] = 1, [a, c] = b , i + j > 2.(Z 2 ) n × Q 8 * D 8 ,(1) (Z 2 ) n × Q 8 * P 2,1,1 . Proof. Let G be a minimal counterexample to the theorem. That is, G is a pgroup with [G : Z(G)] > p 2 that is not isomorphic to one of the above groups. By minimality of G we may assume that G = A 1 * A 2 · · · A k where each A i is minimal non-abelian. Consider the group A * B where A and B are minimal non-abelian. Let z be the generator of G ′ so that z p = 1. We show that we may assume that every non-normal subgroup of either A or B contains z. Suppose not and suppose that A has a non-normal subgroup H and let K 1 , · · · , K p−1 be its distinct conjugates in A. We choose H so that H does not contain z. Let t be a non-central element of B and assume we can pick t such that z / ∈ t . Then there is some b ∈ B such that [t, b] = z, therefore t and tz are conjugate. Clearly the groups H, t , H, tz and K i , t are all distinct and conjugate, a contradiction. If no such t exists, then every non-normal cyclic subgroup, hence every non-normal subgroup of B contains z. So either A or B has the given property and we may assume that p = 2. Without loss of generality, say B does. By 2.7 we have three possibilities for B: ( 1) B ∼ = Q 8 × Z 4 × (Z 2 ) n ,(2)B ∼ = Q 8 × Q 8 × (Z 2 ) n ,(3) B = g, C with C abelian but not elementary abelian, 1 = g 2 ∈ C and a g = a −1 for all a ∈ C. The first two are clearly not minimal non-abelian. By our discussion in the last section, we may assume that B ∼ = P 2,2 . Note that B, hence G, has a normal subgroup N of order 2 besides z such that B/N ∼ = Q 8 . Therefore consider G/N . By minimality of G, we get that AN/N ∼ = D 8 or P 2,1,1 . Since A cannot contain N , we get that A must be one of these two groups. We get the groups [64,201] and [128,1006] neither of which have subgroup breadth 1 by 7.2 and 7.3. Therefore, we may assume that at least one of A or B is Hamiltonian, so assume that A is. Note that the only minimal non-abelian group is Q 8 . A central product of Q 8 with Q 8 is [32,49] and does not have subgroup breadth 1 by 7.1, hence we only must show that Q 8 * B does not have subgroup breadth 1 when B ∼ = P i,j or B ∼ = P i,1,k when (i, 1, k) = (2, 1, 1) or (1, 1, 2). (We note that the groups corresponding to (2,1,1) and (1,1,2) are isomorphic.) Consider the first case so that G ∼ = Q 8 * P i,j . We do not use the standard generators for Q 8 as we have already used the variables i, j and k, therefore we use x, y and z = xy in place of them. Suppose first that i > 2. Now, B has three involutions: a 2 i−1 , b 2 j−1 and a 2 i−1 b 2 j−1 . Since we require that the common involution is also a commutator, we must have that a 2 i−1 = −1. If b has order at least 4, then b 2 is central, hence we may quotient out by b 2 j−1 . Hence we may assume that B is a modular group of order at least 16. So G has a presentation as follows: x, y, a, b | x 4 = 1, x 2 = y 2 , [x, y] = x 2 = −1, a 2 i = 1, b 2 = 1, [a, b] = a 2 i−1 , [x, a] = [x, b] = [y, a] = [y, b], x 2 = a 2 i−1 . If i > 2, we claim that the subgroup H = b, xa 2 i−2 has more than two conjugates. We have H y = b, −xa 2 i−2 and H a = ba 2 i−1 , xa 2 i−2 . Suppose that H = H y . Since both xa 2 i−2 and −xa 2 i−2 are elements of H, we must have that −1 ∈ H. However, H has order 4 and −1 is not the product of the two generators. Now suppose that H = H a . This says that a 2 i−1 ∈ H, which is also impossible. Finally if H y = H a then a 2 i−1 ∈ H y which is also impossible. This proves that we may assume that i = 2. If j ≥ 2 we may quotient out by the central subgroup b 4 and G has a section isomorphic to Q 8 * P 2,2 . This is group 201 of order 64, which does not have subgroup breadth 1 by 7.2. This says that we may assume that i = 2 and j = 1 so that G = Q 8 * D 8 . Next we suppose that B is isomorphic to P i,1,k . Suppose that both i and k are at least 2. Quotienting out by a 4 and c 4 , G must have a section isomorphic to Q 8 * P 2,1,2 . This is group 1008 of order 128, so by 7.3 we may assume that one of i or k is 1. Note also, that structure-wise, the roles of i and k are symmetric, hence we might as well assume that k = 1. Now suppose that i > 2. We quotient out by a 8 so that G has a section isomorphic to Q 8 * P 3,1,1 ; however, this is group 1714 of order 128 which does not have subgroup breadth 1 by 7.3. Hence G = Q 8 * P 2,1,1 . This shows that only in the two exceptional cases of the statement of the theorem can k > 1. We can easily verify that when k = 1, 2-Groups with element breadth 2. In this chapter all groups will be 2-groups. Let G be a group with element breadth 2 and subgroup breadth 1. We aim to show that [G : Z(G)] ≤ 16. We first note that by 2.6 in a minimal counterexample we must have that |G ′ | = 4. In this section, G will be a 2-group with element breadth 2 that is a minimal counterexample to 1.1. Proof. Suppose that z is a central involution that is not a commutator. Since G is a minimal counterexample, the center Z of G/ z has index at most 16. We then have that [G, Z] ≤ z . Since z is not a commutator, we must have that Z is central in G, contradicting that G is a counterexample. Corollary 4.2. G has at most three central involutions. Proof. This follows immediately from |G ′ | = 4 Now let z be a central involution and let π be the natural homomorphism from G to G = G/ z . For g ∈ G, g ∈ Z(G) if and only if [G, g] ≤ z . Suppose that there is no g ∈ G such that [G, g] = z . Then π −1 (Z(G)) = Z(G) and since z ∈ Z(G) we get that [G : Z(G)] = [G : Z(G)] ≤ 16, a contradiction. Therefore there is some element g ∈ G with [G, g] = z . This implies the only conjugates of g are g and gz, so C = C G (g) has index 2 in G. Define X(g) = {h ∈ G | [G, h] = t , C G (h) = C} Using the formula that [a, bc] = [a, c][a, b] c , we see that H(g) = Z(G) ∪ X(g) is a subgroup of C. We also see that if a, b ∈ X(G) then ab ∈ Z(G). Therefore [H(G) : Z(G)] = 2. Since z ∈ C, H(g) = π −1 (C ∩ Z(G)). This implies that |Z(G)| ≥ |Z(G)| 2 . Therefore if [G : Z(G)] ≤ 4 , [G : Z(G)] ≤ 16. Since ebr(G)=1 we get that G must be isomorphic to (Z 2 ) n × Q 8 * D 8 or (Z 2 ) n × Q 8 * P 2,1,1 . For both of these groups let A be the pre-image under π of (Z 2 ) n and let B be the pre-image of the second factor. We use GAP to determine what B can be. We also note that 200, 238 and 245 are the only ones of these that have no subgroup isomorphic to D 8 . Proof. Using GAP, we obtain the following groups of order 128 that are extensions of Q 8 * P 2, 1,1 : 1006, 1008, 1042, 1045, 1048, 1052, 1055, 1059, 1063, 1064, 1068, 1072, 1076, 1083, 1088 1094, 1097, 1103, 1110, 1113, 1114, 1714, 1715, 1716, 1717, 1718, 1719, 2158. By 7.3 the only groups from this list that have subgroup breadth 1 are 1716 and 2158 and both of these have center of index at most 16. Now, since A ′ = 1 we have that A must have element breadth at most 1. Next we consider the possible structures of A. By 3.3, if A is non-abelian, A can either be written as (Z 2 ) n × (Q 8 * D 8 ), (Z 2 ) n × (Q 8 * P 2,1,1 ) or as CZ(A) where C is minimal non-abelian. However A must be elementary abelian. The first case contains dihedral subgroups, which is impossible by 2.15. In the second case A is clearly not elementary abelian as A has rank 4. For the third case, Q 8 is the only possibility for C such that A is elementary abelian. Now Z(A) can have elements of order at most 4. Let t ∈ Z(A) such that t 4 = 1. If t 4 ∈ Q 8 it is easily verified that Q 8 , t ∼ = Q 8 × Z 2 . As z ∈ Q 8 we may, therefore assume that A ∼ = Q 8 × D where D is elementary abelian or that A is abelian with A elementary abelian. Consider the first case. Let t ∈ D. Then the group tB has a center of index at most 16. Also, by 4.1, t is not central. We note that Z(tB) = C B (t) ∩ Z(B). For all of the possible groups B except [128,2158], Z(B) ≤ Φ(B). As C B (t) is maximal in B, we have Z(tB) = Z(B). Therefore tB has a center of index 32, a contradiction. We may, therefore assume that A ∼ = Q 8 in this case. If A is abelian, the same argument shows that A ∼ = Z 4 or Z 2 . This shows that |G| ≤ 512. By 7.1, 7.2, 7.3 and 7.4, we may assume that |G| = 512 so that A ∼ = Q 8 and B is either [128,1716] or [128,2158]. Both contain a subgroup isomorphic to Q 8 , therefore if [A, B] = 1 then G contains a subgroup isomorphic to either Q 8 * Q 8 or Q 8 × Q 8 , neither of which have subgroup breadth 1. Therefore [A, B] = z . If B=[128,1716] we can verify in GAP that this group has the structure (Z 8 × Q 8 ) ⋊ Z 2 and presentation: In this presentation a 4 = z. Therefore, we can find some group of order 256 of the form A(Z 8 × Q 8 ). Using 7.4 we can check that only groups 6648, 26461, 26462, 53175 and 53232 have element breadth 2, subgroup breadth 1 and normal subgroup isomorphic to Z 8 × Q 8 and none of these groups has a normal subgroup isomorphic to Q 8 . (Note that a group with element breadth 1 cannot have this structure by 3.3.) Therefore, we may assume that B is [128,2158] which is isomorphic to a, b, c, d | a 8 = b 4 = d 2 = 1, c 2 = [b, c] = b 2 ,Z 2 × Q 8 * ((Z 4 × Z 2 ) ⋊ Z 2 ). Consider the subgroup iB. It is easily shown that if C = (Z 4 × Z 2 ) ⋊ Z 2 that iB ∼ = Z 4 C. We verify in GAP that no group in 7.4 has this structure. This completes the proof for 2-groups. p-groups with element breadth 2 for odd p Our proof when p is an odd prime will differ greatly from that for p = 2, both in format and in length. There are several underlying reasons for this disparity. It can be shown that every irreducible representation of a 2-group with subgroup breadth 1 has degree at most 4. The tensor product of the respective 2-dimensional irreducible representations of Q 8 and D 8 shows that this bound is sharp for Q 8 * D 8 . Much can be said about groups all of whose irreducible representations have degree dividing p 2 ; however a proof using these facts would likely be fairly complex. By a theorem of Isaacs, much more can be said when all non-linear irreducible representations have degree p (regardless of parity): (1) G has a maximal subgroup which is abelian, (2) [G : Z(G)] = p 3 . By adding a hypothesis, we can show that (1) implies (2). Proof. Let |G| = p n . By Lemma 12.12 in [8], if A is an abelian subgroup with G/A cyclic, we have |A| = |G ′ ||A ∩ Z(G)|. By 2.6 we may assume that |G ′ | = p 2 . Then we have [A : A ∩ Z(G)] = p 2 . This clearly implies that Z(G) has index at most p 3 . Therefore G cannot have an abelian maximal subgroup. We now use a series of smaller results to prove our main theorem: Proof. Let G be a minimal counterexample to the claim that sbr(G) = 1 and [G : Z(G)] ≤ p 3 . By 5.1 and 5.2, G is also a minimal counterexample to the claim that every non-linear irreducible representation of a group with subgroup breadth 1 has degree at most p. Let φ be a representation of degree p i where i > 1. By 3.3, we may assume that G has element breadth 2, and by 2.6 we may assume that |G ′ | = p 2 . (1) Z(G) is cyclic. Proof. Suppose not. By Schur's lemma, the image of any irreducible representation has a cyclic center, therefore φ is not faithful. Hence, G/ker(φ) also has an irreducible representation of degree p i ; however all quotient groups of G must also have subgroup breadth 1, therefore this is a smaller counterexample. (2) Let X = x be a subgroup of G of order p. Then X commutes with all of its G-conjugates. Proof. Since N G (X)/C G (X) is a subgroup of Aut(X) ∼ = Z p−1 , we have N G (X) = C G (X). Also for g ∈ G, we have N G (X g ) = N G (X) g = N G (X) = C G (X) = C G (x) and clearly x g normalizes X g . (The second equality follows from the fact that N G (X) is a maximal subgroup of a nilpotent group, hence is normal.) Therefore, either X ⊳ G or N G (X) is a maximal subgroup of G. (3) If Y = y is another subgroup of G of order p then [X, Y ] = 1. Proof. Suppose not. Then X has p Y -conjugates, call them X 1 , · · · , X p . Since G has subgroup breadth 1, these are all of the G-conjugates of X. Hence N = X 1 , · · · , X p ⊳ G. By (2), N is elementary abelian, of order p k for some k > 0. Let C = C G (N ). Since C G (X g ) = N G (X g ) = N G (X) g = N G (X) = C G (X) we have that C = C G (X). Since [X, Y ] = 1, we have Y is not contained in C, hence G = CY . Therefore, any element of N that centralizes Y is central in G. However, since Z(G) is cyclic, we must have that C N (Y ) has order p. We have G = Y C. Then let y 1 , y 2 ∈ Y , c ∈ C and n ∈ N . Then . Also, since H = C G (Y ) ∩ C G (N ) has index at most p 2 and |Y N | = p 3 we get G = Y N H. Therefore, G is a central product of Y N with H. Now, since Y N has non-normal subgroups if H has non-normal subgroups, they must all contain Z(Y N ). However, by 2.7 this only is possible when p = 2. Therefore, H is abelian. Since X H, and X ≤ Y N , we have HX is also abelian, and, having index p, produces a contradiction. Since Z(G) is cyclic, |Z| = p. If O = Z then G has a unique subgroup of order p, hence is cyclic. So we may assume that Z = O. Let X and Y be subgroups of O of order p both not equal to Z. If N G (X) = N G (Y ) then X × Y has at least p 2 conjugates (neither X nor Y can be normal because then X or Y is central). Therefore, all non-central subgroups of G have the same centralizer, so C = C G (O) = C G (X), so [G : C] = p. Therefore, G acts on O as a linear group L of order p. Let g generate L. If g is in Jordan form with respect to some basis for O then, since C O (g) is contained in Z(G), we have C O (G) = Z. Therefore, g has exactly one Jordan block. Also, this means that if x ∈ G − C then C O (x) = Z. Since G has element breadth 2, we have [O : Z] ≤ p 2 , hence |O| ≤ p 3 . By a result of [12], if |Ω 1 (G)| = p 2 then G is either metacyclic or a 3-group of maximal class. If G is a metacyclic group, [5] says that G must have element breadth 1; however, our classification of such groups shows that this implies [G : Z(G)] = p 2 . If G is a 3-group of maximal class, this implies that [G : G ′ ] = p 2 which implies that G has order p 4 . Since p-groups have non-trivial centers, we clearly have [G : Z(G)] ≤ p 3 , a contradiction. Therefore, O has order p 3 . We now complete the proof of the theorem. Let C = C G (O) and let h ∈ G − C. Let H = h and N = N G (H). As above, h acts with one Jordan block on O, therefore we can find x ∈ O such that x / ∈ N . Let X = x . Therefore, G is a semidirect product of X and N , and we have |Ω 1 (N )| = p 2 . (If |Ω 1 (N )| = p we get that N is cyclic, therefore G is metacyclic and cannot have element breadth 2.) By the result above, N is either metacyclic or a 3-group of maximal class. If N is metacyclic by [5] N has element breadth 1 and by 3.3 we have that N = a, b | a p n−1 = b p = 1, [a, b] = a p n−2 . Therefore, Φ(N ) = Z(N ). Now, since ebr(N )=1, we have that C N (H) has index p, hence is maximal and must therefore contain Z(N ). This shows that Z(N ) = Z(G). Hence We note that the proof of 5.3 does not necessarily imply that every p-group with element breadth 2 and subgroup breadth 1 has an abelian maximal subgroup since we were only examining the structure of a minimal counterexample to the theorem. However, we have no examples of p-groups for odd primes p with subgroup breadth 1 which do not have an abelian maximal subgroup. 6. p-groups with subgroup breadth more than one We conclude with a conjecture regarding p-groups with subgroup breadth more than one: Conjecture 6.1. Let G be a p-group with subgroup breadth k. Then |G : Z(G)| ≤ 2 3k+1 if p = 2, p 3k if p > 2. It can be verified in GAP that there exist no groups of order 2 9 with subgroup breadth 2 and center of order 2 and similarly there exist no groups of order 3 8 with subgroup breadth 2 and center of order 3. It should be mentioned that both of these are substantial computations. The groups of order 512 are in the Small Groups Library, however, there are 10,494,213 such groups. Using trivial parallelization, approximately 50 processors were used to get a list of the sizes of the centers of all groups of order 512. There are 5,327 groups with center of size 2, only 10 of which have cyclic breadth 2. We rule these out case-by-case. The groups of order 3 8 were obtained using the p-group generation algorithm [13] implemented in the GAP package ANUPQ. It is clear that the methods of the proof of 1.1 in this paper cannot be extended to even the case k = 2. It should be noted that classifications of p-groups with element breadth 3 exist (see, for instance, [14] or [17]). However, no generic classification of p-groups with element breadth 4 exist. Computational Results In this section we provide a complete list of all 2-groups which have subgroup breadth 1, up to order 256. These calculations were all made in GAP. Theorem 7.1. A non-abelian group of order 32 has subgroup breadth at most 1 if and only if it is one of the following: 2, 4, 5, 8, 12, 15, 17, 22, 23, 24, 25, 26, 29, 32, 35, 37, 38, 41, 46, 47, 48, 50. Theorem 7.2. A non-abelian group of order 64 has subgroup breadth at most 1 if and only if it is one of the following: 3, 17, 22, 27, 29, 44, 45, 51, 56, 57, 58, 59, 84, 85, 86, 87, 88, 93, 103, 104, 105, 110, 112, 113, 115, 126, 127, 184, 185, 193, 194, 195, 196, 197, 198, 200, 204, 208, 212, 214, 230, 238, 245, 247, 248, 252, 261, 262, 263, 265. Theorem 7.3. A non-abelian group of order 128 has subgroup breadth at most 1 if and only if it is one of the following : 5, 43, 44, 106, 108, 129, 131, 153, 154, 160, 164, 180, 181, 182, 183, 184, 457, 458, 459, 460, 469, 476, 477, 480, 481, 483, 498, 499, 501, 509, 838, 839, 840, 843, 844, 881, 882, 883, 884, 894, 895, 899, 914, 915, 989, 990, 998, 999, 1000, 1001, 1002, 1003, 1004, 1602, 1603, 1604, 1606, 1608, 1609, 1618, 1634, 1635, 1636, 1646, 1649, 1650, 1652, 1658, 1690, 1691, 1692, 1696, 1716, 2137, 2138, 2151, 2152, 2153, 2154, 2155, 2156, 2158, 2165, 2169, 2173, 2175, 2198, 2208 40, 124, 126, 317, 319, 453, 455, 498, 500, 531, 532, 538, 827, 828, 829, 830, 831, 835, 1119, 1131, 1247, 3680, 3681, 3683, 3692, 4385, 4386, 4387, 4388, 4389, 4390, 4395, 4396, 4398, 4399, 5526, 5527, 5531, 5532, 5536, 5578, 5579, 5586, 5587, 5598, 5640, 5641, 5643, 5649, 6535, 6536, 6537, 6540, 6541, 6614, 6615, 6616, 6617, 6627, 6628, 6632, 6647, 6648, 6724, 6725, 6733, 6734, 6735, 6736, Definition 2 . 3 . 23The subgroup breadth of a subgroup H in a p-group G, sbr(H), is defined to be the integer such that [G : N G (H)] = p sbr(H) . The subgroup breadth of G, sbr(G), is the maximum value that sbr(H) takes over all the subgroups of G. Proposition 2. 5 . 5[10] If G is a p-group, then ebr(G) = 1 if and only if |G ′ | = p. Corollary 2 . 10 . 210Let G be a 2-group as in 2.9 with ebr(G)=2 and sbr(G)=1. Then | a ∩ b | ≤ 4. Proposition 2 . 15 . 215If G is a 2-group with subgroup breadth 1 and two involutions of G do not commute, then the center of G has index at most16. Theorem 2. 16 . 16If sbr(G)=1, Z(G) is cyclic and all involutions of G commute with each other, then G contains at most three involutions. Theorem 3. 2 . [ 1 ] (4. 2 ) 212Let G be a p-group with element breadth 1. Then G = (A 1 * A 2 * · · · * A k )Z(G) where '*' denotes a central product where the isomorphic central subgroups are the derived subgroups of the A i 's. Theorem 3. 3 . 3Let G be a p-group with ebr(G)=sbr(G)=1. Then [G : Z(G)] = p 2 unless p = 2 and G is isomorphic to one of the following: [G : Z(G)] ≤ 4. Therefore it remains to show that if [G : Z(G)] = 16 then Z(G) is elementary abelian. Suppose that Z(G) has some element t of order 4. If t2 / ∈ Q 8 * A 2 where A 2 is D 8 or P 2,1,1 , we get groups [128,2162] and [256,26990] respectively, neither of which have subgroup breadth 1 by 7.3 and 7.4. Therefore t 2 ∈ Q 8 * A 2 . Specifically, t 2 ∈ Z(Q 8 * A 2 ). When A 2 = D 8 there is a unique non-identity element of Z(Q 8 * D 8 ) therefore we only have one choice and we get [64,266] which does not have subgroup breadth 1 by 7.2. When A 2 = P 2,1,1 we get three choices for t 2 which result in two different isomorphic classes of groups, [128,2160] and [128,2162], neither of which have subgroup breadth 1 by 7.3. Lemma 4 . 1 . 41If G has element breadth 2, then Ω 1 (Z(G)) ≤ G ′ . Theorem 4. 3 . 3Let B be an extension of the group Q 8 * D 8 by z . Then Z(B) has index at most 16. Proof. Using GAP, we obtain the following groups of order 64 that are extensions of Q 8 * D 8 : 200, 201, 217, 218, 220, 222, 223, 225, 228, 229, 230, 233, 237, 238, 243, 244, 245, 265. By 7.2 the only groups from this list that have subgroup breadth 1 are 200, 230, 238, 245, 265 and all of these have center of index at most 16. Theorem 4 . 4 . 44Let B be an extension of the group Q 8 * P 2,1,1 by z . Then Z(B) has index at most16. [a, b] = [a, c] = [b, d] = [c, d] = 1, [a, d] = a 4 b 2 . Theorem 5.1. [7] (Theorem 12.11) A p-group, G, has irreducible representations of only degrees 1 and p if and only if one of the following holds: Theorem 5. 2 . 2Let G be a p-group with subgroup breadth 1 and element breadth 2 that is a minimal counterexample to [G : Z(G)] ≤ p 3 . Then G does not have an abelian maximal subgroup. Theorem 5. 3 . 3Let G be a p-group with subgroup breadth 1, where p is an odd prime. Then [G : Z(G)] ≤ p 3 . Since N Y is a semidirect product with kernel N , we have that N N (Y ) = C N (Y ). Since G has subgroup breadth 1, we have [N : N N (Y )] ≤ p, hence we have k ≤ 2. If k = 1, then [X, Y ] = 1, a contradiction. Hence assume k = 2. Then N Y = x, y is an extraspecial group of order p 3 . Now, both X and Y have p conjugates in N Y , hence N Y ⊳ G. We claim that [G, Y N ] = [Y N, Y N ] = Z(Y N ). [y 1 1c, y 2 n] = [y 1 c, n][y 1 c, y 2 ] n = [y 1 , n] c [c, n][y 1 , y2 ] nc [c, y 2 ] n = [c, n][c, y 2 ] n Now, since Y N ⊳ G, we have [Y N, Y N ] ⊳ G hence we only must show that if c ∈ C and y ∈ Y , then [c, y 2 ] ∈ [Y N, Y N ]. Now, since C G (y 2 ) has index p in G and C C (y 2 ) has index p in C we have that D = C C (y 2 ) has index p. Then C = DX. We therefore have [c, y] = [dx 1 , y 2 ] = [d, y 2 ] x1 [x 1 , y 2 ] = [x 1 , y 2 ] which is clearly a commutator of Y N . So [G, Y N ] ≤ [Y N, Y N ].Clearly we have the other inclusion, so [G, Y N ] = [Y N, Y N ]. Since Y N is extraspecial, we have that [G, Y N ] = Z(Y N ). Let H = C G (Y N ). Then [H, Y N ] = 1 and H ∩ Y N = Z(Y N ) ( 4 ) 4Ω 1 (G) is an elementary abelian group of size p 3 . Proof. Let O = Ω 1 (G). By (3), O is elementary abelian. Let Z = Z(G)∩O. [G : Z(G)] = [G : Z(N )] = [G : N ][N : Z(N )] = p 3 .If N is a 3-group of maximal class, then, as above, we have that N has order 3 4 , hence G has order 35 . Now, N G (O) = C G (O) which has index 3. Clearly O ≤ Z(C G (O))and since a p-group cannot have a center of index p, we get that C G (O) must be an abelian subgroup. (Note that this argument does not produce an abelian maximal subgroup in the generic case since we have no guarantee that all elements of C G (O) must commute with each other.) This final contradiction completes the proof. , 2262, 2302, 2303, 2308, 2320, 2321, 2322, 2324. Theorem 7.4. A non-abelian group of order 256 has subgroup breadth at most 1 if and only if it is one of the following: Groups of prime power order. Yakov Berkovich, de Gruyter Expositions in Mathematics. Zvonimir Janko1Walter de Gruyter GmbH & Co. KGYakov Berkovich. Groups of prime power order. Vol. 1, volume 46 of de Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin, 2008. With a foreword by Zvonimir Janko. Groups of prime power order. Yakov Berkovich, Zvonimir Janko, de Gruyter Expositions in Mathematics. 2Walter de Gruyter GmbH & Co. KGYakov Berkovich and Zvonimir Janko. Groups of prime power order. Vol. 2, volume 47 of de Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin, 2008. A millennium project: constructing small groups. Bettina Hans Ulrich Besche, E A Eick, O&apos;brien, Internat. J. Algebra Comput. 125Hans Ulrich Besche, Bettina Eick, and E. A. O'Brien. A millennium project: constructing small groups. Internat. J. Algebra Comput., 12(5):623-644, 2002. Finite groups in which the nonnormal subgroups have nontrivial intersection. Norman Blackburn, J. Algebra. 3Norman Blackburn. Finite groups in which the nonnormal subgroups have nontrivial intersection. J. Algebra, 3:30-37, 1966. The nilpotency class of p-groups in which subgroups have few conjugates. Giovanni Cutolo, Howard Smith, James Wiegold, J. Algebra. 3001Giovanni Cutolo, Howard Smith, and James Wiegold. The nilpotency class of p-groups in which subgroups have few conjugates. J. Algebra, 300(1):160-170, 2006. Gap The, Group, GAP -Groups, Algorithms, and Programming. Version 4.4.12The GAP Group. GAP -Groups, Algorithms, and Programming, Version 4.4.12, 2008. A characterization of groups in terms of the degrees of their characters. I M Isaacs, D S Passman, Pacific J. Math. 15I. M. Isaacs and D. S. Passman. A characterization of groups in terms of the degrees of their characters. Pacific J. Math., 15:877-903, 1965. Corrected reprint of the 1976 original. I , Martin Isaacs, Academic Press0460423Providence, RI; New YorkCharacter theory of finite groupsI. Martin Isaacs. Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. Presentations of metacyclic groups. Bruce W King, Bull. Austral. Math. Soc. 8Bruce W. King. Presentations of metacyclic groups. Bull. Austral. Math. Soc., 8:103-131, 1973. Über den Frobenius'schen Klassenbegriff in nilpotenten Gruppen. Hans-Georg Knoche, Math. Z. 55Hans-Georg Knoche.Über den Frobenius'schen Klassenbegriff in nilpotenten Gruppen. Math. Z., 55:71-83, 1951. Some explicit bounds in groups with finite derived groups. I D Macdonald, Proc. London Math. Soc. 113I. D. Macdonald. Some explicit bounds in groups with finite derived groups. Proc. London Math. Soc. (3), 11:23-56, 1961. On finite nearly uniform groups. Izabela Malinowska, Publ. Math. Debrecen. 691-2Izabela Malinowska. On finite nearly uniform groups. Publ. Math. Debrecen, 69(1-2):155-169, 2006. The p-group generation algorithm. E A O&apos;brien, J. Symbolic Comput. 9Computational group theory. Part 1E. A. O'Brien. The p-group generation algorithm. J. Symbolic Comput., 9(5- 6):677-698, 1990. Computational group theory, Part 1. p-groups of small breadth. Gemma Parmeggiani, Bernd Stellmacher, J. Algebra. 2131Gemma Parmeggiani and Bernd Stellmacher. p-groups of small breadth. J. Algebra, 213(1):52-68, 1999. A course in the theory of groups. Derek John Scott Robinson, Graduate Texts in Mathematics. 80Springer-VerlagDerek John Scott Robinson. A course in the theory of groups, volume 80 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. Michio Suzuki, of Grundlehren der Mathematischen Wissenschaften. IIGroup theory. Fundamental Principles of Mathematical SciencesMichio Suzuki. Group theory. II, volume 248 of Grundlehren der Mathema- tischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. . Springer-Verlag, New YorkSpringer-Verlag, New York, 1986. Translated from the Japanese. 2-groups of breadth 3. B Wilkens, J. Algebra. 3181B. Wilkens. 2-groups of breadth 3. J. Algebra, 318(1):202-224, 2007.

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{'abstract': 'We examine p-groups with the property that every non-normal subgroup has a normalizer which is a maximal subgroup. In particular we show that for such a p-group G, when p = 2, the center of G has index at most 16 and when p is odd the center of G has index at most p 3 .', 'arxivid': '0905.3790', 'author': ['Joseph P Bohanon '], 'authoraffiliation': [], 'corpusid': 10517089, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18649, 'n_tokens_neox': 15619, 'n_words': 10164, 'pdfsha': 'b60572d67031670810a5771452323ecbc76feb00', 'pdfurls': ['https://arxiv.org/pdf/0905.3790v2.pdf'], 'title': ['FINITE GROUPS WITH MAXIMAL NORMALIZERS I', 'FINITE GROUPS WITH MAXIMAL NORMALIZERS I'], 'venue': []}

arxiv

Answering Complex Questions over Text by Hybrid Question Parsing and Execution Ye Liu Salesforce Research Semih Yavuz Salesforce Research Rui Meng Salesforce Research Dragomir Radev [email protected] Yale University Caiming Xiong [email protected] Salesforce Research Yingbo Zhou Salesforce Research Answering Complex Questions over Text by Hybrid Question Parsing and Execution The dominant paradigm of textual question answering systems is based on end-to-end neural networks, which excels at answering natural language questions but falls short on complex ones. This stands in contrast to the broad adaptation of semantic parsing approaches over structured data sources (e.g., relational database, knowledge graphs), that convert natural language questions to logical forms and execute them with query engines. Towards combining the strengths of neural and symbolic methods, we propose a framework of question parsing and execution on textual QA. It comprises two central pillars:(1)We parse the question of varying complexity into an intermediate representation, named Hexpression, which is composed of simple questions as the primitives and symbolic operations representing the relationships among them; (2) To execute the resulting H-expressions, we design a hybrid executor, which integrates the deterministic rules to translate the symbolic operations with a drop-in neural reader network to answer each decomposed simple question. Hence, the proposed framework can be viewed as a top-down question parsing followed by a bottom-up answer backtracking. The resulting H-expressions closely guide the execution process, offering higher precision besides better interpretability while still preserving the advantages of the neural readers for resolving its primitive elements. Our extensive experiments on MuSiQue, 2WikiQA, HotpotQA, and NQ show that the proposed parsing and hybrid execution framework outperforms existing approaches in supervised, few-shot, and zero-shot settings, while also effectively exposing its underlying reasoning process. Introduction End-to-end neural models that transductively learn to map questions to their answers have been the dominating paradigm for textual question answering (Raffel et al., 2020;Yasunaga et al., 2021) ing to their flexibility and solid performance. However, they often suffer from a lack of interpretability and generalizability. Symbolic reasoning, on the other hand, relies on producing intermediate explicit representations such as logical forms or programs, which can then be executed against a structured knowledge base (e.g., relational database, knowledge graph, etc.) to answer questions (Gu et al., 2022;Baig et al., 2022). These methods naturally offer better interpretability and precision thanks to the intermediate symbolic representations and their deterministic executions. However, they might be limited in expressing a broad range of questions in the wild depending on the semantic coverage of the underlying symbolic language and grammar employed. Neural Module Networks (Andreas et al., 2016;Gupta et al., 2019;Khot et al., 2020) have been proposed to combine neural and symbolic modality together. However, they require a symbolic language and a corresponding model that only covers limited scenarios in the specific task or domain. To apply this approach on new tasks or domains, new languages and neural modules have to be introduced. Therefore, designing a generalizable framework that use a high-coverage symbolic expression and a flexible neural network that can be versatilely used in various scenarios becomes our goal. Recent chain-of-thought prompting works on large language model Drozdov et al., 2022) have the insight of solving complex questions by iteratively decomposing the unsolved question into simpler sub-questions that can be solved. Inspired by them, we adapt the decomposition into our framework to make the model generalizable to complex questions. In this work, we propose a Hybrid question Parser and Execution framework, named HPE, for textual question answering, which combines neural and symbolic reasoning. To this end, H-expression is introduced as a simple explicit representation of original complex questions, which in the fashion of only contain primitives and operations as in (Liu et al., 2022). As shown in Figure 1, we define a primitive in the H-expression as a single-hop question and the operation is used to express the connections between primitives. H-parser, a semantic parsing process, is used to convert questions into H-expressions. To execute H-expression, we design a hybrid execution (H-executor), which not only utilizes the neural network to answer each single-hop question but also contains deterministic rules to combine the answers from single-hop questions into the final answer. H-executor can be detachable and replaced by any reader model (Izacard and Grave, 2020;Beltagy et al., 2020) and the training of this network can be done globally with the massive single-hop question answering data. Our contributions can be summarized as follows: • Architecture: We propose to combine the advantages of both symbolic and neural reasoning paradigms by parsing questions into hybrid intermediate expressions that can be iteratively executed against the text to produce the final answer. Extensive experiments on MuSiQue (Trivedi et al., 2022b) and 2WikiMultiHopQA (Min et al., 2019) show that our proposed approach achieves state-ofthe-art performance. • Generalizability: End-to-end neural approaches are data hungry and may significantly suffer from poor generalization to unseen data, especially in limited resource scenarios. Our design, on the other hand, naturally splits the reasoning process into Hparser and H-executor, through which it intends to disentangle learning to parse complex questions structurally from learning to resolve simple questions therein. Our few-shot experiments on MuSiQue and 2WikiMultiHopQA and zero-shot experiments on HotpotQA (Yang et al., 2018) and NQ suggest that even with less training data, our approach can generalize better to unseen domains. • Interpretability: The execution process of our model is the same as its reasoning process. Transparency of our approach facilitates spotting and Approach We formulate textual question answering as the task of answering a question q given the textual evidence provided by passage set P . We assume access to a dataset of tuples {(q i , a i , P i )|i = 1 · · · n} where a i is a text string that defines the correct answer to question q i . Previous works take this tuple as input and directly generate the predicted answer. In this work, we cast this task as question parsing with hybrid execution. Given a question q i , a question parser is tasked to generate the corresponding H-expression l i . The generated H-expression l i and the supporting passage set P i are given to the execution model to generate the predicted answer. Our framework consists of two modules, namely H-parser and H-executor. In Section 2.1, we first define H-expression as a hybrid expression consisting of primitives and operations, which serves as an intermediate representation connecting Hparser and H-executor. H-parser, a Seq2Seq model, is introduced to take the question as the input and output an H-expression as an explicit representation of the question. In Section 2.2, H-executor first compiles the H-expression into a tree structure by symbolic rules. Then H-executor delegates a singlehop reader for simple question answering and executes the symbolic operations based on answers in a bottom-up manner to get the final prediction. H-parser To improve the compositional generalization, we follow (Liu et al., 2022) to define H-expression as the composition of primitives and operations. Grammars and rules of H-expression In this framework, we define a primitive as a singlehop question, which is the atomic element con- In the execution step, q1 will be executed first, then q2. Those operations can be combined into more complex H-expression. For example, JOIN (q3, JOIN (q2, q1 ) ) or JOIN (q3, AND (q2, q1)). And for a single-hop question, its H-expression will be itself. In Table 1, we listed operation definitions, which will be executed in the H-executor. More specifically, the JOIN operation is used for linear-chain type reasoning. q1 is the complete question that can be answered directly, but q2 is an incomplete question with a placeholder (a1) inside. In the execution step, This operation will be executed in s serial way, q2 will be executed first and the answer of q2 will be used to replace the placeholder in the q1. AND operation is used for intersection reasoning, which will return the intersection of the answer of q2 and q1. COMPARE_= is used to determine if the answers of q2 and q1 are equal. The return value should be "Yes" or "No". COMPARE_< and COM-PARE_> operations will select the question entity corresponding to the smaller or the bigger answer of q2 and q1. MINUS and ADDITION operations are for subtractions and additions that involve the answers of q2 and q1. H-expression Generation The semantic parsing process on KB and DB usually needs to interact with the background context to match natural questions to logical forms with the specified schema, which is a necessary condition to execute in knowledge base (Ye et al., 2021) or table (Lin et al., 2020). However, in textual question answering, the question parsing process is context-independent because we want the meaning of the original question and the H-expression to be equivalent without any additional information from the context. Our parser is a seq-to-seq model that takes a natural question q as input and generates the H-expression l as the output. We delegate a T5 model (Raffel et al., 2020) as the basis of our question parser, as it demonstrates strong performance on various text generation tasks. We train the model by teacher forcing -the target H-expression is generated token by token, and the model is optimized using cross-entropy loss. At inference time, we use beam search to decode the top-k target Hexpression in an auto-regressive manner. H-executor Unlike the execution in DB and KB, which is fully program-based, our execution has both a neural and a symbolic component. The advantage of symbolic representation is that the resulting execution process is deterministic and robust. H-expression to Tree Structure H-expression is in the nested structure and the linear sequence cannot represent it well. H-executor first interprets the linear H-expression into a binary tree structure, where primitives are leaf nodes and operations are non-leaf nodes of the tree. All primitives will be executed by the neural network and the non-leaf nodes will be executed by the deterministic symbolic rules to generate new primitives or answers. The interpreter of H-executor will traverse from the rightmost primitive, followed by its parent node and the left branch recursively, which is similar to in-order traversal with opposite leaf order. As shown in Figure 2, the question who is winner of 1894-95 FA Cup is the first primitive to be executed and the single-hop reader will be called to answer this question and give the answer Aston Villa. Then non-leaf operation AND will be visited, which will store the A1 as Aston Villa. The left primitive what is member of sports team of Duane Courtney will be visited. The single-hop reader will be called to answer this question and return Birminghan City, which will be stored as A2. Then operation JOIN will be visited, which will replace the placehold A1 and A2 with the stored answer and produce a new primitive When was the last time Birminghan City beat Aston Villa. This primitive will be answered by the single-hop reader and predict the final answer 1 December 2010. Training of Reader Network We use FiD (Izacard and Grave, 2020) as our reader network, which is a generative encoder-decoder model. Each supporting passage is concatenated with the question, and processed independently from other passages by the encoder. The decoder takes attention over the concatenation of all resulting representations from the encoder. To distinguish different components, we add the special tokens question:, title: and context: before the question, title and text of each passage. Note that the reader network is detachable and may be replaced by any generative or extractive reader model at your choice. We have an assumption that the single-hop question is much easier to answer and it's feasible to have a global single-hop reader, which can be adapted to any unseen dataset in a Wikipedia domain. To get a single-hop global reader, we leverage the large-scale QA pairs from Probably-Asked Questions/PAQ (Lewis et al., 2021). To reduce the training computational cost, we first pre-train the model with few passages to get a reasonable checkpoint and then finetune using all supporting passages. In detail, we first train the Seq2Seq model T5-large in the reading comprehension setting (with one positive passage) using PAQ data. Then we use the trained T5-large from PAQ to initialize the FiD model and further train the model using training set of TriviaQA (Joshi et al., 2017), SQuAD (Rajpurkar et al., 2016), BoolQ (Clark et al., 2019) in a multiple passage setting (with one positive passage and nineteen negative passages). A global reader can be zero-shot used to unseen questions and also boost the performance of fine-tune setting as the pre-training weights. Experiments We conduct experiments on two multi-hop multipassage textual QA datasets, MuSiQue and 2Wikimulti-hopQA, which contain complex questions and corresponding decomposed simple questions for the supervised setting. We also test models' generalization on the few shot setting using 5-20% of training data. In real scenarios, neither decomposed questions nor the complexity of questions is known. Therefore, we also investigate our models under the zero-shot setting on both complex (HotpotQA) and simple (NQ) QA datasets. In the end, we carry out a case study to show the interpretability of our framework. Supervised Experiments Datasets We describe each dataset and then explain how to convert original data into the training format for both question parsing and execution. MuSiQue (Trivedi et al., 2022b) contains multihop reasoning questions with the mixed number of hops and question entities which can be asked from 20 supporting passages. It contains 19,938/2,417/2,459 for train, dev and test sets, with 2hop1 (questions with 2 hops and 1 entity), 3hop1, 4hop1, 3hop2, 4hop2 and 4hop3 reasoning types. 2Wikimulti-hopQA (2WikiQA) (Ho et al., 2020) requires models to read and perform multihop reasoning over 10 multiple passages. Three types of reasoning are included, namely comparison, bridge, and bridge-comparison. It contains 167,454/12,576/12,576 for train, dev and test sets. Reconstruction MuSiQue contains complex questions, decomposed single question with answers and the reasoning type for each complex question. We use JOIN operation to combine linearly-chain type questions together and use AND operation to combine intersection type questions. In 2WikiQA, we use evidences (in form of triplet <subject, relation, object>) and reasoning type to create the H-expression. In detail, we first convert the subject and relation into natural questions using templates and the object is the answer of this natural question. Then, we use the operation to combine those singlehop questions into an H-expression based on their reasoning type. From Table 2 amples of complex question and its corresponding H-expression. An example with all reasoning type H-expressions is shown in Appendix A. Evaluation Metrics. We use official evaluation scripts for each dataset with two metrics to measure answer exact match accuracy (EM) and answer token-level accuracy (F1). Baselines Press et al. (2022); Trivedi et al. (2022a) make use of large language models like GPT-3 (Brown et al., 2020). They iteratively generate an answerable question, use retrieval to get supporting passages, and answer the question based on the retrieved passages. SA (Trivedi et al., 2022b) is the stateof-the-art model on the MuSiQue dataset, which first uses a RoBERTa based (Liu et al., 2019) ranking model to rank supporting passages and then uses an End2End reader model to answer complex questions using the top-k ranked passages. EX(SA) (Trivedi et al., 2022b) decomposes a complex question into single-hop questions and builds a directed acyclic graph (DAG) for each singlehop reader (SA) to memorize the answer flow. NA-Reviewer (Fu et al., 2022) proposes a reviewer model that can fix the error prediction from incorrect evidence. We include FiD (Izacard and Grave, 2020) as the baseline End2End reader model. In the original FiD, it takes the question as well as the supporting passages as input, and generates the answer as a sequence of tokens. Moreover, we propose two variants of FiD to compare the influence using Hexpression: one uses H-expressions as the input, instead of original questions, to generate answers (referred to as FiD LF−>Ans ), and the other uses questions as input to generate both H-expressions and answers (referred to as FiD CQ−>LF+Ans ). Implementation Details We describe fine-tuning details for question parsing and single-hop reader models in Appendix B. Pre-training (PT) To pretrain the single-hop reader, we use a subset of PAQ (Lewis et al., 2021) consisting of 20M pairs, which is generated based on named entities and the greedy decoded top-1 sequence with the beam size of 4. We train a T5-large model for 400k steps, with one gold passage, maximum length of 256 and batch size of 64. Then we initialize FiD with the PAQ pre-trained model and further train it for 40k steps, with batch size of 8 and 20 supporting passages, on the combined training sets of TriviaQA (Joshi et al., 2017), SQuAD (Rajpurkar et al., 2016) and BoolQ (Clark et al., 2019). Our code is based on Huggingface Transformers (Wolf et al., 2019). All the experiments are conducted on a cloud instance with eight NVIDIA A100 GPUs (40GB). Fine-tuning Results We present our main results on MuSiQue and 2Wik-iQA in Table 3. We observe that Self-ask and IRCoT, which are based on large language models and search engines, underperform most supervised models. This indicates that multi-hop multiparagraph question answering is a difficult task, and there still has an evident gap between supervised small models and large models with few-shot or zero-shot. Moreover, our framework outperforms the previous SOTA methods on both datasets. We notice that the baseline EX(SA) underperforms SA by a large margin, but our HPE outperforms FiD by 5.3% on MuSiQue EM. This shows the difficulty to build a good H-expression and executor. Moreover, EX(SA) gets a bad performance on 2Wik-iQA, which shows that using DAG to represent the logical relationship between sub-questions is not adaptable to any reasoning type. Compared with the End2End baseline (FiD) that our model is built on, our framework with an explicit representation performs much better. Few-shot Results To illustrate the generalization ability of our framework, we show the analysis of our method under the few-shot setting in Table 4. We run three experiments, random sampling 5, 10, and 20 percentage of the training data. We use the End2End FiD model as the baseline, which inputs complex questions and generates the answers as token sequences. In 5% of MuSiQue dataset, it shows that our framework obtains a 4.9% absolute gain on MuSiQue EM score in comparison to the FiD model. Moreover, with 20% MuSiQue training data, our framework achieves 36.4 EM, which is a comparable performance with FiD trained on full-data (37.6 EM). Similar trends are also observed on 2Wik-iQA. In summary, the overall experiment shows that our model has better generalization ability than the End2End model, which is obtained by decomposing complex questions into single-hop ones and representing in H-expressions. Zero-shot Results We expect the H-parser to work well on questions of varying levels of complexity. To verify this, we test the models on two benchmarks HotpotQA and NQ without any tuning. The former does not contain any decomposed questions, and the latter contains common questions in the real world. Dataset HotpotQA we use the distractor setting (Yang et al., 2018) that a model needs to answer each question given 10 passages. To produce correct answer for a question, the dataset requires the model to reason across two passages. Note that two main reasoning types bridge and comparison in HotpotQA are included in MuSiQue and 2WikiQA. NQ contains opendomain questions collected from Google search queries. Usually, NQ is treated as a simple question dataset and previous works usually use End2End multi-passage reader like FiD. However, we argue that certain questions in NQ involve multi-hop reasoning and the model performance can be improved by decomposing them into single-hop questions. Global Question Parser To seamlessly generate H-expressions on unseen questions, we need a global question parser. This question parser can understand the complexity of the question, which means it can decompose a complex question into several simple questions and keep the simple question as is. To get a global question parser, we train a pretrained generative model T5 (Raffel et al., 2020) to convert questions to H-expressions using MuSiQue and 2Wikimulti-hopQA datasets. As the two datasets are not the same size, we categorize the complex question based on their reasoning type and sample the same amount of data for each category. To endow the model with the ability of understanding question complexity, we also use the simple questions in those datasets (the H-expression of a simple question is itself). Moreover, we decouple the composition of complex H-expressions into a few of simple H-expressions to ensure the coverage of all levels of complexity. Zero-Shot Results on HotpotQA We show the HotpotQA results in Table 5. We use FiD pre-trained on PAQ and TriviaQA, SQuAD and BoolQ as our zero-shot reader. Our framework outperforms both Standard and CoT, using promptbased large language models. This shows that with the hybrid question parsing and execution framework, a small language model is generalizable on unseen questions. Compared with FiD (PT), we get a comparable performance. But checking the union of HPE and FiD, which takes the correct predictions from both methods, we find 15% absolute gain can be obtained. This shows that HPE correctly answers around 15% of questions that FiD predicts incorrectly, with the help of the question decomposition and symbolic operation. On the other hand, we conjecture that the reason that HPE wrongly predicts some questions is that the global question parser fails to generate H-expression correctly. Hence, it is worth exploring how to design a generalizable global question parser in future work. Results on NQ We use the global question parser to decompose NQ question in a zero-shot manner. If a question is recognized as single-hop reasoning and cannot be further decomposed, the parser will keep the question unchanged. We use the DPR model (Karpukhin et al., 2020) to retrieve the Top-20 documents from Wikipedia as the supporting documents. Among the 8k dev set examples, 32 questions have been decomposed into single-hop questions with the logical operations and the rest are left as is. For ex-ample, a question "when did the last survivor of the titanic die" is converted into the H-expression "JOIN [when did A1 die, who was the last person to survive the titanic]". The result in Table 6 shows that HPE can handle questions of different complexity levels and will not degenerate on simple questions. FiD (FT) 51.4 56.2 HPE (FT) 51.7 56.3 Table 6: Answer exact match (EM) and F1 scores on dev split of the simple QA NQ. EM F1 Ablation Study Impact of H-parser We show the performance of different H-parsers. Table 7 shows using T5large rather than T5-base, we can get around 2 to 4 percent performance improvement on both datasets. Compared to the result using gold Hexpression, there is more room for improvement on the MuSiQue dataset. This might also be the case as the questions in MuSiQue are generally more complex than 2WikiQA. Impact of H-executor Our hybrid executor is combined with symbolic operations and replaceable reader network. We analyze the influence of different reader networks to the final performance and experiment with different versions of FiD. Support-FiD generates both answers and the supporting document titles. SelectFiD is a two-step method that first uses a RoBERTa-based (Liu et al., 2019) Case Study In this section, we analyze the error cases. Moreover, we show the performance under each reasoning type on MuSiQue and 2WikiQA in Appendix C. In the end, we show a case of how our model reasons on a complex question in Appendix D. Error Analysis There are two types of errors of our model prediction. One is the error from the semantic parsing of the H-expression. The other is the error from the single-hop question answer. The percentage of the first type of error is 67% and the second type is 33% on the MuSiQue dataset. When the number of hops gets larger, our model could suffer from exposure bias (Bengio et al., 2015). Due to the chain reasoning, the next step question depends on the previous answers. This problem becomes acute if the model predicts a bad output at a certain step which in turn affects the final answer. However, one additional advantage of our work is that once we know where the error come from. We can fix the issue and get the correct final answer. To fix a wrong prediction, we can check whether the generated H-expression is correct. If it has some problem, like generating a bridge type H-expression of the comparison type complex question, we can fix it with the correct one. Otherwise, if we find out one single-step answer wrongly predicts, we can correct this single-hop answer. Moreover, this exposure bias can be solved by beam search (Wiseman and Rush, 2016) meaning that rather than generating one answer at each step, we can generate multiple answers and the final answer is the highest-scoring one. 2020) introduces a NMN variant that decomposes a question into a sequence of simpler ones answerable by different task-specific models. Systematic question decomposition is also explored in (Talmor and Berant, 2018;Min et al., 2019;Wolfson et al., 2020). Although our framework shares some similarities with this line of works, there is an essential difference in that we keep both symbolic and neural representations coincide, whereas they delegate the neural model to replace the non-differentiable symbolic representation in order to end-to-end train the model. Explainable QA A series of recent works focus on generating the explanation, which can be viewed as reasoning chain. (Yavuz et al., 2022;Latcinnik and Berant, 2020;Jhamtani and Clark, 2020) formulate the multi-hop question answering as single sequence generation, which contains an answer along with its reasoning path. Even though the generated reasoning path may provide some explanation on how the question being solved, there is no guarantee that the answer is indeed generated by the predicted reasoning path. Recently, large Language Models (LLMs) have show its capability to answer complex questions by producing step-by-step reasoning chain (chainsof-thought, or CoT) when prompted with a few examples or even without any examples (Kojima et al., 2022;Kadavath et al., 2022). Conclusion We propose HPE for answering complex questions over text, which combines the strengths of neural network approaches and symbolic approaches. We parse the question into H-expressions followed by the hybrid execution to get the final answer. Our extensive empirical results demonstrate that HPE has a strong performance on various datasets under supervised, few-shot, and zero-shot settings. Moreover, our model has a strong interpretability exposing its underlying reasoning process, which facilitates understanding and possibly fixing its errors. By replacing our text reader with KB or Limitations We acknowledge that our work could have the following limitations: • Even if the defined H-expression can be used on various reasoning types and different text question answering datasets, it is not mature to be used to any type of reasoning. When the new reasoning type comes, we need to retrain the question parser. To solve the new reasoning type question, we plan to take advantage of in-context learning in a large language model to generate H-expression as future work. It's worth mentioning that our executor can be easily adapted to new reasoning types by adding new symbolic rules and the reader network doesn't need to be retrained. • As mentioned in the error analysis section, the bottom-up question answering process could suffer from exposure bias since the next step question answering may depend on the previous predicted answers. To deal with this limitation, we anticipate that generating multiple answers using beam search in each step may greatly solve this issue. Since predicted candidates by current reader models have a strong lexical overlap, general beam search needs to be revised to provide a sufficient coverage of semantic meanings. We leave it for future work. B Supervised Training Details To train the question parser, we initiate H-parser using T5-large model. We trained it with batch size of 32 with a learning rate of 3e-5 for 20 epochs on both MuSiQue and 2WikiQA. We selected the model weight based on evaluating the H-expression exact match. We base our reader network FiD on T5-large. We use 20 passages with maximum length of 256 tokens for input blocks on MuSiQue dataset and use 10 passages with 356 tokens as text length on the 2WikiQA dataset. We trained the reader model with a batch size of 8 with a learning rate of 5e-4 for 40k steps. C Performance of each Different Reasoning Type We represent the Answer F1 performance under different reasoning types on both MuSiQue and 2WikiQA in Figure 4. Our hybrid question parsing and execution model performs significantly better than directly getting the answer model in both QA showing that the advantage of delegating semantic parsing to solve complex textual questions. In MuSiQue, for the relevant simple reasoning types (2hop, 3hop1), our model outperform FiD by a great margin. For complex reasoning types (3hop2, 4hop1, 4hop2 and 4hop3), our model gets lower performance compared with the simple reasoning types because the exposure bias issue becomes worse with the step of reasoning increase. But it still has a equivalent or better perform comparing End-to-End FiD. In 2WikiQA, our model performs best on all four reasoning type. Especially on the most complex type bridge comparison, our framework greatly outperform, which shows using deterministic symbolic representation is more robust to produce a correct answer. D A case study of how HPE reasoning In Figure 3, we show an example that FiD predicts a wrong answer but our model correctly predicts. Given a complex question, our framework first parses the complex question into H-expression. Then hybrid executor will convert the binary tree from the H-expression, where operation and natural sub-question as its nodes. H-executor parses the binary tree from the rightmost left node to the left and upper layer with considering the operation. At each question node, the reader neural network will take sub-questions and multiple paragraphs as input to generate the sub-answers. We store the sub-answer in memory for later substitution of the placeholder. For example, Q3 will be rewritten in A1 with the answer of Q1 (the Republicans) and A2 with the answer of Q2 (Senate) as the new question Q3' "when did Senate take control of the Republicans". The final answer is obtained by answering Q3'. Figure 1 : 1An illustration of H-expression. ( Gupta et al., 2019) introduces a neural module network (NMN), which solves QA using different modules by performing text span extraction and arithmetic operations. Khot et al. ( Figure 3 : 33hop2-type MuSiQue question example and how our framework finds the final answer. Figure 4 : 4Answer F1 score on each reasoning type on MuSiQue and 2WikiQA. Type: 2-Hop Question: Who is the deputy prime minister of the country that encompasses Inagua National Park? H-expression: JOIN [ who is the deputy prime minister of the #1 , What is country of Inagua National Park ] Tree: JOIN who is the deputy prime minister of the #1 What is country of Inagua National Park Type: 3-Hop 1-Entity Question: When did the greek orthodox church split from the religious institution located in the city where the creator of The Last Judgment died? H-expression: JOIN [ when did the greek orthodox church split from #2 , JOIN [ In what city did #1 die? , Who is creator of The Last Judgment ] ] Tree: JOIN when did the greek orthodox church split from #2 JOIN In what city did #1 die? Who is creator of The Last Judgment Type: 3-Hop 2-Entity Question: When did the capitol of Virginia move from Robert Banks' birthplace to the town WTVR-FM is licensed in? H-expression: JOIN [ when did the capital of virginia moved from #2 to #1 , AND [ What town is WTVR-FM liscensed in? , What is place of birth of Robert Banks ] ] Tree: JOIN when did the capital of virginia moved from #2 to #1 AND What town is WTVR-FM liscensed in? What is place of birth of Robert Banks Type: 4-Hop 1-Entity Question: When did the civil war start in the country whose capitol was home to the man after whom Korolyov was named? H-expression: JOIN [ when did the civil war in #3 start , JOIN [ #2 is the capital city of which country , JOIN [ What is residence of #1 , Korolyov is named after What ] ] ] Tree: JOIN when did the civil war in #3 start JOIN #2 is the capital city of which country JOIN What is residence of #1 Korolyov is named after What Type: 4-Hop 3-Entity Question: When did Muslim armies invade the country containing Al-Mastumah and the country of the man who followed the reign of Al-Mu'tamid? H-expression: JOIN [ when did muslim armies invade #3 and #2 , AND [ What is country of Al-Mastumah , JOIN [ What is country of citizenship of #1 , Al-Mu'tamid is followed by What ] ] ] Tree: JOIN when did muslim armies invade #3 and #2 AND What is country of Al-Mastumah JOIN What is country of citizenship of #1 Al-Mu'tamid is followed by What Table 1 : 1Operations defined in our H-expressions; q2 and q1 are single-hop natural questions. fixing erroneous cases. Question : QuestionWhen was the last time Duane Courtney's team beat the winner of the 1894-95 FA Cup? H-Parser H-Expression: JOIN [ When was the last time #2 beat #1 , AND [What is member of sports team of Duan Courtney, Who is winner of 1894-95 FA Cup]]Q + JOIN When was the last time A2 beat A1 AND What is member of sports team of Duan Courtney Who is winner of 1894-95 FA Cup Single-hop Reader A 1 December 2010 JOIN Q3: When was the last time A2 beat A1 AND Q2: What is member of sports team of Duane Courtney Q1: Who is winner of 1894-95 FA Cup Q3': When was the last time Birminghan City beat Aston Villa A2: Birminghan City A1: Aston Villa Execution Figure 2: An overview of the HPE pipeline of two stages: H-parser and H-executor. (1) H-parser first maps the input question to the H-expression. (2) Followed by H-executor uses the Reader network to return answer feedback for each question and the deterministic symbolic interpreter executes the expression to derive the final answer. sisting of the complex question. We use the op- eration to represent the relation between primi- tives. H-expression contains seven types of op- erations, which are JOIN, AND, COMPARE_=, COMPARE_>, COMPARE_<, MINUS and AD- DITION. Each operation is a binary function that takes two primitives q2 and q1 as input, written as OP[q2, q1], where OP ∈ {JOIN, AND, COM- PARE_=, COMPARE_>, COMPARE_<, MINUS and ADDITION} and q1, q2 are format-free single- hop question. , we shows a few ex-Bridge Where was the place of film The Iron Man director JOIN [Where is A1's place of death, Who is director of The Iron Man] Intersection Which team is Bernard Lowe was a member of beat the winner of the 1894-95 FA Cup AND [What is member of sports team of Bernard Lowe, Who won the 1894-95 FA Cup] Comparison Did Lenny Mchallister and Ken Xie have the same nationality COMPARE_= [What is country of citizenship of Lenny Mchallister, What is the country of citizenship of Ken Xie] Table 2 : 2Examples of question and corresponding H-expression under three basic reasoning types. For more complex reasoning types, their questions and H-expressions are shown in the Appendix A. Table 3 : 3Answer Exact match (EM) and F1 scores on dev/test split of MuSiQue and 2WikiQA. PT represents pre-training on reader network. The methods in Large LM and SOTA are reported from the previous work. The methods in End2End is implemented by us following the training details in the paper. Table 4 : 4Few-shot setting Exact match (EM) and F1 scores on test/dev split of the MuSiQue and 2WikiQA. Table 5 : 5Zero-shot performance on HotpotQA. Standard and CoT are prompted method using large language model like GPT3(Brown et al., 2020). Table 7 : 7Different question parsers and the gold Hexpression impact on Answer EM and F1 on MuSiQue and 2WikiQA under same FiD as the single-hop reader. ranking model to predict the Top-5 relevant documents and feeds them into FiD to generate the answer. From results inTable 8, we can see that a better single-hop reader produces better performance on MuSiQue. The improvement on single-hop reader translates to a significant performance boost on complex questions.FiD(T5-base) FiD(T5-large) FiD(PT) SupportFiD(PT) SelectFiD(PT) Ans EM/F1 Ans EM/F1 Ans EM/F1 Ans EM/F1 SP EM/F1 Ans EM/F1 SP EM/F1 MuSiQue SQ 64.9/70.7 68.5/74.9 73.3/79.5 72.3/78.5 78.6/92.2 76.8/82.6 73.8/90.2 CQ 34.9/44.6 42.9/50.1 45.5/53.7 43.8/53.4 41.7/72.1 45.9/54.8 39.2/70.5 Table 8 : 8EM and F1 scores of Answer and Support Passage on MuSiQue using different reader models. SQ represents simple question and CQ represents complex question. Table based neural network, our framework can be extended to solve KB and Table QA. Xanh Ho, Anh-Khoa Duong Nguyen, Saku Sugawara, and Akiko Aizawa. 2020. Constructing a multi-hop qa dataset for comprehensive evaluation of reasoning steps. arXiv preprint arXiv:2011.01060.In table 9 and 10, we show the H-expression examples and parsing tree under each reasoning type of Musique and 2WikiQA.Yu Gu, Vardaan Pahuja, Gong Cheng, and Yu Su. 2022. Knowledge base question answering: A semantic parsing perspective. arXiv preprint arXiv:2209.04994. Nitish Gupta, Kevin Lin, Dan Roth, Sameer Singh, and Matt Gardner. 2019. Neural module net- works for reasoning over text. arXiv preprint arXiv:1912.04971. Gautier Izacard and Edouard Grave. 2020. Lever- aging passage retrieval with generative models for open domain question answering. arXiv preprint arXiv:2007.01282. Harsh Jhamtani and Peter Clark. 2020. Learning to explain: Datasets and models for identifying valid reasoning chains in multihop question-answering. arXiv preprint arXiv:2010.03274. Mandar Joshi, Eunsol Choi, Daniel S Weld, and Luke Zettlemoyer. 2017. 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FiD predicted answer: 1931 Q3:When did A2 take control of the A1JOIN AND Q2: Upon whom does the president call on for support in his appointments to the USAF? Q1: Who hold the majority in the house of representatives Q1 + A1: the Republicans A2: Senate Q2 + A3: January 2015 Q3' + Table 9 : 9Examples of H-expression and parsing tree under each reasoning types in MuSiQue. : Comparision Question: Which film was released first, And Who Is Kissing Me? or Bush Christmas? H-expression: COMPARE < [ What is publication date of And Who Is Kissing Me? , What is publication date of Bush Christmas] COMPARE_< What is publication date of And Who Is Kissing Me? What is publication date of Bush Christmas? Type: Bridge Comparison Question: Which film has the director who was born later, Sleepers East or Leaving Fear Behind? H-expression: COMPARE > [ JOIN [ When is date of birth of #3 , Who is director of Sleepers East ] , JOIN [ When is date of birth of #1 , Who is director of Leaving Fear Behind ] ] COMPARE_> JOIN When is date of birth of #3 Who is director of Sleepers East JOIN When is date of birth of #1 Who is director of Leaving Fear Behind Type: Inference Question: Who is the sibling-in-law of Favila Of Asturias? H-expression: JOIN [ Who is spouse of #1 , Who is sibling of Favila Of Asturias ] JOIN Who is spouse of #1 Who is sibling of Favila Of Asturias Type: Compositional Question: Where did the founder of University Of Piura die? H-expression: JOIN [ Where is #1's place of death , The Universidad De Piura is founded by Who ] JOIN Where is #1's place of death The Universidad De Piura is founded by WhoTypeTree: Tree: Tree: Tree: Table 10 : 10Examples of H-expression and parsing tree under each reasoning types in 2WikiQA. Neural module networks. 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{'abstract': 'The dominant paradigm of textual question answering systems is based on end-to-end neural networks, which excels at answering natural language questions but falls short on complex ones. This stands in contrast to the broad adaptation of semantic parsing approaches over structured data sources (e.g., relational database, knowledge graphs), that convert natural language questions to logical forms and execute them with query engines. Towards combining the strengths of neural and symbolic methods, we propose a framework of question parsing and execution on textual QA. It comprises two central pillars:(1)We parse the question of varying complexity into an intermediate representation, named Hexpression, which is composed of simple questions as the primitives and symbolic operations representing the relationships among them; (2) To execute the resulting H-expressions, we design a hybrid executor, which integrates the deterministic rules to translate the symbolic operations with a drop-in neural reader network to answer each decomposed simple question. Hence, the proposed framework can be viewed as a top-down question parsing followed by a bottom-up answer backtracking. The resulting H-expressions closely guide the execution process, offering higher precision besides better interpretability while still preserving the advantages of the neural readers for resolving its primitive elements. Our extensive experiments on MuSiQue, 2WikiQA, HotpotQA, and NQ show that the proposed parsing and hybrid execution framework outperforms existing approaches in supervised, few-shot, and zero-shot settings, while also effectively exposing its underlying reasoning process.', 'arxivid': '2305.07789', 'author': ['Ye Liu \nSalesforce Research\n\n', 'Semih Yavuz \nSalesforce Research\n\n', 'Rui Meng \nSalesforce Research\n\n', 'Dragomir Radev [email protected] \nYale University\n\n', 'Caiming Xiong [email protected] \nSalesforce Research\n\n', 'Yingbo Zhou \nSalesforce Research\n\n'], 'authoraffiliation': ['Salesforce Research\n', 'Salesforce Research\n', 'Salesforce Research\n', 'Yale University\n', 'Salesforce Research\n', 'Salesforce Research\n'], 'corpusid': 258685562, 'doi': '10.48550/arxiv.2305.07789', 'github_urls': [], 'n_tokens_mistral': 15995, 'n_tokens_neox': 13908, 'n_words': 8370, 'pdfsha': '2b8a1c29a65bbc2c2cf857eaed788ec782041ce8', 'pdfurls': ['https://export.arxiv.org/pdf/2305.07789v1.pdf'], 'title': ['Answering Complex Questions over Text by Hybrid Question Parsing and Execution', 'Answering Complex Questions over Text by Hybrid Question Parsing and Execution'], 'venue': []}

arxiv

Dice Semimetric Losses: Optimizing the Dice Score with Soft Labels Zifu Wang ESAT-PSI, KU Leuven LeuvenBelgium Teodora Popordanoska ESAT-PSI, KU Leuven LeuvenBelgium Jeroen Bertels ESAT-PSI, KU Leuven LeuvenBelgium Robin Lemmens Department of Neurosciences KU Leuven LeuvenBelgium Department of Neurology UZ Leuven LeuvenBelgium Matthew B Blaschko ESAT-PSI, KU Leuven LeuvenBelgium Dice Semimetric Losses: Optimizing the Dice Score with Soft Labels Dice Score · Dice Loss · Soft Labels · Model Calibration The soft Dice loss (SDL) has taken a pivotal role in many automated segmentation pipelines in the medical imaging community. Over the last years, some reasons behind its superior functioning have been uncovered and further optimizations have been explored. However, there is currently no implementation that supports its direct use in settings with soft labels. Hence, a synergy between the use of SDL and research leveraging the use of soft labels, also in the context of model calibration, is still missing. In this work, we introduce Dice semimetric losses (DMLs), which (i) are by design identical to SDL in a standard setting with hard labels, but (ii) can be used in settings with soft labels. Our experiments on the public QUBIQ, LiTS and KiTS benchmarks confirm the potential synergy of DMLs with soft labels (e.g. averaging, label smoothing, and knowledge distillation) over hard labels (e.g. majority voting and random selection). As a result, we obtain superior Dice scores and model calibration, which supports the wider adoption of DMLs in practice. Code is available at https://github.com/zifuwanggg/JDTLosses. Introduction Image segmentation is a fundamental task in medical image analysis. One of the key design choices in many segmentation pipelines that are based on neural networks is the loss function. In fact, the choice of loss function goes hand in hand with the metrics chosen to quantify the quality of the predicted segmentation [46]. The intersection-over-union (IoU) and the Dice score are commonly used metrics because they reflect both size and localization agreement, and they are more in line with perceptual quality compared to, e.g., pixel-wise accuracy [10,28]. Hence, directly optimizing the IoU or the Dice score using differentiable surrogates as (part of) the loss function has become standard practice in semantic segmentation dealing with natural or medical images, respectively [2,21,10,47]. In medical imaging in particular, the Dice score and the soft Dice loss (SDL) [42] have become the standard practice, and some reasons behind its superior functioning have been uncovered and further optimizations have been explored [10,3,45]. Another mechanism to further improve the predicted segmentation that has gained much attention in recent years, is the use of soft labels during training. Soft labels can be the result of data augmentation tricks such as label smoothing (LS) [43,22] and are involved in regularization techniques such as knowledge distillation (KD) [18,35]. Their role is to provide additional regularization to make the model less prone to overfitting [43,18] and to combat overconfidence [15], e.g., providing superior model calibration [31]. In medical imaging, soft labels not only arise due to LS or KD, but are also present inherently due to significant intra-and inter-rater variability. For example, multiple annotators often disagree on organ and lesion boundaries, and one can average their annotations to obtain soft label maps [13,24,41,25]. This work investigates how the medical imaging community can combine the use of SDL with soft labels to reach a state of synergy. While the original SDL surrogate was posed as a relaxed form of the Dice score, naively inputting soft labels to SDL is possible (e.g. in open-source segmentation libraries [20,6,21,51]) but will push the predictions towards 0-1 outputs rather than make them similar to the soft labels [3,47]. Consequently, the use of SDL when dealing with soft labels might not align with a user's expectations, with potential adverse effects on the Dice score, model calibration and volume estimation [3]. More recently [47], this incompatibility to soft labels has been assigned to many regional losses, e.g. [32,36,2,49,50]. Motivated by this observation, firstly (Sect. 2), we propose two probabilistic extensions of SDL, i.e., Dice semimetric losses (DMLs), which satisfy the conditions of a semimetric and are fully compatible with both hard and soft labels. In a setting with hard labels, DMLs are identical to SDL and can safely replace SDL in existing implementations. Secondly (Sect. 3), we perform extensive experiments on the public QUBIQ, LiTS and KiTS benchmarks to empirically confirm the potential synergy of DMLs with soft labels (e.g. averaging, LS, KD) over hard labels (e.g. majority voting, random selection). Methods We adopt the notation of [47]. In particular, we denote the predicted segmentation asẋ ∈ {1, ..., C} p and the ground-truth segmentation asẏ ∈ {1, ..., C} p , where C is the number of classes and p the number of pixels. For a class c, we define the set of predictions as x c = {ẋ = c}, the set of ground-truth as y c = {ẏ = c}, the union as u c = x c ∪y c , the intersection as v c = x c ∩y c , the symmetric difference (i.e., the set of mispredictions) as m c = (x c \ y c ) ∪ (y c \ x c ), the Jaccard index as IoU c = |v c | |u c | , and the Dice score as Dice c = 2IoU c 1+IoU c = 2|v c | |x c |+|y c | . In what follows, we will represent sets as binary vectors x c , y c , u c , v c , m c ∈ {0, 1} p and denote |x c | = p i=1 x c i the cardinality of the relevant set. Moreover, when the context is clear, we will drop the superscript c. Existing Extensions If we want to optimize the Dice score, hence, minimize the Dice loss ∆ Dice = 1 − Dice in a continuous setting, we need to extend ∆ Dice with ∆ Dice such that it can take any predicted segmentationx ∈ [0, 1] p as input. Hereinafter, when there is no ambiguity, we will use x andx interchangeably. The soft Dice loss (SDL) [42] extends ∆ Dice by realizing that when x, y ∈ {0, 1} p , |v| = x, y , |x| = x 1 and |y| = y 1 . Therefore, SDL replaces the set notation with vector functions: ∆ SDL : x ∈ [0, 1] p , y ∈ {0, 1} p → 1 − 2 x, y x 1 + y 1 .(1) The soft Jaccard loss (SJL) [32,36] can be defined in a similar way: ∆ SJL : x ∈ [0, 1] p , y ∈ {0, 1} p → 1 − x, y x 1 + y 1 − x, y .(2) A major limitation of loss functions based on L 1 relaxations, such as SDL, SJL, the soft Tversky loss [39] and the focal Tversky loss [1], and loss functions that rely on the Lovasz extension, such as the Lovasz hinge loss [49], the Lovasz-Softmax loss [2] and the PixIoU loss [50], is that they cannot handle soft labels [47]. That is, when y is also in [0, 1] p . In particular, both SDL and SJL are not minimized at x = y, but rather pushing x towards the vertices {0, 1} p [3,47]. For example, consider y = 0.5, it is easy to verify that SDL is minimized at x = 1, which is clearly erroneous. Therefore, Wang and Blaschko [47] proposed two variants of SJL that they call Jaccard metric losses (JMLs). ∆ JML,1 and ∆ JML,2 : [0, 1] p × [0, 1] p → [0, 1] are defined as ∆ JML,1 = 1 − x + y 1 − x − y 1 x + y 1 + x − y 1 , ∆ JML,2 = 1 − x y 1 x y 1 + x − y 1 .(3) JMLs are shown to be a metric in [0, 1] p , according to the definition below. Definition 1 (Metric [8] (i) (Reflexivity). f (a, a) = 0. (ii) (Positivity). If a = b, then f (a, b) > 0. (iii) (Symmetry). f (a, b) = f (b, a). (iv) (Triangle inequality). f (a, c) ≤ f (a, b) + f (b, c). Note that reflexivity and positivity jointly imply x = y ⇔ f (x, y) = 0, hence, a loss function that satisfies these conditions will be compatible with soft labels. Dice Semimetric Losses We focus here on the Dice loss. For a derivation of the Tversky loss, please refer to Appendix E. Since Dice = 2IoU 1+IoU ⇒ 1 − Dice = 1−IoU 2−(1−IoU) , we have ∆ Dice = ∆ IoU 2−∆ IoU . There exist several alternatives to define ∆ IoU , but not all of them are feasible, e.g., SJL. Generally, it is easy to verify the following proposition: Proposition∆ DML,1 = 1 − x + y 1 − x − y 1 x + y 1 , ∆ DML,2 = 1 − 2 xy 1 2 xy 1 + x − y 1 . (4) ∆ Dice that is defined over integers does not satisfy the triangle inequality [12], which is shown to be helpful in KD [47]. Nonetheless, we can consider a weaker form of the triangle inequality: f (a, c) ≤ ρ(f (a, b) + f (b, c)).(5) Functions that satisfy the relaxed triangle inequality for some fixed scalar ρ and conditions (i)-(iii) of a metric are called semimetrics. ∆ Dice is a semimetric in {0, 1} p [12]. We show DMLs are semimetrics in [0, 1] p in the following theorem: The proof can be found in Appendix A. Moreover, DMLs have properties that are similar to JMLs and they are presented as follows: Theorem 2. ∀x ∈ [0, 1] p , y ∈ {0, 1} p and x ∈ {0, 1} p , y ∈ [0, 1] p , ∆ SDL = ∆ DML,1 = ∆ DML,2 . ∃x, y ∈ [0, 1] p , ∆ SDL = ∆ DML,1 = ∆ DML,2 . Theorem 3. ∀x, y ∈ [0, 1] p , ∆ DML,1 ≤ ∆ DML,2 . The proofs are similar to those given in [47]. Importantly, Theorem 2 indicates that we can safely replace the existing implementation of SDL with DMLs and no change will be incurred since they are identical when only hard labels are presented. Experiments In this section, we demonstrate how models can benefit from soft labels. In particular, using QUBIQ [30], which contains multi-rater information, we show that models trained with averaged annotation maps can significantly surpass those trained with majority votes and random selections. Leveraging LiTS [4] and KiTS [17], we illustrate the synergy with LS and KD. Datasets QUBIQ is a recent challenge held at MICCAI 2020 and 2021 specifically for the evaluation of inter-rater variability. Following [24,41], we use QUBIQ 2020, which contains 7 segmentation tasks in 4 different CT and MR datasets, including prostate (55 cases, 2 tasks, 6 raters), brain growth (39 cases, 1 task, 7 raters), brain tumor (32 cases, 3 tasks, 3 raters) and kidney (24 cases, 1 task, 3 raters). For each dataset, we calculate the averaged Dice score between each rater and majority votes in Table 6 (Appendix C). We can see that for certain datasets such as brain tumor task 2, the inter-rater disagreement can be very high. [5] as the decoder. The model is trained with SGD with an initial learning rate of 0.01, momentum of 0.9, weight decay of 0.0005, and the learning rate is decayed with a poly policy. The batch size is set to 8 and the number of epochs is 150 for QUBIQ, 60 for LiTS and KiTS. We leverage a mixture of CE and DMLs weighted by 0.25 and 0.75, respectively. In this paper, we focus on how models can benefit from soft labels. Since the superiority of SDL over CE has been widely proven in the medical imaging community [10,21] and also in our preliminary experiments as shown in Table 7 (Appendix C), we no longer compare our methods with CE in the experimental sections. Evaluation Metrics We report both the Dice score and the expected calibration error (ECE) [15]. For QUBIQ experiments, we additionally present binarized Dice score (BDice), which is the official evaluation metrics used in QUBIQ challenge. To compute BDice, both predictions and soft labels are thresholded at different probability levels (0.1, 0.2, ..., 0.8, 0.9). Then we compute the Dice score at each level and average these scores with all thresholds. For all experiments, we conduct 5-fold cross validation, making sure that each case is presented in exactly one validation set and report the mean values in the aggregated validation set. Results on QUBIQ In Table 1, we compare different training methods on QUBIQ using UNet-ResNet50, including hard labels obtained through (i) majority votes [25] (ii) random sampling each rater's annotation [23]; and soft labels derived from (i) averaging across all annotations [13,41,25] (ii) label smoothing [43]. In the literature [13,41,25], annotations are usually averaged with uniform weights. We additionally consider weighting each rater's annotation by its Dice score with respect to majority votes, so that a rater who deviates far from majority votes receives a low weight. Note that for all methods, the Dice score and ECE are computed with respect to majority votes while BDice is calculated as illustrated in Section 3.3. Generally, models trained with soft labels not only become more accurate, but also more calibrated. The rich multi-rater information can be captured by simply averaging their annotations. In particular, averaging annotations with uniform weights obtains the highest BDice, while weighted average achieves the highest Dice score. Overall, we find weighted average outperforms other methods, except on Brain tumor T2 where there exists a high degree of disagreement among raters (see also Table 6 in Appendix C). We hypothesize that improved results could be obtained with more sophisticated weighting schemes such as learned weights [14], which is beyond the scope of this paper. We compare our method with state-of-the-art (SOTA) methods in Table 2. In our method, we average annotations with uniform weights for Brain tumor T2 and with each rater's Dice score for all other datasets. Our method that simply averages annotations to produce soft labels obtains superior results compared to methods that adopt complex architectures or training techniques. Results on LiTS and KiTS Wang and Blaschko [47] empirically found that a calibrated teacher can distill a more accurate student. In parallel, Menon et al. [29] argued the reason why KD works is because the teacher provides an estimation of Bayes class-probabilities p * (y|x) and this can lower the variance of the student's empirical loss. In Ap- pendix B, we prove |E[p * (y|x)−f (x)]| ≤ E[|E[y|f (x)]−f (x)|]. That is, the bias of the estimation is bounded above by the calibration error and this explains why calibration of the teacher would be important for the student. Inspired by this, we apply a recent kernel density estimator (KDE) [34] that provides consistent estimation of E[y|f (x)]. We then adopt it as a post-hoc calibration method to replace the temperature scaling to calibrate the teacher in order to improve the performance of the student. For more details of KDE, please refer to Appendix D. In Table 3, we compare models trained with hard labels, LS [43] and KD [18] on LiTS and KiTS, respectively. For all KD experiments, we use UNet-ResNet50 as the teacher. Again, we obtain noticeable improvements in both the Dice score and ECE. Also note that for UNet-ResNet18 and UNet-EfficientNetB0 on LiTS, the student's Dice score exceeds that of the teacher. Ablation Studies In models are trained with a mixture of CE and SDL, and CE is compatible with soft labels (ii) although SDL pushes predictions towards vertices, in a binary segmentation setting, this can still add some regularization effects. However, SDL is significantly outperformed by DMLs. As for DMLs, we find ∆ DML,1 is superior to ∆ DML,2 , and therefore we suggest the use of ∆ DML,1 in practice. In Table 5, we ablate the contribution of each KD term on LiTS and KiTS with a UNet-ResNet18 student. CE and DML represents adding the CE and DML term between the teacher and the student, respectively. Results shown in the table verify the effectiveness of the proposed loss and KDE method. In Table 8 (Appendix C), we illustrate the effect of bandwidth that controls the smoothness of the kernel density estimation. Conclusion In this work, we introduce the Dice semimetrics losses (DMLs), which are identical to the soft Dice loss (SDL) in a standard setting with hard labels, but are fully compatible with soft labels. Our extensive experiments on the public QUBIQ, LiTS and KiTS benchmarks support that incorporating soft labels leads to higher Dice score and lower calibration error, indicating that these losses can find wide application in diverse medical image segmentation problems. Hence, we suggest to replace the existing implementation of SDL with DMLs. Proof. We omit the subscript since the proof for ∆ DML,1 and ∆ DML,2 are identical. Note that 0 ≤ ∆ DML = ∆ JML 2−∆ JML ≤ ∆ JML ≤ 1 and ∆ JML satisfies the triangle inequality. ∆ DML (a, c) ≤ ρ(∆ DML (a, b) + ∆ DML (b, c)) (6) ⇒ ∆ JML (a, c) 2 − ∆ JML (a, c) ≤ ρ∆ JML (a, b) 2 − ∆ JML (a, b) + ρ∆ JML (b, c) 2 − ∆ JML (b, c) (7) ⇒ ∆ JML (a, b) + ∆ JML (b, c) 2 − ∆ JML (a, b) − ∆ JML (b, c) ≤ ρ∆ JML (a, b) 2 − ∆ JML (a, b) + ρ∆ JML (b, c) 2 − ∆ JML (b, c) (8) ⇒∆ JML (a, b) 1 2 − ∆ JML (a, b) − ∆ JML (b, c) − ρ 2 − ∆ JML (a, b) ≤ ∆ JML (b, c) ρ 2 − ∆ JML (b, c) − 1 2 − ∆ JML (a, b) − ∆ JML (b, c) .(9) Assume that ∆ DML (a, b) + ∆ DML (b, c) ≤ 1 ρ , otherwise ρ-relaxed triangle inequality trivially holds. Now we show that the left-hand side of Eq. (9) is no greater than 0. ρ 2 − ∆ JML (a, b) ≥ 1 2 − ∆ JML (a, b) − ∆ JML (b, c) (10) ⇒ ρ + ∆ DML (a, b) 1 + ∆ DML (a, b) ≥ 1 + (1 + ρ)∆ DML (b, c) 1 + ∆ DML (b, c) (11) ⇒ ρ + ∆ DML (a, b) 1 + ∆ DML (a, b) ≥ 1 + (1 + ρ)( 1 ρ − ∆ DML (a, b)) 1 + ( 1 ρ − ∆ DML (a, b))(12)⇒ρ 2 − ρ − 1 + ρ 2 ∆ 2 DML (a, b) ≥ 0 (13) ⇒ρ ≥ 1 + √ 5 2 ≈ 1.62.(14) where in Eq. (12), we use the fact the right hand side of Eq. (11) is an increasing function of ∆ DML (b, c). Similarly, we can show that the right-hand side of Eq. (9) is no less than 0 when ρ ≥ 1+ B Theorem 4 Theorem 4. |E[p * (y|x) − f (x)]| ≤ E[|E[y|f (x)] − f (x)|]. Proof. Subscripts are omitted for simplicity and all expectations are over x, y ∼ P x,y , the (unknown) data generating distribution. Note that E[|E[y|f (x)]−f (x)|] is the definition of the calibration error. The proof is similar to [33]. |E[p * (y|x) − f (x)]| (15) =|E[p * (y|x) − E[y|f (x)] + E[y|f (x)] − f (x)]| (16) =|E[p * (y|x)] − E[E[y|f (x)]] + E[E[y|f (x)] − f (x)]| (17) =| E[p * (y|x) − y] =0 +E[E[y|f (x)] − f (x)]| (18) ≤E[|E[y|f (x)] − f (x)|].(19) In Eq. (19), we apply Jensen's inequality due to the convexity of the absolute value. C Tables ? We believe after the teacher is trained with LS, the optimal temperature in KD should generally be lower in order to achieve a low calibration error. If one still uses a high temperature, the teacher might be overly-smoothed and become under-confident. Inspired by the fact that we can improve KD by decreasing the teacher's calibration error, we propose to use E[y|f (x)] as a distillation signal to supervise the student. Note that by the definition of the calibration error, E[y|f (x)] is the optimal recalibration mapping of f (x) that will give 0 calibration error. Moreover, given only access to f (x), predicting E[y|f (x)] is the Bayesian optimal classifier. Nevertheless, E[y|f (x)] cannot be computed exactly because it depends on the unknown data generating distribution. Hence, we adopt a kernel density estimator (KDE) that is proven to be consistent [34]: E[y|f (x)] ≈ E[y|f (x)] = n i=1 k kernel (f (x), f (x i ))y i n i=1 k kernel (f (x), f (x i )) ,(21) For binary classification, k kernel is a Beta kernel: k Beta (f (x j ), f (x i )) = f (x j ) αi−1 (1 − f (x j )) βi−1 Γ (α i + β i ) Γ (α i )Γ (β i ) ,(22) and for the multiclass setting, k kernel is a Dirichlet kernel: k Dir (f (x j ), f (x i )) = Γ ( K k=1 α ik ) K k=1 Γ (α ik ) K k=1 f (x j ) α ik −1 k(23) with Γ (·) the gamma function, α i = f (xi) h + 1 and β i = 1−f (xi) h + 1, where h ∈ R >0 is a bandwidth parameter that controls smoothness of the kernel density estimation. Like the temperature in temperature scaling, the estimation becomes smoother when we increase h. The complexity of KDE is O(n) for a single pixel, leading to O(n 2 ) overall complexity. With gradient descent training, KDE is estimated using a minibatch. Nonetheless, in semantic segmentation, n = B × H × W and it can be a large number, where B is the batch size, H and W are height and width, respectively. To decrease the amount of computation, for each training data batch, we randomly sample n key key points such that n key n. Consequently, for each pixel x, we only perform the kernel computation with respect to these key points. Hence, the complexity is considerably reduced from O(n 2 ) to O(n×n key ), resulting a negligible extra cost. Another consideration in semantic segmentation is that labels are usually highly unbalanced. When sampling n key key points from a n batchsize × H × W data batch, it is quite likely that some classes are omitted. Consequently, during kernel computation, all predictions with respect to these classes will become 0. To overcome this, if the number of unique classes within a training data batch is n unique , we sample each unique class with n key nunique . Besides, since medical images are dominated by background pixels, we empirically find that it suffices to only apply KDE to misclassified pixels as well as pixels that are near the boundary. Through the kernel computation, we broadcast the information contained in these key points to all pixels. We utilize both labels and predictions of these key points to adjust each pixel's predictions. In Table 5, we ablate the role of KDE. In Table 8, we show the effect of bandwidth on LiTS and KiTS with a UNet-ResNet18 student. We notice the performance of KDE is sensitive to the bandwidth. E The Compatible Tversky Loss and the Compatible Focal Tversky Loss The Tversky index is defined as |v| |v| + α(|x| − |v|) + β(|y| − |v|)(24) where α, β ≥ 0 controls the magnitude of penalties for false-positives and falsenegatives, respectively. With α = β = 0.5, the Tversky index becomes the Dice score; with α = β = 1, it simplifies to the Tanimoto coefficient. Adopting the same idea as SDL and SJL, the soft Tversky loss (STL) [39] is written as ∆ STL : x ∈ [0, 1] p , y ∈ {0, 1} p → 1 − x, y α x 1 + β y 1 + (1 − α − β) x, y . Since SDL as a special case of STL is incompatible with soft labels, so is STL. The key is to write |v| as 1 2 ( x + y 1 − x − y 1 ), so we can define the compatible Tversky loss (CTL) as ∆ CTL : x ∈ [0, 1] p , y ∈ [0, 1] p (26) → 1 − x + y 1 − x − y 1 2α x 1 + 2β y 1 + (1 − α − β)( x + y 1 − x − y 1 ) .(27) Due to the asymmetric nature of the Tversky index, ∆ CTL does not satisfy symmetry, thus cannot be a semimetric. However, it is compatible with soft labels. Theorem 5. ∆ CTL satisfies reflexivity and positivity. Proof. Let S 1 = {i : x i ≥ y i } and S 2 = {i : x i < y i }. ∆ CTL (x, y) = 0 (28) ⇒(α − β) x 1 + (−α + β) y 1 + (α + β) x − y 1 = 0 (29) ⇒(α − β) i∈S1 x i + (α − β) i∈S2 x i + (−α + β) i∈S1 y i + (−α + β) i∈S2 y i (30) (α + β) i∈S1 x i − (α + β) i∈S2 x i − (α + β) i∈S1 y i + (α + β) i∈S2 y i = 0 (31) ⇒α i∈S1 (x i − y i ) + β i∈S2 (y i − x i ) = 0(32) where the last equality holds if and only if x = y. Again, following a similar proof as given in [47], we can show that CTL is identical to STL in a standard setting with hard labels. Theorem 6. ∀x ∈ [0, 1] p , y ∈ {0, 1} p and x ∈ {0, 1} p , y ∈ [0, 1] p , ∆ STL = ∆ CTL . ∃x, y ∈ [0, 1] p , ∆ STL = ∆ CTL . We can add a focal term as in the focal Tversky loss [26,1] and we call it the compatible focal Tversky loss (CFTL): ∆ CFTL : x ∈ [0, 1] p , y ∈ [0, 1] p → ∆ γ CTL (33) where γ is the focal term. With γ > 1, CFTL focuses more on less accurate predictions that have been misclassified. In Figure 1, we plot the loss value of CFTL with α = 0.7, β = 0.3 when the soft label is 0.8. With increased γ, the loss value flattens when the prediction is near the ground-truth and increases more rapidly when the prediction is far away from the soft label. Note that we can also add the focal term to JMLs and DMLs. Together with [47], variants of commonly used regional losses, e.g. the soft Jaccard loss, the soft Dice loss, the soft Tversky loss and the focal Tversky loss, are proposed. These new variants are identical to the original versions in a standard setting with hard labels, but are fully compatible with soft labels. Given that soft labels are pervasive in machine learning, we recommend to replace the existing implementation of these losses with our new variants. Theorem 1 . 1∆ DML,1 and ∆ DML,2 are semimetrics in [0, 1] p . 13 . 13Gros, C., Lemay, A., Cohen-Adad, J.: SoftSeg: Advantages of soft versus binary training for image segmentation. MIA (2021) 14. 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Fig. 1 . 1The loss value of CFTL with α = 0.7, β = 0.3 when the soft label is 0.8. 1. ∆ Dice satisfies reflexivity and positivity if and only if ∆ IoU does.Among the definitions of ∆ IoU , Wang and Blaschko [47] found only two can- didates as defined in Eq. (3) that satisfy reflexivity and positivity. Following Proposition 1, we transform these two IoU losses and define Dice semimetric losses (DMLs) ∆ DML,1 , ∆ DML,2 : [0, 1] p × [0, 1] p → [0, 1] as Table 1 . 1Results on QUBIQ with UNet-ResNet50.Dataset Metric Majority Random Uniform Weighted LS Prostate T1 Dice (%) 95.65 95.80 95.74 95.99 95.71 BDice (%) 94.72 95.15 95.19 95.37 94.91 ECE (%) 0.51 0.39 0.22 0.20 0.36 Prostate T2 Dice (%) 89.39 88.87 89.57 89.79 89.82 BDice (%) 88.31 88.23 89.35 89.66 88.85 ECE (%) 0.52 0.47 0.26 0.25 0.41 Brain growth Dice (%) 91.09 90.65 90.94 91.46 91.23 BDice (%) 88.72 88.81 89.89 90.40 89.88 ECE (%) 1.07 0.85 0.27 0.34 0.41 Brain tumor T1 Dice (%) 86.46 87.24 87.74 87.78 87.84 BDice (%) 85.74 86.59 86.67 86.92 86.91 ECE (%) 0.62 0.55 0.38 0.36 0.37 Brain tumor T2 Dice (%) 58.58 48.86 52.42 61.01 61.23 BDice (%) 38.68 49.19 55.11 44.23 40.61 ECE (%) 0.25 0.81 0.74 0.26 0.22 Brain tumor T3 Dice (%) 53.54 54.64 53.45 56.75 57.01 BDice (%) 52.33 53.53 51.98 53.90 55.26 ECE (%) 0.17 0.17 0.14 0.09 0.11 Kidney Dice (%) 62.96 68.10 71.33 76.18 71.21 BDice (%) 62.47 67.69 70.82 75.67 70.41 ECE (%) 0.88 0.78 0.67 0.53 0.62 All Dice (%) 76.80 76.30 77.31 79.85 79.15 BDice (%) 72.99 75.59 77.00 76.59 75.26 ECE (%) 0.57 0.57 0.38 0.29 0.35 Table 2. Comparing with SOTA methods on QUBIQ using UNet-ResNet50. All results are BDice (%). Dataset Dropout [11] Multi-head [14] MRNet [24] SoftSeg [13,25] Ours Prostate T1 94.91 95.18 95.21 95.02 95.37 Prostate T2 88.43 88.32 88.65 88.81 89.66 Brain growth 88.86 89.01 89.24 89.36 90.40 Brain tumor T1 85.98 86.45 86.33 86.41 86.92 Brain tumor T2 48.04 51.17 51.82 52.56 55.11 Brain tumor T3 52.49 53.68 54.22 52.43 53.90 Kidney 66.53 68.00 68.56 69.83 75.67 All 75.03 75.97 76.18 76.34 78.14 Table 4 , 4we compare SDL with DMLs. For QUBIQ, we train UNet-ResNet50 with soft labels obtained from weighted average and report BDice. For LiTS and KiTS, we train UNet-ResNet18 with KD and present the Dice score. For a fair comparison, we disable KDE in all KD experiments. We find models trained with SDL can still benefit from soft labels to a certain extent, this is because (i) Table 3 . 3Results on LiTS and KiTS.Model Metrics LiTS KiTS Hard LS KD Hard LS KD UNet-R50 Dice (%) 59.79 60.59 -72.66 73.92 - ECE (%) 0.51 0.49 - 0.39 0.33 - UNet-R18 Dice (%) 57.92 58.60 60.30 67.96 69.09 71.34 ECE (%) 0.52 0.48 0.50 0.44 0.38 0.44 UNet-EB0 Dice (%) 56.90 57.66 60.11 70.31 71.12 71.73 ECE (%) 0.56 0.47 0.52 0.39 0.35 0.39 UNet-MB2 Dice (%) 56.16 57.20 58.92 67.46 68.19 68.85 ECE (%) 0.54 0.48 0.50 0.42 0.38 0.41 DL3+-R18 Dice (%) 56.10 57.07 59.12 69.95 70.61 70.80 ECE (%) 0.53 0.50 0.52 0.40 0.38 0.40 Table 4 . 4Comparing SDL with DMLs.Dataset Hard ∆SDL ∆DML,1 ∆DML,2 QUBIQ 72.99 73.79 76.59 76.42 LiTS 57.92 58.12 59.31 59.12 KiTS 67.96 68.26 69.29 69.07 Table 5. Evaluating each KD term on LiTS and KiTS with a UNet-ResNet18 student. All results are the Dice score (%). Dataset Hard CE DML KDE LiTS 57.92 58.23 59.31 60.30 KiTS 67.96 68.14 69.29 71.34 Table 6 .Table 8 . 68The averaged Dice score between each rater and majority votes. D1: Prostate T1, D2: Prostate T2, D3: Brain growth T1, D4: Brain tumor T1, D5: Brain tumor T2, D6: Brain tumor T3, D7: Kidney T1.Table 7. Comparing CE with DMLs on QUBIQ, LiTS and KiTS using UNet-ResNet50. All results are the Dice score (%). Comparing different bandwidths on LiTS and KiTS with a UNet-ResNet18 student. All results are the Dice score (%). Why can label smoothing (LS) increase a model's calibration but a teacher trained with LS could hurt the performance of the student [31]Dataset D1 D2 D3 D4 D5 D6 D7 # Raters 6 6 7 3 3 3 3 Dice (%) 96.49 92.17 91.20 95.44 68.73 92.71 97.41 Dataset QUBIQ LiTS KiTS CE 74.97 57.76 68.52 DML 76.80 59.79 72.66 Bandwidth 0 5e-5 1e-4 5e-4 1e-3 5e-3 1e-2 LiTS 59.31 59.05 59.07 59.97 60.30 59.56 59.62 KiTS 69.29 69.05 69.80 70.41 71.34 68.75 69.18 AcknowledgementsWe acknowledge support from the Research Foundation -Flanders (FWO) through project numbers G0A1319N and S001421N, and funding from the Flemish Government under the Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen programme. The resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation -Flanders (FWO) and the Flemish Government. A Novel Focal Tversky Loss Function with Improved Attention U-Net for Lesion Segmentation. N Abraham, N M Khan, ISBI. 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Gragera, A., Suppakitpaisarn, V.: Relaxed triangle inequality ratio of the Sørensen-Dice and Tversky indexes. TCS (2018)

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{'abstract': 'The soft Dice loss (SDL) has taken a pivotal role in many automated segmentation pipelines in the medical imaging community. Over the last years, some reasons behind its superior functioning have been uncovered and further optimizations have been explored. However, there is currently no implementation that supports its direct use in settings with soft labels. Hence, a synergy between the use of SDL and research leveraging the use of soft labels, also in the context of model calibration, is still missing. In this work, we introduce Dice semimetric losses (DMLs), which (i) are by design identical to SDL in a standard setting with hard labels, but (ii) can be used in settings with soft labels. Our experiments on the public QUBIQ, LiTS and KiTS benchmarks confirm the potential synergy of DMLs with soft labels (e.g. averaging, label smoothing, and knowledge distillation) over hard labels (e.g. majority voting and random selection). As a result, we obtain superior Dice scores and model calibration, which supports the wider adoption of DMLs in practice. Code is available at https://github.com/zifuwanggg/JDTLosses.', 'arxivid': '2303.16296', 'author': ['Zifu Wang \nESAT-PSI, KU Leuven\nLeuvenBelgium\n', 'Teodora Popordanoska \nESAT-PSI, KU Leuven\nLeuvenBelgium\n', 'Jeroen Bertels \nESAT-PSI, KU Leuven\nLeuvenBelgium\n', 'Robin Lemmens \nDepartment of Neurosciences\nKU Leuven\nLeuvenBelgium\n\nDepartment of Neurology\nUZ Leuven\nLeuvenBelgium\n', 'Matthew B Blaschko \nESAT-PSI, KU Leuven\nLeuvenBelgium\n'], 'authoraffiliation': ['ESAT-PSI, KU Leuven\nLeuvenBelgium', 'ESAT-PSI, KU Leuven\nLeuvenBelgium', 'ESAT-PSI, KU Leuven\nLeuvenBelgium', 'Department of Neurosciences\nKU Leuven\nLeuvenBelgium', 'Department of Neurology\nUZ Leuven\nLeuvenBelgium', 'ESAT-PSI, KU Leuven\nLeuvenBelgium'], 'corpusid': 257805104, 'doi': '10.48550/arxiv.2303.16296', 'github_urls': ['https://github.com/zifuwanggg/JDTLosses.', 'https://github.com/rwightman/', 'https://github.com/open-mmlab/mmsegmentation'], 'n_tokens_mistral': 14122, 'n_tokens_neox': 12164, 'n_words': 6427, 'pdfsha': 'a4ad1f58595c56e615f6f1ac426c961ddd454631', 'pdfurls': ['https://export.arxiv.org/pdf/2303.16296v2.pdf'], 'title': ['Dice Semimetric Losses: Optimizing the Dice Score with Soft Labels', 'Dice Semimetric Losses: Optimizing the Dice Score with Soft Labels'], 'venue': []}

arxiv

A mean-field version of Bank-El Karoui's representation of stochastic processes * 13 Mar 2023 March 14, 2023 Xihao He Xiaolu Tan Jun Zou A mean-field version of Bank-El Karoui's representation of stochastic processes * 13 Mar 2023 March 14, 2023Stochastic processBank-El Karoui's representation theoremMean-field game MSC2010 subject classification: 60G4093E2060G0793E15 We investigate a mean-field version of Bank-El Karoui's representation theorem of stochastic processes. Under different technical conditions, we established some existence and uniqueness results. As motivation and first applications, the results of mean-field representation provide a unified approach for studying various mean-field games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing and the MFG with singular control, etc. As a crucial technical step, a stability result was provided on the classical Bank-El Karoui's representation theorem. It has its own interests and other applications, such as deriving the stability results of optimizers (in the strong sense) for a class of optimal stopping and singular control problems. Introduction Let (Ω, F, P) be a complete probability space, equipped with the filtration F = (F t ) t≥0 satisfying the usual conditions. For T ∈ [0, ∞], T denotes the space of all stopping times taking values in [0, T ]. Let f : [0, T ] × Ω × R −→ R be such that ℓ −→ f (t, ω, ℓ) is continuous and strictly increasing from −∞ to ∞ for every (t, ω) ∈ [0, T ] × Ω, and (t, ω) −→ f (t, ω, ℓ) is progressively measurable with E T 0 |f (t, ℓ)|dt < ∞ for every ℓ ∈ R. Bank and El Karoui [3] proved that any optional process Y , satisfying Y T = 0 and some further integrability and path regularity conditions, can be represented with an optional process L by the following expression: Y τ = E T τ f t, sup τ ≤s<t L s dt F τ , a.s., for all τ ∈ T .(1) Such a representation result provides a powerful tool in stochastic analysis, particularly for solving various stochastic optimization problems, such as the optimal consumption problem in Bank and Riedel [6], the optimal stopping problem (see, e.g., Bank and Föllmer [5]), and the singular control problem (see, e.g., Bank [1], Bank and Kauppila [4], Chiarolla and Ferrari [23]). It has also been used to solve an exit contract design problem by He, Tan, and Zou [30]. For extensions of this classical representation theorem, let us mention the work of Bank and Besslich [2] for the case with Meyer-measurable processes, and the work of Ma and Wang [37], Qian and Xu [39] for some variations of the reflected stochastic backward differential equation. In this study, we study a mean-field version of the representation result (1) in the form: Y τ = E T τ f t, L(L), sup τ ≤s<t L s dt F τ , a. s., for all τ ∈ T , a.s., (2) where the new added mean-field term L(L) represents the distribution of the process L. Motivated by its applications, we will in fact study a more general version, by considering the conditional distribution L(L|G) of L knowing a sub-σ-filed G ⊂ F and by letting the process Y depend also on conditional distribution L(L|G). To establish the mean-field representation result (2), we consider it as a fixed-point problem. Given a distribution m, one replaces L(L) by m in (2) and then applies the classical representation (1) to obtain a representation process L m , which induce a distributionm := L(L m ). The problem then reduces to find a fixed point of the map m −→m. In a first approach, we applied the Schauder fixed-point theorem to find a fixed point or equivalently to obtain a mean-field representation result. In this approach, a key step is to derive the continuity of m −→ L m in some sense. Therefore, we establish a novel stability result of Bank-El Karoui's representation theorem (2). In a second approach, we applied Tarski's fixed-point theorem to establish the meanfield representation result, in which a partial order was introduced on the space for L(L) (or the conditional law) and formulate some technical conditions on f to verify the required conditions in Tarski's fixed-point theorem. In a third approach, we consider a special structure condition which reduces the fixed point problem to the one-dimensional case satisfying a monotonicity condition, and then obtain an existence and uniqueness result. The main motivation to introduce and to study the mean-field representation in (2) is its applications in the mean-field game (MFG) theory. The MFG consists of studying the limit behavior of the Nash equilibrium of a stochastic differential game with a large number of agents in a symmetric setting. In this symmetric game, each agent interacts with others through the empirical distribution induced by the states (and/or optimal actions) of all agents, which converges to the distribution (or conditional distribution in the case with common noise) of the state (and/or optimal actions) of one representative agent as the population number increases. This (conditional) distribution term is called the mean-field interaction term. The MFG was first introduced by Lasry and Lions [35] and by Huang, Caines, and Malhamé [31]. Since then, MFG has been attracting considerable research interest from various communities because of its widespread applications in fields, such as economics, biology, and finance. In particular, we refer to Cardaliaguet [19] for a basic introduction and to Carmona and Delarue [22] for recent advances. Although the initial formulation of the MFG is obtained by using a classical optimal control problem for each agent, other stochastic optimization problems can be considered to obtain other versions of the MFG problem. For example, when agents solve the optimal stopping problems with interaction in terms of the (conditional) distribution of the optimal stopping times (or stopped processes), the MFG of timing is obtained. The MFG of timing has been initially studied by Nutz [38] and by Carmona, Delarue, and Lacker [21] by using mainly a probabilistic approach. In the setting with the underlying process being given by a stochastic differential equation (SDE), the MFG of timing has been studied by Bertucci [7] with a PDE approach, by Bouveret, Dumitrescu, and Tankov [13] and by Dumitrescu, Leutscher, and Tankov [26] with a linear programming approach. Next, when the agents in the game solve the singular control problem, the MFG with singular control is obtained. Such a formulation of the MFG has been studied by Fu and Horst [29], Fu [28], Cao, Guo, and Lee [18], where the main results consist in the existence of solution, its approximation by those of regular controls, and its approximation by the corresponding N -player games, etc. Other formulations, such as the ergodic MFG with singular control was studied by Cao, Dianetti, and Ferrari [16]. Dianetti, Fischer, and Nendel [25] studied a submodular MFG with singular control, and Campi, De Angelis, Ghio, and Livieri [15] studied a MFG of finite-fuel capacity expansion with singular control. The mean-field representation result (2) provides a unified method for studying various versions of the MFG. In fact, for various classical stochastic optimization problems, the optimal solution can be derived from process L in the classical Bank-El Karoui's representation (1) for an appropriate process Y . In the mean-field setting, the (conditional) distribution of L induces the (conditional) distribution of the optimizer, which would be exactly the required interaction term in the corresponding MFG problem. Consequently, by specifying the dependence of the generator function f in L(L) in the mean-field representation result (2) according to various applications, a solution to (2) would induce solutions to some MFGs. Concretely, similar to the applications of (1) in the optimal stopping problem, singular control problem, optimal consumption problem, one can apply (2) to study the MFG of timing, MFG with singular control, and MFG of optimal consumption. Crucially, for a given process Y , the same induced process L in (1) provides simultaneously optimal solutions to various optimization problems. Based on this result, the mean-field representation result (2) can be used to study the MFG with multiple populations. Our first main contribution consists in providing a nontrivial extension of the classical Bank-El Karoui's representation theorem to the case in which the process Y and generator function f depend on the (conditional) distribution of the representation process L, as in (2). As first applications, the novel representation results provide a unified approach for studying various MFGs (e.g., MFG of timing and MFG with singular control), in a setting with (possibly) multiple populations and common noise. In the literature, these MFG problems are typically studied case by case, and in the one population setting. Furthermore, because our representation theorem is formulated on a fixed filtered probability space, the derived solutions (or ε-solution) of the MFGs are strong solutions. This would be a major difference of our results compared with existing MFG literature (see more discussions in Remarks 2.19 and 2.25). As a potential application, one can consider an optimal stopping version of Lions' [36] MFG planning problem, that is, design a MFG (of timing) such that the equilibrium solution follows the given marginal distributions. Similar to Ren, Tan, Touzi, and Yang [40], the principal-agent type arguments can be used in [30] with our mean-field representation results to solve a MFG planning problem of timing. Moreover, as a crucial technical step, a stability result was established on Bank-El Karoui's representation (1); that is, when (Y, f ) changes slightly, the representation process L changes slightly in some sense. In its applications to various stochastic optimization problems, such as the optimal stopping, singular control, and optimal consumption problems, process L explicitly induces the optimal solutions. Consequently, stability of L implies the stability of the corresponding optimal solutions. We will provide some stability results of the optimal stopping times (resp. the optimal control) for a class of optimal stopping (resp. singular control) problems, which should also be novel in the optimal stopping/control theory. The rest of the paper is organized as follows. In Section 2, we formulate our mean-field version of Bank-El Karoui's representation theorem and then establish some existence and uniqueness results under different technical conditions. Then, we demonstrate how the mean-field representation results can be applied to solve different MFG problems. In Section 3, we provide a stability result of the classical Bank-El Karoui's representation theorem, which constitutes a key technical step for establishing the representation theorem in the first approach, and induces stability of the optimal solutions of a class of optimal stopping and singular control problems. Finally, some technical proofs of the results are reported in Section 4. A mean-field version of Bank-El Karoui's representation of stochastic processes and its applications in MFGs We first describe some notations and then formulate our mean-field version of Bank-El Karoui's representation of stochastic processes in Section 2.1. In Section 2.2, we provide some existence and uniqueness results of the mean-field representation under different technical conditions. In Section 2.3, we show how the representation results can be applied to study different MFGs. Preliminaries and formulation of the mean-field representation Let (Ω, F, P) be a complete probability space equipped with a filtration F := (F t ) t∈[0,T ) satisfying the usual conditions, that is, t −→ F t is right-continuous, and F 0 contains all P-null sets in F. d L (v 1 , v 2 ) := inf ε ≥ 0 : v 1 (t − ε) ∨ 0 − ε ≤ v 2 (t), v 2 (t − ε) ∨ 0 − ε ≤ v 1 (t), ∀t ∈ (0, T ) , when T < +∞ and d L (v 1 , v 2 ) := +∞ n=1 2 −n (d L (v 1 | [0,n) , v 2 | [0,n) ) ∧ 1), when T = +∞, so that it is a Polish space (see Appendix C for a detailed proof). Finally, let us define V + • := ∪ η∈R V + η , and V + := V + −∞ . Let E be a (nonempty) Polish space, we denote by P(E) the space of all (Borel) probability measures on E. The space P(E) is equipped with the weak convergence topology, under which it is also a Polish space. We denote by L 0 F (Ω, E) (resp. L 0 G (Ω, E) the space of all F-measurable (resp. G-measurable)) E-valued random variables, which is equipped with the topology induced by the convergence in probability. Similarly, L 0 G (Ω, P(E)) denotes the space of all G-measurable random measures, and L 0 F (Ω, D × V + ) denotes the space of all random variables taking values in D × V + . We are given a family (X m , Y m , f m ) m∈L 0 G (Ω,P(E)) , where for each m ∈ L 0 G (Ω, P(E)), f m : [0, T ] × Ω × R −→ R, X m is an adapted D-valued càdlàg process, and Y m is a R-valued optional process. Throughout the paper, we assume the following conditions on (Y m , f m ) m∈L 0 G (Ω,P(E)) . 4 Assumption 2.1. For each m ∈ L 0 G (Ω, P(E)), Y m is a R-valued optional process of class (D) and upper semi-continuous in expectation (in the sense of Definition A.1) such that Y m T = 0, and f m satisfies that ℓ −→ f m (t, ω, ℓ) is continuous and strictly increasing from −∞ to ∞ for all (t, ω), and (t, ω) −→ f m (t, ω, ℓ) is progressively measurable with E T 0 f m (t, ℓ) dt < ∞, for all ℓ ∈ R.(3) Given a fixed measurable function Ψ : Ω × D × V + −→ E, our mean-field version of Bank-El Karoui's representation theorem consists of finding a couple (L, m), where L is an optional process and m ∈ L 0 G (Ω, P(E)), satisfying the following: Y m τ = E T τ f m t, sup s∈[τ,t) L s dt F τ , and m = L Ψ(X m , L) G ,(4) for all τ ∈ T and L being the running maximum process of L defined by, with the convention that sup ∅ = −∞, L t := sup s∈[0,t) L s , t ∈ [0, T ). Throughout the paper, L Ψ(X m , L) G denotes the conditional distribution of Ψ(X m , L) knowing G. In particular, L Ψ(X m , L) G ∈ L 0 G (Ω, P(E)). Remark 2.2. (i) For the mean-field term m in (4), the (conditional) distribution of the running maximum process L is considered rather than that of L itself. The technical reason is that, given an optional process Y , the solution L in the classical Bank-El Karoui's representation (1) may not be unique, but the corresponding running maximum process L is unique. Moreover, because process L does not have a priori path regularity, its induced mean-field term would be a probability measure on R [0,T ] . By considering L, the mean-field term becomes a probability measure on V + , which has better topological (metric) and order structure than R [0,T ] . Crucially, in all the wellknown applications of Bank-El Karoui's representation theorem in stochastic optimization, the optimal solutions can be characterized by L. (ii) We include the processes X m in the formulation of the mean-field representation (4). In practice, X m can be considered to be an underlying process, and typically (Y m , f m ) are functionals of X m . It is then desirable to include the (conditional) law of X m in the mean-field term m. For example, X m can be given as a diffusion process, with D = R d , defined by X m t = X m 0 + t 0 b m (s, X m s )ds + t 0 σ m (s, X m s )dW s ,(5)where (b m , σ m ) : Ω × R + × R d −→ R d × S d are coefficient functions such that (ω, t) −→ (b m , σ m )(ω, t, x) is progressively measurable for all x ∈ R d , and W is a standard Brownian motion. (iii) In some applications of the classical Bank-El Karoui's representation (1) in stochastic optimization problems, the optimal solution is given by a transformation of the running maximum process L. This is our main reason to consider the (conditional) law of Ψ(X, L) in the meanfield term, for some functional Ψ : Ω × D × V + −→ E. For example, Ψ can be defined by, with 5 E = D × V + and the given processes (X + , X − , L + , L − ) : Ω −→ D × D × V + × V + such that X − ≤ X + and L − ≤ L + , a.s., Ψ(ω, x, l) := X − t (ω) ∨ x t ∧ X + t (ω), L − t (ω) ∨ l t ∧ L + t (ω) t∈[0,T ) . (iv) In many of the mean-field problems, a common noise filtration G = (G t ) t∈[0,T ] is considered in place of common noise σ-field G. In this case, (X m , Y m , f m ) depends on the mean-field term m in an adaptive way; that is, at each time t ≥ 0, (X m t , Y m t , f m (t, ·)) depends on a conditional distribution of L knowing G t . However, in most situations, G is a Brownian filtration or a subfiltration satisfying the (H)-hypothesis so that the conditional distribution knowing G t at time t ≥ 0 is equivalent to that knowing σ-field G T . Here, we choose to use a unique σ-field G, but require that (X m , Y m , f m ) is F-adapted, but m ∈ L 0 G (Ω, P(E)) may not be F-adapted (see Assumption 2.1). In particular, when G = {∅, Ω}, the conditional distribution knowing G becomes a deterministic probability measure. Thus, any G-measurable random variable is a constant, and L 0 G (Ω, P(E)) can be identified as P(E). Remark 2.3. In Assumption 2.1, it is assumed that Y m T = 0. When (Y m T ) m∈L 0 G (Ω,P(E)) is a family of integrable random variables, one can define the processes Y m by Y m t := Y m t − E[ξ m |F t ] , so that Y m T = 0 and the corresponding representation result for Y m in (4) is equivalent to the following: Y m τ = E Y m T + T τ f m t, sup s∈[τ,t) L s dt F τ , and m = L Ψ(X m , L) G . Existence and uniqueness results We now provide some existence and uniqueness results on the mean-field representation (4) under different (abstract) technical conditions. A setting with continuity conditions Recall that we are given a family (X m , Y m , f m ) m∈L 0 G (Ω,P(E)) , P(E) is equipped with the weak convergence topology, and L 0 G (Ω, P(E)) is equipped with the topology of convergence in probability. Theorem 2.1. Let Assumption 2.1 hold true, and for each m ∈ L 0 G (Ω, P(E)), t −→ Y m t has almost surely upper semi-continuous paths. Suppose also that • for a countable partition (A i ) i≥1 of Ω such that A i ∈ F and P[A i ] > 0 for each i ≥ 1, ∪ i≥1 A i = Ω and A i ∩ A j = ∅ for all i = j, one has G = σ(A i : i ≥ 1),(6) so that L 0 G (Ω, P(E)) can be identified as the space (P(E)) N ; • the map (x, l) −→ Ψ(ω, x, l) is continuous for all ω ∈ Ω, and for some convex compact subset K ⊂ L 0 G (Ω, P(E)), we have L Ψ(X m , L) G : m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + ) ⊆ K; 6 • for all ℓ ∈ R, {m n } n∈N ⊂ K with m n −→ m ∞ in probability for some m ∞ ∈ K, one has lim n→∞ E d D (X m n , X m ∞ ) + sup t∈[0,T ] Y m n t − Y m ∞ t + T 0 f m n (s, ℓ) − f m ∞ (s, ℓ) ds = 0. Then there exists a couple (L, m) as solution to the mean-field representation (4). The proof of Theorem 2.1 is reported in Section 4.2, which is mainly based on the Schauder fixed-point theorem (recalled in Appendix B). Furthermore, the σ-field G is assumed to be generated by a countable partition of Ω so that the space L 0 G (Ω, P(E)) can be identified as the space (P(E)) N , for which obtaining a convex compact subset K is easy as required in the Schauder fixed-point theorem. This condition is restrictive and excludes the case in which G is generated by a Brownian motion. However, it is not surprising as our representation result (4) stays in a strong formulation such that the probability space (Ω, F, P) with the filtration F and the sub-σ-field G is fixed. This is also the main difficulty in the literature of the MFG with common noise. Nevertheless, our setting can be considered as a first step for studying mean-field problems with general common noise. For more general countably generated σ-field G, a further approximation technique can be applied to obtain existence result. See, for example, Carmona, Delarue, and Lacker [20] for a MFG with common noise. However, this usually leads to a weak formulation of the mean-filed problem at the limit, which is out of the scope of this paper. Example 2.5. Let E = D × V + , with L − , L + being two stochastic processes with paths in V + such that L − t ≤ L + t , a.s. for all t ∈ [0, T ), we define Ψ : Ω × D × V + −→ E by Ψ(ω, x, l) := x t , L − t (ω) ∨ l t ∧ L + t (ω) t∈[0,T ) . which satisfies the continuity condition of Ψ in Theorem 2.1. Furthermore, let D = R d , and {X m } m∈L 0 G (Ω,P(E)) be the diffusion processes defined as the unique solution of (5) with coefficient functions (b m , σ m ) : Ω × R + × R d −→ R d × S d , which are uniformly bounded and uniformly Lipschitz continuous in x. Assume also that (X m 0 ) m∈L 0 G (Ω,P(E)) is uniformly bounded, and for m n −→ m ∞ in probability, one has (b m n , σ m n ) −→ (b m ∞ , σ m ∞ ) uniformly. Then it is standard to check that P • (X m ) −1 : m ∈ L 0 G (Ω, P(E)) is tight in P(D), and E d D (X m n , X m ∞ ) −→ 0. Next, we can further obtain that (see also Section 4.1 for a brief proof ) L Ψ(X m , L) G : m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + ) is tight in L 0 G (Ω, P(E)), so that finding a convex compact set K as required in Theorem 2.1 is easy. Because φ is uniformly bounded by C, for any processes (X, L) ∈ L 0 F (Ω, D) × L 0 F (Ω, V + ), the induced process Ψ(X, L) is increasing and uniformly bounded by C. Under the Lévy metric on V, we can check that L Ψ(X m , L) G : m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + ) is tight. A setting with monotonicity conditions In a second setting, where E is a partially ordered Polish space, one can apply Tarski's fixedpoint theorem to obtain an existence result under some monotonicity condition. Notice that the definition of the partial order, partially ordered Polish space, the corresponding complete lattice as well as Tarski's fixed-point theorem are recalled in Appendix B. Let E be a partially ordered Polish space with partial order ≤ E , a function φ : E −→ R is said to be increasing if φ(e 1 ) ≤ φ(e 2 ) for all e 1 , e 2 ∈ E such that e 1 ≤ E e 2 . We introduce a partial order ≤ p on L 0 G (Ω, P(E)) as follows: for all m 1 , m 2 ∈ L 0 G (Ω, P(E)), we say m 1 ≤ p m 2 if and only if E φ(x)m 1 (dx) ≤ E φ(x)m 2 (dx), a.s.,(7) for all R-valued bounded increasing measurable functions φ : E −→ R. Theorem 2.2. Let Assumption 2.1 hold true, D be equipped with a partial order ≤ D such that (D, ≤ D ) becomes a partially ordered Polish space, E be a partially ordered Polish space with partial order ≤ E , and Ψ : Ω × D × V + −→ E satisfy that, x 1 ≤ D x 2 , l 1 t ≤ l 2 t , t ∈ [0, T ) =⇒ Ψ(ω, x 1 , l 1 ) ≤ E Ψ(ω, x 2 , l 2 ). Suppose in addition that there exists a complete lattice (K, ≤ p ) as subset of L 0 G (Ω, P(E)) such that, for all m 1 , m 2 ∈ K, m 1 ≤ p m 2 =⇒ X m 1 ≤ D X m 2 , a.s., Y m 1 −Y m 2 is supermartingale and f m 1 (·) ≥ f m 2 (·). (8) Then there exists a couple (L, m) being a solution to the mean-field representation (4). Remark 2.7. The technical supermartingale condition in (8) is adapted from Assumption B in Carmona, Delarue and Lacker [21]. Similar conditions are typically used in submodular games, see, e.g., Topkis [42]. E := {(v 1 , v 2 ) ∈ V + × V + : |v 1 (t)| ≤ C, |v 2 (t)| ≤ C t ∈ [0, T )}, we introduce a partial order ≤ E by l 1 ≤ E l 2 if l 1 t ≤ l 2 t , for all t ∈ [0, T ). Then (E, ≤ E ) is a complete lattice and the corresponding space L 0 G (Ω, P(E)) with the induced order ≤ p in (7) is also a complete lattice. Furthermore, given an increasing and left continuous function φ : R −→ R uniformly bounded by C > 0, we define Ψ(ω, x, l) := sup s∈[0,t) φ(x s ) t∈[0,T ) , φ(l t ) t∈[0,T ) . Let us consider the partial order ≤ D on D by x 1 ≤ D ≤ x 2 if x 1 t ≤ x 2 t for all t ∈ [0, T ] , then it is clear that Ψ satisfies the monotonicity condition in Theorem 2.2. Example 2.9. Let E = V + , L − , L + be two stochastic processes with paths in V + such that L − t ≤ L + t , a.s. for all t ∈ [0, T ), and Ψ : Ω × D × V + −→ E be defined by Ψ(ω, x, l) := L − t (ω) ∨ l t ∧ L + t (ω) t∈[0,T ) . Then it is direct to check that Ψ satisfies the monotonicity condition in Theorem 2.2. Further, on the space V + and L 0 F (Ω, V + ), let us introduce respectively the partial order ≤ v and ≤ l as follows: l 1 ≤ v l 2 if l 1 t ≤ l 2 t for all t ∈ [0, T ), and L 1 ≤ l L 2 if and only if L 1 t ≤ v L 2 t , a.s. for all t ∈ [0, T ). Let L 0 := L ∈ L 0 F (Ω, V + ) : L − ≤ l L ≤ l L + , . Then, (L 0 , ≤ l ) is a complete lattice. Moreover, let K := L( L|G) : L ∈ L 0 . Then, with the partial order ≤ p induced by (E = V + , ≤ v ) as in (7), the space (K, ≤ p ) is also a complete lattice. ] × Ω × R × R −→ R be such that x −→ f (t, ω, ℓ, x) is decreasing, g : [0, T ] × Ω × R −→ R be such that g(·, x) is F-adapted for all x ∈ R, and x −→ g(·, x) is decreasing, h : [0, T ] × Ω × P(R) −→ R be such that h(·, m) is F-adapted for all m ∈ L 0 G (Ω, P(R)). Moreover, assume that E T 0 |g(s, x)| + h 2 (s, m) ds < +∞. Let f m (·) and Y m be given as follows: f m (·) := f (·, φ, m ), Y m t := Y m 0 + t 0 g(s, φ, m )ds + t 0 h(s, m)dW s , t ∈ [0, T ]. Then it is easy to verify that f m and Y m satisfy the property (8). A setting with the dimension-reduction structure condition We finally consider a case with special structure conditions on f , such that one can obtain existence and uniqueness result of the representation (4). Let E = V + , G = {∅, Ω}, so that L 0 G (Ω, P(E) can be identified as the space P(E). Let us take E = V + as canonical space with canonical process L, for m ∈ P(E), define m t (dx) := m • L −1 t ∈ P(R), for all t ∈ [0, T ). Theorem 2.3. Let Assumption 2.1 hold true, and Ψ : Ω × D × V + be defined by Ψ(ω, x, l) := l, for all (ω, x, l) ∈ Ω × D × V + . Further, for some stochastic process X and Y , together with f : Ω × [0, T ) × R −→ R, one has (X m , Y m ) = ( X, Y ), and f m (ω, t, ℓ) := f ω, t, ℓ − m t (φ) , for all m ∈ P(E), where φ : R −→ R is bounded differential satisfying φ ′ (x) ∈ [0, 1) for all x ∈ R. The optional process L in the representation theorem (1) of Y w.r.t.f has almost surely non-decreasing paths. Then there exists a unique solution (L m , m) to the mean-field representation (4). Remark 2.11. With the above special structure, the representation problem (4) can be reduced to the following fixed point problem: for each t ∈ [0, T ], one looks for y t ∈ R such that y t = E[φ(L t + y t )]. By considering the above fixed-point problem, one can deduce the existence and uniqueness result (see its proof in Section 4.2). Applications in the mean-field games We now present some first applications of the above mean-field representation results in the mean-field games (MFGs) to obtain some existence results. In particular, we stay in a strong formulation in the sense that we fix a filtered probability space (Ω, F, F, P) and a common noise σ-field G ⊂ F. A MFG of timing with multiple populations We consider a MFG with possibly infinitely number of populations indexed by ℓ ∈ R, where each agent solves an optimal stopping problem with time horizon T < ∞. The mean-field (interaction) term is then a measure on [0, T ] R , which is equipped with the product topology. For a given interaction term µ ∈ L 0 G Ω; P D × [0, T ] R , there is an underlying process X µ having paths in D. Further, let T denote the collection of all F-stopping times taking value in [0, T ], G : [0, T ]×Ω×L 0 G Ω; P D×[0, T ] R −→ R and g ℓ : [0, T ]×Ω×L 0 G Ω; P D×[0, T ] R −→ R, ℓ ∈ R, be the reward functions. In each population ℓ ∈ R, a representative agent solves the following optimal stopping problem: sup τ ∈T J ℓ (τ, µ), with J ℓ (τ, µ) := E τ 0 g ℓ (t, µ)dt + G τ (µ) .(9)Definition 2.12. (i) Let ε ≥ 0, a ε-solution of the MFG of timing is a couple {µ * , (τ * ℓ ) ℓ∈R }, such that µ * = L X µ * , (τ * ℓ ) ℓ∈R G , and for each ℓ ∈ R, τ * ℓ ∈ T satisfies that J ℓ (τ * ℓ , µ * ) ≥ sup τ ∈T J ℓ (τ, µ * ) − ε. (ii) When ε = 0, a ε-solution is also called a solution to the MFG of timing. Let us refer to Nutz [38], and Carmona, Delarue and Lacker [21] for a detailed interpretation/justification of the formulation of the MFG of timing, as well as its solution as a Nash equilibrium. Assumption 2.13. (i) For all (t, ω, µ) ∈ [0, T ] × Ω × P D × [0, T ] R , the map ℓ −→ g ℓ (t, ω, µ) is strictly increasing and continuous. (ii) For each µ ∈ L 0 G Ω, P D×[0, T ] R and ℓ ∈ R, the process (t, ω) −→ g ℓ (t, ω, µ) is progressively measurable and E T 0 |g ℓ (t, µ)|dt < +∞, and the process t −→ G t (µ) is F-optional of class(D) and u.s.c. in expectation (see Definition A.1). Remark 2.14. In the applications of the mean-field representation theorem, we will set Y · = G(·). In particular, as discussed in Remark 2.3, we do not assume that G T (·) = 0 in this setting. In the setting with continuity conditions of X µ , g ℓ (t, µ), G t (µ) in µ, we obtain an existence result based on Theorem 2.1. Proposition 2.15. Let Assumption 2.13 hold true, G be generated by a countable partition of Ω as in (6), and the map t −→ G t (µ) has almost surely left upper semi-continuous paths for each µ ∈ L 0 G Ω, P D × [0, T ] R . Suppose in addition that (i) The map ℓ −→ g ℓ is uniformly continuous in the sense that, for any ε > 0, there exists a δ ε > 0 independent of ω and µ such that |g ℓ 1 (t, ω, µ) − g ℓ 2 (t, ω, µ)| ≤ ε, whenever |ℓ 1 − ℓ 2 | ≤ δ ε . (ii) The set L X µ G : µ ∈ L 0 G Ω, P D × [0, T ] R is tight in the space L 0 G (Ω, P(D)). (iii) for µ n , µ ∞ ∈ L 0 G Ω; P D × [0, T ] R with lim n→∞ µ n = µ ∞ almost surely, we have for all ℓ ∈ R, lim n→∞ E d D (X µ n , X µ ∞ ) + sup t∈[0,T ] G t (µ n ) − G t (µ ∞ ) + T 0 g ℓ (t, µ n − g ℓ (t, µ ∞ ds = 0. Then for any ε > 0, there exists a ε-solution to the MFG of timing. The proof will be reported in Section 4.3. Remark 2.16. The above result gives only existence of ε-solution, but not solution to the MFG. The main reason is that the first hitting time τ ℓ of a process L to some level ℓ ∈ R is generally not continuous w.r.t. the paths of L. We therefore need to modify the process L slightly to make its paths be strictly increasing, so that the corresponding hitting time becomes continuous w.r.t. the path. Consequently, an equilibrium in terms of the distribution of L can only induce a εequilibrium in terms of the distribution of τ ℓ . Nevertheless, as a ε-solution in the strong sense, it consists still of a novel result, see more discussions in Remark 2.19. We next consider a setting with monotonicity conditions. Let ≤ D be a partial order on D such that (D, ≤ D ) is a partially ordered Polish space (see Appendix B for its definition and Example 2.8 for an example). We first introduce a partial order on the space L 0 G (Ω, P(D × [0, T ] R )), for µ 1 , µ 2 ∈ L 0 G Ω, P D × [0, T ] R , we say µ 1 ≤ st µ 2 if for all Π ℓ∈R [t ℓ , T ] ⊂ Π ℓ∈R [0, T ] with t ℓ ∈ [0, T ], ℓ ∈ R, and t ℓ = 0 for only finitely many ℓ, D φ(e)µ 1 de, Π ℓ∈R [t ℓ , T ] ≤ D φ(e)µ 2 de, Π ℓ∈R [t ℓ , T ] , a.s., for all bounded increasing functions φ : D −→ R. Proposition 2.17. Let Assumption 2.13 holds true, and assume in addition that, (i) there exists a bounded increasing map ϕ : D −→ I c , with I c being a closed interval of R, such that for any µ 1 , µ 2 ∈ L 0 G (Ω; P(D × [0, T ] R )), (ϕ ⊗ Id)#µ 1 = (ϕ ⊗ Id)#µ 2 =⇒ X µ 1 , G(µ 1 ), g · (·, µ 1 ) = X µ 2 , G(µ 2 ), g · (·, µ 2 ) . (10) (ii) for all µ 1 , µ 2 ∈ L 0 G (Ω; P(D × [0, T ] R )), (ϕ ⊗ Id)#µ 1 ≤ st (ϕ ⊗ Id)#µ 2 implies X µ 1 ≤ D X µ 2 , G(µ 1 ) − G(µ 2 ) is a supermartingale and g · (·, µ 1 ) ≥ g · (·, µ 2 ). (iii) for any m ∈ L 0 G (Ω; P(I c × [0, T ] R )), there exists a µ ∈ L 0 G (Ω; P(D × [0, T ] R )) such that m = (ϕ ⊗ Id)#µ. Then there exists a solution to the MFG of timing. We also refer to Example 2.10 for an example satisfying Condition (ii) in Proposition 2.17. Remark 2.18. Notice that the formulation of our MFG of timing with infinitely many population in Definition 2.12 as well as the results in Propositions 2.15 and 2.17 cover the case with finitely many populations (or one population). Indeed, given finitely many (g i ) n i=1 , let us define a family of functionals (g ℓ ) ℓ∈R as follows: for all ℓ ∈ R, (t, ω, µ) ∈ [0, T ] × Ω × L 0 G P D × [0, T ] n , g ℓ (t, ω, µ) :=      g 1 (t, ω, µ) + (ℓ − 1), ℓ ∈ (−∞, 1], (i + 1 − ℓ)g i (t, ω, µ) + (ℓ − i)g i+1 (t, ω, µ), ℓ ∈ [i, i + 1], i = 1, · · · , n − 1, g n (t, ω, µ) + (ℓ − n), ℓ ∈ [n, +∞).(11) Further, a function defined on L 0 G P D × [0, T ] n can be easily extended on L 0 G P D × [0, T ] R by using the projection π((t ℓ ) ℓ∈R ) = (t i ) n i=1 . Remark 2.19. The MFG of timing has already been studied considerably in the literature. In the first works, Nutz provided some examples of the MFG of timing with explicit solution, Carmona, Delarue and Lacker [21] general MFG of timing with common noise, and obtained some existence as well as limit theory results. Further, Bertucci [7,8] study the problem with PDE arguments. In [13], Bouveret, Dumitrescu and Tankov studied the existence and uniqueness of the MFG of timing in a setting with diffusion underlying process, by a relaxed formulation approach. This has been further extended to the setting with control and optimal stopping in [26]. Recently, Burzoni and Campi [14] studied a MFG with control and optimal stopping in a setting with absorption and common noise. All of these work stay in a setting with one population, our mean-field representation results allows one to consider automatically a setting with multiple population indexed by ℓ ∈ R. Moreover, most of above works studied the weak solutions of the MFG, in the sense that the probability space may not be fixed, except Theorem 3.5 in [21] under the monotonicity condition. Under the monotonicity conditions, our Proposition 2.17 can be considered as an extension of Theorem 3.5 in [21] to the case with infinitely many populations. Without the monotonicity condition, but under continuity conditions, our Proposition 2.15 provides the existence of strong ε-solutions, in the sense that the probability space, as well as the filtration, is fixed. A MFG with singular control We next study a MFG problem with singular control (based on the singular control problem in Bank [1]), in a setting with possibly many populations. Let (θ i ) i∈N ∈ R N be a sequence of real constants, for each i ≥ 1, we denote by V + θ i the space of all increasing and left continuous functions v on [0, T ) such that v(0) = θ i . Recall also that D denotes the space of all D-valued càdlàg paths on [0, T ] if T < ∞, or on [0, T ) if T = ∞. The mean-field interaction in this MFG is a G-measurable random measure µ ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )). Next, for each µ ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )) , there exists a D-valued càdlàg processes X µ . Then for each population i ∈ N, a representative agent solves the following singular control problem: inf Θ∈A i J(Θ, µ), with J(Θ, µ) := E T 0 c(t, µ, Θ t )dt + T 0 k(t, µ)dΘ t ,(12) where c : [0, T ] × Ω × L 0 G (Ω, P(D × Π i∈N V + θ i )) × R −→ R represents the running cost of the problem, k : [0, T ] × Ω × L 0 G (Ω, P(D × Π i∈N V + θ i )) −→ R is the cost related to the control Θ, and with an optional process Θ i having paths in V + θ i , A i := Optional processes Θ having paths in V + θ i such that Θ t ≤ Θ i t for all t ≥ 0, a.s. . Let us refer to [1] for the the motivation and application of the above singular control problem. Notice that for different populations, the cost functions c and k are common, but the constraints process Θ i are different. Definition 2.20. A solution to the MFG with singular control is a pair (µ * , (Θ * i ) i∈N ), such that µ * ∈ L 0 G Ω, P(D × Π i∈N V + θ i ) and Θ * i ∈ A i , i ∈ N, satisfy µ * = L X m * , (Θ * i ) i∈N G and J(Θ * i , µ * ) = inf Θ∈A i J(Θ, µ * ), for each i ∈ N. Now we recall one of the main theorems in [1], which shows how the Bank-El Karoui's representation theorem can be used to construct the unique solution to the singular control problem (12) with given m. Assumption 2.21. (i) For any (t, ω, µ) ∈ [0, T ]×Ω×P(D×Π i∈N V + θ i ), the map ℓ −→ c(t, ω, µ, ℓ) is strictly convex with continuous derivative c ′ (t, ω, m, ℓ) = ∂ ∂ℓ c(t, ω, m, ℓ) strictly increasing from −∞ to +∞, i.e. c ′ (t, ω, m, −∞) = −∞ and c ′ (t, ω, m, +∞) = +∞. (ii) For each (µ, ℓ) ∈ L 0 G (Ω, P(D×Π i∈N V + θ i ))×R, the process (t, ω) −→ c(ω, t, µ, ℓ) is progressively measurable and E T 0 |c(t, µ, ℓ)|dt < +∞, E T 0 inf ℓ∈[θ i ,(θ i )t] |c(t, µ, ℓ)|dt < +∞, i ∈ N. (iii) For each µ ∈ L 0 G Ω, P(Π i∈N V + θ i ) , the process (t, ω) −→ k(t, ω, µ) is an optional process of Class (D), continuous in expectation with k(T, µ) = 0, and sup Θ∈A i E T 0 k(t, µ) dΘ t < +∞, i ∈ N. Theorem 2.4 (Bank [1]). Let Assumption 2.21 hold true. Then each fixed µ ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )), the unique minimizer for the problem (12) is given by Θ µ,i, * t := sup s∈[0,t) L µ t ∧ Θ i t ∨ θ i , t ∈ [0, T ], i ∈ N, where L µ is the optional process solving the representation problem − k(τ, µ) = E T τ c ′ t, µ, sup s∈[τ,t) L µ s dt F τ , a.s., for all τ ∈ T .(13) Remark 2.22. Theorem 2.4 reveals that one can reduce the singular control problem (12) to the corresponding representation problem (13), thus one can also reduce the corresponding MFG with singular control to a mean-field version of the representation (4). First, based on the mean-field representation results in Theorem 2.1, one can deduce the following existence result for the above MFG with singular control. Proposition 2.23. Let Assumption 2.21 hold true and G be generated by a countable partition of Ω as in (6), and t −→ k(t, ·, µ) have almost surely left upper semi-continuous paths for each µ ∈ L 0 G Ω, P(Π i∈N V + θ i ) . Suppose in addition that (i) for µ n , µ ∞ ∈ L 0 G Ω; P D × Π i∈N V + θ i with lim n→∞ µ n = µ ∞ in probability, we have for all ℓ ∈ R, lim n→∞ E d(X µ n , X µ ∞ ) + sup t∈[0,T ] k(t, ω, µ n ) − k(t, ω, µ ∞ ) + T 0 c ′ (t, ω, µ n , ℓ) − c ′ (t, ω, µ ∞ , ℓ) ds = 0. (ii) the collection of probability measures L(X µ |G) : µ ∈ L 0 G Ω; P D × Π i∈N V + θ i is tight in L 0 G (Ω, P(E)) by identifying L 0 G (Ω, P(E) ) as (P(E)) N under the weak convergence topology. Then there exists a solution to the MFG with singular control. We next provide an existence result based on Theorem 2.2 under some monotonicity conditions. Let ≤ D be a partial order on D such that (D, ≤ D ) is a partially ordered Polish space. Recall that the partial order ≤ v on V + θ i is defined by l 1 ≤ v l 2 if l 1 t ≤ l 2 t for all t ∈ [0, T ), i ∈ N. Then, by Example B.3, V + θ i , for all i ∈ N, is a partially ordered Polish space, and Π i∈N V + θ i is also a partially ordered Polish space with the product order. Moreover, there exists a partial order ≤ p on L 0 G Ω; P D × Π i∈N V + θ i defined by (7) by letting E = D × Π i∈N V + θ i . Proposition 2.24. Suppose that the function c and k satisfy Assumption 2.21. Moreover, we assume that (i) there exists a bounded increasing map ϕ : D −→ I c , with I c being a closed interval of R, such that for any µ 1 , µ 2 ∈ L 0 G Ω; P D × Π i∈N V + θ i , (ϕ ⊗ Id)#µ 1 = (ϕ ⊗ Id)#µ 2 =⇒ X µ 1 , k(·, µ 1 ), c ′ (·, µ 1 , ·) = X µ 2 , k(·, µ 2 ), c ′ (·, µ 2 , ·) , (14) (ii) for all µ 1 , µ 2 ∈ L 0 G Ω; P D × Π i∈N V + θ i , (ϕ ⊗ Id)#µ 1 ≤ p (ϕ ⊗ Id)#µ 2 implies k(·, µ 2 ) − k(·, µ 1 ) is a supermartingale, c ′ (·, µ 1 , ·) ≥ c ′ (·, µ 2 , ·), (iii) for any m ∈ L 0 G Ω; P I c × Π i∈N V + θ i , there exists a µ ∈ L 0 G Ω; P D × Π i∈N V + θ i such that m = (ϕ ⊗ Id)#µ. Then there exists a solution to the MFG with singular control. Remark 2.25. The MFG with singular control has been investigated in several works during the last years. In Fu and Horst [29], the authors studied a general mean field game with both regular control and singular control, where the interaction takes place only in states, and provided existence of solutions as well as approximation results by regular controls. Fu [28] extended the model in [29] to the case with jumps, where the interaction takes place in both states and controls. In both [29] and [28], the solutions of the MFG are in the relaxed form. In Campi, De Angelis and Ghio [15], the authors studied a MFG with special interaction terms, and obtained existence of solution as well as convergence rate of the corresponding N -players symmetric game. Cao and Guo [17] considered a MFG with singular control with special dynamic and reward function, and obtained an explicit solution. Cao, Dianetti and Ferrari [16] studied an ergodic MFG with singular control, where the interaction takes a special form, and proved the existence and uniqueness of solution in the strong sense. In Dianetti, Ferrari, Fischer and Nendel [25], the authors provided a unifying framework for several different submodular games including the MFG with singular control, and obtained existence of strong mean field equilibrium by Tarski's fixed point theorem based on order structure assumptions. Let us also mention the work of Bertucci [9] where a MFG with singular control problem has been studied by PDE approaches. Our framework of MFG with singular control is quite closed to that in [29] and [25]. Compared with [29], we consider the MFG with only singular control but not regular control by adapting the framework of Bank [1]. While MFG in [29] is a one population model and the solutions are in relaxed sense, our Proposition 2.23 gives existence results for MFG of multiple populations and the solution is in strong sense, i.e. the solutions are in a fixed filtered probability space. Our Proposition 2.24 can be considered as a multiple population extension of the results in [25]. Technically, our Proposition 2.24 is based on the representation results in Theorem 2.2, whose proof is based on Tarski's fixed point theorem under some order structure conditions, which is similar to the technique used in [25]. A mean field game of optimal consumption We next introduce a MFG of optimal consumption with one population, based on the infinite time horizon (i.e. T = ∞) model in Bank and Riedel [6]. Let us denote by D − the space of all R-valued làdcàg functions on R + equipped with the Skorokhod topology, so that it is a Polish space. In this MFG, the mean-field interaction term is a G-measurable random measure µ ∈ L 0 G Ω, P(D × D − ) . Given µ, the market interest rate is a progressively measurable process (r µ s ) s≥0 , and a 15 consumption process C is left-continuous and right-limit (làdcàg) adapted increasing process. Next, with the total budget b ∈ R, the set of all admissible consumption processes is defined by A(b) := C is làdcàg adapted increasing process s.t. E +∞ 0 e − t 0 r µ s ds dC t ≤ b . Further, given a consumption process C, with the initial satisfaction level η > 0 and discount constant β > 0, a satisfaction process Y C is defined by Y C t := ηe −βt + t 0 βe −β(t−s) dC s , t ≥ 0. Then with the utility function u : R + × Ω × R × P(D × D − ) −→ R, a representative agent in the MFG solves the optimal consumption problem sup C∈A(b) U (C, µ), with U (C, µ) := E +∞ 0 u(t, Y C t , µ)dt .(15) Let us refer to Bank and Riedel [6] for an economic interpretation of the above optimal consumption model. Moreover, in [5], a characterization of the optimal consumption process C * is derived by using Bank-El Karoui's representation (1). Assumption 2.26. (i) For every (t, ω, µ) ∈ [0, T ]×Ω×L 0 G (P(D×D − )), the map ℓ −→ u(t, ω, µ, ℓ) is strictly concave, increasing in the variable ℓ ∈ [0, +∞) with continuous derivative u ′ (t, ω, µ, ℓ) := ∂ ∂ℓ u(t, ω, µ, ℓ) decreasing from +∞ to 0, i.e. u ′ (t, ω, µ, 0) = +∞ and u ′ (t, ω, µ, +∞) = 0. (ii) For all b ∈ R + , and (µ, ℓ) ∈ L 0 G Ω, P(D × D − ) × R, the process (t, ω) → u(t, ω, µ, ℓ) and e − · 0 rs(µ)ds are progressively measurable, and E +∞ 0 |u(t, µ, ℓ)|dt < + ∞, sup C∈A(b) U (C, µ) < +∞. Theorem 2.5 (Bank and Follmer [5]). Let Assumption 2.26 hold true and µ ∈ L 0 G (Ω; P(D × D − )) be fixed. Then for any Lagrange multiplier λ > 0, the discounted price deflator process λe −β·− · 0 r µ s ds admits the representation, λe −βτ − τ 0 r µ s ds = E +∞ τ βe −βt u ′ t, µ, −e −βt sup s∈[τ,t) L µ,λ s dt F τ a.s. for all τ ∈ T ,(16) for some F-progressively measurable process L µ,λ with upper-right continuous paths and the consumption plan C µ,λ for the problem (15) such that Y C µ,λ t = e −βt η ∨ −1 sup s∈[0,t) L µ,λ s , t ∈ R + , is optimal for its cost b µ,λ = E    − u ′ (t, ω, µ, −e −βt ℓ ) if ℓ < 0, ℓ if ℓ ≥ 0. Under Assumption 2.26, the classical Bank-El Karoui's representation theorem (1) ensures the existence of a representation process L m,λ . Then it remains to verify that the stopping time τ 0 := inf{t ≥ 0 : L µ,λ t ≥ 0} is equal to +∞ a.s. The claim holds true if we observe that 0 ≥ −λe −βτ 0 − τ 0 0 r µ s ds = E T τ 0 f µ sup s∈[τ 0 ,t) L µ,λ s dt F τ 0 = E T τ 0 sup s∈[τ 0 ,t) L µ,λ s dt F τ 0 ≥ 0. (ii) Notice that for different Lagrange multiplier λ ∈ R + , the induced solution C µ,λ solves an optimal consumption problem with budget b µ,λ , which is defined posterior. For the constructive results with an a priori given budget, we may refer to Bank and Kauppila [4] for details. We do not adopt the framework in [4] to formulate our MFG with optimal consumption, because it requires good understanding on how the Lagrange multiplier λ µ and the process λ µ e −βt− t 0 r µ s ds change when µ changes. Let us now define the solution to our MFG of optimal consumption. Definition 2.28. Let b ∈ R + , a mean field equilibrium to the MFG of the optimal consumption problem with total budget b is a pair (µ * , C * ), where µ * ∈ L 0 G Ω, P(D × D − ) , and C * ∈ A(b) such that µ * := L (r µ * , Y C * ) G and C * is the optimal consumption process in the sense that U (C * , µ * ) = sup C∈A(b) U (C, µ * ). First, based on the mean-field representation results in Theorem 2.1, one can deduce the following existence result for the above MFG of optimal consumption. Recall that η > 0 is the given initial satisfaction level. Proposition 2.29. Let Assumption 2.26 hold true and G be generated by a countable partition of Ω as in (6). Assume in addition that (i) there exists a constant η > η such that u ′ (·, µ 1 , ℓ) = u ′ (·, µ 2 , ℓ), r µ 1 (·) = r µ 2 (·), for all ℓ ∈ R,(17) whenever µ 1 (A × B) = µ 2 (A × B), a.s. for all A ∈ B(D), B ∈ B({D ∈ D − : ηe −β· ≤ D · ≤ ηe −β· }). (ii) for µ n , µ ∞ ∈ L 0 G (Ω; P(D × D − )) with lim n→∞ µ n = µ ∞ in probability, we have for all ℓ ∈ R + \ {0}, lim n→∞ E d D (r µ n , r µ ∞ ) + sup t∈R + e − t 0 r µ n s ds − e − t 0 r µ ∞ s ds + ∞ 0 u ′ (t, ω, µ n , ℓ) − u ′ (t, ω, µ ∞ , ℓ) ds = 0. (iii) the collection of probability measures L(r µ |G) : µ ∈ L 0 G (Ω; P(D × D − ))} is tight. Then there exists a solution (µ * , C * ) to the MFG of optimal consumption with budget b * := E +∞ 0 e − t 0 r µ * s ds dC * t .(18) We next provide an existence result based on Theorem 2.2 under some monotonicity conditions. Let ≤ D be a partial order on D such that (D, ≤ D ) is a partially ordered Polish space, and define a partial order ≤ D − on D − by l 1 ≤ D − ≤ l 2 if l 1 t ≤ l 2 t for all t ∈ [0, +∞). Then both D and D − are partially ordered Polish spaces, and there exists a partial order ≤ p on L 0 G (Ω; P(D × D − )) defined as (7), if we let E = D × D − . 1 , µ 2 ∈ L 0 G (Ω; P(D × D − )), (ϕ ⊗ Id)#µ 1 = (ϕ ⊗ Id)#µ 2 =⇒ r µ 1 , u ′ (·, µ 1 , ·) = r µ 2 , u ′ (·, µ 2 , ·) .(19) (ii) for any µ 1 , µ 2 ∈ L 0 G (Ω; P(D × D − )), (ϕ ⊗ Id)#µ 1 ≤ p (ϕ ⊗ Id)#µ 2 =⇒ u ′ (·, µ 1 , ℓ) ≤ u ′ (·, µ 2 , ℓ), for all ℓ ∈ R, (iii) for any m ∈ L 0 G (Ω; P(I c × D − )), there exists a µ ∈ L 0 G (Ω; P(D × D − )) such that m = (ϕ ⊗ Id)#µ. (iv) there exists some constant η > η such that Then there exists a solution (µ * , C * ) to the MFG of optimal consumption with budget b * given by (18). u ′ (·, µ 1 , ℓ) = u ′ (·, µ 2 , ℓ), r µ 1 (·) = r µ 2 (·), for all ℓ ∈ R,(20) Finally, we provide an existence result based on the mean-field representation results in Theorem 2.3. In this setting, r µ is assumed to be independent of µ and G = {∅, Ω}, then all the arguments concerned with D × D − reduce to those with D − . Let us take D − as canonical space with canonical process L, for µ ∈ P(D − ), define µ t (dx) := µ • L −1 t ∈ P(R) , for all t ∈ [0, T ). Proposition 2.31. Let Assumption 2.26 hold true. Assume in addition that (i) for some functionũ ′ : R + × Ω × R −→ R, one has u ′ (t, ω, µ, − e −βt ℓ ) =ũ ′ t, ω, − e −βt ℓ − µ t (φ(− e −βt · )) , for all µ ∈ L 0 G (Ω, P(D − )), where the function φ : (−∞, 0) −→ R is bounded and differentiable, satisfying φ ′ (x) ∈ [0, 1) for all x ∈ (−∞, 0), (ii) for some λ > 0, the optional process L to the representation 2.5 of λe −β·− · 0 rsds w.r.t.ũ ′ has almost surely non-decreasing paths. Then there exists a solution (µ * , C * ) to the MFG of optimal consumption with budget b * given by (18). Remark 2.32. In the setting of Proposition 2.31, we consider a special structure of the dependence of the utility function u in µ by letting the utility of the consumer depends not only its own consumption, but also the average consumption of the whole population. Namely, its satisfaction increases in its own consumption, but decreases w.r.t. the average consumption of the whole population. More concretely, the interaction term is µ t (φ(− e −βt · )). Notice that, by Theorem 2.5, the optimal consumption plan C of an agent is that leads to the satisfaction level Y C t = e −βt η ∨ −1 sup s∈[0,t) L s , t ∈ R + , for a strictly negative process L. The process η ∨ −1 sup s∈[0,t) L s t≥0 can be interpreted as a consumption intensity process, and the interaction term is given by µ t φ − e −βt · = E φ − 1 η ∨ sup s∈[0,t) L s . A stability analysis on Bank-El Karoui's representation theorem In this section, we provide a stability result on Bank-El Karoui's representation theorem. This plays a crucial technical step in the proof of Theorem 2.1 for our mean-field representation (4). At the same time, it would have its own interests and other applications. Recall that T ∈ [0, ∞], (Ω, F, P) is a complete probability space, equipped with a filtration satisfying the usual conditions. Let (f n ) n≥0 and f be a sequence of functions defined on [0, T ] × Ω × R, and (Y n ) n≥0 and Y be a sequence of R-valued optional processes. Let (f n ) n≥1 and f all satisfies all the technical conditions in Bank-El Karoui's representation theorem (see Assumption A.2 and Theorem A.1), and (Y n ) n≥1 and Y are all optional processes of class (D) and are u.s.c in expectation (Definition A.1) such that Y n T = Y T = 0. Then by Bank-El Karoui's representation theorem (Theorem A.1), for a couple (f n , Y n ) (resp. (f, Y )), there exists a representation optional process L n (resp. L) such that, for all τ ∈ T , Y n τ = E T τ f n t, sup τ ≤s<t L n s dt F τ resp.Y τ = E T τ f t, sup τ ≤s<t L s dt F τ , a.s. While the representation processes L n and L may not be unique, the corresponding running maximum L n and L defined below is unique for the given (f n , Y n ) and (f, Y ), L n t := sup 0≤s<t L n s , L t := sup 0≤s<t L s , t ∈ [0, T ]. Theorem 3.1. Assume that (f n , Y n ) n≥0 converges to (f, Y ) in the sense that, for all ℓ ∈ R, lim n→∞ E T 0 |f n (t, ℓ) − f (t, ℓ)|dt + sup t∈[0,T ] |Y n t − Y t | = 0,(21) and, in addition, Y has almost surely left upper semi-continuous paths. f n (t) = f (t) = t, Y n t = (t − 1 2 )1 [ 1 2 , 1 2 + 1 n ) (t), Y t = 0, t ∈ [0, 1]. We observe that (f n , Y n ) −→ (f, Y ) uniformly. In this deterministic setting, L n t can be interpreted as the derivative of the convex envelop of t −→ −Y n t on [t, 1] (see [3,Theorem 2]). By direct computation, L n t = − 2 2 + n(1 − 2t) 1 [0, 1 2 + 1 n ) (t), L t = 0, t ∈ [0, 1). For t = 1 2 , we observe that L n 1/2 does not converge to L 1/2 . But for the running maximum process, we have L n t −→ L t for all t ∈ [0, 1]. (ii) We next provide a counter-example, showing that the convergence Y n t −→ Y t , a.s. for every t ∈ [0, T ] is not enough to ensure the convergence of L n −→ L. We still stay in the deterministic setting, with T = 1, and f n (t) = f (t) ≡ t, Y n t = n(t − 1 2 )1 [ 1 2 , 1 2 + 1 n ) (t), Y t = 0, t ∈ [0, 1]. We observe that Y n t −→ Y t , for all t ∈ [0, 1]. Similarly, we compute that L n t = − 2n 2 + n(1 − 2t) 1 [0, 1 2 + 1 n ) (t), L t = 0, t ∈ [0, 1). In particular, L n 1/2 = − 2n n + 2 −→ − 2 = 0 = L 1/2 . Remark 3.2. In the applications of Bank-El Karoui's representation theorem in the optimal stopping problem, the singular control problem with monotone follower type, and the optimal consumption problem, one can characterize the optimizers with the corresponding representation process L. Consequently, the stability result L n −→ L should imply the corresponding stability results of the optimizers in concrete applications. More precisely, Proposition 3.3 below provides a stability results for a family of optimal stopping problems. Similarly, for a class of singular control problems as in Bank [1] (see Theorem 2.4), the optimal singular control is directly given by L with truncation. The stability result in Theorem 3.1 induces naturally the stability of the corresponding optimal singular controls. In preparation of the Proof of Theorem 3.1, let us define, for each ℓ ∈ R, n ≥ 0, τ ℓ := inf t ≥ 0 : L t ≥ ℓ , τ ′ ℓ := inf t ≥ 0 : L t− > ℓ , and τ n ℓ := inf{t ≥ 0 : L n t ≥ 0}. (22) By Bank and El Karoui [3] (see Theorem A.1), τ ℓ and τ ′ ℓ are respectively the smallest and the biggest solution to the optimal stopping problem: sup τ ∈T J ℓ (τ ), with J ℓ (τ ) := E Y τ + τ 0 f (t, ℓ)dt .(23) Similarly, τ n ℓ is the smallest solution to the optimal stopping problem We also introduce a filtration F = (F t ) t∈[0,T ] , together with a sub-filtration F 0 = (F 0 t ) t∈[0,T ] and a sub-σ-field F 0 , by sup τ ∈T J n ℓ (τ ), with J n ℓ (τ ) := E Y n τ + τ 0 f n (t, ℓ)dt .(24)F t := F t ⊗ σ{[0, s], s ≤ t}, F 0 t := F t ⊗ {∅, [0, T ]}, t ∈ [0, T ], and F 0 := F ⊗ {∅, [0, T ]}. Notice that Θ is clearly a F-stopping time. Let P(Ω) denote the space of all probability measures on (Ω, F ), we equip P(Ω) with the stable convergence topology of Jacod and Mémin [33], that is, the coarsest topology making P −→ E P [ξ] continuous for all bounded random variable ξ : Ω −→ R such that θ → ξ(ω, θ) is continuous for all ω ∈ Ω (see Section D in Appendix for more details). Then for each ℓ ∈ R, n ≥ 0, let us define P ℓ,n ∈ P(Ω) by Proposition 3.3. Let the conditions in Theorem 3.1 hold true. Then there exists a countable set L 0 ⊂ R such that, for all ℓ ∈ R \ L 0 , the following holds true: (i) τ ℓ = τ ′ ℓ , a.s., so that it is the unique solution to the optimal stopping problem (23); (ii) for all N > 0 and ε > 0, one has lim n−→∞ P τ n ℓ ∧ N − τ ℓ ∧ N > ε = 0. Proof (i) We first observe that, for a.e. ω ∈ Ω, ℓ −→ τ ℓ (ω) is left-continuous and increasing. By Jacod and Shiryaev [32, Lemma IV.3.12], there exists a countable subset L 0 ⊂ R such that P τ ℓ = τ ℓ+ = 1, for all ℓ ∈ R \ L 0 . Next, for any h > 0, we observe that τ ℓ ≤ τ ′ ℓ ≤ τ ℓ+h , a.s. Letting h −→ 0, it follows that τ ℓ ≤ τ ′ ℓ ≤ τ ℓ+ , a.s. Therefore, one has τ ℓ = τ ′ ℓ = τ ℓ+ , a.s., for all ℓ ∈ R \ L 0 , so that τ ℓ is the unique solution to the optimal stopping problem (23). (ii) The assumption that Y has almost surely left upper semi-continuous path, and is u.s.c. in expectation implies that (see e.g. Bismut P ℓ,n Θ ≤ t F 0 (ω) = P ℓ,n Θ ≤ t F 0 t (ω) = 1 {τ n ℓ (ω)≤t} , for P ℓ,n -a.e.ω ∈ Ω. Next, we observe that T < ∞ and P ℓ,n | Ω = P for all n ≥ 0, then (P ℓ,n ) n≥0 is relatively compact in P(Ω) under the stable convergence topology (see e.g. Theorem D.1). Then there exists a subsequence {n k } k≥0 of {n} n≥0 , such that P ℓ,n k −→ P ℓ for some P ℓ ∈ P(Ω) under the stable convergence topology as k −→ +∞. In particular, by Proposition D.1, one has P ℓ Θ ≤ t F 0 (ω) = P ℓ Θ ≤ t F 0 t (ω), for P ℓ -a.e.ω ∈ Ω. Since F 0 = F ⊗ {∅, [0, T ]}, one can consider P ℓ Θ ≤ t F 0 as a function of ω ∈ Ω. Let us then define F ω (s) := E P ℓ Θ ≤ s F 0 (ω), for all s ∈ [0, T ] ∩ Q, and F ω (t) := lim sup Q∩[0,T ]∋sցt F ω (s), for all t ∈ [0, T ]. Since t → 1 {Θ≤t} is right-continuous, by dominated convergence theorem, it follows that, • for all t ∈ [0, T ], F ω (t) = P ℓ Θ ≤ t F 0 t (ω), for P ℓ -a.e.ω = (ω, θ) ∈ Ω; • for P-a.e. ω ∈ Ω, the map t → F ω (t) is right-continuous, increasing and takes value in [0, 1]. For each ω ∈ Ω, let F ω (u) ≤ t} = {ω ∈ Ω : u ≤ F ω (t)} = {ω ∈ Ω : u ≤ P[τ ≤ t| F t ](ω)} ∈ F t . In particular, for all u ∈ [0, 1], ω −→ F −1 ω (u) is a stopping time w.r.t. F. We next claim that, for each ℓ ∈ R, sup τ ∈T J ℓ (τ ) = lim k→∞ sup τ ∈T J n k ℓ (τ ) = lim k→∞ E P ℓ,n k Y n k Θ + Θ 0 f n k (t, ℓ)dt = E P ℓ Y Θ + Θ 0 f (t, ℓ)dt = 1 0 E Y F −1 · (u) + F −1 · (u) 0 f (t, ℓ)dt du.(26) Notice that F −1 · (u) is a F-stopping time, then (26) implies that F −1 · (u) is an optimal stopping time to the optimal stopping problem (23) for a.e. u ∈ [0, 1]. When ℓ ∈ R \ L 0 , τ ℓ is the unique solution to the optimal stopping problem (23), so that E P ℓ [ξ] = E P ℓ E P ℓ ξ F 0 = 1 0 Ω ξ(ω, F −1 ω (u))P(dω)du = Ω ξ(ω, τ ℓ (ω))P(dω). Let us consider the bounded random variable ξ : Ω −→ [0, 1] define by ξ(ω, θ) := |τ ℓ (ω) − θ| ∧ 1. As θ −→ ξ(ω, θ) is continuous, and P ℓ,n k −→ P ℓ under the stable convergence topology, it follows that lim k→∞ E τ ℓ − τ n k ℓ ∧ 1 = lim k→∞ E P ℓ,n k [ξ] = E P ℓ [ξ] = E τ ℓ − τ ℓ ∧ 1 = 0. In fact, the above arguments show that, for all ℓ ∈ R \ L 0 , any subsequence {n k } k≥0 has a subsequence {n k i } i≥0 such that lim i→∞ E τ ℓ − τ n k i ℓ ∧ 1 , which proves the Item (ii) in the statement when T < ∞. To conclude the proof in the case where T < ∞, it is enough to prove the claim in (26). For the first equality in (26), we notice that sup τ ∈T J n k ℓ (τ ) − sup τ ∈T J ℓ (τ ) ≤ E T 0 |f n k (t, ℓ) − f (t, ℓ)|dt + sup t∈[0,T ] |Y n k t − Y t | −→ 0 as k −→ ∞. The second equality in (26) follows by the definition of P ℓ,n k , together with the fact that τ n k ℓ is an optimal solution to the optimal stopping problem (24) with n = n k . For the third equality in (26), we first notice that E P ℓ,n k Y n k Θ + Θ 0 f n k (t, ℓ)dt − E P ℓ,n k Y Θ + Θ 0 f (t, ℓ)dt ≤ E T 0 |f n k (t, ℓ) − f (t, ℓ)|dt + sup t∈[0,T ] |Y n k t − Y t | −→ 0 as k −→ ∞. Then, whenever the limit exists, lim k→∞ E P ℓ,n k Y n k Θ + Θ 0 f n k (t, ℓ)dt = lim k→∞ E P ℓ,n k Y Θ + Θ 0 f (t, ℓ)dt . Further, recall that t → Y t is almost surely upper semi-continuous, and P ℓ,n k −→ P ℓ under the stable convergence topology. Then for every K > 0, one has lim k→∞ E P ℓ,n k − K ∨ Y Θ + Θ 0 f (t, ℓ)dt ∧ K ≤ E P ℓ − K ∨ Y Θ + Θ 0 f (t, ℓ)dt ∧ K . Since Y is in class (D), together with the integrability condition of f , it follows that sup k≥0 E P ℓ,n k − K ∨ Y Θ + Θ 0 f (t, ℓ)dt ∧ K − E P ℓ,n k Y Θ + Θ 0 f (t, ℓ)dt ≤ sup τ ∈T E |Y τ | + T 0 |f (t, ℓ)|dt 1 {(|Yτ |+ T 0 |f (t,ℓ)|dt)>K} −→ 0, as K −→ ∞, 23 and E P ℓ − K ∨ Y Θ + Θ 0 f (t, ℓ)dt ∧ K − E P ℓ Y Θ + Θ 0 f (t, ℓ)dt −→ 0, as K −→ ∞. This leads to lim k→∞ E P ℓ,n k Y n k Θ + Θ 0 f n k (t, ℓ)dt ≤ E P ℓ Y Θ + Θ 0 f (t, ℓ)dt . Finally, notice that, for all (integrable) random variable ξ : Ω −→ R, one has E P ℓ ξ = E P ℓ E P ξ F = Ω T 0 ξ(ω, θ)F ω (dθ)P(dω) = Ω 1 0 ξ(ω, F −1 ω (u))duP(dω). With ξ(ω, θ) := Y θ (ω) + θ 0 f (t, ω, ℓ)dt, it follows that E P ℓ Y Θ + Θ 0 f (t, ℓ)dt = 1 0 E Y F −1 · (u) + F −1 · (u) 0 f (t, ℓ)dt du. Since F −1 · (u) are stopping times w.r.t. F, it follows that sup τ ∈T J ℓ (τ ) = lim k→∞ E P ℓ,n k Y n k Θ + Θ 0 f n k (t, ℓ)dt ≤ E P ℓ Y Θ + Θ 0 f (t, ℓ)dt = 1 0 E Y F −1 · (u) + F −1 · (u) 0 f (t, ℓ)dt du ≤ sup τ ∈T J ℓ (τ ), which concludes Claim (26). (ii.b) Let us now consider the case where T = +∞. We fix an arbitrary constant N > 0. For n ≥ 0, let us define the random variables: Z N := ess sup τ ∈T , τ ≥N E Y τ + τ N f (t, ℓ)dt F N , Z n N := ess sup τ ∈T , τ ≥N E Y n τ + τ N f n (t, ℓ)dt F N , and then processes Y and Y n : [0, N ] × Ω −→ R by Y t := Y t 1 {t∈[0,N )} + Z N 1 {t=N } , and Y n t := Y n 1 {t∈[0,N )} + Z n N 1 {t=N } , t ∈ [0, N ]. Under condition (21), it is clear that E sup 0≤t≤N Y n t − Y t −→ 0, as n −→ ∞. Moreover, it is clear that Y and ( Y n ) n≥1 are all optional processes of Class (D), are u.s.c. in expectation and have left upper semi-continous paths. Let us denote by T [0,N ] the collection of all F-stopping times τ taking value in [0, N ], we consider the following optimal stopping problems: sup τ ∈T [0,N] J ℓ (τ ), with J ℓ (τ ) := E Y τ + τ 0 f (t, ℓ)dt ,(27) and sup τ ∈T [0,N] J n ℓ (τ ), with J n ℓ (τ ) := E Y n τ + τ 0 f n (t, ℓ)dt , n ≥ 0.(28) In fact, the above optimal stopping problems have the same Snell envelops as that for (23) and (24). More concretely, by the dynamic programming principal of the optimal stopping problem (see e.g. El Karoui [27]), it follows that, for any σ ∈ T [0,N ] , one has Z ℓ σ := ess sup τ ∈T [0,N] ,τ ≥σ E Y τ + τ 0 f (t, ℓ)dt F σ = ess sup τ ∈T ,τ ≥σ E Y τ + τ 0 f (t, ℓ)dt F σ =: Z ℓ σ , and Z ℓ,n t := ess sup τ ∈T [0,N] ,τ ≥σ E Y n τ + τ 0 f n (t, ℓ)dt F σ = ess sup τ ∈T ,τ ≥σ E Y n τ + τ 0 f n (t, ℓ)dt F σ =: Z ℓ,n t . Recall that τ ℓ (resp. τ n ℓ ) is the smallest optimal stopping times to the problems (23) (resp. (24)), then they can be also given by τ ℓ = inf t ≥ 0 : Y t + t 0 f n (s, ℓ)ds = Z ℓ t , τ n ℓ = inf t ≥ 0 : Y n t + t 0 f n (s, ℓ)ds = Z ℓ,n t , a.s. It follows then the smallest optimal stopping timesτ ℓ,N (resp.τ n ℓ,N ) of (27) (resp. (28)) satisfieŝ τ ℓ,N := inf t ≥ 0 : Y t + t 0 f (s, ℓ)ds = Z t = τ ℓ ∧ N, a.s., andτ n ℓ,N := inf t ≥ 0 : Y n t + t 0 f n (s, ℓ)ds = Z n t = τ n ℓ ∧ N, a.s. We thus reduces the problem to the case T = N < ∞ by considering the optimal stopping problems (27) and (28). Since for P-a.e. ω ∈ Ω, the maps ℓ −→ τ n ℓ and ℓ −→ τ ℓ are left-continuous and non-decreasing, it follows that, for P-a.e. ω ∈ Ω, one has lim m→∞ τ n km ℓ (ω) = τ ℓ (ω), whenver τ ℓ (ω) = τ ℓ+ (ω). L(L − ∨ L ∧ L + )|G) : L ∈ L 0 F (Ω, V + ) is tight in P(V + ). In details, for all n ∈ N, we first define a subset V + −n,n of V + by V + −n,n := {l ∈ V + : −n ≤ l t ≤ n, for all t ∈ (0, T ).}. Clearly for all n ∈ N, V + −n,n is compact and V + = ∪ +∞ n=1 V + −n,n . Then for any ε > 0, there exists some N ε ∈ N, such that P • (L − ) −1 V + −Nε,Nε , P • (L + ) −1 V + −Nε,Nε ≥ 1 − ε 2 . Thus, for any P • (L) −1 with L ∈ L 0 F (Ω, V + ), we have P • (L) −1 V + −Nε,Nε = P {−N ε ≤ L ≤ N ε } ≥ P {−N ε ≤ L − ≤ N ε } ∩ {−N ε ≤ L + ≤ N ε } = 1 − (1 − P {−N ε ≤ L − ≤ N ε } ) − (1 − P {−N ε ≤ L + ≤ N ε } ) = 1 − (1 − P • (L − ) −1 V + −Nε,Nε ) − (1 − P • (L + ) −1 V + −Nε,Nε ) ≥ 1 − ε. Finally, when G is generated by countably many disjoint sets {A i } i∈N with P(A i ) > 0, i ∈ N and Ω = ∪ ∞ i=1 A i , L 0 G (Ω, P(E)) can be identified as the space (P(E)) N by the map ψ : L 0 G (Ω, P(E)) −→ (P(E)) N defined as ψ(m) = (m(ω i )) i∈N , where ω i ∈ A i is arbitrarily chosen. Then, since for each i ∈ N, on the probability space (Ω, F, P i ), where P i := 1 P(A i ) P(· ∩ A i ), we can argue that P i • (Ψ(X m , L)) −1 : m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + ) is tight in P(E). On the other hand, for any m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + ), one has ψ(L(Ψ(X m , L)|G)) = (P i • (Ψ(X m , L)) −1 ) i∈N . Hence the set ψ {L(Ψ(X m , L)|G) : m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + )} is tight in (P(E)) N . Thus, we conclude our proof since ψ is a homeomorphism. Then both of (L 0 , ≤ l ) and (K, ≤ p ), where ≤ p is a partial order induced by (E = V + , ≤ v ) as in (7), are complete lattices. 26 Proof For any subset Γ of L 0 , we define a process as follows, for any t ∈ [0, T ], L t := esssup L∈Γ L t , and the corresponding process L with L t := lim s∈Q→t− L s for all t ∈ (0, T ] with L 0 := −∞. Then L has left-continuous paths by its definition, and for any t ∈ [0, T ], L t is F t -measurable, L t ≤ L t a.s. In fact, for any s, t ∈ (0, T ] with s < t, there exist two increasing sequences (s n ) n∈N , (t n ) n∈N of constants in [0, T ] ∈ Q with s n < t n for all n ∈ N such that lim n→∞ s n = s, lim n→∞ t n = t. Moreover, there exist a P-null set N such that on N C , L sn ≤ L tn for all n ∈ N. Letting n tends to ∞, we have that L s ≤ L t . In other words, L admits increasing paths almost surely. Thus we have L ∈ L 0 . On the other hand, we fix some t ∈ [0, T ] \ Q, for any s ∈ Q with s < t and L ∈ Γ, the inequality L s ≤ L s holds a.s. By the left-continuity of L and definition of L, we have L t ≤ L t holds a.s. Thus the definition of L and arbitrariness of L, imply that L t ≤ L t a.s. and we can conclude that for any t ∈ [0, T ], L t = L t a.s., L is the least upper bound of Γ. Similarly we can prove there exists some L ∈ L 0 is the greatest lower bound of Γ. Then we can conclude that (L 0 , ≤ l ) is a complete lattice. Finally, we observe that ξ −→ L(ξ|G) can be seen as an increasing map from L 0 to K, then the partially ordered set (K, ≤ p ) is also a complete lattice. Corollary 4.3. Let E := {l ∈ V + : |l(t)| ≤ 2C, t ∈ [0, T )}, we consider the partial order ≤ E by l 1 ≤ E l 2 if l 1 t ≤ l 2 t , for all t ∈ [0, T ). Then (E, ≤ E ) is a complete lattice and the corresponding space L 0 G (Ω, P(E)) with the induced order ≤ p is also a complete lattice. Let {m n } n∈N ⊂ K be such that lim n→∞ m n = m ∞ in probability for some m ∞ ∈ K. By the conditions in the theorem, one has Proofs of the main theorems lim n→∞ E d D (X m n , X m ∞ ) + sup t∈[0,T ] Y m n t − Y m ∞ t + sup ℓ∈R T 0 f m n (s, ℓ) − f m +∞ (s, ℓ) ds = 0. Then it follows by Theorem 3.1 that lim n→∞ P d D (X m n , X m ∞ ) ≥ ε + P d L L m n , L m ∞ ≥ ε = 0, or equivalently, lim n→∞ P d D (X m n , X m ∞ ) 2 + d L L m n , L m ∞ 2 ≥ ε = 0, 27 The above implies the continuity of m → (X m , L m ) from K to L 0 F (Ω, D × V + ). By the continuity of the map Ψ, one obtains the continuity of L(Ψ(·)|G) : L 0 F (Ω, D × V + ) −→ K, which leads to the continuity of ψ. We can then conclude with Schauder fixed-point theorem for the existence of a fixed point m * ∈ K such that m * = ψ(m * ). In particular, the couple L(Ψ(X m * , L m * )|G), L m * gives a solution to the mean-field representation in (4). Then ψ 2 is order-preserving by order-preserving property of Ψ. We claim that ψ 1 is also orderpreserving. Indeed, by Theorem A.1 (ii), for any m ∈ K, we have L m t = ess inf σ∈T t,+ ℓ m t,σ ,(30) where ℓ m t,σ is the unique F t -measurable random variable such that E σ t f m (s, ℓ m t,σ )ds F t = E[Y m (t) − Y m (σ)|F t ]. For any m 1 , m 2 ∈ K with m 1 ≤ p m 2 , the conditions in the theorem implies that the process Y m 1 (·) − Y m 2 (·) is a F-supermartingale, then for any σ ∈ T t,+ , we have Y m 1 (t) − Y m 2 (t) ≥ E[Y m 1 (σ) − Y m 2 (σ)|F t ], i.e. E[Y m 1 (t) − Y m 1 (σ)|F t ] ≥ E[Y m 2 (t) − Y m 2 (σ)|F t ]. Then since f is order-reversing w.r.t. m (see (8)), we get the estimate E σ t f m 1 (s, ℓ m 1 t,σ )ds F t = E[Y m 1 (t) − Y m 1 (σ)|F t ] ≥ E[ Y m 2 (t) − Y m 2 (σ)|F t ] = E σ t f m 2 (s, ℓ m 2 t,σ )ds F t ≥ E σ t f m 1 (s, ℓ m 2 t,σ )ds F t a. s. Since f m 1 (·, ℓ) is strictly increasing in ℓ, one can then conclude that ℓ m 1 t,σ ≥ ℓ m 2 t,σ , hence L m 1 t ≥ L m 2 t a. s. for all t ∈ [0, T ) by (30). This means that ψ 1 is also order-preserving. Consequently, ψ 2 • ψ 1 : K → K is order-preserving. Finally, by the Tarski's fixed point theorem, there exists some m * ∈ K such that ψ 2 •ψ 1 (m * ) = m * . Let L * be the solution of the representation theorem (1) of Y m * w.r.t. f m * , L * := ψ 1 (m * ), m * = L Ψ(X m * , L * )|G , then (L * , m * ) is a solution to the mean-field representation problem (4). Proof of Theorem 2.3. (i) Let L be the unique solution to the representation (1) problem with Y andf . For each m ∈ P(V + ), let us define the left-continuous increasing process L m by L m · := L · + m · (φ). We observe that L m is the unique solution to the representation problem (1) with Y and generator f m . Then, in the setting of Theorem 2.3, the mean-field representation formula (4) reduces to m = P • L + m · (φ) −1 .(32) (ii) Given a solution m to the fixed-point problem (32), by integrating φ w.r.t. the marginal law of both sides in (32) at any time t ∈ [0, T ], one has that m t (φ) = E φ L t + m t (φ) , for all t ∈ [0, T ]. In other words, for each t ∈ [0, T ], m t (φ) is a solution to the fixed point problem Φ(t, y) := E[φ(L t + y)] = y, y ∈ R.(33) (iii) Let (y * t ) t∈[0,T ] be a solution to the fixed point problem (33), then L + y * = L + E[φ(L · + y * · )] = L + Let E = I c × [0, T ] R . Then for each m ∈ L 0 G (Ω, P(I c × [0, T ] R )), we define X m : [0, T ] × Ω −→ R, Y m : [0, T ] × Ω −→ R and f m : [0, T ] × Ω × R −→ R by X m (t, ω) := X µ (t, ω), Y m (t, ω) := G(t, ω, µ), f m (t, ω, ℓ) := g ℓ (t, ω, µ), where µ ∈ L 0 G (Ω, P(I c × [0, T ] R ) ) is such that (ϕ, Id)#µ = m. Then it is easy to verify that {(X m , Y m , f m )} m∈L 0 G (Ω,P(Ic×[0,T ] R )) satisfy Assumption 2.1. Step 2. Define the set K, the map Ψ and verify their properties. Let us define K and Ψ as follows, K := L (X, (τ ℓ ) ℓ∈R ) G : X ∈ L 0 F (Ω, I c ), τ ℓ ∈ T , ℓ ∈ R , Ψ : Ω × D × V + −→ I c × [0, T ] R (ω, x, l) −→ (ϕ(x), (τ l ℓ ) ℓ∈R ), where τ l ℓ := inf{t ≥ 0; l t ≥ ℓ}. Since the space T equipped with the almost sure partial order is a complete lattice, when T < +∞, the product space T R equipped with the product order is still a complete lattice. On the other hand, since I c is a closed interval on R, by the definition of essential supremum and essential infimum, L 0 F (Ω, I c ) is a complete lattice with almost sure order. Then by the order-preserving property of the conditional expectation map L(·|G), one has that K is a complete lattice. The map Ψ is increasing, for all ω ∈ Ω, in the sense that for any (x 1 , l 1 ), (x 2 , l 2 ) ∈ D × V + , with l 1 ≤ v l 2 and x 1 ≤ D x 2 , one has that τ l 1 ℓ ≤ τ l 2 ℓ for all ℓ ∈ R and ϕ(x 1 ) ≤ ϕ(x 2 ). Now by Theorem 2.2, one obtains the existence of some m * ∈ L 0 G (Ω, P(I c × [0, T ] R )) such that m * = L(Ψ(X m * , L m * )|G), where L m is the running maximum process defined by L m t := sup s∈[0,t) L m s with the convention that sup ∅ = −∞ and L m is the solution to the representation theorem (1) of Y m w.r.t. f m . Step 3. Existence of the mean field equilibrium. Finally, we claim that L((X m * , (τ L m * ℓ ) ℓ∈R )|G), (τ L m * ℓ ) ℓ∈R is a mean field equilibrium. In fact, by Theorem A.1 (iii), (10) and m * = L ϕ(X m * ), (τ L m * ℓ ) ℓ∈R ) G , one has that τ L m * ℓ is the smallest optimal stopping time that maximizes the objective functional J ℓ w.r.t. L((X m * , (τ L m * ℓ ) ℓ∈R )|G), i.e. for any τ ∈ T , it holds that J ℓ τ L m * ℓ , L((X m * , (τ L m * ℓ ) ℓ∈R )|G) ≥ J ℓ τ, L((X m * , (τ L m * ℓ ) ℓ∈R )|G) , and this concludes the proof. Proof of Proposition 2.23 Step Then it is easy to verify that {(X m , Y m , f m )} m∈L 0 G (Ω,P(D×Π i∈N V + θ i )) satisfy Assumption 2.1. 1. Define the tuple (X m , Y m , f m ) m . For each m ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )), we define X m : [0, T ] × Ω −→ R, Step 2. Define the tuple (E, K, Ψ) and verify their properties. We specify the tuple (E, K, Ψ) as follows: E := D × Π i∈N V + θ i K := conv{L((X m , (L i ) i∈N )|G) : m ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )), L i ∈ A i , i ∈ N}, Ψ : Ω × D × V + −→ D × Π i∈N V + θ i ) (ω, x, l) −→ (x, (θ i ∨ l ∧ Θ i (ω)) i∈N ), where K is endowed with the topology of weak convergence, K ⊂ L 0 G (Ω, P(D × Π i∈N V + θ i )), D × Π i∈N V + θ i is endowed with the product metric. The space K is obviously nonempty, convex and closed. Then the tightness of {L(X m |G) : Step 3. Existence of the mean field equilibrium. m ∈ L 0 G (Ω, P(D × Π i∈N V + θ i ))} and {L((L i ) i∈N |G) : L i ∈ A i , i ∈ N} Finally, we claim that (m * , (θ i ∨ L m * ∧ Θ i ) i∈N ) is a mean field equilibrium. Indeed, the consistency condition clearly holds true because m * = L(Ψ(X m * , L m * )|G) = L((X m * , (θ i ∨ L m * ∧ Θ i ) i∈N )|G). Moreover, Theorem 2.4 in Bank [1] implies the required optimality of θ i ∨ L m * ∧ Θ i in the definition of MFG in Definition 2.20. Thus, the proof is concluded. Proof of Proposition 2.24 Step 1. Define the space E and the tuple (X m , Y m , f m ) m . Let E = I c × Π i∈N V + θ i . Then for each m ∈ L 0 G (Ω, P(I c × Π i∈N V + θ i )), we define X m : [0, T ] × Ω −→ R, Y m : [0, T ] × Ω −→ R and f m : [0, T ] × Ω × R −→ R by X m (t, ω) := X µ (t, ω), Y m (t, ω) := G(t, ω, µ), f m (t, ω, ℓ) := g ℓ (t, ω, µ), where µ ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )) is such that (ϕ ⊗ Id)#µ = m. Then it is easy to verify that {(X m , Y m , f m )} m∈L 0 G (Ω,P(Ic×Π i∈N V + θ i )) satisfy Assumption 2.1. Step 2. Define the set K, the map Ψ and verify their properties. Now one may specify the couple (K, Ψ) as follows: K := {L((X, (L i ) i∈N )|G) : X ∈ L 0 F (Ω, I c ), L i ∈ A i , i ∈ N}, Ψ : Ω × D × V + −→ I c × Π i∈N V + θ i (ω, x, l) −→ (ϕ(x), (θ i ∨ l ∧ Θ i (ω)) i∈N ). In Proposition 4.2, we proved that given two stochastic processes L − , L + ∈ L 0 F (Ω, V + ) such that L − ≤ l L + , the set {L ∈ L 0 F (Ω, V + ) : L − ≤ L ≤ L + a.s.} is a complete lattice endowed with the order ≤ l . Then the partially ordered set {(L i ) i∈N : L i ∈ A i , i ∈ N} is still a complete lattice endowed with the product order. On the other hand, since I c is a closed interval on R, by the definition of essential supremum and essential infimum, L 0 F (Ω, I c ) is a complete lattice with almost sure order. Then by the order-preserving property of the conditional expectation map L(·|G), we have K is a complete lattice. Then by Theorem 2.2, one obtains the existence of some m * ∈ L 0 G (Ω, P(I c × Π i∈N V + θ i )) such that m * = L(Ψ(X m * , L m * )|G), where L m is the running maximum process of L m , with L m being the solution to the representation theorem (1) of Y m w.r.t. f m . Step 3. Existence of the mean field equilibrium. Finally, we claim that L((X m * , (θ i ∨ L m * ∧ Θ i ) i∈N |G), (θ i ∨ L m * ∧ Θ i ) i∈N is a mean field equilibrium. In fact, by Theorem 2.4 in Bank [1], (14) and m * = L((ϕ(X m * ), (θ i ∨ L m * ∧ Θ i ) i∈N )|G), one has that θ i ∨ L m * ∧ Θ i maximize J(·, L((X m * , (θ i ∨ L m * ∧ Θ i ) i∈N |G)) over A i , for i ∈ N, thus one can conclude our proof. Proof of Proposition 2.29 Step 1. Define the tuple (X m , Y m , f m ) m . For each m ∈ L 0 G (Ω, P(D × D − )), we define X m : [0, T ] × Ω −→ R, Y m : [0, T ] × Ω −→ R and f m : [0, T ] × Ω × R −→ R by X m (t, ω) := r m (t, ω), Y m (t, ω) := − λe −βt e − t 0 r m s ds , f m (t, ω, ℓ) :=    − u ′ (t, ω, m, −e −βt ℓ ) if ℓ < 0, ℓ if ℓ ≥ 0. Then it is easy to verify that {(X m , Y m , f m )} m∈L 0 G (Ω,P(D×D − )) satisfy Assumption 2.1. Step 2. Define the space E, the set K, the map Ψ and verify their properties. Let us define two maps ψ 1 and ψ 2 as follows: ψ 1 : V + −→ V + ψ 2 : V + − 1 η ,− 1 η −→ D − l −→ − 1 η ∨ l ∧ − 1 η , l −→ −e −β· l . Then one may specify the space E, the set K and the map Ψ as follows: Step 3. Existence of the mean field equilibrium. E = D × D − , Ψ : Ω, D × V + −→ D × D − (ω, Finally, we claim that (L((r m * , Y C * )|G), C * ) is a mean field equilibrium, where C * is the consumption plan such that Y C * = e −βt η ∨ −1 L m * t , and for simplicity, L((r m * , Y C * )|G) is abbreviated as m * . In fact, Theorem 2.5 implies the consumption plan C * is optimal for U (C, m * ) at the cost (14) implies that U (C, m * ) = U (C, m * ) and C * is optimal for U (C, m * ) at the cost E[ Step 1. Define the space E and the tuple (X m , Y m , f m ) m . b * = E[ +∞ 0 e − · 0 r m * s ds dC * t ],Let E := I c × D − so that E is a Polish space. For each m ∈ L 0 G (Ω, P(I c × D − )), we define X m : [0, T ] × Ω −→ R, Y m : [0, T ] × Ω −→ R and f m : [0, T ] × Ω × R −→ R by X m (t, ω) := r µ (t, ω), Y m (t, ω) := − λe −βt e − t 0 r µ s ds , f m (t, ω, ℓ) :=    − u ′ (t, ω, µ, −e −βt ℓ ) if ℓ < 0, ℓ if ℓ ≥ 0, where µ ∈ L 0 G (Ω, P(D × Π i∈N V + θ i )) is such that (ϕ ⊗ Id)#µ = m. Then it is easy to verify that {(X m , Y m , f m )} m∈L 0 G (Ω,P(Ic×D − )) satisfy Assumption 2.1. Step 2. Define the set K, the map Ψ and verify their properties. We define two maps ψ 1 and ψ 2 as follows: ψ 1 : V + −→ V + ψ 2 : V + − 1 η ,− 1 η −→ D − l −→ − 1 η ∨ l ∧ − 1 η , l −→ −e −β· l . Then one may specify the space E, the set K and the map Ψ as follows: E = D × D − , Ψ : Ω, D × V + −→ D × D − (ω, x, l) −→ (ϕ(x), ψ 2 • ψ 1 (l)), K := {L((X, ψ 2 • ψ 1 (L))|G) : X ∈ L 0 F (Ω, I c ), L ∈ L 0 F (Ω, V + )}. In Proposition 4.2, we proved that L 0 := {L ∈ L 0 F (Ω, V + ) : L − ≤ L ≤ L + a.s.} is a complete lattice endowed with the order ≤ l , for some L − , L + ∈ L 0 F (Ω, V + ). As a special case of L 0 , {ψ 1 (L) : L ∈ L 0 F (Ω, V + )} is a complete lattice. Then by the order-preserving property of ψ 2 , the partially ordered set {ψ(L) : L ∈ L 0 F (Ω, V + )} is still a complete lattice endowed with the almost sure order. On the other hand, since I c is a closed interval on R, by the definition of essential supremum and essential infimum, L 0 F (Ω, I c ) is a complete lattice with the almost sure order. Thus, by the order-preserving property of the conditional expectation map L(·|G), it holds that K is a complete lattice. Step 3. Existence of the mean field equilibrium. Finally, we claim that (L((r m * , Y C * )|G), C * ) is a mean field equilibrium, where C * is the consumption plan such that Y C * = e −βt η ∨ −1 L m * t , and for simplicity, L((r m * , Y C * )|G) is abbreviated as m * . In fact, by Theorem 2.5 and (19), one has that the consumption plan C * is optimal for the utility U (C, L(r m * , ψ 1 (Y C * ))|G)) at the cost b (20) implies that U (C, L((r m * , ψ 1 (Y C * ))|G)) = U (C, m * ) and C * is optimal for U (C, m * ) at the cost b * , thus one can conclude the proof. * = E[ +∞ 0 e − · 0 rsds dC * t ], and Proof of Proposition 2.31 Step 1. Define the tuple (Y µ , f µ ) µ . There exists a constant λ ∈ R, such that, for one µ ∈ P(V + f µ (t, ω, ℓ) :=      −ũ ′ t, ω, −e −βt ℓ − µ t (φ) if ℓ < 0, ℓ if ℓ ≥ 0, the corresponding optional process L µ in the representation (1) is almost surely non-decreasing. Then it is easy to verify that {(Y µ , f µ )} µ∈P(D − ) satisfy Assumption 2.1. Step 2. Define the function Ψ and verify its property. We define the map Ψ as the following: Ψ :Ω × V + −→ V + − 1 η (ω, l) −→ −1 η ∨ l Then by Theorem 2.3, one obtains some µ * ∈ P(V + − 1 η ) such that µ * = P • ( −1 η ∨ L µ * ) −1 , where L µ * t := sup s∈[0,t) L µ * s with the convention that sup ∅ = −∞ and L µ * is solution to the representation theorem (1) of Y µ * with generator f µ * . Step 3. Existence of the mean field equilibrium. Finally, we claim that (P • (e −β· (η ∨ −1 L µ * · ) −1 , C * )) is a mean field equilibrium, where C * is the consumption plan such that Y C * = e −β· (η ∨ −1 L µ * · ). Indeed, by Theorem 2.5, one has that the consumption plan C * is optimal at the cost E[ +∞ 0 e − t 0 rsds dC * t ]. A Bank-El Karoui's representation theorem We recall here Bank-El Karoui's representation theorem as well as some first properties of the representation processes. Definition A.1. An optional process Y of class (D) is said to be upper-semicontinuous in expectation if, for any τ ∈ T and any sequence (τ n ) n≥1 ⊂ T satisfying either (τ n ) n≥1 is non-decreasing, τ n < τ on {τ > 0}, for each n ≥ 1, lim n→∞ τ n = τ, or τ n ≥ τ, a.s., for each n ≥ 1, lim n→∞ τ n = τ, one has E[Y τ ] ≥ lim sup n→∞ E[Y τn ]. We are given a function f : [0, T ] × Ω × R −→ R, which satisfies the following condition. Assumption A.2. For all (t, ω) ∈ [0, T ] × Ω, the map ℓ −→ f (t, ω, ℓ) is continuous and strictly increasing from −∞ to +∞. Moreover, for each ℓ ∈ R, the process (t, ω) −→ f (t, ω, ℓ) is progressively measurable, and satisfies E T 0 f (t, ℓ) dt < ∞. Theorem A.1 (Bank-El Karoui's representation, [3,5]). Let f : [0, T ] × Ω × R −→ R satisfy Assumption A.2. Then for every optional process Y of class (D) and u.s.c. in expectation such that Y T = 0, the following holds true. (i) There exists an optional process L : [0, T ] × Ω −→ R, such that E T τ f t, sup v∈[τ,t) L v dt < ∞, for all τ ∈ T , and Y τ = E T τ f t, sup s∈[τ,t) L s dt F τ , a.s., for all τ ∈ T .(35) (ii) Let L be a solution to the representation (35), which is progressively measurable and upperright continuous. Then L τ = ess inf σ∈T τ,+ ℓ τ,σ ,(36) where T τ,+ = {σ ∈ T τ : σ > τ on {τ < T }}, ℓ τ,σ is the unique F τ -measurable random variable such that E σ τ f (t, ℓ τ,σ )dt F τ = E[Y τ − Y σ |F τ ]. In particular, such an upper-right continuous solution L is unique up to optional section. (iii) Let the optional process L be a solution to the representation (35). Then for any ℓ ∈ R, the stopping times τ ℓ := inf t ≥ 0 : L t− ≥ ℓ ,τ ℓ := inf t ≥ 0 : L t− > ℓ are the smallest solution and the biggest solution of the optimal stopping problem: Let us first present Schauder fixed-point theorem, which is frequently used in the fields of differential equations and game theory. sup τ ∈T E Y τ + τ 0 h(t, ℓ)dt .(37) Theorem B.1 (Schauder fixed-point theorem, Theorem 3.2 [12]). Let V be a Hausdorff locally convex topological vector space and K ⊂ V be a nonempty convex closed subset. Let T : K −→ K be continuous, and such that T (K) is precompact. Then T has a fixed point. The second one is Tarski's fixed point theorem, under some order structure condition. Definition B.1. (i) Let L be a set, the relation ≤ is called a partial order if, for all l 1 , l 2 , l 3 ∈ L, one has l 1 ≤ l 1 , l 1 ≤ l 2 , l 2 ≤ l 3 =⇒ l 1 ≤ l 3 , l 1 ≤ l 2 , l 2 ≤ l 1 =⇒ l 1 = l 2 . (ii) A partially ordered set L is called a complete lattice if, for any subset L 0 ⊂ L, there exist some l − , l + ∈ L, such that • l − ≤ l ≤ l + , for all l ∈ L 0 , • if there exists another pair l − , l + ∈ L, such that l − ≤ l ≤ l + for all l ∈ L 0 , then l − ≤ l − and l + ≤ l + . (iii) Suppose that f is a function from a partially ordered set L 1 to a partially ordered set L 2 , we say it is order preserving (order reversing) if for any l 1 , l 2 ∈ L 1 with l 1 ≤ l 2 , f (l 1 ) ≤ f (l 2 ), (f (l 2 ) ≤ f (l 1 )). The Tarski's fixed point theorem gives an existence result, which is based on the iteration method. B.2 Partially ordered Polish space We recall the notion of the partially ordered Polish space, and present a result from Kamae and Krengel [34], that the space of all probability measures on a partially ordered Polish space is still a partially ordered Polish space. Definition B.2. A partially ordered Polish space is a Polish space X equipped with a partial order ≤ such that the graph set {(x, y) ∈ X 2 : x ≤ y} is a closed subset of X 2 . Let (E, ≤ e ) be a partially ordered Polish space, P(E) be the space of all (Borel) probability measures on E equipped with the weak convergence topology. A function f : E −→ R is said to be increasing if f (e 1 ) ≤ f (e 2 ) for all e 1 ≤ e e 2 . Then based on the partial order ≤ e on E, we introduce a partial order ≤ p on P(E) as follows: Let Γ denote the space of all bounded increasing measurable function f : E −→ R, for P, Q ∈ P(E), we say P ≤ p Q, if and only if E f (x)P(dx) ≤ E f (x)Q(dx), for all f ∈ Γ. Theorem B.3 (Kamae and Krengel [34]). Let (E, ≤ e ) be a partially ordered Polish space. Then (P(E), ≤ p ) is also a partially ordered Polish space. Example B.3. On the Polish spaces V + , V + η := {l ∨ η : l ∈ V + } and D, where η ∈ R is a constant, let us introduce the following partial order ≤ e : x 1 ≤ e x 2 if and only if x 1 t ≤ x 2 t , for all t ∈ [0, T ] (or [0, T )). Then all of (V + , ≤ e ), (V + η , ≤ e ) and (D, ≤ e ) are partially ordered Polish spaces. d L (v 1 , v 2 ) := inf ε ≥ 0 : v 1 (t − ε) ∨ 0 − ε ≤ v 2 (t), v 2 (t − ε) ∨ 0 − ε ≤ v 1 (t), ∀t ∈ (0, T ) , when T < +∞, and d L (v 1 , v 2 ) := +∞ n=1 2 −n (d L (v 1 | [0,n) , v 2 | [0,n) ) ∧ 1), when T = +∞. Proposition C.1. The space (V + , d L ) is a complete separable metric space. Moreover, for v n , v ∈ V + , n ∈ N, the sequence {v n } +∞ n=1 converges to v in d L if and only if v n converges point-wise to v on continuity points of v on (0, T ). Proof (i) Let us first consider the case where T < +∞. Step 1. d L is a metric on V + . First, for any v 1 , v 2 ∈ V + , as the set {ε ≥ 0 : v 1 (t − ε) ∨ 0 − ε ≤ v 2 (t), v 2 (t − ε) ∨ 0 − ε ≤ v 1 (t), ∀ t ∈ (0, T )} = ∅, it follows that d L (v 1 , v 2 ) ≥ 0. Next, by its definition, it is easy to check that d L (v 1 , v 2 ) = 0 if and only if v 1 = v 2 , and that d L (v 1 , v 2 ) = d L (v 2 , v 1 ). For the triangle inequality, let v 1 , v 2 , v 3 ∈ V + , there exist two sequences of nonnegative constants (ε i n ) n≥1 with lim n→∞ ε i n = d L (v i , v i+1 ) such that, for each n ≥ 1, v i (t − ε i n ) ∨ 0 − ε i n ≤ v i+1 (t), v i+1 (t − ε i n ) ∨ 0 − ε i n ≤ v i (t) , for all t ∈ (0, T ), i = 1, 2. Then v 1 (t − ε 1 n − ε 2 n ) ∨ 0 − ε 1 n − ε 2 n = v 1 ((t − ε 2 n ) − ε 1 n ) ∨ 0 − ε 1 n − ε 2 n ≤ v 2 (t − ε 2 n ) ∨ 0 − ε 2 n ≤ v 3 (t) , for all t ∈ (0, T ). Similarly, one has v 3 (t − ε 1 n − ε 2 n ) ∨ 0 − ε 1 n − ε 2 n ≤ v 1 (t), for all t ∈ (0, T ). By the definition of d L , it follows that, for all n ≥ 1, d L (v 1 , v 3 ) ≤ ε 1 n + ε 2 n . Let n tends to ∞, one obtains that d L (v 1 , v 3 ) ≤ d L (v 1 , v 2 ) + d L (v 2 , v 3 ). Step 2. Let v ∈ V + , we denote by T v the set of all continuity points t ∈ [0, T ) of v. We next proe that lim n→∞ d L (v n , v) = 0 =⇒ lim n→∞ v n (t) = v(t), for all t ∈ T v . First, for the sufficient condition part (=⇒), let lim n→∞ d L (v n , v) = 0, then there exists a sequence of constants ε n with lim n→∞ ε n = 0 such that v n (t − ε n ) ∨ 0 − ε n ≤ v(t), v (t − ε n ) ∨ 0 − ε n ≤ v n (t), for all t ∈ (0, T ). For fixed t, we can find N large enough such that for any n ≥ N , t + ε n < T , then we can rewrite the above inequalities as v (t − ε n ) ∨ 0 − ε n ≤ v n (t) ≤ v n (t ∨ ε n ) ≤ v t + ε n + ε n , ∀ t ∈ (0, T ). When t ∈ T v , letting n −→ ∞, it follows that v(t) ≤ lim inf n→∞ v n (t) ≤ lim sup n→∞ v n (t) ≤ v(t). Next, for the necessary condition part (⇐=), we fix an ε > 0, then there exists a finite partition π ε = {t 1 , · · · , t Kε } of [0, T ) with 0 = t 0 < t 1 < · · · < t Kε < T =: t Kε+1 and its norm |π ε | := sup 0≤i≤Kε |t i+1 − t i | < ε. Let N = N (ε, t 1 , · · · , t Kε ) be large enough such that for any n ≥ 1, |v n (t i ) − v(t i )| ≤ ε, for all i = 1, · · · , K ε . Therefore, for t ∈ [t i , t i+1 ], i = 0, · · · , K ε , one has the estimation v n (t) ≥ v n (t i ) ≥ v(t i ) − ε ≥ v (t − ε) ∨ 0 − ε, v(t) ≥ v(t i ) ≥ v n (t i ) − ε ≥ v n (t − ε) ∨ 0 − ε, which implies that d L (v n , v) ≤ ε. Step 3. (V + , d L ) is complete and separable. Let (v m ) ∞ m=1 be a Cauchy sequence in V + , we aim to find v ∈ V + such that lim m→∞ d L (v m , v) = 0. We first define a sequence of maps I n : V + −→ V + for each n ≥ 1 by Let V + n := I n (V + ), so that (V + n , d L ) is a compact Polish space. For each n ≥ 1, we notice that d L (I n (v), I n (w)) ≤ d L (v, w), for all v, w ∈ V + and n ≥ 1, so that I n (v m ) ∞ m=1 is alaso a Cauchy sequence in the compact Polish space V + n , then there exists some v n ∈ V + n such that I n (v m ) −→ v n in V + n . In particular, one has I n (v n+1 ) = v n . For any ε > 0, choose m = m(ε, v n , v n+1 ) large enough such that d L v n , I n (v m ) ≤ ε and d L v n+1 , I n+1 (v m ) ≤ ε. Then we obtain the estimation We next check that v is R-valued on (0, T ) to show that v ∈ V + . Assume that v is not Rvalued, then one has, for some s ∈ (0, T ), v(t) = +∞ and v n (t) = n, for all t ∈ [s, T ). Let ε 0 := (T − s)/5 and fix s 0 := (s + T )/2, then there exists some N = N (ε 0 ), such that for any l, m ≥ N , d L (v l , v m ) ≤ ε 0 . Thus we have for any n ∈ N, d L (I n (v l ), I n (v m )) ≤ ε 0 , and d L (I n (v l ), v n ) ≤ ε 0 . Now we focus on v N and choose n large enough such that v n (s 0 ) = n > v N (s 0 + 2ε 0 ) + 2ε 0 , which contradicts with the inequality d L (I n (v l ), v n ) ≤ ε 0 . Therefore, we conclude that v ∈ V + . d L v n , I n (v n+1 ) ≤ d L v n , I n (v m ) + d L I n (v m ), I n (v n+1 ) ≤ d L v n , I n (v m ) + d L I n+1 (v m ), v n+1 ≤ 2ε. Finally, for any continuity point t ∈ T v of v ∈ V + n , one has v(t) = v n (t) and t is also a continuity point of v n , hence v(t) = v n (t) = lim m→∞ I n (v m (t)) = lim m→∞ v m (t). This implies that v m converges to v in V + as m tends to +∞, and hence (V + , d L ) is complete. To prove it is also separable, one can apply similar arguments in Step 2. (ii) For the case T = ∞, it is obvious that d L here is still a metric. The space (V + , d L ) is clearly separable, we finally prove the completeness of (V + , d L ). Let (v m ) ∞ m=1 be a Cauchy sequence in V + . Then there exists some v n ∈ V + such that lim m→∞ d L (v m | [0,n) , v n ) = 0, for all n ≥ 1. We now verify that for all n ∈ N, v n+1 | [0,n) = v n . Since lim m→∞ d L (v m | [0,n+1) , v n+1 ) = 0, we have lim m→∞ d L (v m | [0,n) , v n+1 | [0,n) ) = 0 and uniqueness of the limit of the Cauchy sequence {v m | [0,n) } +∞ m=1 implies our claim. Then v := +∞ n=1 v n 1 [n−1,n) is a well-defined increasing function on R and belongs to V + . D Enlarged space Ω × [0, T ] and the stable convergence topology Given the (abstract) filtered probability space (Ω, F, P), we introduce an enlarged measurable space (Ω, F ) := (Ω × [0, T ], F ⊗ B([0, T ])). Let P(Ω) denote the collection of all probability measures on (Ω, F ), we equip it with the stable convergence topology of Jacod and Mémin [33], and recall some basic facts here. Let P(Ω) (resp. P(Ω), P([0, T ])) denote the collection of all probability measures on (Ω, F ) (resp. (Ω, F), ([0, T ], B([0, T ]))). Further, let B mc (Ω) (resp. B mu (Ω)) denote the collection of all bounded F-measurable functions ξ : Ω −→ R such that for every ω ∈ Ω, the mapping θ ∈ [0, T ] −→ ξ(ω, θ) is continuous (resp. upper semi-continuous). The stable convergence on P(Ω) is the coarsest topology making P −→ E P [ξ] continuous for all ξ ∈ B mc . We also equip P([0, T ]) with the weak convergence topology and P(Ω) with the coarsest topology such that P → E P [ξ] is continuous for all bounded measurable functions on (Ω, F). Let us first recall a result from Jacod and Mémin [33]. Theorem D.1. (i) Let T < ∞, a subset P ⊂ P(Ω) is relatively compact w.r.t. the stable topology if and only if {P| Ω : P ∈ P} is relatively compact in P(Ω). (ii) Let (P n ) n∈N ⊂ P(Ω) be a sequence such that P n converges to some P ∈ P(Ω) under the stable topology, then one has that Proof Notice that [0, T ] is compact and P| Ω = P for all P ∈ P 0 , then P 0 is relatively compact under the stable convergence topology. It remains to prove that the set P 0 is closed. Let {P n } n∈N ⊂ P 0 be such that P n converges to some P ∈ P(Ω) under the stable convergence topology. Let h t , g t , g T be bounded measurable functions on ([0, T ], σ{Θ ∧ t}), (Ω, F t ) and (Ω, F), respectively. We assume further that the function θ → h t (θ) is continuous. Then, by Theorem D.1, Ω h t (θ)g t (ω)g T (ω)P(dω, dθ) = lim n→∞ Ω h t (θ)g t (ω)g T (ω)P n (dω, dθ) = lim n→∞ Ω h t (θ)g t (ω)E Pn [g T | F t ](ω)P n (dω, dθ) = lim n→∞ Ω h t (θ)g t (ω)E P [g T | F t ](ω)P n (dω, dθ) = Ω h t (θ)g t (ω)E P [g T | F t ](ω)P(dω, dθ). Thus, it holds that Ω h t (θ)g t (ω)g T (ω)P(dω, dθ) = Ω h t (θ)g t (ω)E P [g T | F t ](ω)P(dω, dθ), for all bounded measurable functions h t , g t , g T on ([0, T ], σ{θ ∧ t}), (Ω, F t ) and (Ω, F), respectively. Finally, by the definition of conditional expectation, one can conclude the proof. For T ∈ [0, +∞], let us denote by T the collection of all F-stopping times taking values in [0, T ]. Let D be a (nonempty) Polish space, we denote by D the space of all D-valued càdlàg paths on [0, T ] when T < ∞, or the space of all D-valued càdlàg paths on [0, ∞) when T = ∞. The space D is equipped with the Skorokhod metric d D . For a constant η ∈ R ∪ {−∞}, let V + η denote the space of all increasing and left-continuous functions v : [0, T ) −→ [η, ∞) such that v(0) = η and v is R-valued on (0, T ). The space V + η is equipped with the Lévy metric d L defined as follows: Remark 2. 4 . 4The continuity condition on Ψ and m −→ (X m , Y m , f m ) is used to verify the continuity condition in Schauder fixed-point theorem. Furthermore, we prove a continuous property of (Y, f ) −→ L, where L is the running maximum of L in (1), which is a stability result of Bank-El Karoui's representation (1) (see Section 3). Example 2. 6 . 6Let φ : R −→ R be an increasing, continuous function, uniformly bounded by some constant C > 0, and E = V • be the space of all R-valued bounded increasing paths on [0, T ), we define Ψ : Ω × D × V + −→ E by Ψ(ω, x, l) := (φ(l t )) t∈[0,T ) . We next provide two examples of Ψ and the complete lattice (K, ≤ p ) in the setting of Theorem 2.2, where the corresponding proofs are reported in Section 4.1. Example 2. 8 . 8Let C > 0 be a constant, Example 2 . 10 . 210Let T < ∞, E = R, and φ : R −→ R be a monotone function, f : [0, T Recall that D represents the space of all D-valued càdlàg paths on [0, T ], with a Polish space D. We denote by P(D × [0, T ] R ) the collection of all Borel probability measures on D × [0, T ] R , and L 0 G Ω; P D × [0, T ] R denotes the space of all G-measurable random measures m : Ω −→ P D × [0, T ] R . s∈[0,t) L µ,λ s . Remark 2.27. (i) The problem (16) reduces to the representation (1) with Y µ,λ (t, ω) := − λe −βt Proposition 2 . 30 . 230Suppose that the functions r and u satisfy Assumption 2.26. Assume in addition that (i) there exists a bounded increasing map ϕ : D −→ R, such that for any µ whenever µ 1 1(A × B) = µ 2 (A × B), a.s. for all A ∈ B(D), B ∈ B({D ∈ D − : ηe −β· ≤ D · ≤ ηe −β· }). Then, with the Lévy metric d L on V + , one has for every ε > 0,lim n→∞ P d L L n , L ≥ ε = 0.Consequently, with the Lévy-Prokhorov metric d LP on the space P(V + ) one haslim n→∞ P d LP L( L n |G), L( L|G) ≥ ε = 0, lim n→∞ E d LP L( L n |G), L( L|G) = 0.Remark 3.1. (i) In general, one can only expect a stability result on the running process L but not the process L itself. Let us illustrate it by the following counter-example. We consider a deterministic setting with F = {Ω, ∅} and T = 1, so that (Y n , f n ) n≥0 and (Y, f ) are all deterministic. Let Further , based on the (abstract) probability space (Ω, F, P), we introduce an enlarged measurable space (Ω, F) together with a canonical element Θ : Ω −→ [0, T ], by Ω := Ω × [0, T ], F := F ⊗ B([0, T ]), and Θ(ω) := θ, for allω = (ω, θ) ∈ Ω. 1 A ω, τ n ℓ (ω) P(dω), for all A ∈ F ⊗ B([0, T ]). −→ [0, T ] denote the right-continuous inverse function of t −→ F ω (t). It follows that, for all u ∈ [0, 1] and t ∈ [0, T ], F −1 (u) = τ ℓ , a.s. for a.e. u ∈ [0, 1]. This implies that, for any bounded and F ⊗ B([0, T ])-measurable function ξ : Ω −→ R, one has Proof of Theorem 3. 1 1Let us consider an arbitrary subsequence {n k } k∈N of {n} n∈N . By Proposition 3.3 and with the corresponding countable set L 0 ⊂ R, one can find a countable and dense subset L 1 ⊂ R \ L 0 and a subsequence {n km } m∈N of {n k } k∈N , such that lim m→∞ τ n km ℓ = τ ℓ , a.s., for all ℓ ∈ L 1 . Then by Billingsley[10, Theorem 25.6], for P-a.e. ω ∈ Ω) = L t (ω), whenever L t (ω) = L t+ (ω).Hence, by Proposition C.1, one haslim k→∞ d L ( L n km , L) = 0.Finally, since the subsequence {n k } k∈N is arbitrary, one can conclude that, for any ε > 0, lim n→∞ P d L ( L n , L) ≥ ε = 0. In the context of Example 2.5, the admissible set {L(Ψ(X m , L)|G) : m ∈ L 0 G (Ω, P(E)), L ∈ L 0 F (Ω, V + )} is tight in L 0 G (Ω, P(E)). Proof Let us consider the setting G = {∅, Ω}, where it is sufficient to check that Proposition 4 . 2 . 42In the setting of Example 2.9, let us consider two fixed stochastic processesL − , L + ∈ L 0 F (Ω, V + ) such that L − ≤ l L + . LetL 0 := {L ∈ L 0 F (Ω, V + ) : L − ≤ l L ≤ l L + , }, and K := {L( L|G) : L ∈ L 0 }. Proof of Theorem 2.1. First, let us define a map ψ : K → K by ψ(m) = L(ψ(X m , L m )|G), where L m := sup t∈[0,·) L m t and L m is the solution of the representation theorem (1) of Y m w.r.t. f m . Proof of Theorem 2.2. Let us define ψ 1 : K −→ L 0 F Ω, D × V + , and ψ 2 : L 0 F Ω, D × V + −→ K by ψ 1 (m) := (X m , L m ), ψ 2 (X, L) := L(Ψ(X, L)|G). Y m : [0, T ] × Ω −→ R and f m : [0, T ] × Ω × R −→ R by X m (t, ω) := X m (t, ω), Y m (t, ω) := k(t, ω, m), f m (t, ω, ℓ) := c ′ (t, ω, m), ℓ ∈ R. implies the compactness of K. Besides, K is a subset of the Hausdorff locally convex topological vector space L 0 G (Ω, P(D × Π i∈N V + θ i )) endowed with the topology of convergence in probability.Further, the map Ψ is clearly continuous, for all ω ∈ Ω. Hence, one can apply Theorem 2.1 to obtain the existence of some m * ∈ L 0 G (Ω, P(D × Π i∈N V +θ i )) such that m * = L(Ψ(X m * , L m * )|G), where L m t := sup s∈[0,t) L m s with the convention that sup ∅ = −∞ and L m is the solution to the representation theorem (1) of Y m w.r.t. f m . x, l) −→ (x, ψ 2 • ψ 1 (l)), K := conv{L(Ψ(X m , L)|G) : m ∈ L 0 G (Ω, P(D × D − )), L ∈ L 0 F (Ω, V + )}, where K ⊂ L 0 G (Ω, P(D × D − )) is endowed with the topology of weak convergence. 34 The space K is obviously nonempty, convex and closed. The tightness of {L(ψ 1 (L)|G) : L ∈ L 0 F (Ω, V + )} is given by Proposition 4.1. Then by the definition of Ψ, continuity, injectivity of ψ 2 , one has the tightness of {L(ψ(L)|G) : L ∈ L 0 F (Ω, V + )}. Thus, together with the tightness of {L(X m |G) : m ∈ L 0 G (Ω, P(D × D − ))}, the set K is compact. Further, the map Ψ is clearly continuous, for all ω ∈ Ω. Hence, one can apply Theorem 2.1 to obtain the existence of some m * ∈ L 0 G (Ω, P(D × D − )) such that m * = L(Ψ(X m * , L m * )|G), where L m t := sup s∈[0,t) L m s with the convention that sup ∅ = −∞ with L m being the solution in (1) of Y m w.r.t. f m . dC * t ], thus one can conclude the proof. Proof of Proposition 2.30 Then by Theorem 2.2, one obtains the existence of some m * ∈ L 0 G (Ω, P(I c × D − )) such that m * = L(Ψ(X m * , L m * )|G), where L m t := sup s∈[0,t) L m s with the convention that sup ∅ = −∞ and L m being the solution to (1) of Y m w.r.t. f m . Y : [0, T ] × Ω −→ R and f µ : [0, T ] × Ω × R −→ R defined by Y (t, ω) := − λe −βt e − t 0 rsds , Remark A. 3 . 3In the deterministic setting with F = {Ω, ∅}, so that f (·, ℓ) and Y can be seen as deterministic functions on [0, T ]. Let f (·, ℓ) = ℓ for all ℓ ∈ R. Then given Y : [0, T ] −→ R, with the representation process L : [0, T ) −→ R, the function T · sup r∈[t,s) L r ds turns to be the convex envelope of −Y | [t,T ] , and L t is the derivative of the convex envelope of −Y | [t,T ] at time t. B Fixed-point theorems and partially ordered Polish space B.1 Some fixed-point theorems We recall in this section three fixed-point theorems which are used in the paper. It seems that most of the fixed-point theorems in the literature can be considered as variations or extensions of them. Theorem B.2 (Tarski's fixed point theorem). Let (L, ≤) be a complete lattice and let T : L → L be a monotonic function w.r.t. ≤. Then the set of all fixed points of T is also a (nonempty) complete lattice under ≤. Moreover, there exists a unique least fixed point and a unique largest fixed point.38 C On the space V + It is wellknown that the space of all left-continuous and increasing R-valued paths on a closed interval is a Polish space under the Lévy metric. Here, our space V + is defined as the space of all increasing and left-continuous paths v : [0, T ) −→ R ∪ {−∞}, such that v(0) = −∞ and v is R-valued on (0, T ). For completeness, we show that it is also a Polish space under the Lévy metric d L . Recall that the Lévy metric d L defined by I n (v)(0) := v(0) = −∞, and I n (v)(t) := −n ∨ v(t) ∧ n, for all t ∈ (0, T ). Now, let us defineṽ bỹv 0 := −∞, andṽ(t) := lim n→∞ v n (t), for all t ∈ [0, T ), and let v be the left-continuous version ofṽ, then v is obviously increasing as the limit of increasing function on [0, T ). Moreover, for any sequence {vi } ∞ i=0 ⊂ V + , we have lim i→∞ d L (v i , v) = 0 if and only if lim i→∞ d L (v i | [0,n) ,v| [0,n) ) = 0 for all n ∈ N + and we can conclude that lim i→∞ d L (v i , v) = 0 if and only if lim i→∞ v i (t) = v(t) for all continuity points of v by Step 2, . E Pn [ξ] = E P [ξ], for all ξ ∈ B mc (Ω), and lim sup n→∞ E Pn [ξ] ≤ E P [ξ], for all ξ ∈ B mu (Ω).Let F = (F t ) t∈[0,T ] be a filtration on the probability space (Ω, F, P), we introduce F 0 andF 0 = (F 0 t ) t∈[0,T ] by F 0 := F ⊗ {∅, [0, T ]}, F 0 t := F t ⊗ {∅, [0, T ]}, t ∈ [0, T ].Further, we introduce a canonical element Θ : Ω −→ [0, T ] by Θ(ω) := θ, for allω = (ω, θ) ∈ Ω. The following result is mainly a direct adaptation to our context from Carmona, Delarue and Lacker [21, Theorem 6.4], we nevertheless provide a proof for completeness. Proposition D.1. Let T < ∞, then the set P 0 := P ∈ P(Ω) : P| Ω = P, P[Θ ≤ t|F 0 ] = P[Θ ≤ t|F 0 t ], for all t ∈ [0, T ]is compact in P(Ω) under the stable convergence topology. and Skalli[11, Theorem II.1]) Y has upper semicontinuous paths, i.e., for a.e. ω ∈ Ω, t −→ Y t (ω) is u.s.c.(ii.a) Let us first consider the case where T < ∞.Recall that F 0 := F ⊗ {∅, [0, T ]}, by the definition of P ℓ,n in (25), it is clear that, for all t ∈ [0, T ], Step 1. Define the space E and the tuple (X m , Y m , f m ) m . where the last equality follows by the fact that L m in(31)provides the unique solution solution to the representation problem(1). Consequently, (P • (L + y * ) −1 , L + y * ) is a solution to the fixed point problem(32).(iv) Finally, we prove that, for each t ∈ [0, T ], there exists a unique solution to(33). In fact, since φ ′ ∈ [0, 1), one has ∂Φ ∂y (t, y) = E[φ ′ (L t + y)] ∈ [0, 1), for all (t, y) ∈ [0, T ] × R.Together with the boundedness of φ, one obtains that the fixed point problem(33)has a unique solution y * t ∈ R for each t ∈ [0, T ]. Moreover, as L t is increasing in t, this implies that Φ(t, y) = E[φ(L t + y)] is also increasing in t, and so is y * t in t.Proofs of the results in Section 2.3Proof of Proposition 2.15.Step 1. Define the Polish space E and the tuple (X m , Y m , f m ). Let us fix ε > 0. Recall that there exists a δ ε > 0 independent of ω and µ such thatand denote by E ε the collection of elements (t ℓ ) ℓ∈R in [0, T ] R such that the map ℓ ∈ R −→ t ℓ ∈ [0, T ] is increasing and Lipschitz continuous with Lipschitz constant 1.Then we equip E ε with the Lévy metric d L so that E ε is a Polish space and its topology induced by d L coincides with the product topology of Then it is easy to verify that {(X m , Y m , f m )} m∈L 0 G (Ω,P(E)) satisfy Assumption 2.1. 29Step 2. Define the couple (K, Ψ) and verify their properties.Let Ψ :where τ l ℓ := inf{t ≥ 0; l t ≥ ℓ}, ℓ ∈ R, for any l ∈ V + and I is the identity function on [0, T ]. It is easy to see that Ψ takes value in E.Further, let the space K be defined bywhere conv(P ) denote the convex hull of the set P in L 0 G (Ω, P(E)), i.e. the intersection of all convex sets in L 0 G (Ω, P(E)) that contain P . Then the space K is obviously nonempty, convex and closed. Tychonoff's theorem implies the compactness of [0, T ] R , endowed with product metric. By the tightness of {L(X m |G) :) is a Hausdorff locally convex topological vector space. Next, for µ n , µ ∞ ∈ K, n ∈ N with lim n→∞ m n = m ∞ in probability, since E ε is a closed subset of [0, T ] R , it holds that lim n→∞ m n = m ∞ almost surely. By assumption (iii) in the statement, we have for any ℓ ∈ R,Then it remains to verify that the map Ψ is continuous from D × V + to E, for all ω ∈ Ω. We claim that if the elements {l n } n∈N , l ∞ in V + are such that lim n→∞ d L (l n , l ∞ ) = 0 and l n , l ∞ are strictly increasing, then lim n→∞ τ l n ℓ = τ l ∞ ℓ , a.s., for all ℓ ∈ R.Indeed, since lim n→∞ d L (l n , l ∞ ) = 0, by Proposition C.1, we have l n · converges to l ∞ · on every continuity points of L ∞ · , and then τ l n ℓ converges to τ l ∞ ℓ on every continuity points of ℓ −→ τ l ∞ ℓ , as n tends to ∞. At the same time, as inverse function of the strictly increasing function t −→ l ∞ t , ℓ −→ τ l ∞ ℓ is continuous for all ℓ ∈ R. This proves the claim in(34).Further, by Proposition C.1, Ψ is clearly continuous. Hence, we can apply Theorem 2.1 to obtain the existence of some m * ε ∈ L 0 G (Ω, P(E)) such that m * ε = L(Ψ(X m * ε , L m * ε )|G), where L m is the running maximum process defined by L m t := sup s∈[0,t) L m s with the convention that sup ∅ = −∞ and L m is the solution to the representation theorem (1) of Y m w.r.t. f m .Step 3. Existence of the ε-mean field equilibrium.We claim that the pairis a ε-mean field equilibrium. The stopping time τ L m * ε +δ ε/(3T ) I ℓ is hereinafter abbreviated as τ * ,ε ℓ and one may define a map ψ : V + −→ V + by ψ(l) := l + δ ε/(3T ) I.30First, the consistency conditionholds by the definition of Ψ.Then it is sufficient to verify that, for any τ ∈ T , one hasBy Theorem A.1 (iii), we know that τ L m * ε ℓ (hereinafter abbreviated as τ * ℓ ) is the smallest optimal stopping time that maximizes the objective functional J ℓ w.r.t. m * ε , i.e. for any τ ∈ T , we haveThen we have the estimationIn above, the first two equalities follow by the definition of J ℓ , f m and Y m , the third equality holds by Theorem A.1 and that τ * ℓ , is a hitting time of L m * ε , the fourth equality holds by the strictly increasing property of ℓ −→ f m (t, ω, ℓ), the first inequality is trivial. For the last inequality, it is enough to notice that on {s < τ * ,ε ℓ }, L Optimal control under a dynamic fuel constraint. P Bank, SIAM Journal on Control and Optimization. 444Bank P. Optimal control under a dynamic fuel constraint. SIAM Journal on Control and Optimization. 2005;44(4):1529-41. On a stochastic representation theorem for Meyer-measurable processes. P Bank, D Besslich, Annales de l'Institut Henri Poincaré-Probabilités et Statistiques. 573Bank P, Besslich D. On a stochastic representation theorem for Meyer-measurable processes. Annales de l'Institut Henri Poincaré-Probabilités et Statistiques. 2021;57(3):1336-68. A stochastic representation theorem with applications to optimization and obstacle problems. The Annals of Probability. P Bank, El Karoui, N , 32Bank P, El Karoui N. A stochastic representation theorem with applications to optimization and obstacle problems. The Annals of Probability. 2004 Jan;32(1B):1030-67. Convex duality for stochastic singular control problems. The Annals of Applied Probability. P Bank, H Kauppila, 27Bank P, Kauppila H. Convex duality for stochastic singular control problems. 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Monotone solutions for mean field games master equations: finite state space and optimal stopping. C Bertucci, 8Journal de l'École polytechnique-MathématiquesBertucci C. Monotone solutions for mean field games master equations: finite state space and optimal stopping. Journal de l'École polytechnique-Mathématiques. 2021 Jun 3;8:1099-132. Fokker-Planck equations of jumping particles and mean field games of impulse control. Annales de l'Institut Henri Poincaré C. C Bertucci, 37Bertucci C. Fokker-Planck equations of jumping particles and mean field games of impulse control. Annales de l'Institut Henri Poincaré C. 2020 Oct 1;37(5):1211-44. Probability and measure. P Billingsley, John Wiley& SonsBillingsley P. Probability and measure. John Wiley& Sons. 2008 Aug 4. Temps d'arrêt optimal, théorie générale des processus et processus de Markov. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. J M Bismut, B Skalli, Bismut J. M, Skalli B. Temps d'arrêt optimal, théorie générale des processus et proces- sus de Markov. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete. 1977 Lectures on some fixed point theorems of functional analysis. Bombay: Tata Institute of Fundamental Research. F F Bonsall, K B Vedak, Bonsall F. F, Vedak K. B. Lectures on some fixed point theorems of functional analysis. Bombay: Tata Institute of Fundamental Research. 1962. Mean-field games of optimal stopping: a relaxed solution approach. G Bouveret, R Dumitrescu, P Tankov, SIAM Journal on Control and Optimization. 584Bouveret G, Dumitrescu R, Tankov P. Mean-field games of optimal stopping: a relaxed solution approach. SIAM Journal on Control and Optimization. 2020;58(4):1795-821. Mean field games with absorption and common noise with a model of bank run. M Burzoni, L Campi, arXiv:2107.00603arXiv preprintBurzoni M, Campi L. Mean field games with absorption and common noise with a model of bank run. arXiv preprint arXiv:2107.00603. 2021 Jul 1. Mean-field games of finite-fuel capacity expansion with singular controls. The Annals of Applied Probability. L Campi, T De Angelis, M Ghio, G Livieri, 32Campi L, De Angelis T, Ghio M, Livieri G. Mean-field games of finite-fuel capacity expansion with singular controls. The Annals of Applied Probability. 2022 Oct;32(5):3674-717. Stationary discounted and ergodic mean field games with singular controls. Mathematics of Operations Research. H Cao, J Dianetti, G Ferrari, Cao H, Dianetti J, Ferrari G. Stationary discounted and ergodic mean field games with singular controls. Mathematics of Operations Research. 2022 Oct 28. MFGs for partially reversible investment. Stochastic Processes and their Applications. H Cao, X Guo, 150Cao H, Guo X. MFGs for partially reversible investment. Stochastic Processes and their Applications. 2022 Aug 1;150:995-1014. Approximation of N-player stochastic games with singular controls by mean field games. H Cao, X Guo, Lee J , arXiv:2202.06835arXiv preprintCao H, Guo X, Lee J.S. Approximation of N-player stochastic games with singular controls by mean field games. arXiv preprint arXiv:2202.06835. 2022 Feb 14. Notes on mean field games. P Cardaliaguet, Technical reportCardaliaguet P. Notes on mean field games. Technical report, 2010. Mean field games with common noise. R Carmona, F Delarue, D Lacker, 44Annals of ProbabilityCarmona R, Delarue F, Lacker D. Mean field games with common noise. Annals of Proba- bility. 2016 Nov;44(6):3740-803. Mean field games of timing and models for bank runs. R Carmona, F Delarue, D Lacker, Applied Mathematics & Optimization. 761Carmona R, Delarue F, Lacker D. Mean field games of timing and models for bank runs. Applied Mathematics & Optimization. 2017 Aug;76(1):217-60. Probabilistic theory of mean field games with applications I-II. R Carmona, F Delarue, Springer Nature. Carmona R, Delarue F. Probabilistic theory of mean field games with applications I-II. Springer Nature; 2018. Identifying the free boundary of a stochastic, irreversible investment problem via the Bank-El Karoui representation theorem. M B Chiarolla, G Ferrari, SIAM Journal on Control and Optimization. 522Chiarolla M.B, Ferrari G. Identifying the free boundary of a stochastic, irreversible invest- ment problem via the Bank-El Karoui representation theorem. SIAM Journal on Control and Optimization. 2014;52(2):1048-70. Submodular mean field games: Existence and approximation of solutions. The Annals of Applied Probability. J Dianetti, G Ferrari, M Fischer, M Nendel, 31Dianetti J, Ferrari G, Fischer M, Nendel M. Submodular mean field games: Existence and approximation of solutions. The Annals of Applied Probability. 2021 Dec;31(6):2538-66. A unifying framework for submodular mean field games. 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J Jacod, A Shiryaev, Springer Science & Business MediaJacod J, Shiryaev A. Limit theorems for stochastic processes. Springer Science & Business Media. 2013 Mar 9. Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. InSéminaire de Probabilités XV. J Jacod, J Mémin, Springer80Berlin, HeidelbergJacod J, Mémin J. Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. InSéminaire de Probabilités XV 1979/80 1981:529-46. Springer, Berlin, Heidelberg. Stochastic partial ordering. The Annals of Probability. T Kamae, U Krengel, Kamae T, Krengel U. Stochastic partial ordering. The Annals of Probability. 1978 Dec 1:1044-9. Mean field games. J M Lasry, P Lions, Japanese Journal of Mathematics. 21Lasry J.M, Lions P.L. Mean field games. Japanese Journal of Mathematics. 2007 Mar; 2(1):229-60. P.-L Lions, &apos; Cours Au Coll, France De, Lectures on Nov 27th. Lions P.-L. Cours au Coll'ege de France (Lectures on Nov 27th, Dec 4th-11th, 2009). www.college-de-france.fr, 2009. On variant reflected backward SDEs, with applications. J Ma, Y Wang, International Journal of Stochastic Analysis. Ma J, Wang Y. On variant reflected backward SDEs, with applications. International Journal of Stochastic Analysis. 2009;2009. A mean field game of optimal stopping. M Nutz, SIAM Journal on Control and Optimization. 562Nutz M. A mean field game of optimal stopping. SIAM Journal on Control and Optimization. 2018;56(2):1206-21. Reflected backward stochastic differential equations with resistance. Z Qian, M Xu, The Annals of Applied Probability. 282Qian Z, Xu M. Reflected backward stochastic differential equations with resistance. The Annals of Applied Probability. 2018 Apr 1;28(2):888-911. Entropic optimal planning for path-dependent mean field games. Z Ren, X Tan, N Touzi, J Yang, SIAM Journal on Control and Optimization. to appearRen Z, Tan X, Touzi N, Yang J. Entropic optimal planning for path-dependent mean field games. SIAM Journal on Control and Optimization, to appear. Ecole d'été de probabilités de Saint-Flour XIX-1989. A S Sznitman, 1464Topics in propagation of chaosSznitman A.S. Topics in propagation of chaos. Ecole d'été de probabilités de Saint-Flour XIX-1989. 1991 Jan 1;1464:165-251. . D M Topkis, Supermodularity, Complementarity, Princeton University PressTopkis D.M. Supermodularity and complementarity. Princeton University Press; 1998 May 3.

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{'abstract': "We investigate a mean-field version of Bank-El Karoui's representation theorem of stochastic processes. Under different technical conditions, we established some existence and uniqueness results. As motivation and first applications, the results of mean-field representation provide a unified approach for studying various mean-field games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing and the MFG with singular control, etc. As a crucial technical step, a stability result was provided on the classical Bank-El Karoui's representation theorem. It has its own interests and other applications, such as deriving the stability results of optimizers (in the strong sense) for a class of optimal stopping and singular control problems.", 'arxivid': '2302.03300', 'author': ['Xihao He ', 'Xiaolu Tan ', 'Jun Zou '], 'authoraffiliation': [], 'corpusid': 256627384, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 45230, 'n_tokens_neox': 41106, 'n_words': 24501, 'pdfsha': '72f8519fad1ba93cc115ded8f816be3082af835c', 'pdfurls': ['https://export.arxiv.org/pdf/2302.03300v2.pdf'], 'title': ["A mean-field version of Bank-El Karoui's representation of stochastic processes *", "A mean-field version of Bank-El Karoui's representation of stochastic processes *"], 'venue': []}

arxiv

The q-neighbor Ising model on multiplex networks with partial overlap of nodes A Krawiecki Faculty of Physics Warsaw University of Technology Koszykowa 75PL-00-662WarsawPoland T Gradowski Faculty of Physics Warsaw University of Technology Koszykowa 75PL-00-662WarsawPoland The q-neighbor Ising model on multiplex networks with partial overlap of nodes The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate Master equations as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results; predictions based on the approximate Master equations are usually quantitatively more accurate, but yet not exact. Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation. In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition. arXiv:2301.03107v1 [cond-mat.stat-mech] 8 Jan 2023 I. INTRODUCTION Investigation of the opinion formation process by means of nonequilibrium models has become a firmly established research field in statistical physics in the last decades [1]. Many results in this area were obtained using models with agents' opinions represented by spins with discrete (in most cases two) states obeying stochastic dynamics described by various rates at which agents change (e.g., flip) their opinions, e.g., the majority-vote model [2][3][4][5][6][7], the noisy voter model [8][9][10], different versions of the noisy nonlinear and q-voter model [11][12][13][14][15][16][17][18][19][20] and the q-neighbor Ising model [21][22][23][24]. In particular, much effort was devoted to determining conditions under which the above-mentioned models exhibit phase transition from a disordered paramagnetic (PM) state in which each opinion appears with the same probability to an ordered ferromagnetic (FM) state with one dominant opinion as the parameter controlling the level of stochastic noise in the model is varied, measuring the agents' uncertainty in decision making. In this context the presence of the first-order FM transition, or even transition to a frozen FM phase is of prime importance, with abrupt occurrence of a dominant opinion as well as possible hysteresis and bistability of the PM and FM phases [4,5,[12][13][14][15][16][17][18][19][20][21]24]. Following the growing interest in the dynamical processes on complex networks [25] agents in the models for the opinion formation are often located in the nodes and interact via edges of complex networks reflecting a complicated structure of social interactions [3-7, 9, 13-16, 18-20, 23, 24]. In this case analytic predictions concerning the critical behavior of the models based on the mean-field approximation (MFA) need not exhibit quantitative agreement with results of Monte Carlo (MC) simulations, hence, more accurate approaches based on the pair approximation (PA) [26][27][28][29][30] and approximate Master equations (AMEs) [28][29][30] were applied to describe theoretically the observed phase transitions [10, 14-16, 18-20, 24]. Recently much attention has been devoted to combining complex networks in order to create even more complicated and heterogeneous structures known in general as "networks of networks" [31]. An important class of such structures is formed by multiplex networks (MNs) which consist of a fixed set of nodes connected by various sets of edges called layers [31][32][33]. In the simplest case, the layers are independently generated random networks with a full overlap of nodes, i.e., with each node belonging to all layers, which means it has at least one attached edge from each layer. In turn, in MNs with partial overlap of nodes, there are nodes belonging only to some rather than all layers. In particular, in the case of MNs with two layers (duplex networks) and partial overlap of nodes, the nodes are divided into a class of nodes belonging to both layers and forming the overlap, and two other classes, each consisting of nodes belonging only to one of the two layers [34][35][36] (the node overlap should not be confused with the link overlap [37][38][39] which is negligible in the case of independently generated layers). FM phase transition in equilibrium models on MNs was studied, e.g., in the Ising model [40,41] and a related Ashkin-Teller model [42]. Analogously, FM transition in nonequilibrium models for the opinion formation on MNs was studied, e.g., in the majority vote model [44,45], the q-voter model [46][47][48] and the q-neighbor Ising model [49]. As expected, the critical properties of the nonequilibrium models, in particular the extension or confinement of the range of parameters for which the first-order transition occurs, strongly depend on the way in which the multiplexity affects the spin-flip rate. In this respect, very interesting seems the q-neighbor Ising model with LOCAL&AND spin update rule [50], which so far has been studied by MC simulations and in the MF approximation on duplex networks with full and partial overlap of nodes and with layers in the form of fully connected graphs [49]. In this model, the flip probability per unit time for the spins in nodes belonging to only one layer (i.e., outside the overlap) is given by a Metropolis-like rate, but with a local field produced only by a subset of q randomly chosen neighboring spins (q-neighborhood), and for the spins in nodes belonging to both layers (i.e., within the overlap) it is given by a product of two above-mentioned rates evaluated separately for each layer. With the increase of the relative size of the overlap, and depending on the size of the q-neighborhood, suppression of the first-order transition, appearance of a tricritical point separating first-and second-order FM transition, and possible coexistence of the PM and FM phases even in zero temperature were observed in the model [49]. In this paper, the q-neighbor Ising model on MNs with partial overlap of nodes, with layers in the form of complex networks and with the LOCAL&AND spin update rule is studied by means of MC simulations and theoretically in the framework of the PA and AMEs. It should be noted that the q-neighbor Ising model is used here as a convenient example since the results can be readily compared with the above-mentioned ones for the limiting case of the model on MNs with layers in the form of complete graphs [49], and the PA and AMEs used here can be easily generalized to other models for the opinion formation with similar structure of interactions. In order to make large systems of AMEs numerically tractable in this paper only the case of duplex networks with layers in the form of homogeneous random networks is considered; nevertheless, such MNs exhibit certain overlap-induced inhomogeneity since the nodes within and outside the overlap form distinct classes characterized by different degrees within the individual layers (both non-zero or one zero and one non-zero). Thus also the flip rates for the spins located in nodes belonging to distinct classes are different; a related q-voter model with quenched disorder, with agents divided into subpopulations according to different rates of the opinion change, has been recently considered [15]. The aim of this paper is first to provide a general formulation of the PA and AMEs, which take into account to a different extent the above-mentioned inhomogeneity of nodes, for models on MNs with partial overlap of nodes. For this purpose, first, the homogeneous PA for models on MNs with a full overlap of nodes [47] is extended to the case with partial overlap. For nodes belonging to different classes this simplest form of the PA takes into account the inhomogeneity of the average directions of spins (opinions) but neglects possible inhomogeneity of the distributions of directions of neighboring spins within each layer. For the q-neighbor Ising model predictions of this approximation concerning the FM phase transition show surprisingly good agreement with results of MC simulations for a wide range of the size of the q-neighborhood, the mean degrees of nodes within layers and the size of the overlap. Then, the most advanced approximation based on the AMEs for models on MNs with the full overlap of nodes [45] and weighted networks [51] is extended to the case of models on MNs with partial overlap of nodes. Finally, two kinds of heterogeneous PA, the fully heterogeneous PA [15] and the AMEs-based heterogeneous PA [28][29][30] are applied to models on MNs with partial overlap of nodes. Both versions of the PA take into account, to a different extent, the above-mentioned inhomogeneity of distributions of directions of neighboring spins within each layer and are in general intermediate with respect to the accuracy of predictions between the homogeneous PA and the AMEs. For the q-neighbor Ising model under study, it turns out that their predictions are only marginally different or even identical with these of the homogeneous PA. On the other hand, predictions based on the AMEs show slightly better quantitative agreement with the results of MC simulations, in particular for smaller mean degrees of nodes within layers. In general, predictions of all approximations concerning the first-order FM transition (e.g., location and width of the hysteresis loop) are quantitatively worse than those concerning the second-order transition (e.g., location of the critical point). Besides, the aim of this paper is also to study in detail the phase diagram for the q-neighbor Ising model on MNs with partial overlap of nodes and with layers with a finite mean degree of nodes. It is shown that the critical behavior of this model resembles qualitatively that of the analogous model on MNs with layers in the form of fully connected graphs [49]. However, as the mean degree of nodes is decreased, the first-order FM transition, the tricritical point separating it from the second-order transition, and the possible coexistence of the PM and FM phases occur for smaller relative sizes of the overlap, while the range of the occurrence of the second-order FM transition is broadened correspondingly. II. THE MODEL A. Multiplex networks with partial overlap of nodes MNs consist of a fixed set of nodes connected by several sets of edges; the set of nodes with each set of edges forms a network which is called a layer of a MN [32,33]. Henceforth, the nodes are indexed by i, i = 1, 2, . . . N , and the subsequent layers are denoted as G (L) , L = A, B, . . . L max . In the case of MNs with a full overlap of nodes each node belongs to all layers, i.e., each node has at least one edge from each layer attached to it. In general, MNs with partial overlap of nodes are defined as MNs in which nodes may belong to (i.e., may have attached edges from) some rather than all layers, given that each node belongs to at least one layer. Henceforth, the number of nodes belonging to the layer G (L) is denoted as N (L) . In this paper, it is assumed that the sets of edges for the subsequent layers G (L) are generated independently and form complex random networks with N (L) nodes. As a result, multiple connections between nodes are not allowed within the same layer, but the same nodes belonging to several layers can be accidentally connected by multiple edges belonging to different layers. A simple example of the MN with partial overlap of nodes is that with only two layers G (A) , G (B) , called a duplex network, and with n nodes belonging to both layers which form the overlap (0 ≤ n ≤ N ); then, N = N (A) + N (B) − n. Furthermore, if both layers contain the same number of nodes N (A) = N (B) =Ñ it is possible to introduce a single parameter r = n/Ñ , also called the overlap. Then, the nodes are divided into three subsets:Ñ − n = N (1 − r)/(2 − r) nodes belonging only to the layer G (A) , N (1 − r)/(2 − r) nodes belonging only to the layer G (B) and n = N r/(2 − r) nodes belonging both to G (A) and G (B) . The numbers of edges attached to the node i (degrees) within the individual layers G (L) are denoted as k k i = k (A) i , k (B) i , . . . k (Lmax) i , with possible zero components in the case of MNs with partial overlap of nodes, is called a multidegree of the node. The multidegree distribution P (k) = P k (A) i , k (B) i , . . . k (Lmax) i characterizes the MN as a complex "network of networks"; in the case of MNs with the full overlap of nodes and independently generated layers, it is obviously P (k) = Lmax L=A P (k (L) ). In the formulas below, averages are evaluated over the multidegree distribution, e.g., k (L) = N −1 N i=1 k (L) i = k P (k)k (L) is the mean degree of nodes within the layer G (L) (note that the average is over all N nodes rather than N (L) nodes belonging to the layer G (L) ). As a simple example, in this paper the q-neighbor Ising model is considered on a duplex network with partial overlap of nodes and with the two independently generated layers in the form of random regular graphs (RRGs) with K edges attached to each node belonging to the layer, the same numbers of nodes N (A) = N (B) =Ñ and the overlap r, for which the multidegree distribution is P (k) = P k (A) , k (B) = 1 − r 2 − r δ k (A) ,K δ k (B) ,0 + r 2 − r δ k (A) ,K δ k (B) ,K + 1 − r 2 − r δ k (A) ,0 δ k (B) ,K ,(1) and k (A) = k (B) =Ñ K/N = K/(2 − r). B. The q-neighbor Ising model on multiplex networks with partial overlap of nodes The q-neighbor Ising model [21][22][23][24]49] is a nonequilibrium variant of the Ising model used to investigate the process of opinion formation. In this paper the above-mentioned model is considered on MNs with partial overlap of nodes and layers in the form of complex networks; the MF version of this model, on MNs with layers in the form of fully connected graphs, was studied in Ref. [49]. The main interest is in the FM transition which can occur in the q-neighbor Ising model with decreasing effective temperature T , which measures the level of internal noise (uncertainty in agents' decision making). In order to introduce the model under study, it is convenient to start with the q-neighbor Ising model on (monoplex) networks which can be regular, complex, or fully connected graphs [21][22][23][24]49]. In this model agents with two possible opinions on a given subject are represented by two-state spins σ i = ±1, i = 1, 2, . . . N placed in the nodes and interacting via edges of the network. It is assumed that these interactions prefer identical orientations of spins in the connected nodes, which is reflected in the spin-flip rate. Thus, interactions between spins with opposite directions in general increase the probability that one of the spins flips, i.e., the corresponding agent changes opinion, and edges representing these interactions are called active links. The dynamics of the q-neighbor Ising model on networks is a modification of that of the kinetic Ising model with the Metropolis spin-flip rate in which, at each time step, each spin interacts only with its q randomly chosen neighbors. MC simulations of the model are performed using random asynchronous updating of spins, with each MC simulation step (MCSS) corresponding to updating all N spins. Nodes are picked randomly and for each picked node q its neighbors are chosen randomly and without repetitions, which form the q-neighborhood of the picked node. Then, the spin in the picked node is flipped with probability given by a Metropolis-like formula, E (l; T, q) = min {1, exp[−2(q − 2l)/T ]} ,(2) where l is the number of nodes belonging to the q-neighborhood occupied by spins with a direction opposite to that of the spin in the picked node, i.e., the number of active links attached to the picked node leading to nodes within the chosen q-neighborhood (notation in Eq. (2) emphasizes that T , q, are parameters of the model). As a result, the flip rate for a picked spin given that it is placed in a node with degree k which has in total i active links attached (0 ≤ i ≤ k) is f (i; T |k) = 1 k q q l=0 i l k − i q − l E (l; T, q) = 1 k i q l=0 k − q i − l q l E (l; T, q) .(3) The q-neighbor Ising model on complete graphs for q = 3 exhibits second-order FM transition, while for q ≥ 4 first-order FM transition occurs with a clearly visible hysteresis loop. Width of the hysteresis loop in general increases with q, though for q > 4 there are oscillations superimposed on this trend such that loops for the consecutive odd values of q are narrower than for the neighboring even values of q [21]. The same is true for the model on networks with finite mean degree k provided that q k . However, as q is increased and becomes comparable with k the hysteresis loop becomes narrower and eventually disappears, and the FM transition becomes second-order [24]. In the q-neighbor Ising model on MNs with full or partial overlap of nodes, interactions take place within individual layers with respective, independently chosen q-neighborhoods. Then, spins flip according to a probabilistic rule which combines the effect of the above-mentioned interactions. In this paper the LOCAL&AND spin update rule is used [50] according to which the spin in the picked node flips if interaction with every q-neighborhood from every layer suggests flip; consequently, the probability of the spin-flip is given by a product of the Metropolis-like factors (2) corresponding to all layers containing the picked node. The LOCAL&AND rule is assumed in this paper since it usually leads to richer phase diagrams than other methods of including the multiplex character of the network of interactions in the spin-flip rate [46][47][48][49]. Eventually, in numerical simulations of the q-neighbor Ising model on MNs with partial overlap of nodes and the LOCAL&AND spin update rule, each MCSS is performed as follows. (i.) A node i, 1 ≤ i ≤ N , with multidegree k i is picked randomly. (ii.) From each layer G (L) containing the picked node a set of its q neighbors (q-neighborhood) is chosen randomly and without repetitions; it is assumed that 0 < q ≤ k (L) i . Sets from different layers are chosen independently, thus the same node can by chance belong to two or more q-neighborhoods if it is a neighbor of the picked node within two or more layers. (iii.) The Metropolis-like factor for the picked node is evaluated separately for each layer G (L) , E l (L) ; T, q = min 1, exp[−2(q − 2l (L) )/T ](4) where l (L) is the number of nodes in the q-neighborhood in the layer G (L) occupied by spins with direction opposite to that of the spin in the picked node; note that if a node does not belong to G (L) then q = l (L) = 0 and E(T, 0, 0) = 1. (iv.) Due to the LOCAL&AND spin update rule, the spin σ i in the picked node flips with probability E (l; T, q) = Lmax L=A E l (L) ; T, q ,(5) where l = l (A) , l (B) , . . . l (Lmax) ; and obviously l (L) = 0 if the picked node does not belong to the layer G (L) (i.e., l is a vector of numbers of active links from the individual layers attached to the picked node which lead to nodes within the respective q-neighborhoods). f (i; T |k ) = Lmax L=A f i (L) ; T |k (L)(6) (note that if a node does not belong to the layer G (L) there is k (L) = i (L) = 0 and f (0; T |0) ≡ 1). The q-neighbor Ising model on a duplex network with layers in the form of complete graphs and partial overlap of nodes, and with the LOCAL&AND spin update rule exhibits FM phase transition already for q ≥ 1 [49]. This transition is in general second-order, with some exceptions. For q = 2 the transition is first-order for 1/2 < r < 1, with a clearly visible hysteresis loop, and for r c < r ≤ 1/2, where r c = 2(3 √ 2 − 4) = 0.4853 . . ., the coexistence of the FM and PM phases is observed as the temperature is decreased below a critical value down to T = 0; for r < r c there is no phase transition and the PM phase remains the only stable phase down to T = 0. For q ≥ 4 the transition for small r is first-order and for larger r is second-order. The first-and second-order transitions are separated by a tricritical point at r = r T CP (q) which for q = 4 occurs at a particularly high value of r, and for q > 4 is an increasing function of q, but again with oscillations between the consecutive odd and even values of q superimposed on this trend. Remarkably, for r = 1 the FM transition is always second-order for any q, i.e., full overlap of nodes suppresses discontinuous transition. In this paper, it is investigated how the phase diagram of the model changes if the layers of the MN are complex networks with a finite mean degree of nodes. III. THEORY A. Pair approximation In the case of spin models on networks, the effect of the network topology (e.g, of the degree distribution or the mean degree of nodes) on the observed phase transitions often can be more accurately described in the framework of the PA than by the usual MFA [26][27][28][29][30]. In particular, this was demonstrated for the q-neighbor Ising model on complex networks [24] and a sort of stochastic q-voter model on MNs with a full overlap of nodes [47]. In both above-mentioned studies the networks, or the layers of the MNs, were homogeneous complex networks (e.g., RRGs), thus the simplest homogeneous PA was enough to reproduce quantitatively results of MC simulations in a wide range of the parameters of the models. As mentioned in Sec. I & II MNs with partial overlap of nodes retain some multiplexity-induced inhomogeneity even if the layers are homogeneous complex networks. Nevertheless, in this section the homogeneous PA derived in Ref. [47] for a wide class of models with various spin update rules on MNs with the full overlap of nodes is presented in a more general form which makes it applicable to models on MNs with partial overlap of nodes, in order to find, inter alia, to what extent it can be used to explain critical behavior of systems with multiplicity-induced inhomogeneity. The advantage of the PA consists in that it takes into account dynamical correlations between pairs of interacting agents (spins). In the framework of the homogeneous PA, macroscopic quantities characterizing a model with twostate spins on MNs are concentrations c k of spins directed up located in nodes with multidegree k (with possible zero components in the case of MNs with partial overlap of nodes) as well as concentrations b (L) of active links within separate layers G (L) . The homogeneous character of the PA allows for the simplification that the latter concentrations are averaged over all nodes belonging to a given layer and do not depend on the multidegrees of the connected nodes. Consequently, it is assumed that conditional probabilities θ (L) j , j ∈ {↑, ↓}, that an active link within the layer G (L) is attached to a node given that it is occupied by spin with direction j are also independent of the multidegree of the node. These probabilities can be evaluated as ratios of the number of attachments of active links to nodes with spins with direction j, independently of their multidegrees, within the layer G (L) , which is N k (L) b (L) /2, and the number of attachments of all links within G L to such nodes, which is k N P (k) k (L) c k,j , where c k,↑ = c k , c k,↓ = 1 − c k , thus θ (L) ↑ = b (L) 2 k P (k) k (L) c k,↑ / k (L) = b (L) 2 k P (k) k (L) c k / k (L) ,(7)θ (L) ↓ = b (L) 2 k P (k) k (L) c k,↓ / k (L) = b (L) 2 1 − k P (k) k (L) c k / k (L)(8) The core approximation made in the PA for models on MNs is that the numbers of active links i (L) attached to a node with degrees k (L) within individual layers G (L) (0 ≤ i (L) ≤ k (L) ) occupied by spin with direction j obey independent binomial distributions with parameters θ (L) j given by Eq. (7,8). Then, the rates at which the concentration c k increases or decreases are given by averages of the spin-flip rate, Eq. (6), over the appropriate joint distributions of the number of active links within all layers which have a multiplicative form P (j, i|k) = Lmax L=A B k (L) ,i (L) θ (L) j ,(9) where B k,i (θ) = k i θ i (1 − θ) k−i denotes the binomial factor and, formally, B 0,0 (θ) ≡ 1. Hence, the equation for the time dependence of c k can be written as a rate equation, ∂c k ∂t = j∈{↑,↓} (−1) δ j,↑ c k,j i Lmax L=A B k (L) ,i (L) θ (L) j f (i; T |k ) ,(10)where i ≡ k (A) i (A) =0 . . . k (Lmax) i (Lmax) =0 . In order to obtain an equation for the time dependence of the concentrations of active links b (L) one should observe that each flip of a spin (irrespective of its direction) in a picked node with multidegree k with the numbers of active links attached given by the components of the vector i results in the change of the numbers of active links within the individual layers G (L) by k (L) − 2i (L) , since then i (L) previously active links become inactive and k (L) − i (L) previously inactive links become active. The corresponding changes in the concentrations of active links b (L) are thus k (L) − 2i (L) /(N k (L) /2). As in Eq. (10), such changes connected with the flip of a spin with direction j occur at a rate given by the average of the spin-flip rate, Eq. (6), over the appropriate joint distributions of the number of active links attached to the picked node, Eq. (9). Due to the homogeneous character of the PA, in order to obtain time dependence of b (L) further averaging over all nodes occupied by spins with direction j should be performed, which is equivalent to averaging over the probability distribution P (k)c k,j that a node with multidegree k is occupied by a spin with direction j. Eventually, taking into account that nodes are picked and spins are updated within time intervals 1/N , for a given layer G (L ) it is obtained that ∂b (L ) ∂t = 2 k (L ) j∈{↑,↓} k P (k) c k,j i Lmax L=A B k (L) ,i (L) θ (L) j f (i; T |k ) k (L ) − 2i (L ) ,(11) where L = A, B . . . L max . In particular, let us consider the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial overlap of nodes, with the multidegree distribution given by Eq. = c (K,0) , b (A) = b (B) ≡ b; moreover, according to Eq. (7,8) there is θ (6), performing summations in Eq. (10), (11) as in Ref. [24] and introducing functions R(θ; T, q) and S(θ; T, K, q) to shorten notation, (A) j = θ (B) j ≡ θ j . Using Eq. (1), (3),R(θ; T, q) = q l=0 B q,l (θ) E(l; T, q),(12)S(θ; T, K, q) = q l=0 B q,l (θ) [(K − q)θ + l]E(l; T, q),(13) the following system of three equations for the time dependence of the macroscopic quantities in the homogeneous PA is obtained, dc (K,0) dt = 1 − c (K,0) R (θ ↓ ; T, q) − c (K,0) R (θ ↑ ; T, q)(14)dc (K,K) dt = 1 − c (K,K) [R (θ ↓ ; T, q)] 2 − c (K,K) [R (θ ↑ ; T, q)] 2 (15) db dt = 2 K (1 − r) 1 − c (K,0) [KR (θ ↓ ; T, q) − 2S (θ ↓ ; T, K, q)] + c (K,0) [KR (θ ↑ ; T, q) − 2S (θ ↑ ; T, K, q)] + 2 K r 1 − c (K,K) [KR (θ ↓ ; T, q) − 2S (θ ↓ ; T, K, q)] R (θ ↓ ; T, q) + c (K,K) [KR (θ ↑ ; T, q) − 2S (θ ↑ ; T, K, q)] R (θ ↑ ; T, q) ,(16) where θ ↑ = b 2 (1 − r)c (K,0) + rc (K,K) ,(17)θ ↓ = b 2 1 − (1 − r)c (K,0) − rc (K,K) .(18) Other macroscopic quantities of interest are the concentration of spins directed up in each layer, i.e., the fraction ofÑ nodes occupied by such spins, which isc = (1 − r)c (K,0) + rc (K,K) , the concentration of spins directed up in the whole MN, i.e., the fraction of N nodes occupied by such spins, which is c = 2(1−r) 2−r c (K,0) + r 2−r c (K,K) , and the resulting magnetization of the MN m = 2c − 1. Note that in the limiting case of layers in the form of fully connected graphs there is b =Ñ 2c (1 −c)/[Ñ (Ñ − 1)/2] ≈ 2c(1 −c) and θ ↓ =c, θ ↑ = 1 −c; after inserting this into Eq. (14) and (15) equations for the concentrations c (K,0) , c (K,K) in the MF approximation are reproduced [49], as expected. Natural extension of the homogeneous PA consists in taking into account heterogeneity of the concentrations of the (possibly active) links connecting classes of nodes with different multidegrees, so that, instead of the average concentration b (L) of active links within the layer G (L) , e.g., concentrations of classes of active links connecting spins in nodes with multidegrees k, k within the layer G (L) become separate macroscopic quantities characterizing the model. This leads to the most advanced and accurate version of the PA called fully heterogeneous PA [15,27]; corresponding equations for the macroscopic quantities for spin models on MNs with partial overlap of nodes, in particular for the q-neighbor Ising model under study, are given in Appendix A. In the latter case solutions of these equations show that in the stationary state concentrations of active links (strictly speaking, of their ends called bonds) belonging to different classes indeed show noticeable heterogeneity; nevertheless, this does not lead to the values of magnetization noticeably different from these predicted by the homogeneous PA. Thus, magnetization curves and phase diagrams for the model under study obtained from the fully heterogeneous PA are practically indistinguishable from those obtained from the homogeneous PA and do not show better agreement with the results of MC simulations. dc k,m dt = −R k,m c k,m + F k,m s k,m + Lmax L=A −β (L) i k (L) − m (L) c k,m + β (L) i k (L) − m (L) + 1 c k,m−e (L) + Lmax L=A −γ (L) i m (L) c k,m + γ (L) i m (L) + 1 c k,m+e (L) .(19) In Eq. (19), (20) the first two terms account for the effect of a flip of a spin in a node with multidegree k and the remaining terms account for the average effect of the flips of spins in the neighboring nodes, irrespective of their multidegrees. In terms of Sec. II B the flip rate for a spin directed down occupying a node with multidegree k with m neighboring spins directed up is F k,m = f (m; T |k ) and that for a spin directed up R k,m = f (k − m; T |k ). The remaining average rates can be estimated by evaluating the ratios (at a given time step) of the average number of edges connecting spins with a given direction such that one of these spins flips to the average total numbers of these edges [28,29]; in the case of models on MNs this should be done separately for each layer [45,51]. R (0,K),(0,m (B) ) = f K − m (B) ; T |K , R (K,K),(m (A) ,m (B) ) = f K − m (A) ; T |K f K − m (B) ; T |K , with f (m; T |K ) given by Eq. (3). Hence, the system (19), (20) consists of 2(K +1) 2 +4(K +1) equations and can be solved numerically for moderate K. The quantities of interest, e.g., the concentration c of spins directed up in the MN and the magnetization m = 2c−1 can be evaluated as in Sec. III A using c ( m (B) ) . The AMEs are a starting point for a more elaborate approximation representing another formulation of the heterogeneous PA [28-30, 45, 51] which takes into account the possible heterogeneity due to different multidegrees k of nodes of both the concentrations c k of spins directed up and of the conditional probabilities that a link attached to a node is active or, equivalently, leads to a spin with a given (say, up) direction. A general formulation of such AMEs-based heterogeneous PA for spin (two-state) models on (monoplex) networks by Gleeson [28,29] was extended to the case of weighted networks [51] and, partly, MNs [45]. It is believed that due to the approximations made the AMEs-based heterogeneous PA is in general more accurate than the homogeneous PA and less accurate than the fully heterogeneous PA mentioned in Sec. III A. In this paper the AMEs-based heterogeneous PA is applied to spin models on MNs with partial overlap of nodes, in particular to the q-neighbor Ising model under study; equations for the macroscopic quantities are given in Appendix B. Surprisingly, it turns out that in the stationary state the above-mentioned conditional probabilities that a node has a link leading to a spin directed up do not depend on whether the node belongs or not to the overlap. Hence, predictions of the AMEs-based heterogeneous PA concerning the FM transition in the model under study are identical to those of the homogeneous PA from Sec. III A, so they are not further discussed. K,0) = K m (A) =0 c (K,0),(m (A) ,0) , c (0,K) = K m (B) =0 c (0,0),(0,m (B) ) , c (K,K) = K m (A) =0 K m (B) =0 c (K,K),(m (A) , IV. RESULTS The main results concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of complete graphs have been summarized in Sec. II B. These results were obtained in the MF approximation and confirmed by MC simulations [49]. In this section first predictions of the homogeneous PA of Sec. III A concerning the FM transition in the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of RRGs are presented and compared with results of MC simulations. In this case, noticeable discrepancies occur between theoretical and numerical results, in particular concerning the first-order FM transition. As pointed out in Sec. III, the more advanced fully and AMEs-based heterogeneous PA yield results practically indistinguishable or even identical to the homogeneous PA, thus their predictions are only briefly mentioned in the Appendix. Finally, it is verified in which cases and to what extent theoretical predictions are improved by using the AMEs of Sec. III B. In the framework of the homogeneous PA of Sec. III A stationary values of the magnetization m vs. T , corresponding to different thermodynamic phases, are given by stable fixed points of the system of equations (14-16) withċ (K,0) = c (K,K) =ḃ = 0; for certain ranges of parameters r, q, K many stable fixed points can coexist for given T , and their basins of attraction are then separated by stable manifolds of unstable fixed points. The homogeneous PA predicts various critical behavior of the model under study as the temperature T is varied, depending on r, q, K which are fixed: first-and second-order FM phase transition, the coexistence of the PM and FM phases for T → 0 and absence of the FM transition. At high temperatures, the only stable fixed point is that with m = 0 corresponding to the PM phase. In the case of the second-order FM transition, this fixed point loses stability as the temperature is decreased below the critical value T c , and simultaneously a pair of symmetric stable fixed points with m > 0 and m < 0 occurs via a supercritical pitchfork bifurcation, corresponding to the two symmetric FM phases. In the case of the first-order transition two symmetric pairs of stable and unstable fixed points with m > 0 and m < 0 occur simultaneously via two saddle-node bifurcations as the temperature is decreased below the upper critical value T (2) c , and the two above-mentioned stable fixed points correspond to the two symmetric FM phases. As the temperature is further decreased both the FM and PM fixed points remain stable (coexist) until the PM point loses stability via a subcritical as in the case of the first-order transition, but these FM points, as well as the PM fixed point, remain stable (coexist) as T → 0. Finally, it can also happen that fixed points corresponding to the FM phase do not exist for any T > 0, thus the FM transition is absent and the only stable phase for T → 0 is the PM one. Exemplary curves m(T ) predicted by the homogeneous PA for the model under study with different K and selected values of r are shown in Fig. 1 for the most interesting cases q = 2 and q = 4; in the former case, the MFA (valid for the model on MNs with layers in the form of fully connected graphs with K → ∞) predicts occurrence of all above-mentioned kinds of the critical behavior for different ranges of r, while in the latter one it predicts occurrence of the first-order transition for a particularly wide range of small r. The curves m(T ) for q = 2 are drawn in Fig. 1(a) for r = 0.49, and in Fig. 1(b) for r = 0.5, i.e., for the values of r within or at the border of the interval r c < r < 0.5 where the MFA predicts coexistence of the FM and PM phases for T → 0. In contrast, for the model on MNs with layers in the form of RRGs the homogeneous PA for r = 0.49 ( Fig. 1(a)) predicts second-or first-order FM transition for small and moderate K, respectively; the critical temperature(s) decrease and the width of the hysteresis loop increases with K. Only for large K coexistence of the FM and PM phases for T → 0 is predicted by the PA, and the curves m(T ) approach those resulting from the MFA, as expected. For r = 0.5 ( Fig. 1(b)) only second-or first-order FM transitions for finite K are predicted by the PA, with the lower critical temperature for the first-order transition Fig. 1(c) for r = 0.05, and in Fig. 1(d) for r = 0.15, i.e., for the values of r where the MFA predicts first-order FM transition with a wide and narrow hysteresis loop, respectively. For the model on MNs with layers in the form of RRGs the homogeneous PA for small r = 0.05 (Fig. 1(c)) similarly predicts the first-order FM transition for moderate and large K, while for larger r = 0.15 ( Fig. 1(d)) it predicts the second-order FM transition already for moderate K and the first-order FM transition only for large K. Again, the critical temperature(s) decrease, and the width of the hysteresis loop increases with K, and the curves m(T ) eventually approach these resulting from the MFA. It may be inferred from Fig. 1 that the homogeneous PA predicts for the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of RRGs with finite K the same critical behavior as the MFA for the model on analogous MNs with layers in the form of complete graphs, only for different ranges of the overlap r. This conclusion is supported by Fig. 2 where the critical behavior predicted by the PA is summarized for the former model with fixed q = 2 and q = 4 and different K, r. For all K and r = 1 (full overlap of nodes), both PA and MFA predict continuous FM transition with decreasing T , i.e., the first-order transition observed in the model on monoplex networks is suppressed. However, for both q = 2, 4 and finite K the PA predicts the second-order FM transition also for a range of r below r = 1 which is broadened with decreasing K. As a consequence, for q = 2 (Fig. 2, left and middle panels) the PA predicts that the range of the occurrence of the first-order FM transition is shifted toward smaller values of r. Similarly, for a narrow range of still smaller values of the overlap the PA predicts the coexistence of the FM and PM phases for T → 0, but for small K this kind of critical behavior is expected at r significantly below the interval r c < r < 0.5 obtained from the MFA. Finally, it is predicted that the range of small r for which the FM transition is absent for decreasing K is narrowed. Eventually, for very small K = 4 comparable with q only continuous FM transition is expected for any r, and all other kinds of critical behavior are suppressed. For q = 4 (Fig. 2, right panel) the range of small r for which the PA predicts the first-order FM transition is substantially diminished with decreasing K. In order to verify predictions of the homogeneous PA, MC simulations of the q-neighbor Ising model with q = 2, 4 on large MNs with various parameters r, K were performed and the magnetization curves m(T ) were obtained for random PM initial conditions σ i = ±1, i = 1, 2, . . . N and decreasing temperature as well as for FM initial conditions σ i = +1, i = 1, 2 . . . N and increasing temperature. Comparison with MC simulations shows that the homogeneous PA qualitatively captures modification of the critical behavior of the model under study due to finite values of the mean degree of the layers K, but its quantitative predictions, though much improved in comparison with those from the MFA valid for large K, are not exact (Fig. 3, 4). For fixed q, K the PA approximately predicts the ranges of the overlap r where different kinds of critical behavior should occur. However, as a rule, these predictions are overestimated and in MC simulations the particular kinds of critical behavior appear for smaller values of r than estimated from the PA. For example, for q = 2 the ranges of appearance of the coexistence of the FM and PM phases for T → 0 ( Fig. 3(a,b)) and of the second-order FM transition (Fig. 3(e,f)) in MC simulations are, respectively, shifted and extended toward smaller values of r than expected from the PA. Consequently, in the case of the first-order FM transition for q = 2 (Fig. 3(c,d)) and q = 4 ( Fig. 4(a,b,e,f)) the lower and upper critical temperatures T (1) c , T (2) c are underestimated and the width of the hysteresis loop is overestimated by the PA in comparison with these obtained from MC simulations; it is interesting to note that discrepancies between the theoretical and numerical values of T (2) c are usually smaller than those for T (1) c . Similarly, in the case of the second-order FM transition for q = 2 (Fig. 3(f)) and q = 4 ( Fig. 4(c,d)) the critical temperature T c is underestimated by the PA in comparison with that obtained from MC simulations. In general, the curves m(T ) evaluated from the PA show better agreement with those obtained from MC simulations in the case of the second-order than the first-order FM transition. In order to investigate the critical behavior of the model under study by means of appropriate AMEs as defined in Sec. III B, Eq. (19,20) were solved numerically with various initial conditions and the curves m(T ) were obtained using long-time asymptotic values of the concentrations of spins directed up c (K,0),(m (A) ,0) , etc., to evaluate stationary values of the magnetization. As expected, predictions of the AMEs usually show comparable or better agreement with the results of MC simulations than those of the homogeneous PA. This is particularly visible in the case of the second-order FM transition in the model under study with small K and q = 2 ( Fig. 3(f)) and q = 4 ( Fig. 4(d)), where the theoretical and numerical curves m(T ) coincide very well and the critical temperature T c is predicted correctly. However, the ranges of the overlap predicted by the AMEs for which different kinds of critical behavior occur are still shifted toward slightly higher values of r than obtained from MC simulations (see Fig. 3(a), where the AMEs predict the absence of the FM transition rather than coexistence of the FM and PM phases observed in MC simulations, and Fig. 3(e), where the AMEs predict the first-order FM transition with a narrow hysteresis loop rather than the second-order transition). In the case of the coexistence of the FM and PM phases for T → 0 for q = 2 ( Fig. 3(b)) and the first-order FM transition for q = 2 ( Fig. 3(c,d)) and q = 4 ( Fig. 4(a,b,e,f)) predictions of the AMEs concerning the upper critical temperature T (2) c are usually better than those of the homogeneous PA, but the lower critical temperature T (1) c is again usually underestimated and the width of the hysteresis loop is overestimated. In general, some improvement of theoretical predictions by the AMEs in comparison with the homogeneous PA can be seen for small and moderate K; for large K the curves m(T ) obtained from the AMEs and PA coincide (Fig. 4(e)). V. DISCUSSION AND CONCLUSIONS In this paper the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of random networks was investigated; as an example, the model on MNs with two layers in the form of RRGs was studied in detail. Both theoretical considerations based on the homogeneous PA and AMEs as well as MC simulations show that for given q ≥ 1 and finite mean degree of nodes K, and for varying overlap r and temperature T the model exhibits qualitatively similar critical behavior as the q-neighbor Ising model on MNs with partial overlap of nodes and layers in the form of complete graphs. In particular, for any q and full overlap of nodes r = 1 the first-order FM transition is suppressed and only the second-order transition appears with decreasing T . Besides, for decreasing K continuous rather than discontinuous FM transition is observed for an increasing range of large (for q = 2) and large and moderate (for q > 2) values of r below r = 1. As a consequence, for decreasing K the ranges of r for which the model exhibits the first-order FM transition (for q ≥ 2) and the coexistence of the FM and PM phases for T → 0 (for q = 2) are shifted toward smaller values. It should be mentioned that in the q-neighbor Ising model on (monoplex) networks the first-order FM transition is also suppressed for small K comparable with q [24]; in contrast, in the model on MNs this suppression is due to the overlap of nodes and occurs for any K. The q-neighbor Ising model was used here as an example, and related models for the opinion formation on MNs with partial overlap of nodes can be studied using similar numerical and analytic methods; however, the expected qualitative changes of the observed critical behavior with r will be probably less spectacular, since, e.g., in the case of the q-voter model even for r = 1 the first-order FM transition is not suppressed [47]. For the model under study with large K predictions of the simple homogeneous PA and more advanced system of AMEs converge to these of the MFA and agree quantitatively with the results of MC simulations. For finite K the predicted curves m(T ) and critical temperature(s) differ quantitatively from the numerically obtained ones; usually, the particular kinds of critical behavior are predicted to occur for smaller values of the overlap than observed in MC simulations. In general, predictions of both PA and AMEs show better agreement with the results of MC simulations in the case of the continuous than discontinuous FM transition. Predictions based on the AMEs are comparable to or better than those of the PA; in particular, the critical temperature for the second-order FM transition and the upper critical temperature for the first-order transition are more accurately predicted. Nevertheless, both PA and AMEs qualitatively correctly capture changes of the critical behavior of the model with varying parameters K, r characterizing the underlying MN. Two versions of the heterogeneous PA were derived for the model under study, the more accurate fully heterogeneous PA and the less accurate AMEs-based heterogeneous PA, which take to a different extent into account heterogeneity of the distribution of the active links due to inhomogeneity of the nodes. In both cases, systems of equations for the macroscopic quantities characterizing the model are significantly larger and more complicated than that in the homogeneous PA but do not lead to a noticeable improvement of theoretical predictions concerning the magnetization curves and phase diagrams for the model. This suggests that the simple homogeneous PA is as reliable as more advanced versions of the PA in the study of the critical behavior of systems with multiplexity-induced inhomogeneity. Only using much larger systems of AMEs in certain cases quantitatively improves agreement between theoretical predictions and results of MC simulations of the above-mentioned models. APPENDIX A. Fully heterogeneous pair approximation The following outline of the fully heterogeneous PA for models on MNs is an extension of that for the q-voter models with quenched disorder on networks, with two populations of agents differing by the spin-flip rates [15]. The fully heterogeneous PA uses the assumption that the probability that a spin directed up or down in a node with multidegree k has a given number of attached edges leading to spins directed up or down in nodes with multidegreek obeys a binomial distribution; this assumption is valid for each layer and for any pair k,k separately, and the related binomial distributions are assumed to be independent. The macroscopic quantities characterizing a model with two-state spins on a MN are concentrations c k of spins directed up in nodes with multidegree k and concentrations e k,k,(L) j,j (= ek ,k,(L) j,j ) of bonds (ends of edges) attached within the layer G (L) to nodes with multidegree k containing spins with direction j ∈ {↓, ↑} such that at the other end of the edge there is a node with multidegreek containing spin with directioñ j (normalized to the total number of bonds N k (L) within G (L) ). According to the above-mentioned assumptions the joint probability that a node with multidegree k containing spin with direction j has i = i (A) , i (B) , . . . i (Lmax) active bonds (ends of active links) attached within the consecutive layers, pointing at nodes with arbitrary multidegree containing spins with opposite direction −j, has a multiplicative form P (j, i|k) = Lmax L=A B k (L) ,i (L) α k,(L) j ,(21) where α k,(L) j = k e k,k ,(L) j,−j / k j ∈{↓,↑} e k,k ,(L) j,j are conditional probabilities that an active bond is attached to a node with multidegree k and spin with direction j (similar to θ (L) ↑ , θ (L) ↓ given by Eq. (7,8)). In order to evaluate the change in the concentration e k,k,(L) j,j due to, e.g., flipping the spin with direction j in a node with multidegree k, it is necessary to know the numbers of bonds y, z attached to this node within the layer G (L) pointing at nodes with multidegreek given that these bonds are active (j = −j) or inactive (j = j), respectively. These numbers ∂c k ∂t = j∈{↑,↓} (−1) δ j,↑ c k,j i Lmax L=A B k (L) ,i (L) α k,(L) j f (i; T |k ) ,(22) while the rate equations for the concentrations of active and inactive bonds within the layers contain terms such as d dt e k,k,(L ) j,j = 1 k (L ) P (k)c k,j i Lmax L=A B k (L) ,i (L) α k,(L) j k (L ) −i (L ) z=0 B k (L ) −i (L ) ,z γ k,k,(L ) j,j (−z)f (i; T |k ) + . . .(24) It can be seen that Eq. (22) resembles Eq. (10) in the homogeneous PA. Concerning the equations for the concentrations of bonds, e.g., Eq. (23) states that a flip of the spin with direction j in node with multidegree k, which has i (L ) active bonds attached within the layer G (L ) , out of which y bonds point at nodes with multidegreesk, decreases the concentration e k,k,(L ) j,−j by y/ N k (L ) ; such a flip occurs with probability P (k)c k,j f (i; T |k ) within a time interval 1/N ; and the final input to the rate equation (23) dη (L ) k dt = m η (L ) k − m (L ) k (L ) R k,m Lmax L=A B k (L) ,m (L) η (L) k − 1 − c k c k F k,m Lmax L=A B k (L) ,m (L) ϑ (L) k +β (L ) i 1 − η (L ) k −γ (L ) i η (L ) k ,(41) where L = A, B . . . L max . The above equations are very similar to those obtained in the AMEs-based heterogeneous PA for the spin models on (monoplex) networks [28,29]; in particular, terms containing β since the respective terms from Eq. (19), (20) sum up to zero in the derivation. It should be mentioned that the AMEs can also be a starting point to obtain the homogeneous PA from Sec. III A by assuming that the probability that a spin directed down has within the layer G (L) a neighboring spin directed up does not depend on k and can be expressed as the average θ dϑ (K,K) dt = ϑ (K,K) R ϑ (K,K) ; T, q 2 − c (K,K) 1 − c (K,K) R 1 − η (K,K) ; T, q 2 − 1 K S ϑ (K,K) ; T, K, q R ϑ (K,K) ; T, q node i does not belong to the layer G (L) then k (L) i = 0. In the case of MNs with independently generated layers the degrees of nodes belonging to the individual layers G (L) , i.e., these with k (L) i > 0, are drawn from probability distributions P k (L) which characterize the layers as complex networks. For a given node i a vector of its degrees within the individual layers (v.) Steps (i.)-(iv.) are repeated until all N spins are updated without repetition.Hence, the flip rate for a spin placed in a node with multidegree k = k (A) , k (B) , . . . k (Lmax) and with the numbers of attached active links within the individual layers i (L) , 0 ≤ i (L) ≤ k (L) , given by the corresponding components of the vector i = i (A) , i (B) , . . . i (Lmax) assumes a multiplicative form, (1). Then, the nodes are divided into three classes, these belonging only to the layer G(A) with multidegree k = (K, 0), only to the layer G (B) with k = (0, K) and to the overlapping part of G (A) and G (B) , with k = (K, K). The macroscopic quantities to be used in the homogeneous PA are thus concentrations of spins directed up in the nodes belonging to the subsequent classes c (K,0) , c (0,K) , c (K,K) and concentrations of active links in the two layers b (A) , b (B) . Since both layers are identical, with N (A) = N (B) =Ñ , stable solutions of the system of equations (10), (11) are limited to the subspace with c (0,K) k accurate approximation for the study of spin models on MNs with partial overlap of nodes is based on approximate Master equations (AMEs) for the densities of spins directed up c k,m and down s k,m which are located in nodes with multidegree k and have m (L) neighboring spins directed up within the consecutive layers G (L) , which is denoted as m = m (A) , m (B) . . . m (Lmax) . In the thermodynamic limit and for mutually uncorrelated layers in the form of random networks with finite mean degrees k (L) possibility that a pair of nodes is connected simultaneously by edges within different layers can be neglected. Thus, in the AMEs it is assumed that in a single simulation step for a given node the allowed changes of the number of neighboring spins directed up are m → m ± e (L) , where e (L) is a unit vector with L max components and only L-th component equal to one, while simultaneous changes of many components of m, e.g., m → m ± e (L) ± e (L ) , L = L , etc., cannot occur. Under the above-mentioned assumptions, the AMEs in a general form are[45,51] ds k,m dt = −F k,m s k,m + R k,m c (L) − m (L) s k,m + β (L) s k (L) − m (L) + 1 s k,m−e (L) + Lmax L=A −γ (L) s m (L) s k,m + γ (L) s m (L) + 1 s k,m+e (L) , k (L) − m (L) F k,m s k,m / m k (L) − m (L) s k,m , γ (L) s = m k (L) − m (L) R k,m c k,m / =0 and . .. denotes average over the multidegree distribution P (k), as usually. Natural initial conditions for the system of equations(19), in the case of the q-neighbor Ising model on a MN with two layers in the form of RRGs and partial overlap of nodes, with the multidegree distribution P (k) given by Eq.(1), there are three classes of nodes with k = (0, K), k = (K, 0) and k = (K, K). The corresponding spin flip rates are F (K,0),(m (A) ;0) = f m (A) ; T |K , F (0,K),(0;m (B) ) = f m (B) ; T |K , F (K,K),(m (A) ,m (B) ) = f m (A) ; T |K f m (B) ; T |K and R (K,0),(m (A) ,0) = f K − m (A) ; T |K , FIG. 1 . 1The curves show magnetization m vs. temperature T obtained from the homogeneous PA for different K (green solid lines, both stable and unstable fixed points of the system of equations(14)(15)(16) are shown) and from the MFA of Ref.[49] (black solid lines), for q = 2, K = 200, 100, 50, 20, 10, 4 (from left) and (a) r = 0.49, (b) r = 0.5, as well as for q = 4, K = 500, 200, 100, 50, 20, 10 and (c) r = 0.05, (d) r = 0.15.pitchfork bifurcation at the lower critical temperature T by colliding simultaneously with the two above-mentioned unstable fixed points; coexistence of the PM and FM phases for T the occurrence of the hysteresis loop in the magnetization curves m(T ). Eventually, for T < T (1) c the only stable fixed points remain these corresponding to the two symmetric FM phases. In the case of the coexistence of the FM and PM phases for T → 0 a pair of symmetric stable FM fixed points occurs at T = T(2) c FIG. 2 . 2Critical behavior predicted by the homogeneous PA for the model with q = 2 (left and middle panels) and q = 4 (right panel) and different r, K; filled circles -continuous FM transition, open circles -discontinuous FM transition, filled squares -coexistence of the FM and PM phases for T → 0, crosses -absence of the transition. FIG. 3 . 3Results of MC simulations, predictions of the PA and AMEs for the model with q = 2, K = 20 and (a) r = 0.45, (b) r = 0.46, (c) r = 0.47, (d) r = 0.50, (e) r = 0.60, (f) r = 0.70; blue dots -results of MC simulations with FM initial conditions and increasing temperature, red dots -results of MC simulations with PM initial conditions and decreasing temperature, black dots -predictions of the AMEs for both FM (c(0) = 1) and PM (c(0) = 0.5) initial conditions and increasing or decreasing temperature, respectively, green solid lines -predictions of the PA as in Fig. 1. FIG. 4 .> 0 . 40As in Fig. 3 but for q = 4. (a) K = 20, r = 0.05, (b) K = 20, r = 0.10, (c) K = 20, r = 0.15, (d) K = 10, r = 0.10, (e) K = 50, r = 0.10, (f) k = 10, r = 0.05. The curves m(T ) for q = 4 are drawn in . obey binomial distributions B i (L) ,y β k,k,(L) j,−j , B k (L) −i (L)Then the rate equations for the macroscopic concentrations of spins directed up are (L ) P (k)c k,j i Lmax L=A B k (L) ,i (L) α k,(L) j i (L ) y=0 B i (L ) ,y β k,k,(L ) j,−j (−y)f (i; T |k ) + . . . FIG. 5 . 5is obtained by averaging the above-mentioned change over the probability distributions B k (L ) ,i (L ) α k,(L) j for i (L ) and B i (L ) ,y β k,k,(L ) j,−j for y; etc. In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with two layers in the form of RRGs, with the multidegree distribution P (k) given by Eq. (1), there are three classes of nodes with k = (K, 0), k = (0, K) and k = (K, K), and two layers G (L) , L = A, B. Taking into account the symmetry of the model under study and general symmetry conditions for the concentrations e The curves show magnetization m vs. temperature T obtained from the homogeneous PA (solid lines) and from the heterogeneous PA (symbols) for q = 4, K = 20 and r = 0.1, 0.15, 0.2 (from left to right). to time and using Eq.(19),(20) with the above-mentioned approximations yields the following system of equations for the time dependence of the macroscopic quantities in the heterogeneous PA, m (L) s k,m / k (L) (1 − c k )[28,29]. In the case of the q-neighbor Ising model on MNs with partial overlap of nodes and with layers in the form of RRGs, with the multidegree distribution P (k) given by Eq. (1), there are three classes of nodes with k = (K, 0), k = (0, K) and k = (K, K), and two layers G (L) , L = A, B, thus the system of equations (40-42) is 11-dimensional. Due to the symmetry of the model solutions of these equations should be constrained to a subspace c (0,K) = c (K,K) ≡ η (K,K) which reduces the number of equations to six. Performing summations in Eq.(40)(41)(42) as in Ref.[24] the following system of equations for the macroscopic quantities is obtained in the AMEs-based heterogeneous PA for the model under study,dc (K,0) dt = −c (K,0) R 1 − η (K,0) ; T, q + 1 − c (K,0) R ϑ (K,0) ; T, q ,(43)dϑ (K,0) dt = ϑ (K,0) R ϑ (K,0) ; T, q − c (K,0) 1 − c (K,0) R 1 − η (K,0) ; T, q − 1 K S ϑ (K,0) ; T, K, q − c (K,0) 1 − c (K,0) KR 1 − η (K,0) ; T, q − S 1 − η (K,0) ; T, K, q +β s 1 − ϑ (K,0) −γ s ϑ (K,0) ,(44)dη (K,0) dt = η (K,0) R 1 − η (K,0) ; T, q − 1 − c (K,0) c (K,0) R ϑ (K,0) ; T, q − 1 K KR 1 − η (K,0) ; T, q − S 1 − η (K,0) ; T, K, q − 1 − c (K,0) c (K,0) S ϑ (K,0) ; T, K, q +β i 1 − η (K,0) −γ i η (K,0) ,(45)dc (K,K) dt = −c (K,K) R 1 − η (K,K) ; T, q 2 + 1 − c (K,K) R ϑ (K,K) ; T, q 2 , ↑ ; T, K, q , j, j ∈ {↓, ↑}. Thus, as in Ref.[15], there are effectively only two classes of agents located in nodes with k = (K, 0) and k = (K, K), differing by the spin-flip rates(6). Besides, the distributions of the number of links pointing at nodes belonging to each class given that these links are active or inactive are fully determined by the conditional probabilities β k,k j,−j , γ k,k j,j for the links within each class. Taking this into account and performing summations in Eq.(22 -24)as in Ref.[24]the following system of equations for the macroscopic quantities is obtained in the fully heterogeneous PA for the model under study,d dt ed dt e (K,0),(K,0)d dt ewhere the significant conditional probabilities are As mentioned in Sec. III A the magnetization curves obtained from the fully heterogeneous PA are practically indistinguishable from those obtained from the homogeneous PA. This is illustrated by examples inFig. 5.B. AMEs-based heterogeneous pair approximationThe AMEs-based heterogeneous PA again uses the assumption that the probability that a spin directed up or down in a node with multidegree k has a given number of neighboring spins directed up obeys a binomial distribution; for models on MNs this assumption is made for each layer separately, and the related binomial distributions are assumed to be independent. Hence, in contrast with the homogeneous PA, in the AMEs-based heterogeneous PA it is taken into account that for a node with multidegree k occupied by a spin with downward or upward direction the respective probabilities ϑ (L) k , η (L) k that a randomly chosen neighboring node within the layer G (L) is occupied by a spin directed upward can depend on k. However, in contrast with the fully heterogeneous PA developed in Appendix A, all active or inactive edges attached to a given node within a given layer are treated in the same way and obey common binomial distributions[28,29]. As mentioned in Sec. III B, these two assumptions should make the AMEs-based heterogeneous PA more accurate than the homogeneous PA and less accurate than the fully heterogeneous PA. Eventually, in the AMEs-based heterogeneous PA the time-dependent macroscopic quantities are the density c k of spins directed up in nodes with multidegree k as well as the above-mentioned probabilities ϑ KR 1 − η (K,0) ; T, q − S 1 − η (K,0) ; T, K, q R 1 − η (K,K) ; T, qwhere the average rates arē0); T, K, q + r 2 − r 1 − c (K,K) S ϑ (K,K) ; T, K, q R ϑ (K,K) ; T, q ,Concentrationc of spins directed up within each layer and concentration c of spins directed up in the MN are defined in the same way as in Sec. III A. Natural initial conditions for the system of equations(43)(44)(45)(46)(47)(48)are ϑ (K,0) (0) = η (K,0) (0) = ϑ (K,K) (0) = η (K,K) (0) =c(0), while c (K,0) (0), c (K,K) (0) can be chosen arbitrarily. . K) , (k , K) ↑ , ↑ , ,K),(K,K) ↑ , ↑ Statistical physics of social dynamics. 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{'abstract': 'The q-neighbor Ising model for the opinion formation on multiplex networks with two layers in the form of random graphs (duplex networks), the partial overlap of nodes, and LOCAL&AND spin update rule was investigated by means of the pair approximation and approximate Master equations as well as Monte Carlo simulations. Both analytic and numerical results show that for different fixed sizes of the q-neighborhood and finite mean degrees of nodes within the layers the model exhibits qualitatively similar critical behavior as the analogous model on multiplex networks with layers in the form of complete graphs. However, as the mean degree of nodes is decreased the discontinuous ferromagnetic transition, the tricritical point separating it from the continuous transition and the possible coexistence of the paramagnetic and ferromagnetic phases at zero temperature occur for smaller relative sizes of the overlap. Predictions of the simple homogeneous pair approximation concerning the critical behavior of the model under study show good qualitative agreement with numerical results; predictions based on the approximate Master equations are usually quantitatively more accurate, but yet not exact. Two versions of the heterogeneous pair approximation are also derived for the model under study, which, surprisingly, yield predictions only marginally different or even identical to those of the simple homogeneous pair approximation. In general, predictions of all approximations show better agreement with the results of Monte Carlo simulations in the case of continuous than discontinuous ferromagnetic transition. arXiv:2301.03107v1 [cond-mat.stat-mech] 8 Jan 2023', 'arxivid': '2301.03107', 'author': ['A Krawiecki \nFaculty of Physics\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland\n', 'T Gradowski \nFaculty of Physics\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland\n'], 'authoraffiliation': ['Faculty of Physics\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland', 'Faculty of Physics\nWarsaw University of Technology\nKoszykowa 75PL-00-662WarsawPoland'], 'corpusid': 255546297, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23349, 'n_tokens_neox': 20878, 'n_words': 13325, 'pdfsha': 'b6d598c480e9e26e30b39d5dbbf3fbb169ca37b7', 'pdfurls': ['https://export.arxiv.org/pdf/2301.03107v1.pdf'], 'title': ['The q-neighbor Ising model on multiplex networks with partial overlap of nodes', 'The q-neighbor Ising model on multiplex networks with partial overlap of nodes'], 'venue': []}

arxiv

Robust optimal policies for team Markov games Feng Huang Ming Cao Long Wang Robust optimal policies for team Markov games Index Terms-Robust optimal controlteam Markov gamesmulti-agent learningsequential social dilemmas In stochastic dynamic environments, team Markov games have emerged as a versatile paradigm for studying sequential decision-making problems of fully cooperative multi-agent systems. However, the optimality of the derived policies is usually sensitive to model parameters, which are typically unknown and required to be estimated from noisy data in practice. To mitigate the sensitivity of optimal policies to these uncertain parameters, we propose a "robust" model of team Markov games in this paper, where agents utilize robust optimization approaches to update strategies. This model extends team Markov games to the scenario of incomplete information and meanwhile provides an alternative solution concept of robust team optimality. To seek such a solution, we develop a robust iterative learning algorithm of team policies and prove its convergence. This algorithm, compared with robust dynamic programming, not only possesses a faster convergence rate, but also allows for using approximation calculations to alleviate the curse of dimensionality. Moreover, some numerical simulations are presented to demonstrate the effectiveness of the algorithm by generalizing the game model of sequential social dilemmas to uncertain scenarios. I. INTRODUCTION M ARKOV games [1], also known as stochastic games [2], as a general framework of multi-agent reinforcement learning (MARL) have long been an active research topic across the fields of stochastic optimal control, operations research, and artificial intelligence (AI) [3], [4], [5]. Especially, it has attracted much attention in recent years due to some ground-breaking advances made by the MARL in conjunction with deep neural networks in achieving humanlevel performance in several long-standing sequential decisionmaking conundrums [6], [7]. Compared with the normal-form matrix games [8], [9] and evolutionary games [10], [11], one of its most distinctive features is the introduction of game states. In each stage of the play, the game is in a state taken from a given set, and each player then relies on its decision rule as a function of the current state to choose actions. The collection of players' actions, together with the current state, subsequently determine the probability distribution of the state that the play will visit at next stage. As the consequence of the joint actions and state transitions, every player will receive an immediate payoff. Therefore, Markov games are usually deemed as the generalization of one-shot matrix games to the dynamic multi-stage and multi-state situations, in which game states change in response to players' decisions. Also, they extend the Markov decision processes (MDPs) [12] from only involving a single agent to the scenario involving multiple decision-makers [13]. In contrast to the non-cooperative setting of Markov games, team Markov games [14], [15] are a fully cooperative multiagent system, in which a group of players works together, through interactions, coordinations, and information-sharing, to collectively accomplish a task or achieve a goal [16], [17]. Hence, it is formally defined as the Markov games where all players have a common payoff function [3], [5]. In particular, such a game model has recently sparked increasing research interest across various disciplines due to its extensive realworld applications, such as multi-robot systems, unmanned aerial vehicles, and communication networks (see [3], [5], [18] for an overview), and its close connection to the theory of team decisions [19], [20], [21]. However, most of the existing work is based on the complete information setting where the structural information of the game, such as players' payoff functions and state transition probabilities, is the common prior knowledge to all players. Under such an assumption, the notion of the optimal Nash equilibrium [14], i.e. the joint decision rule achieving the maximal expected discounted sum of the gains received by the team, naturally provides a plausible solution to predicting the outcome of the game evolution. This is because at this equilibrium state, no player can improve the long-term expected return of the team by unilaterally deviating from its policy. Although there are some debates about the predictive or prescriptive role of the equilibrium concept in practice (see [22] and its commentaries), Nash equilibria have played a central role in the corresponding studies [23], [24]. While the Nash equilibrium solution always exists for the Markov games with complete information [25], in the real world, the problem related to the model uncertainty of games is pervasive. For example, in many realistic applications associated with reinforcement learning (RL), the reward functions of agents are usually required to be designed or learned from interactions, and their properties strongly affect the success of targeted tasks [26]; also, the transition probability distribution of states is generally estimated from historical data, thereby influenced by statistical errors [27]. In particular, such an issue has given rise to well-grounded concerns, such as in the field of AI safety [28], [29] and uncertain robotic systems [30], and accordingly it prompts the research priorities of robust AI [31]. A. Related work In game theory, the problem related to model uncertainty has a long research tradition and usually refers to the games with "incomplete information" [32], in which part of game parameters is unspecified. The first analytical framework developed to study this class of games is from the seminal three-part essay by Harsanyi [32] where a new model named "Bayesian games" is constructed and the notion of Nash equilibrium is extended to the incomplete information scenario, termed "Bayesian equilibrium." Utilizing robust optimization [33], Aghassi and Bertsimas then relax the prior distribution assumption of Harsanyi's model for uncertain parameters and open a new avenue to robust games [34], where players take a best response depending on a worst-case payoff matrix in one-shot normal-form games. Via a similar technique, these results are later extended to Markov games [35] and some deep MARL algorithms have been developed to find the equilibria [36]. Parallel to the game-theoretical research, robust dynamic programming (rDP) [37], [38] is another domain devoting to coping with the decision-making problems with data uncertainty. To mitigate the sensitivity of optimal policies to the ambiguity of transition probabilities in MDPs, two algorithms, the so-called robust value iteration (rVI) and robust policy iteration (rPI), are proposed to find robust optimal policies. Inspired by these work, Kaufman and Schaefer [39] then develop a robust version of modified policy iteration (rMPI) [12] as the generalization of these two algorithms. In particular, recent years have witnessed some applications of these methods in the temporal logic control of stochastic systems [40] and the continuous control of physical systems [41]. However, one common shortcoming of these algorithms is that the convergence rate may be quite slow when the discount factor used to compute the long-term expected return is close to one. As the policy space is very large and/or continuous, moreover, these algorithms may become less applicable in general due to the curse of dimensionality, and thus require to resort to function approximations or data-driven techniques [27]. Motivated by this fact, a large body of robust RL algorithms has recently been proposed by incorporating diverse approaches, such as adversarial training [42], [43], online policy search [44], [45], and least squares policy iteration [46]. B. Contributions In this work, we aim to deal with the sequential decisionmaking problem of a cooperative team in stochastic uncertain environments, and accordingly propose a robust model of team Markov games. This model relaxes the complete information assumption in team Markov games and meanwhile provides an alternative solution concept termed robust team-optimal policy. To seek such a solution, we develop a robust iterative learning algorithm of team policies, which we call robust approximate Team Policy Iteration (raTPI). Compared with rDP, this algorithm allows for using approximation computations to alleviate the curse of dimensionality. Furthermore, we present the convergence analysis of the algorithm within mild approximation tolerance and accordingly calculate its convergence rates. The results manifest that at a near exponential convergence rate, this algorithm can effectively find the robust team-optimal policy with sufficient accuracy after a finite number of iterations. To demonstrate the effectiveness of the algorithm, we also carry out numerical simulations by generalizing the canonical game model of social dilemmas to sequential uncertain scenarios. Using them as benchmarks, simulations show that, as compared with rVI and rMPI, our algorithm admits substantially less iteration time to find the robust team-optimal policy. Notation: Throughout the paper, we use R and N to represent the set of real numbers and the set of non-negative integers, respectively. For n sets A i , i = 1, 2, . . . , n, let n i=1 A i be their Cartesian product. For a scalar x, we use |x| to denote its absolute value. The space of probability distributions on the set S = {s 1 , s 2 , ..., s m } is denoted by ∆(S), and the set of bounded real valued functions on S is represented by V. For a vector v = [v(s 1 ), v(s 2 ), . . . , v(s m ))] T ∈ V, its vector norm is defined by v := sup s∈S |v(s)|, where v(s) represents the component of v corresponding to s ∈ S and T represents transpose. As such, (V, · ) forms a normed linear space, and it is also a Banach space. For a matrix P = [p kl ] ∈ R m×m with element p kl in row k and column l, its matrix norm and spectral radius are defined by P := sup k∈{1,2,...,m} m l=1 |p kl | and σ(P ) := lim sup t→∞ P t 1/t , respectively. Alternatively, we also use P (l | k) to represent the element of P in row k and column l. We use I and 1 to denote the identity matrix and the all-ones vector with appropriate dimensions, respectively. The order, maximum, and minimum of vectors and matrices refer to componentwise order, maximum, and minimum. II. PROBLEM FORMULATION AND PRELIMINARIES In this section, we first introduce the sequential decisionmaking problem in team Markov games and then present one of its robust counterparts in uncertain environments. A. Team Markov games We consider the decision-making problem of a cooperative team with n players, which is modelled by a team Markov game [14] (also known as multi-agent MDPs [16]) with the infinite decision horizon T := {1, 2, . . .}. A team Markov game is characterized by a tuple N , S, A, {r i } i∈N , p , where N := {1, 2, . . . , n} is the finite set of the indices of n players; S := {s 1 , s 2 , ..., s m } is the finite set consisting of m states shared by all players; A := n i=1 A i represents the set of the joint action of all players, which is the Cartesian product of the discrete finite set A i of actions available to player i ∈ N ; r i (s, a, s ) : S × A × S → R defines the payoff function of player i, which depends on the current state s, the joint action a of all players, and the state s at the next epoch; and p(·|s, a) : S × A → ∆(S) represents the transition probability function of states, which maps from the current state s to the probability distribution over the state space, given the joint action a. Without loss of generality, we assume in this paper that both the payoff functions of players and the transition probability functions of states are stationary. That is, for any triple (s, a, s ) ∈ S × A × S, r i (s, a, s ) for ∀i ∈ N and p(s |s, a) do not change with time. Moreover, r i (s, a, s ) is assumed to be bounded, i.e. |r i (s, a, s )| ≤ R max < ∞ for ∀i ∈ N and ∀(s, a, s ) ∈ S × A × S, where R max is a constant. Moreover, in this paper, we restrict our attention to stationary and deterministic policies due to their mathematical tractability [12], [47]. Specifically, at each decision epoch, we consider that given the current state s ∈ S, each player i ∈ N in the team selects an action a i ∈ A i deterministically according to its individual decision rule d i (s) : S → A i , which is a mapping from state s to a deterministic action a i . These decision rules adopted by player i at all decision epochs then constitute its policy or strategy used in the game, which is a time sequence (d i , d i , . . .) of decision rules. For ease of notations, we denote the policy of player i by π i := (d i ) ∞ = (d i , d i , . . .) . As such, the team decision rule and the team policy can be represented by the joint decision rule of all players, d := (d 1 , . . . , d n ), and the joint policy of all players, π := (π 1 , . . . , π n ) = (d) ∞ , respectively. Accordingly, we denote the space of team decision rules and the space of team policies by D and Π, respectively. From the definition of team decision rules, one can see that every joint action of all players is completely determined by the team decision rule d. In response to the joint action a τ given by d at decision epoch τ ∈ T , the game system will transit from the current state s τ to the state s τ +1 at the next epoch with probability p(s τ +1 |s τ , a τ ). As the consequence of the joint action and the state transition, every player i ∈ N will then get an immediate payoff r i (s τ , a τ , s τ +1 ). Starting from an initial state s 1 = s ∈ S, when the team adopts a specific joint policy π ∈ Π to play the game, the system will induce a probability measure on the trajectory of states and the sequence of joint actions. To evaluate the performance of the joint policy, we adopt the expected total discounted return of the team-average payoff as the criterion, which is calculated by v π (s) = E π ∞ τ =1 λ τ −1 r(s τ , a τ , s τ +1 ) s 1 = s ,(1) where E π {·} represents the expectation over the stochastic process {(s τ , a τ )} τ ∈T induced by the joint policy π; λ ∈ [0, 1) is the discount factor used to evaluate the present value of future payoffs; and r(s τ , a τ , s τ +1 ) = 1 n n i=1 r i (s τ , a τ , s τ +1 ) is the team-average payoff at epoch τ ∈ T . As a solution to the team decision-making problem, we consider that all players aim to seek a team-optimal policy π ∈ Π such that for ∀π ∈ Π, v π (s) ≥ v π (s), ∀s ∈ S. To find π , one can see from (1) that all players will rely on the common utility r to make decisions at each epoch. In other words, all players in effect have a common payoff function r in the game. Thus, the team decision-making problem is in a fully cooperative setting [3], [5]. Moreover, it is easy to verify that every team-optimal policy is a Markov perfect equilibrium (see [5], [35] for its definition) of Markov games. For ease of the exposition of our main results, we further introduce some vector and matrix notations, and rewrite (1) in a vector form. Given a joint action a ∈ A, let P a be the transition probability matrix of states, where the element of P a in row k and column l is given by p(s l |s k , a), ∀s k , s l ∈ S. Accordingly, we denote the transition probability matrix induced by the joint decision rule d ∈ D by P d , where the entry of P d in row k and column l is given by p(s l |s k , d(s k )). As such, given a joint policy π = (d) ∞ , the probability P r(s τ = s l |s 1 = s k ; π) that the game system transits from the initial state s k ∈ S to the state s l ∈ S at epoch τ ∈ T can be calculated by P r(s τ = s l |s 1 = s k ; π) = (P d P d · · · P d ) (l|k) = (P d ) τ −1 (l|k),(2) where (P d ) 0 = I. For a given P d , moreover, one can calculate the expected value r (d,P d ) (s) of the team-average payoff for ∀s ∈ S by r (d,P d ) (s) = s ∈S r(s, d(s), s )p(s |s, d(s)).(3) It follows from (2) and (3) that for s 1 = s k ∈ S, (1) can be rewritten by v π (s k ) = ∞ τ =1 λ τ −1 E π {r(s τ , a τ , s τ +1 )|s 1 = s k } = ∞ τ =1 λ τ −1 m l=1 P r(s τ = s l |s 1 = s k ; π)r (d,P d ) (s l ) = ∞ τ =1 λ τ −1 m l=1 (P d ) τ −1 (l|k)r (d,P d ) (s l ).(4) Let v (d,P d ) := [v π (s 1 ), v π (s 2 ), . . . , v π (s m )] T and r (d,P d ) := [r (d,P d ) (s 1 ), r (d,P d ) (s 2 ), . . . , r (d,P d ) (s m )] T . Then, from (4), the vector form of (1) can be given by v (d,P d ) = ∞ τ =1 λ τ −1 P d τ −1 r (d,P d ) = r (d,P d ) + λP d r (d,P d ) + λP d r (d,P d ) + · · · = r (d,P d ) + λP d v (d,P d ) .(5) Since r (d,P d ) ≤ R max in view of the assumption |r i (s, a, s )| ≤ R max < ∞ for ∀i ∈ N and ∀(s, a, s ) ∈ S × A × S and P d is a row stochastic matrix, v π (s) ≤ R max /(1 − λ) for ∀s ∈ S and ∀π ∈ Π. It implies that v (d,P d ) ∈ V for ∀d ∈ D and any P d . Moreover, note that σ(λP d ) ≤ λP d = λ < 1. It then follows that v (d,P d ) = (I − λP d ) −1 r (d,P d ) = ∞ τ =1 (λP d ) τ −1 r (d,P d ) is the unique solution to the equation v = r (d,P d ) + λP d v, where v ∈ V is the variable. For the sake of convenience, we will call any vector v ∈ V the value function afterwards. B. Robust team Markov games The aforementioned model of team Markov games is based on the assumption of complete information, i.e. the parameter information of the game is explicitly known to players. Here, we propose one of its robust counterparts by relaxing this assumption. We assume that players do not know the true transition probabilities of game states, but rather commonly perceive an uncertainty set of their possible values. More specifically, given a joint action a ∈ A, the transition probability matrix P a is not predetermined but lies in an uncertainty set P a ⊂ R m×m . Since players' payoff functions are defined to depend on the state at the next epoch in team Markov games, the fluctuation of transition probabilities will lead to the uncertainty of players' payoffs. Therefore, we refer to a team Markov game as a robust team Markov game if players' payoffs and/or the transition probabilities of states are uncertain. Following the convention, we assume that for ∀a ∈ A, the uncertainty set P a satisfies the so-called "rectangularity" property [37], [38]. Namely, P a has the form of P a = m k=1 P a (·|k) and is independent of historically visited states and actions, where P a (·|k) ⊆ ∆(S) for ∀k ∈ {1, 2, . . . , m} characterizes the uncertainty of the k-th row of P a and is assumed to be discrete and finite for ease of exposition. The extension to the continuous case is straightforward, which will not significantly bring new insights but rather require more complex notations and additional assumptions for the existence of optimal policies [12], [39]. Moreover, in many practical applications, such as video games, board games, and autonomous vehicles, a robust learning system acquires all possible transition probabilities of different scenarios usually by massively sampling from the realistic physical system, even though the underlying uncertainty set may be continuous. Therefore, in practice, the learning system generally operates in the discrete and finite sample set. Also, such a discrete and finite set is more practical for the numerical evaluation of an algorithm. Under this assumption, the admissible set of P d for a given d ∈ D can be then represented by P d := m k=1 P d(s k ) (·|k). To systematically mitigate the influence of uncertain transition probabilities on the performance of the team-optimal policy, we consider the robust team-optimal policy π * = (d * ) ∞ ∈ Π as an alternative solution such that the value function v (d * ,P * d * ) is maximal with respect to the worst-case P * d * in the uncertainty set P d * . Definition 1. A value function v * := v (d * ,P * d * ) is said to be robust team-optimal if v (d * ,P * d * ) = max d∈D min P d ∈P d v (d,P d ) , and accordingly π * = (d * ) ∞ is called a robust team-optimal policy. Similar to the team-optimal policy, it is also easy to verify that every robust team-optimal policy is a robust Markov perfect equilibrium (see [35] for its definition) of robust stochastic games. Note that both D and P d for ∀d ∈ D are discrete and finite. Therefore, such a robust team-optimal policy can always be attained. Moreover, in contrast to the robust team-optimal policy, its near-optimal counterpart may be more preferred in practice. Definition 2. A policy (d ) ∞ is said to be -robust team- optimal for > 0 if min P d ∈P d v (d ,P d ) ≥ v * − 1. In the following section, we will develop an algorithm to seek the robust team-optimal policy and meanwhile present its convergence analysis. III. ROBUST TEAM-OPTIMAL POLICY SEEKING To seek the robust team-optimal policy, we here develop an algorithm, named robust approximate Team Policy Iteration (raTPI). This algorithm is built on the structure of the so-called "classical information pattern" [48], [49] or the framework of centralized learning with decentralized execution, under which a central controller is used to compute the desired team policy π and then each individual policy of π is communicated to the corresponding player for execution. A. The raTPI Algorithm The basic idea of the raTPI algorithm is to utilize rMPI to unify the advantages of both rVI and rPI, and meanwhile to improve the convergence rate by splitting methods [12]. Its fundamental architecture is to generate a sequence of value functions, {v t }, t = 0, 1, . . . by an iterative process such that v t can sufficiently approach the robust team-optimal value function v * after a large number of iterations. The iterative process primarily encompasses four parts. First, it starts with a given initial value function v 0 := [v 0 s 1 , v 0 s 2 , . . . , v 0 s m ] T ∈ V 0 at t = 0, where for ease of notations, we adopt v 0 s j , ∀j ∈ {1, 2, . . . , m} to represent the component of v 0 corresponding to state s j ∈ S and V 0 to represent the feasible set of v 0 , respectively. Subsequently, the algorithm applies a process of policy improvement to obtain an improved team policy. Specifically, given an arbitrary state s k ∈ S, for every joint action a ∈ A, it first computes an approximationρ ( s k ,a) (v t ) of ρ (s k ,a) (v t ) with respect to v t := [v t s 1 , v t s 2 , . . . , v t s m ] T at t, where v t s j , ∀j ∈ {1, 2, . . . , m} is the element of v t corresponding to state s j ∈ S, and ρ (s k ,a) (v t ) is the quantity obtained by running one step of the Gauss-Seidel value iteration [12] under the worst case of transition probability distributions, i.e. ρ (s k ,a) (v t ) = min p(·|s k ,a)∈Pa(·|k) m l=1 r(s k , a, s l )p(s l | s k , a) +λ   l<k p(s l | s k , a)u t 0 (s l ) + l≥k p(s l | s k , a)v t s l      ,(6) and u t 0 (s l ) = max a∈Aρ(s l ,a) (v t ). One reason for using approximations therein is that an exact calculation is usually impractical when the state and/or action space is very large, and thus it requires to resort to approximation techniques. By selecting d t+1 (s k ) ∈ arg max a∈Aρ(s k ,a) (v t ) for every s k ∈ S, one can then obtain an improved team policy (d t+1 ) ∞ , and subsequently each individual decision rule of d t+1 is communicated to the corresponding player. Using the acquired intermediate value u t 0 := [u t 0 (s 1 ), u t 0 (s 2 ), . . . , u t 0 (s m )] T , the algorithm next implements a partial performance evaluation of the improved team policy (d t+1 ) ∞ by running multiple steps of the value iteration similar to (6). In this process, two new notations ς and M t of non-negative integers are introduced to represent the current iteration step and the total numbers of iterations, respectively. Specifically, for each given state s k ∈ S, every player first executes an action according to its received individual decision rule from d t+1 , and subsequently the algorithm collects the joint action a = d t+1 (s k ) ∈ A of all players. Next, an approximationρ (s k ,a ) (u t ς ) of ρ (s k ,a ) (u t ς ) is calculated with respect to u t ς := [u t ς (s 1 ), u t ς (s 2 ), . . . , u t ς (s m )] T at iteration ς (ς = 0, 1, . . . , M t ). Therein, ρ (s k ,a ) (u t ς ) is the quantity obtained by running one step of the Gauss-Seidel value iteration under the worst-case transition probability distribution used in (6), i.e. ρ (s k ,a ) (u t ς ) = m l=1 r(s k , a , s l )p * (s l | s k , a ) (7) +λ   l<k p * (s l | s k , a )u t ς+1 (s l ) + l≥k p * (s l | s k , a )u t ς (s l )   ,and u t ς+1 (s l ) =ρ (s l ,a ) (u t ς ), where p * (· | s k , a ) is the worstcase transition probability distribution used in (6) for the given state s k and action a . This iteration repeats by increment ς = ς + 1 until ς = M t . Lastly, if the termination condition u t 0 − v t < (1 − λ) /2λ − δ is satisfied at t, the algorithm stops and returns a team policy (d ) ∞ , where d = d t+1 , δ ∈ [0, (1 − λ) 2 /2λ(1 + λ)) is the parameter used to confine the maximal approximation tolerance between ρ (s,a) (·) andρ (s,a) (·) for ∀(s, a) ∈ S × A, and > 0 is the small constant used to control the precision of the acquired -robust team-optimal policy. The complete computational process is illustrated in Algorithm 1. B. The convergence analysis Since directly proving the convergence of raTPI is difficult, we here divide the convergence analysis of raTPI into two parts. The first part demonstrates the convergence of raTPI in a degenerated situation, i.e. M t = 0 for ∀t ∈ N. Since in this case Algorithm 1 primarily relies on the value iteration to work and the partial policy evaluation will not come into effect, we refer to this degenerated form of raTPI as the robust approximate Team Value Iteration (raTVI). Based on the results of raTVI, the second part establishes the convergence of raTPI in its general form where {M t } t∈N is an arbitrary sequence of non-negative integers. 1) The degenerated form raTVI: This part establishes the convergence of the degenerated form raTVI by setting M t = 0 for ∀t ∈ N. We first consider the exact case where the tolerance of approximation computations is zero, i.e. δ = 0. Note that the partial policy evaluation in Algorithm 1 will not work due to M t = 0 for ∀t ∈ N, and ρ (s,a) (·) =ρ (s,a) (·) holds for ∀(s, a) ∈ S × A due to δ = 0. Thus, from (9) and in view of v t+1 s = u t 0 (s) for ∀s ∈ S in the step 3(f), one can get the iterative scheme of raTVI by v t+1 s = max a∈A ρ (s,a) (v t ), ∀s ∈ S.(12) Utilizing the regular splitting in matrix iterative analysis [50], we proceed to derive the vector form of (12). Letď ∈ D be the decision rule such that for ∀s ∈ S, d(s) ∈ arg max a∈A ρ (s,a) (v t ), and Pď ∈ Pď be the worstcase transition probability matrix with element Pď(l | k) in Algorithm 1 robust approximate Team Policy Iteration (raTPI) Input: v 0 ∈ V 0 , > 0, λ ∈ [0, 1), δ : 0 ≤ δ < (1 − λ) 2 /2λ(1 + λ) , and {M t } t∈N . 1: Set t = 0. 2: (Policy improvement) Set k = 1 and go to 2(a). (a) For ∀a ∈ A, first compute an approximatioñ ρ (s k ,a) (v t ) of ρ (s k ,a) (v t ) such that |ρ (s k ,a) (v t ) − ρ (s k ,a) (v t )| ≤ λδ; (8) then set u t 0 (s k ) = max a∈Aρ (s k ,a) (v t ),(9) and select d t+1 (s k ) ∈ arg max a∈Aρ(s k ,a) (v t ). (b) If k = m, go to 3; otherwise set k = k + 1 and then return to 2(a). 3: (Partial policy evaluation) (a) If u t 0 − v t < (1 − λ) /2λ − δ where u t 0 := [u t 0 (s 1 ), u t 0 (s 2 ), . . . , u t 0 (s m )] T , go to 4; otherwise set ς = 0 and then go to 3(b). (b) If ς = M t , go to 3(f); otherwise set k = 1 and then go to 3(c). (c) For u t ς := [u t ς (s 1 ), u t ς (s 2 ), . . . , u t ς (s m )] T , compute an approximationρ (s k ,dt+1(s k )) (u t ς ) of ρ (s k ,dt+1(s k )) (u t ς ) such that |ρ (s k ,dt+1(s k )) (u t ς ) − ρ (s k ,dt+1(s k )) (u t ς )| ≤ λδ, (10) and set u t ς+1 (s k ) =ρ (s k ,dt+1(s k )) (u t ς ).(11) (d) If k = m, go to 3(e); otherwise set k = k + 1 and then return to 3(c). (e) Set ς = ς + 1 and then return to 3(b). (f) Set v t+1 = u t Mt and t = t + 1, and then return to 2. 4: Set d (s) = d t+1 (s) for ∀s ∈ S and then stop. Output: -robust team-optimal policy (d ) ∞ . row k and column l when computing ρ (s,ď(s)) (v t ) by (6). We then split Pď into Pď = Pď L + Pď U , where Pď L and Pď U are the lower triangular matrix of Pď and the upper triangular matrix including the diagonal elements of Pď, respectively. As such, from (6) and v t+1 s = u t 0 (s) for ∀s ∈ S in the step 3(f), the vector form of (12) can be given by v t+1 = r (ď,Pď) + λ Pď L v t+1 + Pď U v t . By re-arrangement, it yields v t+1 = (I − λPď L Qď ) −1 r (ď,Pď) + (I − λPď L ) −1 λPď U Rď v t = Qď −1 r (ď,Pď) + Qď −1 Rďv t ,(13) where (I − λPď L ) −1 exists because σ(λPď L ) ≤ λPď L < 1. According to the definition of regular splitting (i.e. a pair of matrices (B, C) is a regular splitting of matrix A if A = B − C, B −1 ≥ 0, and C ≥ 0) [50], one can see that (Qď, Rď) is a regular splitting of I − λPď because Qď − Rď = I − λPď, Qď −1 = ∞ ı=0 (λPď L ) ı ≥ 0, and Rď ≥ 0. For convenience, we refer to this specific regular splitting as the GS regular splitting afterwards, due to the use of the Gauss-Seidel (GS) value iteration. That is, given a transition probability matrix P and a constant λ ∈ [0, 1), the regular splitting (Q, R) of I − λP is said to be a GS regular splitting if Q = I − λP L and R = λP U , where P L and P U are the lower triangular matrix of P and the upper triangular matrix including the diagonal elements of P , respectively. Since in (13),ď ∈ D is chosen such that the summation of the two terms on the righthand side of (13) is maximized with respect to the worse-case transition probability matrix Pď ∈ Pď, (13) is equivalent to v t+1 = max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v t ,(14) where (Q d , R d ) is the GS regular splitting of I − λP d . For any v ∈ V, let the operator Y : V → V be defined by Y v := max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v .(15) Then, (14) can be rewritten by v t+1 = Y v t . Although the definition of Y is based on the GS regular splitting, we will show that whenever (Q d , R d ) is a regular splitting of I − λP d , Y is a contraction mapping on V and the convergence rate of the sequence generated by it is less than 1. Before formally presenting this result, we first introduce the definition of the convergence rate of a sequence [12]. Definition 3. Let {v t }, ∀t ∈ N be a sequence that con- verges to v * . {v t } is said to converge at the order c > 0 if there exists a constant L such that ||v t+1 − v * || ≤ L||v t − v * || c for ∀t ∈ N, and its convergence rate is defined to be the smallest L satisfying this inequality. The asymptotic average rate of convergence (AARC) of {v t } is defined as lim sup t→∞ ||v t − v * ||/||v 0 − v * || 1/t . Given a non-negative function f (t) ∈ V, {v t } is said to be O(f (t)) if lim sup t→∞ ||v t − v * ||/f (t) is finite. Any one of the above results is said to be global if it holds for ∀v 0 ∈ V; otherwise, it is said to be local. Having this definition, we now show in the following lemma that the sequence generated by Y will converge in norm to the robust team-optimal value function and its convergence rate is no greater than the constant α (α < 1). Lemma 1. For any given d ∈ D and P d ∈ P d , let (Q d , R d ) be a regular splitting of I −λP d . If (Q d , R d ) satisfies α := sup d∈D,P d ∈P d Q −1 d R d < 1, then (a) the sequence {v t } generated by v t+1 = Y v t converges in norm to the robust team-optimal value function v * for t → ∞ and v * is the unique fixed point of Y ; and (b) {v t } converges globally at order 1 at a rate no greater than α; its global AARC is no greater than α, and it is globally O(β t ), β ≤ α. The detailed proof of this lemma is given in Appendix A. From Lemma 1, one can see that only if the regular splitting (Q d , R d ) of I −λP d satisfies α := sup d∈D,P d ∈P d Q −1 d R d < 1 , the sequence generated by the iterative scheme (14) of raTVI will converge to the robust team-optimal value function. In reality, however, for the GS regular splitting, α < 1 will always hold. This fact is guaranteed by the following lemma. (14) is the GS regular splitting of I − λP d , and Q d = I and R d = λP d are a trivial regular splitting of I − λP d . Moreover, since R d is the upper triangular matrix including the diagonal elements of λP d from the definition of the GS regular splitting, one can get R d ≤ R d . It then follows from Lemma 2 that (14) is always satisfied. From Lemma 1, one can then obtain the part (a) of the following theorem. Lemma 2. (Proposition 6.3.5 in [12]) Given a transition probability matrix P , let (Q 1 , R 1 ) and (Q 2 , R 2 ) be the regular splittings of I − λP for λ ∈ [0, 1). If R 2 ≤ R 1 ≤ λP , then ||Q −1 2 R 2 || ≤ ||Q −1 1 R 1 ||. Note that for any given d ∈ D and P d ∈ P d , the (Q d , R d ) inQ −1 d R d ≤ Q d −1 R d = λ < 1. Since d ∈ D and P d ∈ P d are arbitrary, the condition α < 1 for the (Q d , R d ) inTheorem 1. For M t = 0, ∀t ∈ N, if the approximation calculations are exact, i.e. δ = 0, in Algorithm 1, then for any v 0 ∈ V, (a) the sequence {v t } generated by the iterative scheme (14) of raTVI converges in norm to the robust teamoptimal value function v * for t → ∞; it converges globally at order 1 at a rate no greater than λ; its global AARC is no greater than λ, and it is globally O(β t ), β ≤ λ; and (b) Algorithm 1 terminates within a finite number of iterations with an -robust team-optimal policy (d ) ∞ and its corresponding value function v under the worst-case transition probability matrix satisfies v − v * < . Proof: The result in part (a) is immediate from the above analysis. We now prove part (b). Since {v t } converges to v * , it is a Cauchy sequence. Hence, the termination condition in the step 3(a) of Algorithm 1 will be satisfied for any > 0 after a finite number of iterations in view of v t+1 = u t 0 for M t = 0 in the step 3(f) of Algorithm 1. Without loss of generality, suppose that the algorithm terminates at t = N , i.e. v N +1 − v N < (1 − λ) /2λ, and meanwhile it returns a decision rule d . Then, based on the setup in the step 2(a) and the step 4 of Algorithm 1, one can get from the iterative scheme (14) that d ∈ arg max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v N , where (Q d , R d ) is the GS regular splitting of I − λP d . Next, for the decision rule d , select P d ∈ arg min P d ∈P d Q −1 d r (d ,P d ) + Q −1 d R d v N , where (Q d , R d ) is the GS regular splitting of I − λP d . Moreover, given d ∈ D, P d ∈ P d , and the GS regular splitting (Q d , R d ) of I − λP d for λ ∈ [0, 1), we define the operator T (d,P d ) : V → V for v ∈ V by T (d,P d ) v = Q −1 d r (d,P d ) + Q −1 d R d v.(16) It is easy to check that T (d,P d ) is a contraction mapping on V because for any u, v ∈ V, P d ) . Also, from the definitions of d and P d , one can see that T (d,P d ) u − T (d,P d ) v = Q −1 d R d (u − v) ≤ Q −1 d R d u − v ≤ I −1 (λP d ) u − v = λ u − v , where Q −1 d R d ≤ I −1 (λP d ) follows from Lemma 2. Then, by solving the fixed point equation v = T (d ,P d ) v for variable v ∈ V, one can get that the unique fixed point of T (d ,P d ) is v (d ,T (d ,P d ) v N = Y v N = v N +1 . It follows that v (d ,P d ) − v N +1 = T (d ,P d ) v (d ,P d ) − v N +1 ≤ T (d ,P d ) v (d ,P d ) − T (d ,P d ) v N +1 + T (d ,P d ) v N +1 − T (d ,P d ) v N ≤ λ v (d ,P d ) − v N +1 + λ v N +1 − v N . By re-arrangement, one can get v (d ,P d ) − v N +1 ≤ λ v N +1 − v N /(1 − λ) < /2. Moreover, by applying the contraction property of Y , we have v N +1 − v * ≤ v N +1 − Y v N +1 + Y v N +1 − v * ≤ λ v N +1 − v N + λ v N +1 − v * . By re-arrangement, v N +1 −v * ≤ λ v N +1 −v N /(1−λ) < /2. Consequently, v (d ,P d ) − v * ≤ v (d ,P d ) − v N +1 + v N +1 − v * < . Since v (d ,P d ) is the value function corresponding to the policy (d ) ∞ under the worst-case transition probability matrix, we have the result in part (b). One important precondition for the establishment of Theorem 1 is that those approximation calculations for ρ (s,a) (·), ∀(s, a) ∈ S × A in Algorithm 1 are exact, i.e. δ = 0. In contrast, if the error is inevitable for approximations, the establishment of the convergence will become quite intractable. This is because the approximation error will be continuously accumulated along with the iteration of the algorithm and eventually it will lead the iterative sequence to deviate from the desired value function. Fortunately, however, we find that if the parameter δ is bounded by 0 ≤ δ < (1−λ) 2 /2λ(1+λ), every elementṽ t of the sequence generated by Algorithm 1 will always lie within a small interval, which is centered at the element v t of the sequence generated by Algorithm 1 for δ = 0. That is, for the sequence {ṽ t } generated by raTVI with δ ∈ [0, (1 − λ) 2 /2λ(1 + λ)) and the sequence {v t } generated by raTVI with δ = 0, the following lemma holds. Lemma 3. If v 0 =ṽ 0 , then {v t } and {ṽ t } satisfy v t − θ1 ≤ṽ t ≤ v t + θ1, ∀t ∈ N,(17)where θ := λδ/(1 − λ) ∈ [0, /2). See Appendix B for its proof. From this lemma, one can immediately get lim t→∞ṽ t = v * by noticing that lim t→∞ v t = v * from Theorem 1 and θ ∈ [0, /2). We display this result in the following theorem. Theorem 2. For M t = 0, ∀t ∈ N, if the tolerance of approximation calculations is within λδ for δ ∈ [0, (1 − λ) 2 /2λ(1 + λ)) in Algorithm 1, then for any v 0 ∈ V, (a) the sequence {ṽ t } generated by Algorithm 1 with the initializatioñ v 0 = v 0 converges in norm to the robust team-optimal value function v * for t → ∞; it converges globally at order 1 at a rate no greater than λ; its global AARC is no greater than λ, and it is globally O(β t ), β ≤ λ; and (b) those results shown in the part (b) of Theorem 1 still hold. Proof: Since we have got in Theorem 1 that the sequence {v t } generated by Algorithm 1 for M t = 0, ∀t ∈ N and δ = 0 converges to v * , there is a positive integer N for any > 0 such that v t − v * < /2 holds for ∀t ≥ N . Moreover, from Lemma 3, one can obtain that for ∀t ∈ N, ṽ t − v * ≤ ṽ t − v t + v t − v * ≤ v t − v * + θ. (18) Therefore, we have ṽ t − v * ≤ /2 + θ < for ∀t ≥ N , which implies thatṽ t converges to v * . We now calculate the convergence rates of {ṽ t }. From Theorem 1, we first note that {v t } satisfies v t − v * ≤ λ v t−1 − v * for ∀t ∈ N. Moreover, from Lemma 3, v t−1 −v * ≤ v t−1 −ṽ t−1 + ṽ t−1 −v * ≤ ṽ t−1 −v * +θ holds for all t. Based on these two inequalities, it follows from (18) that ṽ t − v * ≤ v t − v * + θ ≤ λ v t−1 − v * + θ ≤ λ ṽ t−1 − v * + (λ + 1)θ. Since θ ∈ [0, /2) can be arbitrarily close to zero in view of the arbitrariness of , ṽ t −v * ≤ λ ṽ t−1 −v * for all t. It means that {ṽ t } converges globally at order 1 and the convergence rate is no greater than λ. Similarly, from (18), we also have ṽ t −v * ≤ v t −v * . It follows that ṽ t − v * / v 0 − v * ≤ v t − v * / v 0 − v * and ṽ t − v * /β t ≤ v t − v * /β t for β > 0 and ∀t ∈ N. Therefore, based on the results in Theorem 1, one can get that the global AARC of {ṽ t } is no greater than λ and it is globally O(β t ), β ≤ λ. We next prove part (b). Since {ṽ t } converges to v * from the result in part (a), the termination condition in the step 3(a) of Algorithm 1 will be satisfied after a finite number of iterations. Moreover, since when δ = 0, it is shown from Theorem 1(b) that {v t } will satisfy the termination condition within a finite number of iterations, there exists a positive integer N for any 1 > 0 such that v N +1 − v N < (1 − λ) 1 /2λ. Then, from Lemma 3, we have ṽ N +1 −ṽ N ≤ ṽ N +1 − v N +1 + v N +1 − v N + v N −ṽ N ≤ v N +1 − v N + 2θ < (1 − λ) 1 /2λ + 2λδ/(1 − λ). Given that 1 is arbitrary, we therefore select a specific 1 such that 1 ≤ − 2λ(1 + λ)δ/(1 − λ) 2 for any given > 0. As such, ṽ N +1 −ṽ N ≤ (1 − λ) /2λ − δ holds. It implies that for M t = 0, ∀t ∈ N, Algorithm 1 with approximation errors will also terminate at t = N . Moreover, for any given s ∈ S, one can find that there exists a constant gap between ρ (s,a) (v t ) and ρ (s,a) (ṽ t ) (see the proof of Lemma 3) and an approximation error between ρ (s,a) (ṽ t ) andρ (s,a) (ṽ t ) from (8) for ∀a ∈ A and ∀t ∈ N, and they do not affect the selection of the maximizing actions. Thus, arg max a∈A ρ (s,a) (v N ) = arg max a∈Aρ(s,a) (ṽ N ). Without loss of generality, we choose the same decision rule for Algorithm 1 in the case with and without approximation errors at t = N . Thus, the result v − v * < shown in Theorem 1(b) also holds for δ ∈ [0, (1 − λ) 2 /2λ(1 + λ)). 2) The general form: We have presented the convergence of the degenerated form raTVI by the aforementioned analysis. Based on the results in this specific case, we now proceed to demonstrate the convergence of raTPI in its general form, i.e. M t is a non-negative integer for ∀t ∈ N. Via a similar analytical process, we first give the results in the exact case where the approximation calculations in Algorithm 1 can be exactly obtained, i.e. δ = 0, and then extend them to the inexact case where the tolerance of approximation calculations is within λδ for 0 ≤ δ < (1 − λ) 2 /2λ(1 + λ). We begin with considering the exact case, i.e. δ = 0. Let the operator B : V → V for v ∈ V be defined by Bv := Y v − v = max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v − v ,(19) where (Q d , R d ) is the GS regular splitting of I − λP d . From this definition, it is easy to see that the fixed-point of Y is the same as the zero-point of B. Moreover, for any given v ∈ V, we denote the set of v-improving decision rules by D v := arg max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v , and accordingly denote the set of d v -decreasing transition probability matrices for a given d v ∈ D v by P * dv := arg min P dv ∈P dv Q dv −1 r (dv,P dv ) + Q dv −1 R dv v . Having these concepts at hand, one can immediately obtain the following proposition. Proposition 1. For any given u, v ∈ V and d v ∈ D v , there exists a P dv ∈ P dv such that Bu ≥ Bv+(Q dv −1 R dv −I)(u− v), where (Q dv , R dv ) is the GS regular splitting of I − λP dv . See Appendix C for its proof. Intuitively, this proposition implicitly suggests that B is a Lipschitz continuous mapping by selecting appropriate norms. On the other hand, one can conveniently derive the vector formulation of the iterative scheme of raTPI. For any v t , ∀t ∈ N, let d v t be the decision rule satisfying d v t (s) ∈ arg max a∈A ρ (s,a) (v t ) for ∀s ∈ S, and P * d v t be the transition probability matrix with element p * (s l | s k , d v t (s k )) in row k and column l for s k , s l ∈ S, where p * (· | s k , d v t (s k )) is the worst-case transition probability distribution used to calculate ρ (s k ,d v t (s k )) (v t ) by (6). Then, from (8) and (9), by applying an analogous argument to the derivation of (13), one can get the vector form of (9) for δ = 0 by u t 0 = (I − λP * d v t L Q * d v t ) −1 r (d v t ,P * d v t ) + (I − λP * d v t L ) −1 λP * d v t U R * d v t v t = Q * d v t −1 r (d v t ,P * d v t ) + Q * d v t −1 R * d v t v t ,(20) where P * d v t L and P * d v t U are the lower triangular matrix of P * d v t and the upper triangular matrix including the diagonal elements of P * d v t , respectively. Clearly, one can see ) v t . Also, via an analogous derivation from (7), (10), and (11) for δ = 0, one can get the vector form of (11) by u t that (Q * d v t , R * d v t ) is the GS regular splitting of I − λP * d v t , d v t ∈ D v t , and P * d v t ∈ P * d vς+1 = T (d v t ,P * d v t ) u t ς . In view of v t+1 = u t Mt in the step 3(f) of Algorithm 1, one can then obtain the iterative scheme of raTPI for δ = 0 by v t+1 = T (d v t ,P * d v t ) Mt+1 v t = Mt ς=0 Q * d v t −1 R * d v t ς Q * d v t −1 r (d v t ,P * d v t ) + Q * d v t −1 R * d v t Mt+1 v t = v t + Mt ς=0 Q * d v t −1 R * d v t ς Q * d v t −1 r (d v t ,P * d v t ) +Q * d v t −1 R * d v t v t − v t = v t + Mt ς=0 Q * d v t −1 R * d v t ς Bv t ,(21) where the second equality follows from the definition of (16) and the fourth equality follows from (19) (21), one can see that raTPI incorporates raTVI as a special case, because when M t = 0 for ∀t ∈ N, (21) will degenerate to the iterative scheme of raTVI, v t+1 = Y v t . Moreover, for another extreme case M t → ∞, (21) will reduce to the performance evaluation of the improved decision rule T (d v t ,P * d v t ) inin view of d v t ∈ D v t and P * d v t ∈ P * d v t . Fromd v t ∈ D v t for v t under the worst-case transition probability matrix P * d v t ∈ P * d v t , i.e. v t+1 = v t + ∞ ς=0 Q * d v t −1 R * d v t ς (Bv t ) = v t + I − Q * d v t −1 R * d v t −1 Q * d v t −1 r (d v t ,P * d v t ) − I − Q * d v t −1 R * d v t v t = I − Q * d v t −1 R * d v t −1 Q * d v t −1 r (d v t ,P * d v t ) = I − λP * d v t −1 r (d v t ,P * d v t ) = v (d v t ,P * d v t ) ,(22) where the second equality follows from the third equality in (21) by letting M t → ∞ and using ∞ ς=0 (Q * d v t −1 R * d v t ) ς = (I − Q * d v t −1 R * d v t ) −1 . To analyze the convergence of the iterative scheme (21), we construct two auxiliary operators. Given a non-negative integer M ∈ N, for v ∈ V, we define the operator W M : V → V by W M v := (T (dv,P * dv ) ) M +1 v,(23) and the operator U M : V → V by U M v := max d∈D max P d ∈P d Φ(d, P d , v),(24) where d v ∈ D v , P * dv ∈ P * dv , and Φ(d, P d , v) := M ς=0 Q −1 d R d ς Q −1 d r (d,P d ) + Q −1 d R d M +1 v for the GS regular splitting (Q d , R d ) of I − λP d . In particular, based on (23), the iterative scheme (21) of raTPI for δ = 0 can be rewritten by v t+1 = W Mt v t . In the following, we show some properties of these two operators, which will become the basis of proving the convergence of raTPI. This lemma (see Appendix D for its proof) guarantees that the sequence generated by U M will converge to the robust team-optimal value function v * with a convergence rate at least λ M +1 . Also, note from Lemma 1 that v * is the limit point of convergence for the sequence generated by Y . Therefore, it motivates us to postulate the following fact that for any M ∈ N and ∀t ∈ N, (U M ) t v 0 ≥ (W M ) t v 0 ≥ (Y ) t v 0(25) holds for some initial value functions v 0 . This is because once such a presumption holds, the sequence generated by W M (i.e. the iterative scheme (21) of raTPI) will also converge to v * for t → ∞. As such, the convergence of raTPI for δ = 0 to the robust team-optimal value function will be established. We show by the following two lemmas that such a postulation is indeed sound for v 0 ∈ V B := {v ∈ V | Bv ≥ 0}. Lemma 5. For any u, v ∈ V satisfying u ≥ v, U M u ≥ W M v holds for any M ∈ N. Moreover, if u ∈ V B , then W M u ≥ Y v. Lemma 6. If v ∈ V B , then W M v ∈ V B for any M ∈ N. See Appendix E and F for their proofs, respectively. Based on the above three lemmas, one can then get the convergence result of raTPI for δ = 0, which is shown in the following theorem. Theorem 3. For any non-negative integer sequence {M t } t∈N , if the approximation calculations are exact, i.e. δ = 0, in Algorithm 1, then for any v 0 ∈ V B , (a) the sequence {v t } generated by the iterative scheme (21) of raTPI converges monotonically and in norm to the robust team-optimal value function v * ; (b) those results shown in the part (b) of Theorem 1 still hold; and (c) let d v t and d v * be the v timproving and v * -improving decision rules, respectively. Then, there exists a P * d v t ∈ P * d v t and a P d v * ∈ P d v * for which v t+1 − v * ≤ v t − v * × Q * d v t −1 R * d v t − Q d v * −1 R d v * 1 − λ Mt 1 − λ + λ Mt+1 , where (Q * d v t , R * d v t ) and (Q d v * , R d v * ) are the GS regular splitting of I −λP * d v t and I −λP d v * , respectively. In particular, if lim t→∞ P * d v t − P d v * = 0, then there exists a K ∈ N for any > 0 such that v t+1 − v * ≤ + λ Mt+1 v t − v * for t ≥ K. Proof: We first prove part (a). Let {y t } and {ω t } be the sequences generated by y t+1 = Y y t and ω t+1 = U Mt ω t , respectively, where y 0 = ω 0 = v 0 . In the following, we show by induction that v t ∈ V B , v t+1 ≥ v t , and ω t ≥ v t ≥ y t for ∀t ∈ N. First, when t = 0, v 0 ∈ V B and y 0 = ω 0 = v 0 follow from the assumption. Moreover, from Lemma 5, we have v 1 = W M0 v 0 ≥ Y v 0 ≥ v 0 . Consequently, the induction hypothesis holds for t = 0. Suppose now that it is satisfied for t = κ. Then, applying Lemma 6 leads to v κ+1 = W Mκ v κ ∈ V B . From the definition of W M in (23) and the fourth equality in (21), moreover, v κ+2 = W Mκ+1 v κ+1 can further be given by v κ+2 = v κ+1 + Mκ+1 ς=0 Q * d v κ+1 −1 R * d v κ+1 ς Bv κ+1 ≥ v κ+1 . Since ω κ ≥ v κ ≥ y κ from the hypothesis, it follows from Lemma 5 that ω κ+1 = U Mκ ω κ ≥ W Mκ v κ = v κ+1 ≥ Y y κ = y κ+1 . Thus, the induction hypothesis is satisfied for t = κ + 1. Note from Lemma 1 and Lemma 4 that both y t and ω t converge in norm to v * for t → ∞. Therefore, we have v * ≤ lim t→∞ v t ≤ v * , which means that v t will converge in norm to v * . Part (a) is established. We proceed to prove part (b). Since part (a) has shown that the sequence {v t } generated by the iterative scheme of raTPI for δ = 0 is convergent, the termination condition in the step 3(a) of Algorithm 1 will be satisfied after a finite number of iterations by noticing v t+1 = (T (d v t ,P * d v t ) ) Mt+1 v t = W Mt v t ≥ u t 0 = T (d v t ,P * d v t ) v t = Y v t ≥ v t , where the first line follows from (21) and (23); the first inequality follows from Lemma 5 and (20); and the second inequality follows from v t ∈ V B . Suppose now that Algorithm 1 terminates at t = N and returns a policy (d ) ∞ . From the step 2(a) and 4 in Algorithm 1, it is known that d is the v N -improving decision rule. Let P * d ∈ P * d be the d -decreasing transition probability matrix with respect to v N . Then, from the definition of Y , we have T (d ,P * d ) v N = Y v N . Moreover, by applying the contraction property of Y , one can get T (d ,P * d ) v N − v * = Y v N − v * ≤ Y v N − (Y ) 2 v N + (Y ) 2 v N − v * ≤ λ v N − Y v N + λ Y v N − v * = λ v N − T (d ,P * d ) v N + λ T (d ,P * d ) v N − v * . Since Algorithm 1 terminates at t = N , i.e. u N 0 − v N = T (d ,P * d ) v N − v N < (1 − λ) /2λ , by re-arrangement the above inequality can lead to T (d ,P * d ) v N − v * ≤ v N − T (d ,P * d ) v N λ 1 − λ < /2. (26) On the other hand, since (T (d ,P * d ) ) ∞ v N − v N = (T (d ,P * d ) v N − v N ) + ((T (d ,P * d ) ) 2 v N − T (d ,P * d ) v N ) + · · · , applying the contraction property of T (d ,P * d ) yields (T (d ,P * d ) ) ∞ v N − T (d ,P * d ) v N ≤ λ (T (d ,P * d ) ) ∞ v N − v N ≤ λ ∞ ı=0 λ ı T (d ,P * d ) v N − v N < /2. Note from (22) and the definition of d and P * d that the value function v := v (d ,P * d ) corresponding to d under the worst-case transition probability matrix P * d can be given v = (T (d ,P * d ) ) ∞ v N . Therefore, v − v * ≤ (T (d ,P * d ) ) ∞ v N − T (d ,P * d ) v N + T (d ,P * d ) v N − v * < . Finally, we derive the result shown in part (c). From Proposition 1, one can first get that there exists a P d v * ∈ P d v * for the v * -improving decision rule d v * ∈ D v * such that Bv t ≥ Bv * +(Q d v * −1 R d v * −I)(v t −v * ), where (Q d v * , R d v * ) is the GS regular splitting of I − λP d v * . By applying this inequality and noticing Bv * = 0, it follows from (21) and the definition of W Mt in (23) that 0 ≤ v * − v t+1 = v * − W Mt v t = v * − v t − Mt ς=0 Q * d v t −1 R * d v t ς Bv t ≤ v * − v t + Mt ς=0 Q * d v t −1 R * d v t ς Q d v * −1 R d v * − I (v * − v t ) = Q * d v t −1 R * d v t − Q d v * −1 R d v * Mt−1 ς=0 Q * d v t −1 R * d v t ς × (v t − v * ) − Q d v * −1 R d v * Q * d v t −1 R * d v t Mt (v t − v * ). Taking norms on both sides leads to v * − v t+1 ≤ Q * d v t −1 R * d v t − Q d v * −1 R d v * 1 − λ Mt 1 − λ × v t − v * + λ Mt+1 v t − v * = v t − v * × Q * d v t −1 R * d v t − Q d v * −1 R d v * 1 − λ Mt 1 − λ + λ Mt+1 , where Q * d v t −1 R * d v t ≤ λ and Q d v * −1 R d v * ≤ λ follow from Lemma 2. In particular, if lim t→∞ P * d v t − P d v * = 0, then lim t→∞ Q * d v t −1 R * d v t − Q d v * −1 R d v * = 0 in view of the definition of the GS regular splitting. As such, there exists a K ∈ N for any > 0 such that v t+1 − v * ≤ + λ Mt+1 v t − v * for any t ≥ K. From this theorem, one can see that the algorithm of raTPI for δ = 0 can be guaranteed to find a robust team-optimal policy at a convergence rate of near λ Mt+1 . Compared with the convergence rate λ of raTVI in the degenerated case (see Theorem 1), the improvement is in exponential order. Using a similar argument to that in the degenerated case, we next extend the results for δ = 0 to the inexact case where 0 ≤ δ < (1 − λ) 2 /2λ(1 + λ). To distinguish from the notations used in the exact case δ = 0, we denote the sequence generated by Algorithm 1 in the inexact case by {ṽ t }. Moreover, for those intermediate variables given in (9) and (11), we useũ t 0 (s) andũ t ς+1 (s) to representũ t 0 (s) = max a∈Aρ(s,a) (ṽ t ) and u t ς+1 (s) =ρ (s,dt+1(s)) (ũ t ς ) in the inexact case, respectively, whered t+1 (s) ∈ arg max a∈Aρ(s,a) (ṽ t ) for ∀s ∈ S. In contrast, we still use those notations {v t }, u t 0 (s), u t ς+1 (s), and d t+1 (s) for ∀s ∈ S given in Algorithm 1 to represent the corresponding notions in the exact case. As such, we present the convergence result of raTPI in its general form by the following theorem. Theorem 4. For any non-negative integer sequence {M t } t∈N , if the tolerance of approximation calculations is within λδ for δ ∈ [0, (1 − λ) 2 /2λ(1 + λ)) in Algorithm 1, then for any v 0 ∈ V B , (a) ifṽ 0 = v 0 , then v t − θ1 ≤ṽ t ≤ v t + θ1 and u t ς − θ1 ≤ũ t ς ≤ u t ς + θ1, ς = 0, 1, . . . , M t hold for ∀t ∈ N, where θ := λδ/(1 − λ) ∈ [0, /2); and (b) those results shown in the part (b) of Theorem 1 still hold. The proof of this theorem is similar to those of Lemma 3 and Theorem 2. See Appendix G for more details. From this theorem, we have demonstrated our final result that the algorithm of raTPI in its general form can be guaranteed to converge to the robust team-optimal value function, and accordingly can terminate within a finite number of iterations with an -robust team-optimal policy. Finally, we give a remark to elaborate on the choice of initial iterative value functions and approximation methods. Remark 1. The initialization condition v 0 ∈ V B can be satisfied easily in practice. For example, selecting v 0 s ≤ (1 − λ) −1 min s k ,s l ∈S,a∈A r(s k , a, s l ) for ∀s ∈ S will ensure that v 0 ∈ V B . Moreover, to implement the approximation computations in Algorithm 1, many state-of-the-art techniques can be used, such as deep neural networks or empirical samples [27], [51]. In particular, when the number of samples is large enough, it has been proven that a good enough approximate solution can be obtained by sampling-based methods [52]. To validate these theoretical results, we present some numerical simulations in the following section. IV. SIMULATIONS To numerically demonstrate the effectiveness of Algorithm 1, we here generalize the game model of sequential social dilemmas in [53] to the scenario of incomplete information, and we refer to it as Robust Sequential Social Dilemmas (RSSDs). Formally, we consider an n-player Markov game where every player can only select one of the two actions, cooperation (C) and defection (D), from the action set A = {C, D}. In response to players' actions, the game system will transit from the current state to a new one at the next time. More specifically, if there are ∈ {0, 1, 2, . . . , n} players choosing action C in the state s k ∈ S, the state at the next time will change to s l ∈ S with probability p(s l | s k , ), where p(s l | s k , ) is not predetermined but rather lies in a discrete and finite uncertain set P (l | k). As the consequence of players' actions and state transitions, those players who choose action C (resp. D) will get a bounded payoff a (s l | s k ) ∈ R (resp. b (s l | s k ) ∈ R) as a function of , s k , and s l . To adhere to the existence of dilemmas, we assume as in the canonical multi-player social dilemmas [54] that (i) a +1 (s l | s k ) ≥ a (s l | s k ) and b +1 (s l | s k ) ≥ b (s l | s k ); (ii) b (s l | s k ) > a (s l | s k ); and (iii) a n (s l | s k ) > b 0 (s l | s k ) for all and ∀s k , s l ∈ S. Condition (i) states that players' payoffs increase with the number of C players in the group, whereas condition (ii) implies that within any mixed group, those C players always have a strictly lower payoff than that of D players. These two conditions indicate that taking action C is an altruistic behavior because C players entail a potential cost to improve other agents' benefits. In contrast, condition (iii) shows that mutual cooperation is more beneficial than mutual defection. Hence, for maximizing the gains of the whole group, all players should uniformly choose C. However, under the hypothesis of Homo economicus, each rational player will be tempted by myopic interests to take action D, thereby leading to the existence of social dilemmas. By extending social dilemmas to the scenario of incomplete information, the current model incorporates the prototypical multi-player social dilemmas [54] as an extreme case where the set of states is a singleton. Also, since the uncertainty of transition probabilities implies that there are some parameters of games unknown to players, it generalizes social dilemmas to the scenario of robust games [34], [35] In contrast to the underlying non-cooperative setting in conventional social dilemmas, we here assume that although every player has its individual payoff, all players aim to maximize the long-term benefit of the whole group. Such an assumption is to some extent reasonable in some social dilemmas encountered by humans, such as the coalition of nations when facing global warming. Under this assumption, the game model of RSSDs then becomes a specific instance of robust team Markov games. Therefore, we can apply Algorithm 1 to seek the robust team-optimal policy. Specifically, we consider a game of RSSDs with the state set of three elements, S = {s 1 , s 2 , s 3 }. In state s 1 , players play a public goods game and players' payoffs are calculated by a (s l | s 1 ) = r s l c/n − c and b (s l | s 1 ) = r s l c/n for ∀s l ∈ S, where c is the cost of cooperation and r s l ∈ (c, n) is the synergy factor dependent on the state s l at the next time. In state s 2 , an n-player stag-hunt game is played, where players' payoffs are calculated by a (s l | s 2 ) = r s l c/n − c and b (s l | s 2 ) = r s l c/n if is no less than the threshold Z; and otherwise a (s l | s 2 ) = −c and b (s l | s 2 ) = 0. In state s 3 , an n-player snow-drift game is played, and players' payoffs are calculated by a (s l | s 3 ) = ϑ s l −c/ and b (s l | s 3 ) = ϑ s l if > 0, and otherwise a (s l | s 3 ) = b (s l | s 3 ) = 0, where ϑ s l , as a function of the state s l at the next time, is the benefit of players when there exists at least one player choosing action C in the group. (see [53] and references therein for more details on these three games.) The simulation results are shown in Fig. 1 and Table I, where the model parameters of games are given by r s 1 = 1.5, r s 2 = 1.8, r s 3 = 2.2, ϑ s l = r s l for ∀s l ∈ S, n = 3, c = 1.0, and P (l | k) = {1 − µ } µ∈{0.1,0.2,0.3} if k = l and otherwise P (l | k) = {µ /2} µ∈{0.1,0.2,0.3} for ∀ ∈ {0, . . . , n}. Moreover, the algorithm parameters are = 10 −5 and δ ≈ (1 − λ) 2 /2λ(1 + λ), where a fixed λ = 0.97 is adopted in Fig. 1 while different λ values are used in Table I. From Fig. 1 and Table I, one can see that both raTVI and raTPI are able to effectively find the robust team-optimal policy, and their convergence rates are faster than rVI and rMPI, respectively. Although the computational model adopted here is minimal, the improvement is noticeable. In problems with large action and state sets, moreover, the performance of Algorithm 1 will be in general more competitive. V. CONCLUSIONS AND FUTURE WORK In this work, we have addressed a sequential decisionmaking problem of a cooperative team in stochastic uncertain environments, and accordingly proposed a robust version of team Markov games by relaxing the complete information assumption, in which players do not know the accurate transition probabilities of states, but rather are commonly aware of an uncertainty set. Without assuming a prior probability distribution over the uncertainty set, we consider that players adopt robust optimization methods to update their strategies. To characterize the optimality of team decisions, we have also proposed a solution concept named robust team-optimal policy, and meanwhile developed a robust iterative learning algorithm to seek it. Under mild conditions, we have presented the convergence analysis of the algorithm and calculated its convergence rates. By numerical simulations, moreover, we have demonstrated the effectiveness of these theoretical results in a game of RSSDs. However, the paper has left some open questions, which point to future directions: how to achieve decentralized or individual robust learning with theoretical guarantees, especially in network/graphic games; and how to achieve sample efficiency by data-driven robust learning or design robust policy search algorithms by function approximation. Also, we will be interested in applying RSSDs to study real-world issues, such as the evolution of social power [55] and human decisions in social diffusion [56], in the future. APPENDIX In this section, we provide some proof details of the results shown in the main text. A. Proof of Lemma 1 We establish part (a) by first proving that Y is a contraction mapping on V. For any given u, v ∈ V, we start with considering those states s ∈ S satisfying (Y v − Y u)(s) ≥ 0, where (Y v − Y u)(s) is the com- ponent of Y v − Y u corresponding to s ∈ S. Select d v ∈ arg max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v . It then follows from the definition of Y in (15) that (Y v − Y u)(s) ≤ min P dv ∈P dv Q −1 dv r (dv,P dv ) + Q −1 dv R dv v − min P dv ∈P dv Q −1 dv r (dv,P dv ) + Q −1 dv R dv u (s).(27) LetP dv ∈ arg min P dv ∈P dv Q −1 dv r (dv,P dv ) + Q −1 dv R dv u . Then, the right-hand side of (27) will be no larger than Q −1 dv r (dv,P dv ) +Q −1 dvR dv v − Q −1 dv r (dv,P dv ) +Q −1 dvR dv u (s) = Q −1 dvR dv (v − u) (s) ≤ Q −1 dvR dv (v − u) ≤ Q −1 dvR dv v − u ≤ α v − u , where (Q dv ,R dv ) is the corresponding regular splitting of I − λP dv . Consequently, 0 ≤ (Y v − Y u)(s) ≤ α v − u . Similarly, for those states s ∈ S satisfying (Y v−Y u)(s) ≤ 0, one can obtain 0 ≤ (Y u − Y v)(s) ≤ α u − v by the same argument. Therefore, Y v − Y u ≤ α v − u . Since α < 1, Y is a contraction mapping on V. From the Banach fixedpoint theorem (see Theorem 6.2.3 in [12]), it follows that {v t } converges in norm to the unique fixed pointṽ * of Y . We next show thatṽ * = v * . Note thatṽ * is the unique fixed point of Y . Then, arbitrarily given a d ∈ D, we havẽ (15). Also, for any > 0, there exists a P d ∈ P d such that v * = Yṽ * ≥ min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R dṽ * from the definition of Y inmin P d ∈P d Q −1 d r (d,P d ) + Q −1 d R dṽ * ≥ Q d −1 r (d,P d ) + Q d −1 R dṽ * − 1, where (Q d , R d ) is the cor- responding regular splitting of I − λP d . As a result,ṽ * ≥ Q d −1 r (d,P d ) + Q d −1 R dṽ * − 1. By re-arrangement, it leads toṽ * ≥ (I − Q d −1 R d ) −1 Q d −1 r (d,P d ) − (I − Q d −1 R d ) −1 1 = (Q d − R d ) −1 r (d,P d ) − (I − Q d −1 R d ) −1 1 = (I − λP d ) −1 r (d,P d ) − ∞ ı=0 (Q d −1 R d ) ı 1 = ∞ ı=0 (λP d ) ı r (d,P d ) − ∞ ı=0 (Q d −1 R d ) ı 1 = v (d,P d ) − ∞ ı=0 (Q d −1 R d ) ı 1,(28) where (I − λP d ) −1 = ∞ ı=0 (λP d ) ı and (I − Q d −1 R d ) −1 = ∞ ı=0 (Q d −1 R d ) ı hold because σ(λP d ) ≤ λP d = λ < 1 and σ(Q d −1 R d ) ≤ Q d −1 R d ≤ α < 1 ; and the last equality follows from the first equality of (5). Since for any nonnegative integer ı, (Q d −1 R d ) ı 1 ≤ 1 sup k∈{1,2,...,m} m l=1 |(Q d −1 R d ) ı (l|k)| = 1 (Q d −1 R d ) ı ≤ 1 Q d −1 R d ı ≤ α ı 1,(29) the inequality (28) can be further given byṽ * ≥ v (d,P d ) − ∞ ı=0 (Q d −1 R d ) ı 1 ≥ v (d,P d ) − ∞ ı=0 α ı 1 = v (d,P d ) − 1/(1 − α). As > 0 is arbitrary,ṽ * ≥ v (d,P d ) ≥ min P d ∈P d v (d,P d ) . In addition, note that d ∈ D is arbitrary. Thus,ṽ * ≥ v * = max d∈D min P d ∈P d v (d,P d ) . On the other hand, since from the definition of Y , v * = Yṽ * = max d∈D min P d ∈P d Q −1 d r (d,P d ) + Q −1 d R dṽ * , there exists ad ∈ D for any > 0 such thatṽ * ≤ min Pd∈Pd Q −1 d r (d,Pd) + Q −1 d Rdṽ * + 1. Thus, for anỹ Pd ∈ Pd,ṽ * ≤Q −1 d r (d,Pd) +Q −1 dRdṽ * + 1,(30) where (Qd,Rd) is the corresponding regular splitting of I − λPd. Using a similar argument to (28) and (29), one can then getṽ * ≤ v (d,Pd) + 1/(1 − α) from (30). Since both > 0 andPd ∈ Pd are arbitrary for the givend ∈ D,ṽ * ≤ min Pd∈Pd v (d,Pd) . As such, ṽ * ≤ max d∈D min P d ∈P d v (d,P d ) = v * . Therefore, we haveṽ * = v * . Finally, we establish part (b). Note first from part (a) that Y is a contraction mapping on V with constant α and its unique fixed point is v * . Thus, v t+1 − v * = Y v t − Y v * ≤ α v t − v * for ∀t ∈ N. Then, using this inequality recursively yields v t −v * ≤ α t v 0 −v * . Dividing both sides by v 0 − v * and taking the t-th root show that β := lim sup t→∞ v t − v * / v 0 − v * 1/t ≤ α. From this inequality, lim sup t→∞ v t − v * /β t ≤ v 0 − v * < ∞ can be immediately obtained. From Definition 3, the proof of part (b) is completed. B. Proof of Lemma 3 We establish the proof by induction. For t = 0, (17) follows from the assumption v 0 =ṽ 0 . Suppose now that (17) is satisfied for t = κ ≥ 0, and we derive the result for t = κ + 1. Note first that for any given a ∈ A in state s k ∈ S, the term inside the curly brace in (6) is monotonically increasing with respect to v t for any given p(·|s k , a) ∈ P a (·|k). Moreover, since the minimization operation in (6) over the probability distribution will not change the monotonicity, ρ (s k ,a) (v t ) is monotonically increasing with respect to v t . For any given a ∈ A in state s 1 , substituting each term of (17) into ρ (s 1 ,a) (·) then yields ρ (s 1 ,a) (v κ − θ1) = ρ (s 1 ,a) (v κ ) − λθ ≤ ρ (s 1 ,a) (ṽ κ ) ≤ ρ (s 1 ,a) (v κ ) + λθ = ρ (s 1 ,a) (v κ + θ1). Taking the maximum of each term of this inequality over a ∈ A, one can get v κ+1 s 1 − λθ ≤ max a∈A ρ (s 1 ,a) (ṽ κ ) ≤ v κ+1 s 1 + λθ,(31) where v κ+1 s 1 = max a∈A ρ (s 1 ,a) (v κ ) follows from (12). Since from (8),ρ (s 1 ,a) (ṽ κ ) − λδ ≤ ρ (s 1 ,a) (ṽ κ ) ≤ρ (s 1 ,a) (ṽ κ ) + λδ forṽ κ , taking the maximum of each term subject to a ∈ A yieldsṽ κ+1 s 1 − λδ ≤ max a∈A ρ (s 1 ,a) (ṽ κ ) ≤ṽ κ+1 s 1 + λδ,(32) whereṽ κ+1 s 1 = max a∈Aρ(s 1 ,a) (ṽ κ ) follows from (9) and the step 3(f) of Algorithm 1 for δ ∈ [0, (1 − λ) 2 /2λ(1 + λ)) in the case of M t = 0. It then follows from (31) and (32) that v κ+1 s 1 − λ(δ + θ) ≤ṽ κ+1 s 1 ≤ v κ+1 s 1 + λ(δ + θ).(33) Since λ(δ + θ) = θ, (33) is changed to v κ+1 s 1 − θ ≤ṽ κ+1 s 1 ≤ v κ+1 s 1 + θ. Combing this inequality with the hypothesis v κ s j − θ ≤ṽ κ s j ≤ v κ s j + θ, j = 2, 3, . . . , m, and from (6), one can further obtain ρ (s 2 ,a) (v κ ) − λθ ≤ ρ (s 2 ,a) (ṽ κ ) ≤ ρ (s 2 ,a) (v κ ) + λθ for ∀a ∈ A by a similar argument to the above process. Taking the maximum of each term subject to a ∈ A then leads to v κ+1 s 2 − λθ ≤ max a∈A ρ (s 2 ,a) (ṽ κ ) ≤ v κ+1 s 2 + λθ.(34) Likewise, for s 2 ∈ S, sinceρ (s 2 ,a) (ṽ κ ) − λδ ≤ ρ (s 2 ,a) (ṽ κ ) ≤ ρ (s 2 ,a) (ṽ κ ) + λδ forṽ κ from (8), taking the maximum of each term over a ∈ A leads tõ v κ+1 s 2 − λδ ≤ max a∈A ρ (s 2 ,a) (ṽ κ ) ≤ṽ κ+1 s 2 + λδ.(35) Using (34) and (35), one can similarly obtain v κ+1 s 2 − θ ≤ v κ+1 s 2 ≤ v κ+1 s 2 + θ. Then, applying the same argument for s j , j = 3, 4, . . . , m by recursion, one can get v κ+1 s j − θ ≤ṽ κ+1 s j ≤ v κ+1 s j + θ. Therefore, v κ+1 − θ1 ≤ṽ κ+1 ≤ v κ+1 + θ1. The proof is completed. C. Proof of Proposition 1 For any given u, v ∈ V and d v ∈ D v , from the definition of B in (19), one can get Bu ≥ min P dv ∈P dv Q −1 dv r (dv,P dv ) + Q −1 dv R dv u − u(36) and Bv = min P dv ∈P dv Q −1 dv r (dv,P dv ) + Q −1 dv R dv v − v .(37) Moreover, for any > 0, there exists a P dv ∈ P dv such that the term on the right-hand side of (36) is no less than Q dv −1 r (dv,P dv ) + Q dv −1 R dv u − u − 1, and meanwhile the term on the right-hand side of (37) is no larger than Q dv −1 r (dv,P dv ) +Q dv −1 R dv v−v, where (Q dv , R dv ) is the GS regular splitting of I − λP dv . Then, subtracting Bv from Bu yields Bu−Bv ≥ (Q dv −1 R dv −I)(u−v)− 1. Note that > 0 is arbitrary. Therefore, Bu ≥ Bv +(Q dv −1 R dv −I)(u−v). D. Proof of Lemma 4 Since for any given d ∈ D and P d ∈ P d , the (Q d , R d ) given in the definition (24) of U M is the GS regular splitting of I − λP d , Q d −1 R d ≤ I −1 (λP d ) = λ < 1 from Lemma 2. It implies that sup d∈D,P d ∈P d Q −1 d R d ≤ λ. Thus, we can adopt a similar argument to the proof of Lemma 1(a) to prove part (a). For any given u, v ∈ V, we begin with considering those states s ∈ S satisfying (U M v − U M u)(s) ≥ 0. Select d v ∈ arg max d∈D {max P d ∈P d Φ(d, P d , v)} . Then, from the definition of U M in (24), we have U M v(s) = { max P d v ∈P d v Φ(d v , P d v , v)}(s), and U M u(s) ≥ { max On the other hand, define the operator M : V → V for v ∈ V by M v := max d∈D max P d ∈P d Q −1 d r (d,P d ) + Q −1 d R d v . It is easy to show by a similar argument to the proof of Lemma 1(a) that M is a contraction mapping on V and the sequence generated by it converges in norm to v * . Moreover, from the definition of U M in (24), ω * = U M ω * ≤ M M ω * holds for ∀M ∈ N and especially it is true for M → ∞. Consequently, ω * ≤ v * . Therefore, ω * = v * . The proof is completed. E. Proof of Lemma 5 Given a v-improving decision rule d v ∈ D v ⊆ D and a d v -decreasing transition probability matrix P * dv ∈ P * dv ⊆ P dv for v ∈ V, we first have (21), the definition of W M in (23), and the definition of Φ(·) in (24). Moreover, from the definition of U M in (24), one have U M u ≥ Φ(d v , P * dv , u). Then, subtracting W M v from U M u and utilizing u ≥ v lead to W M v = Φ(d v , P * dv , v) fromU M u − W M v ≥ Φ(d v , P * dv , u) − Φ(d v , P * dv , v) = (Q * dv −1 R * dv ) M +1 (u − v) ≥ 0, where (Q * dv , R * dv ) is the GS regular splitting of I − λP * dv . Furthermore, if Bu ≥ 0, then from the definition of W M in (23) and the fourth equality in (21), one can get W M u = u + M ς=0 Q * du −1 R * du ς (Bu) ≥ u + Bu = Y u, where (Q * du , R * du ) is the GS regular splitting of I − λP * du and P * du ∈ P * du . Since for any given d ∈ D and P d ∈ P d , Q −1 d r (d,P d ) + Q −1 d R d u ≥ Q −1 d r (d,P d ) + Q −1 d R d v holds for any regular splitting (Q d , R d ) of I − λP d in view of u ≥ v, and the maximin operation will not change the direction of the inequality, Y u ≥ Y v. Consequently, W M u ≥ Y v. F. Proof of Lemma 6 Let u = W M v and d v be the v-improving decision rule. Then, from Proposition 1, there exists a P dv ∈ P dv such that Bu ≥ Bv + (Q dv −1 R dv − I)(u − v),(38) where (Q dv , R dv ) is the GS regular splitting of I − λP dv . Moreover, let P * dv ∈ P * dv ⊆ P dv be the d v -decreasing transition probability matrix. From the definition of W M in (23) and the fourth equality in (21), one can then get u − v = W M v − v = v + M ς=0 Q * dv −1 R * dv ς (Bv) − v = M ς=0 Q * dv −1 R * dv ς (Bv) ≥ 0,(39) where (Q * dv , R * dv ) is the GS regular splitting of I − λP * dv . Substituting (39) into (38) Q −1 R − I min P ∈P dv M ς=0 Q −1 R ς (Bv) ,(40) where (Q, R) is the GS regular splitting of I − λP for P ∈ P dv . Note that for any given v ∈ V, both Q −1 R − I v and M ς=0 Q −1 R ς v have the same minimum point subject to P ∈ P dv . Thus, there exists â P dv ∈ P dv such that the second term on the right-hand side of (40) is equal to Q −1 dvR dv − I (Bv) − Bv ≥ 0. G. Proof of Theorem 4 We establish part (a) by a similar induction argument to Lemma 3. Note first that v 0 =ṽ 0 follows from the assumption, and u 0 0 (s) andũ 0 0 (s) for ∀s ∈ S are calculated by the same equations (6), (8), and (9) as in the degenerated case raTVI. Therefore, applying Lemma 3 leads to u 0 0 − θ1 ≤ũ 0 0 ≤ u 0 0 + θ1.(41) Moreover, since ρ (s,a) (v 0 ) = ρ (s,a) (ṽ 0 ) for ∀(s, a) ∈ S × A follows from (6) in view of v 0 =ṽ 0 , and for any given s ∈ S, the approximation error betweenρ (s,a) (ṽ 0 ) and ρ (s,a) (ṽ 0 ) for ∀a ∈ A in (8) does not affect the selection of maximizing actions, arg max a∈Aρ(s,a) (ṽ 0 ) = arg max a∈A ρ (s,a) (v 0 ) for ∀s ∈ S. Without loss of generality, we selectd 1 (s) = d 1 (s) for ∀s ∈ S. Then, from (6), one can get that the set of minimizing transition probability distributions are the same for computing ρ (s,a) (v 0 ) and ρ (s,a) (ṽ 0 ), ∀s ∈ S because a =d 1 (s) = d 1 (s) and v 0 =ṽ 0 . As such, utilizing (7) and the monotonicity of ρ (s,a) (·) shown in Appendix B, one can get by substituting each term of (41) that, ρ (s 1 ,a) (u 0 0 − θ1) = ρ (s 1 ,a) (u 0 0 ) − λθ ≤ ρ (s 1 ,a) (ũ 0 0 ) ≤ ρ (s 1 ,a) (u 0 0 ) + λθ = ρ (s 1 ,a) (u 0 0 + θ1), for a =d 1 (s 1 ) = d 1 (s 1 ). In view of u 0 1 (s 1 ) = ρ (s 1 ,a) (u 0 0 ), a = d 1 (s) from (10) and (11) for δ = 0, it follows from (42) that u 0 1 (s 1 ) − λθ ≤ ρ (s 1 ,a) (ũ 0 0 ) ≤ u 0 1 (s 1 ) + λθ. On the other hand, in the inexact case, one can get ρ (s 1 ,a) (ũ 0 0 ) − λδ ≤ ρ (s 1 ,a) (ũ 0 0 ) ≤ρ (s 1 ,a) (ũ 0 0 ) + λδ for a =d 1 (s) from (10), andũ 0 1 (s 1 ) =ρ (s 1 ,a) (ũ 0 0 ) from (11). Then, we further havẽ u 0 1 (s 1 ) − λδ ≤ ρ (s 1 ,a) (ũ 0 0 ) ≤ũ 0 1 (s 1 ) + λδ. Consequently, combining (43) with (44) yields u 0 1 (s 1 ) − θ ≤ũ 0 1 (s 1 ) ≤ u 0 1 (s 1 ) + θ, in view of λ(θ+δ) = θ. Using this inequality and u 0 0 (s j )−θ ≤ u 0 0 (s j ) ≤ u 0 0 (s j ) + θ, j = 2, 3, . . . , m from (41), one can further get u 0 1 (s 2 ) − θ ≤ũ 0 1 (s 2 ) ≤ u 0 1 (s 2 ) + θ, from (7), (10), and (11) by a similar argument. By recursion, one can then obtain u 0 1 (s j ) − θ ≤ũ 0 1 (s j ) ≤ u 0 1 (s j ) + θ, j = 3, 4, . . . , m by applying the aforementioned argument for s j . Therefore, u 0 1 − θ1 ≤ũ 0 1 ≤ u 0 1 + θ1. Similarly, leveraging this inequality and applying the same argument for ς = 2, 3, . . . , M 0 by recursion, one can accordingly get u 0 ς − θ1 ≤ũ 0 ς ≤ u 0 ς + θ1. Consequently, the induction hypothesis is satisfied for t = 0. Suppose that the induction hypothesis is satisfied for t = κ, and we now show that it also holds for t = κ + 1. First, from the hypothesis for t = κ, we have u κ ς − θ1 ≤ũ κ ς ≤ u κ ς + θ1, ς = 0, 1, . . . , M κ . Moreover, according to the step 3(f) in Algorithm 1, one can get v κ+1 = u κ Mκ in the exact case andṽ κ+1 =ũ κ Mκ in the inexact case. Therefore, (45) implies v κ+1 − θ1 ≤ṽ κ+1 ≤ v κ+1 + θ1. Applying a similar argument process to the above case for t = 0, one can first obtain u κ+1 0 − θ1 ≤ũ κ+1 0 ≤ u κ+1 0 + θ1 based on Lemma 3, and then by recursion, one can further show that u κ+1 ς − θ1 ≤ũ κ+1 ς ≤ u κ+1 ς + θ1 holds for ς = 1, 2, . . . , M κ+1 . As a result, the induction hypothesis holds for t = κ + 1. Thus, part (a) is established. We next prove part (b). First, when δ = 0, one can obtain from the proof of Theorem 3 that there exists a positive integer N for any 1 > 0 such that the termination condition u N 0 − v N < (1 − λ) 1 /2λ is satisfied, and meanwhile v N +1 − v * = (T (d ,P * d ) ) M N +1 v N − v * ≤ T (d ,P * d ) v N − v * < 1 /2,(46) where the first inequality follows from v N ≤ Y v N = T (d ,P * d ) v N ≤ (T (d ,P * d ) ) 2 v N ≤ · · · ≤ (T (d ,P * d ) ) M N +1 v N = v N +1 ≤ v * by applying Theorem 3 and v N ∈ V B , and the second inequality follows from (26). Moreover, from part (a), we have ũ N 0 −ṽ N ≤ ũ N 0 − u N 0 + u N 0 − v N + v N −ṽ N ≤ u N 0 − v N + 2θ < (1 − λ) 1 /2λ + 2λδ/(1 − λ) . Given that 1 is arbitrary, we therefore select a specific 1 such that 1 ≤ − 2λ(1 + λ)δ/(1 − λ) 2 ,(47) for any > 0. As a result, ũ N 0 −ṽ N < (1 − λ) /2λ − δ. It implies that the raTPI algorithm in the inexact case will also terminate at t = N . Suppose that the algorithm terminates with returning a policy (d ) ∞ , and denote its corresponding value function under the worst-case transition probability matrix bỹ v . From (22), one can see thatṽ can be regard as the valuẽ v N +1 generated by Algorithm 1 at t = N + 1 by selecting M N → ∞ and δ = 0 when t = N . Therefore, from part (a), (46), and (47), one can get v − v * = ṽ N +1 − v * ≤ ṽ N +1 − v N +1 + v N +1 − v * < θ + 1 /2 < . The proof is completed. t from their definitions. Moreover, based on the definition of T (d,P d ) in (16), one can rewrite (20) by u t 0 = T (d v t ,P * d v t Lemma 4 . 4For the operator U M , (a) it is a contraction mapping on V with constant λ M +1 ; and (b) the sequence {ω t } generated by ω t+1 = U M ω t for any ω 0 ∈ V converges in norm to the robust team-optimal value function v * , which is the unique fixed point of U M . Fig. 1 : 1The iterative process of the value functions generated by raTVI and raTPI. Each point on lines is drawn by the iterative data. The dashed lattice illustrates the values of ρ (s,a) (v t ) for ∀(s, a) ∈ S × A at the termination time. ( Bv) − Bv, where (Q dv ,R dv ) is the GS regular splitting of I − λP dv . As a result, Bu ≥ Bv + Q TABLE I : IThe number of iterations for different algorithms to find the -robust team-optimal policy.Algorithms the magnitude of λ 0.95 0.96 0.97 0.98 0.99 rVI 298 380 519 802 1679 raTVI 258 328 446 690 1442 rMPI 7 9 12 17 34 raTPI 7 8 10 15 30 leads to Bu ≥ Bv + (Q dv −1 R dv − I)M ς=0 Q * dv −1 R * dv ς (Bv) ≥ Bv + min P ∈P dv Moreover, since there exists a P d v ∈ P d v for any > 0 such thatNote that > 0 is arbitrary. Therefore, from the definition of Φ(·) in (24), we further haveApplying the same argument for those states s ∈ S satisfyingSince λ M +1 < 1, U M is a contraction mapping on V. 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{'abstract': 'In stochastic dynamic environments, team Markov games have emerged as a versatile paradigm for studying sequential decision-making problems of fully cooperative multi-agent systems. However, the optimality of the derived policies is usually sensitive to model parameters, which are typically unknown and required to be estimated from noisy data in practice. To mitigate the sensitivity of optimal policies to these uncertain parameters, we propose a "robust" model of team Markov games in this paper, where agents utilize robust optimization approaches to update strategies. This model extends team Markov games to the scenario of incomplete information and meanwhile provides an alternative solution concept of robust team optimality. To seek such a solution, we develop a robust iterative learning algorithm of team policies and prove its convergence. This algorithm, compared with robust dynamic programming, not only possesses a faster convergence rate, but also allows for using approximation calculations to alleviate the curse of dimensionality. Moreover, some numerical simulations are presented to demonstrate the effectiveness of the algorithm by generalizing the game model of sequential social dilemmas to uncertain scenarios.', 'arxivid': '2105.07405', 'author': ['Feng Huang ', 'Ming Cao ', 'Long Wang '], 'authoraffiliation': [], 'corpusid': 248496929, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 34211, 'n_tokens_neox': 30594, 'n_words': 19234, 'pdfsha': 'a4e476031ccdc489e4bddd6f86ff663b11c18df3', 'pdfurls': ['https://arxiv.org/pdf/2105.07405v2.pdf'], 'title': ['Robust optimal policies for team Markov games', 'Robust optimal policies for team Markov games'], 'venue': []}

arxiv

STOCHASTIC GRADIENT DESCENT WITH NOISE OF MACHINE LEARNING TYPE PART I: DISCRETE TIME ANALYSIS Stephan Wojtowytsch STOCHASTIC GRADIENT DESCENT WITH NOISE OF MACHINE LEARNING TYPE PART I: DISCRETE TIME ANALYSIS Stochastic gradient descent (SGD) is one of the most popular algorithms in modern machine learning. The noise encountered in these applications is different from that in many theoretical analyses of stochastic gradient algorithms. In this article, we discuss some of the common properties of energy landscapes and stochastic noise encountered in machine learning problems, and how they affect SGD-based optimization.In particular, we show that the learning rate in SGD with machine learning noise can be chosen to be small, but uniformly positive for all times if the energy landscape resembles that of overparametrized deep learning problems. If the objective function satisfies a Lojasiewicz inequality, SGD converges to the global minimum exponentially fast, and even for functions which may have local minima, we establish almost sure convergence to the global minimum at an exponential rate from any finite energy initialization. The assumptions that we make in this result concern the behavior where the objective function is either small or large and the nature of the gradient noise, but the energy landscape is fairly unconstrained on the domain where the objective function takes values in an intermediate regime. References 22 Appendix A. On the non-convexity of objective functions in deep learning 23 Appendix B. Auxiliary observations on objective functions and Lojasiewicz geometry Introduction Stochastic gradient descent algorithms play an important role in convex and non-convex optimization. They are used in machine learning when the computation of the exact gradient of the objective function is computationally costly, but stochastic approximations can be evaluated fairly cheaply. The stochastic noise is furthermore believed to aid the algorithm in non-convex optimization by allowing it to escape 'bad' local minima and saddle points. this work, we analyze toy models for stochastic gradient descent in machine learning applications. These differ from more classical perspectives principally in the fact that the intensity of noise in estimating ∇f (θ) depends on the value f (θ) of the objective function. By comparison, more classical works on SGD assume uniform L p -bounds for p ∈ [2, ∞), independently of θ. Our toy models are inspired by overparametrized supervised learning, i.e. minimization problems where the objective function vanishes on a high-dimensional set. In this setting, SGD with ML noise has the following novel properties: (1) The learning rate has to be small in terms of the smoothness of the objective function and the noise intensity, but can be uniformly positive for all time. (2) If the learning rate remains uniformly positive, we can prove almost sure convergence to a global minimizer, not just a critical point that is not a strict saddle. Let us make these claims more precise. The crucial property of ML noise is that its variance scales at most linearly with the objective function f . In particular, if f is small, so is the noise. If f satisfies a Lojasiewicz inequality (e.g. if f is strongly convex), the learning rate has to be small enough for the noise to not exceed a certain strength, and compared to the Lipschitz constant of ∇f . It does not have to vanish asymptotically to guarantee convergence to the global minimum. In this sense, SGD with ML noise resembles deterministic GD more than it does SGD with classical noise. For details, see Theorem 3.1. If the learning rate is strictly positive, at any time there are two options: • f (θ t ) is small and so is the noise. If f has good properties on the set {f < ε} for some ε > 0, then with large probability we stay inside the set once we enter for the first time. Conditioned on this event, we expect to converge to a minimizer linearly. • f (θ t ) is large and so is the noise. If f has nice properties on the set {f > S} for some S > 0, we expect SGD with ML noise to not let us escape to infinity despite the growing noise intensity. If additionally the set {f ≤ S + 1} is not too spread out away from the set of minimizers and the noise is 'uniformly unbounded' (but with bounded variance), then from any point in {f ≤ S + 1} there is a positive probability of jumping into the set {f < ε} independently of the local gradient. This intuition can be extended and made precise. In particular, under strong conditions on the target function and noise, we find that we converge linearly to a minimizer almost surely from any finite energy initialization. However, since the probability of jumping into the set {f < ε} may be exceedingly small, we note that we do not expect to observe this linear convergence on realistic time scales in complicated real world applications for poor initialization. A precise statement of our main result can be found in Theorem 3.6. Stochastic gradient descent can be studied as a general tool in convex and non-convex optimization or specifically in the context of deep learning applications. In this article, we take a balanced approach by incorporating key features of the energy landscape and stochastic noise in deep learning problems, but without specializing to a specific machine learning model. The article is structured as follows. In Section 2, we review known results on stochastic gradient descent and objective functions in deep learning. We deduce some properties of the energy landscape and stochastic noise which a suitable toy model should satisfy. Discrete time SGD with ML noise is discussed in Section 3. Numerical examples in Section 4 illustrate the difference of SGD with classical and ML noise in a toy problem. Some proofs are postponed to the appendices. Continuous time results on continuous time SGD with noise of ML type are presented in a companion article [Woj21]. The different parts of the article -analysis of energy landscape and noise, discrete time analysis -can be read independently, and a reader only interested in one chapter can easily skip ahead. 1.1. Context. Stochastic gradient descent algorithms have been an active field of research since their inception in the seminal paper [RM51]. Stochastic gradient descent (SGD) and advanced gradient descent-based optimization schemes with stochastic gradient estimates have been the subject of increased attention recently due to their relevance in neural network-based machine learning, see e.g. [DDB17, NWS14, VBS19, WWB19, XWW20, DBBU20, AZ17, MB11, RSS11, JKNvW21, GLZ16, GL13, FGJ20, BCN18, BAWA18, BM13] and many more. A good literature review can be found in [FGJ20]. Theoretic guarantees for the convergence of SGD can be split into two categories: • For convex target functions, SGD converges to the global minimum. • For smooth non-convex target functions, SGD converges to a critical point, which is not a strict saddle. Both types of guarantee can be obtained under different conditions and either in expectation or almost surely, and both can be complemented with sharp rates. We review results of both types in Section 2.1. Results may hold for the final iteration θ t of SGD, the best parameterθ t along the SGD trajectory up to time t, or a weighted averageθ t of previous positions of SGD. We focus on guarantees for the final position since a list of previous values, or even a non-trivially weighted average of previous positions, is expensive to maintain in deep learning, where a model may have millions of parameters. Recently, it has been noted in the SGD literature [KNS16] that convexity can be replaced by the condition that the objective function satisfies a Lojasiewicz-inequality, i.e. (1.1) Λ f (θ) ≤ |∇f | 2 (θ) ∀ θ. This advance is particularly crucial since objective functions in deep learning typically have complicated geometries where the set of global minimizers is a submanifold of high dimension and co-dimension. Such functions are not usually convex, even in a neighbourhood of a minimizer. The inequality (1.1) holds in particular for all strongly convex functions, but may not hold for smooth convex functions, e.g. f (x) = √ x 2 + 1. Like for convex functions, the only critical points of a function satisfying a Lojasiewicz inequality lie in the set of global minimizers. The assumption of a global Lojasiewicz inequality thus remains restrictive, and other models, e.g. functions satisfying local Lojasiewicz inequalities at the critical level sets have been considered [DK21]. Classically, the only assumption on SGD noise is a uniform second moment bound. In this work, we consider more realistic noise model inspired by deep learning applications. In this setting, we can prove convergence to the global minimum for a more general class of functions which are neither required to be convex, nor to satisfy a Lojasiewicz inequality. In this sense, we prove a first type guarantee under weaker second type assumptions. While we obtain fast (linear) convergence to the global minimum f (θ t ) − inf f ≤ Z ρ t rather than the rate f (θ t ) − inf f ≤ C t , which is typical in the stochastic setting, we note that the random variable Z is typically too large to yield meaningful bounds in practice, unless the target function and noise have particularly convenient properties. Nevertheless, the fast convergence allows us to circumvent a common 'staying local' assumption, which is automatic for exponentially decaying sequences. The guarantees we obtain hold almost surely (i.e. with probability 1) over the choice of initial condition and stochastic gradient optimization. The importance of almost sure statements in this context has for example been emphasized in [Pat20] in the context of deriving stopping criteria which are 'triggered' with probability 1. 1.2. Notation and conventions. All random variables are defined on a probability space (Ω, A, P) which remains abstract and is characterized mostly as expressive enough to support a random initial condition and countably many iid copies of a random variable to select a gradient estimator. The dyadic product of two vectors a, b is denoted by a ⊗ b = a · b T , i.e.(a ⊗ b) ij = a i b j . 2. Energy landscapes and stochastic noise in machine learning 2.1. A brief review of stochastic gradient descent. Stochastic gradient descent algorithms are a class of popular algorithms in machine learning to find minimizers of an objective function f . Instead of taking a small step in the direction −∇f in every iteration, we choose the update direction randomly according to a random variable g with expectation −∇f and suitable L pbounds for some p ≥ 2. More formally, we consider the following model. ( 1) f : R m → [0, ∞) is a C 1 -function (2) ∇f satisfies the one-sided Lipschitz-condition ∇f (θ 1 ) − ∇f (θ 2 ), θ 1 − θ 2 ≤ C L |θ 1 − θ 2 | 2 (3) (Ω, A, P) is a probability space and g : R m × Ω → R m is a family of functions such that (a) E ξ∼P g(θ, ξ) = ∇f (θ) for all θ ∈ R m and (b) E ξ∼P |g(θ, ξ) − ∇f (θ)| 2 < ∞ for all θ ∈ R m . The SGD algorithm associated to the family g of gradient estimators is given by the timestepping scheme (2.1) θ t+1 = θ t − η t g(θ t , ξ t ) where (1) the initial condition θ 0 is a random variable in R m , (2) η t > 0 is the learning rate (or time step size) in the t-th time step, and (3) {ξ t } t≥0 is a family of random variables in Ω with law P, which are iid and independent of θ 0 . In theoretical works, we typically consider decaying learning rates. This is rooted in results such as this, originating from [RM51]. Theorem 2.1. Assume that ∇f is Lipschitz-continuous with constant C L > 0 and that • f is convex or • f satisfies the Lojasiewicz inequality Λ(f − inf f ) ≤ |∇f | 2 for some Λ > 0. If (2.2) E ξ |g(θ, ξ) − ∇f (θ)| 2 ≤ σ 2 ∀ θ ∈ R m and the learning rates η t satisfy the Robbins-Monro conditions ∞ t=0 η t = ∞, ∞ n=0 η 2 t < ∞, then lim t→∞ E f (θ t ) = inf f . If f satisfies the Lojasiewicz inequality and η t = 2 Λ(t+1) , then E f (θ t ) − inf f ≤ 2C L σ 2 Λ 2 log(t + 1) t E f (θ 0 ) − inf f A proof for the Lojasiewicz case (which in particular contains the uniformly convex case) can be found in [KNS16,Theorem 4], albeit under the assumption that |∇f | is uniformly bounded, which contradicts the assumption that f satisfies a Lojasiewicz inequality (unless f is constant). This can be weakened to the assumption that the noise has bounded variance [BCN18]. The first summation condition is equivalent to stating that we are solving a gradient-flow type equation on the entire real line. Violating it would introduce a finite time horizon, which would prevent convergence to the minimizer as t → ∞ even in the non-stochastic setting and even for smooth and strongly convex objective functions. The second condition guarantees that the impact of gradient noise diminishes sufficiently as t → ∞ such that the iterates {θ t } t≥0 are not driven away from the global minimum in the long term by random oscillations. Unlike the first condition, it can be weakened [KY03,Chapter 5]. It has been suggested that the learning rate decay η t =η/t is optimal [LTW17, Section 4.1.2]. If the objective function is non-convex and does not satisfy a Lojasiewicz inequality, convergence to a critical point which is not a strict saddle can be guaranteed under suitable conditions on f and g. We only give an imprecise statement. Theorem 2.2. [MHKC20] Under suitable conditions on the objective function f and a condition of the type E |g − ∇f | 2 ≤ σ 2 , the following holds: If η t = η0 (t+t0) p for some p ∈ (1/2, 1], then (1) f (θ t ) converges to a random variable Z almost surely and Z lies in the set of critical values of f almost surely. (2) sup t |θ t | < ∞ almost surely. (3) for every trajectory of SGD, there exists a connected component X of the set of critical points of f such that lim t→∞ dist(θ t , X ) = 0. (4) If S is a connected component of the set of critical points of f such that D 2 f (θ) has a negative eigenvalue for all θ ∈ S (a ridge manifold), then P lim t→∞ dist(θ t , S) = 0 = 1. In particular, an isolated strict saddle point is a zero-dimensional ridge manifold. Stronger statements, including a rate of convergence, are available at isolated local minima. The conditions on f posed in [MHKC20] are neither implied by our assumptions, nor are they more general. While sublevel sets of f have to be compact in [MHKC20] but not for us, we impose stronger conditions on the gradient away from the set of minimizers. In practice, we typically choose learning rates according to a schedule of the form η t ≡ η (1) for 0 ≤ t < T 1 , η t ≡ η (2) for T 1 ≤ t < T 2 , . . . with a discrete set of switching times 0 < T 1 < T 2 < . . . and learning rates η (1) > η (2) > . . . . Generally, the learning rates are chosen as large as possible without inducing numerical instability, and it is realistic to consider the setting where the learning rate is reduced only a finite number of times. We consider the setting where even in the long term limit, the learning rate remains strictly positive. Under the conditions of Theorem 2.1, we find that the energy quickly decays below a threshold depending on the learning rate. Theorem 2.3. [KNS16,Theorem 4] Assume that f satisfies the Lojasiewicz inequality Λ(f − inf f ) ≤ |∇f | 2 and that ∇f is Lipschitz-continuous with Lipschitz-constant C L . Assume further that the stochastic noise satisfies (2.2). If η t ≡ η < 1 C L , then E f (θ t ) − inf f ≤ (1 − Λη) t E f (θ 0 ) − inf f + C L σ 2 2Λ η ∀ t ∈ N. Theorems 2.1, 2.2 and 2.3 made very weak assumptions on the nature of the stochastic noise in the estimators g. Under different conditions, we can obtain better results. In particular, the type of noise encountered in the gradient estimators in machine learning is fairly special. On energy landscapes in machine learning. Consider a parameterized function class h : R m × R d → R k where R m is the space of parameters, R d is the space in which the data is given, and R k is a target space. In supervised learning problems, we seek to minimize functions of the form L(θ) = 1 n n i=1 h(θ, x i ), y i where (1) {(x i , y i )} n i=1 is our training set of data/output or data/label pairs in R d × R k , (2) θ denotes the parameters of our function model, and (3) : R k × R k → [0, ∞) is a positive function. We use to establish a concept of similarity on R k and optimize θ in order to make h(θ, x i ) "similar" to y i with respect to , i.e. to make (h(θ, x i ), y i ) small. Remark 2.4. Depending on the context, both L and are referred to as the loss function. At times, L is referred to as 'risk' and denoted by the symbol R instead. Example 2.5. The prototypical example of a loss function in our context is MSE (mean squared error)/ 2 -loss (h, y) = 1 2 h − y 2 R k . Another popular loss function in the setting of classification problems is softmax cross entropy loss (2.3) (h, y) = − log exp(h · y) k i=1 exp(h · e i ) , y ∈ {e 1 , . . . , e k }. The energy landscapes in both types of learning problems are quite different since cross entropy loss is never zero (see 2.11 for more details) and our results do not apply in the second setting. In this article, we focus on 2 -type losses. Note that L(θ) = 0 ⇔ 1 n n i=1 h(θ, x i ) − y i 2 = 0 ⇔ h(θ, x i ) = y i ∀ 1 ≤ i ≤ n. We give a modified version of a theorem of Cooper [Coo18, Theorem 2.1]. Theorem 2.6. [Coo18] Assume that (1) the function model h is overparametrized, i.e. m > nk, (2) the function model h is so expressive that for any collection of outputs y 1 , . . . , y n there exists θ ∈ R m such that h(θ, x i ) = y i ∀ 1 ≤ i ≤ n, (3) h is C m−nk+1 -smooth in θ. Then for Lebesgue-almost any choice of (y 1 , . . . , y n ) ∈ R nk , the set N = θ ∈ R m : L(θ) = 0 is a closed m − nk-dimensional C 1 -submanifold of R m . If h is Lipschitz-continuous in the first argument for every fixed x and sufficiently overparameterized that it can fit arbitrary values at n + 1 points, then N is non-compact. The conditions of this theorem are satisfied by certain overparametrized neural networks with smooth activation function [Coo18, Lemma 3.3], but also many other suitably expressive function classes. The condition m ≥ nk is implicit in the assumptions on expressivity and smoothness of h. Proof. We can write N = Ψ −1 (y) where Ψ : R m → R nk , Ψ(θ) =    h(θ, x 1 ) . . . h(θ, x n )    , y =    y 1 . . . y n    . By assumption, N is non-empty and if y is a regular value of Ψ, then N is a C 1 -manifold of dimension m − nk. Since Ψ is as smooth as h, we can conclude by Sard's theorem [Sar42] that Lebesgue-almost every y is a regular value of Ψ. As the pre-image of a single point under a continuous map, N is closed. If h is sufficiently overparametrized that it can fit values not only at n but at n+1 points, then for any x n+1 ∈ R d and y n+1 ∈ R k there exists θ ∈ R m such that h(θ, x i ) = y i for 1 ≤ i ≤ n + 1. In particular, for any choice y n+1 ∈ R k we have θ ∈ N . Let y n+1 , y n+1 ∈ R k and θ, θ associated indices in N . Since y n+1 − y n+1 = h(θ, x n+1 ) − h(θ , x n+1 ) ≤ C h |θ − θ |, we see N cannot be compact since achieving very different outputs y n+1 , y n+1 requires very different parameters θ, θ . We observe the following: If L is convex, then the set of minimizers is convex. An m − nkdimensional closed submanifold of R m is a convex set if and only if it is an m − nk-dimensional affine subspace of R m . If the map θ → h(θ, x) is generically non-linear, there is no reason to expect N to be an affine space. Thus, we typically expect that L is non-convex, even close to its set of minimizers. A rigorous statement on the non-convexity of objective functions in deep learning is given in Appendix A. Thus, any toy model for the energy landscape L : R m → [0, ∞) of deep learning in overparametrized regression problems should have the following property: There exists a non-compact closed C 1 -manifold N ⊆ R m such that L(θ) = 0 if and only if θ ∈ N . We can think of both the dimension and co-dimension of N as being large. Remark 2.7. While functions with these properties in general cannot be convex even in a neighbourhood of their set of minimizers, they may satisfy a Lojasiewicz inequality of the form ∇f 2 ≥ c f. One such function is f (x, y) = 1 2 y − h(x) 2 for any h ∈ C 1 (R) since ∇f 2 ≥ |∂ y f | 2 = y − h(x) 2 = 2f. The assumption that a Lojasiewicz inequality holds is standard in non-convex optimization, but may not be realistic in machine learning, where complex energy landscapes with many local minima, maxima, ridges and saddle points are observed. For a fascinating demonstration of how diverse these energy landscapes can be, see [SB19]. However, we note that there are no critical points of high risk in square loss regression problems under very general conditions in the finite data or infinite data case. Lemma 2.8. Let µ be a probability distribution on R d × R k with finite second moments (the data distribution). Assume that h(θ, ·) is a multi-layer perceptron with weights θ which maps R d to R k . Define the risk functional R(θ) = 1 2 R d ×R k h(θ, x) − y 2 dµ (x,y) . Then there exists a constant C > 0 such that |θ| · |∇R(θ)| ≥ R(θ) − C. In particular, there exists S > 0 such that R(θ) > S ⇒ ∇R(θ) = 0. The function R is an empirical risk functional if µ is a finite sum of Dirac deltas and a population risk functional otherwise. Proof. Let h * (x) = E y|x and µ the distribution of x in R d if (x, y) is distributed according to µ. Then R(θ) = 1 2 R d ×R k h(θ, x) − h * (x) 2 dµ (x,y) + 1 2 R d ×R k h * (x) − y 2 dµ (x,y) = 1 2 R d ×R k h(θ, x) − h * (x) 2 dµ x + C µ . For a neural network, we can decompose θ = (W,θ) and h(θ, x) = W σ ĥ (θ, x) whereĥ : R d → R l is a neural network with one layer less than h and W : R l → R k is linear. If R(θ) is larger than R d ×R k |y| 2 dµ (x,y) , then h(θ, x) ≡ 0, so in particular W = 0. We can therefore compute |∇ θ R|(θ) ≥ |W · ∇ W R| |W | = 1 |W | W · R d h(θ, x) − y · ∇ W h(θ, x) dµ (x,y) 2 = 1 |W | R d h(θ, x) − y (W ∇ W h(θ, x)) dµ (x,y) 2 = 1 |W | R d h(θ, x) − h * (x) · h(θ, x) dµ x since the representation of h is linear in W , so W · ∇ W h(θ, x) = h(θ, x). Consequently |W | |∇ θ R|(θ) ≥ R d h(θ, x) − h * (x) 2 − h * (x) · h(θ, x) − h * (x) dµ x ≥ h θ − h * 2 L 2 (P) − h θ − h * L 2 (P) h * L 2 (µ) = 2 R(θ) − C µ − √ 2 R(θ) − C µ h * L 2 (µ) . The claim follows since 2R(θ) − h * L 2 (µ) 2 R(θ) ≥ R(θ) − h * 2 L 2 (µ) . 2. 3. Lojasiewicz landscapes. We briefly discuss local Lojasiewicz conditions in machine learning and beyond. For sufficiently 'nice' objective functions, these can easily be seen to hold. Example 2.9. Assume that f : R m → [0, ∞) is C 2 -smooth and N := f −1 (0) is a C 1 -manifold. Assume that (1) D 2 f is globally Lipschitz-continuous on the set U = {θ : f (θ) ≤ 1} and (2) there exists 0 < λ ≤ Λ such that λ|v| 2 ≤ v T D 2 (θ)v ≤ Λ |v| 2 for all θ ∈ N and v ∈ (T θ N ) ⊥ . Then there exists ε > 0 such that f satisfies a Lojasiewicz inequality on the set U ε = {θ : f (θ) < ε} since f (θ + tv) = t 2 2 v T D 2 f (θ)v + O(t 3 ) ∇f (θ + tv) = t D 2 f (θ) v + O(t 2 ) so if v⊥T θ N and |θ| = 1 we have f (θ + tv) = t 2 2 v T D 2 f (θ)v + O(t 3 ) ≤ Λ 2 t 2 + O(t 3 ) ≤ Λ 2λ 2 λ 2 t 2 + O(t 3 ) ≤ Λ 2λ 2 D 2 f (θ) v 2 t 2 + O(t 3 ) = Λ 2λ 2 |∇f (θ)| 2 + O(t 3 ). Since f (θ + tv) ≥ λ 2 t 2 and due to the uniform Lipschitz bound, the smallness of the correction term is uniform over the set {f < ε}. Example 2.10 (Quadratic regression). For quadratic regression problems L(θ) = 1 2n n i=1 h(θ, x i ) − y i 2 we have ∇L(θ) = 1 n n i=1 h(θ, x i ) − y i ∇ θ h(θ, x i ) D 2 L(θ) = 1 n n i=1 ∇ θ h(θ, x i ) ⊗ ∇ θ h(θ, x i ) + h(θ, x i ) − y i D 2 h(θ, x i ). In particular if L(θ) = 0, the Hessian terms of h drop out and D 2 L(θ) = 1 n n i=1 ∇ θ h(θ, x i ) ⊗ ∇ θ h(θ, x i ) The upper bound D 2 L ≤ C is therefore satisfied on N if the map θ → h(θ, x i ) is Lipschitz- continuous. The lower bound corresponds to the assumption that the n vectors ∇ θ h(θ, x i ) are bounded away from zero and bounded away from becoming linearly dependent. If this is the case, the lower bound λ|v| 2 ≤ v T D 2 L(θ)v holds since v ∈ (T θ N ) ⊥ lies in the n-dimensional space spanned by {∇ θ h(θ, x i ) : 1 ≤ i ≤ n}. These assumptions correspond to a function model in which we can always change the parameters by a small amount and change the model output by a positive amount, and in which the output at different data points x i , x j is principally governed by different parameters. The Lipschitzcondition on D 2 L corresponds to the assumption that h, ∇ θ h, D 2 θ h are Lipschitz-continuous and bounded. Example 2.11 (Cross-entropy classification). For classification problems, the cross-entropy loss function L(θ) = −E (x,i)∼µ log exp h(x) · e i k j=1 exp h(x) · e j never vanishes, so the energy landscape is fundamentally different from that of quadratic regression problems. We show that generally, a Lojasiewicz condition of the type Λf ≤ |∇f | 2 cannot hold even for small values of the objective function. The loss function (h, i) = − log exp(h · e i ) k j=1 exp(h · e j ) = log   k j=1 exp(h · e j )   − h · e i satisfies ∇ h (h, i) = k l=1 exp(h · e l ) k j=1 exp(h · e j ) e l − e i ∇ h (h, i) 2 = exp(h · e i ) k j=1 exp(h · e j ) − 1 2 + l =i exp(h · e l ) k j=1 exp(h · e j ) 2 . With the counting density p l = exp(h·e l ) j exp(h·ej ) , we find that = − log(p i ) ≥ 1 − p i , |∇ | 2 = (1 − p i ) 2 + l =i p 2 l ≤ 2 (1 − p i ) 2 , so 1 2 ∇ 1 h(θ, x), i 2 ≤ (1 − p i ) 2 ≤ min 1, − log(p i ) 2 ≤ min 1, h(θ, x), i 2 . Assume for the sake of contradiction that λ L(θ) ≤ |∇L| 2 (θ) on the set {L < ε}. Then the solution θ of the gradient flow equationθ = −∇L(θ) satisfies d dt L(θ(t)) = −|∇L| 2 (θ(t)) ≤ −λ L(θ(t)) ⇒ L(θ(t)) ≤ e −λt L(θ(0)). In particular θ(t) − θ(0) ≤ t 0 ∇L(θ(s)) ds ≤ 2 t 0 L(θ(s)) ds ≤ 2 L(θ(0)) t 0 e −λs ds ≤ 2 L(θ(0)) λ . if L(θ(0)) < ε. In particular, θ(t) is contained in a compact subset of R m and L(θ(t)) → 0 as t → ∞, leading to a contradiction. Random selection SGD. Consider a more general objective function f : R m → [0, ∞) of the form f (θ) = E ξ∼P φ(θ, ξ) where (Ω, A, P) is a probability space and φ : R m × Ω → [0, ∞) is a non-negative function such that (G1) for any fixed θ, the function ξ → φ(θ, ξ) is A-measurable, (G2) for any fixed ξ, the function θ → φ(θ, ξ) is continuously differentiable, and (G3) for any compact set K ⊆ R m , there exists ψ K ∈ L 1 (P) such that φ(θ, ξ) + ∇ θ φ(θ, ξ) ≤ ψ K (ξ) ∀ θ ∈ K P-almost everywhere. The assumptions guarantee that expectation and parameter gradient commute [Kön13, Section 8.4]. A particular gradient estimator is g(θ, ξ) = (∇ θ φ)(θ, ξ) since E ξ g(θ, ξ) = E ξ (∇ θ φ)(θ, ξ) = ∇ θ E ξ φ(θ, ξ) = ∇f (θ). Denote by Σ the covariance matrix of the estimator g: Σ ij (θ) = E ξ g i (θ, ξ) − ∂ i f (θ) g j (θ, ξ) − ∂ j f (θ) , which we assume is well-defined: (G4) for all θ ∈ R m we have ∇ θ φ(θ, ·) ∈ L 2 (P). This model encompasses machine learning problems as discussed above with f (θ) = R d ×R k h(θ, x), y dµ (x,y) , ξ = (x, y), g(θ, ξ) = ∂ 1 h θ (x), y ∇ θ h(θ, x), if the loss function and the data distribution µ are compatible via growth conditions and moment bounds. 2.5. SGD noise in machine learning. We make an obvious observation. Lemma 2.12. Assume there exists an m−n-dimensional manifold N ⊆ R m such that φ(θ, ξ) = 0 for P-almost all ξ. Then Σ = 0 everywhere on N for the random selection gradient estimator. Proof. Since φ(θ, ξ) = 0 an φ ≥ 0, we find that φ(·, ξ) is minimal at θ, so ∇ θ φ(θ, ξ) = 0 for all ξ. Since there is no oscillation in the gradient estimator, the variance vanishes. Remark 2.13. A popular toy model for SGD is the family of gradient estimators g(x, ξ) = ∇f (x) + σξ where σ > 0 and ξ is a standard Gaussian random variable. Unlike machine learning SGD, the estimators correspond to calculating exact gradients and deliberately perturbing them stochastically. Clearly, this toy model fails to capture key features of random selection SGD at the minimizing manifold. We prove a more quantitative version of Lemma 2.12. Lemma 2.14. Let µ be a data distribution and h a parametrized function model such that |∇ θ h(θ, x)| ≤ C for µ-almost every x. For a quadratic regression problem L(θ) = 1 2 R d ×R k h(θ, x) − y 2 dµ (x,y) the noise satisfies E ξ∼P |g(θ, ξ) − ∇L(θ)| 2 ≤ E ξ∼P g(θ, ξ) 2 ≤ ∇ θ h 2 L ∞ L(θ). For a cross-entropy classification problem L(θ) = − R d ×{1,...,k} log exp h(θ, x) · e i k j=1 exp(h θ, x) · e j dµ (x,i) the noise satisfies E ξ∼P |g(θ, ξ) − ∇L(θ)| 2 ≤ E ξ∼P g(θ, ξ) 2 ≤ 2 ∇ θ h 2 L ∞ min 1, L(θ) . Proof. Quadratic regression. ∇L(θ) = R d ×R k h(θ, x) − y · ∇ θ h(θ, x) dµ (x,y) g(θ, ξ) = h(θ, x) − y · ∇ θ h(θ, x) where ξ = (x, y). Under the assumption that ∇ θ h is uniformly bounded on R m × R d , we observe that E ξ∼P g(θ, ξ) 2 = R d ×R k tr (∇ θ h) T (θ, x) · h(θ, x) − y T h(θ, x) − y · ∇ θ h(θ, x) dµ (x,y) = R d ×R k h(θ, x) − y 2 tr (∇ θ h) T ∇ θ h (θ, x) dµ (x,y) ≤ ∇ θ h 2 L ∞ R d ×R k h(θ, x) − y 2 dµ (x,y) = ∇ θ h 2 L ∞ L(θ). As a Corollary, also E ξ∼µ |g(θ, ξ) − ∇L(θ)| 2 ≤ E ξ∼µ g(θ, ξ) 2 ≤ ∇ θ h 2 L ∞ L(θ) since E ξ∼µ g = ∇L. Classification. As in Example 2.11, we find that ∇ 1 h(θ, x), i 2 ≤ 2 min 1, 2 h(θ, x), i . We set ξ = (x, i) and estimate g(θ, ξ) = (∇ 1 ) h(θ, x), y · ∇ θ h(θ, x) by E ξ∼µ |g(θ, x)| 2 ≤ ∇ θ h 2 L ∞ E ξ∼µ ∇ 1 h(θ, x), y 2 ≤ 2 ∇ θ h 2 L ∞ E ξ∼µ min{1, 2 h(θ, x), y ≤ 2 ∇ θ h 2 L ∞ E ξ∼µ min{1, h(θ, x), y ≤ 2 ∇ θ h 2 L ∞ min{E ξ∼µ h(θ, x), y , 1 = 2 ∇ θ h 2 L ∞ min{L(θ), 1}. Remark 2.15. Neural networks with L layers can be decomposed as h(θ, x) = W σ ĥ (θ, x) , θ = (W,θ) whereĥ(θ, ·) : R d → R M is a neural network with L − 1 layers and σ is a nonlinear map between vector spaces. In particular ∇θh(θ, x) = W σ ĥ (θ, x) ∇θĥ(θ, x), i.e. also the gradient of h with respect to deep layer weights is linear in the final linear weights. The derivative σ has to be interpreted as a diagonal matrix. By the chain rule, the linearity extends to the gradient of the loss function. The assumption that |∇ θ h| ≤ C independently of θ and x is therefore unrealistic in deep learning, but a weaker assumption like |∇ θ h|(θ, x) ≤ C 1 + |θ| 2 , E |g(θ, ξ)| 2 ≤ C f (θ) (1 + |θ| 2 ) holds for two-layer neural networks with ReLU activation and deep networks (ResNets, multilayer perceptra, ...) if the activation function σ and its first derivative σ are bounded. All statements above apply to both the overparametrized and underparametrized setting. A noise intensity which may scale with the objective function is a key feature of machine learning applications in general. The next result is specific to the overparametrized regime, where we show that the stochastic noise has low rank. Lemma 2.16. Under the same conditions as Theorem 2.6, we find that rk(Σ(θ)) ≤ n for all θ ∈ R m , independently of k. Proof. If we set L i (θ) = h(θ, x i ) − y i 2 , then Σ(θ) = 1 n n i=1 ∇L i − ∇L ⊗ ∇L i − ∇L (θ) is a sum of n rank 1 matrices. 3. Stochastic gradient descent with ML noise in discrete time 3.1. Functions satisfying a Lojasiewicz inequality. It is easy to prove that if the noise in SGD is proportional to an objective function f which satisfies a Lojasiewicz condition, SGD reduces f with small, but fixed positive step size η > 0. The SGD scheme converges linearly, much like non-stochastic gradient descent. This is not a realistic model for machine learning, as Lojasiewicz inequalities rule out the presence of all critical points which are not the global minimum, but the same tools can be used in more complex topics below. Theorem 3.1. Assume the following. ( 1) f : R m → [0, ∞) satisfies the Lojasiewicz inequality Λ f (θ) ≤ |∇f | 2 (θ), Λ > 0. (2) f is C 1 -smooth and ∇f satisfies the one-sided Lipschitz-condition ∇f (θ) − ∇f (θ ) · (θ − θ ) ≤ C L |θ − θ | 2 ∀ θ, θ ∈ R m . (3) The family of gradient estimators g : R m × Ω → R m satisfies the following properties: (a) E ξ∼P g(θ, ξ) = ∇f (θ) for all θ ∈ R m . (b) E ξ∼P |g(θ, ξ) − ∇f (θ)| 2 ≤ σf (θ). (4) The initial condition θ 0 is a random variable which satisfies the bound E f (θ 0 ) < ∞. (5) The random gradient selectors {ξ t } are iid with law P and independent of θ 0 . If θ t is generated by the SGD iteration law θ t+1 = θ t − η g(θ t , ξ t ) and 0 < η < Λ Λ+σ 2 C L , then E f (θ t ) ≤ ρ t η E f (θ 0 ) , where ρ η = 1 − Λ η + η 2 C L (1 + σ) 2Λ < 1. Furthermore β t f (θ t ) → 0 almost surely for every β ∈ [1, ρ −1 η ). We note that similar results for SGD with non-standard noise bounds have been proved in the uniformly convex setting in [SK20,Sti19]. Proof. Expected objective value. Take θ t , g t to be fixed for now and θ t+1 = θ t − η g t . Then f (θ t+1 ) − f (θ t ) = 1 0 d ds f θ t − ηs g t ds = 1 0 ∇f θ t − s ηg t · − ηg t ds = η 1 0 1 s ∇f θ t − s ηg t − ∇f (θ t ) · θ t − sη g t − θ t − ∇f (θ t ) · η g t ds ≤ 1 0 1 s C L θ t − sη g t − θ t 2 − η ∇f (θ t ) · g t ds = C L η 2 |g t | 2 1 0 s ds − η ∇f (θ t ) · g t . Now we consider θ t , g t = g(θ t , ξ t ) as the random variables they are in practice. We note that θ t only depends on θ 0 , ξ 0 , . . . , ξ t−1 and is independent of ξ t . Thus in expectation E ∇f (θ t ) · g(θ t , ξ t ) = E E ∇f (θ t ) · g(θ t , ξ t ) θ 0 , ξ 0 , . . . ξ t−1 = E ∇f (θ t ) · E g(θ t , ξ t ) θ 0 , ξ 0 , . . . ξ t−1 = E θ0,ξ0,...,ξt−1 |∇f | 2 (θ t ) E |g t | 2 = E |g t − ∇f (θ t )| 2 + 2 g t · ∇f (θ t ) − |∇f (θ t )| 2 = E |g t − ∇f (θ t )| 2 + E |∇f (θ t )| 2 by the tower identity for conditional expectations. Hence E f (θ t+1 ) − f (θ t ) ≤ C L η 2 2 E |g t | 2 − η E |∇f | 2 (θ t ) = C L η 2 2 E |g t − ∇f (θ t )| 2 + C L η 2 2 E |∇f | 2 (θ t ) − η E |∇f | 2 (θ t ) ≤ C L σ η 2 2 E f (θ t ) + C L η 2 2 − η E |∇f | 2 (θ t ) . If C L η 2 2 − η < 0, i.e. 0 < η < 2 C L , we can estimate further that E f (θ t+1 ) − f (θ t ) ≤ C L σ η 2 2 + Λ C L η 2 2 − η E f (θ t ) or equivalently E f (θ t+1 ) ≤ 1 + C L σ + Λ 2 η 2 − Λη E f (θ t ) . The prefactor ρ η := 1 + C L σ+Λ 2 η 2 − Λη is smaller than 1 if and only if 1−Λη+η 2 C L (Λ + σ) 2 < 1 ⇔ η C L (Λ + σ) 2 η − Λ < 0 ⇔ η ∈ 0, 2Λ C L (Λ + σ) . Almost sure convergence. The argument is standard for series with summable norms. Set E t (ε) = {sup s≥t β s f (θ s ) ≥ ε}. Then E ∞ (ε) := t≥0 E t (ε) = lim sup t→∞ β t f (θ t ) ≥ ε satisfies P E ∞ (ε) ≤ P E t (ε) ≤ ∞ s=t P {β s f (θ s ) ≥ ε} ≤ ∞ s=t E β s f (θ s ) ε ≤ 1 ε ∞ s=t (βρ η ) s E f (θ 0 ) = (βρ η ) t 1 − βρ η E f (θ 0 ) ε . By taking t → ∞, we observe that P E ∞ (ε) = 0, i.e. lim sup t→∞ β t f (θ t ) < ε almost surely. As this holds for all ε > 0, we find that β t f (θ t ) → 0 almost surely. The optimal learning rate if the constants are known satisfies d dη 1 − Λη + C L (Λ + σ) 2 η 2 = 0 ⇒ η = Λ C L (Λ + σ) , ρ η = 1 − Λ 2 2 C L (Λ + σ) . In the noiseless case σ = 0, this boils down to the well-known estimate for deterministic gradient descent. Corollary 3.2. Assume that f, η and g are like in Theorem 3.1. Then θ t converges to a random variable θ ∞ in L 2 and almost surely. The rate of convergence is θ t − θ ∞ L 2 (P) ≤ ρ t/2 η η √ 2C L + σ E f (θ 0 ) 1 − ρ 1/2 η f (θ ∞ ) ≡ 0 almost surely. Proof. Note that θ t = θ 0 + t i=1 θ i − θ i−1 and that ∞ i=1 θ i − θ i−1 L 2 = η ∞ i=1 g(θ i , ξ i ) L 2 ≤ η t i=1 E |∇f (θ i )| 2 + E |g(θ i , ξ i ) − ∇f (θ i )| 2 . Recall that |∇f (θ)| 2 ≤ 2C L f (θ) ∀ θ ∈ R m for non-negative functions with Lipschitz-continuous gradient (see e.g. Lemma B.1 in Appendix C) and that E |g(θ i , ξ i ) − ∇f (θ i )| 2 ≤ σ E f (θ i ) . as above. Thus ∞ i=1 θ i − θ i−1 L 2 ≤ η ∞ i=1 E 2C L f (θ i ) + σ E f (θ i ) ≤ η 2C L + σ E f (θ 0 ) ∞ i=1 ρ 1/2 η < ∞, i.e. the limit θ ∞ := lim t→∞ θ t = θ 0 + ∞ i=1 (θ i − θ i−1 ) exists in the L 2 -sense and θ ∞ −θ t L 2 ≤ ∞ i=t θ i+1 −θ i L 2 ≤ η √ 2CL + σ E f (θ 0 ) ∞ i=t ρ 1/2 η = η √ 2CL + σ E f (θ 0 ) ρ t/2 η 1 − ρ 1/2 η . Since the increments are summable, the convergence is pointwise almost everywhere by the same argument as in Theorem 3.1. 3.2. Convergence close to the minimum. Any function which has a critical point that is not the global minimum does not satisfy a Lojasiewicz inequality. However, if such an inequality holds close to the minimum and the initial condition is very close to the global minimum, then with high probability we converge to the minimum exponentially fast. and that ∇f is C L -Lipschitz continuous on U (ε). Then for every δ > 0 there exists ε > 0 such that the following holds: If θ 0 ∈ U (ε ) almost surely, then with probability at least 1 − δ we have θ t ∈ U (ε) for all t ∈ N. Conditioned on the event that θ t ∈ U (ε) for all t, the estimate (3.1) lim t→∞ β t f (θ t ) = 0 holds almost surely for every (3.2) β ∈ [1, ρ −1 η ) where ρ η = 1 − Λη + η 2 C L (Λ + σ) 2Λ if the learning rate satisfies 0 < η < 2Λ C L (1 + σ) . The proof idea is as follows: In any iteration, we expect the objective value to decrease. While it may increase at times, it can only increase by small amounts as long as the objective value is small, and it is unlikely for errors to accumulate sufficiently for f (θ t ) to exceed ε if f (θ 0 ) ε is small enough. In the event that f (θ t ) < ε for all t ∈ N, we can follow along the lines of Theorem 3.1. The rigorous proof is a variation of that of [MHKC20, Theorem 4] and given in Appendix C. The main difference to the original is that the error is controlled by a decaying learning rate in [MHKC20] and by the low noise intensity for low objective value in our context. If the set U (ε) has multiple connected components, of course the Lojasiewicz inequality and Lipschitz-condition are only required to hold locally as the result is local in nature. Remark 3.4. A version of Corollary 3.2 holds also in this case with virtually the same proof, conditioned on the event that θ t ∈ U (ε) for all t ∈ N. 3.3. On the global convergence of SGD with ML noise. Whether SGD converges to a global minimum even with poor initialization if the target function does not satisfy a Lojasiewicz inequality is quite delicate and requires strong assumptions. In general, this cannot be guaranteed. Example 3.5. Consider the functions h α (x) = (x 2 − 1) 2 4 + αx, f α (x) = h α (x) − inf x ∈R h α (x ). For α = 0, the function f α has two minima of equal depth at x = ±1. If α is small but non-zero, the two local minima do not have equal depth. Assume that the gradient estimators are of the form g(x, ξ) = f α (x) + σ f α (x) ξ where ξ is equal to 1 or −1 with equal probability. If √ σ < min 1/4<|x|<1/2 |f α (x)| f α (x) and η is reasonably small, x can never escape the potential well it started in. The situation is different if the noise is unbounded and the objective function has particularly convenient properties not only at the set of global minimizers. (2) f is C 1 -smooth and ∇f satisfies the one-sided Lipschitz condition (3.3) ∇f (θ) − ∇f (θ ) · θ − θ ) ≤ C L |θ − θ | 2 . (3) A Lojasiewicz inequality holds on the set where f is small, i.e. there exist ε, λ > 0 such that (3.4) f (θ) < ε ⇒ λ f (θ) ≤ |∇f (θ)| 2 . (4) A Lojasiewicz inequality holds on the set where f is large, i.e. there exist S, Λ > 0 such that (3.5) f (θ) ≥ S ⇒ Λ f (θ) ≤ |∇f (θ)| 2 . (5) f grows uniformly away from its minimum, i.e. there exists R > 0 such that (3.6) f (θ) ≤ S ⇒ ∃ θ s.t. f (θ ) = 0 and |θ − θ | < R where S is the same as in (3.5). Assume that the gradient noise satisfies the following: (1) The noise has ML type, i.e. E ξ |g(θ, ξ) − ∇f (θ)| 2 ≤ σ f (θ) ∀ θ. (2) The noise is uniformly unbounded in the sense that g(θ, ξ) = ∇f (θ) + σ f (θ) Y θ,ξ and there exists a continuous function ψ : (0, ∞) 2 → (0, ∞) such that Y satisfies (3.7) EY θ,ξ = 0, P Y θ,ξ ∈ B r (θ) ≥ ψ |θ|, r > 0 ∀θ ∈ R m , r > 0 independently of θ. Finally, assume that the learning rate satisfies 0 < η < min 2λ C L (λ + σ) , 2Λ C L (Λ + σ) . Assume that θ 0 is an initial condition such that E f (θ 0 ) < ∞. Then the estimate lim sup t→∞ f (θ t ) ρ t < ∞ holds almost surely for every (3.8) ρ ∈ (ρ η , 1) , ρ η := 1 − λη + η 2 C L (λ + σ) 2λ . Proof sketch. We use the Lojasiewicz inequality on the set {f > S} to show that SGD iterates visit the set {f ≤ S} infinitely often almost surely. For θ t ∈ {f ≤ S}, there exists a uniformly positive probability that the θ t+1 ∈ {f < ε } due to the condition on the noise to be sufficiently 'spread out' and on the set {f ≤ S} to be sufficiently close to {f = 0}. Thus for infinite time, we visit the set {f < ε } almost surely. Once we enter {f < ε }, we remain trapped in the set {f < ε} with high probability by Theorem 3.3, and in that case, we almost surely observe that f → 0 at a linear rate. Even if we leave from {f < ε}, we almost surely visit again. The probability to escape infinitely often vanishes, so we expect that f (θ t ) ≤ ρ t−T η f (θ T ) for all t ≥ T for almost every realization of SGD for some random time T . The full proof is given in Appendix D. With a variation of the proof of Corollary 3.2, the following can be obtained. Note that the rate of convergence ρ η depends only on the Lojasiewicz constant on the set where f is small. Corollary 3.7. Assume that f, η and g are like in Theorem 3.6. Then θ t converges to a minimizer of f almost surely. Remark 3.8. Our results suggest that, in a fairly general class of functions, SGD with ML noise converges to a global minimizer exponentially fast (or at least that the objective function decays exponentially along SGD iterates). However, it may take a very long time to reach the set in which exponential convergence is achieved, especially if the dimension m of the parameter space and the co-dimension m − n of the minimizer manifold N are both high. We give a few simple examples of situations in which Theorem 3.6 can in fact be applied. Despite its weaknesses, we believe this mechanism to be a major driving factor behind the success of SGD in the machine learning of overparametrized neural networks. Remark 3.9. If the function f satisfies the conditions of Theorem 3.6, noise of the form g(θ, ξ) = ∇f (θ) + σ f (θ) Y ξ is admissible, where where Y ∼ N (0, I) is standard Gaussian noise. Example 3.10. The following functions satisfy the conditions of Theorem 3.6, but do not satisfy a global Lojasiewicz inequality: (1) f (x, y) = sin(x) + sin(y) + 2. The set of minimizers is a lattice, f is bounded, and every minimizer is non-degenerate. (2) f ε (x) = x 2 + 2ε sin 2 (x/ε) for any fixed ε > 0. The only minimizer is at x = 0. Far away from the origin, the oscillatory perturbation becomes negligible in both f ε and f ε . More generally, if f is a perturbation of a quadratic form in the following sense, then Theorem 3.6 applies: • f ≥ 0. • The set {f = 0} is a finite union of disjoint compact C 2 -manifolds N 1 , . . . , N k (potentially of different dimensions). If θ ∈ N k , then D 2 f (θ) has rank m − dim(N k ) at x and the smallest non-zero singular value of D 2 f is bounded away from zero. Also the first example could be generalized to small perturbations of periodic functions with non-degenerate minimizers. Example 3.11. The following functions do not satisfy the conditions of Theorem 3.6 since there are low energy points arbitrarily far away from the set of global minimizers: f 1 , f 2 : R → R, f 1 (x) = 1 + sin 2 (x) − 1 x 2 + 1 , f 2 (x) = 1 + x 2 sin 2 (x) − 1 x 2 + 1 . In this one-dimensional setting, that problem could presumably be solved by appealing to the recurrence of random walks, but the situation is hopeless in analogous constructions in dimension three or higher. Let us consider the conditions placed in Theorem 3.6 in the context of deep learning. Remark 3.12. • As the parameter gradient of a deep neural network with respect to deep layer coefficients is linear in the final layer coefficients (see Remark 2.15), the Lipschitzcondition (3.4) cannot hold globally. If the activation function is smooth, it holds locally. • Generically, the Hessian of L has maximal rank on N due to Theorem 2.6, so the local Lojasiewicz inequality (3.4) should hold at least locally on R m . • Due to Lemma 2.8, the objective function has no critical points where it is large. While (3.5) may not be satisfied as such, a condition of this type does not violate the spirit of the minimization problem under consideration. • It is currently unknown to us whether the condition (3.6) that the set of low objective values is within a bounded tube around N is generically satisfied or not. We offer the following rationale: Let θ such that L(θ) ≤ S. Choose n = m−nk k additional points x n+1 , . . . , x n+n (assuming that n is an integer) and setŷ i = h(θ, x i ) for 1 ≤ i ≤ n + n . If the map Φ : R m → R nk+n k = R m , Φ(θ) = h(θ, x 1 ), . . . , h(θ, x n ) has a Lipschitz-continuous inverse, then there exists θ such that (1) Φ(θ , x i ) = y i for 1 ≤ i ≤ n and Φ(θ , x i ) =ŷ i for n + 1 ≤ i ≤ n + n . (2) θ and θ are somewhat close as |θ − θ | ≤ C |y − y | = C n i=1 |y i −ŷ i | 2 = C n i=1 |y i − h(θ, x i )| 2 = C n L(θ) ≤ C √ nS. Thus a condition of the type (3.6) may hold for nicely parametrized models, but the constant is expected to be somewhat large if the data set is large. • The type of noise specified in (3.7) is unrealistic in overparametrized learning models as it is omni-directional, whereas realistic noise is necessarily low rank due to Lemma 2.16. The fact that the noise is 'spread out' is needed to guarantee that we may randomly 'jump' into the set {f < ε} from anywhere in the set {f ≤ S}. Identifying more realistic geometric conditions with similar guarantees remains an open problem. Remark 3.13. Theorem 3.6, were it to apply, could be viewed as a negative result in the context of implicit regularization in machine learning. We can decompose the mean squared error population risk functional as R(h) = R d h(x) − y 2 P(dx ⊗ dy) = R d h(x) − h * (x) 2 P(dx ⊗ dy) + R d h * (x) − y 2 P(dx ⊗ dy) = R d h(x) − h * (x) 2 P(dx ⊗ dy) + C 0 where h * (x) = E[y|x] is the risk minimizer in the class of measurable functions and C 0 = R(h * ) is the minimum Bayes risk. If the distribution P admits any uncertainty in the output y given observations x, then C 0 > 0. However, in overparametrized learning the empirical risk R n (h) = 1 n n i=1 h(x i ) − y i 2 can be zero. Thus if C 0 is large and the parameters of a model are trained by SGD with small positive learning rate, then after a long time, we expect R and R n to differ greatly at the parameters θ t . To avoid overfitting the training data, we therefore require an early stopping strategy. If the observations are virtually noiseless (which is the case for benchmark image classification problems), then this rationale may not apply, and SGD may perform well without early stopping. A numerical illustration We compare toy models for stochastic gradient descent where η is a learning rate, σ, σ > 0 control the noise level, and Y n is an is a standard Gaussian. For a fair comparison, σ and σ have to be chosen such that the noise has the same magnitude at a certain point of interest. Consider the function (4.1) x n+1 = x n − η ∇f (x n ) + σY n ,x n+1 =x n − η ∇f (x n ) + σ f (x n ) Y nf : R 2 → R, f (x, y) = x 4 4 − x 2 + αx + y 2 + c α where α ≥ 0 is a parameter and c α is chosen such that inf (x,y) f (x, y) = 0. For any α ∈ (0, 1], the function f has two distinct local minima at (x − , 0) and ( x + , 0) where x − < 0 < x + . Clearly, we have f (x − , 0) < f (x + , 0). We considered 1000 realizations of SGD according to the schemes (4.1) with α = 0.25, learning rate η = 0.05 and noise σ = 4.0. The parameter σ was chosen such that the noise at the ridge between the minima of f has the same intensity for both algorithms to give them equal opportunity to escape the local minimum. In all runs, the initial condition was (x 0 , y 0 ) = (3, 0). In Figure 1, we see that SGD with ML type noise and constant learning rate approaches the global minimum rapidly whereas SGD with homogeneous Gaussian noise and constant learning rate forms a cloud around the minima which has higher density at the global minimum. SGD with homogeneous noise and decaying learning rate forms more focussed clouds around the minima, but about 25% of trajectories do not escape the local minimum. The decaying learning rate is chosen as (4.2) η t = t 0 η t + t 1 , t 0 = 20, t 1 = 5. We make similar observations if the learning rate decays of the same order but less rapidly with t 0 = 200 and t 1 = 50, but with less focussed point clouds. After 300 iterations, the point cloud of SGD with homogeneous noise and positive learning rate is comparable to that of SGD with ML noise at the local minimum. This seems to be due to the fact that both start closer to the local minimum and are equally likely to escape the local minimum due to our scaling of the noise. Once a trajectory of SGD with ML noise enters the potential well of the global minimum, it is unlikely to escape, whereas SGD with homogeneous Gaussian noise exchanges particles back and forth between the two wells. Once within the potential well of the global minimum, a trajectory of SGD with ML noise converges to the global minimum rapidly. Therefore, after 1000 iterations, trajectories of SGD with ML noise are almost guaranteed to have found the global minimum, whereas trajectories of SGD with homogeneous noise are somewhat likely to be found in the potential well of the local minimum, independently of whether the learning rate is constant or decaying. Conclusion More realistic abstract models for the noise of stochastic gradient descent in machine learning may explain help explain some of the success that SGD has enjoyed in this specific non-convex optimization task. We believe our results are indicate why SGD often finds global minima/low loss local minima rather than positive loss local minima. Our results give an indication that it may be admissible in these applications to leave the learning rate uniformly positive. This may be particularly relevant for online learning tasks, where new data is added to the model. Decaying learning rates significantly diminish the effect that new data can have in finite time. The reason that the learning rate should be small in these optimization tasks is related both to the roughness of the loss landscape and the presence of stochastic noise. The estimates resemble non-stochastic gradient descent more closely than SGD with homogeneous noise estimates. There are several pressing questions that were not explored in this work, both on the theoretical and practical side. (1) The biggest deficiency of our global convergence result is the reliance on omni-directional noise while realistic noise in machine learning SGD is low rank, at least for overparametrized models. Understanding the geometry of realistic noise and its impact on the convergence of SGD is an important open problem. (2) The objective functions we employ serve as toy models for the energy landscape of L 2regression problems in deep learning. The energy landscape of classification problems and the associated gradient noise are quite different as noted in Example 2.5 and Lemma 2.14. Understanding the behavior of SGD in classification-like loss landscapes and noise models remains an open problem. (3) While our assumptions are fairly general in the context of optimization theory, they likely are too restrictive in the setting of deep learning. Understanding the geometry of deep learning problem, and whether they allow for similar results, is the next important step in this approach to the analysis of SGD. (4) The Lipschitz-constant of the gradient of the objective function is not generally uniformly controlled over the parameter space. If a minimum is too steep, gradient descent may escape from it exponentially fast due to finite step size effects. This mechanism cannot be captured by continuum models and renders the positive step-size analysis invalid in the context of deep learning, unless we assume or enforce confinement to a bounded domain by other means. (5) In practice we do not use random selection SGD (choosing a batch of data samples randomly from the training set), but random pass SGD (passing through the entire training set batch by batch before repeating the same data point). The random directions in consecutive iterations are therefore not truly iid, and the noise may be less oscillatory in practice than random selection estimation would suggest. We believe the impact of this difference to be negligible for large data sets, but a rigorous connection has not been established to the best of our knowledge. (6) Typically, advanced optimizers like gradient descent with momentum, Nesterov's accelerated gradient descent or ADAM are used in deep learning with stochastically estimated gradients. It remains to be seen whether noise of ML type allows for stronger estimates and global convergence guarantees also in that setting. (7) The main goal of this article was to understand SGD in toy models inspired by problems of deep learning. Another interesting direction is whether artificially perturbing (exact or estimated) gradients by omni-directional noise of ML type can improve the convergence of a gradient descent type optimization algorithm. Proof. First claim. Compute ∇L y (θ) = 2 n n i=1 h(θ, x i ) − y i ∇ θ h(θ, x i ) D 2 L y (θ) = 2 n n i=1 ∇h(θ, x i ) ⊗ ∇h(θ, x i ) + h(θ, x i ) − y i D 2 h(θ, x i ) . If D 2 h(θ, x i ) = 0, there exists v ∈ R d such that v T D 2 h(θ, x i )v = 0 since the Hessian matrix is symmetric. Thus lim inf |yi|→∞ v T D 2 L y (θ) v = − lim |yi|→∞ y i v T D 2 h(θ, x i ) v = ±∞, meaning that L y cannot be convex. Thus if L y is convex for all y, by necessity D 2 h(θ, x i ) ≡ 0 for all i, i.e. θ → h(θ, x i ) is linear. Second claim. Set y j = h(θ, x j ) for j = i and y i = h(θ, x i ) ± ε/2. Then by a simple result in linear algebra (see Lemma A.3 below), the matrix D 2 L y (θ) = 2 n n i=1 [∇h(θ, x i ) ⊗ ∇h(θ, x i )] + ε 2 D 2 h(θ, x i ) has a negative eigenvalue. Third claim. Let V = span{∇f (θ, x j ) : j = i}. Since the set of gradients is linearly independent, there exists v such that v⊥V but v T ∇f (θ, x i ) = 0. Thus to leading order D 2 L y (θ + tv) = D 2 L y (θ) = 2 n n i=1 [∇h(θ, x i ) ⊗ ∇h(θ, x i )] + t ∇f (θ, x i ) · v D 2 h(θ, x i ) + o(t). As before, we conclude that D 2 L y (θ + tv) has a negative eigenvalue if t > 0 is small enough. If D 2 ψ is bounded and y is fixed, then for every R > 0 we can choose ε so small that L y is convex on B R (0). Note that the rank of D 2 ψ is at most n in this example. We prove a result which we believe to be standard in linear algebra, but have been unable to find a reference for. Lemma A.3. Let A ∈ R m×m be a symmetric positive definite matrix of rank at most n < m. Let B be a symmetric matrix of at least n + 1. Then, for every s > 0 at least one of the matrices A + sB or A − sB has a negative eigenvalue. Proof. Without loss of generality, we assume that A = diag(λ 1 , . . . , λ n , 0, . . . , 0). First case. The Lemma is trivial if there exists v ∈ span{e n+1 , . . . , e m } such that v T Bv = 0 since then v T A + sB v = s v T Bv is a linear function. Second case. Assume that v T Bv = 0 for all v ∈ span{e n+1 , . . . , e m }. Since B has rank n+1, there exists an eigenvector w for a non-zero eigenvalue µ of B such that w / ∈ span{e 1 , . . . , e n }. Without loss of generality, we may assume that e n+1 · w > 0. Consider g s (t) := e n+1 + tv T A + sB e n+1 + tv g s (0) = e T n+1 (A + sB)e n+1 = 0 g s (0) = 2 v T A + sB e n+1 = 2v T (Ae n+1 ) + 2s e T n+1 Bv = 0 + 2µs e T n+1 v = 0. Thus, we choose the correct sign for t depending on µ and s, then e n+1 + tv T A + sB e n+1 + tv < 0. In this situation, B is indefinite and the sign of s does not matter. Appendix B. Auxiliary observations on objective functions and Lojasiewicz geometry Lemma B.1. Assume that f : R m → R is a non-negative function and ∇f is Lipschitz continuous with constant C L . Then |∇f (θ)| 2 ≤ 2C L f (θ) ∀ θ ∈ R m . Proof. Take θ ∈ R m . The statement is trivially true at θ if ∇f (θ) = 0, so assume that ∇f (θ) = 0. Consider the auxiliary function g(t) = f θ − t ν where ν = ∇f (θ) |∇f (θ)| . Then g (0) = −ν · ∇f (θ) = |∇f (θ)| and g (t) − g (0) = ∇f (θ − tν) − ∇f (θ) · ν ≤ c L θ − tν − θ = c L t. Thus g(t) = g(0) + t 0 g (s) ds ≤ f (θ) + t 0 −|∇f (θ)| + C L s ds = f (θ) − |∇f (θ)|t + C L 2 t 2 . The bound on the right is minimal for t = − |∇f (θ)| C L when f (θ) − |∇f (θ)|t + C L 2 t 2 = f (θ) − |∇f (θ)| 2 2 C L . Since f ≥ 0 also g ≥ 0, so f (θ) − |∇f (θ)| 2 2C L ≥ 0. Remark B.2. In particular, If f satisfies a Lojasiewicz inequality and has a Lipschitz-continuous gradient, then (B.1) Λ f ≤ |∇f | 2 ≤ 2C L f. We show that the class of objective functions which can be analyzed by our methods does not include loss functions of cross-entropy type under general conditions. Corollary B.3. Assume that f : R m → R is a C 1 -function such that • f ≥ 0 and • (B.1) holds. Then there existsθ ∈ R m such that f (θ) = 0. Proof. Choose θ 0 ∈ R m and consider the solution of the gradient flow equation θ = −∇f (θ) t > 0 θ = θ 0 t = 0. Then d dt f (θ(t)) = −|∇f | 2 (θ(t)) ≤ −Λ f (θ(t)) ⇒ d dt log(f (θ(t))) = d dt f (θ(t)) f (θ(t)) ≤ −Λ and thus f (θ(t)) ≤ f (θ(0)) e −Λt . Furthermore θ(t 2 ) − θ(t 1 ) ≤ t2 t1 |∇f |(θ(s)) ds ≤ t2 t1 2C L f (θ(s)) ds ≤ 2C L L(θ(0)) t2 t1 e −Λs/2 ds = 2 2C L L(θ(0)) Λ e −Λt1/2 − e −Λt2/2 whence we find that θ(t) converges to a limiting point θ ∞ as t → ∞. By the continuity of f , we find that f (θ ∞ ) = lim t→∞ f (θ(t)) = 0. Appendix C. Proof of Theorem 3.3: Local Convergence We split the proof up over several lemmas. Our strategy follows along the lines of [MHKC20, Appendix D], which in turn uses methods developed in [HIMM19,HIMM20]. We make suitable modifications to account for the fact that the smallness of noise comes from the fact that the values of the objective function are low, not that the learning rate decreases. Furthermore, we have slightly weaker control since we do not impose quadratic behavior with a strictly positive Hessian at the minimum, but only a Lojasiewicz inequality and Lipschitz continuity of the gradients. While weaker conditions may hold for the individual steps of the analysis, we always assume that the conditions of Theorem 3.3 are met for the remainder section. We decompose the gradient estimators as g(θ, ξ) = ∇f (θ) + σ f (θ) Y θ,ξ , E ξ Y θ,ξ = 0, E ξ |Y θ,ξ | 2 ≤ 1 and interpolate θ t+s = θ t − sη g(θ t , ξ t ) for s ∈ [0, 1]. With these notations, we can estimate the change of the objective in a single time-step as f (θ t+1 ) − f (θ t ) = 1 0 d ds f θ t − sη g(θ t , ξ t ) ds = − 1 0 ∇f (θ t+s ) · η g(θ t , ξ t ) ds = −η 1 0 ∇f (θ t ) · ∇f (θ t ) + σf (θ t ) Y θt,ξt + ∇f (θ t+s ) − ∇f (θ t ) · g(θ t , ξ t ) ds ≤ −η |∇f (θ t )| 2 + η σ f (θ t )∇f (θ t ) · Y θt,ξt + C L η 1 0 |θ t+s − θ t | |g(θ t , ξ t )| ds ≤ −η |∇f (θ t )| 2 + η σ f (θ t )∇f (θ t ) · Y θt,ξt + C L η 1 0 s |g(θ t , ξ t )| 2 ds = −η |∇f (θ t )| 2 + η σ f (θ t )∇f (θ t ) · Y θt,ξt + C L η 2 2 |g(θ t , ξ t )| 2 = −η |∇f (θ t )| 2 + η σ f (θ t )∇f (θ t ) · Y θt,ξt + C L η 2 2 ∇f (θ t ) + σ f (θ t ) Y θt,ξt 2 = C L η 2 − 1 η |∇f (θ t )| 2 + (1 + C L η) √ σ η f (θ t ) ∇f (θ t ) · Y θt,ξt + C L σ η 2 2 f (θ t ) Y θt,ξt 2 = − η |∇f (θ t )| 2 +η ∇f (θ t ) · f (θ t ) Y θt,ξt +η 2 f (θ t ) Y θt,ξt 2 where η = C L η 2 − 1 η,η = √ σ (1 + C L η) η,η = C L σ 2 η. All three variables scale like η and the difference between them can be ignored for the essence of the arguments. As usual, we denote by F t the filtration generated by θ 0 , ξ 0 , . . . , ξ t−1 , with respect to which θ t is measurable. We note that ∇f (θ t ) · f (θ t ) Y θt,ξt is a martingale difference sequence with respect to F t since E ∇f (θ t ) · f (θ t ) Y θt,ξt |F t = ∇f (θ t ) · f (θ t ) E Y θt,ξt |F t = 0. To analyze the θ t over several time steps, we define the cumulative error terms M t =η t i=0 ∇f (θ t ) · f (θ t ) Y θt,ξt S t =η 2 t i=0 f (θ t ) Y θt,ξt 2 R t = M 2 t + S t . We furthermore define the events Ω t (ε ) = {f (θ i ) < ε for all 0 ≤ i ≤ t} = {"objective remains small"} E t (r) = {R t < r for all 0 ≤ i ≤ t} = {"noise remains small until time t"} E t (r) = E t−1 (r) \ E t (r) = {"noise exceeds threshold in t-th step"}. for ε > 0 and 0 < r < 1. The sets E t are useful as they allow us to estimate the measure of E c t = t i=0 E i , where all sets in the union are disjoint. Lemma C.1. The following are true. (1) Ω t+1 ⊆ Ω t and E t+1 ⊆ E t . (2) If θ 0 ∈ Ω 0 (ε ) for ε < ε, then E t−1 (r) ⊆ Ω t (ε) if ε + r + √ r < ε. (3) Under the same conditions, the estimate (C.1) E R t 1 Et−1 ≤ E R t−1 1 Et−2 + Cσ 2c L ε + 1 η 2 E 1 Et−1 f (θ t ) − r P E t holds, where the constant C > 0 incorporates the factors betweenη,η andη. Proof. The first claim is trivial. Second claim. Recall that θ 0 is initialized in Ω(ε ) ⊆ Ω(ε). In particular Ω 0 = E −1 = Ω is the entire probability space since the sum condition for E −1 is empty. We proceed by induction. Proof. First step. The probably that E t does not occur coincides with the probability that there exists some i ≤ t such that R i exceeds r δ at i, but not i − 1. More precisely t−1 i=0 E i = t−1 i=0 (E i−1 \ E i ) = E −1 \ E t = E c t since E 0 is the whole space. Hence P(E c t ) = t−1 i=0 P( E i ) . Using E i = E i−1 ∩ {R i > r} and 1 Ei = 1 Ei−1 1 {Ri>r} , we bound (C.3) P( E i ) = E 1 Ei−1 1 {Ri>r} ≤ E 1 Ei−1 R i r ≤ E R i 1 Ei−1 r since R i ≥ 0 as a sum of squares. Second step. From (C.1), we obtain E R t 1 Et−1 − E R t−1 1 Et−2 ≤ Cση 2 2C L ε + 1 E 1 Et−1 f (θ t ) − r P E t , so by the telescoping sum identity (C.4) E R t 1 Et−1 − E R −1 1 E−2 ≤ Cση 2 2C L ε + 1 t i=0 E 1 Ei−1 f (θ i ) − r P E i . Note that the second term on the right hand side vanishes since the sum defining R −1 is empty. Conclusion. Combining (C.3) and (C.3), we find that P( E t ) ≤ E R t 1 Et−1 r ≤ Cση 2 2C L ε + 1 r t i=0 E 1 Ei−1 f (θ i ) − t i=0 P E i , so P(E c t ) = t−1 i=1 P( E i ) ≤ Cση 2 2C L ε + 1 r t i=0 E 1 Ei−1 f (θ i ) . Finally, we are in a position to prove the local convergence result. Proof of Theorem 3.3. Step 1. Since E i ⊆ E i−1 and Ω i ⊆ E i−1 , we have E 1 Ei−1 f (θ i ) ≤ E 1 Ei−1 E f (θ i )|F i−1 ≤ ρ η E f (θ i−1 1 Ei−1 ≤ ρ η E f (θ i−1 ) 1 Ei−2 where ρ η = 1 − Λη + η 2 C L (Λ + σ) 2 < 1 as previously for functions which satisfy the Lojasiewicz inequality globally. We conclude from (C.4) that P(E c t ) ≤ Cση 2 2C L ε + 1 r E f (θ 0 ) t i=0 ρ i η ≤ Cση 2 2C L ε + 1 r(1 − ρ η ) ε . Thus for every δ > 0, there exists ε > 0 such that P(E t ) ≥ 1 − δ for all t ∈ N if f (θ 0 ) < ε almost surely. Step 2. Consider the event Ω := t≥0 Ω t (ε) which satisfies P( Ω) ≥ 1 − δ since E t−1 ⊆ Ω t and E t ⊆ E t−1 . Then E f (θ t ) 1 Ω ≤ E f (θ t ) 1 Ωt−1 ≤ ρ t η E f (θ 0 ) as in Step 1. We conclude that for every β ∈ [1, ρ −1 η ) the estimate lim sup t→∞ β t f (θ t ) = 0 holds almost surely conditioned on Ω as in the proof of Theorem 3.1. Appendix D. Proof of Theorem 3.6: Global Convergence Again, we split the proof up over several Lemmas. First, we show that P lim inf t→∞ f (θ t ) ≤ S = 1. We always assume that f satisfies the conditions of Theorem 3.6, although weaker conditions suffice in the individual steps. Lemma D.1. If E f (θ 0 ) < ∞, the estimate sup t≥0 E f (θ t ) < ∞. holds. Proof. Recall that for any θ we have E ξ f (θ − ηg(θ, ξ)) ≤ 1 + C L ση 2 2 f (θ) − 1 − C L η 2 η E |∇f (θ)| 2 as in the proof of Theorem 3.1. We distinguish two cases: • If f (θ) ≤ S, then E ξ f (θ − ηg(θ, ξ)) ≤ 1 + C L ση 2 2 f (θ) ≤ 1 + C L ση 2 2 S. • If f (θ) ≥ S, then E ξ f (θ − ηg(θ, ξ)) ≤ 1 − Λη + C L (Λ + σ) 2Λ η 2 f (θ) due to the Lojasiewicz inequality on the set where f is large. In particular, since f is non-negative, we have E f (θ t+1 ) ≤ 1 + C L ση 2 2 E f (θ t ) 1 {f (θt)≤S} +ρ η E f (θ t ) 1 {f (θt)>S} ≤ 1 + C L ση 2 2 S +ρ η E f (θ t ) whereρ η = 1 − Λη + C L (Λ+σ) 2Λ η 2 . If we abbreviate z t = E f (θ t ) , we deduce that z t+1 ≤ 1 + C L ση 2 2 S +ρ η z t ≤ max    1 +ρ η 2 z t , 1 + C L ση 2 2 S 1 − 1+ρη 2    . Thus if z t is large, then z t decays. In fact lim sup t→∞ z t ≤ 1 + C L ση 2 2 S 1 − ρ η = 1 + C L ση 2 2 S Λη − C L (Λ+σ) 2Λ η 2 independently of the initial condition. Note that the finiteness of the bound hinges on the fact that the learning rate remains uniformly positive in this simple proof. In other variants of gradient flow, it can be non-trivial to control the possibility of escape. The trajectories of SGD satisfy stronger bounds than the expectations. Lemma D.2. P lim inf t→∞ f (θ t ) ≤ S = 1. Proof. Let Ω n,N = {f (θ t ) ≥ S for all n < t ≤ N }, Ω n = N ≥n Ω n,N In particular, Ω n ⊆ Ω n,N for all N ≥ n and 1 Ωn ≤ 1 Ωn,t+1 ≤ 1 Ωn,t for all t ≥ n. Thus P(Ω n ) = E 1 Ωn ≤ 1 S E 1 Ωn f (θ t+1 ) ≤ 1 S E 1 Ωn,t f (θ t+1 ) = 1 S E E 1 Ωn,t f (θ t+1 )|F t ≤ ρ η S E 1 Ωn,t f (θ t ) ≤ . . . ≤ ρ t−n η E f (θ n ) S ≤ ρ t−n η sup s∈N E f (θ s ) S for any t ≥ n. Thus P(Ω n ) = 0 for all n ∈ N. Hence also P lim inf t→∞ f (θ t ) > S ≤ P ∞ n=1 Ω n = 0. We have shown that we visit the set {f < S} infinitely often almost surely. We now show that in every visit θ t , the probability that f (θ t+1 ) < ε is uniformly positive. Below, we will use this to show that we visit the set {f < ε } infinitely often with uniformly positive probability (which then implies that SGD iterates approach the set of minimizers almost surely). In this step, we use that the noise is uniformly 'spread out'. Lemma D.3. There exists γ > 0 such that the following holds: If f (θ) ≤ S, then P f (θ − ηg(θ, ξ) < ε ≥ γ. Proof. We consider two cases separately: f (θ) < ε or f (θ) > ε . In the first case, we can argue by considering the gradient descent structure, while we rely on the stochastic noise in the second case. First case. If f (θ) < ε , then E f (θ − ηg(θ, ξ) ≤ ρ η f (θ) since f satisfies a Lojasiewicz inequality in this region. In particular P f (θ − ηg(θ, ξ) ≥ f (θ) ≤ E f (θ − ηg(θ, ξ) f (θ) ≤ ρ η < 1. Second case. By assumption, the set of moderate energy is not too far from the set of global minimizers in Hausdorff distance, i.e. there exists θ such that (1) f (θ ) = 0 and (2) |θ − θ | < R. Due to the Lipschitz-continuity of the gradient of f , there existsr > 0 depending only on C L such that f (θ) < ε for all θ ∈ Br(θ ). We conclude that P f (θ − ηg(θ, ξ) < r ≥ P θ − ηg(θ, ξ) ∈ Br(θ ) = P Y θ,ξ ∈ Br η √ f (θ) θ − θ − η ∇f (θ) η f (θ) . By assumption, the radiusr η f (θ) ≥r η √ S > 0 is uniformly positive and the center of the ball θ −θ − η ∇f (θ) η f (θ) ≤ |θ −θ| + η √ 2C L S η √ ε ≤ R + η √ 2C L S η √ ε is in some large ball independent of θ. Thus the probability of jumping into {f < ε }, albeit small, is uniformly positive with a lower bound P f (θ − ηg(θ, ξ) ∈ B r δ (θ ) ≥ inf ψ(s, r) : s ≤ R + η √ 2C L S η √ ε , r >r η √ S > 0. By Lemma D.2, the sequence of stopping times τ 0 = 0, τ k = inf{n > τ k + 1 : f (θ n ) ≤ S} is well-defined except on a set of measure zero. Consider the Markov process Z 2k = θ τ k , Z 2k+1 = θ τ k +1 . Note that we use τ k + 1 for odd times, not τ k+1 . To show the stronger statement that P lim inf t→∞ f (θ t ) ≤ ε = 1, we use the conditional Borel-Cantelli Lemma D.4. Lemma D.4. [Kle06,Übung 11.2.6] Let F n be a filtration of a probability space and A n a sequence of events such that A n ∈ F n for all n ∈ N. Define A * = ∞ n=1 E 1 An |F n−1 = ∞ , A ∞ = lim sup n→∞ A n = A n for infinitely many n ∈ N . Then P(A * ∆A ∞ ) = 0 where A∆B denotes the symmetric difference of A and B. Corollary D.5. P lim inf t→∞ f (θ t ) ≤ ε = 1. Proof. Consider the filtration F n generated by Z n and the events A n = {f (θ n ) < ε }. Then ∞ n=1 E 1 An |F n−1 ≥ ∞ n=1 E 1 A2n+1 |F 2n ≥ ∞ n=1 γ = +∞ except on the null set where τ k is undefined for some k. Thus P lim sup n→∞ A n = 1, so almost surely there exist infinitely many k ∈ N such that f (Z k ) < ε . In particular, almost surely there exist infinitely many t ∈ N such that f (θ t ) < ε . We are now ready to prove the global convergence result. Proof of Theorem 3.6. Let β ∈ [1, ρ −1 η ) and consider the event Ω β := lim sup t→∞ β t f (θ t ) = 0 . Choose δ ∈ (0, 1) and associated ε > 0. By Corollary D.5, the stopping time τ := inf{t ≥ 0 : f (θ t ) < ε } is finite except on a set of measure zero. Consider θ τ as the initial condition of a different SGD realizationθ and note that the conditional independence properties which we used to obtain decay estimates still hold for the gradient estimators g t = g(θ t+τ , ξ t+τ ) with respect to the σ-algebras F t generated by the random variables θ t+τ for t ≥ 0. By Theorem 3.3, we observe that with probability at least 1 − δ, we have lim sup t→∞ β t f (θ t+τ ) = 0. Thus for any T > 0, with probability at least 1 − δ − P(τ > T ) we have lim sup t→∞ β t f (θ t ) ≤ lim sup t→∞ β t+τ f (θ t+τ ) ≤ β T · 0 = 0. Taking T → ∞ and δ → 0, we find that almost surely lim sup t→∞ β t f (θ t ) = 0. We conclude by proving that not just the function values f (θ t ), but also the arguments θ t converge. Proof of Corollary 3.7. The proof follows the same lines as that of Corollary 3.2. Consider the set U T = f (θ t ) ≤ β t for all t ≥ T }. Evidently E θ t+1 − θ t 2 1 U T = η 2 E 1 U T |g(θ t , ξ t )| 2 ≤ η 2 E 1 U T |∇f (θ t )| 2 + E 1 U T f (θ t ) ≤ η 2 (2C L + σ) β t . As in the proof of Corollary 3.2, we deduce that θ t converges pointwise almost everywhere on the set U T . As this is true for all T and P ∞ T =1 U T = 1. Theorem 3. 3 . 3Assume that there exists ε > 0 such that the local Lojasiewicz inequalityΛ f (θ) ≤ |∇f (θ)| 2 holds on the set U (ε) = {θ : f (θ) < ε} Theorem 3 . 6 . 36Let f : R m → [0, ∞) be a function such that (1) The set N := f −1 (0) is not empty. • The perturbation becomes negligible at infinity, i.e. there exists a positive definite symmetric matrix A Figure 1 . 1SGD with homogeneous noise and constant learning rate (left) or decaying learning rate (middle) vs SGD with ML noise (right) after 300 iterations (top row) and after 1000 iterations (bottom row). For 1000 realizations of SGD, we mark the final position by a red dot in the energy landscape and plot a histogram density estimator of the x-coordinate below. Remark A. 2 . 2If θ → h(θ, x i ) is linear, then D 2 h(θ, x i ) vanishes, i.e. rk(D 2 h) = 0. The large discrepancy between the conditions ensuring local convexity and global convexity is in fact necessary. Consider h(θ, x i ) = θ i for all i ≥ 1 and h(θ, x 1 ) = θ 1 +ε ψ(θ 1 , . . . , θ n ) ⇒ D 2 L y (θ) = I n×n + ε h(θ, x 1 ) − y 1 D 2 ψ 0 n×(m−n) 0 (m−n)×n 0 (m−n)×(m−n) . t i=0 E 1 Ei−1 f (θ i )holds. Appendix A. On the non-convexity of objective functions in deep learning Under general conditions, energy landscapes in machine learning regression problems have to be non-convex. The following result follows from Theorem 2.6.(1) If L y is convex for every y ∈ R n , the map θ → h(θ, x i ) is linear for all x i .(2) Assume that L y (θ) = 0 and that there exists 1 ≤ j ≤ n such that D 2 h(θ, x j ) has rank k > n. Then for every ε > 0, there exists y ∈ R m such that |y − y | < ε and D 2 L y (θ) has a negative eigenvalue.(3) Assume that L y (θ) = 0 and that there exists 1 ≤ j ≤ n such that D 2 h(θ, x j ) has rank k > n. Assume furthermore that the gradients ∇ θ h(θ, x 1 ), . . . , ∇ θ h(θ, x n ) are linearly independent. 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Stochastic gradient descent with noise of machine learning type. part ii: Continuous time analysis. arXiv:2106.02588 [cs.LG], 2021. Adagrad stepsizes: Sharp convergence over nonconvex landscapes. R Ward, X Wu, L Bottou, International Conference on Machine Learning. PMLRR. Ward, X. Wu, and L. Bottou. Adagrad stepsizes: Sharp convergence over nonconvex landscapes. In International Conference on Machine Learning, pages 6677-6686. PMLR, 2019. Linear convergence of adaptive stochastic gradient descent. Y Xie, X Wu, R Ward, International Conference on Artificial Intelligence and Statistics. PMLRY. Xie, X. Wu, and R. Ward. Linear convergence of adaptive stochastic gradient descent. In Inter- national Conference on Artificial Intelligence and Statistics, pages 1475-1485. PMLR, 2020. . Stephan Wojtowytsch, 155 Ireland Street, College Station, TXDepartment of Mathematics, Texas A&M University77840 Email address: [email protected] Wojtowytsch, Department of Mathematics, Texas A&M University, 155 Ireland Street, College Station, TX 77840 Email address: [email protected]

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{'abstract': 'Stochastic gradient descent (SGD) is one of the most popular algorithms in modern machine learning. The noise encountered in these applications is different from that in many theoretical analyses of stochastic gradient algorithms. In this article, we discuss some of the common properties of energy landscapes and stochastic noise encountered in machine learning problems, and how they affect SGD-based optimization.In particular, we show that the learning rate in SGD with machine learning noise can be chosen to be small, but uniformly positive for all times if the energy landscape resembles that of overparametrized deep learning problems. If the objective function satisfies a Lojasiewicz inequality, SGD converges to the global minimum exponentially fast, and even for functions which may have local minima, we establish almost sure convergence to the global minimum at an exponential rate from any finite energy initialization. The assumptions that we make in this result concern the behavior where the objective function is either small or large and the nature of the gradient noise, but the energy landscape is fairly unconstrained on the domain where the objective function takes values in an intermediate regime. References 22 Appendix A. On the non-convexity of objective functions in deep learning 23 Appendix B. Auxiliary observations on objective functions and Lojasiewicz geometry', 'arxivid': '2105.01650', 'author': ['Stephan Wojtowytsch '], 'authoraffiliation': [], 'corpusid': 233714813, 'doi': '10.1007/s00332-023-09903-3', 'github_urls': [], 'n_tokens_mistral': 29956, 'n_tokens_neox': 26051, 'n_words': 16348, 'pdfsha': '55237913974c2967f2966beb692b5aa15718658e', 'pdfurls': ['https://arxiv.org/pdf/2105.01650v2.pdf'], 'title': ['STOCHASTIC GRADIENT DESCENT WITH NOISE OF MACHINE LEARNING TYPE PART I: DISCRETE TIME ANALYSIS', 'STOCHASTIC GRADIENT DESCENT WITH NOISE OF MACHINE LEARNING TYPE PART I: DISCRETE TIME ANALYSIS'], 'venue': []}

arxiv

Comment on "Resilience of gated avalanche photodiodes against bright illumination attacks in quantum cryptography" 19 Jun 2011 Lars Lydersen Department of Electronics and Telecommunications Norwegian University of Science and Technology NO-7491TrondheimNorway University Graduate Center NO-2027KjellerNorway Vadim Makarov Department of Electronics and Telecommunications Norwegian University of Science and Technology NO-7491TrondheimNorway University Graduate Center NO-2027KjellerNorway Johannes Skaar Department of Electronics and Telecommunications Norwegian University of Science and Technology NO-7491TrondheimNorway University Graduate Center NO-2027KjellerNorway M K Lydersen A H Akhlaghi J Majedi V Skaar Makarov Comment on "Resilience of gated avalanche photodiodes against bright illumination attacks in quantum cryptography" 19 Jun 2011[1] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Nat. Photonics 4, 686 (2010). [2] L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Opt. Express 18, 27938 (2010). [3] C. Wiechers, L. Lydersen, C. Wittmann, D. Elser, J. Skaar, C. Marquardt, V. Makarov, and G. Leuchs, New J. Phys. 13, 013043 (2011). [4] L. Lydersen, N. Jain, C. Wittmann, Ø. Marøy, * [email protected] J. Skaar, C. Marquardt, V. Makarov, and G. Leuchs, arXiv:1106.2119 [quant-ph]. [5] L. Quantum key distribution (QKD) has initially been proven secure using ideal devices. However, implementations use imperfect devices available with current technology. Therefore, there are security proofs for QKD which model the devices to allow these imperfection, though at the expense of a lower secure key rate. To achieve provable security, it is crucial that the devices and implementations are verified to be within the models in the security proofs.Security loopholes have been found originating from discrepancies between the actual implementations and the models in the security proofs. For instance, one such discrepancy allows the tailored bright illumination attacks[1][2][3], recently shown also to be applicable against superconducting single-photon detectors[4,5]. In this case the loophole is caused by the response of qubit measurement devices (detectors) to swarms of qubits (bright illumination). The question is how to counter such loopholes.In their paper, Yuan et al. propose to counter these bright illumination attacks by monitoring the avalanche photodiode (APD) current for "anomalously high values"[6]. The robustness of this countermeasure is shown by arguing that previously proposed attacks do not work anymore. First of all, this leaves the challenge of determining what is "anomalously high". In order to achieve provable security, this threshold must originate from a security proof. Secondly, the fundamental issue, namely that the detector response deviates from the models in the security proofs [7], is not solved by this countermeasure.As discussed previously[8,9], practical QKD cannot become provably secure by intuitive countermeasures against known attacks. This approach also requires manufacturers to make frequent, possibly costly upgrades to their systems. Loopholes should instead be countered by modifying the implementation and/or the security proofs such that the devices are within the models of the security proofs. This is the only way practical QKD can obtain the provable security that makes it superior to classical key distribution schemes. This is also how loopholes have been handled previously: for example, the photonnumber splitting attack [10] led to more general security proofs[11]and eventually more efficient protocols to negate the decrease in the key rate[12]. In another example, detector efficiency mismatch [13], enabling for instance the time-shift attack[14,15], is now included in security proofs[16,17]. For the bright illumination attacks, we have proposed a secure detection scheme which integrates with security proofs[18]. In this scheme, a calibrated light source is used to verify the quantum efficiency in the center of the detector gate. Randomizing detection events outside the center of the gate provides a lower bound on the fraction of detections in the center of the gate.In this particular case, we have already shown that an eavesdropper using temporally tailored light of short pulses containing less than 120 photons can threaten the security of QKD[4]. This faint after-gate attack would not be detectable with the countermeasure proposed by Yuan et al., since the pulses would not cause an "anomalously high" current, but rather a current similar to the current caused by a single photon. Therefore, this serves as an example of the risk associated with closing loopholes in an intuitive way. Quantum key distribution (QKD) has initially been proven secure using ideal devices. However, implementations use imperfect devices available with current technology. Therefore, there are security proofs for QKD which model the devices to allow these imperfection, though at the expense of a lower secure key rate. To achieve provable security, it is crucial that the devices and implementations are verified to be within the models in the security proofs. Security loopholes have been found originating from discrepancies between the actual implementations and the models in the security proofs. For instance, one such discrepancy allows the tailored bright illumination attacks [1][2][3], recently shown also to be applicable against superconducting single-photon detectors [4,5]. In this case the loophole is caused by the response of qubit measurement devices (detectors) to swarms of qubits (bright illumination). The question is how to counter such loopholes. In their paper, Yuan et al. propose to counter these bright illumination attacks by monitoring the avalanche photodiode (APD) current for "anomalously high values" [6]. The robustness of this countermeasure is shown by arguing that previously proposed attacks do not work anymore. First of all, this leaves the challenge of determining what is "anomalously high". In order to achieve provable security, this threshold must originate from a security proof. Secondly, the fundamental issue, namely that the detector response deviates from the models in the security proofs [7], is not solved by this countermeasure. As discussed previously [8,9], practical QKD cannot become provably secure by intuitive countermeasures against known attacks. This approach also requires manufacturers to make frequent, possibly costly upgrades to their systems. Loopholes should instead be countered by modifying the implementation and/or the security proofs such that the devices are within the models of the security proofs. This is the only way practical QKD can obtain the provable security that makes it superior to classical key distribution schemes. This is also how loopholes have been handled previously: for example, the photonnumber splitting attack [10] led to more general security proofs [11] and eventually more efficient protocols to negate the decrease in the key rate [12]. In another example, detector efficiency mismatch [13], enabling for instance the time-shift attack [14,15], is now included in security proofs [16,17]. For the bright illumination attacks, we have proposed a secure detection scheme which integrates with security proofs [18]. In this scheme, a calibrated light source is used to verify the quantum efficiency in the center of the detector gate. Randomizing detection events outside the center of the gate provides a lower bound on the fraction of detections in the center of the gate. In this particular case, we have already shown that an eavesdropper using temporally tailored light of short pulses containing less than 120 photons can threaten the security of QKD [4]. This faint after-gate attack would not be detectable with the countermeasure proposed by Yuan et al., since the pulses would not cause an "anomalously high" current, but rather a current similar to the current caused by a single photon. Therefore, this serves as an example of the risk associated with closing loopholes in an intuitive way. . L Lydersen, C Wiechers, C Wittmann, D Elser, J Skaar, V Makarov, Nat. Photonics. 4686L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Nat. Photonics 4, 686 (2010). . L Lydersen, C Wiechers, C Wittmann, D Elser, J Skaar, V Makarov, Opt. Express. 1827938L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Opt. Express 18, 27938 (2010). . C Wiechers, L Lydersen, C Wittmann, D Elser, J Skaar, C Marquardt, V Makarov, G Leuchs, New J. Phys. 1313043C. Wiechers, L. Lydersen, C. Wittmann, D. Elser, J. Skaar, C. Marquardt, V. Makarov, and G. Leuchs, New J. Phys. 13, 013043 (2011). . L Lydersen, N Jain, C Wittmann, Ø Marøy, C Skaar, V Marquardt, G Makarov, Leuchs, arXiv:1106.2119quant-phL. Lydersen, N. Jain, C. Wittmann, Ø. Marøy, * [email protected] J. Skaar, C. Marquardt, V. Makarov, and G. Leuchs, arXiv:1106.2119 [quant-ph]. . L Lydersen, M K Akhlaghi, A H Majedi, J Skaar, V Makarov, arXiv:1106.2396quant-phL. Lydersen, M. K. Akhlaghi, A. H. Majedi, J. Skaar, and V. Makarov, arXiv:1106.2396 [quant-ph]. . Z L Yuan, J F Dynes, A J Shields, Appl. Phys. Lett. 98231104Z. L. Yuan, J. F. Dynes, and A. J. Shields, Appl. Phys. Lett. 98, 231104 (2011). There are security proofs including this detector response in their model of the receiver (for instance Ref. but they predict zero secret key rate for such receiversThere are security proofs including this detector response in their model of the receiver (for instance Ref. [17]), but they predict zero secret key rate for such receivers. . Z L Yuan, J F Dynes, A J Shields, Nat. Photonics. 4800Z. L. Yuan, J. F. Dynes, and A. J. Shields, Nat. Pho- tonics 4, 800 (2010). . L Lydersen, C Wiechers, C Wittmann, D Elser, J Skaar, V Makarov, Nat. Photonics. 4801L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, Nat. Photonics 4, 801 (2010). . G Brassard, N Lütkenhaus, T Mor, B C Sanders, Phys. Rev. Lett. 851330G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, Phys. Rev. Lett. 85, 1330 (2000). . D Gottesman, H.-K Lo, N Lütkenhaus, J Preskill, Quant. Inf. Comp. 4325D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, Quant. Inf. Comp. 4, 325 (2004). . W.-Y Hwang, Phys. Rev. Lett. 9157901W.-Y. Hwang, Phys. Rev. Lett. 91, 057901 (2003). . V Makarov, A Anisimov, J Skaar, Phys. Rev. A. 7422313V. Makarov, A. Anisimov, and J. Skaar, Phys. Rev. A 74, 022313 (2006); Erratum ibid. 7819905Erratum ibid. 78, 019905 (2008). . B Qi, C.-H F Fung, H.-K Lo, X Ma, Quant. Inf. Comp. 773B. Qi, C.-H. F. Fung, H.-K. Lo, and X. Ma, Quant. Inf. Comp. 7, 73 (2007). . Y Zhao, C.-H F Fung, B Qi, C Chen, H.-K Lo, Phys. Rev. A. 7842333Y. Zhao, C.-H. F. Fung, B. Qi, C. Chen, and H.-K. Lo, Phys. Rev. A 78, 042333 (2008). . C.-H F Fung, K Tamaki, B Qi, H.-K Lo, X Ma, Quant. Inf. Comp. 9131C.-H. F. Fung, K. Tamaki, B. Qi, H.-K. Lo, and X. Ma, Quant. Inf. Comp. 9, 131 (2009). . Ø Marøy, L Lydersen, J Skaar, Phys. Rev. A. 8232337Ø. Marøy, L. Lydersen, and J. Skaar, Phys. Rev. A 82, 032337 (2010). . L Lydersen, V Makarov, J Skaar, Phys. Rev. A. 8332306L. Lydersen, V. Makarov, and J. Skaar, Phys. Rev. A 83, 032306 (2011).

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{'abstract': 'Quantum key distribution (QKD) has initially been proven secure using ideal devices. However, implementations use imperfect devices available with current technology. Therefore, there are security proofs for QKD which model the devices to allow these imperfection, though at the expense of a lower secure key rate. To achieve provable security, it is crucial that the devices and implementations are verified to be within the models in the security proofs.Security loopholes have been found originating from discrepancies between the actual implementations and the models in the security proofs. For instance, one such discrepancy allows the tailored bright illumination attacks[1][2][3], recently shown also to be applicable against superconducting single-photon detectors[4,5]. In this case the loophole is caused by the response of qubit measurement devices (detectors) to swarms of qubits (bright illumination). The question is how to counter such loopholes.In their paper, Yuan et al. propose to counter these bright illumination attacks by monitoring the avalanche photodiode (APD) current for "anomalously high values"[6]. The robustness of this countermeasure is shown by arguing that previously proposed attacks do not work anymore. First of all, this leaves the challenge of determining what is "anomalously high". In order to achieve provable security, this threshold must originate from a security proof. Secondly, the fundamental issue, namely that the detector response deviates from the models in the security proofs [7], is not solved by this countermeasure.As discussed previously[8,9], practical QKD cannot become provably secure by intuitive countermeasures against known attacks. This approach also requires manufacturers to make frequent, possibly costly upgrades to their systems. Loopholes should instead be countered by modifying the implementation and/or the security proofs such that the devices are within the models of the security proofs. This is the only way practical QKD can obtain the provable security that makes it superior to classical key distribution schemes. This is also how loopholes have been handled previously: for example, the photonnumber splitting attack [10] led to more general security proofs[11]and eventually more efficient protocols to negate the decrease in the key rate[12]. In another example, detector efficiency mismatch [13], enabling for instance the time-shift attack[14,15], is now included in security proofs[16,17]. For the bright illumination attacks, we have proposed a secure detection scheme which integrates with security proofs[18]. In this scheme, a calibrated light source is used to verify the quantum efficiency in the center of the detector gate. Randomizing detection events outside the center of the gate provides a lower bound on the fraction of detections in the center of the gate.In this particular case, we have already shown that an eavesdropper using temporally tailored light of short pulses containing less than 120 photons can threaten the security of QKD[4]. This faint after-gate attack would not be detectable with the countermeasure proposed by Yuan et al., since the pulses would not cause an "anomalously high" current, but rather a current similar to the current caused by a single photon. Therefore, this serves as an example of the risk associated with closing loopholes in an intuitive way.', 'arxivid': '1106.3756', 'author': ['Lars Lydersen \nDepartment of Electronics and Telecommunications\nNorwegian University of Science and Technology\nNO-7491TrondheimNorway\n\nUniversity Graduate Center\nNO-2027KjellerNorway\n', 'Vadim Makarov \nDepartment of Electronics and Telecommunications\nNorwegian University of Science and Technology\nNO-7491TrondheimNorway\n\nUniversity Graduate Center\nNO-2027KjellerNorway\n', 'Johannes Skaar \nDepartment of Electronics and Telecommunications\nNorwegian University of Science and Technology\nNO-7491TrondheimNorway\n\nUniversity Graduate Center\nNO-2027KjellerNorway\n', 'M K Lydersen ', 'A H Akhlaghi ', 'J Majedi ', 'V Skaar ', 'Makarov '], 'authoraffiliation': ['Department of Electronics and Telecommunications\nNorwegian University of Science and Technology\nNO-7491TrondheimNorway', 'University Graduate Center\nNO-2027KjellerNorway', 'Department of Electronics and Telecommunications\nNorwegian University of Science and Technology\nNO-7491TrondheimNorway', 'University Graduate Center\nNO-2027KjellerNorway', 'Department of Electronics and Telecommunications\nNorwegian University of Science and Technology\nNO-7491TrondheimNorway', 'University Graduate Center\nNO-2027KjellerNorway'], 'corpusid': 118083469, 'doi': '10.1063/1.3658806', 'github_urls': [], 'n_tokens_mistral': 3735, 'n_tokens_neox': 3182, 'n_words': 1743, 'pdfsha': '5d387f5261234a3418a68ba362026d7fff6f74d3', 'pdfurls': ['https://arxiv.org/pdf/1106.3756v1.pdf'], 'title': ['Comment on "Resilience of gated avalanche photodiodes against bright illumination attacks in quantum cryptography"', 'Comment on "Resilience of gated avalanche photodiodes against bright illumination attacks in quantum cryptography"'], 'venue': []}

arxiv

Efficient long distance quantum communication Sreraman Muralidharan Department of Electrical Engineering Yale University 06511New HavenCTUSA Department of Applied Physics Yale University 06511New HavenCTUSA Linshu Li Jungsang Kim Department of Electrical and Computer Engineering Duke University 27708DurhamNCUSA Norbert Lütkenhaus Institute of Quantum computing University of Waterloo N2L 3G1WaterlooCanada Mikhail D Lukin Department of Physics Harvard University 02138CambridgeMAUSA Liang Jiang Department of Applied Physics Yale University 06511New HavenCTUSA Efficient long distance quantum communication (Dated: September 29, 2015) Despite the tremendous progress of quantum cryptography, efficient quantum communication over long distances (≥ 1000km) remains an outstanding challenge due to fiber attenuation and operation errors accumulated over the entire communication distance. Quantum repeaters (QRs), as a promising approach, can overcome both photon loss and operation errors, and hence significantly speedup the communication rate. Depending on the methods used to correct loss and operation errors, all the proposed QR schemes can be classified into three categories (generations). Here we present the first systematic comparison of three generations of quantum repeaters by evaluating the cost of both temporal and physical resources, and identify the optimized quantum repeater architecture for a given set of experimental parameters. Our work provides a roadmap for the experimental realizations of highly efficient quantum networks over transcontinental distances. arXiv:1509.08435v1 [quant-ph] I. INTRODUCTION First developed in the 1970s, fiber-optic communication systems have boosted the rate of classical information transfer and played a major role in the advent of the information age. The possibility to encode information in quantum states using single photons and transmit them through optical channels has led to the development of quantum key distribution (QKD) systems [1]. However, errors induced by the intrinsic channel attenuation, i.e. loss errors, become a major barrier for efficient quantum communication over continental scales, due to the exponential decay of communication rate [2]. In contrast to classical communication, due to the quantum no-cloning theorem [3], quantum states of photons cannot be amplified without any disturbance. In addition to loss errors, depolarization errors introduced by the imperfect optical channel can impair the quality of the single photon transmitted and hence the quantum information encoded. To overcome these challenges, quantum repeaters (QRs) have been proposed for the faithful realization of long-distance quantum communication [4]. The essence of QRs is to divide the total distance of communication into shorter intermediate segments connected by QR stations, in which loss errors from fiber attenuation can be corrected. Active mechanisms are also employed at every repeater station to correct operation errors, i.e. imperfections induced by the channel, measurements and gate operations. As illustrated in Fig. 1, loss errors can be suppressed by either heralded entanglement generation (HEG) [4,5] or quantum error correction (QEC) [6][7][8][9][10]. During HEG, quantum entanglement can be generated with techniques such as two-photon interference conditioned on the click * Equal contribution patterns of the detectors in between. Loss errors are suppressed by repeating this heralded proceDure until the two adjacent stations receive the confirmation of certain successful detection patterns via two-way classical signaling. Alternatively, one may encode the logical qubit into a block of physical qubits that are sent through the lossy channel and use quantum error correction to restore the logical qubit with only one-way signaling. Quantum error correcting codes can correct no more than 50% loss rates deterministically due to the no-cloning theorem [9,11]. To suppress operation errors, one may use either heralded entanglement purification (HEP) [12,13] or QEC [6][7][8][9][10] as listed in Fig. 1. In HEP, multiple lowfidelity Bell pairs are consumed to probabilistically generate a smaller number of higher-fidelity Bell pairs. Like HEG, to confirm the success of purification , two-way classical signaling between repeater stations for exchanging measurement results is required. Alternatively, QEC can correct operation errors using only one-way classical signaling, but it needs high fidelity local quantum gates. Based on the methods adopted to suppress loss and operation errors, we can classify various QRs into three categories as shown schematically in Fig. 2, which we refer to as three generations of QRs [14] [54]. Each generation of QR performs the best for a specific regime of operational parameters such as local gate speed, gate fidelity, and coupling efficiency. In this paper, we consider both the temporal and physical resources consumed by the three generations of QRs and identify the most efficient architecture for different parameter regimes. The results can guide the design of efficient long distance quantum communication links that act as elementary building blocks for future quantum networks. The paper is organized as follows: In the following section, we will briefly review the chacracterstics of three generations of QRs. In section III, we use the cost coefficient as an optimization metric to compare the QR per- formance, and study its dependence on the individual operational parameters including coupling efficiencies, gate fidelities, and gate speed. In section IV, we present a holistic view of the optimization and illustrate the parameter regions where each generation of QRs perform more efficiently than others. In section V, we analyze the advantages and challenges of each generation of QRs and discuss the experimental candidates for their realizations. II. THREE GENERATIONS OF QUANTUM REPEATERS The first generation of QRs uses HEG and HEP to suppress loss and operation errors, respectively [4,5]. This approach starts with purified high-fidelity entangled pairs with separation L 0 = L tot /2 n created and stored in adjacent stations. At k-th nesting level, two entangled pairs of distance L k−1 = 2 k−1 L 0 are connected to extend entanglement to distance L k = 2 k L 0 [15]. As practical gate operations and entanglement swapping inevitably cause the fidelity of entangled pairs to drop, HEP can be incorporated at each level of entanglement extension [12,13]. With n nesting levels of connection and purification, a high-fidelity entangled pair over distance L n = L tot can be obtained. The first generation of QRs reduces the exponential overhead in direct state transfer to only polynomial overhead, which is limited by the two-way classical signaling required by HEP between non-adjacent repeater stations. The communication rate still decreases polynomially with distance and thus becomes very slow for long distance quantum communication. The communication rate of the first generation of QRs can be boosted using temporal, spatial, and/or frequency multiplexing associated with the internal degrees of freedom for the quantum memory [5,16]. The second generation of QRs uses HEG to suppress loss errors and QEC to correct operation errors [6,7,17]. First, the encoded states |0 L and |+ L are fault-tolerantly prepared using the Calderbank-Shor-Steane (CSS) codes [55] and stored at two adjacent stations. Then, an encoded Bell pair |Φ + L = 1 √ 2 (|0, 0 L + |1, 1 L ) between adjacent stations can be created using teleportation-based non-local CNOT gates [18,19] applied to each physical qubit in the encoded block using the entangled pairs generated through HEG process. Finally, QEC is carried out when entanglement swapping at the encoded level is performed to extend the range of entanglement. The second generation uses QEC to replace HEP and therefore avoids the time-consuming two-way classical signaling between non-adjacent stations. The communication rate is then limited by the time delay associated with two-way classical signaling between adjacent stations and local gate operations. If the probability of accumulated operation errors over all repeater stations is sufficiently small, we can simply use the second generation of QRs without encoding. The third generation of QRs relies on QEC to correct both loss and operation errors [8][9][10]20]. The quantum information can be directly encoded in a block of physical qubits that are sent through the lossy channel. If the loss and operation errors are sufficiently small, the received physical qubits can be used to restore the whole encoding block, which is retransmitted to the next repeater station. The third generation of QRs only needs one-way signaling and thus can achieve very high communication rate, just like the classical repeaters only limited by local operation delay. It turns out that quantum parity codes [21] with moderate coding blocks (˜200 qubits) can efficiently overcome both loss and operation errors [9,20]. Note that the second and third generations of QRs can achieve communication rate much faster than the first generation over long distances, but they are technologically more demanding. For example, they require high fidelity quantum gates as QEC only works well when operation errors are below the fault tolerance threshold. The repeater spacing for the third generation of QRs is smaller compared to the first two generation of QRs because error correction can only correct a finite amount of loss errors. Moreover, quantum error correcting codes can correct only up to 50% loss error rates deterministically, which restricts the applicable parameter range for the third generation of QRs [9]. III. COMPARISON OF THREE GENERATIONS OF QRS To present a systematic comparison of different in terms of efficiency, we need to consider both temporal and physical resources. The temporal resource depends on the rate, which is limited by the time delay from the two-way classical signaling (first and second generations) and the local gate operation (second and third generations) [22]. The physical resource depends on the total number of qubits needed for HEP (first and second generations) and QEC (second and third generations) [9,23]. We propose to quantitatively compare the three generations of QRs using a cost function [9] related to the required number of qubit memories to achieve a given transmission rate. Suppose a total of N tot qubits are needed to generate secure keys at R bits/second, the cost function is defined as C(L tot ) = N tot R = N s R × L tot L 0 ,(1) where N s is the number of qubits needed per repeater station, L tot the total communication distance, and L 0 the spacing between neighboring stations. Since the cost function scales at least linearly with L tot , to demonstrate the additional overhead associated with L tot , the cost coefficient can be introduced as C (L tot ) = C/L tot ,(2) which can be interpreted as the resource overhead (qubits × time) for the creation of one secret bit over 1km (with target distance L tot ). Besides the fiber attenuation (with L att = 20 km), the cost coefficient also depends on other experimental parameters, in particular the coupling efficiency η c (see supplementary material), the gate error probability G , and the gate time t 0 [56]. For third generation QRs, we restrict the search only up to 200 qubits per logical qubit considering the complexity involved in the production of larger codes and for a fair comparison with second generation of QRs. For simplicity, we will assume that t 0 is independent of code size for small encoded blocks for second and third generation QRs. We will now investigate how C varies with these parameters for three generations of QRs, and identify the optimum generation of QR depending on the technological capability. A. Coupling efficiency The coupling efficiency η c accounts for the emission of photon from the memory qubit, coupling of the photon into the optical fiber and vice versa, and the final detection of photons. The first and second generations of QRs use HEG compatible with arbitrary coupling efficiency, while the third generation relies on QEC requiring the overall transmission (including the coupling efficiency η c and the channel transmission) to be at least above 50% [9,24]. As illustrated in Fig. 3, for high coupling efficiency (η c 90%) the third generation of QRs has an obvious advantage over the other generations due to the elimination of two-way classical signaling. As the coupling efficiency is reduced and approaches (∼ 90%) for quantum parity codes, the size of the coding block quickly increases and it becomes less favorable to use this approachsince we restrict the size of the encoded block for third generation of QRs. For coupling efficiency below ∼ 90%, the optimization chooses the first and second generations of QRs, and then C is proportional to η −2 c for HEG protocols heralded by two-photon detector click patterns. If the gate error becomes large (e.g., G = 10 −2 ), the capability of correcting loss errors will be compromised for the third generation QRs. Similar trends can be observed as we fix G and increase L tot . In contrast to the third generation QRs, the first and second generation QRs with HEG works well even for low coupling efficiencies. First Genera+on QR Second Genera+on QR Third Genera+on QR Schema+c Architecture Loss Error HEG (two--way signaling) HEG (two--way signaling) QEC (one--way signaling) Opera+on Error HEP (two--way signaling) QEC (one--way signaling) QEC (one--way signaling) Procedure 1. Create entangled pairs over L 0 between adjacent sta?ons 2. At k--th level, connect two pairs over L k and extend to L k+1 =2L k , followed by HEP. 3. AGer n nes?ng levels, obtain high-fidelity pair over L tot =2 n ×L 0 Inverse Rate (1/R) Poly(L tot /c) L 0 /c × PolyLog(L tot ) t 0 × PolyLog(L tot ) Cost Coefficient (C') Poly(L tot ) PolyLog(L tot ) PolyLog(L tot ) Ri Ri+1 Ri+2 Ri Ri+1 Ri+2 Ri Ri+1 Ri+2 Ri Ri+1 Ri+2 Ri+1 Ri+2 Ri+3 Ri Ri+1 Ri+2 Ri+3 Ri Ri+3 Ri Ri Ri+1 Ri+2 Ri+3 Ri+4 Physical qubit Encoded qubit block B. Speed of quantum gates We investigate the performance of different generations of QRs for different gate times in the range 0.1µs ≤ t 0 ≤ 100µs. As shown in Fig. 4, for high speed quantum gates (t 0 1µs) the third generation of QRs provides a very fast communication rate, which makes it the most favorable protocol, with C ∝ t 0 . For slower quantum gates (t 0 10µs), the gate time becomes comparable or even larger than the delay of two-way classical signaling between adjacent stations (t 0 L0 c ≈ Latt c ); as the third generation of QRs loses its advantage in communication rate, the second generation of QRs with less physical resources becomes the optimized QR protocol, with almost constant C for a wide range of t 0 . We notice that for small gate error and intermediate distance (e.g., G = 10 −4 and L tot = 1000km appeared in Fig. 3a and 4a), encoding might not even be necessary for the second generation of QRs, because the accumulated errors over the entire repeater network are within the tolerable range for quantum communication ( G Ltot Latt 0.1). However, for larger error probability or longer distances ( G Ltot Latt 0.1), encoding is required for the second generation QRs[57]. The cost coefficient for the first generation of QRs (C > 1 qubit×sec sbit×km ) lies beyond the scope of Fig. 4, with little dependence on t 0 that is mostly negligible compared to the two-way classical signaling between non-adjacent stations ( Ltot c > 10ms). C. Gate fidelity The three generations of QRs have different thresholds in terms of gate error probability G . The first generation relies on HEP with the highest operation error threshold up to about 3% [4]. The second and third generations both use QEC to correct operation errors, with error correction thresholds of approximately 1% [25]. The gate error threshold of the second generation is slightly lower than that of the third generation, because of the extra gates required for teleportation-based non-local CNOT gates and entanglement swapping in the second generation of QRs (See supplementary material). However, since we restrict the size of the encoded block for third generation of QRs, C' increases exponentially with G slightly below the theoretical threshold of quantum parity codes. As illustrated in Fig. 5, for almost perfect coupling efficiency (e.g. η c = 100%) and fast local operation (t 0 = 1µs), the third generation using QEC to correct both fiber attenuation loss and operation errors is the optimized protocol for moderate gate errors. For lower coupling efficiencies (e.g. η c = 30% and 80%) with too many loss errors for the third generation to tolerate, the first and second generations with HEG yield good performance. As G increases, there is a transition at about 0.8%(0.6%) below which the second generation is more favorable for 1000km (10000km). ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ϵ G =10 -2 ▲ ϵ G =10 -3 ■ ϵ G =10 -4 0.1 0.2 0.5 1 10 -4 10 -3 10 -2 0.1 1 10 η c C' ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ϵ G =10 -2 ▲ ϵ G =10 -3 ■ ϵ G =10 -4 0.1 0.2 0.5 1 10 -4 10 -2 1 100 η c C' 1G 2G(NC) 2G(C) 3G ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0.80 0.85 0.90 0.95 1.00 10 -4 10 -3 10 -2 ηc C' ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0. IV. OPTIMUM GENERATION OF QRS Based on the above analysis of the cost coefficient that depends on the coupling efficiency η c , the gate time t 0 , and the gate infidelities G , we may summarize the results using the bubble plot and the region plot in the three-dimensional parameter space, as shown in Fig. 6. The bubble color indicates the associated optimized QR protocol, and the bubble diameter is proportional to the 1% and (η c 90% or t 0 1µs)], the second generation with encoding is more favorable; (II.B) For low gate error probability, but low coupling efficiency or slow local operation [ G 0.1 Latt Ltot and (η c 90% or t 0 1µs)], the second generation without encoding is more favorable; (III) For high coupling efficiency, fast local operation, and low gate error probability (η c 90%, t 0 1µs, G 1%), the third generation becomes the most favorable scheme in terms of the cost coefficient. ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ϵ G =10 -3 , η c =90% ▲ ϵ G =10 -4 , η c =90% 10 -7 10 -6 10 -5 10 -4 10 -3 10 -5 10 -4 10 -3 10 -2 t 0 (s) C' ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ϵ G =10 -3 , η c =90% ▲ ϵ G =10 -4 , η c =90% 10 -7 10 -6 10 -5 10 -4 10 -3 10 -5 10 -4 10 -3 10 -2 t 0 (s) C' 1G 2G(NC) 2G(C) 3G (a) (b) V. DISCUSSIONS So far, we have mostly focused on the standard pro-ceDure of HEG and HEP [4,5,12,13], the CSS-type quantum error correcting codes, and the teleportationbased QEC, which all can be improved and generalized. We have also assumed the simple cost function that scales linearly with the communication time and the total num- ber of qubits. In practice, however, the cost function may have a more complicated dependence on various resources. Nevertheless, we may extend our analysis by using more realistic cost functions to compare various QR protocols. As we bridge the architectural design of QRs and the physical implementations, we may include more variations of HEG, HEP as well as QEC (such as all optical schemes [14,26]) and use more realistic cost functions, while the general trend and different parameter regions should remain mostly insensitive to these details. The classification of QR protocols with different performance in the parameter space also provides a guideline for optimized architectural design of QRs based on technological capabilities, which are closely related to physical implementations, including atomic ensembles, trapped ions, NV centers, quantum dots, nanophotonic devices, etc. (1) The atomic ensemble can be used as quantum memory with high coupling efficiency (> 80% [27,28]) and compatible with HEG for the first generation of QRs [29]. An important challenge for ensemblebased QRs is the use of non-deterministic quantum gates, which can be partly compensated by multiplexing various internal modes of the ensemble memory [5,16]. Alternatively, the atomic ensemble approach can be sup-plemented by deterministic atom-photon and atom-atom gates using Rydberg blockade, which can dramatically improve the performance of atomic ensemble approaches and make them compatible with both first and second generations of QRs [30,31]. (2) The trapped ions, NV centers, and quantum dots all can implement local quantum operations deterministically [32][33][34][35][36], as well as HEG [37][38][39][40]. In principle, they are all compatible with the first and second generations of QRs. Although the coupling efficiency is relatively low for single emitters compared to ensembles, it can be boosted with cavity Purcell enhancement [41] (by two orders of magnitude). With high coupling efficiency [42,43], these systems can also be used for the third generation of QRs. (3) The system of nanophotonic cavity with individual trapped neutral atoms has recently demonstrated quantum optical switch controlled by a single atom with high coupling efficiency [44,45], which can be used for deterministic local encoding and QEC for the third generation of QRs. Realization of similar techniques with atom-like emitters are likewise being explored. (4) The opto-electromechanical systems have recently demonstrated efficient coherent frequency conversion between optical and microwave photons [46,47] and can potentially enable using superconducting systems [48] for reliable fast local quantum gates for QRs. ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ η c =30% η c =80% ■ η c =100% 0.10 -2 0.1 1 ϵ G C' ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ η c =30% η c =80% ■ η c =100% 0. VI. CONCLUSION In this work, we have classified various QR protocols into three generations based on different methods for suppressing loss and operation errors. Introducing the cost function to characterize both temporal and physical resources, we have systematically compared three generations of QRs for various experimental parameters, including coupling efficiency, gate fidelity, and gate times. There are different parameter regions with drastically different architectural designs of quantum repeaters with different possible physical implementations. Our work will provide a guideline for the optimal design of quantum networks and help in the extension of quantum network of clocks [49], interferometric telescopes [50] and distributed quantum computation [19,51] to global scales. In the future, the integration of different generations of QRs will enable the creation of a secure quantum internet [52]. [54] Note that the combination of QEC for loss errors and HEG for operation errors is sub-optimum compared to the other three combinations. [55] CSS codes are considered because of the fault tolerant implementation of preparation, measurement, and encoded CNOT gate [6,53] [56] For simplicity, we assume that the fidelity of physical Bell pairs F0 = 1− 5 4 g achieved with entanglement purification and measurement error probability m = g 4 through a verification proceDure. [57] When G increases from 10 −4 to 10 −3 , the cost coefficient for the second generation of QR without encoding increases by almost a factor of 10 ( Fig. 4a), while the change is less significant for the second and third generations of QRs with encoding (Fig. 4). This is because at the logical level, the change in the effective logical error probability is suppressed for the given set of parameters. Supplementary Material Descriptions of error models a) Two-qubit gate error Local two-qubit gates, e.g. CNOT gate, are characterized by the gate infidelity G . With probability 1 − G the desired two-qubit gate is applied, while with probability G the state of the two qubits becomes a maximally mixed state. Mathematically the imperfect two-qubit operation on qubit i and j can be expressed as U ρU † = (1 − G )U ij ρU † ij + G 4 T r ij [ρ] ⊗ I ij ,(3) where U ij stands for perfect two-qubit operation on qubit i and j, T r ij [ρ] the partial trace over qubit i and j, and I ij the identity operator for qubits i and j. b) Measurement error Qubit measurement error is described by the measurement infidelity ξ, which is the probability of a wrong measurement. The error models for projective measurements of states |0 and |1 are P 0 = (1 − ξ)|0 0| + ξ|1 1| P 1 = (1 − ξ)|1 1| + ξ|0 0|.(4) The measurement error can be suppressed by introducing an ancillary qubit for measurement and measuring both the data and the ancillary qubits. If the measurement outcomes don't match, it can be considered as a loss error on that qubit. The contribution of the measurement error to the overall loss error is negligible given the range of the gate error rates (10 −4 − 10 −2 ) we are considering; if they match, then the effective measurement error is given by G 4 [8]. c) Memory life-times In the calculations, the memory qubits are assumed to be perfect, i.e. their life-time is tremendously longer than any characteristic times involved in each scheme. In this sense the most demanding scheme is the first generation, which is optimum at long communication distances, e.g. L tot = 10 4 km, and high gate errors. The required coherence time τ for memory qubits is at least limited by the fundamental two-way classical communication time between Alice and Bob t c1 = L tot c ∼ 50ms,(5) where c = 2 × 10 5 km/s is the speed of light in telecom-wavelength optical fiber. Recent experiments with trapped ions, superconducting qubits, solid state spins and neutral atoms have demonstrated quantum memory life-times approaching or exceeding this characteristic value. For the second generation, the characteristic communication time is t c2 = L att c ∼ 100µs(6) where L att = 20km at telecomm wavelength. The corresponding coherence times are far less demanding than that of the first generation, which relieves the strong life-time requirements on memory qubits and makes the two generations more plausible in practice. Note that when the operation time t 0 becomes comparable or larger than the characteristic communication time, it is then the operation time t 0 that puts limits to the coherence time τ . Third generation QRs are not limited by the two way communication time because it is a fully one way communication scheme. Quantum entanglement generation, purification and connection a) Generation of elementary entangled pairs Heralded entanglement generation with two-photon detection Using two photon-detection in the middle [1,2], the success probability of one trial of generating entanglement between two memory qubits in neighboring stations is p = 1 2 η 2 c e −L0/Latt ,(7) where L 0 is the spacing between neighboring stations and η c is the coupling efficiency accounting for the emission of the photon from the memory qubit, "upload" of the photon into the optical fiber, "download" of the photon from the fiber and the final detection of photons. Fidelity Practically an entangled pair generated between neighboring stations may not be a perfect Bell pair and its state is characterized by a density matrix ρ = a|ϕ + ϕ + | + b|ϕ − ϕ − | + c|ψ + ψ + | + d|ψ − ψ − |,(8) where |ϕ ± = 1 √ 2 (|00 ± |11 ) and |ψ ± = 1 √ 2 (|01 ± |10 ) are the four Bell states. The fidelity of the pair is thus defined as F ≡ a = ϕ + |ρ|ϕ + ,(9) Both the first and second generations QRs rely on generating elementary entangled pairs between neighboring repeater stations and then extending the entanglement to longer distances. With the technique of HEP (see the next section), the fidelity of entangled pairs can be boosted to near-unity at the cost of reducing the total number of them and purified pairs can be connected to obtain longer entangled pairs or used as resources for the implementation of remote quantum gates. However, with imperfect quantum operations and measurements, there is an upper bound on the fidelity of entangled pairs even with entanglement purification. It is in general, a function of the density matrix of raw Bell pairs ρ, gate infidelity G and measurement infidelity ξ, and depends on the specific purification protocol one uses. Using Deutsch purification protocol (see the next section), the value of this upper bound can be approximated as F u.b. = 1 − 5 4 G − ( 9 4 ξ + 19 4 G ) G + O( G, ξ) 3 ≈ 1 − 5 4 G ,(10) in which we assume depolarized states for input raw Bell pairs. This approximate expression holds at small G s ( 1%). In our calculations and comparison, the temporal resources and physical resources consumed in obtaining purified pairs at the elementary level are not accounted; elementary entangled pairs generated between neighboring stations are assumed to directly take this asymptotic value. The associated additional cost in the purification can be easily added as an overhead into the cost function. b) Deutsch and Dür purification protocols In general, despite differences in experimental requirements and efficiencies, the choice of purification protocols will not change the big picture. In this paper, we mainly consider two widely used entanglement purification protocols: Deutsch protocol [3] and Dür [5] protocol. Compared to other purification schemes, the Deutsch protocol reaches higher fidelities with fewer rounds of purification so its upper bound is used as the fidelity of elementary pairs between neighboring stations. The Dür purification protocol is very similar to the Deutsch protocol, except that one of the two pairs, call auxiliary pair, is never discarded and will be prepared in the same state in each round of purification. This is sometimes also call "entanglement pumping". The Dür purification protocol saves qubit resources by keeping making use of the auxiliary pair, while the state preparation in each purification round is costly in time as a trade-off and if one round fails, the whole purification needs to be started over again. Here we study the purification with two input pairs characterized by density matrices ρ 1 and ρ 2 (ρ 1 = ρ 2 in the case of Deutsch protocol). As mentioned in the previous section, we express the density matrices in the Bell basis {|ϕ + , |ϕ − , |ψ + , |ψ − }. With input states{a 1 , b 1 , c 1 , d 1 } and {a 2 , b 2 , c 2 , d 2 }, in the presence of gate infidelity G and measurement infidelity ξ, the success probability P and the purified state characterized by the diagonal elements {a, b, c, d} are the following P = (1 − G ) 2 {[ξ 2 + (1 − ξ) 2 ][(a 1 + d 1 )(a 2 + d 2 ) + (b 1 + c 1 )(c 2 + b 2 )] + 2ξ(1 − ξ)[(a 1 + d 1 )(b 2 + c 2 ) + (b 1 + c 1 )(a 2 + d 2 )]} + 1 2 [1 − (1 − G ) 2 ] a = 1 P {(1 − G ) 2 [(ξ 2 + (1 − ξ) 2 ](a 1 a 2 + d 1 d 2 ) + 2ξ(1 − ξ)(a 1 c 2 + d 1 b 2 )} + 1 8 [1 − (1 − G ) 2 )] b = 1 P {(1 − G ) 2 [(ξ 2 + (1 − ξ) 2 ](a 1 d 2 + d 1 a 2 ) + 2ξ(1 − ξ)(a 1 b 2 + d 1 c 2 )} + 1 8 [1 − (1 − G ) 2 )] c = 1 P {(1 − G ) 2 [(ξ 2 + (1 − ξ) 2 ](b 1 b 2 + c 1 c 2 ) + 2ξ(1 − ξ)(b 1 d 2 + c 1 a 2 )} + 1 8 [1 − (1 − G ) 2 )] d = 1 P {(1 − G ) 2 [(ξ 2 + (1 − ξ) 2 ](b 1 c 2 + c 1 b 2 ) + 2ξ(1 − ξ)(b 1 a 2 + c 1 d 2 )} + 1 8 [1 − (1 − G ) 2 )](11) c) Entanglement Swapping Entanglement swapping is used in the first generation and second generation without encoding to extend the distance of entanglement. With imperfect CNOT operation and measurements, the diagonal elements {a, b, c, d} in the Bell basis of the resulting state obtained from connecting deterministically the input pairs {a 1 , b 1 , c 1 , d 1 } and {a 2 , b 2 , c 2 , d 2 } is a = (1 − G ){(1 − ξ) 2 (a 1 a 2 + b 1 b 2 + c 1 c 2 + d 1 d 2 ) + ξ(1 − ξ)[(a 1 + d 1 )(b 2 + c 2 ) + (b 1 + c 1 )(a 2 + d 2 )] +ξ 2 (a 1 d 2 + d 1 a 2 + b 1 c 2 + c 1 b 2 )} + G 4 b = (1 − G ){(1 − ξ) 2 (a 1 b 2 + b 1 a 2 + c 1 d 2 + d 1 c 2 ) + ξ(1 − ξ)[(a 1 + d 1 )(a 2 + d 2 ) + (b 1 + c 1 )(b 2 + c 2 )] +ξ 2 (a 1 c 2 + c 1 a 2 + b 1 d 2 + d 1 b 2 )} + G 4 c = (1 − G ){(1 − ξ) 2 (a 1 c 2 + c 1 a 2 + b 1 d 2 + d 1 b 2 ) + ξ(1 − ξ)[(a 1 + d 1 )(a 2 + d 2 ) + (b 1 + c 1 )(b 2 + c 2 )] +ξ 2 (a 1 b 2 + b 1 a 2 + c 1 d 2 + d 1 c 2 )} + G 4 d = (1 − G ){(1 − ξ) 2 (a 1 d 2 + d 1 a 2 + c 1 b 2 + b 1 c 2 ) + ξ(1 − ξ)[(a 1 + d 1 )(b 2 + c 2 ) + (b 1 + c 1 )(a 2 + d 2 )] +ξ 2 (a 1 a 2 + b 1 b 2 + c 1 c 2 + d 1 d 2 )} + G 4(12) Note that deterministic entanglement swapping is crucial for long distance quantum communication using QRs. Otherwise the success probability of entangling two qubits separated by L tot drops exponentially as L tot increases. Implementation and optimization a) First generation The first generation of QRs corrects photon loss and operation errors with HEG and HEP, respectively. To overcome the exponential decay in key generation rate induced by photon loss, the total distance L tot is divided into 2 n segments (n is called nesting level [1,5]) and elementary entangled pairs are generated within each segment, i.e. over repeater spacing L 0 = Ltot 2 n . An entangled pair covering the total distance L tot can be generated via n levels of entanglement swapping: at each level, two adjacent entangled pairs are connected so that an entangled pair over twice the distance is produced. However, entanglement swapping necessarily reduces the fidelity of the entangled pairs due to the following two reasons: 1) entanglement swapping involves CNOT operation and measurements, which themselves are imperfect in reality and will introduce noise. 2) Despite imperfect operations, the connection of two imperfect Bell pairs gives a pair with lower fidelity. So multiple rounds of entanglement purification may need to be incorporated at each level to maintain the fidelity, so that the final pair covering L tot is sufficiently robust for secure key distribution. In the optimization, to determine the best scheme from the first generation QRs, we first fix • Total distance L tot , • Coupling efficiency η c In carrying out the time resource consumed in generating one remote pair, we adopted similar approximations and derivations as in a previous work [1], with three major changes: 1. We allow arbitrary number of rounds of purification at each level in the optimization, and hence schemes selected could potentially be better optimized. 2. Without losing generality, we use four-state protocol, instead of six-state protocol, in calculating the secure fraction at the asymptotic limit. 3. For long-distance quantum communication, we only consider deterministic entanglement swapping and take into account the gate operation time t 0 . For a given set of N and − → M , detailed derivations of expressions of temporal and physical resource consumed in Deutsch and Dür are given below. Deutsch et al. entanglement purification protocol The temporal resource to generate one bit of raw key, T Deu , can be calculated as follows: T Deu = T 0 · { ( 3 2 ) N ( N −1 x=0 A Deu [N-x])( 1 P 0 A Deu [0] + B Deu [0]) + N y=1 ( 3 2 ) N −y B Deu [y] N −(y+1) x=0 A Deu [N-x] + t 0 T 0 N y=1 ( 3 2 ) N −y N −y x=0 A Deu [N-x]},(13) where A Deu [i] ≡ ( 3 2 ) Mi Mi−1 x=0 1 P Deu (M i − x, i) B Deu [i] ≡ ( t 0 T 0 + 2 i ) Mi−1 y=0 ( 3 2 ) y y x=0 1 P Deu (M i − x, i) .(14) Here, A Deu [i] accounts for the time consumed in the preparation of Bell pairs for purification, and B Deu [i] includes gate operation time and the two-way classical signaling time associated with confirming the success of purification. P Deu (i, j) is the success probability of the i th -round of purification at the j th nesting level with Deustch purification protocol. T 0 = L0 c is the time unit for the two-way classical signaling between neighboring stations, where c = 2 × 10 5 km/s in optical fiber. The secure key generation rate, R secure (sbit/s), can be written as R Deu secure = r secure · 1 T Deu ,(15) where r secure is the asymptotic secure fraction and in the four-state protocol can be approximately expressed as r secure = M ax[1 − 2h(Q), 0],(16) where Q = Q X +Q Z 2 is the average quantum bit error rate (QBER) and h(Q) = −Q log 2 Q − (1 − Q) log 2 (1 − Q) is the binary entropy function. Q X/Z can be calculated from the density matrix of the entangled shared by Alice and Bob in the end. The physical resource, in terms of the number of memory qubits, consumed at half a station can be written as Z Deu = 2 N +1 i=0 Mi ,(17) and the cost function becomes C = 2 N +1 · Z Deu R Deu secure .(18) Dür et al. entanglement purification protocol The temporal resource to generate one bit of raw key, T Dür , can be calculated as follows: T Dür = T 0 · { ( 3 2 ) N ( 1 P 0 A Dür [0] + B Dür [0])( N −1 x=0 A Dür [N-x]) + N y=1 ( 3 2 ) N −y B Dür [y] N −(y+1) x=0 A Dür [N-x] + t 0 T 0 N y=1 ( 3 2 ) N −y N −y x=0 A Dür [N-x]},(19) where, A Dür [i] ≡ Mi−1 x=0 1 P Dür (M i − x, i) + Mi−1 y=0 y x=0 1 P Dür (M i − x, i) B Dür [i] ≡ ( t 0 T 0 + 2 i ) Mi−1 y=0 y x=0 1 P Dür (M i − x, i) .(20) Here, A Dür [i] accounts for the time consumed in the preparation of Bell pairs for purification (notice the extra term due to entanglement pumping), and B Dür [i] includes gate operation time and the two-way classical signaling time associated with confirming the success of purification. P Dür (i, j) is the success probability of the i th -round of purification at the j th nesting level with Dür purification protocol. Because of the unique entanglement pumping mechanism, the physical resource consumed at half a station is reduced compared to Deustch purification protocol and expressed as Z Dür = N + 2 − |{M i : M i = 0}|.(21) The derivations of secure key generation rate and hence cost function are similar to those in the previous section. b) Second generation without encoding The second generation of QRs relies on generating encoded Bell pairs between neighboring stations and performing error correction during entanglement swapping at the encoded level. With encoding and error correction, physical gate error rates and imperfections in raw Bell pairs are suppressed to higher orders and thus entanglement can be extended to very long distances with high fidelity. However, if we are interested in the best schemes at low gate error rate 10 −3 and total distance L tot ∼ 10 3 km, the encoding may turn out unnecessary and resources can be saved by simply generating elementary pairs between neighboring stations and implementing entanglement swapping. Fixing the same parameters as the previous section, we vary the following parameters The secure key generation rate, R secure (sbit/s), can be written as R secure = [1 − P rob(0, n E.G. )] L tot L 0 · r secure n E.G. · (T 0 + t 0 ) ,(22) where basic communication time T 0 = L0 c and r secure follows the same definition above. P rob(i, n 0 ) = M n 0 p n0 (1 − p) M −n0 is the probability to generate i elementary pairs with M qubits in the two half nodes after n 0 rounds of entanglement generation. Therefore, 1 − P rob(0, n E.G. ) means the probability to have at least one entangled pair between two neighboring stations. Note that frequency and spatial multiplexing may needed to be incorporated during the transmission of flying qubits and entanglement swapping, respectively. The cost function that will be optimized can be written as C = 2M · Ltot L0 R secure .(23) c) Second generation with encoding For second generation QRs with encoding, encoded Bell pairs are created between neighboring repeater stations and later an encoded entanglement swapping operation is performed at every QR station to generate an encoded Bell pair between distant stations. As in the case of first generation QRs, Bell pairs are generated using HEG between neighboring stations with a high fidelity. These Bell pairs are used as a resource to perform teleportation based CNOT gates between neighboring stations, thereby realizing an encoded CNOT operation between neighboring QR stations. The depolarization error on the data qubits can be modeled as ρ = E (ρ) = (1 − d ) ρ + d 4 3 k=0 σ k ρσ k .(24) The probability of an error being detected in any one (X or Z) of the measurements is given by, X/Z = d + G + 2ξ + 2 3 (1 − F 0 ) + O( G , ξ) 2(25) Any [[N, k, 2t + 1]] CSS code can correct up to t X-errors and t Z-errors respectively. Taking this into account, the probability of correctly and incorrectly decoding the qubit are given by, p 2G correct(X/Z) = t k=0 N k k X/Z (1 − X/Z ) (N −k) ,(26)p 2G incorrect(X/Z) = N k=t+1 N k k X/Z (1 − X/Z ) (N −k)(27) respectively. Accounting for logical errors in odd number of repeater stations, quantum bit error rates for X and Z basis after R repeater stations is given by, Q (X/Z) = 1 2 1 − p 2G correct(X/Z) − p 2G incorrect(X/Z) R .(28) Where the effective quantum bit error rate is given byQ = 1 2 (Q X + Q Z ) . The success probability of the protocol is conditioned on having enough Bell pairs between neighboring stations to apply a teleportation based CNOT gate. We can then obtain the key generation rates similar to the case of second generation without encoding. We consider the Steane [ [7,1,3]] code, Golay [ [23,1,7]] code and the QR [[103, 1,19]] codes in our optimization. d) Third generation QRs Third generation QRs rely on encoded qubits to relay data from one repeater station to the next where an error correction operation is performed. Since there is just one round of upload and download between the memory qubit and the fiber for third generation QRs, the probability that the photon reaches the neighboring station is given by η c e −L0/Latt . Unlike second generation QRs with encoding, teleportation based error correction (TEC) is performed within every repeater station locally for third generation QRs. Teleportation based error correction requires an encoded CNOT gate between the incoming encoded qubit block (with loss and operation errors) and encoded qubit block (with no loss errors) at every repeater station. R is the incoming encoded block and S is the encoded block from the encoded Bell pair. The depolarization errors on blocks R and S can be modeled as, ρ R = E R (ρ RS ) = η (1 − d ) ρ R + η d 4 3 k=0 σ k ρσ k + (1 − η) |vac vac| (29) ρ S = E S (ρ RS ) = (1 − d ) ρ S + d 4 3 k=0 σ k ρσ k ,(30) To be consistent with [7], we make the following assumptions in our analysis. 1) Errors are not propagated between repeater stations. 2) Each qubit has an independent error. The effective X/Z error detected at the measurement (Y errors are detected in both X and Z measurements) is given by, X/Z = ( d + G 2 + ξ)η + O( G, , ξ) 2 .(31) We consider (n,m) quantum parity codes given by |± L = 1 2 n/2 (|0 ⊗m ± |1 ⊗m ) ⊗n , in our analyses. The outcome of the measurement of the logical operators X L and Z L for TEC can be determined through a majority voting procedure discussed in detail in [7]. There are three possible outcomes of the majority voting procedure: a) Heralded failure leading to the inability to perform a majority voting with probability p 3G unknown(X/Z) . b) Perform a majority voting and correctly decoding the qubit with probability p 3G correct(X/Z) . c) Perform a majority voting and incorrectly decoding the qubit with probability p 3G incorrect(X/Z) . Here, we treat the three events as independent for the measurement of the logical operators for simplicity. The success probability accounting for no heralded failure in any one of the R repeater stations is given by, P succ = (1 − p 3G unknown(X/Z) ) R Accounting for errors in odd number of repeater stations, the quantum bit error rate for X/Z bases is given by, Q (X/Z) = 1 2   1 − p 3G correct(X/Z) − p 3G incorrect(X/Z) p 3G correct(X/Z) + p 3G incorrect(X/Z) R  (34) The asymptotic secure key generation rates is given by, R secure = M ax[ P succ t 0 {1 − 2h(Q)}, 0],(35) where t 0 is the time taken to apply local operations. The cost function for the (n, m) quantum parity codes is given by C = 2m · n Ltot L0 R secure .(36) For a fair comparison with second generation QRs, where a largest code of [[103, 1, 19]] code was used, we restrict the maximum qubits for the third generation quantum repeaters to be 200. We restrict the search of (n, m) quantum parity codes within the range 2 ≤ (m, n) ≤ 20. FIG. 1 : 1A list of methods to correct loss and operation errors. Depending on the methods used to correct the errors, QRs are categorized into three generations. Flying qubit (FIG. 2 : 2Comparison of three generations of QRs. FIG. 3 : 3The optimized cost coefficient C' as a function of ηc for t0 = 1µs, G ∈ 10 −2 , 10 −3 , 10 −4 , and a) Ltot = 1000km, b) Ltot = 10, 000km. The associated optimized QR protocols are indicated in different colors. FIG. 4 : 4The optimized cost coefficient C' as a function of t0 for ηc = 0.9, G ∈ 10 −3 , 10 −4 , and a) Ltot. = 1000km, b) Ltot. = 10, 000km. The associated optimized QR protocols are indicated in different colors. cost coefficient. The parameter space can be divided into the following regions: (I) For high gate error probability ( G 1%), the first generation dominates; (II.A) For intermediate gate error probability, but poor coupling efficiency or slow local operation [0.1 Latt Ltot G FIG. 5 : 5The optimized cost coefficient C' as a function of G for t0 = 1µs, ηc ∈ {30%, 80%, 100%}, and a) Ltot = 1000km, b) Ltot = 10, 000km. The associated optimized QR protocols are indicated in different colors. FIG. 6 : 6The bubble plot comparing various QR protocols in the three-dimensional parameter space spanned by ηc, G, and t0, for a) Ltot = 1000km and b) Ltot = 10, 000km. The bubble color indicates the associated optimized QR protocol, and the bubble diameter is proportional to the cost coefficient. The region plots (c) and (d) showing the distribution of different optimized QR protocol in the three dimensional parameter space for Ltot = 1000km and Ltot = 10, 000km respectively. The region plot (c) contains a yellow region of second generation with encoding, which can be verified in a bubble plot with a finer discretization of G. quantum information (Cambridge University Press, Cambridge, U.K; New York, 2000). •• Gate error rate G , and thus the fidelity of elementary Bell pairs F 0 Gate time t 0 and we vary the following parameters: • Number of nesting level: N • Number of rounds of purification at each level: − → M = (M 0 , M 1 , M 2 , · · · , M N ) • Choice of entanglement purification protocol: Deutsch or Dür • 0 • 0Number of memory qubits per half station: N • Spacing between neighboring stations: L Number of rounds of elementary entanglement generation trial: n E.G. AcknowledgementsThis work was supported by the DARPA Quiness program, ARL CDQI program, ARO, AFOSR, NBRPC (973 program), the Alfred P. Sloan Foundation and the Packard Foundation. We thank Anna Wang, Hong Tang, Ryo Namiki, Prasanta Panigrahi and Steven Girvin for discussions. . 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{'abstract': 'Despite the tremendous progress of quantum cryptography, efficient quantum communication over long distances (≥ 1000km) remains an outstanding challenge due to fiber attenuation and operation errors accumulated over the entire communication distance. Quantum repeaters (QRs), as a promising approach, can overcome both photon loss and operation errors, and hence significantly speedup the communication rate. Depending on the methods used to correct loss and operation errors, all the proposed QR schemes can be classified into three categories (generations). Here we present the first systematic comparison of three generations of quantum repeaters by evaluating the cost of both temporal and physical resources, and identify the optimized quantum repeater architecture for a given set of experimental parameters. Our work provides a roadmap for the experimental realizations of highly efficient quantum networks over transcontinental distances. arXiv:1509.08435v1 [quant-ph]', 'arxivid': '1509.08435', 'author': ['Sreraman Muralidharan \nDepartment of Electrical Engineering\nYale University\n06511New HavenCTUSA\n\nDepartment of Applied Physics\nYale University\n06511New HavenCTUSA\n', 'Linshu Li ', 'Jungsang Kim \nDepartment of Electrical and Computer Engineering\nDuke University\n27708DurhamNCUSA\n', 'Norbert Lütkenhaus \nInstitute of Quantum computing\nUniversity of Waterloo\nN2L 3G1WaterlooCanada\n', 'Mikhail D Lukin \nDepartment of Physics\nHarvard University\n02138CambridgeMAUSA\n', 'Liang Jiang \nDepartment of Applied Physics\nYale University\n06511New HavenCTUSA\n'], 'authoraffiliation': ['Department of Electrical Engineering\nYale University\n06511New HavenCTUSA', 'Department of Applied Physics\nYale University\n06511New HavenCTUSA', 'Department of Electrical and Computer Engineering\nDuke University\n27708DurhamNCUSA', 'Institute of Quantum computing\nUniversity of Waterloo\nN2L 3G1WaterlooCanada', 'Department of Physics\nHarvard University\n02138CambridgeMAUSA', 'Department of Applied Physics\nYale University\n06511New HavenCTUSA'], 'corpusid': 119166141, 'doi': '10.1038/srep20463', 'github_urls': [], 'n_tokens_mistral': 19913, 'n_tokens_neox': 16930, 'n_words': 10375, 'pdfsha': '3e026227198b372c154d249047b596d8c48c58af', 'pdfurls': ['https://arxiv.org/pdf/1509.08435v1.pdf'], 'title': ['Efficient long distance quantum communication', 'Efficient long distance quantum communication'], 'venue': []}

arxiv

Spectral Type and Geometric Albedo of (98943) 2001 CC21, the Hayabusa2# Mission Target 2023 Jooyeon Geem Department of Physics and Astronomy Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea SNU Astronomy Research Center Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea Masateru Ishiguro Department of Physics and Astronomy Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea SNU Astronomy Research Center Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea †mikael Granvik Department of Physics University of Helsinki PO. Box 64FI-00014HelsinkiFinland Asteroid Engineering Laboratory Luleå University of Technology Box 848SE-98128KirunaSweden Hiroyuki Naito Nayoro Observatory 157-1 Nisshin096-0066NayoroHokkaidoJapan Hiroshi Akitaya Planetary Exploration Research Center Chiba Institute of Technology 2-17-1 Tsudanuma275-0016NarashinoChibaJapan Hiroshima Astrophysical Science Center Hiroshima University Kagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan Tomohiko Sekiguchi Asahikawa Campus Hokkaido University of Education 9 Hokumon070-8621AsahikawaHokkaidoJapan Sunao Hasegawa Institute of Space and Astronautical Science (ISAS) Aerospace Exploration Agency (JAXA) 252-5210SagamiharaKanagawaJapan, Japan Daisuke Kuroda Spaceguard Association Bisei Spaceguard Center Bisei-cho1716-3, 714-1411Okura, IbaraOkayamaJapan, Japan Tatsuharu Oono Department of Cosmosciences Graduate School of Science Hokkaido University Kita-ku060-0810SapporoHokkaidoJapan Yoonsoo P Bach Department of Physics and Astronomy Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea SNU Astronomy Research Center Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea Sunho Jin Department of Physics and Astronomy Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea SNU Astronomy Research Center Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea Ryo Imazawa Department of Physics Graduate School of Advanced Science and Engineering Hiroshima University Kagamiyama 1-3-1 Higashi-Hiroshima739-8526HiroshimaJapan Koji S Kawabata Hiroshima Astrophysical Science Center Hiroshima University Kagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan Seiko Takagi Department of Earth and Planetary Sciences Faculty of Science Hokkaido University Kita-ku060-0810SapporoHokkaidoJapan Makoto Yoshikawa Institute of Space and Astronautical Science (ISAS) Aerospace Exploration Agency (JAXA) 252-5210SagamiharaKanagawaJapan, Japan Anlaug A Djupvik Nordic Optical Telescope Rambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain Department of Physics and Astronomy Aarhus University Ny Munkegade 120DK-8000Aarhus CDenmark Julie Thiim Gadeberg Nordic Optical Telescope Rambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain Tapio Pursimo Nordic Optical Telescope Rambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain Department of Physics and Astronomy Aarhus University Ny Munkegade 120DK-8000Aarhus CDenmark PedrosOliver Durfeldt Nordic Optical Telescope Rambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain DTU Space National Space Institute Technical University of Denmark Elektrovej 328DK-2800Kgs. LyngbyDenmark Jeppe Sinkbaek Thomsen Nordic Optical Telescope Rambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain Dipartimento di Fisica e Astronomia Università di Bologna Via Zamboni33 -40126BolognaItalia Zuri Gray Nordic Optical Telescope Rambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain Armagh Observatory and Planetarium College HillBT61 9DGArmaghUK Department of Space & Climate Physics Mullard Space Science Laboratory University College London Holmbury St. Mary RH5 6NTDorkingSurreyUK Spectral Type and Geometric Albedo of (98943) 2001 CC21, the Hayabusa2# Mission Target MNRAS 0002023Accepted 20XX. Received 2023; in original form ZZZPreprint 7 April 2023 Compiled using MNRAS L A T E X style file v3.0minor planets, asteroids: individual: (98943) 2001 CC21 -techniques: polarimetric -techniques: spectroscopic We conducted optical polarimetry and near-infrared spectroscopy of JAXA's Hayabusa2# mission target, (98943) 2001 CC21, in early 2023. Our new observations indicated that this asteroid has a polarimetric inversion angle of ∼21 • , absorption bands around 0.9 and 1.9 m, and a geometric albedo of 0.285±0.083. All these features are consistent with those of S-type but inconsistent with L-type. Based on this evidence, we conclude that JAXA's Hayabusa2# spacecraft will explore an S-type asteroid with albedo and size (0.42-0.56 km when we assume the absolute magnitude of 18.6) similar to (25143) Itokawa. INTRODUCTION (98943) 2001 CC21 (hereafter CC21) is the target of the Hayabusa2 extended mission (the nickname is Hayabusa2#) operated by the Japan Aerospace Exploration Agency (JAXA). After a successful sample return from (162173) Ryugu in 2020, Hayabusa2 spacecraft plans to explore its next targets, CC21 in July 2026 and 1998 KY26 in July 2031. CC21 has a rotation period of 5 hours (Hirabayashi et al. 2021). However, little is known about CC21. Initially, Binzel et al. (2004) ★ E-mail: [email protected] † E-mail: [email protected] reported that CC21 is classified as L-type. On the contrary, Lazzarin et al. (2005) and DeMeo et al. (2009) pointed out the possibility of an S-complex asteroid (either Sk-or Sw-type). Therefore, the taxonomic type of this space mission target still needs to be examined more thoroughly. In addition, it should be emphasized that CC21's albedo has yet to be determined. Knowing albedo is critical for setting appropriate exposure times during the Hayabusa2#'s fast flyby and estimating the size. We present our new observational evidence for the taxonomic type and the geometric albedo of this asteroid, taking advantage of the observation opportunity in 2023 January-March. We conducted optical polarimetry in a wide range of phase angles ( = 21.7 • -99.9 • ), which allows for estimating the taxonomic classification and the geometric albedo. Moreover, we conducted the near-infrared (NIR, 0.7-2.4 m) spectroscopy at an intermediate phase angle ( ∼ 32 • ). We describe our observations and data reduction in Section 2 and the results in section 3. Based on the results, we discuss our results in section 4. OBSERVATIONS AND DATA ANALYSIS Optical Polarimetry The observation circumstances are summarized in Table 1. We obtained the polarimetric data of CC21 by using three instruments: the FAPOL polarimeter of the Alhambra faint object spectrograph and camera (ALFOSC) on the 2.56-m Nordic Optical Telescope at the Observatorio del Roque de los Muchachos, the Hiroshima optical and near-infrared camera (HONIR; Akitaya et al. 2014) on the 1.5-m Kanata Telescope at the Higashi-Hiroshima observatory and the visible multi-spectral imager (MSI; Watanabe et al. 2012) on the 1.6-m Pirka Telescope at the Nayoro Observatory of Hokkaido University. We employed -and -band. All instruments are composed of a rotatable half-wave plate (HWP) and mounted in the Cassegrain focus of each telescope. HONIR and MSI equip a Wollaston prism, and ALFOSC employs a calcite plate mounted in the aperture wheel. To obtain sets of the Stokes parameters, we rotated the HWP in the order of 0 • , 45 • , 22.5 • and 67.5 • for HONIR and MSI and from 0 • to 337.5 • at 22.5 • interval for ALFOSC. The polarimetric data were analyzed in the same manner as described in Ishiguro et al. (2022) for MSI data and in Geem et al. (2022) for ALFOSC and HONIR data. We observed unpolarized standard stars to obtain the instrumental polarization parameters ( inst and inst ) and strongly polarized stars to determine the position angle offset ( off ≡ cat − obs ). Here, cat and obs are the position angle of a star from a catalog and an observation, respectively. We summarize the instrument calibration parameters in Table 2 and the derived polarimetric degrees in Table 1. In Table 1, the polarization degree with respect to the scattering plane ( r ) is given, as is conventionally driven for asteroid polarimetry. We fit the polarization phase curve (PPC) by using the Lumme-Muinonen function (L/M, Lumme & Muinonen 1993) and the linear function by employing the Markov chain Monte Carlo method implemented in PyMC3 (Salvatier et al. 2016). 10,000 samples per chain with four chains are adopted. We used the same boundary conditions to derive the uncertainties of the optimal parameters written in Geem et al. (2022). The initial guesses of each parameter are (ℎ, 0 , 1 , 2 ) = (0.07 percent cent deg −1 , 20 • , 0.1, 0.001). Beyond the 0 , it is known that r of intermediate or high-albedo asteroids, such as S-complex asteroids, pseudo-linearly increases with increasing up to the maximum phase angle (Cellino et al. 2005). Thus, we applied the linear function to derive ℎ and 0 by using the data at < 80 • . The best-fitting results and their uncertainties obtained with the two different fitting functions agree with each other. The results obtained using L/M cover those obtained by the linear fitting. Therefore, we discuss only the results derived from L/M hereafter. Near-infrared Spectroscopy The NIR spectral data (0.7-2.4 m) were obtained during two nights, 2023 February 5-6 UT, using the SpeX instrument at the Mauna Kea Observatory 3.2-m NASA Infrared Telescope Facility (IRTF). We used the 0.8 arcsec width slit aligned with the parallactic angle images with an integration time of 120s each. The observational circumstances are summarized in Table 3. The standard star of SAO 42382 (G2V) was observed, which is closely located from the asteroid (the airmass difference<0.1). We used Spextool, an IDL-based spectral reduction program 1 for data reduction (i.e., flat-fielding, sky subtraction, spectrum extraction, and wavelength calibration.). Figure 1 indicates the nightly-averaged PPC. Similar to other asteroids, it shows an upward trend in the observed phase angles, having an inversion angle ( 0 ) of around 20 • and a maximum polarization phase ( max ) of around 100 • . Table 4 summarizes these key parameters in PPC. For comparison, we plot the typical PPC of the S-type and L-type in low phase angles (Belskaya et al. 2017). We also compare CC21 with S-type asteroids: (1566) Icarus (Ishiguro et al. 2017), (4179) Toutatis (Ishiguro et al. 1997), and (25143) Itokawa (Cellino et al. 2005), in -band and with an L-type asteroid (85989) 1999 JD 6 in -and -bands (Kuroda et al. 2021). A glance at Figure 1 finds that CC21 is closer to these S-type than to the L-type. RESULTS It is known that the polarimetric slope ℎ is a good proxy for V . The relation between ℎ and V is given by log 10 ( V ) = 1 log 10 (ℎ)+ 2 , where 1 and 2 are constants (Geake & Dollfus 1986). We derived the V by substituting ℎ = 0.07 ± 0.02 per cent deg −1 in -band and obtained V = 0.285±0.083 and V = 0.284±0.076 using 1 and 2 values in Lupishko (2018) and Cellino et al. (2015) (for V > 0.08), respectively. Although these two albedo estimates are very close, we will henceforth use the former value ( V = 0.285 ± 0.083), which has the larger error, for safety. We created 0 -ℎ plot to discriminate the polarization properties of L-, S-, and other types of asteroids using databases in Kuroda (4) the polarization efficiency, Unpolarized standard stars, Strongly polarized standard stars, the references of standard stars (1) Turnshek et al. (1990), (2) Whittet et al. (1992), (3) Schmidt et al. (1992), (4) Wolff et al. (1996) Lupishko (2022) (Figure 2). Because ℎ and V are inversely correlated, asteroids with lower V are typically found higher up in the plot. L-type asteroids are known to have distinctively larger 0 than other asteroids. From this comparison, 0 -ℎ of CC21 matches those of S-types rather than L-types. The NIR spectra obtained over the two nights are consistent with each other. No rotational variation of the target spectra is found during the observations. For this reason, we combined all spectral data from two nights. Figure 3 shows the resultant spectrum. The plot compares CC21's reflectance spectrum with typical S-and L-type asteroids from DeMeo et al. (2009). Our CC21 spectrum indicates not only the clear absorption around 0.9 m but also the shallow absorption around 1.9 m associated with pyroxene. The continuum in the visible range at 0.75 m is confirmed. These features are characteristic of S-type asteroids and are not found in L-type. DISCUSSION In the previous studies, CC21 was classified in either L-or S-type (Binzel et al. 2004;Lazzarin et al. 2005;DeMeo et al. 2009). Our observations indicates S-type features and rules out L-type possibility. Although empirical, L-type asteroids are known to have distinctive 0 values compared to other asteroids. The 0 value in this study is predominantly different from that of L-type asteroids but is consistent with S-type asteroids. Another important aspect of our polarimetry is that the geometric albedo was determined. The derived CC21's albedo ( V = 0.285 ± 0.083) is in the range of S-type asteroids ( V = 0.258 ± 0.087, DeMeo & Carry 2013) but is slightly higher than the average L-type asteroids ( V = 0.183 ± 0.089, DeMeo & Carry 2013). We also emphasize that our NIR spectrum with a good S/N provides definitive evidence for the taxonomic type. Our NIR spectrum agrees with the previous CC21's NIR spectra (Lazzarin (Marsset et al. 2020) (Figure 3). The 0.9 and 1.9 m absorption features (typical of S-type) are detected. Using an online-based classification tool 2 , CC21 is further subcategorized into Sq or Q-type. These taxonomic types (less weathered ordinary chondritic asteroids) are common in the near-Earth region. Given all these observational results, we conclude that CC21 is an S-type rather than L-type. We 2 http://smass.mit.edu/cgi-bin/busdemeoclass-cgi expect that a discussion of the scientific case of Hayabusa2# will be conducted with the S-type in consideration. The second significant point of our study is to determine the albedo. Because Hayabusa2# will conduct a fast flyby with CC21, the exposure times should be pre-determined based on the geometric albedo and scattering phase function. The geometric albedo of an S-type asteroid, (25143) Itokawa, was estimated by polarimetry before the launch (Cellino et al. 2005). Notably, the geometric albedo estimated by the polarization method matches the actual value derived by the in-situ observation (Lee & Ishiguro 2018;Tatsumi et al. 2018). Considering that CC21 is an S-type asteroid, our albedo value obtained by the same method as Cellino et al. (2005) is also quite reliable. Our albedo estimate is higher than the assumed value in Hirabayashi et al. (2021) ( V = 0.15). Although we need to refer to the absolute magnitude V =18.6 from the literature (Binzel et al. 2004), the effective diameter of CC21 estimated from our albedo would be 0.42-0.56 km. Note that the size becomes smaller than the value in the previous research Hirabayashi et al. (2021). Polarimetry at large phase angles is also useful for estimating particle size on the asteroid's surface. It is known that the max is correlated with the V and the grain size. Their relationship is given by = 0.03 exp 2.9(log (10 2 ) + 0.845 log (10 max )), where is the grain size in m and is an albedo at = 5 • (Shkuratov & Opanasenko 1992). We calculate CC21's =0.198 ± 0.058 by considering the intensity ratio of (0. • 3)/ (5 • ) = 1.44 ± 0.04 for typical S-type asteroids (Belskaya & Shevchenko 2000). As a result, CC21 would be covered by grain with the size of 100-130 m, like (1566) Icarus, another near-Earth S-type asteroid whose size is similar to CC21 ( 1 km). SUMMARY We conducted optical polarimetric and near-infrared spectroscopic observations of 2001 CC21 in early 2023. The inversion angle ( 0 ∼ 21 • ) and geometric albedo ( V ∼ 0.3) are consistent with those of S-types but significantly different from those of L-types. The nearinfrared spectrum of the target shows clear absorption bands around 0.9 and 1.9 m, which are the typical spectral features of S-type asteroids. Based on the results, we conclude that 2001 CC21 is a near-Earth S-type asteroid with an albedo of V = 0.285 ± 0.083 and a size of 0.42-0.56 km. DATA AVAILABILITY The observational data are available in Zenodo 3 . The source codes and scripts for the data analyses, plots, and resultant data tables are available via the GitHub service 4 . Figure 1 . 1Phase angle ( ) dependence of polarization degree ( r ). Data taken by HONIR, MSI, and FAPOL are shown by the triangle, square, and diamond markers, respectively. The filled and empty markers indicate -and -bands. The solid and dash lines are curves that fit data in -and -bands, respectively, by using the L/M function. The dotted line is the PPC of 1999 JD 6 . Figure 2 .Figure 3 . 23A comparison with other types of asteroids(Kuroda et al. 2021;Lupishko 2022). Each marker indicates the taxonomic type of asteroid (Tholen or Bus-DeMeo type). CC21 in -band is shown as the green empty box. Ltype asteroids with 0 25 • and unknown ℎ are shown below the arrow(Lupishko 2022). The green and red colored boxes show the typical 0 ranges of the S-type and the L-type, respectively, fromBelskaya et al. (2017). The combined spectra of CC21 from two nights' observations is colored red. For the comparison, NIR spectra of CC21 from the previous studies(Lazzarin et al. 2005;DeMeo et al. 2009) by the gray triangle, green cross markets. The coral dotted and blue dashed lines show the typical spectra of S-type and L-type asteroids (DeMeo et al. 2009), respectively. The reflectances are normalized at 1.6 m. et al. 2005; DeMeo et al. 2009) within observational uncertainty Table 1 .Table 2 . 12Summary of Polarimetry UT at exposure start, Instrument, Exposure time, Number of valid images, Median heliocentric distance, Median geocentric distance, Position angle of the scattering plane, ℎ Median solar phase angle, Nightly averaged polarization degree, Uncertainty of , Position angle of the strongest electric vector, Uncertainty of P , Polarization degree referring to the scattering plane, Position angle referring to the scattering plane. The web-based JPL Horizon system (http://ssd.jpl.nasa.gov/?horizons) was used to obtain , Δ, , and in the table. Calibration Parameters of PolarimetryDate in UT Inst Filter Exp N Δ ℎ r r (s) (au) (au) ( • ) ( • ) (%) (%) ( • ) ( • ) (%) ( • ) Jan 24 18:35-18:49 HONI 60 12 1.14 0.17 213.0 21.7 0.38 0.30 83.7 22.1 0.18 -31.3 Jan 25 17:40-18:00 HONI 60 16 1.14 0.17 208.5 22.1 0.00 0.19 31.7 52.0 -0.00 45.2 Jan 29 14:35-15:14 MSI 120 20 1.13 0.16 191.2 24.5 0.31 0.19 -85.0 17.3 0.30 -8.1 Jan 29 15:34-16:45 MSI V 150 20 1.13 0.16 190.9 24.5 0.45 0.25 -89.8 16.2 0.37 -16.7 Feb 03 23:01-23:10 FAPOL 120 8 1.11 0.15 170.3 30.4 1.25 1.15 54.4 26.3 0.75 -26.6 Feb 09 13:39-16:18 MSI 120 16 1.09 0.14 151.5 38.6 1.22 0.53 63.5 12.4 1.19 -6.8 Feb 09 16:32-16:50 MSI V 180 8 1.09 0.14 151.3 38.7 1.62 0.73 73.5 12.9 1.62 -2.8 Feb 11 23:30-23:47 FAPOL 120 12 1.09 0.14 144.3 42.4 1.89 0.43 49.8 6.4 1.87 -4.5 Feb 15 13:05-13:28 MSI 90 8 1.07 0.14 133.4 48.4 2.09 0.50 45.8 6.9 2.06 -5.2 Feb 15 13:55-14:29 MSI V 120,240 12,4 1.07 0.14 133.3 48.5 2.92 0.56 35.9 5.5 2.85 6.0 Feb 17 12:03-14:52 MSI 90,120 32,24 1.07 0.14 127.4 51.9 2.53 0.12 38.0 1.4 2.53 0.9 Feb 17 13:06-16:08 MSI V 90,120,150 4,24,24 1.07 0.14 127.2 52.0 2.76 0.17 36.9 1.8 2.75 -1.8 Feb 26 20:42-20:59 FAPOL 120 12 1.03 0.13 100.1 68.0 4.13 0.19 8.8 1.3 4.13 -1.3 Feb 26 16:40-16:53 MSI 180 8 1.03 0.13 100.5 67.7 3.28 1.18 5.5 10.3 3.24 -4.7 Feb 26 18:28-19:28 MSI V 240 12 1.03 0.13 100.3 67.9 4.61 1.16 22.4 7.2 4.60 -0.9 Mar 04 14:15-15:20 MSI 180 16 1.01 0.13 84.6 78.0 4.66 0.88 8.4 5.4 4.64 -2.5 Mar 04 11:56-13:48 MSI V 180 20 1.01 0.13 84.9 77.8 5.51 1.33 -6.5 6.9 5.50 1.7 Mar 05 20:35-20:43 FAPOL 120 16 1.01 0.13 81.5 80.2 5.69 0.39 -7.7 2.0 5.69 0.8 Mar 13 21:07-21:07 FAPOL 120 4 0.98 0.13 62.6 94.4 6.02 0.48 -27.8 2.3 6.01 -0.5 Mar 16 20:24-20:41 FAPOL 120 12 0.96 0.13 56.1 99.9 6.33 0.63 -33.5 2.9 6.22 -5.4 Inst Filter eff inst inst off UP SP Ref (%) (%) (%) ( • ) HONIR 97.58 0.010 ± 0.050 −0.008 ± 0.037 38.00 ± 0.83 G191B2B HD 29333, HD 251204 (1),(2) ALFOSC 100 (assumed) 0.012 ± 0.065 −0.054 ± 0.055 −87.78 ± 0.10 HD 42182, HD 65629 BD+59 389 (3) MSI 99.59 ± 0.02 0.785 ± 0.020 1.077 ± 0.019 −20.61 ± 0.26 HD 15318 HD 7927 (4) 99.55 ± 0.01 0.584 ± 0.011 0.751 ± 0.011 −17.09 ± 0.52 HD 15318 HD 7927 Table 3 . 3Observation Circumstance of SpectroscopyDate in UT Exp Airmass N Solar Analog (s) ( • ) Feb 05 09:48-12:39 120 32.3 1.18-1.41 38 SAO 42382 Feb 06 06:08-09:38 120 33.7 1.20-1.69 48 SAO 42382 UT at exposure start, solar phase angle, number of valid images Table 4. PPC Fitting Result Filter slope ℎ 0 max max (% deg −1 ) ( • ) (%) ( • ) 0.07 +0.02 −0.02 20.7 +3.3 −2.6 6.7 +1.0 −0.6 114.5 +5.4 −12.2 0.06 +0.01 −0.02 21.2 +2.2 −2.2 4.3 +3.5 −1.2 88.4 +31.5 −8.4 et al. (2021) and © 2023 The Authors http://irtfweb.ifa.hawaii.edu/research/dr_resources/ MNRAS 000, 1-5 (2023) https://doi.org/10.5281/zenodo.XXXXX 4 https://github.com/Geemjy/XXXXX.git MNRAS 000, 1-5 (2023) ACKNOWLEDGEMENTSThis research at SNU was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2023R1A2C1006180). SH was supported by the Hypervelocity Impact Facility (former name: The Space Plasma Laboratory), ISAS, JAXA. The data presented here were obtained with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOT. 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M Lazzarin, S Marchi, S Magrin, J Licandro, 10.1111/j.1365-2966.2005.09006.xMonthly Notices of the Royal Astronomical Society. 3591575Lazzarin M., Marchi S., Magrin S., Licandro J., 2005, Monthly Notices of the Royal Astronomical Society, 359, 1575 . M Lee, M Ishiguro, 10.1051/0004-6361/201832721A&A. 616178Lee M., Ishiguro M., 2018, A&A, 616, A178 K Lumme, K Muinonen, 10.1016/j.icarus.2016.11.003IAU Symp.160: Asteroids, Comets, Meteors. 160194Lumme K., Muinonen K., 1993, IAU Symp.160: Asteroids, Comets, Meteors 1993, 160, 194 . D F Lupishko, 10.1134/S0038094618010069Solar System Research. 5298Lupishko D. F., 2018, Solar System Research, 52, 98 NASA Planetary Data System. D Lupishko, 10.26033/hyf9-47621Lupishko D., 2022, NASA Planetary Data System, p. 1 . M Marsset, 10.3847/1538-4365/ab7b5fThe Astrophysical Journal Supplement Series. 24773Marsset M., et al., 2020, The Astrophysical Journal Supplement Series, 247, 73 . 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{'abstract': "We conducted optical polarimetry and near-infrared spectroscopy of JAXA's Hayabusa2# mission target, (98943) 2001 CC21, in early 2023. Our new observations indicated that this asteroid has a polarimetric inversion angle of ∼21 • , absorption bands around 0.9 and 1.9 m, and a geometric albedo of 0.285±0.083. All these features are consistent with those of S-type but inconsistent with L-type. Based on this evidence, we conclude that JAXA's Hayabusa2# spacecraft will explore an S-type asteroid with albedo and size (0.42-0.56 km when we assume the absolute magnitude of 18.6) similar to (25143) Itokawa.", 'arxivid': '2304.02917', 'author': ['Jooyeon Geem \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Masateru Ishiguro \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', '†mikael Granvik \nDepartment of Physics\nUniversity of Helsinki\nPO. Box 64FI-00014HelsinkiFinland\n\nAsteroid Engineering Laboratory\nLuleå University of Technology\nBox 848SE-98128KirunaSweden\n', 'Hiroyuki Naito \nNayoro Observatory\n157-1 Nisshin096-0066NayoroHokkaidoJapan\n', 'Hiroshi Akitaya \nPlanetary Exploration Research Center\nChiba Institute of Technology\n2-17-1 Tsudanuma275-0016NarashinoChibaJapan\n\nHiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', 'Tomohiko Sekiguchi \nAsahikawa Campus\nHokkaido University of Education\n9 Hokumon070-8621AsahikawaHokkaidoJapan\n', 'Sunao Hasegawa \nInstitute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan\n', 'Daisuke Kuroda \nSpaceguard Association\nBisei Spaceguard Center\nBisei-cho1716-3, 714-1411Okura, IbaraOkayamaJapan, Japan\n', 'Tatsuharu Oono \nDepartment of Cosmosciences\nGraduate School of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan\n', 'Yoonsoo P Bach \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Sunho Jin \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Ryo Imazawa \nDepartment of Physics\nGraduate School of Advanced Science and Engineering\nHiroshima University Kagamiyama\n1-3-1 Higashi-Hiroshima739-8526HiroshimaJapan\n', 'Koji S Kawabata \nHiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', 'Seiko Takagi \nDepartment of Earth and Planetary Sciences\nFaculty of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan\n', 'Makoto Yoshikawa \nInstitute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan\n', 'Anlaug A Djupvik \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDepartment of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark\n', 'Julie Thiim Gadeberg \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n', 'Tapio Pursimo \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDepartment of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark\n', 'PedrosOliver Durfeldt \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDTU Space\nNational Space Institute\nTechnical University of Denmark\nElektrovej 328DK-2800Kgs. LyngbyDenmark\n', 'Jeppe Sinkbaek Thomsen \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDipartimento di Fisica e Astronomia\nUniversità di Bologna\nVia Zamboni33 -40126BolognaItalia\n', 'Zuri Gray \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nArmagh Observatory and Planetarium\nCollege HillBT61 9DGArmaghUK\n\nDepartment of Space & Climate Physics\nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK\n', 'Jooyeon Geem \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Masateru Ishiguro \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', '†mikael Granvik \nDepartment of Physics\nUniversity of Helsinki\nPO. Box 64FI-00014HelsinkiFinland\n\nAsteroid Engineering Laboratory\nLuleå University of Technology\nBox 848SE-98128KirunaSweden\n', 'Hiroyuki Naito \nNayoro Observatory\n157-1 Nisshin096-0066NayoroHokkaidoJapan\n', 'Hiroshi Akitaya \nPlanetary Exploration Research Center\nChiba Institute of Technology\n2-17-1 Tsudanuma275-0016NarashinoChibaJapan\n\nHiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', 'Tomohiko Sekiguchi \nAsahikawa Campus\nHokkaido University of Education\n9 Hokumon070-8621AsahikawaHokkaidoJapan\n', 'Sunao Hasegawa \nInstitute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan\n', 'Daisuke Kuroda \nSpaceguard Association\nBisei Spaceguard Center\nBisei-cho1716-3, 714-1411Okura, IbaraOkayamaJapan, Japan\n', 'Tatsuharu Oono \nDepartment of Cosmosciences\nGraduate School of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan\n', 'Yoonsoo P Bach \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Sunho Jin \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n\nSNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Ryo Imazawa \nDepartment of Physics\nGraduate School of Advanced Science and Engineering\nHiroshima University Kagamiyama\n1-3-1 Higashi-Hiroshima739-8526HiroshimaJapan\n', 'Koji S Kawabata \nHiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', 'Seiko Takagi \nDepartment of Earth and Planetary Sciences\nFaculty of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan\n', 'Makoto Yoshikawa \nInstitute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan\n', 'Anlaug A Djupvik \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDepartment of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark\n', 'Julie Thiim Gadeberg \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n', 'Tapio Pursimo \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDepartment of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark\n', 'PedrosOliver Durfeldt \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDTU Space\nNational Space Institute\nTechnical University of Denmark\nElektrovej 328DK-2800Kgs. LyngbyDenmark\n', 'Jeppe Sinkbaek Thomsen \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nDipartimento di Fisica e Astronomia\nUniversità di Bologna\nVia Zamboni33 -40126BolognaItalia\n', 'Zuri Gray \nNordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain\n\nArmagh Observatory and Planetarium\nCollege HillBT61 9DGArmaghUK\n\nDepartment of Space & Climate Physics\nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics\nUniversity of Helsinki\nPO. Box 64FI-00014HelsinkiFinland', 'Asteroid Engineering Laboratory\nLuleå University of Technology\nBox 848SE-98128KirunaSweden', 'Nayoro Observatory\n157-1 Nisshin096-0066NayoroHokkaidoJapan', 'Planetary Exploration Research Center\nChiba Institute of Technology\n2-17-1 Tsudanuma275-0016NarashinoChibaJapan', 'Hiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', 'Asahikawa Campus\nHokkaido University of Education\n9 Hokumon070-8621AsahikawaHokkaidoJapan', 'Institute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan', 'Spaceguard Association\nBisei Spaceguard Center\nBisei-cho1716-3, 714-1411Okura, IbaraOkayamaJapan, Japan', 'Department of Cosmosciences\nGraduate School of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics\nGraduate School of Advanced Science and Engineering\nHiroshima University Kagamiyama\n1-3-1 Higashi-Hiroshima739-8526HiroshimaJapan', 'Hiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', 'Department of Earth and Planetary Sciences\nFaculty of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan', 'Institute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Department of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Department of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'DTU Space\nNational Space Institute\nTechnical University of Denmark\nElektrovej 328DK-2800Kgs. LyngbyDenmark', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Dipartimento di Fisica e Astronomia\nUniversità di Bologna\nVia Zamboni33 -40126BolognaItalia', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Armagh Observatory and Planetarium\nCollege HillBT61 9DGArmaghUK', 'Department of Space & Climate Physics\nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics\nUniversity of Helsinki\nPO. Box 64FI-00014HelsinkiFinland', 'Asteroid Engineering Laboratory\nLuleå University of Technology\nBox 848SE-98128KirunaSweden', 'Nayoro Observatory\n157-1 Nisshin096-0066NayoroHokkaidoJapan', 'Planetary Exploration Research Center\nChiba Institute of Technology\n2-17-1 Tsudanuma275-0016NarashinoChibaJapan', 'Hiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', 'Asahikawa Campus\nHokkaido University of Education\n9 Hokumon070-8621AsahikawaHokkaidoJapan', 'Institute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan', 'Spaceguard Association\nBisei Spaceguard Center\nBisei-cho1716-3, 714-1411Okura, IbaraOkayamaJapan, Japan', 'Department of Cosmosciences\nGraduate School of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Department of Physics\nGraduate School of Advanced Science and Engineering\nHiroshima University Kagamiyama\n1-3-1 Higashi-Hiroshima739-8526HiroshimaJapan', 'Hiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', 'Department of Earth and Planetary Sciences\nFaculty of Science\nHokkaido University\nKita-ku060-0810SapporoHokkaidoJapan', 'Institute of Space and Astronautical Science (ISAS)\nAerospace Exploration Agency (JAXA)\n252-5210SagamiharaKanagawaJapan, Japan', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Department of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Department of Physics and Astronomy\nAarhus University\nNy Munkegade 120DK-8000Aarhus CDenmark', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'DTU Space\nNational Space Institute\nTechnical University of Denmark\nElektrovej 328DK-2800Kgs. LyngbyDenmark', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Dipartimento di Fisica e Astronomia\nUniversità di Bologna\nVia Zamboni33 -40126BolognaItalia', 'Nordic Optical Telescope\nRambla José Ana Fernández Pérez 7ES-38711Breña BajaSpain', 'Armagh Observatory and Planetarium\nCollege HillBT61 9DGArmaghUK', 'Department of Space & Climate Physics\nMullard Space Science Laboratory\nUniversity College London\nHolmbury St. Mary\nRH5 6NTDorkingSurreyUK'], 'corpusid': 257984994, 'doi': None, 'github_urls': ['https://github.com/Geemjy/XXXXX.git'], 'n_tokens_mistral': 11971, 'n_tokens_neox': 9429, 'n_words': 4118, 'pdfsha': 'c9d261a404f5fd8213e20aa5369141feefed9789', 'pdfurls': ['https://export.arxiv.org/pdf/2304.02917v1.pdf'], 'title': ['Spectral Type and Geometric Albedo of (98943) 2001 CC21, the Hayabusa2# Mission Target', 'Spectral Type and Geometric Albedo of (98943) 2001 CC21, the Hayabusa2# Mission Target', 'Spectral Type and Geometric Albedo of (98943) 2001 CC21, the Hayabusa2# Mission Target', 'Spectral Type and Geometric Albedo of (98943) 2001 CC21, the Hayabusa2# Mission Target'], 'venue': ['MNRAS', 'MNRAS']}

arxiv

Learning from Noisy Labels Generated by Extremely Point Annotations for OCT Fluid Segmentation Tengjin Weng Yang Shen Kai Jin Zhiming Cheng [email protected] Yunxiang Li [email protected] Gewen Zhang Shuai Wang [email protected] Zhejiang Sci-Tech University China Lishui University China Second Affiliated Hospital of Zhejiang University China UT Southwestern Medical Center Hangzhou Dianzi University DallasTXChina, USA Lishui University China Hangzhou Dianzi University China Learning from Noisy Labels Generated by Extremely Point Annotations for OCT Fluid Segmentation Automatic segmentation of fluid in OCT (Optical Coherence Tomography) images is beneficial for ophthalmologists to make an accurate diagnosis. Currently, data-driven convolutional neural networks (CNNs) have achieved great success in OCT fluid segmentation. However, obtaining pixel-level masks of OCT images is time-consuming and requires expertise. The popular weakly-supervised strategy is to generate noisy pseudo-labels from weak annotations, but the noise information introduced may mislead the model training. To address this issue, (i) we propose a superpixel-guided method for generating noisy labels from weak point annotations, called Point to Noisy by Superpixel (PNS), which limits the network from over-fitting noise by assigning low confidence to suspiciously noisy label pixels, and (ii) we propose a Two-Stage Mean-Teacher-assisted Confident Learning (2SMTCL) method based on MTCL for multi-category OCT fluid segmentation, which alleviates the uncertainty and computing power consumption introduced by the real-time characterization noise of MTCL. For evaluation, we have constructed a 2D OCT fluid segmentation dataset. Compared with other state-of-art label-denoising methods, comprehensive experimental results demonstrate that the proposed method can achieve excellent performance in OCT fluid segmentation as well as label denoising. Our study provides an efficient, accurate, and practical solution for fluid segmentation of OCT images, which is expected to have a positive impact on the diagnosis and treatment of patients in the field of ophthalmology. Introduction The intricate anatomical structures and diverse disease symptoms associated with eye diseases often present substantial hurdles in diagnosis. Optical Coherence Tomography (OCT), a non-intrusive imaging technique [1,2], offers high-definition, sectional imaging of the retina and optic nerve. This allows eye specialists to accurately observe and quantify retinal abnormalities. More specifically, retinal fluid in OCT, a key indicator in detecting and diagnosing eye conditions, is categorized into intraretinal fluid (IRF), subretinal fluid (SRF), and pigment epithelial detachment (PED) based on its location of accumulation. These fluids are vital biomarkers for ocular diseases like agerelated macular degeneration (AMD) and retinal vein occlusion (RVO). Identifying the existence and location of these fluids assists eye doctors in diagnosing, monitoring, and planning effective treatment strategies to maintain vision. Nevertheless, manual analysis of OCT images can be time-consuming and prone to errors. Traditional segmentation methods, such as threshold-based [3], graph-based [4,5], and machine learning-based [6] techniques have been used in OCT segmentation, but often fall short due to varying image quality, the necessity for extensive specialized knowledge, and limited generalization abilities. In contrast to traditional segmentation methods that rely on carefully crafted handcrafted features, convolutional neural networks (CNNs) can automatically learn and extract image features from the data itself. Therefore, various CNN-based methods have been developed for performing segmentation tasks, such as FCN [7], SegNet [8], DeepLab [9], and UNet [10]. The utilization of CNNs in medical image segmentation requires substantial amounts of data. Unfortunately, manual segmentation of medical images demands significant expertise and time. Obtaining an adequate quantity of accurately-labeled data from medical experts can be a difficult and challenging task, thereby posing obstacles to developing precise CNN models for medical image segmentation. Without enough clearly labeled pixel-level annotations, CNN-based segmentation methods often struggle to fit, leading to performance degradation. To address this problem, researchers choose to collect additional labeled data without quality control, such as crowdsourcing or noisy pseudo-labeled data generated based on weak supervision. However, directly combining clean labels with noisy labels may confuse the network during training and lead to performance degradation, negating the benefit provided by clean labels [11,12]. Therefore, it is crucial to effectively and robustly utilize the additional information available in large amounts of noisy-labeled data. Initiatives to tackle the challenge of noisy labels, there are a range of label-denoising strategies. These strategies are generally divided into two categories, depending on the way input data is partitioned. The first category comprises methods that gather and amalgamate data from various sources to indiscriminately train the model. The second category is designed for practical scenarios where professionals are asked to label or perform quality checks on a small dataset, distinguishing between clearly-labeled and noisy-labeled data. Our research is centered on this second approach, where prior knowledge is employed to aid the network in differentiating between straightforward and ambiguous labeled data. For this purpose, several techniques have been suggested, one of which is Mean-Teacherassisted Confident Learning (MTCL [13]), a classic method of distinguishing data sources. It can robustly learn segmentation from limited high-quality labeled data and abundant low-quality labeled data. Specifically, the framework of MTCL can leverage the extra dark knowledge in lowquality labeled images based on perturbation-based unsupervised consistency, and effectively exploit the beneficial information in low-quality noisy labels through explicit label refinement. However, the Confident Learning (CL) module based on Cleanlab can only be calculated on the CPU, and it is time-consuming to introduce CL characterization label noise for each iteration during training. Moreover, multi-category segmentation is more challenging than two-category segmentation leading to CL real-time characterization label noise that may be inaccurate. In this work, we illustrate the application of NLL to Figure 1: The Point to Noisy by Superpixel (PNS) method generates noisy labels and label trust graphs from point annotations, guided by superpixels. The superpixel image is obtained using the superpixel algorithm (SLIC). By computing the similarity between the labeled superpixel block and its adjacent superpixel blocks to obtain the noisy label and label trust graph. the OCT fluid segmentation task from the following two aspects: (i) We proposed Point to Noisy by Superpixel (PNS), which can generate noisy labels from weak point annotations via superpixels guidance, and generate label trust graphs to provide a confidence measure for each label pixel in the noisy labels. These label trust graphs can constrain the network from over-fitting noise by assigning lower confidence to suspected noisy label pixels. (ii) We choose MTCL as the NLL framework, which can robustly learn segmentation from limited high-quality labeled data and abundant low-quality labeled data. Considering that multi-category segmentation is more challenging than two-category segmentation, we propose a Two-Stage Mean-Teacher-assisted Confident Learning (2SMTCL) method for multi-category OCT fluid segmentation, which alleviates the uncertainty and computing power consumption introduced by the real-time characterization noise of MTCL. 2SMTCL trains two networks: a noisy network and a denoising network. Specifically, the first-stage noisy network is trained based on the teacher-student architecture, then introduce CL to characterize the pixel-level label noise and refine the noisy labels. Finally, the second-stage denoising network is trained by denoising labels. We evaluate the performance of 2SMTCL on the OCT fluid segmentation dataset employed in this study. The results show that our method can effectively exploit weak point annotations to improve segmentation performance, outperforming other competing methods. The contributions of our research are summarized as follows: • To the best of our knowledge, we are the first researchers to apply NLL to the task of OCT fluid segmentation. • We propose a superpixel-guided method for generating noisy labels from weak point annotations named PNS, which can constrain the network from over- fitting noise by assigning lower confidence to suspected noisy label pixels. (a) (b) (c) (d) • We propose 2SMTCL, a two-stage method based on MTCL for multi-category OCT fluid segmentation, which alleviates the uncertainty and computing power consumption introduced by the real-time characterization noise of MTCL. • We have constructed a 2D OCT image segmentation dataset with corresponding ground truth annotations and point annotations. This dataset can serve as a valuable resource for training and evaluating deep learning models aimed at achieving accurate fluid segmentation. RELATE WORK CNN-Based OCT Fluid Segmentation Many successful OCT fluid segmentation methods use convolutional neural networks (CNNs) based on the UNet [10] architecture. Rashno et al. [14] incorporated a graph shortest path technique as a post-processing step to enhance the predictive results of UNet for OCT fluid segmentation. To exploit the structural relationship between retinal layers and fluids, Xu et al. [15] proposed a two-stage fluid segmentation framework. They first trained a retinal layer segmentation network to extract retinal layer maps which were used to constrain the fluid segmentation network in the second stage. Several other studies, such as [16,17], employed a graph-cut method to generate retinal layer segmentation maps. These maps were then combined to train a UNet for fluid segmentation. Moreover, De et al. [18] proposed a UNet-based architecture that can simultaneously segment retinal layers and fluids, utilizing pixel-level annotations of retinal layer and fluid masks to enhance OCT segmentation performance. Although various methods have been proposed with little difference in performance, the ef-fectiveness of current OCT fluid segmentation methods relies heavily on a large number of datasets with annotations. Weakly-Supervised Segmentation Given the inaccessibility of large amounts of fully annotated data, several researchers have developed various weakly supervised medical image segmentation methods. A prevalent approach involves generating noisy pseudo-labels from weak annotations and subsequently using these to train a segmentation model. Pu et al. [19] proposed a technique that utilizes a graph neural network based on superpixels to create noisy pseudo-labels from weak annotations like points or scribbles. Nevertheless, this method could introduce two sources of error: inaccuracies in the generated noisy pseudo-labels and the subsequent errors in learning segmentation from these labels. To circumvent these issues, some strategies directly train segmentation models on partial annotations. Bearman et al. [20], for instance, combined point-supervised and self-supervised techniques to master object segmentation within images. Other strategies like [21,22] employ prior knowledge of constraint expressions to aid segmentation during training. In the realm of medical image segmentation, this prior knowledge can be incredibly useful, given the frequent availability of information about the target region in advance. While existing weakly supervised methods have demonstrated their potential in reducing manual labor and improving segmentation performance, their application to OCT fluid segmentation hasn't been extensively investigated. He et al. [23] introduced a method dubbed Intra-Slice Contrast Learning Network (ISCLNet) that relies on weak point supervision for 3D OCT fluid segmentation. However, in actual diagnoses, ophthalmologists typically only concentrate on a limited number of OCT images displaying fluid. The inter-image comparison technique deployed by ISCLNet can be challenging when dealing with incomplete OCT data. Motivated by NLL, we believe this method can be adapted to weakly supervised 2D OCT fluid segmenta-tion. Learning Segmentation with Noisy Labels Previous work has pointed out that labeled data with noise can mislead network training and degrade network performance. Most existing noise-supervised learning works focus on image-level classification tasks [24], [25], [26] while more challenging pixel-wise segmentation tasks remain to be studied. Zhang et al. [27] proposed a TriNet based on Co-teaching [25], which trains a third network using combined predictions from the first two networks to alleviate the misleading problem caused by label noise. Li et al. [28] proposed a method that employs superpixels to guide the network for noise-aware training and refinement of noisy labels. Zhang et al. [29] suggested a twostage strategy for pre-training a network using a combination of different datasets, followed by fine-tuning the labels by Confident Learning to train a second network. Zhu et al. [30] proposed a module for assessing the quality of image-level labels to identify high-quality labels for finetuning a network. Xu et al. [13] developed the MTCL framework based on Mean-Teacher architecture and Confident Learning, which can robustly learn segmentation from limited high-quality labeled data and abundant low-quality labeled data. The KDEM [31] method is an extension of the semi-supervised learning approach proposed by [11], which introduces additional techniques such as knowledge distillation and entropy minimization regularization to further improve the segmentation performance. Yang et al. [32] introduce a dual-branch network that can learn efficiently by processing accurate and noisy annotations separately. These methods demonstrate how to improve the network's ability to learn noisy labels and provide insights for future research in this area. Extensive experiments of many NLL methods on datasets such as JSRT [33] and ISIC [34] have achieved promising results, but limitations caused by the lack of OCT fluid segmentation datasets hinder the application of these methods in this field. Therefore, the effectiveness of NLL methods in OCT fluid segmentation remains largely unexplored. METHODOLOGY Framework Overview Our method divides the dataset into two groups: clearlylabeled data (CD) and noisy-labeled data (ND). The noisy labels and label trust graphs of ND are generated using weak point annotations via PNS. To simplify the description of our methodology, we define M samples to represent the CD, while the remaining N − M samples represent the ND. We denote the CD as D c = {(X (i) , Y (i) )} M i=1 and the ND as D n = {(X (i) ,Ỹ (i) , U (i) )} N i=M +1 , where X (i) ∈ R Ωi represents the input 2D OCT images. Y (i) ,Ỹ (i) ∈ {0, 1, 2, 3} Ωi (four types of segmentation tasks) denotes the clean segmentation label and noisy segmentation label of X (i) , respectively. The label trust graph U (i) indicates the degree of trust ofỸ (i) , where U (i) ⊆ {0, 0.1, . . . , 1} Ωi . Fig. 3 illustrates our method that aims to learn OCT fluid segmentation simultaneously from limited CD and abundant ND. The images of the CD are fed to the student model, and the images of the ND are both fed to the student model and teacher model. Simultaneously, the PNS method generates noisy labels and label trust graphs of ND. After obtaining the noisy network (student network) based on MT architecture, CL is used to characterize the label error in ND to obtain estimated error maps. The denoising network is trained with denoised labels and refined label trust graphs, which are obtained guided by the estimated error maps. Our method will be elaborated on the following two aspects: (i) How to generate noisy labels and label trust graphs to constrain the network from over-fitting noise. (ii) How 2SMTCL robustly learns multi-category OCT fluid segmentation from abundant noisy labels. Labels and Label Trust Graphs of ND Generated by PNS Our proposed PNS can generate noisy labels from weak point annotations via superpixels guidance, and generate label trust graphs to provide a confidence measure for each label pixel in the noisy labels. These label trust graphs can constrain the network from over-fitting noise by assigning lower confidence to suspected noisy label pixels. Fig. 1 shows how noisy labels and label trust graphs are generated from point annotations via PNS. Superpixel-guided for Generating Noisy Labels from Weak Point Annotations Our OCT fluid segmentation dataset includes fully annotated labels and weakly annotated labels for three types of fluids: PED, SRF, and IRF. For weak annotation, use points to indicate the center of the fluid accumulation (for SRF and IRF) and lines (consisting of two or more points) to mark the bottom of the PED. This greatly simplifies the annotation process and reduces the required labor. The proposed PNS can generate noisy labels from these point annotations. Formally, give an image X, the weak label is represented by Y ′ = {Y ′ i } n i=1 , Y ′ i ∈ {1, 2..., C} where C is the number of semantic classes and n is the number of pixels. The superpixel image is obtained based on the SLIC [35] algorithm. We denote superpixel image as S = {S i } n i=1 , where S i ∈ {1, 2, ..K} and the K is the number of superpixel blocks. Here S j = k means that the pixel j belongs to the k th superpixel block. We can represent all the pixels j that are included in the k th superpixel block byS = {S k } K k=1 , whereS k = {j : S j=k }. Further, the superpixel label is . The images of the CD are fed to the student model, and the images of the ND are both fed to the student model and teacher model. Simultaneously, the PNS method generates noisy labels and label trust graphs of ND. After obtaining the noisy network (student network) based on MT architecture, CL is used to characterize the label error in ND to obtain estimated error maps X err . The denoising network is trained with denoised labels and refined label trust graphs, which are obtained guided by the estimated error maps. represented byȲ = {Ȳ k } K k=1 and the initial values are zero. The following procedure illustrates how to convert weak label Y ′ to superpixel labelȲ: Y k = c, ∃ (Y ′ j = c),(1) where j ∈S k and c ̸ = 0. From this, we get the initial superpixel labelȲ. Due to a scarcity of pixel annotations in Y ′ , the majority ofȲ k values are equal to zero. Identify and isolate allȲ k not equal to 0 and randomly select a superpixel block labelȲ ms . The corresponding superpixel block isS ms . We select one of the adjacent (all adjacent superpixel blocks for IRF and SRF, upper adjacent superpixel block for PED) superpixel blocksS ns ofS ms and performs the following operations to infectȲ ns : Y ns =Ȳ ms · I(cos dis(S ms ,S ns ) ≥ t),(2) where cos dis(S ms ,S ns ) represents the superpixel blocks similarity and t is the similarity threshold (the similarity threshold of IRF and SRF to 0.6, and the similarity threshold of PED to 0.5). The similarity ofS ms andS ns as follows: cos dis(S ms ,S ns ) = 255 v=0 (O v ms )(O v ns ) 255 v=0 (O v ms ) 2 255 v=0 (O v ns ) 2 ,(3) If cos dis(S ms ,S ns ) ≥ t, we assign the value ofȲ ms tō Y ns and it is obvious that the adjacent superpixel blocks of Y ns are also very likely to be similar toȲ ms . Therefore, the adjacent superpixel blocks ofS ns will be regarded as the adjacent superpixel blocks ofS ms . The processing of Y ms will not end until all the similarity values of adjacent superpixel blocks are less than threshold t. After processing all initialȲ k not equal to 0, the superpixel labelȲ will be converted to the pixel-wise noisy labelỸ = {Ỹ } n i=1 : Fig. 2 shows the visualization of our noisy labels generated from point annotations. Next, we will describe how to generate label trust graphs. Y i =Ȳ Si .(5) Label Trust Graph for Noise-robust Learning In the process of generating noisy labels, it is unreasonable to give the same confidence to all noisy labeled pixels. Therefore, we propose a method to assign suitable confidence by measuring the actual distance of superpixel blocks. We introduce a pixel-wise label trust graph U = 0.1, . . . , 1). The label trust graph is used to adjust the influence of each pixel's label during training, which can help to mitigate the impact of noise on the network. Specifically, all values of U i are set to 0.5 (the U i value of the pixels contained in the initialȲ k not equal to 0 is set to 1) and update U at the same time when updatingȲ. If cos dis(S ms ,S ns ) ≥ t, calculate the superpixel block distance betweenS ms andS ns and assign lower confidence values to the corresponding pixels on U that are farther away from theS ms . {U i } n i=1 where U i ⊆ (0, 2SMTCL for Multi-category OCT Fluid Segmentation We choose MTCL with one-stage two-category segmentation as the NLL framework. Based on MTCL, we propose a two-stage method 2SMTCL for multi-category OCT fluid segmentation. Unlike MTCL, which introduces CL in real-time during training for label noise characterization, 2SMTCL is a two-stage NLL framework. Specifically, after training the first-stage noisy network based on the teacher-student architecture, introduce CL to characterize the pixel-level noisy labels and refine the noisy labels, and then get the second-stage denoising network trained by the denoised label. Whether it is the first stage or the second stage, the network architecture is MT and keeps CD unchanged. Our motivation is as follows: (i) CL real-time characterize noise will mislead the training of the network because multi-category segmentation is more challenging than two-category segmentation. (ii) Under the premise of introducing label trust graphs to constrain network training, CL real-time characterize noise is unnecessary and timeconsuming. More details about the framework of 2SMTCL are explained in the following. Training Noisy Network Based on Mean-Teacher Architecture Previous studies have demonstrated that noisy labels can have a detrimental effect on model training. To address this challenge, we choose MTCL with one-stage two-category segmentation as the NLL framework, which involves partitioning the dataset into two categories: confidently CD and non-confidently ND. The basic network architecture chooses the Mean-Teacher (MT) model, which is effective in Semi-Supervised Learning (SSL). The MT architecture comprises a student model (updated through backpropagation) and a teacher model (updated based on the weights of the student model at different training stages). A great strength of the MT framework is superior in its ability to leverage knowledge from image-only data using perturbation-based consistency regularization. Formally, we denoted the weights of the student model at training step t as θ t . We updated the teacher model's weights θ t using an exponential moving average (EMA) strategy, which can be formulated as follows: θ t = α θ t−1 + (1 − α)θ t ,(6) where α is the EMA decay rate, and it is set to 0.99, as recommended by [36]. Based on the smoothness assumption [37], we encouraged the teacher model's temporal ensemble prediction to be consistent with that of the student model under different perturbations, such as adding random Gaussian noise ξ to the input images. The student network in MT serves as our first-stage noisy network for pixel-level label noise characterization in the next stage. Confident Learning for Multi-Category Pixel-Wise Conditional Label Errors Despite the presence of label trust graphs U, which are designed to limit the impact of noisy labels on model learning, there remains the potential for label noise to be learned by the model. Confident Learning (CL) [24] is able to identify label errors in datasets and enhance training with noisy labels by estimating the joint distribution between the noisy (observed) labelsỹ and the true (latent) labels y * , as assumed by Angluin [38]. This estimation enables CL to assign higher confidence to instances with more reliable labels and lower confidence to instances with more questionable labels, resulting in finding the error labels. Zhang et al. [29] pioneered the application of CL to medical image segmentation and achieved promising results. Moreover, many follow-up studies [12], [13] have proved the effectiveness of CL for medical image segmentation. However, most of the research is based on the segmentation task of binary classification, further research is needed to explore the effectiveness of CL for multi-category medical image segmentation. Specifically, given an ND image X and we denote X = (x,ỹ) n , whereỹ means the label of pixels and n = w × h means the number of pixels in X. We can obtain the predicted probabilitiesP for m classes by the first-stage noisy network. Assuming that a pixel x labeledỹ = i has large enough predicted probabilitiesP j (x) ≥ t j , there is a possibility that the current annotation for x is incorrect, and it may actually belong to the true latent label y * = j (i ∈ C m , j ∈ C m , C m indicates the set of m class label). Here, we set the average predicted probabilitiesP j (x) of all pixels labeledỹ = j as the threshold t j : t j := 1 |Xỹ =j | x∈Xỹ=jP j (x).(7) Based on this assumption, we can construct the confusion matrix Cỹ ,y * by counting the number of pixels x that are labeled asỹ = i and may actually belong to the true latent label y * = j. The Cỹ ,y * [i][j] represents the count of such pixels for which the observed label isỹ = i and the true latent label is y * = j: Cỹ ,y * [i][j] := Xỹ =i,y * =j ,(8) wherê Xỹ =i,y * =j := {x ∈ Xỹ =i :P j (x) ≥ t j , j = arg max k∈M :P k (x)≥t kP k (x)}. (9) After obtaining the confusion matrix Cỹ ,y * it needs to be normalized. Then, the joint distribution Qỹ ,y * between the noisy labels and the true labels can be obtained by dividing each element in the confusion matrix by the total number of pixels: Qỹ ,y * [i][j] =Cỹ ,y * [i][j] i∈Cm,j∈CmCỹ ,y * [i][j] ,(10)whereCỹ ,y * [i][j] = Cỹ ,y * [i][j] j∈Cm Cỹ ,y * [i][j] · |Xỹ =i | .(11) In order to identify label noise, we adopt the prune by class noise rate (PBNR) strategy, which works by removing examples with a high probability of being mislabeled for every non-diagonal in the Qỹ ,y * [i][j] and select n · Qỹ ,y * [i][j] as mislabeled pixels. Considering our task is multi-category segmentation, we sort the returned error labels index by self-confidence (predicted probability of the given label) for each pixel and select the first 80% of error labels to form the binary estimate error map X err , where "1" denotes that this pixel is identified as a mislabeled one. Such pixel-level error map X err can guide the subsequent label refinement and label trust graph refinement process. Label Refinement and Label Trust Graph Refinement MTCL [13] proposes three different label refinement methods and the MTCL Hard has the best performance. We highly trust the accuracy of the estimated error map X err and impose the hard refinement on the given noisy labels Y. The predicted labelŶ = {Ŷ } n i=1 is calculated by the prediction probabilityP: Y i = arg max cP (c, n).(12) We denoteẎ = {Ẏ } n i=1 as the denoised label, which is formulated by: Y i = I(X i err = 0)Ỹ i + I(X i err = 1)Ŷ i .(13) Similar to the noisy labelỸ, the label trust graph U requires modification since the previous graph represented the trustworthiness of the unreliable noisy label. We denotė U = {U } n i=1 as the refined label trust graph, it can be formulated as:U i = I(X i err = 0)U i + I(X i err = 1)δ,(14) where δ ∈ [0, 1] is the trust level of estimated error map X err , we set as 1. Denoising Network Training We update {Ỹ, U} of ND using {Ẏ,U} for the purpose of training the denoising network. The experimental parameters applied during the training of the noisy network were retained for the training of the denoising network. The student network obtained in the second stage of training is used as our final denoising network. Final Loss Function The overall loss function of the model in the first stage is consistent with that in the second stage. In general, Our total loss is divided into three parts: the supervised loss L c = L ce c + L dice c on CD, the perturbation-based consistency loss L con and the supervised loss L n = L ce n · U ′ + L dice n on ND. The total loss is calculated by: L = αL c + β(L n ) + λL con .(15) Here, U ′ in L n is U for training noisy network and will be replaced byU for denoising network. Empirically, α and β are hyper-parameters and we set α = 1, β = 1. The L con are calculated by the pixel-wise mean squared error (MSE), λ is a ramp-up trade-off weight commonly scheduled by the time-dependent Gaussian function [39] λ(t) = w max · e (−5(1− t tmax ) 2 ) , where w max is the maximum weight commonly set as 0.1 [40] and t max is the maximum training iteration. Such a λ weight representation avoids being dominated by misleading targets when starting online training. EXPERIMENTS Datasets and Experimental Setup OCT Fluid Segmentation Dataset The data for our experiments come from the Eye Center at the Second Affiliated Hospital, School of Medicine, Zhejiang University. The dataset consists of OCT images from various patients, taken at different times and with two distinct resolutions of 1476 × 560 and 1520 × 596. Given the extensive background area in the original images, any other fluids appeared comparatively small. To address this, we centrally cropped all images and resized them to a resolution of 600 × 250. A subset of OCT images rich in the fluid was selected for comprehensive and weak point labeling. The total dataset consists of 1704 OCT images, with 1304 images designated for training and 400 for testing. To ensure the reliability of our results, we meticulously partitioned the dataset such that data from a single patient was used exclusively for either training or testing. A detailed overview of our dataset can be found in Table 2. Baseline Approaches Considering the lack of solid research on noisy label learning for OCT fluid segmentation, it is our objective to in- corporate an extensive range of baselines to facilitate comprehensive and purposeful comparisons across diverse scenarios. This will enable us to provide insights for future research in this field. The baselines can be categorized as follows: (a) (b) (c) (d) • Fully supervised baselines: (i) CD-Sup: uses only CD to train the backbone (2D U-Net [10]) network; (ii) ND-Sup: uses only ND to train the backbone network: (iii) CD&ND-Sup: mixes both CD and ND to train the backbone network. • Mix CD and ND: (i) 2SRnT [29]: involves two stages for pre-training a network using a combination of different datasets, followed by fine-tuning the labels using confidence estimates to train a second network; (ii) Co-teaching [25]: a joint teaching method of the double network; (iii) TriNet [27]: a tri-network based noise-tolerant method extended from co-teaching. • Separate CD and ND: (i) MTCL [13]: Mean-Teacherassisted Confident Learning, which can robustly learn segmentation from limited high-quality labeled data and abundant low-quality labeled data; (ii) Dast [32]: a dual-branch network to separately learn from the accurate and noisy annotations. Implementation and Evaluation Metrics Our method is implemented in Python with PyTorch, using an NVIDIA GeForce RTX 3090 GPU with 24GB memory. The network is trained using the RMSprop optimizer (weight decay=1e-8,momentum=0.9). The learning rate is initialized as 1e-5 and divided by 10 every 2000 iterations. We totally trained 4000 iterations as the network converged. The batch size is set to (8,8) for CD and ND separately and 16 when they are not distinguished during input to the network. We scale the size of the image uniformly to 256 × 128 and then directly input it into the student network. To conduct a comprehensive evaluation, we utilize three renowned indicators -dice score, average surface distance (ASD), and 95% Hausdorff distance (95HD). The dice scores for each category, excluding the background, are presented. Table 1 presents the comparison results under 10%, 20%, and 30% CD settings. Firstly, in the typical supervised settings of CD-Sup and CD&ND-Sup, the network performs poorly and can benefit from additional ND, although their labels contain noise. We hypothesize two possibilities: (i) The partially noisy labels generated by PNS are highly accurate and can provide reliable guidance for the network. The effect of ND-Sup can also reach 73.64%, which is also confirmed. (ii) Even with only 30% of the CD data, the network may still be under-fitting and potentially learn valuable features from the ND. When turning to the Mix CD and ND setting, the three baseline methods, 2SRnT, Coteaching, and TriNet have shown effective performance in mitigating the negative effects caused by noisy labels. In contrast, under the Separate CD and ND settings, MTCL has demonstrated a steady improvement ranging from 10% to 30%. Although Dast has also shown improvement, it still falls short of the baseline performance (CD&ND-Sup). One possible explanation for this discrepancy is domain crossing since Dast was originally designed to perform COVID-19 pneumonia lesion segmentation. Under the 10% -20% CD setting, Co-teaching and MTCL achieved highly competitive results, but 2SMTCL still outperforms them in most metrics. When we increased the proportion of CD to 30%, our method substantially surpassed other label-denoising methods. Overall, in the OCT fluid segmentation task, our method achieves satisfactory results, suggesting that the la-bel trust graph and estimated error map can accurately characterize the location of label noise, enabling the network to fully exploit these informative denoised labels. Fig. 5 presents the results of 2SMTCL and other approaches under the 30% CD setting and 70% ND setting. It is evident that the mask predicted by our method is closer to the ground truth, further demonstrating the effectiveness of 2SMTCL. Experiments on OCT Fluid Segmentation Analytical Ablation Study To verify the effectiveness of each component, we propose different variants to perform ablation studies. Table 3, Table 4, and Table 5 shows our ablation experiments. Our ablation experiments are performed on the denoising network and CD accounted for 30%, and ND accounted for 70%. Table 3 demonstrates the impact of perturbation-based consistency learning and label trust graphs on the performance of the denoising network, where δ refers to the set of label trust graphs and MT denotes the Mean-Teacher architecture. Without the MT architecture, the model's average dice score decreased by 2.5% and performed worse than the noisy network. It shows that the network trained on MT architecture can effectively utilize the pure image information of ND to improve performance. Furthermore, Table 3 indicates that the label trust information contained in the label trust graph can help the network avoid over-fitting noise. The average dice score of the final model's result was even less than 80% when trained without the label trust graph. In the Refinement stage, trust estimation error maps (set to 1) can improve the performance of the network compared to discarding the estimation error maps (set to 0) or leaving them unchanged. These experimental results surface that the components (perturbation-based consistency learning and label trust graphs) of the 2SMTCL can effectively mitigate the negative effects of noisy labels on the network for OCT fluid segmentation. Table 4 displays the impact of different hyper-parameters α and β of the loss function (Equation 15) on the denoising network. As we already have the label trust graph to constrain the loss of ND, we chose to set α = 1 and β = 1, which performed optimally in terms of most metrics. 2SMTCL with appropriate hyper-parameters achieved superior results and proved effective in denoising labels for deep learning-based OCT fluid segmentation. As the bulk of the training data consists of noisy labels generated by point annotations, conducting ablation studies on the PNS method to yield the best noisy-labeled data is critical. We set the superpixel block size to 13 in our experiment, meaning each superpixel block encompasses approximately 169 pixels (13 × 13 on average). A pivotal factor to consider is the setting of the similarity threshold. If the threshold for creating noisy labels is excessively high, the labels may convey insufficient information, leading to network under-fitting. Conversely, if the threshold is too low, the noisy labels could introduce an overwhelming amount of noise, adversely affecting network performance. Therefore, careful selection of an appropriate threshold for generating noisy labels is essential to strike a balance between the amount of information and noise in the labels for optimum network performance. Compared to SRF and IRF, we noted that visually distinguishing PED fluids can be more challenging. Therefore, we set a smaller similarity threshold for PED when determining the threshold. Table 5 presents the final impact of our generated noisy labels for network training under different similarity thresholds. We selected SI t = 0.6, P t = 0.5 due to its superior performance across most metrics. DISCUSSION Visualization of Label Denoising The PNS method is heavily dependent on the similarity among superpixel blocks, leading to the generation of noisy labels with noticeable gaps. These gaps can substantially impact the training process of the model. To rectify this, we utilized the label denoising module to mend the noisy labels with the assistance of reliable guidance. The denoised labels generated by our proposed method are more closely aligned with the ground truths compared to the original noisy labels. Our module proficiently fills in the gaps and refines the edges of the labels, resulting in notable improvements. The superior effectiveness of our label-denoising process is highlighted both in the final performance of the model and in visual depictions of the label-denoising process. As evidenced by the results in Fig. 4, our label denoising module exhibits impressive effectiveness. Future Works In this research, we introduce a strategy that leverages point annotations to generate noisy labels, thereby decreas-ing the reliance on pixel-level annotations for training segmentation models. Our approach is capable of robustly learning OCT fluid segmentation from a limited volume of fully annotated data and a substantial amount of weakly annotated data. Although our methodology has demonstrated encouraging results, it currently still relies on a modest amount of fully annotated data. As a future direction, we will try to devise a training framework that is solely dependent on weakly supervised annotations, which would further lessen the model's requirement for high-quality annotations. Conclusion In this study, we explored the efficacy of employing noisy label learning techniques for OCT fluid segmentation. Initially, we introduced a superpixel-guided method for generating noisy labels from weak point annotations, termed Point to Noisy by Superpixel (PNS). This technique restricts the network from over-fitting to noise by assigning low confidence to pixels with potentially noisy labels. Subsequently, we developed a Two-Stage Mean-Teacher-assisted Confident Learning (2SMTCL) method, designed for multi-category OCT fluid segmentation. This method is capable of segmenting OCT fluid utilizing limited clearly-labeled data and a significant quantity of noisylabeled data. To substantiate the robustness and efficiency of our approach, we compiled an OCT fluid segmentation dataset. The empirical results displayed that our technique surpassed other label-denoising methods, delivering superior segmentation performance and demonstrating notable effectiveness in label denoising. Our study provides an efficient, accurate, and practical solution for fluid segmentation of OCT images, which is expected to have a positive impact on the diagnosis and treatment of patients in the field of ophthalmology. Figure 2 : 2Examples of retinal in OCT image with manual annotations. (a) The original OCT image with fluids; (b) The point annotations by points and lines. (c) The noisy labels generated by PNS; (d) The full mask annotations; In (c) and (d), the green, blue, and red contours denote PED, SRF, and IRF, respectively. Figure 3 : 3Illustration of Two-Stage Mean-Teacher-assisted Confident Learning (2SMTCL) I where O v k represents the number of pixel values v contained in the k th superpixel block. The number of each pixel value contained in theS ms andS ns are calculated by the following formula: (X[j] = v), v = [0, 255]. Figure 4 : 4Examples of retinal in OCT image with manual annotations. (a) The original OCT images with fluids; (b) The noisy labels generated by PNS; (c) The denoised labels; (d) The full mask annotations. Figure 5 : 5Visualized segmentation results of different methods under 30% CD setting and 70% ND setting. From top to bottom is Image, Co-teaching, TriNet, 2SRnT, MTCL, Dast, 2SMTCL, GT. Table 1 : 1Oct Fluid Segmentation Studies. Comparison of the Experimental Results of the State-of-Art Methods on Denoising Methods on Different Ratios of CD and ND. The Best Results are in Bold. (Dice Unit: %, ASD And 95HD Unit: mm)Methods Settings Metrics CD ND Separate? DSC D SRF D IRF D P ED ASD 95HD CD-Sup 10% 0% - 60.94 67.32 77.83 37.65 4.066 14.875 CD&ND-Sup 10% 90% × 77.53 82.84 85.90 63.86 2.775 16.488 2SRnT[29] 10% 90% × 78.08 83.43 87.16 63.66 2.511 15.398 Co-teaching [25] 10% 90% × 80.28 86.42 85.14 69.26 2.904 17.895 TriNet[27] 10% 90% × 78.08 85.35 83.70 65.20 3.192 18.063 MTCL [13] 10% 90% ✓ 77.84 89.30 84.65 59.59 1.680 16.897 Dast[32] 10% 90% ✓ 72.11 71.40 83.90 61.06 3.184 15.934 2SMTCL 10% 90% ✓ 80.45 86.86 88.26 66.23 1.970 13.187 CD-Sup 20% 0% - 65.26 73.60 81.57 40.62 3.670 16.184 CD&ND-Sup 20% 80% × 77.84 84.28 85.94 63.31 2.834 19.499 2SRnT[29] 20% 80% × 78.97 84.37 85.92 66.63 1.939 17.576 Co-teaching [25] 20% 80% × 80.07 87.79 83.51 68.91 2.923 19.241 TriNet[27] 20% 80% × 80.00 88.02 83.13 68.86 2.621 16.871 MTCL [13] 20% 80% ✓ 78.67 83.71 87.43 64.88 2.129 14.716 Dast[32] 20% 80% ✓ 75.41 79.09 77.70 69.45 3.807 14.417 2SMTCL 20% 80% ✓ 80.86 88.57 85.87 68.12 1.333 13.184 CD-Sup 30% 0% - 70.16 69.06 86.40 55.02 3.5061 15.111 CD&ND-Sup 30% 70% × 79.32 87.08 83.75 67.13 1.772 15.710 2SRnT[29] 30% 70% × 79.16 88.39 86.67 62.42 2.465 15.436 Co-teaching [25] 30% 70% × 80.66 86.92 86.63 68.43 2.960 15.513 TriNet[27] 30% 70% × 80.00 85.41 83.03 71.54 2.509 15.324 MTCL [13] 30% 70% ✓ 80.35 88.19 83.25 69.59 2.105 16.355 Dast[32] 30% 70% ✓ 76.83 80.05 85.01 65.43 2.114 12.308 2SMTCL 30% 70% ✓ 82.87 89.46 88.78 70.37 1.696 12.896 ND-Sup 0% 100% - 73.64 76.67 85.03 59.22 3.358 20.292 CD-Sup 100% 0% - 83.53 89.40 87.88 73.31 1.339 11.593 Table 2 : 2ALL THE OCT IMAGES IN OUR DATASET CONTAIN THE NUMBER OF DIFFERENT TYPES OF FLUIDS.Dataset\Fluid PED SRF IRF All Negative Train 1064 691 227 29 Test 155 147 148 84 Table 3 : 3Ablation Study On OCT Fluid Segmentation. The Best Results Are In Bold. (Dice Unit: %, ASD And 95HD Unit: mm)Methods Settings Metrics δ MT DSC DSRF DIRF DPED ASD 95HD Noisy network - - 81.24 87.91 86.66 69.16 1.764 14.163 2SMTCL set to 1 ✓ 82.87 89.46 88.78 70.37 1.696 12.896 set to 1 × 80.17 86.87 86.59 67.04 1.593 13.837 unchanged ✓ 82.14 88.41 86.45 71.54 1.301 13.128 set to 0 ✓ 81.63 89.03 87.28 68.59 1.050 13.670 × ✓ 79.73 85.08 84.80 69.29 2.447 16.223 Table 4 : 4Ablation Study Of Different Loss Weight β Of ND. The Best Results Are In Bold. (Dice Unit: %, ASD And 95HD Unit: mm)Methods Settings Metrics α β DSC D SRF D IRF D P ED ASD 95HD 2SMTCL 1 1 82.87 89.46 88.78 70.37 1.696 12.896 1 0.3 81.57 86.10 85.59 72.72 1.602 13.165 1 0.5 80.99 87.60 83.79 71.57 1.469 13.250 Table 5 : 5Ablation Study Of Different t Of PED, IRF, And SRF When Generating Noisy Label By PNS. The Best Results Are In Bold. (Dice Unit: %, ASD And 95HD Unit: mm) SI t P t DSC D SRF D IRF D P ED ASD 95HD 2SMTCL 0.6 0.5 82.87 89.46 88.78 70.37 1.696 12.896 0.7 0.6 79.35 83.51 86.54 69.19 1.549 15.612 0.5 0.4 81.52 90.87 83.55 70.16 1.58 13.526Methods setting Metrics Optical coherence tomography. D Huang, science. 2545035D. Huang et al., "Optical coherence tomography," sci- ence, vol. 254, no. 5035, pp. 1178-1181, 1991. 1 Optical coherence tomography imaging of macular oedema. G Trichonas, P K Kaiser, British Journal of Ophthalmology. 982G. Trichonas and P. K. Kaiser, "Optical coherence to- mography imaging of macular oedema," British Jour- nal of Ophthalmology, vol. 98, no. Suppl 2, pp. ii24- ii29, 2014. 1 Automated segmentation of intraretinal cystoid fluid in optical coherence tomography. G R Wilkins, IEEE Transactions on Biomedical Engineering. 594G. R. 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{'abstract': 'Automatic segmentation of fluid in OCT (Optical Coherence Tomography) images is beneficial for ophthalmologists to make an accurate diagnosis. Currently, data-driven convolutional neural networks (CNNs) have achieved great success in OCT fluid segmentation. However, obtaining pixel-level masks of OCT images is time-consuming and requires expertise. The popular weakly-supervised strategy is to generate noisy pseudo-labels from weak annotations, but the noise information introduced may mislead the model training. To address this issue, (i) we propose a superpixel-guided method for generating noisy labels from weak point annotations, called Point to Noisy by Superpixel (PNS), which limits the network from over-fitting noise by assigning low confidence to suspiciously noisy label pixels, and (ii) we propose a Two-Stage Mean-Teacher-assisted Confident Learning (2SMTCL) method based on MTCL for multi-category OCT fluid segmentation, which alleviates the uncertainty and computing power consumption introduced by the real-time characterization noise of MTCL. For evaluation, we have constructed a 2D OCT fluid segmentation dataset. Compared with other state-of-art label-denoising methods, comprehensive experimental results demonstrate that the proposed method can achieve excellent performance in OCT fluid segmentation as well as label denoising. Our study provides an efficient, accurate, and practical solution for fluid segmentation of OCT images, which is expected to have a positive impact on the diagnosis and treatment of patients in the field of ophthalmology.', 'arxivid': '2306.02582', 'author': ['Tengjin Weng ', 'Yang Shen ', 'Kai Jin ', 'Zhiming Cheng [email protected] ', 'Yunxiang Li [email protected] ', 'Gewen Zhang ', 'Shuai Wang [email protected] ', '\nZhejiang Sci-Tech University\nChina\n', '\nLishui University\nChina\n', '\nSecond Affiliated Hospital of Zhejiang University\nChina\n', '\nUT Southwestern Medical Center\nHangzhou Dianzi University\nDallasTXChina, USA\n', '\nLishui University\nChina\n', '\nHangzhou Dianzi University\nChina\n'], 'authoraffiliation': ['Zhejiang Sci-Tech University\nChina', 'Lishui University\nChina', 'Second Affiliated Hospital of Zhejiang University\nChina', 'UT Southwestern Medical Center\nHangzhou Dianzi University\nDallasTXChina, USA', 'Lishui University\nChina', 'Hangzhou Dianzi University\nChina'], 'corpusid': 259075398, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18293, 'n_tokens_neox': 15015, 'n_words': 9288, 'pdfsha': '410ad1934747c6be2cb51bac2ca3c3437317a7bf', 'pdfurls': ['https://export.arxiv.org/pdf/2306.02582v1.pdf'], 'title': ['Learning from Noisy Labels Generated by Extremely Point Annotations for OCT Fluid Segmentation', 'Learning from Noisy Labels Generated by Extremely Point Annotations for OCT Fluid Segmentation'], 'venue': []}

arxiv

GrASPE: Graph based Multimodal Fusion for Robot Navigation in Outdoor Environments Kasun Weerakoon Jagan Adarsh Jing Sathyamoorthy Tianrui Liang Utsav Guan Dinesh Patel Manocha GrASPE: Graph based Multimodal Fusion for Robot Navigation in Outdoor Environments Video is available at We present a novel trajectory traversability estimation and planning algorithm for robot navigation in complex outdoor environments. We incorporate multimodal sensory inputs from an RGB camera, 3D LiDAR, and the robot's odometry sensor to train a prediction model to estimate candidate trajectories' success probabilities based on partially reliable multi-modal sensor observations. We encode high-dimensional multi-modal sensory inputs to low-dimensional feature vectors using encoder networks and represent them as a connected graph. The graph is then used to train an attention-based Graph Neural Network (GNN) to predict trajectory success probabilities. We further analyze the number of features in the image (corners) and point cloud data (edges and planes) separately to quantify their reliability to augment the weights of the feature graph representation used in our GNN. During runtime, our model utilizes multi-sensor inputs to predict the success probabilities of the trajectories generated by a local planner to avoid potential collisions and failures. Our algorithm demonstrates robust predictions when one or more sensor modalities are unreliable or unavailable in complex outdoor environments. We evaluate our algorithm's navigation performance using a Spot robot in real-world outdoor environments. We observe an increase of 10-30% in terms of navigation success rate and a 13-15% decrease in false positive estimations compared to the state-of-the-art navigation methods. Abstract-We present a novel trajectory traversability estimation and planning algorithm for robot navigation in complex outdoor environments. We incorporate multimodal sensory inputs from an RGB camera, 3D LiDAR, and the robot's odometry sensor to train a prediction model to estimate candidate trajectories' success probabilities based on partially reliable multi-modal sensor observations. We encode high-dimensional multi-modal sensory inputs to low-dimensional feature vectors using encoder networks and represent them as a connected graph. The graph is then used to train an attention-based Graph Neural Network (GNN) to predict trajectory success probabilities. We further analyze the number of features in the image (corners) and point cloud data (edges and planes) separately to quantify their reliability to augment the weights of the feature graph representation used in our GNN. During runtime, our model utilizes multi-sensor inputs to predict the success probabilities of the trajectories generated by a local planner to avoid potential collisions and failures. Our algorithm demonstrates robust predictions when one or more sensor modalities are unreliable or unavailable in complex outdoor environments. We evaluate our algorithm's navigation performance using a Spot robot in real-world outdoor environments. We observe an increase of 10-30% in terms of navigation success rate and a 13-15% decrease in false positive estimations compared to the state-of-the-art navigation methods. I. INTRODUCTION Mobile robots have increasingly been utilized in numerous outdoor applications such as delivery [1], agriculture [2], surveillance [3], exploration [4], rescue missions [5], etc. These applications need the ability for the robots to navigate in challenging outdoor environmental conditions such as low lighting, cluttered vegetation, etc. In this work, we consider such environments as unstructured outdoor environments. The robots' perception could encounter noise, occlusions, or other error modes and failures when navigating in such environments. Especially, cameras undergo motion blur, low lighting, and occlusions [6,7], while LiDAR point clouds experience heavy distortions/scattering in cluttered vegetation [8]. A key issue is developing methods that can perform reliable perception and planning computations by taking into account such sensor uncertainties. In terms of robot perception and multi-modal sensor fusion, many recent deep learning (DL) techniques have demonstrated promising results. These methods combine distinct feature representations from different sensor modalities such as cameras, IMU, and LiDAR [11,12,13,14] or perform only camera-LiDAR fusion [15,16]. Many of these works in sensor fusion incorporate Convolutional Neural Networks (CNN) or other linear operators to encode spatial information. However, such Fig. 1. Spot robot trajectories while navigating in an unstructured outdoor terrain using various methods. We consider that a trajectory is non-traversable/unsuccessful if it leads the robot to collisions, stumbling, or similar modes of failure. Our method GrASPE, utilizes multi-modal sensory inputs from RGB camera, 3D LiDAR, and the robot's odometry to estimate the future success probability of a candidate trajectory to avoid obstacles and other non-traversable regions such as A and C. Multi-modal fusion methods such as PAAD [7] and NMFNet [9] provide inconsistent perception under extremely cluttered settings. LiDAR-based methods such as DWA [10] recognize A, B, and C regions as obstacles due to the cluttered vegetation even though region B is actually traversable. GrASPE identifies trajectories that are traversable by intelligently fusing such partially reliable camera and LiDAR data. operators cannot capture the complex correlations between different modalities to fuse them effectively [17]. Multi-sensor data represented as graphs inherently overcome this limitation since graph edges can be defined to model the complex correlations in the data. Furthermore, recently developed Graph Neural Networks (GNN) [18] provide powerful learning and fusion capabilities on graph data as opposed to traditional neural networks such as CNNs. A key challenge is developing reliable methods for outdoor scenes since most existing DL and GNN methods [19,20] are formulated for indoor scenarios and do not consider the variable reliability of the cameras and LiDAR outdoors. Main Contributions: We present GrASPE(Graph Attention based Sensor fusion for Path Evaluation), a novel trajectory traversability estimation and planning algorithm for legged robot navigation in unstructured outdoor environments. We incorporate multi-modal observations from an RGB camera, 3D LiDAR, and robot odometry to train a prediction model to estimate candidate trajectories' navigation success probabilities. Our model learns the correlation between multi-sensor data under unstructured outdoor conditions where the camera undergoes occlusions, motion-blur, and low-lighting, while the LiDAR point cloud experiences scattering and distortions. We further explicitly evaluate the reliability of the camera and LiDAR data for achieving reliability-aware multi-modal fusion in our prediction model. The key contributions of our approach include: • A novel trajectory success prediction model that estimates a candidate trajectory's probability of avoiding navigation failures such as collisions, getting stuck, etc. based on partially reliable multi-modal sensor observations in unstructured outdoor environments. We project the highdimensional sensor inputs (RGB images and point clouds) into lower dimensional feature vectors and represent them as a connected undirected feature graph to train an attention-based GNN (A-GNN) for accurate sensor fusion. Our model accurately estimates trajectories' success probabilities even in poorly-lit, densely vegetated environments in real-time which results in a 13-15% decrease in false positive rate compared to other methods. • A novel reliability estimation method to quantify the usefulness of the image and 3D point cloud data for sensor fusion. Our formulation counts the number of features such as corners (for RGB images), and edges and planes (for point clouds) to compute reliability scores and uses it to weigh the edges in the feature graph. Weighing based on reliability suppresses the correlations between unreliable sensor modalities leading to success predictions only based on reliable sensors at any time instant. Including reliability in feature graph improves success rate by 20-40% compared to the predictions made without the reliability measures. • We show that our feature graph's construction using the reliability metric is undirected and non-negatively weighted. We prove that its Laplacian is spectrally decomposable, thereby allowing graph convolution operations to be utilized in our GNN. This helps to learn complex node embeddings from our feature graph. • A local planner that computes dynamically feasible, collision-free, and traversable velocities accounting for their success probabilities. We demonstrate our algorithm on a Boston Dynamics Spot robot to evaluate its performance in real-world outdoor scenarios with variable lighting and vegetation density. Our method results in an 10-50% improvement in terms of success rate compared to state-of-the-art methods. II. RELATED WORK In this section, we discuss the existing work on multi-modal sensor fusion and anomaly detection in the context of robot navigation. A. Multi-modal Sensor Fusion for Navigation Robot perception in real-world environments can be challenging due to varying conditions such as lighting, noise, occlusions, motion blur, etc. To this end, prior algorithms incorporate uncertainty modeling and adaptation techniques for robot navigation using a single sensor modality [21,22]. However, such methods perform well only under controlled outdoor settings, where the sensor inputs experience limited environmental perturbation or noise. Therefore, multi-sensor fusion methods are used to mitigate unreliable perception from individual sensors [23,24,25]. For instance, different variants of the Kalman Filter [26] are widely used to fuse odometry (from wheels or visual/LiDAR estimations), IMU, and GPS sensor data to compute improved localization for navigation on slippery and complex terrains [27]. However, these techniques require the different sensors to provide the same type of feature observations (e.g., odometry), which limits their applicability to fuse inputs that provide distinct features (e.g., images and point clouds). Deep learning methods have also been widely used for sensor fusion [23,25,28,29]. Particularly, the camera-lidar fusion is employed in navigation and simultaneous localization and mapping (SLAM) [30] literature to obtain combined perception from visual and geometric features [12,15]. However, these methods typically assume that both camera and LiDAR perception are reliable in terms of feature availability. In contrast, our proposed algorithm deals with real unstructured outdoor environments where this assumption is inapplicable. B. Anomaly Detection based Prediction Models To navigate in unstructured outdoor environments, the robot must compute trajectories that avoid collisions and prevent the robot from getting stuck (e.g., in vegetation such as bushes, vines, etc.). To this end, several recent works have incorporated anomaly detection algorithms to identify collisions or failures during navigation as anomalies based on multisensory observations [31,32,33,34]. Such algorithms can be trained using simple positively (traversable) and negatively (non-traversable, collision, etc) labeled observations, which can be created trivially, as opposed to the extensive labeling required for training supervised learning methods [35,36,37]. Wellhausen et al. [35] utilize RGB and depth images to train a predictive model that can generate an anomaly mask that reflects "known" and "unknown" (anomaly) regions in the scene. However, they assume that the observations have constant illumination and are feature-rich for accurate predictions. However, since these observations do not account for potential future navigation failures and collisions, such methods could lead to catastrophic accidents during navigation. C. Proactive Anomaly Detection To deal with possible future navigation failures due to sensor uncertainties, proactive anomaly detection methods have been utilized to estimate the probability of such failures using predicted trajectories, and multi-sensor fusion [7,38]. A supervised variational autoencoder (SVAE) is used [39] for failure identification in unstructured environments using 2D LiDAR data. This SVAE is used in [7] as a LiDAR feature encoder to identify navigation anomalies proactively in crop fields using multi-modal fusion. However, the 2D LiDAR observations are heavily distorted in cluttered outdoor environments such as tall grass where 3D point clouds can provide richer information. Other navigation models [40,36] execute actions by avoiding undesirable maneuvers in cluttered environments based on the current visual sensory observations. However, such visionbased systems lead to erroneous predictions during camera occlusions and low light conditions. Our proposed method uses a camera, 3D LiDAR, and odometry sensor fusion strategy to achieve better perception in such conditions for robot navigation. III. BACKGROUND AND PROBLEM FORMULATION In this section, we state our problem formulation, provide some background to the Graph Neural Networks (GNNs) used in our approach, and the Dynamic Window Approach (DWA) [10]. A. Notations, and Definitions We present the essential symbols and notations in Table I. B. Problem Formulation Our formulation for GrASPE can be stated as follows: Formulation III.1. To predict the success probability vector E t ∈ [0, 1] T of a robot trajectory using its local multi-modal fused observations O t while accounting for the variable reliability (r img , r point ) of the camera and LiDAR observations respectively. We define success probability as a trajectory's probability of not encountering collisions, stumbling, or similar modes of failure. The success probability is then used to evaluate trajectories that the robot can use for navigation. The overall system architecture of the prediction model is presented in Fig. 2. GrASPE System Architecture for candidate trajectories' success probability estimation. We utilize RGB images, 3D point clouds, the robot's velocity history, and the predicted trajectory of the robot as an image to train a novel graph-based prediction model to estimate the given trajectory's probability of success in terms of navigation. C. Graphs and Graph Neural Networks Real-world data modalities (say images and point clouds) used for sensor fusion exhibit complex correlations between them. For instance, a rock captured in an image from a robot is represented by RGB values, as XYZ values in point clouds, and a spikes in the acceleration and angular velocity in a 6-DOF IMU when the robot runs over it. For accurate sensor fusion, these complex correlations cannot be represented by regular data structures (i.e. in the Euclidian domain). Representing the data streams as a graph offers the capability of modeling the complex correlations between them as weighted edge connections in the graph. Therefore, in our work, sensor data is processed and represented as a weighted graph G := (V, W), where V = 1, 2, . . . , N is a set of N nodes/vertices associated with elements of the sensors' feature vectors. W is a weighted adjacency matrix with entries W i,j representing the strength of the connection (edge) between feature elements i and j. Graph Neural Networks (GNNs) are a class of neural networks that operate on data represented as graphs. We incorporate Graph Convolution Network (GCN) [41] and Graph Attention Network (GAT) [42,43] operators in our GNN to encode and pay attention to the local graph structure and node embeddings. These networks are general forms of convolution and attention mechanisms and are capable of encoding node features and correlations between the graph nodes respectively. Especially, the GAT operator performs graph attention in four steps: Linear transformation, Leaky ReLU activation, Softmax normalization, and Multi-head attention which often outperforms traditional multi-head attention approaches on Euclidean data structures. IV. OUR ALGORITHM: GRASPE In this section, we present the details of our proposed path evaluation algorithm. We first discuss the components of the prediction model trained for the trajectory's success probability estimation: 1. Acquiring multi-modal data and reducing its dimensionality; 2. Estimating camera and LiDAR's reliability; 3. Graph-based prediction model. Finally, we explain the details of how robust perception is used for planning. A. Multi-modal Observations We design our prediction model (i.e., success probability estimation model) using the following multi-sensor inputs as observations: RGB image I rgb t ∈ R w×h×3 , 3D point cloud P lidar t ∈ R 3×Np , robot's velocity history V t = {(v, ω) t−T +1 , . .., (v, ω) t } and predicted trajectory from the robot's current velocity as an image I traj t ∈ R w×h×1 (see Fig. 2). To obtain I traj t , we first extrapolate the robot's trajectory for the next T = 10 times steps based on the current velocity (v, ω) t . Then the resulting trajectory is projected to a blank image using homography projection. The RGB image and 3D point cloud capture visual and geometric features of the environment respectively. The robot's velocity history is used to account for the robot's recent behavior. The ground truth labels for predicting a trajectory's success are represented by a vector E t of 0's (for portions of the trajectory with failures) and 1's (for portions with successes) of length T. Finally, the prediction model uses the observations O t = I rgb t , P lidar t , V t , I traj t , and the ground truth labels to train to estimate the traversability of the predicted trajectoryÊ t = [p t , p t+1 , .., p t+T −1 ] (i.e., success probability vector). Details of the observation data collection and ground truth labeling are described in Section V-B. B. Multi-modal Feature Vector Generation Different sensor modalities capture different environmental features (images capture visual features, point clouds capture edges and surfaces), and fusing them effectively is non-trivial. Moreover, raw image and point cloud data processing are computationally expensive due to the high dimensionality. Therefore, we pre-process each sensor input using separate feature encoding networks to obtain dimension-reduced feature vector representations. The sizes of these feature vectors are chosen empirically considering the trade-off between the resulting graph size (effects on the computation complexity) and the feature encoding quality (smaller feature vectors may not properly encode the input data). 1) Visual Feature Extraction: We utilize a ResNet [44] based pipeline to extract image features f img from the camera RGB image I rgb t . We incorporate a CBAM [45] module between the network layers to perform spatial and channel attention towards the important visual features. The input image I rgb t is of size 320 × 240 (i.e., w = 320 and h = 240). The ImgNet branch in Fig. 2 includes a ResNet18 backbone with CBAM layers added similar to [45]. The output feature vector f img of dimension 40×1 is obtained by passing through a Sigmoid() activation layer. 2) LiDAR Feature Extraction: A Pointnet [46] based network is incorporated to extract point cloud features f point from the LiDAR data P lidar t . To reduce the data size, we restrict LiDAR point cloud to [−π/2, π/2] field of view w.r.t. the robot's heading direction (see C. Sensor Reliability Estimation RGB images and the point cloud data become unreliable in certain outdoor conditions (e.g. low luminance, scattering of laser rays, etc.). Even though the feature encoding networks proposed in the literature and in Section IV-B1 are capable of feature extraction from images and point clouds, they do not account for their reliability. Therefore, we estimate the reliability of the image and point cloud inputs at each time step quantitatively using classical image and point processing methods that execute in real-time [47,48]. We further assume that the instantaneous robot velocities obtained from its odometry are not significantly affected by the environment. Therefore, it is considered reliable during operation. 1) Image Reliability Estimation: We consider that the images captured by the camera have high reliability if they have: 1. High image brightness, and 2. Availability of visual features such as corners, edges, etc [49]. See Fig. 4 and 5 for sample reliability comparisons. To estimate the image brightness, we first convert the input RGB image I rgb To estimate the availability of useful features in the image, we calculate the number of corner features n c in the input image I rgb t using FAST (Features from Accelerated Segment Test) algorithm [50,47]. These corner features are fast to compute and inherently reflect the non-blurriness and well-lit condition, therefore a good measure of the image's reliability. We consider the image is feature rich if the n c is higher than a threshold. Hence, r corners = nc (w×h) . Here, w and h are the height and width of the input RGB image. The final image reliability measure is obtained as a normalized scalar value r img ∈ [0, 1] using the weighted sum of r bright and r corners : r img = α b r bright + α c r corners(2) 2) Point Cloud Reliability: We observe that the lidar point cloud is heavily distorted in the presence of unstructured vegetation leading to poor estimation of the surrounding objects' geometries. Therefore, to evaluate the point cloud reliability, we perform edge and planar feature extraction to calculate the number of 3D features available in the point cloud P lidar t at a given time t. High numbers of edges and planes denote low scattering (i.e., structured) in the point cloud. Let X l be the l th 3D point in the input point cloud P lidar t and let M be the set of points in the neighborhood of X l acquired from an instance of the point cloud. Then, we can obtain the local surface smoothness evaluation factor c l of the point l as, c l = 1 ||X l ||.|M| k∈M,k =l X l − X k ,(3) where X k with k = 1, 2, .., |M| are the coordinates of the 3D points in the set M. Zhang et al. [48] demonstrate that the points with higher and lower c l values obtained from equation 3 belong to edge (non-smooth) and planar (smooth) features respectively. Hence, we define two threshold values c max and c min as the minimum and maximum smoothness thresholds to consider a given point on a surface as an edge or a plane respectively. The resulting edge and planar feature point sets can be denoted as: S edge = {l|c l ≥ c max , l ∈ P lidar t } and S planar = {l|c l ≤ c min , l ∈ P lidar t } respectively. r edge = |S edge | |P lidar t | , r planar = |S planar | |P lidar t | .(4) Finally, the point cloud reliability measure r point is derived as a weighted combination of r edge and r planar using two tunable parameters β e and β p . r point = β e r edge + β p r planar . The weighted adjacency matrix W represents the strength between the graph nodes (i.e. edge weights). i.e., the connectivity between the encoded feature elements in f vec . Typically, the edge weights are calculated using a distance measure (absolute difference or L2 norm). However, we modify these edge weights (especially between the nodes corresponding to the image and point cloud features) based on the reliability measures (Section IV-C) as follows. Let W i,j be the edge weight between the feature elements i and j in f vec . W i,j = exp −{λ·|fvec(i)−fvec(j)|·(2−ri,j )} , if i = j 0, otherwise,(6) where, λ > 0 is a tunable scalar and r i,j ∈ [0, 1] represents the reliability measure between the node i and j as follows, r i,j =          r img , if {i, j|i ∈ f img , j / ∈ f point } r point , if {i, j|i / ∈ f img , j ∈ f point } 1 2 r point + r img , if {i, j|i ∈ f img , j ∈ f point } 1, otherwise. (7) Hence, we refer to the derived graph representation G t of the feature vector f vec as a reliability-aware feature graph. E. A-GNN Architecture We design a light-weight Attention-based Graph Neural Network (A-GNN) presented in Fig.3 to predict the success probability vectorÊ t for a given trajectory. We first pass our reliability-aware feature graph G t through a 16-channel graph convolutional layer (GCN) to encode the node features. Secondly, a graph attention network (GAT) is utilized to identify the important neighbors (i.e., strongly correlating neighbors) of each node of the 16-channel graphs. Next, a dropout layer is incorporated for regularization to minimize the risk of network over-fitting. The output graph is obtained after passing through another GCN layer. We use the ReLU activation layer after each GCN layer. The output graph is concatenated into a vector and passed through two dense layers and sigmoid activation to obtain the output prediction vector E t . Hence, the overall prediction pipeline can be treated as a mapping function ψ GrASP E : O t →Ê t . F. Theoretical Validation of Graph Construction Lemma 1. GrASPE's reliability aware feature graph G t generated using the weight matrix W ensures that G t is an undirected graph with non-negative weights. Proof: Let w i,j ∈ W. By construction, r i,j = r j,i ∀ i, j, and i = j =⇒ w i,j = w j,i ∀ i, j, and i = j and w i,j = 0 ∀ i, j, and i = j. This implies that W is symmetric and G t is undirected. Let m i,j = λ|f vec (i) − f vec (j)|r i,j from the Eq.6. Then, m i,j ≥ 0, ∀ i, j because λ > 0 and r i,j ∈ [0, 1]. Therefore, exp −mi,j > 0 ∀ i, j, i = j =⇒ w i,j ≥ 0 ∀ i, j =⇒ W has non-negative weights. GCN layers in our A-GNN perform spectral graph convolution which requires the input graph's Laplacian L to be spectrally decomposable [41] (i.e., eigen decomposition on real symmetric matrices). We prove that our graph construction leads to a spectrally decomposable graph Laplacian matrix using the Lemma 2. From Graph theory, we consider the graph Laplacian L = D − W, where D is a diagonal matrix with D i,i = deg(v i ). Here, deg(v i ) is the degree of a vertex which is a measure of the number of edges terminating at that vertex. In this context, we consider D i,i = j w i,j . By construction and from Lemma 1, we can observe that D is real and W is hermitian (i.e., w i,j = w * j,i ). Lemma 2. Laplacian matrix L of the graph G t is spectrally decomposable. Proof: From Lemma 1, Re(w i,j ) ≥ 0 ∀ i, j. Consider a real valued x, x T Lx = i,j w i,j (x i − x j ) 2 ∵ L = D − W = i<j w i,j (x i − x j ) 2 + i>j w i,j (x i − x j ) 2 = i<j w i,j (x i − x j ) 2 + i>j w * j,i (x i − x j ) 2 = i<j w i,j (x i − x j ) 2 + i<j w * i,j (x j − x i ) 2 = i<j w i,j (x i − x j ) 2 + i<j w * i,j (x i − x j ) 2 = i<j (w i,j + w * i,j )(x i − x j ) 2 = 2 i<j Re(w i,j )(x i − x j ) 2 ≥ 0 =⇒ L is Positive semi-definite.(8) ∴ Symmetric matrix L has all real eigenvalues. Further, the corresponding eigenvectors u 1 , .., u N can be taken to be orthonormal by, u T i u j = 1, if i = j 0, if i = j,(9) Therefore, from the Spectral theorem [51], L can be spectrally decomposed as, L = U ΛU T , where U is the eigenvector matrix, and Λ denotes the diagonal matrix of sorted eigenvalues. Hence, having an undirected graph with non-negative weights and a spectrally decomposable Laplacian enables us to perform the graph convolution operation on our feature graph G t to encode and learn its complex node embedding (i.e., the correlation between adjacent feature nodes) significantly better compared to CNNs and other architectures (see Benefits of A-GNN in Section V). G. Reliability-aware Planning Our local planner adapts the Dynamic Window Approach (DWA) [10] to perform navigation while evaluating the success probabilities of the generated trajectories using GrASPE. Our overall algorithm is presented in Algorithm 1. We explain the algorithm and the symbols used here. V s (line 4) is defined as the space of all the possible linear and angular robot velocities (v, ω). While the robot's distance to its goal (d goal ) is greater than a threshold (line 5), DWA computes V a , the set of collision-free velocities, and V d , the set of reachable velocities based on the robot's acceleration constraints (lines 6 and 7). Next, the reliability of the point cloud r point obtained from the robot is calculated and compared against a threshold (lines 8 and 9). If the point cloud is unreliable, implying that the detected obstacles and the calculated collision-free set V a are erroneous, the search space V r for choosing the robot's velocity is restricted to the intersection of V s and V d . Algorithm 1 GrASPE based Outdoor Navigation 1: Input: goal, obs, d goal , I rgb t , P lidar t , V t 2: Output: (v (t+1) , ω (t+1) ) 3: Initialize : T, ∆t, v max , ω max ,v max ,ω max , α b , α c , β e , β p , γ 1 , γ 2 , γ 3 , r th , d th 4: V s = {(v, ω)|v ∈ [0, v max ], ω ∈ [−ω max , ω max ]} 5: while d goal ≥ d th do 6: V d = {(v, ω)|v ∈ [v −v max ∆t, v +v max ∆t], ω ∈ [ω − ω max ∆t, ω +ω max ∆t]} 7: r point = P ointcloudRelibaility(P lidar t ) using the Eq.5. 8: if r point ≤ r th then 9: V r = V s ∩ V d 10: Q(v, ω) = σ γ 1 .heading(v, ω) + γ 3 .vel(v, ω)V r = V s ∩ V d ∩ V a 14: Q(v, ω) = σ γ 1 .heading(v, ω) + γ 2 .dist(v, ω) + γ 3 .vel(v, ω) 15: end if 16: (v * t , ω * t ) = argmax(Q(v, ω)) 17: I traj t = GenerateT rajectoryImage(v * t , ω * t , T ) 18: O t = I rgb t , P lidar t , V t , I traj t 19:Ê t = ψ GrASP E (O t ) 20: if min(Ê t ) ≥ e th then 21: return (v * t , ω * t ) 22: else 23: V r = V r \ (v * t , ω * t ) 24: Goto to step 8 25: end if 26: end while DWA chooses the velocity belonging to V r that maximizes an objective function Q(v, ω) as the final velocity for navigation. Q(v, ω) consists of three terms: heading(.), dist(.) and vel(.) which quantify the robot's heading towards the goal, distance to the closest obstacle in the trajectory, and the forward velocity of the robot, respectively. If the point cloud is unreliable, the dist(.) function in Q(v, ω) is unreliable. Therefore, we omit it from the objective function (line 11). Otherwise, DWA's original search space and objective function are used (lines 13 and 14) to compute the optimal/maximizing velocities (v * t , ω * t ) (line 16). Lines 17-19 use GrASPE to evaluate the optimal trajectory's success probability. If the optimal trajectory has a high success probability, the robot uses it for navigation (lines 20-21). Otherwise, it is removed from the search space and a new optimal velocity is computed (lines [22][23][24]. V r typically has around 12 to 16 (v, ω) pairs making the search tractable. V. RESULTS AND ANALYSIS We detail our method's implementation and experiments conducted on a Spot robot. Then, we perform ablation studies and comparisons to highlight the benefits of our algorithm. A. Implementation GrASPE is implemented using PyTorch and PyTorch Geometric (PyG). The prediction model is trained in a workstation with an Intel Xeon 3.6 GHz processor and an Nvidia Titan GPU using real-world data collected from a Spot robot. The robot is equipped with an Intel NUC 11 (a mini-PC with Intel i7 CPU and NVIDIA RTX 2060 GPU), a Velodyne VLP16 LiDAR, and an Intel RealSense L515 camera. B. Dataset The multi-modal dataset used in this work is collected by operating the Spot robot under different lighting conditions in an outdoor field that includes bushes, small trees, hanging leaves, and grass regions of different heights and densities. We incorporate a randomized planner to collect I rgb t , P lidar t and V t from the robot to minimize human effort. The ground truth labels are created part manually and part automatically. C. Evaluations We use the following evaluation metrics to compare our method's performance against: PAAD [7], NMFNet [9], GA-Nav [52], and DWA [10] algorithms. PAAD is a proactive anomaly detection-based navigation method that fuses images, 2D LiDAR scans, and the robot's future trajectory using Multi-head attention. NMFNet is a deep multi-modal fusion network that has three branches: 2D laser, RGB images, and point cloud data for action prediction. GA-Nav is a multi-modal navigation framework that combines image semantic segmentation with elevation maps for traversability estimation. DWA is a local planner that utilizes a 2D LiDAR scan for obstacle avoidance. We further compare the GrASPE model without reliability estimations (GrASPE w/o reliability) and without the A-GNN model (GrASPE w/o A-GNN) for ablation studies. PAAD, NMFNet, and the two ablation models are trained using the same dataset (Section V-B) for evaluations. GA-Nav is trained on publicly available outdoor datasets (RUGD [54] and RELLIS-3D [55]) since it requires pixel-level segmentation labels. Success Rate -The number of successful goal-reaching attempts (while avoiding non-pliable vegetation and collisions) over the total number of trials. Normalized Trajectory Length -The ratio between the robot's trajectory length and the straight-line distance to the goal in both successful and unsuccessful trajectories. False Positive Rate (FPR) -The ratio between the number of false positive predictions (i.e., actually non-traversable trajectories predicted as traversable) and the total number of [7], NMFNet [9], GA-Nav [52], DWA [10], GrASPE w/o reliability, and GrASPE w/o A-GNN. The test environments are different from the training scenarios. GrASPE is able to successfully continue navigation in varying environmental conditions. actual negative (non-traversable) samples encountered during a trial. We report the average over all the trials. False Negative Rate (FNR) -The ratio between the number of false negative predictions (i.e., actually traversable trajectories as non-traversable) and the total number of actual positive samples encountered during a trial. We report the average over all the trials. D. Test Scenarios We evaluate our navigation method's performance in the following four outdoor test scenarios that differ from the training environments. At least 10 trials are conducted in each scenario. • Scenario 1 -Contains trees, pliable and non-pliable plants and bushes (Fig. 1). • Scenario 2 -Contains hanging leaves and trees where the sensors can undergo partial and complete occlusions ( Fig. 7(a)). • Scenario 3 -Contains open ground and pliable tall grass regions ( Fig. 7(b)). • Scenario 4 -Contains trees and vegetation under low light conditions (Fig. 7(c)). E. Analysis and Discussion The qualitative and quantitative results are presented in Fig. 6, 7, and Table II respectively. Qualitative results in Fig. 6 demonstrate that our method is able to differentiate trajectories with high and low success probabilities even during camera occlusions and low light conditions (see Fig. 6(c), (d)). GrASPE also provides accurate success prediction in cluttered regions in Fig. 6(a), (b) where point cloud encounters heavy distortions. We further observe that GrASPE outperforms all the comparison methods in terms of success rate, false positive rate, and false negative rate when navigating in highly unstructured outdoor regions. This is primarily due to the improved trajectory success probability predictions generated by our algorithm using reliability-aware sensor fusion. PAAD and NMFNet demonstrate comparative perception and navigation performance in moderately complex environments since they are designed for multi-modal fusion. Especially, PAAD is capable of predicting potential navigation failures under slightly distorted sensor observations which result in reasonably lower FPR and FNR for Scenarios 2 and 3. However, PAAD fails to continue with successful navigation when the robot enters to extremly challenging regions such as cluttered vegetation (i.e., tall grass in Scenario 3) and hanging leaves in Scenario 2. NMFNet shows severe navigation performance degradation in all scenarios due to lack of potential failure prediction capabilities (i.e., only performs reactive prediction based on current observation). 2D LiDAR based methods such as DWA perform well in less cluttered regions such as scenario 4. However, thin, cluttered, and pliable vegetation are also detected as obstacles by DWA. Hence, it could not succeed in Scenarios 1 and 3 even once during testing. Instead, the robot gets stuck near the cluttered region and attempts to find free space by rotating. In scenario 4, DWA generates longer trajectories to avoid thin pliable grass regions while GrASPE traverses through them to generate relatively shorter trajectories. Further, PAAD, NMFNet and GA-Nav result in incomplete trajectories mostly under visually challenging scenarios 2 and 4, which leads to normalized trajectory length values of less than 1. We report them since they measure the progress toward the goal. GA-Nav uses a point cloud based elevation map along with RGB images to estimate terrain traversability. We observe that the distorted point clouds generate misleading estimations of terrain elevation. This results in a significantly lower success rate for GA-Nav in scenario 2. However, GrASPEbased navigation is robust even when the robot's camera is 100% occluded, and the LiDAR is partially occluded. It uses the partially reliable LiDAR and robot's odometry to avoid collisions and navigate to its goal. Failure cases: There are trials in scenarios 1 and 2 where both the camera and LiDAR are highly occluded and unreliable. In such cases, GrASPE-based navigation does not have any feedback to avoid obstacles and non-traversable regions and could cause collisions or get stuck. We note, however, that our method still significantly outperforms the most recent and related works such as PAAD and NMFNet. Benefits of Attention-based GNN (A-GNN) : In this ablation, we feed f vec to a set of 1D convolution layers, instead of using the graph representation and our A-GNN. We observe this GrASPE w/o A-GNN pipeline cannot outperform our GrASPE system qualitatively or quantitatively after training and testing with our dataset. This is primarily due to the strong feature correlation learning capabilities of our A-GNN. Ablation Study on Reliability Measures: We compare the navigation and prediction performance of GrASPE with and without using the reliability terms in equation 6. GrASPE without reliability leads to erroneous predictions when one or more sensory inputs are heavily distorted. Hence, even though the GrASPE model without reliability can navigate reasonably well compared to other methods, it still could not complete the navigation tasks consistently which result in a lower success rate and higher false positive and negative rates. For example in Scenario 3 in Fig. 7, the robot navigates successfully until the point cloud and camera are distorted in the tall vegetation region. Inference Rate: Method Inference Rate (Hz) GA-Nav [52] 14.832 NMFNet [9] 10.725 PAAD [7] 13.578 GrASPE without A-GNN 19.415 GrASPE without Reliability 9.167 GrASPE (ours) 8.873 rate with the other methods which use RGB images and LiDAR scan or point clouds. The rates are calculated from the instant sensor data is obtained to when a velocity is calculated. We observe that GrASPE has a low inference rate due to the computationally heavy GNN backbone. However, the design choices in our GNN architecture (Section IV-E) such as dropout and GAT layers (instead of deep GCN backbones that are computationally expensive) help keep the rates high enough for navigation. To the best of our knowledge, our GNN architecture is the first to achieve real-time execution on a robot mounted computer. VI. ACKNOWLEDGEMENT This work was supported in part by ARO Grants W911NF2110026, and Army Cooperative Agreement W911NF2120076. We acknowledge the support of the Maryland Robotics Center. VII. CONCLUSIONS, LIMITATIONS AND FUTURE WORK We present a novel multi-modal fusion algorithm to navigate a legged robot in unstructured outdoor environments where the sensors experience distortions. We utilize a graph attention-based prediction model along with a sensor reliability estimator to obtain a given trajectory's navigation success probabilities. This prediction model is combined with a local planner to generate navigation actions while avoiding actions corresponding to unsuccessful trajectories from the prediction model. We validate our method's performance in different unstructured environments and compare it with the other methods qualitatively and quantitatively. Our algorithm has a few limitations. The method assumes non-holonomic robot dynamics to ensure that the trajectories can be projected into the RGB image. However, the legged robot dynamics are holonomic and further investigation is required to extend our fusion strategy to relax the action constraints. Our method could cause collisions in extreme cases where all the perception sensors become unreliable. Adding a haptic sensor modality to navigate more cautiously could help reduce the impact of such failures. A new sensor modality can be easily added to our pipeline to enhance GrASPE's capabilities. However, its effect on the real-time execution needs to be analyzed. Fig. 2 . 2Fig. 2. Fig. 3 . 3Architecture of the attention-based graph neural network (A-GNN) used in our GrASPE system. The reliability-aware feature graph Gt is fed into this A-GNN to encode and attend to the useful multi-modal sensory feature interactions. 3 × N p , where N p = 10000 is the number of points captured from the LiDAR at each time t. The output f point of dimensions 40 × 1 is obtained by passing through a Sigmoid() activation layer at the end similar to ImgNet. 3) Velocity Feature Extraction: A velocity vector V t of size 100 × 1 includes linear and angular velocities of the previous 50 time steps (i.e. v t , ω t data of size 50 × 2 is reshaped to a vector V t of size 100 × 1). This V t is fed into 4 linear convolutional layers with dilation size {3, 2, 2, 2} in VelNet to obtain the velocity features f vel of length 20. 4) Trajectory Feature Extraction: The predicted trajectory image I traj t ∈ R 320×240×1 passes through 3 pooling and 3 convolutional layers with kernel size 3, 2, 2, a flatten layer and Sigmoid() activation layer to obtain the trajectory feature vector f traj of length 20. Hence, the final feature vector f vec is of dimensions 120 × 1 after concatenating f img , f point , f vel and f traj . Fig. 4 . 4Image Reliability Estimation: We estimate input image reliability based on two factors: 1. Overall lighting condition 2. Feature richness. (a) and (b) demonstrate images of the same scene under two different lighting conditions. (c) and (d) presents the FAST [50, 47] features extracted from the images (a) and (b). We observe that the regions in images with poor lighting and motion blur (yellow rectangle in (a) and (c)) have a significantly lower number of features. Similarly, the motion blur and camera occlusions result in images with low reliability measures. Fig. 5 . 5Point cloud Reliability Estimation: We estimate point cloud reliability based on the edge and planar feature availability. (a) Top view of a point cloud observation with many edge and planar features (blue rectangle) and the corresponding RGB image (top right) (b) Top view of a distorted point cloud observed in tall grass environments. Point clouds are reliable in open and structured environments and become unreliable in cluttered vegetation and adverse weather. ( 5 ) 5Fig. 5(c) and (d) presents an example comparison of r edge and r planar in real outdoor scenarios.D. Reliability-aware Graph ConstructionOnce the sensor inputs are encoded into feature vectors (Section IV-B), we concatenate them to obtain a combined vector f vec = [f img , f point , f traj , f vel ]. The i th element of f vec ∈ [0, 1] N (N = 120) becomes the i th node of the graph representation for fusion. The non-traversable and traversable portions in a trajectory are marked with 0's and 1's manually. Trajectories that are completely traversable during data collection are marked automatically with 1's. The final dataset includes a set of observations and label pairs (O t , E t ) for each time step t. The training set includes 28721 positive samples and 8463 negative samples whereas the test set contains 4537 positive samples and 1096 negative samples. Fig. 6 . 6Navigability predictions from GrASPE(Ours), PAAD[7], NMFNet[9], GA-Nav[52], DWA[10], GrASPE w/o reliability, GrASPE w/o A-GNN and PSPNet[53] under different environmental conditions compared to the Ground Truth. (a) Hanging leaves; (b) Pliable tall grass; (c) Camera occlusion; (d) Low-light condition. The gradient on the right denotes the success probabilities. GrASPE predictions outperform the other methods under varying conditions that are critical for robot navigation in unstructured outdoor environments. Our method results in a 13-15% decrease in the false positive rate during navigation. Fig. 7 . 7Spot robot navigation in challenging outdoor scenarios using GrASPE (ours), PAAD TABLE I ILIST OF SYMBOLS USED IN OUR APPROACH.Symbol Definition I rgb t RGB image of size w × h from the camera at time t P lidar t 3D point cloud from the LiDAR at time t Vt Robot's velocity history of the past T time steps I traj t Extrapolated trajectory from the robot's current velocity as a binary image of size w × h Ot Input observations to the GrASPE model at time t Et Ground truth label vector for a trajectory's suc- cess probabilitŷ Et Predicted success probability vector from the GrASPE model G(V, W) A connected graph represented by a set of ver- tices V and an adjacency matrix W rimg Image reliability measure ∈ [0, 1] rpoint LiDAR point cloud reliability measure ∈ [0, 1] Table III compares GrASPE's inferenceMetrics Method Scenario 1 Scenario 2 Scenario 3 Scenario 4 Success Rate (%) DWA [10] 0 20 0 70 GA-Nav [52] 10 10 30 10 NMFNet [9] 30 10 40 20 PAAD [7] 40 40 50 40 GrASPE without A-GNN 10 20 10 0 GrASPE without Reliability 50 40 40 60 GrASPE(ours) 70 70 80 80 Norm. Traj. Length DWA [10] 0.38 0.56 0.33 1.32 GA-Nav [52] 0.35 0.59 0.52 0.49 NMFNet [9] 0.69 0.58 0.53 0.72 PAAD [7] 0.81 0.95 0.76 0.93 GrASPE without A-GNN 0.42 0.63 0.38 0.43 GrASPE without Reliability 0.89 1.25 0.96 0.77 GrASPE(ours) 1.11 1.09 1.15 1.23 False Positive Rate DWA [10] - - - - GA-Nav [52] 0.33 0.39 0.30 0.61 NMFNet [9] 0.38 0.41 0.56 0.59 PAAD [7] 0.21 0.29 0.32 0.42 GrASPE without A-GNN 0.42 0.58 0.53 0.55 GrASPE without Reliability 0.28 0.31 0.25 0.36 GrASPE(ours) 0.15 0.18 0.12 0.21 False Negative Rate DWA [10] - - - - GA-Nav [52] 0.41 0.34 0.46 0.58 NMFNet [9] 0.46 0.38 0.49 0.57 PAAD [7] 0.34 0.23 0.31 0.39 GrASPE without A-GNN 0.48 0.55 0.59 0.61 GrASPE without Reliability 0.22 0.25 0.29 0.19 GrASPE(ours) 0.09 0.16 0.17 0.19 TABLE II NAVIGATION IIPERFORMANCE OF OUR METHOD COMPARED TO OTHER METHODS ON VARIOUS METRICS. GRASPE OUTPERFORMS OTHER METHODS CONSISTENTLY IN TERMS OF SUCCESS RATE, FALSE POSITIVE RATE, AND FALSE NEGATIVE RATE IN DIFFERENT UNSTRUCTURED OUTDOOR SCENARIOS INCLUDING LOW-LIGHT CONDITIONS. TABLE III INFERENCE IIIRATES (HIGHER IS BETTER) OF THE METHODS IN COMPARISONS. GRASPE HAS A LOW INFERENCE RATE DUE TO THE COMPUTATION-HEAVY NATURE OF GNNS. HOWEVER, WE NOTE THAT TO THE BEST OF OUR KNOWLEDGE, GRASPE IS THE FIRST GNN-BASED FORMULATION THAT EXECUTES IN REAL-TIME WHICH IS A PREREQUISITE FOR ROBOT NAVIGATION. t to a gray-scale image I gray t . Then we calculate the Root Mean Square(RMS) value of the histogram distribution of I gray t as follows to obtain brightness estimation, r bright = RM S(hist(I gray t ))(1)Here, hist is the histogram operator and RM S is the root mean square operator where RM S(x) = 1 n n i=1 x 2 i . From Graph theory, we consider the graph Laplacian L = D − W, where D is a diagonal matrix with D i,i = deg(v i ). Here, deg(v i ) is the degree of a vertex which is a measure of the number of edges terminating at that vertex. In this context, we consider D i,i = j w i,j . By construction and from Lemma 1, we can observe that D is real and W is hermitian (i.e., w i,j = w * j,i ). Lemma 2: Laplacian matrix L of the graph G t is spectrally decomposable.Proof: From Lemma 1, Re(w i,j ) ≥ 0 ∀ i, j. Consider a real valued x,∴ Symmetric matrix L has all real eigenvalues. Further, the corresponding eigenvectors u 1 , .., u N can be taken to be orthonormal by,Therefore, from the Spectral theorem[51], L can be spectrally decomposed as, L = U ΛU T , where U is the eigenvector matrix, and Λ denotes the diagonal matrix of sorted eigenvalues.B. Data Collection Process (extending Section V.B)The multi-modal dataset used in this work is collected by operating the Spot robot under different lighting conditions in an outdoor field that includes bushes, small trees, hanging leaves, and grass regions of different heights and densities. We chose 8 different locations in the outdoor field and collected data at least for 6 runs including high, medium, and low lighting conditions. The data streams were collected at around 10-15Hz rate since sensors such as LiDAR can provide data only up to 20Hz even though camera images can be collected at 30Hz. The total duration of data collection was 90 minutes.C. Training SetupGrASPE is implemented using PyTorch and PyTorch Geometric (PyG). 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{'abstract': "We present a novel trajectory traversability estimation and planning algorithm for robot navigation in complex outdoor environments. We incorporate multimodal sensory inputs from an RGB camera, 3D LiDAR, and the robot's odometry sensor to train a prediction model to estimate candidate trajectories' success probabilities based on partially reliable multi-modal sensor observations. We encode high-dimensional multi-modal sensory inputs to low-dimensional feature vectors using encoder networks and represent them as a connected graph. The graph is then used to train an attention-based Graph Neural Network (GNN) to predict trajectory success probabilities. We further analyze the number of features in the image (corners) and point cloud data (edges and planes) separately to quantify their reliability to augment the weights of the feature graph representation used in our GNN. During runtime, our model utilizes multi-sensor inputs to predict the success probabilities of the trajectories generated by a local planner to avoid potential collisions and failures. Our algorithm demonstrates robust predictions when one or more sensor modalities are unreliable or unavailable in complex outdoor environments. We evaluate our algorithm's navigation performance using a Spot robot in real-world outdoor environments. We observe an increase of 10-30% in terms of navigation success rate and a 13-15% decrease in false positive estimations compared to the state-of-the-art navigation methods.", 'arxivid': '2209.05722', 'author': ['Kasun Weerakoon ', 'Jagan Adarsh ', 'Jing Sathyamoorthy ', 'Tianrui Liang ', 'Utsav Guan ', 'Dinesh Patel ', 'Manocha '], 'authoraffiliation': [], 'corpusid': 252211736, 'doi': '10.48550/arxiv.2209.05722', 'github_urls': [], 'n_tokens_mistral': 20770, 'n_tokens_neox': 18051, 'n_words': 10952, 'pdfsha': '6302822c7ee5716664410196d99dd1b16bf7989b', 'pdfurls': ['https://export.arxiv.org/pdf/2209.05722v3.pdf'], 'title': ['GrASPE: Graph based Multimodal Fusion for Robot Navigation in Outdoor Environments', 'GrASPE: Graph based Multimodal Fusion for Robot Navigation in Outdoor Environments'], 'venue': []}

arxiv

SOLVING N=2 SYM BY REFLECTION SYMMETRY OF QUANTUM VACUA ‡ 25 Feb 1997 Giulio Bonelli [email protected]@[email protected] Department of Physics "G. Galilei" -Istituto Nazionale di Fisica Nucleare University of Padova Via Marzolo 8 -35131PadovaItaly Marco Matone Department of Physics "G. Galilei" -Istituto Nazionale di Fisica Nucleare University of Padova Via Marzolo 8 -35131PadovaItaly Mario Tonin Department of Physics "G. Galilei" -Istituto Nazionale di Fisica Nucleare University of Padova Via Marzolo 8 -35131PadovaItaly SOLVING N=2 SYM BY REFLECTION SYMMETRY OF QUANTUM VACUA ‡ 25 Feb 1997DFPD96/TH/29 hep-th/9610026 To be published in Phys. Rev. D The recently rigorously proved nonperturbative relation u = πi(F − a∂ a F /2), underlying N = 2 SYM with gauge group SU(2), implies both the reflection symmetry u(τ ) = u(−τ ) and u(τ + 1) = −u(τ ) which hold exactly. The relation also implies that τ is the inverse of the uniformizing coordinate u of the moduli space of quantum vacua M SU (2) , that is τ : M SU (2) → H where H is the upper half plane. In this context, the above quantum symmetries are the key points to determine M SU (2) . It turns out that the functions a(u) and a D (u), which we derive from first principles, actually coincide with the solution proposed by Seiberg and Witten. We also consider some relevant generalizations. ‡ Work supported by the European Commission TMR programme ERBFMRX-CT96-0045, to which M.M. and M.T. are associated. The exact results about N = 2 SUSY Yang-Mills obtained by Seiberg and Witten [1] concern the low-energy Wilsonian effective action with at most two derivatives and four fermions. These terms are completely described by the prepotential F whose instanton contributions can be determined by recursion relations [2]. In [3] it has been shown that the relation between F and u = tr φ 2 derived in [2], is connected to the nonperturbative Renormalization Group Equation. Related results concern the appearance of the WDVV equation [4,5] indicating that there are topological structures underlying N = 2 SYM. In [4] it was argued that these aspects are connected to associativity which arises in considering divisors on moduli spaces and therefore to quantum cohomology (see also [6]). An interesting point is that Seiberg-Witten theory can be described in the framework of uniformization theory [2,7] and the related Picard-Fuchs equations [8]. This aspect is also useful in considering the critical curve C = {u|Im a D (u)/a(u) = 0} [1,7,9,10]. In [11,12,13,14,15] nonperturbative investigations in the framework of instanton theory and superconformal Ward identities have been performed. The aspects concerning integrability have been considered in [16] whereas other related field theoretical structures have been considered in [17] including some generalizations [18]. All these results are a consequence of the Seiberg-Witten derivation of the low-energy dynamics of N = 2 SYM. One of the interesting aspects of the Seiberg-Witten results, both from a physical and mathematical point of view, is that their solution implies that all the instanton contributions are determined by recursion relations [2]. As a consequence, the important problem of evaluating the relevant integrals defining F k , has been done in a elegant way. We observe that to understand the explicit structure of the integrals defining k ≥ 3 is still an open problem which is of interest also for other QFT's. These observations, while indicating that the full consequences of the Seiberg-Witten results should be further investigated, also make evident the necessity of proving what still remains at the conjectural level. Even if there is evidence supporting the Seiberg-Witten results, a clear proof is still lacking. In this letter we will show that actually the basic structures underlying N = 2 SYM with gauge group SU(2) are the asymptotic analysis which implies u(τ + 1) = −u(τ ), a consequence of the presence of the Θ-angle, and the property of reflection symmetry u(τ ) = u(−τ ) of the quantum vacua. Of course, reflection symmetry is related to the CPT symmetry, which actually, together with the holomorphicity of the prepotential, turns out to be the crucial nonperturbative information. Therefore, what we will prove is that the one-loop approximation and CPT arguments are sufficient to solve the theory in the proper low-energy limit. Whereas the T 2 symmetry arises from the asymptotic analysis, the other generators of Γ(2) turns out to be fixed by T 2 itself together with the reflection symmetry, which can be seen as an alternative way to define some subgroups of the discrete group SL(2, Z). Our method of characterizing some discrete groups from the symmetry properties of their fundamental domain is quite general and may be also useful to shed new light not only in N = 2 SYM but also in other quantum field theories. An important point in our construction is that the relation [2] u = πi F (a) − a 2 ∂F (a) ∂a ,(1) has been proved in the framework of multiinstanton calculations up to two instanton contributions by Fucito and Travaglini [12] and at all orders by Dorey, Khoze and Mattis [13]. Furthermore, it has been derived in the framework of superconformal Ward identities by Howe and West [14] proving also its generalization obtained in [19]. In this letter, we will use the relation (1) in order to derive the uniformizing equation for the u-moduli space M SU (2) of quantum vacua. We will also show that two important consequences of the relation are the reflection symmetry u(τ ) = u(−τ ),(2) and u(τ − n) = (−1) n u(τ ). We stress the important point that as the relation between u and F has been derived both from multiinstanton calculations and superconformal Ward Identities, it follows that we can exclude other unknown nonperturbative effects besides the instanton contributions. As a consequence, both (2) and (3) hold exactly. Eqs. (2)(3) turns out to be the key points to determine both M SU (2) and its fundamental domain. Indeed we shall prove from (2) and (2) is the Riemann sphere with punctures at u = ∞ and u = ±Λ 2 , the main conjecture in [1]. In particular, it turns out that the functions a(u) and a D (u), which we derive from first principles, actually coincide with those obtained by Seiberg and Witten. (3) that M SU Let us consider the chiral part of the low-energy effective action for N = 2 SYM with gauge group SU (2). In N = 2 superspace notation the part with at most two derivatives and four fermions reads 1 4π Im d 4 xd 2 θd 2θ F (Ψ),(4) where F is the prepotential and Ψ the N = 2 chiral superfield. The effective Θ-angle and the gauge coupling constant enter in the effective coupling constant τ = ∂ 2 a F (a) in the form τ = ∂ 2 a F = Θ 2π + 4πi g 2 . The asymptotic expansion of F has the structure [20] F = a 2 i π log a Λ + ∞ k=0 F k a Λ −4k ,(5) where Λ is the dynamically generated scale. Another important result in [20] is that actually at least F 1 is non-vanishing. For further purpose we write down the asymptotic expansion of τ τ = 2i π log a Λ + 3i π + ∞ k=0 F k (1 − 4k)(2 − 4k) a Λ −4k .(6) By making some assumptions, Seiberg and Witten argued that the u-quantum moduli space is the thrice punctured Riemann sphere. In particular, in [1] the exact form of the functions a = a(u) and a D = a D (u) = ∂ a F has been obtained. This solution determines all the F k 's implicitly. In [2] it has been shown that the results in [1] imply the relation (1). This relation will be useful in determining both M SU (2) and the functions a(u) and a D (u). Eq.(1) has been checked in the framework of multiinstanton calculations up to two instanton contributions in [12] and at all orders [13]. Furthermore, it has ben derived in the framework of superconformal Ward identities in [14]. The fact that (1) is rigorously proved is essential for our construction. By (5) and (1) it follows that the asymptotic expansion for u = G(a) is G(a) = a 2 ∞ k=0 G k a Λ −4k , G 0 = 1 2 ,(7) where G k = 2πikF k . Furthermore, by instanton theory [12,13] we have Re F k = 0. Differentiating (1) with respect to u we get aa ′ D − a D a ′ = 2i π . This implies that a D and a are linearly independent solutions of a second-order linear differential equation with respect to u, that is [∂ 2 u + V (u)]a = 0 = [∂ 2 u + V (u)]a D ,(8) for some unknown V (u) = −a ′′ /a = −a ′′ D /a D . Inverting (8) we obtain a differential equation for u with respect to a ∂ 2 a G − a (∂ a G) 3 V (G) = 0,(9) which implies recursion relations for G k = 2πikF k . The full Seiberg-Witten solution follows from (9) once one proves that V (u) = 1 4(u 2 − Λ 4 ) ,(10) so that Eq. (9) becomes Λ 4 − G 2 G ′′ + 1 4 aG ′ 3 = 0,(11) and by (7) [2] G n+1 = 1 8G 2 0 (n + 1) 2 · ·    (2n − 1)(4n − 1)G n + 2G 0 n−1 k=0 G n−k G k+1 c(k, n) − 2 n−1 j=0 j+1 k=0 G n−j G j+1−k G k d(j, k, n)    ,(12) where n ≥ 0, G 0 = 1/2 and c(k, n) = 2k(n − k − 1) + n − 1, d(j, k, n) = [2(n − j) − 1][2n − 3j − 1 + 2k(j − k + 1)]. Let us set T /2 = V − V 1/2 ∂ 2 u V −1/2 .(13) From the identity V 1/2 (u)∂ u V −1 (u)∂ 2 u + 1 = ∂ 2 u + T (u)/2 V −1/2 (u)∂ u , and by (8) we obtain ∂ 2 u + T (u)/2 V −1/2 (u)∂ u a = 0 = ∂ 2 u + T (u)/2 V −1/2 (u)∂ u a D ,(14) where T is the Schwarzian derivative (here ′ ≡ ∂ u ) T (u) = τ ′′′ /τ ′ − 3(τ ′′ /τ ′ ) 2 /2. Since τ lives in the upper half plane H (except that at the possible singularities where Im τ = 0), the polymorphic function τ (u) = ∂ u a D /∂ u a may be seen as the inverse of the uniformizing coordinate u : H → M SU (2) . From the monodromy transformation properties of τ (u) we know that M SU (2) ∼ = H/Γ where Γ is the uniformizing group to be determined. We now observe that some information about the structure of M SU (2) already comes in considering the physical role of u. For each value of u, that is for each choice of representation of the vacuum (which fixes the Hilbert space of states), we should determine the functions a(u) and a D (u). Therefore, one has to consider the theory for each value of u ∈ C = C∪{∞}. In other words, defining M SU (2) as the u-moduli space means that u is the uniformizing coordinate of M SU (2) itself. In this context, by 'singularities' we mean the values of u which cause non trivial monodromies for a(u) and a D (u). Geometrically this means that the u-space is the Riemann sphere C with n-punctures. Therefore, the unique non-trivial topological complications that we can expect are those induced from some particular values of u. In this context we observe that singularities imply symmetries. Actually, from the above discussion it follows that u(γ · τ ) = u(τ ), γ ∈ Γ. That is, for any choice of u there are infinitely many equivalent prepotentials [2] γ · F (a) = F (a) + a 11 a 21 2 a 2 D + a 12 a 22 2 a 2 + a 12 a 21 aa D , In the following we will prove that the number of punctures is 3 and will determine a(u) and a D (u) explicitly. As we will see we will do not need to make assumptions about the finiteness of the number of punctures. Since the F k 's are purely imaginary, it follows by (6) that τ (a) = −τ (ā), so that a(τ ) = a(−τ ) and by (7) we have the reflection symmetry (2) which is crucial for our construction. Let us now consider the effect of the transformation a → e iπn/2 a, n ∈ Z, on F , τ and u. By (5) and (1) we have F (a) → e πin F (a) − e πin n 2 a 2 , τ → τ − n,(16) and by (7) G(e iπn/2 a) = (−1) n G(a) which is equivalent to (3). We observe that since from multiinstanton calculations both F 1 and F 2 are non-vanishing [20,12], we can exclude that G(e iπn/m a) ∝ G(a) for m > 2. We also stress that as a consequence of [12,13,14] both (2) and (3) hold exactly. As Eq. (16) shows, the group elements acting on (a D , a) have phases which do not appear in the projective transformations of τ . By asymptotic analysis we already know that there is a puncture at u = ∞. We denote the other punctures and their image in the closure of a given fundamental domain in H by u k and τ k = τ (u k ), k = 0, . . . , n − 2, respectively. As well known from uniformization theory the τ k correspond to cusps on the real axis, the boundary of H. We fix the labelling of the punctures u 0 , . . . , u n−2 , in such a way that τ k+1 > τ k , k = 0, . . . , n − 3. Let us denote by F the closure of the fundamental domain in H which has non empty intersection with the imaginary axis and byḞ its interior. By (3) the width of F is 2 whereas from the asymptotic behavior τ ∼ i π log(2u/Λ 2 ), it follows that the τ -image of the puncture at u = ∞ corresponds to the point at infinity. This implies that the left and right parts of the boundary ∂F of F are two half-infinite vertical lines. We extend to ∀k ∈ Z the definition of the τ k 's by setting τ k+j(n−1) = τ k + 2j, j ∈ Z.(17) We also set τ 0 ≤ 0 and τ 1 > 0. Let τ (0) be the image of the point u = 0 in F such that Re τ (0) > 0. Observe that if τ (0) ∈ ∂F , then there is another point τ (−1) in ∂F such that u(τ (−1) ) = 0. We require Re τ (−1) < 0, so that by construction Re τ (0) < Re τ (k) , ∀τ (k) ∈ {τ |τ ∈ G, u(τ ) = 0, Re τ > 0},(18) where G = ∪ k∈Z F (n) and F (n) = {τ + 2n|τ ∈ F }. One can check that by (2) and (3) the above choices do not imply lost of generality. We now start to determine both M SU (2) and the fundamental domain. The starting point is to observe that Eqs. (2)(3) in the u = 0 case yield 0 = u(τ (0) ) = u(τ (0) − 1) = u(−τ (0) ).(19) We now show that by (19) the point u = 0 cannot be a puncture and that Re τ (0) = 1/2. Actually, τ (0) / ∈Ḟ otherwise either (τ (0) + 1) ∈Ḟ or (τ (0) − 1) ∈Ḟ , which is excluded by the one-to-one nature of the covering. For the same reason τ (0) is neither the image of a puncture nor belongs to the half-infinite vertical lines in ∂F . Hence, τ (0) can be only on a half-circle corresponding to the Poincaré geodesic connecting two cusps τ k and τ k+1 for some k ∈ Z. Furthermore, by (18) and (19) we have Reτ (0) = 1 2 .(20) Let us denote by R and I the loci Im u = 0 and Re u = 0 respectively. We also set τ y = Im τ . In order to determine the image of R in G we observe that reflection symmetry implies Im u(τ = iτ y ) = 0 that by (3) extends to Im u(τ = k + iτ y ) = 0, k ∈ Z. A similar reasoning yields Re u(τ = k + 1/2 + iτ y ) = 0, k ∈ Z. Summarizing, we have Im u(τ = k + iτ y ) = 0, Re u(τ = k + 1/2 + iτ y ) = 0, k ∈ Z.(21) From the above investigation it follows that the points τ (u = 0) in G correspond to the end-points of the vertical lines belonging to the τ -image of the I locus. Let us denote by V k+1/2 , k ∈ Z these vertical lines. Since the holomorphicity of F (a) [21,20] implies the holomorphicity of τ (u), it follows that the angles on the u-space are preserved on the fundamental domains except that at the possible punctures. As u = 0 is not a puncture, for each k ∈ Z, the line V k+1/2 is perpendicular to the curve image of R in G at the point τ (0) + k. We note that the locus V ∈Ḟ corresponding to points in the τ -image of R is the intersection of the imaginary axis withḞ , that is V =Ḟ ∩ {τ |Re τ = 0}. It follows that the only possibility for the τ -image of R to be a line in F which is continuous, except that at the cusps, is that the full boundary ∂F itself be in the τ -image of R. Therefore, we have τ : R → ∂F ∪ V.(22) Furthermore, since τ k ∈ ∂F , it follows that the punctures are real Im u k = 0. By reflection symmetry and (23) we have u k = u(τ k ) = u(−τ k ), and by (3) τ k ∈ Z. Since τ 0 ≤ 0, τ 1 > 0 and τ k+1 > τ k , we have τ k = k, k ∈ Z.(24) It follows that M SU (2) is the Riemann sphere with punctures at u 0 = u(τ = 0), u 1 = u(τ = −1) = u(τ = 1) = −u 0 and ∞. As well known this surface is uniformized by Γ (2). In order to find the value of u 0 and the explicit form of a(u) and a D (u), we follow [2] by first considering the explicit expression of the projective connection T (u) = {τ, u}. For a n-punctured Riemann sphere with a puncture at infinity we have (see for example [22]) T (u) = n−2 i=0 1 2(u − u i ) 2 + c i u − u i .(25) where the c i 's, called accessory parameters, satisfy the constraints n−2 i=0 c i = 0, n−2 i=0 c i u i = 1 − n 2 .(26) In our case n = 3, so that c 1 = −c 0 = 1/4u 0 , and Eq. (14) becomes ∂ 2 u + 3u 2 0 + u 2 4(u 2 0 − u 2 ) 2 ψ = 0.(27) Note that by (13) we have V (u) = 1/4(u 2 − u 2 0 ). To find u 0 , a(u) and a D (u) we first note that (27) is solved by the Legendre functions P −1/2 and Q −1/2 . This fact and the asymptotic analysis imply [2] a D (u, Λ) = √ 2 π u Λ 2 dx √ x − u √ x 2 − Λ 4 , a(u, Λ) = √ 2 π Λ 2 −Λ 2 dx √ x − u √ x 2 − Λ 4 ,(28) which actually coincides with the solution proposed by Seiberg and Witten [1]. Observe that u 0 = u(τ = 0) = Λ 2 . We also note that as a consequence of the nonperturbative quantum symmetries (2) and (3) of N = 2 SYM, although the width of F is 2, the tessellation of H by Γ (2) has an automorphism under the shift τ → τ + 1. In this context we observe that the asymptotic behavior fixes the sign ambiguity. In particular, by (28), whose normalization is fixed by (6) and (7), it follows that the positive imaginary axis of H is in the τ -image of the real points u > Λ 2 . We observe that in our construction we did not make any assumptions about finiteness of the number of punctures. Let us now shortly consider how the approach works in other cases. As the relation between u and the prepotential is the crucial point, we have to see if there is an extension of it. In the SU (2) case with N f = 0 the relation has a very simple generalization. In the case of rank group r the relation has the form [19] u = 1 4πb 1 F − 1 2 r k=1 a k ∂F ∂a k ,(29) where b 1 is the one-loop contribution to the beta-function. Eq.(29) has been proved in [14]. Of course, besides u there are other (r − 1)-moduli coordinates. In the SU (3) case, the relation between v = tr φ 3 and F has been derived in [4]. Other informations concerning the structure of the theory may be obtained from the asymptotic analysis. As in the SU (2) case, this asymptotic fixes the monodromy at infinity. Then again, using asymptotic analysis and CPT arguments (reflection symmetry), will uniquely fix the structure of the fundamental domain for the monodromy group. The fact that the situation still involves uniformization, follows by the geometrical analysis considered in [4] where it was proved that M SU (3) ∼ = S/Γ, with S the genus two τ ij -space, a subvariety of the genus 2 Siegel upper-half space of complex codimension 1 covering M SU (3) . Therefore, the uniformization arguments still work in the higher rank case. Actually, this was also of mathematical interest as considers the problem of formulating the uniformization theorem in the case of moduli space of Riemann surfaces rather than the Riemann surfaces themselves. However the uniformization in the case of the quantum moduli space seems to be easier than the general case as the kind of Riemann surfaces described by points in M SU (n) actually have a higher symmetry coming just from the asymptotic analysis and from CPT arguments. This reflects in the fact that the position of the branching points in the Riemann sphere is highly symmetric. 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{'abstract': 'The recently rigorously proved nonperturbative relation u = πi(F − a∂ a F /2), underlying N = 2 SYM with gauge group SU(2), implies both the reflection symmetry u(τ ) = u(−τ ) and u(τ + 1) = −u(τ ) which hold exactly. The relation also implies that τ is the inverse of the uniformizing coordinate u of the moduli space of quantum vacua M SU (2) , that is τ : M SU (2) → H where H is the upper half plane. In this context, the above quantum symmetries are the key points to determine M SU (2) . It turns out that the functions a(u) and a D (u), which we derive from first principles, actually coincide with the solution proposed by Seiberg and Witten. We also consider some relevant generalizations. ‡ Work supported by the European Commission TMR programme ERBFMRX-CT96-0045, to which M.M. and M.T. are associated.', 'arxivid': 'hep-th/9610026', 'author': ['Giulio Bonelli [email protected]@[email protected] \nDepartment of Physics "G. Galilei" -Istituto Nazionale di Fisica\nNucleare University of Padova Via Marzolo\n8 -35131PadovaItaly\n', 'Marco Matone \nDepartment of Physics "G. Galilei" -Istituto Nazionale di Fisica\nNucleare University of Padova Via Marzolo\n8 -35131PadovaItaly\n', 'Mario Tonin \nDepartment of Physics "G. Galilei" -Istituto Nazionale di Fisica\nNucleare University of Padova Via Marzolo\n8 -35131PadovaItaly\n'], 'authoraffiliation': ['Department of Physics "G. Galilei" -Istituto Nazionale di Fisica\nNucleare University of Padova Via Marzolo\n8 -35131PadovaItaly', 'Department of Physics "G. Galilei" -Istituto Nazionale di Fisica\nNucleare University of Padova Via Marzolo\n8 -35131PadovaItaly', 'Department of Physics "G. Galilei" -Istituto Nazionale di Fisica\nNucleare University of Padova Via Marzolo\n8 -35131PadovaItaly'], 'corpusid': 18477415, 'doi': '10.1103/physrevd.55.6466', 'github_urls': [], 'n_tokens_mistral': 10704, 'n_tokens_neox': 9088, 'n_words': 4867, 'pdfsha': 'f6a31e9f6765de76f41d6f1419f722718b206786', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9610026v3.pdf'], 'title': ['SOLVING N=2 SYM BY REFLECTION SYMMETRY OF QUANTUM VACUA ‡', 'SOLVING N=2 SYM BY REFLECTION SYMMETRY OF QUANTUM VACUA ‡'], 'venue': []}

arxiv

Bivariate Bannai-Ito polynomials 25 Sep 2018 Jean-Michel Lemay [email protected] Centre de Recherches Mathématiques Université de Montréal Succ. Centre-villeC.P. 6128, H3C 3J7MontréalQCCanada Luc Vinet [email protected] Centre de Recherches Mathématiques Université de Montréal Succ. Centre-villeC.P. 6128, H3C 3J7MontréalQCCanada Bivariate Bannai-Ito polynomials 25 Sep 2018 A two-variable extension of the Bannai-Ito polynomials is presented. They are obtained via q → −1 limits of the bivariate q-Racah and Askey-Wilson orthogonal polynomials introduced by Gasper and Rahman. Their orthogonality relation is obtained. These new polynomials are also shown to be multispectral. Two Dunkl shift operators are seen to be diagonalized by the bivariate Bannai-Ito polynomials and 3and 9-term recurrence relations are provided. Introduction In their classification of P-and Q-polynomial association schemes [1], Bannai and Ito identified a new 4parameter family of orthogonal polynomials that now bear their names. They provided the explicit expressions of these polynomials and observed that they correspond to a q → −1 limit of the q-Racah polynomials. The understanding of the Bannai-Ito polynomials has considerably increased in recent years. Of particular relevance to the present study is the fact that they have been shown [2] to also arise as q → −1 limits of the Askey-Wilson polynomials. The Bannai-Ito polynomials are now known to be multispectral : they are eigenfunctions of the most general first order shift operator of Dunkl type that preserves the space of polynomials of a given degree [2]. They have been identified with the non-symmetric Wilson polynomials [3] and are essentially the Racah coefficients of the Lie superalgebra osp(1|2) [4]. They have moreover found various applications beyond algebraic combinatorics especially in the context of superintegrable and exactly solvable models [5,6,7]. The Bannai-Ito polynomials and their kernel partners, the complementary Bannai-Ito polynomials admit various multispectral families of orthogonal polynomials as descendants and special cases and thus sit at the top of a q = −1 analog of the Askey-scheme [8,9,10,11,2,12,13]. The extension to many variables of the theory of univariate orthogonal polynomials is obviously of great interest. There are two major directions in this broad topic (see [14,15] for instance). One involves the theory of symmetric functions [16] and has the Macdonald and Koornwinder polynomials associated to root systems as main characters. The other works through the coupling of univariate polynomials and features the multivariable extension of the Racah and Wilson polynomials and their descendants introduced by Tratnik [17,18]. We shall focus on the latter area in the following. A key feature of Tratnik's construction is that the multivariate orthogonality relation is obtained by induction on the univariate one. Iliev and Geronimo have shown that these Tratnik polynomials are multispectral [19,20]. Their q-generalizations have been discovered by Gasper and Rahman who thus provided multivariable extension of the q-Racah and Askey-Wilson polynomials and in so doing of the entire q-scheme [21,22]. The goal of the present paper is to initiate a multivariable extension for the q = −1 scheme. Specifically, we introduce a bivariate extension of the Bannai-Ito polynomials and provide various structure relations. These polynomials are defined in formula (2.1). The paper will be comprised of three main sections. We begin with a review of the Bannai-Ito polynomials and their structure relations. In section 2, we define the bivariate Bannai-Ito polynomials from a q → −1 limit of Gasper and Rahman's two-variables q-Racah polynomials. The truncation conditions are examined and the orthogonality relation is obtained. In section 3, we obtain an untruncated definition for the bivariate Bannai-Ito polynomials from a q → −1 limit of two-variable Askey-Wilson polynomials. This has the benefit of expressing the multispectrality relations in terms of operators which act directly on the variables instead of on the points of the orthogonality grid. A connection between the two definitions is established and the multispectrality relations for the polynomials are derived. Remarks and open questions are discussed in the conclusion. Univariate Bannai-Ito polynomials The monic Bannai-Ito polynomials B n (x; ρ 1 , ρ 2 , r 1 , r 2 ), or B n (x) for short, depend on 4 parameters ρ 1 , ρ 2 , r 1 , r 2 and are symmetric with respect to the Z 2 × Z 2 group of transformations generated by ρ 1 ↔ ρ 2 and r 1 ↔ r 2 . Explicitly, this means that the BI polynomials verify B n (x; ρ 1 , ρ 2 , r 1 , r 2 ) = B n (x; ρ 2 , ρ 1 , r 1 , r 2 ) = B n (x; ρ 1 , ρ 2 , r 2 , r 1 ) = B n (x; ρ 2 , ρ 1 , r 2 , r 1 ). (1.1) We denote by g the combination of parameters g = ρ 1 + ρ 2 − r 1 − r 2 . (1.2) Throughout this section, it will be convenient to write integers as follows n = 2n e + n p , n p ∈ {0, 1}, n ∈ N. (1.3) With these notations, the Bannai-Ito polynomials can be expressed in terms of two generalized hypergeometric series 1 η n B n (x; ρ 1 , ρ 2 , r 1 , r 2 ) = 4 F 3 −ne, ne+g+1, x−r1+ 1 2 , −x−r1+ 1 2 1−r1−r2, ρ1−r1+ 1 2 , ρ2−r1+ 1 2 ; 1 (1.4) + (−1) n (n e +n p +gn p )(x−r 1 + 1 2 ) (ρ 1 − r 1 + 1 2 )(ρ 2 − r 1 + 1 2 ) 4 F 3 −ne−np+1, ne+np+g+1, x−r1+ 3 2 , −x−r1+ 1 2 1−r1−r2, ρ1−r1+ 3 2 , ρ2−r1+ 3 2 ; 1 where the normalization coefficient is given by η n = (−1) n (ρ 1 − r 1 + 1 2 ) ne+np (ρ 2 − r 1 + 1 2 ) ne+np (1 − r 1 − r 2 ) ne (n e + g + 1) ne+np . (1.5) The expression (1.4) can be obtained from a q → −1 limit of the q-Racah polynomials [1] and also from a q → −1 limit of the Askey-Wilson polynomials [2]. Note that the two hypergeometric functions appearing in (1.4) are almost identical except for two +1 shifts in the upper parameter row and two in the lower row. The B n (x) satisfy the three-term recurrence relation xB n (x) = B n+1 (x) + (ρ 1 − A n − C n )B n (x) + A n−1 C n B n−1 (x),(1.6) with the initial conditions B −1 (x) = 0 and B 0 (x) = 1. The recurrence coefficients A n and C n are given by A n =        (n + 2ρ 1 − 2r 1 + 1)(n + 2ρ 1 − 2r 2 + 1) 4(n + g + 1) , n even, (n + 2g + 1)(n + 2ρ 1 + 2ρ 2 + 1) 4(n + g + 1) , n odd, C n =        − n(n − 2r 1 − 2r 2 ) 4(n + g) , n even, − (n + 2ρ 2 − 2r 2 )(n + 2ρ 2 − 2r 1 ) 4(n + g) , n odd. (1.7) It can be seen from the above relations that the positivity conditions u n = A n−1 C n > 0 cannot be satisfied for all n ∈ N. This comes from the fact that C n becomes negative for large n. It follows that the Bannai-Ito polynomials can only form a finite set of orthogonal polynomials for which the conditions u n > 0, n = 1, 2, . . . , N are verified. This requires that the parameters realize a truncation condition for which u 0 = u N +1 = 0. (1.8) We call the integer N the truncation parameter. If these conditions are fulfilled, the BI polynomials B n (x) satisfy the discrete orthogonality relation N k=0 w k B n (x k )B m (x k ) = h n δ nm , (1.9) with respect to a positive set of weights w k . The orthogonality grid x k corresponds to the simple roots of the polynomial B N +1 (x). The explicit formulas for the weight function w k and the grid points x k depend on the parity of N and more explicitly on the realization of the truncation condition u N +1 = 0. If N is even, it follows from (1.7) that the condition u N +1 = 0 is tantamount to one of the following requirements associated to all possible values of j and ℓ : i) r j − ρ ℓ = N + 1 2 , j, ℓ ∈ {1, 2}. (1.10) Note that the four possibilities coming from the choices of j and ℓ are equivalent since the polynomials B n (x) are invariant under the exchanges ρ 1 ↔ ρ 2 and r 1 ↔ r 2 . To make the formulas explicit, fix j = ℓ = 1. Then the grid points have the expression x k = (−1) k (k/2 + ρ 1 + 1/4) − 1/4,(1.11) for k = 0, . . . , N and using (1.3) the weights take the form w k = (−1) k k e ! (ρ 1 − r 1 + 1/2) ke+kp (ρ 1 − r 2 + 1/2) ke+kp (ρ 1 + ρ 2 + 1) ke (2ρ 1 + 1) ke (ρ 1 + r 1 + 1/2) ke+kp (ρ 1 + r 2 + 1/2) ke+kp (ρ 1 − ρ 2 + 1) ke , (1.12) where (a) n = a(a + 1) · · · (a + n − 1) denotes the Pochhammer symbol. The normalization factors are h n = n e !N e !(1 + 2ρ 1 ) Ne (1 + ρ 1 + ρ 2 ) ne (1 + n e + g) Ne−ne ( 1 2 + ρ 1 − r 2 ) ne+np ( 1 2 + ρ 2 − r 2 ) ne+np (N e − n e − n p )!( 1 2 + ρ 1 + r 2 ) Ne−ne ( 1 2 + n e + n p + ρ 2 − r 1 ) Ne−ne−np (1 + n + g) 2 ne+np . (1.13) The formulas for other values of j and ℓ can be obtained by using the appropriate substitutions ρ 1 ↔ ρ 2 and r 1 ↔ r 2 in (1.10)-(1.13). If N is odd, it follows from (1.7) that the condition u N +1 = 0 is equivalent to one of the following restrictions: ii) ρ 1 + ρ 2 = − N + 1 2 , iii) r 1 + r 2 = N + 1 2 , iv) ρ 1 + ρ 2 − r 1 − r 2 = − N + 1 2 . (1.14) We refer to the possible truncation conditions as type i) to type iv). Note however that type iv) leads to a singularity in u n when n = (N + 1)/2 and is therefore not admissible 1 . For type ii), the formulas (1.11) and (1.12) hold and the normalization factors are given by h n = n e !N e !(1 + 2ρ 1 ) Ne+1 (1−r 1 −r 2 ) ne (1+n e +g) Ne+1−ne ( 1 2 +ρ 1 −r 1 ) ne+np ( 1 2 +ρ 1 −r 2 ) ne+np (N e −n e )!( 1 2 + ρ 1 + r 1 ) Ne+1−ne−np ( 1 2 + n e + n p + ρ 2 − r 2 ) Ne+1−ne−np (1 + n + g) 2 ne+np . (1.15) For type iii), the spectral points are x k = (−1) k (r 1 − k/2 − 1/4) − 1/4,(1.16) for k = 0, . . . , N , the weight function is given by (1.12) with the substitutions (ρ 1 , ρ 2 , r 1 , r 2 ) → −(r 1 , r 2 , ρ 1 , ρ 2 ) and the normalization factors read hn = ne!Ne!(2r2 −N )N e +1(ρ1 +ρ2 −Ne)N e+1+ne (ρ1 +ρ2 −Ne)n e (r2 +ρ1 − 1 2 −Ne)n e+np (r2 +ρ2 − 1 2 −Ne)n e +np (Ne − ne)!(ρ1 + ρ2 − Ne) 2 n (r2 − ρ1 − 1 2 − Ne)N e +1−ne−np (r2 − ρ2 + 1 2 + ne + np)N e+1−ne −np . (1.17) 1 It might be possible to absorb this singularity with some fine-tuning of the parameters as has been done for the Racah and q-Racah polynomials [23,24] but this has not been explored yet and goes beyond the scope of this paper. To construct a bivariate extension of the Bannai-Ito polynomials, the different truncation conditions for different parities of N will play an important role. The BI polynomials also verify a difference equation : LB n (x) = λ n B n (x) (1.18) with L = (x − ρ 1 )(x − ρ 2 ) 2x (1 − R x ) + (x − r 1 + 1 2 )(x − r 2 + 1 2 ) 2x + 1 (T 1 x R x − 1) (1.19) where 1 is the identity operator, R x f (x) = f (−x) denotes the reflection operator and T m x f (x) = f (x + m) is a shift operator. The eigenvalues are given by λ n = n 2 n even, r 1 + r 2 − ρ 1 − ρ 2 − n+1 2 n odd. (1.20) It was shown in [2] that L is in fact the most general first order Dunkl difference operator with orthogonal polynomials as eigenfunctions. B n1,n2 (z 1 , z 2 ) = B n1 z 1 − 1 4 ; ρ (1) 1 , ρ (1) 2 , r(1)1 , r (1) 2 B n2 (−1) n1 z 2 − 1 4 ; ρ (2) 1 , ρ(2) 2 , r 1 , r (2) 2 (2.1) where the B n and η n are as in (1.4) and (1.5) and the parameters are given by ρ (1) 1 = c − p 1 + 1 2 , r(1)1 = 1 2 − p 1 , ρ(1)2 = z 2 + p 2 − 1 4 , r(1)2 = z 2 − p 2 + 1 4 (2.2) and ρ (2) 1 = n1+1 2 + c + p 2 − p 1 , r(2)1 = 1−n1 2 − p 1 − p 2 , ρ(2)2 = p 3 − (−1) n1+N ( N 2 + p 1 + p 2 + p 3 ), r(2)2 = −p 3 − (−1) n1+N ( N 2 + p 1 + p 2 + p 3 ). (2.3) It is useful to denote the first and second BI polynomial in the definition by B (1) n1 (z 1 ) and B (2) n2 (z 2 ). Note that B (1) n1 (z 1 ) contains the variable z 2 , while B (2) n2 (z 2 ) contains the degree n 1 in their respective parameters. We shall see that the B n1,n2 (z 1 , z 2 ) are orthogonal polynomials of degree n 1 + n 2 ≤ N in the variables z 1 and z 2 which depend on four parameters p 1 , p 2 , p 3 and c. Let us motivate this definition. In a spirit similar to the one that led to the discovery of the Bannai-Ito polynomials, we look at a q → −1 limit of q-Racah polynomials in two-variable introduced by Gasper and Rahman in [21] as a q-generalization of Tratnik's multivariable Racah polynomials. We start here by specializing their q-Racah polynomials to two variables. Consider the product of q-Racah polynomials R (1) n1 × R (2) n2 depending on four parameters a 1 , a 2 , a 3 and b where R (1) n1 and R (2) n2 are defined by R (1) n1 = 4 φ 3 q −n1 , ba 2 q n1 , q −x1 , a 1 q x1 bq, a 1 a 2 q x2 , q −x2 q; q , R (2) n2 = 4 φ 3 q −n2 , ba 2 a 3 q 2n1+n2 , q n1−x2 , a 1 a 2 q n1+x2 ba 2 q 2n1+1 , a 1 a 2 a 3 q N +n1 , q n1−N q; q (2.4) in terms of the usual basic hypergeometric function r φ s (see e.g. [25]). Up to normalization of the polynomials, those corresponds to the q-Racah polynomials introduced by Gasper and Rahman [21]. Our goal is to take a q → −1 limit of these polynomials in such a way that each generalized q-hypergeometric function reduces to a Bannai-Ito polynomials and that each parameter survives the limit. Let us first write the hypergeometric functions as series : R (1) n1 = ∞ k=0 A (1) k q k , R (2) n2 = ∞ k=0 A (2) k q k (2.5) where the coefficients are given by A (1) k = k−1 i=0 (1 − q −n1+i )(1 − ba 2 q n1+i )(1 − q −x1+i )(1 − a 1 q x1+i ) (1 − q 1+i )(1 − bq 1+i )(1 − a 1 a 2 q x2+i )(1 − q −x2+i ) , A (2) k = k−1 i=0 (1 − q −n2+i )(1 − ba 2 a 3 q 2n1+n2+i )(1 − q n1−x2+i )(1 − a 1 a 2 q n1+x2+i ) (1 − q 1+i )(1 − ba 2 q 2n1+1+i )(1 − a 1 a 2 a 3 q N +n1+i )(1 − q n1−N +i ) . (2.6) Note that the coefficients of an hypergeometric series are usually written in terms of Pochhammer symbols, but for our purpose, it is essential to expand them as products because the parity of the dummy index i will play an important role. Now, achieve the q → −1 limit with the following parametrization q → −e t , t → 0, q x1 → (−1) s 1 2 e ty1 , q x2 → (−1) s 2 2 e ty2 , a 1 → (−1) s 3 2 e tα1 , a 2 → (−1) s 4 2 e tα2 , a 3 → (−1) s 5 2 e tα3 , b → (−1) s 6 2 e tβ (2.7) where the s i ∈ {0, 1}, i = 1, 2, . . . , 6 are integers to be determined. Let us now sketch how one chooses a proper set of s i . First, insert the parametrization (2.7) in the coefficients (2.6) to obtain A (1) k = k−1 i=0 (1 − (−1) b1+i e tB1 )(1 − (−1) b2+i e tB2 )(1 − (−1) b3+i e tB3 )(1 − (−1) b4+i e tB4 ) (1 − (−1) b5+i e tB5 )(1 − (−1) b6+i e tB6 )(1 − (−1) b7+i e tB7 )(1 − (−1) b8+i e tB8 ) , A (2) k = k−1 i=0 (1 − (−1) b9+i e tB9 )(1 − (−1) b10+i e tB10 )(1 − (−1) b11+i e tB11 )(1 − (−1) b12+i e tB12 ) (1 − (−1) b13+i e tB13 )(1 − (−1) b14+i e tB14 )(1 − (−1) b15+i e tB15 )(1 − (−1) b16+i e tB16 ) (2.8) where the b j are linear combinations of n 1 , n 2 , N and the s i , while the B j are linear combinations of the dummy index i, the degrees n 1 , n 2 , N , the variables y 1 , y 2 and the parameters α 1 , α 2 , α 3 , β. The choice of s i should be such that the b j are integers. Thus, depending on the parity of the b j , each factor in the limit t → 0 will alternate between 0 and 2 for incrementing values of i. It is straightforward to see that the parities of the b j must be chosen in such a way that there are the same number of zeroes and twos in the numerator and in the denominator. Otherwise, the limit would diverge or become zero. Now, ratios of 2 will simply cancel out while ratios of 0 will give a non-trivial limit : 1 + e tB l 1 + e tBm t→0 − −− → 1, 1 − e tB l 1 − e tBm t→0 − −− → B l B m . (2.9) The key to obtaining Bannai-Ito polynomials in the limit t → 0 is to chose the s i in such a way that, for each i, there are always 2 zeroes and 2 twos in both numerators and denominators. Such a choice is not unique, but it can be verified by enumeration that all possible choices yield results that are equivalent under affine transformations of the parameters. This implies that (2.8) will reduce to A (1) k t→0 − −− → i even B j1 B j2 B j5 B j6 × i odd B j3 B j4 B j7 B j8 , A (2) k t→0 − −− → i even B j9 B j10 B j13 B j14 × i odd B j11 B j12 B j15 B j16 (2.10) for some permutation π ∈ S 16 of the integers j k = π(k), k = 1, . . . , 16 depending on the choice of the s i . The explicit computation of the limit requires to consider separately all possible parities of the degrees n 1 , n 2 , N and also of the dummy indices k and i. Some notable features arise : First, the products in (2.10) can be written in terms of Pochhammer symbols. However, the products over even values of i will get additional factors when k is odd. Thus, the sums in (2.5) must be split between even and odd values of k. Each sum can be expressed as an hypergeometric 4 F 3 , but the additional factors for k odd have to be pulled in front. One obtains a linear combination of two similar 4 F 3 with some +1 shifts. It is then possible to compare the result with (1.4) to express the result in terms of Bannai-Ito polynomials. Finally, let us consider without loss of generality one possible parametrization for the limit q → −1 : q → −e t , t → 0, q x1 → e ty1 , q x2 → e ty2 , a 1 → −e tα1 , a 2 → e tα2 , a 3 → e tα3 , b → −e tβ . (2.11) For convenience, we also use a different set of parameters : α 1 = 4p 1 − 1, α 2 = 4p 2 , α 3 = 4p 3 + 1, β = 2c, y 1 = 1 2 − 2z 1 , y 2 = 1 2 − 2z 2 − 2p 1 − 2p 2 . (2.12) Using (2.11) and (2.12), a straightforward computation yields R (1) n1 → 1 η n1 B n1 z 1 − 1 4 ; ρ (1) 1 , ρ (1) 2 , r (1) 1 , r (1) 2 R (2) n2 → 1 η n2 B n2 (−1) n1 z 2 − 1 4 ; ρ (2) 1 , ρ (2) 2 , r (2) 1 , r (2) 2 where the parameters are given by (2.2) and (2.3). Omitting the normalization factors, this corresponds to Definition 1 of the bivariate Bannai-Ito polynomials given above. There are two reasons for removing the factors η ni : It is more natural to define the bivariate polynomials as a product of two monic Bannai-Ito polynomials and more importantly, the normalization factor η n1 being a rational function in z 2 would break the polynomial structure. Truncation conditions and orthogonality relation Given our definition of the bivariate Bannai-Ito polynomials, the most important property to verify is orthogonality. We begin by stating the result. r,N −s w (2) s,N B n1,n2 (z 1 (r), z 2 (s))B m1,m2 (z 1 (r), z 2 (s)) = H n1,n2,N δ n1,m1 δ n2,m2 (2.13) where the grids are given by z 1 (r) = 1 2 (−1) r+s+N (r + s − N − 2p 1 + 1 2 ) r = 0, . . . , N (2.14) z 2 (s) = 1 2 (−1) s+N (s − N − 2p 1 − 2p 2 + 1 2 ) s = 0, . . . , N (2.15) the weights by 2.17) and the normalization coefficients H n1,n2,N are given in the appendix. w (1) 2r,2s = (2p 2 ) r (−s) r (1 − 2p 1 − 2s) r ( 3 2 + c − 2p 1 −s) r r!( 1 2 − c −s) r (1 − 2p 1 −s) r (1 − 2p 1 − 2p 2 − 2s) r w (1) 2r+1,2s = − (2p 2 ) r+1 (−s) r+1 (1 − 2p 1 − 2s) r ( 3 2 + c − 2p 1 −s) r r!( 1 2 − c −s) r (1 − 2p 1 −s) r+1 (1 − 2p 1 − 2p 2 − 2s) r+1 w (1) 2r,2s+1 = (2p 2 ) r (−s) r (−2p 1 − 2s) r ( 1 2 + c − 2p 1 −s) r r!(− 1 2 − c −s) r (1 − 2p 1 −s) r (−2p 1 − 2p 2 − 2s) r w (1) 2r+1,2s+1 = − (2p 2 ) r+1 (−s) r (−2p 1 − 2s) r ( 1 2 + c − 2p 1 −s) r+1 r!(− 1 2 − c −s) r+1 (1 − 2p 1 −s) r (−2p 1 − 2p 2 − 2s) r+1 (2.16) and w (2) 2s,2N = (−1) s N s ( 3 2 + c − 2p1 − N )s(1 − 2p1 − 2p2 − 2N )s( 1 2 + 2p3)s N !( 1 2 −c−N )s(1−2p1 −2p2 −2N )2s( 1 2 −2p1 −2p2 −2p3 −2N )s(1−2p1 −2N +2s)N−s w (2) 2s+1,2N = (−1) s N−1 s ( 3 2 + c − 2p1 − N )s(1 − 2p1 − 2p2 − 2N )s( 1 2 + 2p3)s+1 (N −1)!( 1 2 −c−N )s(1−2p1 −2p2 −2N )2s+1( 1 2 −2p1 −2p2 −2p3 −2N )s+1(2−2p1 −2N +2s)N−s w (2) 2s,2N+1 = (−1) s N s ( 1 2 + c − 2p1 − N )s(−2p1 − 2p2 − 2N )s( 1 2 + 2p3)s N !(− 1 2 −c−N )s(−2p1 −2p2 −2N )2s(− 1 2 −2p1 −2p2 −2p3 −2N )s(−2p1 −2N +2s)N+1−s w (2) 2s+1,2N+1 = (−1) s+1 N s ( 1 2 + c − 2p1 − N )s+1(−2p1 − 2p2 − 2N )s( 1 2 + 2p3)s+1 N !(− 1 2 −c−N )s+1(−2p1 −2p2 −2N )2s+1(− 1 2 −2p1 −2p2 −2p3 −2N )s+1(1−2p1 −2N +2s)N−s ( Proof. Our main tool will be the orthogonality relation of the univariate BI polynomials which depends on the truncation conditions (1.10) and (1.14). Notice that definition (2.1) involves a truncation parameter N inherited from the q-Racah polynomials in the limit process and which appears in the parameters (2.3). This implies that the definition comes with truncation conditions that are already built-in. Indeed, one can easily check that B (2) n2 (z 2 ) satisfies a mixture of type i) (1.10) and type iii) (1.14) truncation conditions :    r (2) 1 − ρ (2) 2 = N −n1+1 2 if N − n 1 even, r (2) 1 + r (2) 2 = N −n1+1 2 if N − n 1 odd (2.18) with truncation parameters N − n 1 . Both conditions impose grid points for the variable z 2 : (−1) n1 z 2 − 1 4 = x s (2.19) where x s is given by (1.11) 1 − ρ (1) 2 = N −s+1 2 if N − s even, r(1)1 + r (1) 2 = N −s+1 2 if N − s odd.(1) (2.20) Thus, the polynomial B n1 also satisfies the mixed type i) and type iii) truncation conditions with parameter N − s. Again, both conditions impose grid points for the variable z 1 : z 1 − 1 4 = x r (2.21) s,N B n1,n2 (z 1 (r), z 2 (s))B m1,m2 (z 1 (r), z 2 (s)) = H n1,n2,N δ n1,m1 δ n2,m2 . r,N −s B (1) n1 (z 1 (r))B (1) m1 (z 1 (r)) = h(1)2n 1 ,2s =s ! n1!(2p2)n 1 ( 1 2 +c+n1 +2p2)s−n 1 ( 1 2 +c+2p2 +s)n 1 ( 3 2 +c−2p1 −s)n 1 (1−2p1 −2s)s (s − n1)!( 1 2 + c + n1)s−n 1 ( 1 2 + c + n1 + 2p2) 2 n 1 (1 − 2p1 − 2p2 − 2s)s−n 1 , h(1)2n 1 +1,2s =s ! n1!(2p2)n 1 +1( 1 2 +c+n1 +2p2)s−n 1 ( 1 2 +c+2p2 +s)n 1 +1( 3 2 +c−2p1 −s)n 1 (1−2p1 −2s)s (s − n1 − 1)!( 1 2 + c + n1 + 1)s−n 1 −1( 1 2 + c + n1 + 2p2) 2 n 1 +1 (1 − 2p1 − 2p2 − 2s)s−n 1 , h(1)2n 1 ,2s+1 =s ! n1!(2p2)n 1 ( 1 2 +c+n1 +2p2)s+1−n 1 ( 3 2 +c+2p2 +s)n 1 ( 1 2 +c−2p1 −s)n 1 (−2p1 −2s)s+1 (s − n1)!( 1 2 + c + n1)s+1−n 1 ( 1 2 + c + n1 + 2p2) 2 n 1 (−2p1 − 2p2 − 2s)s+1−n 1 , h(1)2n 1 +1,2s+1 =s ! n1!(2p2)n 1 +1( 1 2 +c+n1 +2p2)s+1−n 1 ( 1 2 +c+2p2 +s)n 1 +1( 1 2 +c−2p1 −s)n 1 (−2p1 −2s)s+1 (s − n1)!( 3 2 + c + n1)s−n 1 ( 1 2 + c + n1 + 2p2) 2 n 1 +1 (−2p1 − 2p2 − 2s)s−n 1 . (2.24) Using (2.23) in (2.22), one gets that s h (1) n1,N −s w (2) s,N B (2) n2 (z 2 (s))B (2) m2 (z 2 (s)) = H n1,n2,N δ n2,m2 (2.25) should be the orthogonality relation satisfied by the univariate BI polynomials B (2) n2 (z 2 ). Indeed, using (2.18) and the corresponding weights from section 1, one readily checks that the w (2) s,N are given by (2.17) and the normalization coefficients are those provided in the appendix. Hence the bivariate Bannai-Ito polynomials obey the orthogonality relation N −n1 s=0 N −s r=0 w (1) r,N −s w (2) s,N B n1,n2 (z 1 (r), z 2 (s))B m1,m2 (z 1 (r), z 2 (s)) = H n1,n2,N δ n1,m1 δ n2,m2 (2.26) with the weights, the grids and the normalization coefficients given. It is not hard to verify that both sums can be extended from 0 to N without changing the results. Indeed, one can check that all the extra terms are in fact zero because of the weights. Limit from the bivariate Askey-Wilson polynomials In this section, a different definition for the bivariate Bannai-Ito polynomials via a q → −1 limit of the Askey-Wilson polynomials is investigated. While very similar to the approach from q-Racah polynomials, the main difference lies in the fact that the Askey-Wilson polynomials do not have truncation conditions. Hence, no truncation parameter N is carried through the limit and a definition for untruncated bivariate Bannai-Ito polynomials is obtained. This definition has the advantage that its multispectrality relations can be expressed in terms of operators acting directly on the variables instead of acting on the orthogonality grids. The connection between both approaches is established. Untruncated bivariate Bannai-Ito polynomials Definition 2. The untruncated bivariate Bannai-Ito polynomials are defined by B n1,n2 (z 1 , z 2 ) = B n1 z 1 − 1 4 ; β, z 2 + ǫ − 1 4 , α, z 2 − ǫ + 1 4 (3.1) × B n2 (−1) n1 z 2 − 1 4 ; β + ǫ + n1 2 , (1 − π n1 )γ + π n1 δ, α − ǫ − n1 2 , (π n1 − 1)δ − π n1 γ in terms of the monic BI polynomials B n (x) and where π n = 1 + (−1) n 2 = 1 n even, 0 n odd, is the indicator function of even numbers. Note that Definition 2 reduces to Definition 1 (2.1) of section 2 if we let α → 1 2 − p 1 , β → 1 2 + c − p 1 , ǫ → p 2 , γ = − N 2 − p 1 − p 2 N even, N −1 2 + p 1 + p 2 + p 3 N odd, δ = N −1 2 + p 1 + p 2 + p 3 N even, − N 2 − p 1 − p 2 N odd. (3.3) To motivate Definition 2, let us first consider the Askey-Wilson polynomialŝ p n (x; a, b, c, d) = 4 φ 3 q −n , abcdq n−1 , az, az −1 ab, ac, ad q; q (3.4) in the variable x = 1 2 (z + z −1 ). The bivariate Askey-Wilson polynomials depending on five parameters a, b, c, d, a 2 as introduced by Gasper and Rahman in [22] arê P n1,n2 (x 1 , x 2 ) =p n1 (x 1 ; a, b, a 2 z 2 , a 2 z −1 2 )p n2 (x 2 ; aa 2 q n1 , ba 2 q n1 , c, d). (3.5) The limiting procedure will be the same as in the previous section. Briefly, the choice of parametrization amounts to a selection of phases in front of each parameter defined as exponentials. Expanding the hypergeometric functions as series and expanding the Pochhammer symbols as products, one obtains an expression of the form (2.8) and must select the phases in such a way that there are always 2 zeroes and 2 twos in both numerators and denominators for each value of the dummy index i. Again, this choice is not unique, but all possibilities can again be shown to yield equivalent Bannai-Ito polynomials under affine transformations of the parameters. We take q → −e t , t → 0, z 1 → e ty1 , z 2 → e ty2 , a → ie tã , b → −ie tb , c → ie tc , d → −ie td , a 2 → e tã2 (3.6) with the reparametrization y 1 = −2z 1 , y 2 = −2z 2 ,ã = −2α + 1 2 ,b = 2β + 1 2 ,c = 2γ + 1 2 ,d = 2δ + 1 2 ,ã 2 = 2ǫ. (3.7) This gives P (1) n1 (cos θ 1 ) → 1 η n1 B n1 z 1 − 1 4 ; β, z 2 + ǫ − 1 4 , α, z 2 − ǫ + 1 4 P (2) n2 (cos θ 2 ) → 1 η n2 B n2 (−1) n1 z 2 − 1 4 ; β + ǫ + n1 2 , (1−π n1 )γ + π n1 δ, α − ǫ − n1 2 , (π n1 −1)δ − π n1 γ . (3.8) The definition for the untruncated bivariate Bannai-Ito polynomials is thus obtained by taking the product of the corresponding monic Bannai-Ito polynomials, again dropping the normalization factors. Multispectrality of the bivariate BI polynomials Iliev demonstrated the multispectrality of the bivariate Askey-Wilson polynomials (3.5) (with a different normalization) in [20]. This section examines how the multispectrality relations of these polynomials are carried in the q → −1 limit. Recalling the definition of the shift operator T m x and the reflection operator R x given after (1.19), we have the following equations in the variables z 1 and z 2 . L 1 B n1,n2 (z 1 , z 2 ) = µ n1 B n1,n2 (z 1 , z 2 ) (3.9) L 2 B n1,n2 (z 1 , z 2 ) = λ n1,n2 B n1,n2 (z 1 , z 2 ) (3.10) for the operators L 1 = (ǫ − z 1 + z 2 )(z 1 − β − 1 4 ) 2(z 1 − 1 4 ) (T −1/2 z1 R z1 − 1) + (ǫ + z 1 − z 2 )(z 1 − α + 1 4 ) 2(z 1 + 1 4 ) (T 1/2 z1 R z1 − 1) (3.11) and L 2 = 1 i,j=−1 c i,j T i/2 z1 R i z1 T j/2 z2 R j z2 (3.12) with coefficients c −1,−1 = (z 1 − α + 1 4 )(z 2 + γ + 1 4 )(ǫ + z 1 + z 2 + 1 2 ) 4(z 1 + 1 4 )(z 2 + 1 4 ) c −1,0 = (z 1 − α + 1 4 )(ǫ + z 1 − z 2 )(δ(z 2 + 1 4 ) − γ(z 2 − 1 4 )) 4(z 1 + 1 4 )(z 2 − 1 4 )(z 2 + 1 4 ) c −1,1 = (z 1 − α + 1 4 )(z 2 − δ − 1 4 )(ǫ + z 1 − z 2 ) 4(z 1 + 1 4 )(z 2 − 1 4 ) c 0,−1 = (z 2 + γ + 1 4 )(ǫ − z 1 + z 2 )(α(z 1 − 1 4 ) + β(z 1 + 1 4 )) 4(z 1 − 1 4 )(z 1 + 1 4 )(z 2 + 1 4 ) c 0,0 = α(γ − δ + 1 2 ) + β(γ − δ − 1 2 ) − 1 2 (γ + δ + 1 2 ) − ǫ + (ǫ + 4z 1 z 2 − 1 4 )(α(z 1 − 1 4 ) + β(z 1 + 1 4 ))(δ(z 2 + 1 4 ) − γ(z 2 − 1 4 )) 4(z 1 − 1 4 )(z 1 + 1 4 )(z 2 − 1 4 )(z 2 + 1 4 ) c 0,1 = (z 2 − δ − 1 4 )(ǫ + z 1 − z 2 )(α(z 1 − 1 4 ) + β(z 1 + 1 4 )) 4(z 1 − 1 4 )(z 1 + 1 4 )(z 2 − 1 4 ) c 1,−1 = (z 1 − β − 1 4 )(z 2 + γ + 1 4 )(ǫ − z 1 + z 2 ) 4(z 1 − 1 4 )(z 2 + 1 4 ) c 1,0 = (z 1 − β − 1 4 )(ǫ − z 1 + z 2 )(δ(z 2 + 1 4 ) − γ(z 2 − 1 4 )) 4(z 1 − 1 4 )(z 2 − 1 4 )(z 2 + 1 4 ) c 1,1 = (z 1 − β − 1 4 )(z 2 − δ − 1 4 )(ǫ − z 1 − z 2 + 1 2 ) 4(z 1 − 1 4 )(z 2 − 1 4 ) . (3.13) The eigenvalues are given by µ n1 = n1 2 n 1 even, − n1 2 + α − β − 2ǫ n 1 odd, (3.14) and λ n1,n2 = n1+n2 2 n 1 + n 2 even, where ξ n (a, b, c, d) = (ab, ac, ad; q) n a n (3.17) and their corresponding bivariate extension P n1,n2 (x 1 , x 2 ) = ζ n1,n2 p n1 (x 1 ; a, b, a 2 z 2 , a 2 z −1 2 )p n2 (x 2 ; aa 2 q n1 , ba 2 q n1 , c, d) (3.18) with normalization ζ n1,n2 = c n1+n2 a n1 2 (a 2 2 ; q) n1 (aca 2 ; q) n1+n2 (bca 2 ; q) n1+n2 (cd; q) n2 . n1+n2+1 2 − α + β + γ + δ + 2ǫ n 1 + n 2 odd. (3.19) They obey the q-difference equation [20] LP n1,n2 (x 1 , x 2 ) = Λ n1,n2 P n1,n2 (x 1 , x 2 ) (3.20) where L = 1 i,j=−1 C i,j E i q,z1 E j q,z2 (3.21) in terms of the shifts operators E q,z which send z → qz. The explicit expression for the coefficients C i,j can be found in the appendix and the eigenvalues are given by Λ n1,n2 = q −n1−n2 − 1 1 − aa 2 2 bcdq n1+n2−1 . (3.22) The difference equation for the bivariate Bannai-Ito polynomials is found as a limit of this relation. The operator L will correspond to the operator L 2 given by (3.12) in the limit (3.6) with the reparametrization (3.7). The coefficients c i,j are obtained by the limits c i,j = lim q→−1 C i,j 4(1 + q) (3.23) and the eigenvalues by λ n1,n2 = lim q→−1 Λ n1,n2 4(1 + q) = n1+n2 2 n 1 + n 2 even, n1+n2+1 2 − α + β + γ + δ + 2ǫ n 1 + n 2 odd. (3.24) The factor 1 4 is just for convenience. The bivariate BI polynomials (3.1) will thus satisfy the difference equation (3.10). The difference equation (3.9) follows directly from the univariate Dunkl difference equation given by (1.18), (1.19) and (1.20). It is also possible to obtain it from a q → −1 limit of the bivariate Askey-Wilson polynomials second q-difference equation [20]. Let us now turn to the recurrence relations. Proposition 3. The untruncated bivariate Bannai-Ito polynomials B n1,n2 (z 1 , z 2 ) defined in (3.1) verify the 3-term recurrence relation (−1) n1 z 2 − 1 4 B n1,n2 (z 1 , z 2 ) = B n1,n2+1 (z 1 , z 2 ) + (β + ǫ + n1 2 − A n2 − C n2 )B n1,n2 (z 1 , z 2 ) (3.25) + A n2−1 C n2 B n1,n2−1 (z 1 , z 2 ) where the coefficients A n2 and C n2 are given by (1.7) with the parameters ρ 1 , ρ 2 , r 1 , r 2 being those of the second BI polynomial of (3.1). They also satisfy the 9-term recurrence relation (z 1 − α 2 + β 2 )B n1,n2 (z 1 , z 2 ) = θ (1) n1,n2 B n1+1,n2 + θ (2) n1,n2 B n1+1,n2−1 + θ (3) n1,n2 B n1+1,n2−2 + θ (4) n1,n2 B n1,n2+1 + θ (5) n1,n2 B n1,n2 + θ (6) n1,n2 B n1,n2−1 + θ (7) n1,n2 B n1−1,n2+2 + θ (8) n1,n2 B n1−1,n2+1 + θ (9) n1,n2 B n1−1,n2 (3.26) where the explicit expression for the coefficients θ (i) n1,n2 are given in the appendix. Proof. The polynomials P n1,n2 (x 1 , x 2 ) verify [20] ca 2 (a+b)(ab+q) ab(1+q) −z 1 −z −1 1 P n1,n2 (x 1 , x 2 ) = τ (1) n1,n2 P n1+1,n2 + τ (2) n1,n2 P n1+1,n2−1 + τ (3) n1,n2 P n1+1,n2−2 + τ (4) n1,n2 P n1,n2+1 + τ (5) n1,n2 P n1,n2 + τ (6) n1,n2 P n1,n2−1 + τ (7) n1,n2 P n1−1,n2+2 + τ (8) n1,n2 P n1−1,n2+1 + τ (9) n1,n2 P n1−1,n2 . (3.27) The expression for the coefficients τ (i) n1,n2 can be found in the appendix. This will become a 9-term recurrence relation for the bivariate Bannai-Ito polynomials in the q → −1 limit. The only tricky part is to keep track of all the changes in normalization of the various polynomials in play. Denote by N n1,n2 = ζ n1,n2 ξ n1 (x 1 ; a, b, a 2 z 2 , a 2 z −1 2 )ξ n2 (x 2 ; aa 2 q n1 , ba 2 q n1 , c, d) (3.28) the normalization factors that appear in (3.18) and by M n1,n2 = η (1) n1 η (2) n2 (3.29) the normalization coefficients in the monic BI OPs (3.1) given by (1.5). Now, the recurrence coefficients are obtained by the following limits : θ (1) n1,n2 = M n1,n2 M n1+1,n2 lim q→−1 N n1+1,n2 N n1,n2 τ (1) n1,n2 4(1+q) , θ (6) n1,n2 = M n1,n2 M n1,n2−1 lim q→−1 N n1,n2−1 N n1,n2 τ (6) n1,n2 4(1+q) , θ (2) n1,n2 = M n1,n2 M n1+1,n2−1 lim q→−1 N n1+1,n2−1 N n1,n2 τ (2) n1,n2 4(1+q) , θ (7) n1,n2 = M n1,n2 M n1−1,n2+2 lim q→−1 N n1−1,n2+2 N n1,n2 τ (7) n1,n2 4(1+q) , θ (3) n1,n2 = M n1,n2 M n1+1,n2−2 lim q→−1 N n1+1,n2−2 N n1,n2 τ (3) n1,n2 4(1+q) , θ (8) n1,n2 = M n1,n2 M n1−1,n2+1 lim q→−1 N n1−1,n2+1 N n1,n2 τ (8) n1,n2 4(1+q) , θ (4) n1,n2 = M n1,n2 M n1,n2+1 lim q→−1 N n1,n2+1 N n1,n2 τ (4) n1,n2 4(1+q) , θ (9) n1,n2 = M n1,n2 M n1−1,n2 lim q→−1 N n1−1,n2 N n1,n2 τ (9) n1,n2 4(1+q) , θ (5) n1,n2 = lim q→−1 τ (5) n1,n2 4(1+q) . The limits are assumed to be parametrized by (3.6) and (3.7). The results of these limits can be found in the appendix. Moreover, the recurrence relation operator is obtained via lim q→−1 ca 2 4(1 + q) (a+b)(ab+q) ab(1+q) −z 1 −z −1 1 = z 1 − α 2 + β 2 . (3.30) Combining all these results, the recurrence relation (3.27) reduces to the desired 9-term recurrence relation for the bivariate Bannai-Ito polynomials. The 3-term recurrence relation simply follows from the the recurrence relation of the univariate Bannai-Ito polynomials (1.6) applied to the second polynomial of (3.1). These two propositions establish the full multispectrality of the bivariate Bannai-Ito polynomials. Importantly, the recurrence relations prove that the B n1,n2 are polynomials and not simply rational functions of z 1 and z 2 . Conclusion This paper has enlarged the catalogue of orthogonal polynomials in two variables with the construction of bivariate polynomials of Bannai-Ito type. Their identification and characterization made use of the q → −1 limits of both the bivariate q-Racah and Askey-Wilson polynomials of Gasper and Rahman. The first instance led to a truncated version (Definition 1) equipped with a set of positive-definite weights on a twodimensionnal lattice against which the BI polynomials are orthogonal. The q → −1 limit of the bivariate Askey-Wilson polynomials yielded untruncated Bannai-Ito polynomials in two variables (Definition 2) out of which the finite ones (Definition 1) can be obtained by the choice of parameters (3.3). This latter approach allowed for the identification of the difference equations and recurrence relations obeyed by the resulting functions showing in particular that they are indeed polynomials. Let us remark that the finite bivariate Bannai-Ito polynomials that have been found do only make use of the truncation conditions i) and iii) that the univariate polynomials admit. The question of whether there are other bivariate extensions that rely on different reduction mixtures and in particular condition ii) is open and certainly worth exploring. In another vein, one may wonder if there are natural multivariate generalizations of the Bannai-Ito polynomials along the symmetric function direction. In this respect, the examination of the q → −1 limit of the Koornwinder polynomials of BC 2 type could prove illuminating and is envisaged. We have initiated this exploration of the Bannai-Ito polynomials in many variables within the Tratnik framework because of the expected occurence of extensions of that type in the representation theory of the higher rank Bannai-Ito algebra [26] as well as in certain superintegrable models that have been constructed [27,28]. Let us mention the following to be concrete. A Hamiltonian system on the 3-sphere whose symmetries realize the Bannai-Ito algebra of rank 2 has been constructed in [27] and various bases of wavefunctions have been explicitly obtained using the Cauchy-Kovalevskaia extension theorem. It is expected that bivariate Bannai-Ito polynomials arise in the interbasis connection coefficients. Do these overlaps coincide with the two-variable polynomials constructed here or do they belong to another extension yet to be found? We plan on looking into this in the near future. Another related question is to determine the algebra underscoring the multispectrality of the two variable BI polynomials we have defined, that is the algebra generated by L 1 , L 2 , x 1 and x 2 . How does the resulting algebra compare with the rank 2 Bannai-Ito algebra? We hope to report on most of these questions soon. Acknowledgments JML holds an Alexander-Graham-Bell PhD fellowship from the Natural Science and Engineering Research Council (NSERC) of Canada. LV is grateful to NSERC for support through a discovery grant. A Appendix To make the article more reader friendly, some cumbersome formulas have been relegated to this appendix. The normalization coefficients appearing in the orthogonality relation (2.13) are given by H 2n1,2n2,2N = n 1 !n 2 !(2p 2 ) n1 (2p 3 + 1 2 ) n2 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 3 2 ) n1+n2 (N −n 1 −n 2 )!(c+n 1 + 1 2 ) N−n1 (c+n 1 +2p 2 + 1 2 ) 2 n1 (c+2n 1 +n 2 +2p 2 + 1 2 ) N−n1−n2 × (c+N +n 1 +2p 2 +2p 3 +1) n2 (c+2n 1 +n 2 +2p 2 +2p 3 +1) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +1) 2 n2 (−2N −2p 1 −2p 2 −2p 3 + 1 2 ) N−n1−n2 H 2n1+1,2n2,2N = n 1 !n 2 !(2p 2 ) n1+1 (2p 3 + 1 2 ) n2 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 3 2 ) n1+n2 (N −n 1 −n 2 −1)!(c+n 1 + 3 2 ) N−n1−1 (c+n 1 +2p 2 + 1 2 ) 2 n1+1 (c+2n 1 +n 2 +2p 2 + 3 2 ) N−n1−n2−1 × (c+N +n 1 +2p 2 +2p 3 +2) n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) 2 n2 (−2N −2p 1 −2p 2 −2p 3 + 1 2 ) N−n1−n2 H 2n1,2n2+1,2N = n 1 !n 2 !(2p 2 ) n1 (2p 3 + 1 2 ) n2+1 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 3 2 ) n1+n2 (N −n 1 −n 2 −1)!(c+n 1 + 1 2 ) N−n1 (c+n 1 +2p 2 + 1 2 ) 2 n1 (c+2n 1 +n 2 +2p 2 + 3 2 ) N−n1−n2−1 × (c+N +n 1 +2p 2 +2p 3 +1) n2+1 (c+2n 1 +n 2 +2p 2 +2p 3 +1) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +1) 2 n2+1 (−2N −2p 1 −2p 2 −2p 3 + 1 2 ) N−n1−n2 H 2n1+1,2n2+1,2N = n 1 !n 2 !(2p 2 ) n1+1 (2p 3 + 1 2 ) n2+1 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 3 2 ) n1+n2+1 (N −n 1 −n 2 −1)!(c+n 1 + 3 2 ) N−n1−1 (c+n 1 +2p 2 + 1 2 ) 2 n1+1 (c+2n 1 +n 2 +2p 2 + 5 2 ) N−n1−n2−2 × (c+N +n 1 +2p 2 +2p 3 +2) n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) 2 n2+1 (−2N −2p 1 −2p 2 −2p 3 + 1 2 ) N−n1−n2−1 H 2n1,2n2,2N+1 = n 1 !n 2 !(2p 2 ) n1 (2p 3 + 1 2 ) n2 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 1 2 ) n1+n2 (N −n 1 −n 2 )!(c+n 1 + 1 2 ) N−n1+1 (c+n 1 +2p 2 + 1 2 ) 2 n1 (c+2n 1 +n 2 +2p 2 + 1 2 ) N−n1−n2 × (c+N +n 1 +2p 2 +2p 3 +2) n2 (c+2n 1 +n 2 +2p 2 +2p 3 +1) N−n1−n2+1 (c+2n 1 +n 2 +2p 2 +2p 3 +1) 2 n2 (−2N −2p 1 −2p 2 −2p 3 − 1 2 ) N−n1−n2+1 H 2n1+1,2n2,2N+1 = n 1 !n 2 !(2p 2 ) n1+1 (2p 3 + 1 2 ) n2 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 1 2 ) n1+n2+1 (N −n 1 −n 2 )!(c+n 1 + 3 2 ) N−n1 (c+n 1 +2p 2 + 1 2 ) 2 n1+1 (c+2n 1 +n 2 +2p 2 + 3 2 ) N−n1−n2−1 × (c+N +n 1 +2p 2 +2p 3 +2) n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) 2 n2 (−2N −2p 1 −2p 2 −2p 3 − 1 2 ) N−n1−n2 H 2n1,2n2+1,2N+1 = n 1 !n 2 !(2p 2 ) n1 (2p 3 + 1 2 ) n2+1 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 1 2 ) n1+n2+1 (N −n 1 −n 2 )!(c+n 1 + 1 2 ) N−n1+1 (c+n 1 +2p 2 + 1 2 ) 2 n1 (c+2n 1 +n 2 +2p 2 + 3 2 ) N−n1−n2−1 × (c+N +n 1 +2p 2 +2p 3 +1) n2+1 (c+2n 1 +n 2 +2p 2 +2p 3 +1) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +1) 2 n2+1 (−2N −2p 1 −2p 2 −2p 3 − 1 2 ) N−n1−n2 H 2n1+1,2n2+1,2N+1 = n 1 !n 2 !(2p 2 ) n1+1 (2p 3 + 1 2 ) n2+1 (c+n 1 +2p 2 + 1 2 ) N (c−N −2p 1 + 1 2 ) n1+n2+1 (N −n 1 −n 2 −1)!(c+n 1 + 3 2 ) N−n1 (c+n 1 +2p 2 + 1 2 ) 2 n1+1 (c+2n 1 +n 2 +2p 2 + 5 2 ) N−n1−n2−2 × (c+N +n 1 +2p 2 +2p 3 +2) n2+1 (c+2n 1 +n 2 +2p 2 +2p 3 +2) N−n1−n2 (c+2n 1 +n 2 +2p 2 +2p 3 +2) 2 n2+1 (−2N −2p 1 −2p 2 −2p 3 − 1 2 ) N−n1−n2 τ (7) n1,n2 = aa 4 2 bq 2n1+1 (1−q n1 ) (q−abq n1 ) (1−cdq n2 ) 1−cdq n2+1 (1−aa 2 cq n1+n2 ) (1−a 2 bcq n1+n2 ) (q−aa 2 2 bq 2n1 ) (q 2 −aa 2 2 bq 2n1 ) q−aa 2 2 bcdq 2(n1+n2) aa 2 2 bcdq 2(n1+n2) −1 τ (8) n1,n2 = a 3 2 cq n1 (q n1 −1) (q−abq n1 ) (cdq n2 −1) q−aa 2 2 bq 2n1+n2 (a + b)(q + aa 2 2 bcdq 2(n1+n2) ) − aa 2 b(c+d)(1+q)q n1+n2 (q−aa 2 2 bq 2n1 ) (q 2 −aa 2 2 bq 2n1 ) q 2 −aa 2 2 bcdq 2(n1+n2) 1−aa 2 2 bcdq 2(n1+n2) τ (9) n1,n2 = − a 2 2 c 2 (q n1 −1) (abq n1 −q) aa 2 2 bq 2n1+n2 −q aa 2 2 bq 2n1+n2 −q 2 (aa 2 dq n1+n2 −q) (a 2 bdq n1+n2 −q) (q−aa 2 2 bq 2n1 ) (q 2 −aa 2 2 bq 2n1 ) q−aa 2 2 bcdq 2(n1+n2) q 2 −aa 2 2 bcdq 2(n1+n2) The coefficients for the 9-term recurrence relation (3.26) satisfied by the bivariate Bannai-Ito polynomials are θ (1) n1,n2 = (−1) n2 θ (2) n1,n2 =                  n2 4 2β+2γ+2ǫ+n1+n2+1 −α+β+γ+δ+2ǫ+n1+n2+1 − −2α+2γ+2ǫ+n1+n2 −α+β+γ+δ+2ǫ+n1+n2 n 1 even, n 2 even, 1 4 (2γ + 2δ + n 2 ) −2α+2γ+2ǫ+n1+n2+1 −α+β+γ+δ+2ǫ+n1+n2+1 − 2β+2γ+2ǫ+n1+n2 −α+β+γ+δ+2ǫ+n1+n2 n 1 even, n 2 odd, n2 4 2β+2γ+2ǫ+n1+n2 −α+β+γ+δ+2ǫ+n1+n2 − −2α+2γ+2ǫ+n1+n2+1 −α+β+γ+δ+2ǫ+n1+n2+1 n 1 odd, n 2 even, 1 4 (2γ + 2δ + n 2 ) −2α+2γ+2ǫ+n1+n2 −α+β+γ+δ+2ǫ+n1+n2 − 2β+2γ+2ǫ+n1+n2+1 −α+β+γ+δ+2ǫ+n1+n2+1 n 1 odd, n 2 odd, θ (3) n1,n2 =                − n2(2γ+2δ+n2−1)(−2α+2γ+2ǫ+n1+n2)(2β+2δ+2ǫ+n1+n2) 16(−α+β+γ+δ+2ǫ+n1+n2) 2 n 1 even, n 2 even, (n2−1)(2γ+2δ+n2)(−2α+2δ+2ǫ+n1+n2)(2β+2γ+2ǫ+n1+n2) 16(−α+β+γ+δ+2ǫ+n1+n2) 2 n 1 even, n 2 odd, − n2(2γ+2δ+n2−1)(−2α+2δ+2ǫ+n1+n2)(2β+2γ+2ǫ+n1+n2) 16(−α+β+γ+δ+2ǫ+n1+n2) 2 n 1 odd, n 2 even, n 1 odd, n 2 odd, θ (5) n1,n2 =                                                − 1 4 n1 −2α+2β+4ǫ+2n1−1 + 4ǫ+n1 −2α+2β+4ǫ+2n1+1 − 1 2α+2β − 2β+2γ+2ǫ+n1+n2+1 −α+β+γ+δ+2ǫ+n1+n2+1 +1 × −2α+2β −θ (6) n1,n2 =                                            n2 2α−2β+θ (8) n1,n2 =                       θ (9) n1,n2 =                                        Proposition 1 . 1The bivariate Bannai-Ito polynomials defined in (2.1) satisfy the orthogonality relation when N + n 1 is even and by (1.16) when N + n 1 is odd. Inspecting this equation for each possible parities of n 1 and N , one obtains (2.15). Substituting this relation for z 2 in the parameters (2.2), one can now check that the truncation conditions satisfied by B n1 are with x r given by ( 1 . 111) for N − s even and (1.16) for N − s odd. This amounts to(2.14). Recall that we are looking for an orthogonality relation of the form Proposition 2 . 2The untruncated bivariate Bannai-Ito polynomials (3.1) obey the difference equations (3. 15 ) 15Proof. Consider the renormalized Askey-Wilson polynomials p n (x; a, b, c, d) = ξ n (a, b, c, d)p n (x; a, b, c, d)(3.16) +4ǫ+2n 1 −1 n 1 odd, n 2 odd, odd, n 2 odd. Definition 1. The bivariate Bannai-Ito polynomials are defined by2 Limit from the bivariate q-Racah polynomials 2.1 Defining the bivariate Bannai-Ito polynomials The coefficients of the differential operator L (3.21) of section 3 readThe recurrence coefficients for the Askey-Wilson polynomials appearing in (3.27) have the following expressions(a+b)(q+aa 2 2 bcdq 2(n 1 +n 2 ) ) aa2bdq n 1 +n 2 (q+1) 1+ cd q − (q+aa 2 2 bq 2n 1 )(q+aa 2 2 bcdq 2(n 1 +n 2 ) ) aa 2 2 bq 1+2n 1 +n 2 (q+1)1− q 2−2n 1(1−aa 2 2 bq 2n1 ) 1− q −2(n 1 +n 2 −1) aa 2 2 bcd 1−aa 2 2 bcdq 2(n1+n2) τ (6) n1,n2 = a 2 2 c 2 q n1 (q n2 −1) (q−aa 2 2 bq 2n1+n2 ) (q−aa 2 dq n1+n2 ) (q−a 2 bdq n1+n2 ) (abq n1 (1+q)(a 2 2 +q)−(ab+q)(q+aa 2 2 bq 2n1 )) (q 2 −aa 2 2 bq 2n1 ) (aa 2 2 bq 2n1 −1) q−aa 2 2 bcdq 2(n1+n2) q 2 −aa 2 2 bcdq 2(n1+n2) Algebraic combinatorics. E Bannai, T Ito, Benjamin/Cummings Menlo ParkE. Bannai, T. Ito, Algebraic combinatorics, Benjamin/Cummings Menlo Park, 1984. 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F. Dunkl, Y. Xu, Orthogonal polynomials of several variables, no. 155, Cambridge University Press, 2014. L Fernández, F Marcellán, T E Pérez, M A Pinar, Differential algebra, complex analysis and orthogonal polynomials. RIThe American Mathematical Society Providence509Recent trends on two variable orthogonal polynomialsL. Fernández, F. Marcellán, T. E. Pérez, M. A. Pinar, Recent trends on two variable orthogonal poly- nomials, in: Differential algebra, complex analysis and orthogonal polynomials, Vol. 509, The American Mathematical Society Providence, RI, 2010, pp. 59-86. I G Macdonald, Affine Hecke algebras and orthogonal polynomials. Cambridge University Press157I. G. Macdonald, et al., Affine Hecke algebras and orthogonal polynomials, Vol. 157, Cambridge Uni- versity Press, 2003. Some multivariable orthogonal polynomials of the askey tableau-discrete families. M Tratnik, Journal of mathematical physics. 329M. Tratnik, Some multivariable orthogonal polynomials of the askey tableau-discrete families, Journal of mathematical physics 32 (9) (1991) 2337-2342. M Tratnik, Some multivariable orthogonal polynomials of the askey tableau-continuous families. 32M. Tratnik, Some multivariable orthogonal polynomials of the askey tableau-continuous families, Journal of mathematical physics 32 (8) (1991) 2065-2073. J S Geronimo, P Iliev, Bispectrality of multivariable racah-wilson polynomials. 31J. S. Geronimo, P. Iliev, Bispectrality of multivariable racah-wilson polynomials, Constructive Approx- imation 31 (3) (2010) 417-457. P Iliev, Bispectral commuting difference operators for multivariable askey-wilson polynomials. 363P. Iliev, Bispectral commuting difference operators for multivariable askey-wilson polynomials, Trans- actions of the American Mathematical Society 363 (3) (2011) 1577-1598. Some systems of multivariable orthogonal q-racah polynomials. G Gasper, M Rahman, The Ramanujan Journal. 131-3G. Gasper, M. Rahman, Some systems of multivariable orthogonal q-racah polynomials, The Ramanujan Journal 13 (1-3) (2007) 389-405. Some systems of multivariable orthogonal askey-wilson polynomials. G Gasper, M Rahman, Theory and applications of special functions. SpringerG. Gasper, M. Rahman, Some systems of multivariable orthogonal askey-wilson polynomials, in: Theory and applications of special functions, Springer, 2005, pp. 209-219. The para-racah polynomials. J.-M Lemay, L Vinet, A Zhedanov, Journal of Mathematical Analysis and Applications. 4382J.-M. Lemay, L. Vinet, A. Zhedanov, The para-racah polynomials, Journal of Mathematical Analysis and Applications 438 (2) (2016) 565-577. A q-generalization of the para-racah polynomials. J.-M Lemay, L Vinet, A Zhedanov, Journal of Mathematical Analysis and Applications. 4621J.-M. Lemay, L. Vinet, A. Zhedanov, A q-generalization of the para-racah polynomials, Journal of Mathematical Analysis and Applications 462 (1) (2018) 323-336. R Koekoek, P A Lesky, R F Swarttouw, Hypergeometric orthogonal polynomials and their qanalogues. Springer Science & Business MediaR. Koekoek, P. A. Lesky, R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q- analogues, Springer Science & Business Media, 2010. The Z n 2 Dirac-Dunkl operator and a higher rank Bannai-Ito algebra. H De Bie, V X Genest, L Vinet, Advances in Mathematics. 303H. De Bie, V. X. Genest, L. Vinet, The Z n 2 Dirac-Dunkl operator and a higher rank Bannai-Ito algebra, Advances in Mathematics 303 (2016) 390-414. H De Bie, V X Genest, J.-M Lemay, L Vinet, arXiv:1601.07642A superintegrable model with reflections on S 3 and the rank two Bannai-Ito algebra. arXiv preprintH. De Bie, V. X. Genest, J.-M. Lemay, L. Vinet, A superintegrable model with reflections on S 3 and the rank two Bannai-Ito algebra, arXiv preprint arXiv:1601.07642. A superintegrable model with reflections on S n−1 and the higher rank Bannai-Ito algebra. 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{'abstract': 'A two-variable extension of the Bannai-Ito polynomials is presented. They are obtained via q → −1 limits of the bivariate q-Racah and Askey-Wilson orthogonal polynomials introduced by Gasper and Rahman. Their orthogonality relation is obtained. These new polynomials are also shown to be multispectral. Two Dunkl shift operators are seen to be diagonalized by the bivariate Bannai-Ito polynomials and 3and 9-term recurrence relations are provided.', 'arxivid': '1809.09705', 'author': ['Jean-Michel Lemay [email protected] \nCentre de Recherches Mathématiques\nUniversité de Montréal\nSucc. Centre-villeC.P. 6128, H3C 3J7MontréalQCCanada\n', 'Luc Vinet [email protected] \nCentre de Recherches Mathématiques\nUniversité de Montréal\nSucc. Centre-villeC.P. 6128, H3C 3J7MontréalQCCanada\n'], 'authoraffiliation': ['Centre de Recherches Mathématiques\nUniversité de Montréal\nSucc. Centre-villeC.P. 6128, H3C 3J7MontréalQCCanada', 'Centre de Recherches Mathématiques\nUniversité de Montréal\nSucc. Centre-villeC.P. 6128, H3C 3J7MontréalQCCanada'], 'corpusid': 119136803, 'doi': '10.1063/1.5063468', 'github_urls': [], 'n_tokens_mistral': 23791, 'n_tokens_neox': 20327, 'n_words': 10208, 'pdfsha': '07b3407055b4b84e8c100bf3dddc5605df989edd', 'pdfurls': ['https://arxiv.org/pdf/1809.09705v1.pdf'], 'title': ['Bivariate Bannai-Ito polynomials', 'Bivariate Bannai-Ito polynomials'], 'venue': []}

arxiv

Stochastic Langevin Differential Inclusions with Applications to Machine Learning January 4, 2023 Fabio V Difonzo Vyacheslav Kungurtsev Jakub Mareček Stochastic Langevin Differential Inclusions with Applications to Machine Learning January 4, 2023 Stochastic differential equations of Langevin-diffusion form have received significant attention, thanks to their foundational role in both Bayesian sampling algorithms and optimization in machine learning. In the latter, they serve as a conceptual model of the stochastic gradient flow in training over-parametrized models. However, the literature typically assumes smoothness of the potential, whose gradient is the drift term. Nevertheless, there are many problems, for which the potential function is not continuously differentiable, and hence the drift is not Lipschitz continuous everywhere. This is exemplified by robust losses and Rectified Linear Units in regression problems. In this paper, we show some foundational results regarding the flow and asymptotic properties of Langevin-type Stochastic Differential Inclusions under assumptions appropriate to the machine-learning settings. In particular, we show strong existence of the solution, as well as asymptotic minimization of the canonical free-energy functional. Introduction In this paper, we study the following stochastic differential inclusion, dX t ∈ −F (X t )dt + √ 2σ dB t(1) wherein F (x) : R n ⇒ R n is a set-valued map. We are particularly interested in the case where F (x) is the Clarke subdifferential of some continuous tame function f (x). This is motivated by the recent interest in studying Langevintype diffusions in the context of machine learning applications, both as a scheme for sampling in a Bayesian framework (as spurred by the seminal work [Welling and Teh, 2011]) and as a model of the trajectory of stochastic gradient descent, with a view towards understanding the asymptotic properties of training deep neural networks [Hu et al., 2017]. It is typically assumed that F (x) above is Lipschitz, and as such its potential f (x) is continuously differentiable. In many problems of relevance, including empirical risk minimization with a robust (e.g., l1 or Huber) loss and neural networks with ReLU activations, this is not the case, and yet there is at least partial empirical evidence suggesting the long-run behavior of a numerically similar operation is similar in its capacity to generate a stochastic process which minimizes a Free Energy associated with the learning problem. This paper is organized as follows. In Section 2 we study the functional analytical properties of F (x) as it appears in (1) when it represents a noisy estimate of a subgradient element of an empirical loss function that itself satisfies the conditions of a definable potential, especially as it appears in the context of deep learning applications. Note that the research program undertaken relates to a recent conjecture of Bolte and Pauwels [Bolte and Pauwels, 2021, Remark 12] which suggests the strong convergence of iterates in a stochastic subgradient type sequence to stationary points for this class of potentials. Subsequently, in Section 3 we prove that there exists a strong solution to (1), confirming the existence of a trajectory in the general case. In this sense, we extend the work of Szölgyenyi, 2017, Leobacher andSteinicke, 2021] studying diffusions with discontinuous drift to set-valued drift. Next in Section 4 we prove the correspondence of a Fokker-Planck type equation to modeling the probability law associated with this stochastic process, and show that it asymptotically minimizes a free-energy functional corresponding to the loss function of interest, extending the seminal work of [Jordan et al., 1998] which had proven the same result in the case of continuous F (x). We present some numerical results confirming the expected asymptotic behavior of (1) in Section 5 and summarize our findings and their implications in Section 6. Related Work Stochastic differential inclusions are the topic of the monograph [Kisielewicz, 2013], which presents the associated background of stochastic differential equations (SDEs) as well as set-valued analysis and differential inclusions, providing a notion of a weak and strong solution to equations of the form (1), and ones with more general expressions, especially with respect to the noise term. In this work, and others studying stochastic differential inclusions in the literature, such as, e.g., [Kisielewicz, 2009, Kisielewicz, 2020, it is assumed that the set-valued drift term F (x) is Lipschitz continuous. In this paper we are interested in the more general case, where F (x) may not be Lipschitz. Langevin diffusions have had three distinct significant periods of development. To begin with, the original elegant correspondence of SDEs with semi-groups associated with parabolic differential operators was explored in depth by [Stroock and Varadhan, 2007] (first edition published in the 1970s), following the seminal paper of [Itô, 1953]. See also [Kent, 1978]. Later, they served as a canonical continuous Markovian process with the height of activity on ergodicity theory, the long run behavior of stochastic processes, associated with the famous monograph of [Meyn and Tweedie, 2012], first appearing in the 1990's. See, e.g. [Meyn and Tweedie, 1993]. Most recently, the Langevin diffusion has been a model to study the distributional dynamics of noisy training of contemporary machine learning models [Hu et al., 2017]. At this point, we have the works closest to ours. In Deep Neural Networks (DNNs), Rectified Linear Units (ReLUs), which involve a component-wise maximum of zero and a linear expression, are standard functional components in predictors appearing in both regression and classification models. Mean-field analyses seek to explain the uncanny generalization abilities of DNNs by considering the distributional dynamics of idealized networks with infinitely many hidden layer neurons by considering both the stochastic equations corresponding to said dynamics, as well as Fokker-Planck-type PDEs modeling the flow of the distribution of the weights in the network. Analyses along these lines in the contemporary literature include [Luo et al., 2021, Shevchenko et al., 2021. Although this line of works does use limiting arguments involving stochastic and distributional dynamics, they do not directly consider the Langevin differential inclusion and the potential PDE solution for the distribution, as we do here. We finally note that the Langevin diffusion as minimizing free energy even in the case of nonsmooth potentials should not be surprising. In fact, in the seminal book of [Ambrosio et al., 2005], it is shown that a Clarke subgradient of a regular function still exhibits a locally minimizing flow structure for a functional potential on a probability space. Thus, at least locally, the necessary properties appear to have some promise to exist. Tame functions, o-minimal structures and the exceptional set Deep learning raises a number of important questions on the interface of optimization theory and stochastic analysis [Bottou et al., 2018]. In particular, there are still major gaps in our understanding of the applications of plain stochastic gradient descent (SGD) in deep learning, leaving aside the numerous related recent optimization algorithms for deep learning [Schmidt et al., 2021, Davis et al., 2020. A particularly challenging aspect of deep learning is the composition of functions that define the objective landscape. These functions are typically recursively evaluated piecewise non-linear maps. The nonlinearities are due to the sigmoidal function, exponentiation and related operations, which are neither semialgebraic nor semianalytic, and the piece-wise nature of the landscape is due to the common appearance of component-wise max. We consider Euclidean space R n with the canonical Euclidean scalar product ·, · and a locally Lipschitz continuous function f : R n → R. For any x ∈ R n , the Clarke subgradient of f [Clarke, 1990]: ∂ c f (x) = conv v ∈ R n : ∃y k −→ k→∞ x with y k ∈ R, v k = ∇f (y k ) −→ k→∞ v . Following a long history of work [Macintyre and Sontag, 1993, e.g.], we utilize a lesser known function class of definable functions, which are known [van den Dries et al., 1994] to include restricted analytic fields with exponentiation. We refer to Macintyre, McKenna, and van den Dries [Macintyre et al., 1983] and Knight, Pillay, and Steinhorn [Knight et al., 1986] for the original definitions, and to van den Dries-Miller [Van den Dries andMiller, 1996, van den Dries, 1998] and Coste [Coste, 1999] for excellent book-length surveys. In particular, we use the following: Definition 2.1 (Structure, cf. [Pillay and Steinhorn, 1986]). A structure on (R, +, ·) is a collection of sets O = (O p ) p∈N , where each O p is a family of subsets of R p such that for each p ∈ N: 1. O p contains the set {x ∈ R p : g(x) = 0}, where g is a polynomial on R p , i.e., g ∈ R[X 1 , X 2 , . . . , X p ], and the set is often known as the family of real-algebraic subsets of R p ; A subset of R n is called tame if there exists an o-minimal structure such that the subset is definable in the o-minimal structure. Notice that this notion of tame geometry goes back to topologie modérée of [Grothendieck, 1997]. if any Next, we consider two more regularity properties. First, we consider functions that have conservative set-valued fields of [Bolte and Pauwels, 2021], or equivalently, are path differentiable [Bolte and Pauwels, 2021]. This includes convex, concave, Clarke regular, and Whitney stratifiable functions. Definition 2.3 (Conservative set-valued fields of [Bolte and Pauwels, 2021]). Let D : R n ⇒ R n be a set-valued map. D is a conservative (set-valued) field whenever it has closed graph, nonempty compact values and for any absolutely continuous loop γ : [0, 1] → R n , that is γ(0) = γ(1), we have 1 0 max v∈D(γ(t)) γ(t), v dt = 0 in the Lebesgue sense. Definition 2.4 (Potential function of [Bolte and Pauwels, 2021]). Let D be a conservative (set-valued) field. For any γ absolutely continuous with γ(0) = 0 and γ(1) = x, any function f defined as f (x) = f (0) + 1 0 max v∈D(γ(t)) γ(t), v dt (2) = f (0) + 1 0 min v∈D(γ(t)) γ(t), v dt (3) = f (0) + 1 0 γ(t), D(γ(t)) dt (4) is called a potential function for D. We shall also say that D admits f as a potential or that D is a conservative field for f . Let us note that a particular structure D admits f that is unique up to a constant. The second notion of regularity, which we consider, is the notion of piecewise Lipschitzianity on R n of [Leobacher and Szölgyenyi, 2017]. The associated exception(al) set [Leobacher and Steinicke, 2021] is the subset of that function's domain where the function is not Lipschitz. We recall the definition here for convenience, and we state it for set-valued maps. Definition 2.5. A function f : R n → R n is piecewise Lipschitz continuous if there exists a hypersurface Θ, which we call the exceptional set for f , such that Θ has finitely many connected components θ i , i = 1, 2, . . ., such that f R n \Θ is intrinsic Lipschitz (cf. [Leobacher and Szölgyenyi, 2017, Definition 3.2]). A set-valued function F : R n ⇒ R n is piecewise Lipschitz continuous if there exists an exceptional set Θ, defined analogously as for single-valued functions, such that F is Lipschitz on R n \ Θ with respect to Haudsdorff H-metric (cf. [Baier and Farkhi, 2013]). Let us recall that, given a set-valued map with compact convex values F : R n ⇒ R n , we say that F is H-Lipschitz, or just Lipschitz, if and only if there exists a constant l > 0 such that for any x, y ∈ R n we have ρ H (F (x), F (y)) ≤ l x − y , where, for arbitrary compact sets A, B ⊆ R n , ρ H (A, B) := max{max a∈A dist(a, B), max b∈B dist(b, A)}, with dist(a, B) := min b∈B a − b 2 . Definition 2.6 ([Van den Miller, 1996, Bolte et al., 2007]). A C r stratification of a closed (sub)manifold M of R n is a locally finite partition (M i ) i∈I of M into C r submanifolds (called strata) having the property that for i = j, cl(M i )∩M j = ∅ implies that M j is entirely contained in cl(M i )\M i (called the frontier of M i ) and dim(M j ) < dim(M i ). Moreover, a C r stratification (M i ) i∈I of M has the Whitney-(a) property if, for each x ∈ M i ∩ M j (with i = j) and for each sequence {x k } ⊆ M i we have lim k→∞ x k = x, lim k→∞ T x k M i = T , =⇒ T x M j ⊆ T , where T x M j (respectively, T x k M i ) denotes the tangent space of the manifold M j at x (respectively, of M i at x k ) and the convergence in the second limit is in the sense of the standard topology of the Grassmannian bundle of the [Mather, 2012]). If, moreover, a Whitney stratification (M i ) i∈I satisfies for all i ∈ I and x ∈ M i the transversality condition dim M i −planes in T x M , x ∈ M i (seee n+1 / ∈ T x M i ,(5) where e n+1 = 0 · · · 0 1 ∈ R n+1 , then it is called a nonvertical Whitney stratification. A function f : R n → R is said to be stratifiable, in any of its connotations, if the graph of f , denoted by Graph(f ), admits a corresponding C r stratification. Remark 2.1. Let us note that, from Definition 2.6, if (M i ) i∈I is a stratification of a stratifiable submanifold M , then it follows that M = i∈I M i . Therefore, by the Hausdorff maximality principle, there must exist i ∈ I such that dim M i = max i∈I dim M i and, thus, M = cl M i since all the other subspaces M i , i = i have zero Lebesgue measure. The same argument holds, a fortiori, for every finite subset J ⊆ I. In this paper, we establish the existence guarantees of strong solutions to the equation (1) and study the PDE describing the flow of the probability mass of X(t). To set up the subsequent exposition, we must first establish the groundwork of linking Whitney stratifiable potential functions, which describe the loss landscape of neural networks and other statistical models on the one hand, to set valued piecewise Lipschitz continuous maps, which describe the distributional flow of stochastic subgradient descent training, modeled by (1), on the other. However, in order to do so, we must make an additional assumption that limits the local oscillation and derivative growth of the potential; specifically, we assume that f , the potential of F , has a bounded variation. It can be seen that the standard activation and loss functions that appear in neural network training satisfy this condition. Theorem 2.1. Let F : R n ⇒ R n be a definable conservative field that admits a tame potential f : R n → R with bounded variation. Let F be Lipschitz on R n \B δ (0) with constant L, for some L, δ > 0. Then F is piecewise Lipschitz continuous. Proof. As f is tame, by definition, it is also definable. Since F is conservative, from [Bolte and Pauwels, 2021], f is locally Lipschitz and, since it is tame, [Bolte et al., 2009, Theorem 1] implies that f is semismooth. Therefore, by [Bolte et al., 2007, Corollary 9], letting r ≥ 2 be an arbitrarily fixed integer, f admits a nonvertical C r Whitney stratification (M i ) i∈I . With abuse of notation, let (M i ) i∈I denote the stratification relative to the domain of f . Therefore, due to local finiteness, there must exist a maximal finite subset of indices J ⊆ I such that [Bolte et al., 2007]. Let us note that the Clarke [Bolte et al., 2007, Proposition 4]. Now, on the account of Remark 2.1, for some j B δ (0) ∩ M j = ∅, j ∈ J. Since f is semismooth we deduce that its directional derivative f (·; v) is C r−1 for all v ∈ R n and, since f ∈ BV(R n ), f (·; v) is bounded on B δ (0) and for all j for all x ∈ B δ (0) ∩ M j and v ∈ T x (B δ (0) ∩ M j ) it holds that f (x, v) is Lipschitz continuous with respect to x, restricted to B δ (0) ∩ M j ; hence, the Riemannian gradient ∇ R f (x) is Lipschitz on B δ (0) ∩ M j , x ∈ M j , j ∈ J, since ∇ R f (x) is the restriction of the directional derivative to tangent directions onto M jsubgradient of f at x, denoted by ∂ • f (x), is such that Proj TxM j ∂ • f (x) ⊆ {∇ R f (x)}, where M j is the stratum such that x ∈ M j , j ∈ J. Seex ∈ J, we have dim M j x = dim B δ (0) = n. By compactness, there exists a finite covering of B δ (0) of maximal dimension, denoted by (M j ) j∈J for some J ⊆ J, such that B δ (0) ⊆ M , where M := j∈J M j . Therefore ∇ R f (x) = ∇f (x) on B δ (0). Then, as a consequence of [Bolte and Pauwels, 2021, Theorem 1], there exists a zero-measure set S ⊆ R n such that F (x) = {∇f (x)} for all x ∈ B δ (0) \ S. Now, for x ∈ B δ (0) \ S,∂ • f (x) ⊆ convF (x) = conv{∇f (x)} = {∇f (x)}, so that Proj TxM j ∂ • f (x) ⊆ Proj TxM j {∇f (x)} = {∇f (x)}. It then follows that ∇f (x), and so F , is Lipschitz on B δ (0)\S. Since B δ (0) is compact, there exists a finite family {θ k } k∈K such that B δ (0) ∩ S ⊆ ∪ k∈K θ k . Thus F is Lipschitz on B δ (0) \ ∪ k∈K θ k , and the claim is proved. Example 2.1. In case F : R n ⇒ R n is a definable conservative field such that none of its tame potentials f : R n → R has bounded variation, then Theorem 2.1 does not hold. In fact, let us consider F (x) := x sin 1 x , x ∈ R \ {0}, [−1, 1], x = 0. A simple computation provides that F is a definable conservative field. Moreover, its unique potential, up to constants, is f (x) := sin 1 x − 1 x cos 1 x , that is not of bounded variation. In this case, Theorem 2.1 does not hold, as it is clear that F is not piecewise Lipschitz. Existence and Uniqueness of Solution to the SDI In this section we will prove that (1) admits a strong solution. More precisely, it will be proven that there exists a suitable selection of F (X t ) such that the corresponding SDE has a strong solution: this will in turn imply that the original stochastic differential inclusion has a solution as well. Our result will rely on an existence and uniqueness pertaining to SDEs with discontinuous drift in [Leobacher and Szölgyenyi, 2017]. Piecewise Lipschitz selections of upper semicontinuous set-valued maps In our setting, assuming that F is the Clarke subdifferential of some continuous tame function guarantees that F is an upper semi-continuous set-valued map. We further assume that F is bounded with compact convex values. Our aim is to prove that, under these assumptions, F has a piecewise Lipschitz selection. We are going to need the following results: Theorem 3.1 (Theorem 9.4.3 in [Aubin and Frankowska, 1990]). Consider a Lipschitz set-valued map F from a metric space to nonempty closed convex subsets of R n . Then F has a Lipschitz selection, called Steiner selection. Theorem 3.2 (Kirszbraun's Theorem, cf. [Federer, 1996]). If S ⊆ R n and f : S → R n is Lipschitz, then f has a Lipschitz extension g : R n → R n with the same Lipschitz constant. In the next Theorem, which is the main result of this section, we prove that, under some mild assumptions, the set-valued map F in (1) has a piecewise Lipschitz selection for any suitable compact covering of R n . Theorem 3.3. Let F : R n ⇒ R n be an upper semi-continuous set-valued map with closed convex values, and piecewise Lipschitz continuous, with exceptional set Θ. Let us assume that there exists b > 0 such that F (x) ⊆ bB n for all x ∈ R n . Then F has a piecewise Lipschitz selection with exceptional set Θ, arbitrarily smooth on the interior of each connected component. Proof. Let {R i } r i=1 be the finite family of closed subsets of R n such that r i=1 R i = R n \ Θ. For each i = 1, . . . , r, Theorem 3.1 on F R i implies that there exists a finite sequence of equi-Lipschitz, and thus continuous, selection functions {f i } where each f i is defined on R i , that is f i : R i → R n . Let us note that, since F (x) ⊆ bB n for all x ∈ R n then {f i } r i=1 is uniformly bounded. We can now extend each f i on the whole R n , to some function, still denoted by f i , which is Lipschitz with the same constant as the original function's on the account of Kirszbraun's Theorem 3.2; moreover, the family {f i } can be assumed to retain uniform boundedness. Let now ε > 0 be given and let i = 1, . . . , r be fixed. Let us consider a partition of unity {ϕ ε } as in [Shubin, 1990, Lemma 1.2]. Now, following a classical construction (e.g., see [Azagra et al., 2007]), we let g ε i (x) := R n f i (x)ϕ ε (y − x) dy. It then follows that g ε i ∈ C ∞ (R n ) is a Lipschitz function, with the same Lipschitz constant as f i 's, and is such that g ε i ∈ F (x) by straightforward computations. Let us stress that the Lipschitz constant of each g ε i is independent of ε, so that {g ε i } ε>0 is equi-Lipschitz on R i . Let now x ∈ R n and i x ∈ N be unique index such that x ∈ R ix . We then define f (x) := g ε ix (x), x ∈ R n . It then follows that f : R n → R n is countably piecewise Lipschitz on R n according to Definition 2.5 and its exceptional set is Θ. Obviously f (x) ∈ F (x), and this proves the claim. Remark 3.1. Theorem 3.3 still holds if we replace R n with any A ⊆ R n : in fact, we can always extend F to an upper semi-continuous set-valued map defined on the whole R n by virtue of [Smirnov, 2002, Theorem 2.6]. Finally, we have the following. Corollary 3.1. Let F : R n ⇒ R n be an upper semi-continuous set-valued map with closed convex values, and piecewise Lipschitz continuous with a C 3 exceptional set Θ. Assume that there exists b > 0 such that F (x) ⊆ bB n for all x ∈ R n . Then the SDI (1) admits a strong solution. In particular, for every piecewise Lipschitz selection of F , there exists a unique strong solution to the SDI (1). Proof. From Theorem 3.3 there exists a drift µ : R n → R n , piecewise Lipschitz selection of the set-valued map F , which satisfies Assumptions 3.4 − 3.6 from [Leobacher and Szölgyenyi, 2017]. Therefore, on the account of [Leobacher and Szölgyenyi, 2017, Theorem 3.21] , we obtain that the SDE dX t = −µ(X t )dt + √ 2σ dB t(6) has a unique global strong solution, which in turn represents a strong solution to the stochastic differential inclusion (1), and this proves the claim. Remark 3.2. Let us stress that the Lipschitz selection of the set-valued map F may not be Lipschitz on Θ s \Θ. This does not affect the result of Corollary 3.1 since, in order for it to hold, it suffices to provide a strong solution to (6). Fokker-Planck Equation and Free Energy Minimization 4.1 Fokker-Planck Equation The Fokker-Planck (FP) equation describes the evolution of the probability density associated with the random process modeled by the diffusion flow. Classical results deriving the Fokker-Planck equation from SDEs can be reviewed in [Eklund, 1971, Itô, 1953, Kent, 1978, Stroock and Varadhan, 2007. In particular, for a drift diffusion of the form dX t = −∇f (X t ) dt + √ 2σdB t it holds that, from any initial distribution ρ(0) = ρ 0 , the density ρ(x, t) of X t is given by ∂ρ ∂t = −∇ · (∇f (x)ρ) + 1 2 ∆ (σρ)(7) and has a limiting stationary distribution defined by the Gibbs form exp {−f (x)/σ} /Z. In [Jordan et al., 1998], it was shown that the Fokker-Planck evolution corresponds to the gradient flow of a variational problem of minimizing a freeenergy functional composed of the potential f (x) and an entropy regularization. These classical results in order to even concern well-defined objects, require the smoothness of f . The FP equation can be derived from applying integration by parts after Itô's Lemma on the diffusion process. Follow-ing [Gardiner, 2009, 4.3.5], for arbitrary g, we have, g(x(t)) dt = d dt g(X(t)) = ∇f (x)∂ x f + 1 2 σ∂ 2 xx g = dx ∇f (x)∂ x g + 1 2 σ∂ 2 xx g ρ(x, t|x 0 , t 0 ) = dxg(x)∂ρ(x, t|x 0 , t 0 ) Replacing the expression ∇f (x) with an arbitrary coefficient function of form a(x), we can see that in the case that a(x) is a selection of the Clarke subdifferential it may not be a continuous function of x. In this case, even the weak distributional sense derivative of a(x) does not exist and integration by parts cannot be applied. Generic Solution Existence Results Nevertheless, a solution ρ(x, t) can be shown to exist for the system as stated in weak form. To this end, we follow [Bogachev et al., 2001]. Let Ω T ⊆ R n × [0, T ] be open, and L A,b ϕ = n i=1 a i ∂ϕ ∂x i + n i,j=1 σ ij ∂ϕ ∂x i ∂x j , ϕ ∈ C ∞ 0 (Ω T ). Moreover, let n be such that 1 n + 1 n = 1. The following holds: Theorem 4.1 (Corollary 3.2, [Bogachev et al., 2001]). Let µ be a locally finite Borel measure on Ω such that a i , σ ij ∈ L 1 loc (Ω T , µ) and, Ω T   ∂φ ∂t + n i=1 a i ∂φ ∂x + n i,j=1 σ ij ∂ϕ ∂x i ∂x j   dµ = 0 for all nonnegative ϕ ∈ C ∞ 0 (Ω T ) . Furthermore let σ ij be uniformly bounded, nondegenerate and Hölder continuous, then µ = ρ(x, t) dx dt with ρ ∈ L p loc (Ω T ) for every p ∈ [1, (n + 2) ). Remark 4.1. As observed in [Bogachev et al., 2001], one cannot expect that the density of µ is continuous even for infinitely differentiable σ ij under these conditions. However, we note that in [Portenko, 1990] a continuous solution is shown under the assumption that a i ∈ L p (Ω T , µ), i.e. it is globally integrable. Again, however, this is very restrictive in the case of studying the evolution of diffusion operators on tame nonsmooth potentials of interest. Existence of an invariant measure µ for the probability flow, i.e., a solution for the purely elliptic part, is guaranteed by [Bogachev and Röckner, 2001 The strong regularity conditions, and the bounded open set Ω T , however, clearly limit both the applicability and informativeness of these results for our SDI. Fokker-Planck With Boundary Conditions Note, however, that we have an expectation as to what the stationary distribution for (1), namely, of Gibbs form proportional to exp{−f (x)/σ}. This measure is even absolutely continuous, and thus has higher regularity than Theorem 4.1 suggests. To this end, consider [Gardiner, 2009, Chapter 5.1.1] which considers boundary conditions at a discontinuity for the FP equation. We can consider that the space is partitioned into connected components R i , and in each region, the continuous SDE and associated FP (7) holds. However, for boundaries between regions S ij ⊆ Θ we have, n · J i (x, t) S i ij = n · J j (x, t) S j ij , ρ(x, t) S i ij = ρ(x, t) S j ij (8) where the probability current J is defined as, J i (x, t) = F i (x)ρ(x, t) + 1 2 σ ∂ρ(x, t) ∂x i as defined on region R i Note that in one dimension this is simple: we have, at a point of nondifferentiability x c , ρ(x, t) x + c = ρ(x, t) x − c , F (x)ρ(x, t) + 1 2 σ ∂ρ(x, t) ∂x i x + c = − F (x)ρ(x, t) + 1 2 σ ∂ρ(x, t) ∂x i x − c where we abuse notation to indicate the partial directional derivative of ρ from either side of x c . We can consider writing a stationary solution π(x) to this system as we have a suspected ansatz, similarly as done in [Gardiner, 2009, Section 5.3]. Generically stationary implies ∇ · J(x) = 0, or that J is divergence-free with respect to x. We have, n i=1 ∂ ∂x i F i (x)ρ(x, t) + 1 2 σ ∂ρ(x, t) ∂x i = 0 (9) Indeed, let π(x) = exp {−f (x)/σ} /Z. On the domain R n \ Θ where ∇f (x) is well-defined, this can be seen immediately to solve (9). Since Θ is of measure zero with respect to the ambient space, we can define, Z = R n π(x) dx = ∪ i R i π(x) dx Of course, a constructive ansatz for the stationary solution, the elliptic form, neither shows its uniqueness, nor the existence, uniqueness and regularity of the parabolic evolution of the probability mass flow ρ(x, t). To the best of our knowledge, no such result exists for the network of parabolic firstorder systems under consideration, and at the same time, we shall see that the continuity of ρ(x, t) is important in the next section for the variational conception of the FP equation as a minimizing flow for the free energy. To this end, we can consider two lines of work as a foundation to formulate the requisite results. In particular, [Nittka, 2011] shows regularity conditions of solutions to second-order parabolic equations defined on a Lipschitz domain. As seen in Section 2, for the problems of interest, the exceptional set can be parameterized in a smooth way, which implies that for compact sets, the boundary is Lipschitz. We must, however, take care to translate the results appropriately to the potentially unbounded domains a potential R i could correspond to. The closest to our line of work is considering graph-structured networks of PDEs, such as modeling Kirchhoff's laws. A prominent and representative work along these lines is [von Below, 1988]. In this setting, there is a network of one-dimensional paths embedded in some ambient space in R n , with the paths connected at a series of vertices, with boundary conditions connecting a collection of linear parabolic PDEs governing the flow of a quantity across the network. Thus, the spirit of a network structure of PDEs with connecting boundary connections is analogous to our problem, however, with the caveat that they are one-dimensional embedded domains, even if embedded in a larger space. Existence and smoothness regularity conditions are shown. We consider an approach using domain decomposition methods for the solutions of PDEs based on optimal control theory [Gunzburger et al., 1999]. See also, e.g., [Dolean et al., 2015]. We consider reformulating the boundary conditions as a control to establish a variational formulation. First let δ > 0 be a regularization parameter and consider the optimization problem min {ρ i ∈H 1 (R i )}{g ij ∈L 2 (S ij )} J δ ({ρ i }, {g ij }) subject to ∂ρ i ∂t + F (x) · ∇ρ i + σ∇ · ∇ρ i = 0, on R i , ∀i, n · J i = g ij , on S ij , n · J j = −g ij , on S ij ,(10) where J δ ({ρ i }, {g ij }) := ij S ij (ρ i − ρ j ) 2 dS ij + δ 2 S ij g 2 ij dS ij . Let us also note the weak form of the PDE constraints, R i [∂ t ρ i v + σ∇ρ i · ∇v + F · ∇ρv] dx = j S ij g ij dS ij − j S ji g ji dS ji , ∀v ∈ H 1 (R i ).(11) We have the following, akin to [Gunzburger et al., 1999, Theorem 2.1]. Let U = {ρ i ∈ H 1 (R i )}{g ij ∈ L 2 (S ij )} satisfying (11), J δ ({ρ i }, {g ij }) < ∞ Theorem 4.2. There exists a unique solution ({ρ i ∈ H 1 (R i )}{g ij ∈ L 2 (S ij )}) to (10) in U. Proof. Let {ρ (n) i , g (n) ij } be a minimizing sequence in U, i.e., lim n→∞ J δ {ρ (n) i , g (n) ij } = inf {ρ i ,g ij }∈U J δ ({ρ i , g ij }) By the definition of U we have that g (n) ij are uniformly bounded in L 2 (S ij ). Now, we argue that by [Ladyzhenskaya et al., 1967, Theorem IV.5.3] the PDEs given by (11) have unique solutions ρ i continuous with respect to the inputs, i.e., ρ i H l+2,l/2+1 ≤ C j g ij H l+1,(l+1)/2 (S ij ) + g ji H l+1,(l+1)/2 (S ji )(12) for some non-integral l, when the norms on the right-hand side are well defined. When we consider the conditions for the application of this Theorem, we see that the only unsatisfied assumption is the integrability of the coefficients with respect to an appropriate dual Sobolev space. However, we can see from the proof of the result, in particular [Ladyzhenskaya et al., 1967, Equation (IV.7.1)], that this is used only to show that the operators, ∂ t + σ∇ 2 + F · ∇ and [σ∇ + F ·] are bounded. However, with σ constant and F ∈ L ∞ (R i ), this also clearly holds and these are operators from H l+2,l/2+1 to H l,l/2+1 on R i and H l+1,l/2+1 on S ij respectively. For any l, we can apply the Poincaré inequality on the left and Sobolev embedding on the right of (12) to obtain, ρ i H 1 (R i ) ≤ C j g ij L 2 (S ij ) + g ji L 2 (S ji )(13) Thus, by the uniform boundedness of g (n) ij from the definition of U, we get the boundedness of ρ (n) i and the existence of a convergent subsequence {{ρ (n k ) i }, {g (n k ) ij }} convergent to ({ρ i },ĝ ij } with everyρ i in H 1 (R i ) andĝ ij in L 2 (S ij ). Passing to the limit we see that they also satisfy (11) and by the lower semicontinuity of J δ we have that inf {ρ i },{g ij } J δ (ρ i , g ij ) = lim inf k J δ (ρ (n k ) i , g (n k ) ij ) ≥ J δ ({ρ i , g ij }) and ({ρ i },ĝ ij } is optimal. Since J δ is convex and U is linear, it is uniquely optimal. Now consider the following weak system of PDEs, now for a unique ρ, R i [∂ t ρv + σ∇ρ · ∇v + F · ∇ρv] dx = j S ij (n · (F ρ + 1 2 σ∇ρ))v dS ij − j S ji (n · (F ρ + 1 2 σ∇ρ))v dS ji , ∀v ∈ H 1 (R i )(14) Then, we prove the following. Theorem 4.3. For each δ > 0, denote ({ρ δ i }{g δ ij }) the solutions to (10) as given by Theorem 4.2. We have that for any convergent subsequence as δ → 0, it holds that there exists ρ such that with ρ i = ρ| R i and g ij = n · (F ρ + 1 2 σ∇ρ)| S ij , i ρ δ i − ρ i H 1 (R i ) + ij g δ ij − g ij L 2 (S ij ) → 0 and ρ solves (14). Proof. From the definition of ({ρ δ i }{g δ ij }) we have that J δ ({ρ δ i }{g δ ij }) ≤ J δ ({ρ i }{g ij }), which is, ij S ij (ρ δ i − ρ δ j ) 2 dS ij + δ 2 S ij (g δ ij ) 2 dS ij ≤ δ 2 S ij g 2 ij dS ij , implying, by the uniform boundedness of g ij in L 2 (S ij ) that ρ δ i −ρ δ j L 2 (S ij ) → 0 as δ → 0. Furthermore, from (13) we get that ρ δ i H 1 (R i ) are uniformly bounded. Thus, with δ → 0 there is a subsequence converging to {ρ * ij , g * ij } over {H 1 (R i ), L 2 (S ij )} and passing to the limit implies they satisfy (11). Furthermore ρ δ i − ρ δ j H 1 (S ij ) → 0 implies that ρ * i | S ij = ρ j | S ij . Defining ρ ∈ H 1 (R n ) by ρ| R i ∪(∪ j {S ij }) = ρ i we obtain the unique solution to (14). We have now proven the existence of ρ ∈ H 1 (R n ) satisfying the weak form of the PDE (7) over ∪ i R i , corresponding to where the coefficients are smooth almost everywhere, and with the boundary conditions (8). Variational Flow for the Free Energy In [Jordan et al., 1998], the FP equation (7) is shown to be the gradient flow of the functional, F(ρ) = E(ρ) + S(ρ; σ) = f (x)ρ dx + ρ log ρ dx(15) when F (x) = ∇f (x) and the stationary distribution is given by its minimizer. To this effect, one can consider a scheme, ρ (k) := arg min ρ 1 2 W (ρ (k−1) , ρ) 2 + hF(ρ)(16) for some small h, where W refers to the Wasserstein distance. Let M be defined as M := ρ : R n → [0, ∞) measurable, and ρ(x) dx = 1, |x| 2 ρ(x) dx < ∞ We have the following Proposition, whose proof is unchanged in our setting, Proposition 4.1. [Jordan et al., 1998, Proposition 4.1] Given ρ ∈ M , there exists a unique solution to (16). Now we extend the classical main result, Theorem 5.1 in [Jordan et al., 1998], to our setting, which requires a few modifications to account for the nonsmooth potential in the free energy. ρ h (t) = ρ (k) h for t ∈ [kh, (k + 1)h) and k ∈ N ∪ {0} Then as h → 0, ρ h (t) → ρ(t) weakly in L 1 (R n ) for all t ∈ (0, ∞) where ρ ∈ H 1 ((0, ∞) × R n ) solves (9) with initial condition ρ(t) → ρ 0 strongly in L 1 (R n ) for t → 0. Proof. Let ξ ∈ C ∞ 0 (R n , R n ) be a smooth vector field with bounded support, and its flux Φ τ as ∂ τ Φ τ = ξ • Φ τ for all τ ∈ R and Φ 0 = id. The measure ρ τ (y) dy is the push forward of ρ (k) (y) dy under Φ τ . This means that R n ρ τ (y)ζ(y) dy = R n ρ (k) (y)ζ(Φ τ (y)) dy for all ζ ∈ C 0 0 (R n ), which implies that det ∇Φ τ ρ t • Φ τ = ρ (k) . By the properties of (16) we have that 1 τ 1 2 W (ρ (k−1) , ρ τ ) 2 + hF(ρ τ ) − 1 2 W (ρ (k−1) , ρ (k) ) 2 + hF(ρ (k) ) ≥ 0. (17) Now, consider ζ = f and so R n ρ τ (y)f (y) dy = R n ρ (k) (y)f (Φ τ (y)) dy and also 1 τ E(ρ τ ) − E(ρ (k) ) = R n 1 τ (f (Φ τ (y)) − f (y)) ρ (k) (y) dy Recalling the consideration of a conservative vector field, cf. Definition 2.3 above. In the original, the term ∇f (y) · ξ(y) appears, instead, we have a vector χ(y) that represents directional change in f . Specifically, we can write that d dτ [E(ρ τ )] τ =0 = R n χ(y) · ξ(y)ρ (k) (y) dy. We can continue similarly as in the proof of [Jordan et al., 1998, Theorem 5.1] to obtain, d dτ [S(ρ τ )] τ =0 = − R n ρ (k) divξ dy and subsequently the following a priori estimates, wherein for any T < ∞ there exists C such that for all N ∈ N and h ∈ [0, T ] with N h ≤ T there holds, M (ρ (N ) h ) ≤ C, R n max{ρ (N ) h log ρ (N ) h , 0} dx ≤ C, E(ρ (N ) h ) ≤ C, N k=1 W (ρ (k−1) h , ρ (k) h ) 2 ≤ Ch(18) which implies the existence of a convergent subsequence, ρ h ρ weakly in L 1 ((0, T ) × R n ) for all T < ∞ with ρ(t) ∈ M and E(ρ) ∈ L ∞ ((0, T )), and ρ satisfying − (0,∞)×R n ρ(∂ t ζ −χ·∇ζ + ζ) dx dt = R n ρ 0 ζ(0) dx for all ζ ∈ C ∞ 0 (R×R n )(19) We can proceed similarly as in the proof of Theorem 5.1 in [Jordan et al., 1998] using appropriate test functions to conclude that ρ ∈ L p loc ((0, ∞), R n ). Now, clearly this ρ solves (14), which has a unique solution in H 1 (R n , (0, T )) by Theorem 4.3. The final statements of the Theorem follow as in Theorem 5.1 in [Jordan et al., 1998]. Numerical Illustration One-Dimensional Example For the first illustration, we took a one-dimensional function f (x) :=        −x − 1 x < −1 x + 1 −1 ≤ x < 0 1 − x 0 ≤ x < 1 x − 1 x ≥ 1 , F (x) :=                    −1 x < −1 [−1, 1] x = −1 1 −1 < x < 0 [−1, 1] x = 0 −1 0 < x < 1 [−1, 1] x = 1 1 x > 1(20) The probability density function associated with minimizing the free energy is shown in Figure 1 and the result of one hundred thousand samples generated by the Metropolis Algorithm in Figure 2. As an illustration, we ran unadjusted Langevin dynamics with the Euler-Maruyama discretization, i.e., generated samples with the following iteration, x k+1 = x k − g k + √ 2 B k , g ∈ F (x k ), B k ∼ N (0, 1).(21) We generated ten million samples with ∈ {0.01, 0.001, 0.0001}. We plot the histograms of the final sample count in Figure 3 and plot the Wasserstein distance (computed with scipy) in Figure 4. We can observe that indeed the iteration does recreate the posterior, i.e., it appears to be ergodic, although the quantity of samples required is fairly large, indeed it seems to converge in probability distance after about 3 million samples, suggesting geometric ergodicity is unlikely. Bayesian ReLU Neural Network The use of gradient-based samplers had been introduced due to their improved mixing rate with respect to dimensionality dependence. Although we do not derive quantitative mixing rates in this work, it would be naturally suspected that such behavior could carry over to the nonsmooth case. For this purpose, we perform Metropolis-Hastings, (subgradient) unadjusted Langevin, and Metropolis-corrected Langevin on a ReLU network. To avoid complications associated with inexactness, we used moderately sized datasets and performed backpropagation on the entire data sample, rather than a stochastic variant. Specifically, we consider the E2006 and YearPredictionMSD datasets from the UCI LIBSVM datset repository [Chang and Lin, 2011]. Both datasets have high parameter dimension, 150000 and 90, respectively, and with 16k or 460k as the number of training samples. We used a ReLU network with three hidden layers, each with 10 neurons. In this case, with the high dimension, rather than attempting to visualize the posterior, we plot the averaged (across 20 runs) of the loss on the test set. See Figure 5 for the results of the Langevin and Metropolis-adjusted Langevin approach on the test accuracy for dataset E2006. In this case, Metropolis resulted in a completely static mean high variance test loss across the samples, being entirely uninformative. Next, Figure 6 shows the test loss for a ReLU network using unadjusted Langevin. In this case, note the noisy initial plateau is followed by decrease. For both Metropolis as well as the Metropolis-corrected variant of Langevin there is no improvement due to repeated step rejection of the initial sample or minimal change in the error. Figure 6: Test Loss on the sampled parameters generated by the unadjusted Langevin method on a ReLU Neural Nework on a regression task for the dataset YearPredictionMSD. The discretization rate is 1e-05. Discussion and Implications The standard potential gradient diffusion process has featured prominently in the theoretical analysis meant to give insight as to the approximation and generalization performance associated with the long-term behavior of SGD as applied to neural networks, for example, [Hu et al., 2017]. It has also featured prominently in algorithms for sampling high-dimensional data sets, e.g. [Bussi and Parrinello, 2007]. It is standard for these studies to require that the drift term is Lipschitz and, as such, that the potential is continuously differentiable. In many contemporary applications, for example in (Bayesian, in the case of sampling) Neural Networks with ReLU or convolutional layers, this is not the case, and the presence of points wherein the loss function is not continuously differentiable is endemic. Therefore, the primary results of this paper, the existence of a solution to the stochastic differential inclusion drift as well as the existence of a Fokker-Planck equation characterizing the evolution of the probability distribution which asymptotically converges to a Gibbs distribution of the free energy, provide some basic insights into these processes without unrealistic assumptions. Specifically, even with these nonsmooth elements as typically arising in Whitney-stratifiable compositions of model and loss criteria, the overall understanding of the asymptotic macro behavior of the algorithmic processes remains as expected. Additional insights gathered from studying an approximating SDE are not as straightforward to extend to the differential-inclusion setting. The issues of wide and shallow basins around minima seem minor, when one considers that a Hessian of a loss function may not exist at certain points. Thus, considerations of mixing rate for sampling and qualitative properties of limiting distributions are interesting topics for further study in specific cases of specific stochastic differential inclusions. Theorem 4 . 4 . 44Let ρ 0 ∈ M with F (ρ 0 ) < ∞ and ρ (k)h the solution for (16) and the interpolation ρ h : (0, ∞) × R n → [0, ∞) by, Figure 1 : 1Stationary Distribution e −f (x) Associated with (20). Figure 2 : 2Histogram Figure 3 : 3Histogram for (21) at ∈ {0.01, 0.001, 0.0001} in this order. Figure 4 : 4Wasserstein Distance of the Langevin-type iteration (21) and Metropolis-generated samples for ∈ {0.01, 0.001, 0.0001}, in this order. Figure 5 : 5Test Loss on the sampled parameters generated by the unadjusted as well as the Metropolis-adjusted Langevin method on a ReLU Neural Nework on a regression task for the dataset E2006. The discretization rate is 1e-05. A belongs to O p , then A × R and R × A belong to O p+1 ;3. if any A belongs to O p+1 , then π(A) ∈ O p , where π : R p+1 → R p is the coordinate or "canonical" projection onto R p , i.e., the projection to the first p coordinates; 4. O p is stable by complementation, finite union, finite intersection and contains R p , which defines a Boolean algebra of subsets of R p . Definition 2.2 (Definable functions, cf. [Van den Dries and Miller, 1996]). A structure on (R, +, ·) is called o-minimal when the elements of O 1 are exactly the finite unions of (possibly infinite) intervals and points. The sets A ⊆ R p belonging to an o-minimal structure O p , for some p ∈ N, are called definable in the o-minimal structure O. One often shortens this to definable if the structure O is clear from the context. A set-valued mapping is said to be definable in O, whenever its graph is definable in O. from [Bolte and Pauwels, 2021, Corollary 1], we have that , Theorem 1.6] and [Albeverio et al., 1999, Theorem 1.2]. Acknowledgements We'd like to thank Thomas Surowiec for helpful discussions regarding PDE theory as related to the arguments in this paper. FVD has been supported by REFIN Project, grant number 812E4967 funded by Regione Puglia; he is also part of the INdAM-GNCS research group. 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{'abstract': 'Stochastic differential equations of Langevin-diffusion form have received significant attention, thanks to their foundational role in both Bayesian sampling algorithms and optimization in machine learning. In the latter, they serve as a conceptual model of the stochastic gradient flow in training over-parametrized models. However, the literature typically assumes smoothness of the potential, whose gradient is the drift term. Nevertheless, there are many problems, for which the potential function is not continuously differentiable, and hence the drift is not Lipschitz continuous everywhere. This is exemplified by robust losses and Rectified Linear Units in regression problems. In this paper, we show some foundational results regarding the flow and asymptotic properties of Langevin-type Stochastic Differential Inclusions under assumptions appropriate to the machine-learning settings. In particular, we show strong existence of the solution, as well as asymptotic minimization of the canonical free-energy functional.', 'arxivid': '2206.11533', 'author': ['Fabio V Difonzo ', 'Vyacheslav Kungurtsev ', 'Jakub Mareček '], 'authoraffiliation': [], 'corpusid': 249953734, 'doi': '10.48550/arxiv.2206.11533', 'github_urls': [], 'n_tokens_mistral': 19916, 'n_tokens_neox': 17084, 'n_words': 10169, 'pdfsha': '161ac7bae95c31661e5137fdcc78e472b86ccda5', 'pdfurls': ['https://export.arxiv.org/pdf/2206.11533v2.pdf'], 'title': ['Stochastic Langevin Differential Inclusions with Applications to Machine Learning', 'Stochastic Langevin Differential Inclusions with Applications to Machine Learning'], 'venue': []}

arxiv

Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole 20 Oct 2022 Yiqian Chen *[email protected]†[email protected]‡[email protected]§[email protected] Center for Theoretical Physics College of Physics Sichuan University 610064ChengduChina Peng Wang Center for Theoretical Physics College of Physics Sichuan University 610064ChengduChina Houwen Wu Center for Theoretical Physics College of Physics Sichuan University 610064ChengduChina Department of Applied Mathematics and Theoretical Physics University of Cambridge Wilberforce RoadCB3 0WACambridgeUK Haitang Yang Center for Theoretical Physics College of Physics Sichuan University 610064ChengduChina Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole 20 Oct 20222 CONTENTS It has been recently reported that, at late times, the total luminosity of a star freely falling in black holes decays exponentially with time, and one or two series of flashes with decreasing intensity are seen by a specific observer, depending on the number of photon spheres. In this paper, we examine observational appearances of an infalling star in a reflection-asymmetric wormhole, which has two photon spheres, one on each side of the wormhole. We find that the late-time total luminosity measured by distant observers gradually decays with time or remains roughly constant due to the absence of the event horizon. Moreover, a specific observer would detect a couple of light flashes in a bright background at late times. These observations would offer a new tool to distinguish wormholes from black holes, even those with multiple photon spheres. I. INTRODUCTION The Event Horizon Telescope (EHT) collaboration released images of the supermassive black holes M87* [1][2][3][4][5][6][7][8] and Sgr A* [9][10][11][12][13][14], which provides a new method to test general relativity in the strong field regime. The main feature displayed in these images is a central brightness depression, namely black hole shadow, surrounded by a bright ring. The edge of black hole shadow involves a critical curve in the sky of observers, which is closely related to some unstable bound photon orbits. For static spherically symmetric black holes, unstable photon orbits form photon spheres outside the event horizon. Since light rays undergo strong gravitational lensing near photon spheres, black hole images encode valuable information of the geometry in the vicinity of photon spheres. Therefore, black hole images have been widely studied in the context of different theories of gravity, e.g., nonlinear electrodynamics [15][16][17][18][19][20][21], the Gauss-Bonnet theory [22][23][24][25], the Chern-Simons type theory [26,27], f (R) gravity [28][29][30], string inspired black holes [31][32][33][34] and other theories [35][36][37][38][39][40][41][42][43][44][45][46]. On the other hand, testing the nature of compact objects in the universe has been an important question in astrophysics for decades. Although the black hole images captured by EHT are in good agreement with the predictions of Kerr black holes, the black hole mass/distance and EHT systematic uncertainties still leave some room within observational uncertainty bounds for black hole mimickers. Among all black hole mimickers, ultra compact objects (UCOs), e.g., boson stars, gravastars and wormholes, which are horizonless and possess light rings (or photon spheres in the spherically symmetric case), are of particular interest since their observational signatures can be quite similar to those of black holes [47][48][49][50]. Nevertheless, it is of great importance to seek observational signals to distinguish UCOs from black holes. For example, due to a reflective surface or an extra photon sphere, echo signals associated with the post-merger ringdown phase in the binary black hole waveforms can be found in various ECO models [51][52][53][54][55][56][57][58][59][60][61]. In addition, asymmetric thin-shell wormholes with two photon spheres were found to have double shadows and an additional photon ring in their images [62][63][64][65][66]. For black holes with one photon sphere, there is one shadow and one photon ring in black hole images, and no echo signal in late-time waveforms. These observational features would allow us to distinguish wormholes from black holes with one photon sphere. Intriguingly, more than one photon sphere has been reported to exist outside the event horizon for a class of hairy black holes in certain parameter regions [67][68][69][70][71]. Multiple photon spheres can introduce distinctive features in black hole images, e.g., double shadows [71], extra photon rings [72] and tripling higher-order images [73]. Furthermore, late-time echo signals were also observed since the effective potential of a scalar perturbation possesses a multiple-peak structure [74,75]. Can we distinguish black holes with multiple photon spheres from UCOs? To answer this question, we investigate dynamic observations of a luminous object freely falling in an asymmetric thin-shell wormhole in this paper. Lately, observational appearances of a star freely falling onto black holes with a single or double photon spheres have been numerically simulated [76,77]. Particularly, the total observed luminosity fades out exponentially with a declining tail, which is caused by photons orbiting around the photon sphere, in the single-photon-sphere case. In contrast, when there exist two photon spheres, the total luminosity exhibits two exponential decays and a sharp peak between them. In addition, due to photons trapped between two photon spheres, a specific observer can detect one more cascade of flashes in the double-photon-sphere case. Recently, luminous matter falling onto a black hole has been reported to occur periodically near the Cyg X-1 [78] and the Sgr A* source [79,80]. Moreover, a new way to measure the spin of Sgr A* was proposed by simulating an infalling gas cloud [81]. In practice, detecting photons circling around photon spheres several times at late times could be a challenging task due to the scarcity of these photons. Interestingly, it showed that precise measurements of photon rings, which are formed of photons circling around photon spheres more than once, may be feasible with a very long baseline interferometry [82][83][84]. Therefore, it is timely to study observational appearances of a freely-falling star in the wormhole background, which provides a new way to detect wormholes. The rest of the paper is organized as follows. In Section II, we briefly review the asymmetric thin-shell wormhole and introduce our observational settings. Numerical results are presented in Section III. Finally, we conclude with a brief discussion in Section IV. We set G = c = 1 throughout this paper. II. SETUP As introduced in [62,65,85], an asymmetric thin-shell wormhole has two distinct spacetimes, M 1 and M 2 , which are glued together by a thin shell at its throat. The metric of the wormhole is described as ds 2 i = −f i (r i )dt 2 i + dr 2 i f i (r i ) + r 2 i dΩ 2 ,(1) where i = 1 and 2 indicate quantities in M 1 and M 2 , respectively. Focusing on the Schwarzschild spacetime, we have f i (r i ) = 1 − 2M i r i for r i ≥ R,(2) where M i are the mass parameters, and R is throat radius. Without loss of generality, we set M 1 = 1 and M 2 = k in the rest of this paper. For more details of the asymmetric thin-shell wormhole, refer to [62]. In M 1 and M 2 , the local tetrads are e t i = f − 1 2 i (r i ) ∂ ∂t i , e r i = f 1 2 i (r i ) ∂ ∂r i , e θ i = 1 r i ∂ ∂θ i , e φ i = 1 r i sin(θ) ∂ ∂φ i .(3) At the throat, one has e t 1 = e t 2 , e r 1 = −e r 2 , e θ 1 = e θ 2 and e φ 1 = e φ 2 , which yields the relations between the bases of the tangent space of M 1 and M 2 , ∂ ∂t 1 = Z −1 ∂ ∂t 2 , ∂ ∂r 1 = −Z ∂ ∂r 2 , ∂ ∂θ 1 = ∂ ∂θ 2 , ∂ ∂φ 1 = ∂ ∂φ 2 ,(4) where Z ≡ f 2 (R)/f 1 (R). Therefore, the components of a vector at the throat in M 1 and M 2 are related by V t 1 = ZV t 2 , V r 1 = −Z −1 V r 2 , V θ 1 = V θ 2 , V φ 1 = V φ 2 .(5) In this paper, we study a point-like star freely falling along the radial direction at θ i = π/2 and ϕ i = 0, which emits photons isotropically in its rest frame. With spherical symmetry, we can confine ourselves to emissions on the equatorial plane. The geodesics on the equatorial plane are described by the Lagrangian L = − 1 2 f i (r i )ṫ 2 i + 1 f i (r i )ṙ 2 i + r 2 iφ i 2 ,(6) where dots stand for derivative with respect to an affine parameter τ . Since the Lagrangian L does not depend on coordinates t i and ϕ i , the geodesics can be characterized by their conserved energy E i and angular momentum l i in M i , E i = −p t i = f i (r i )ṫ i , l i = p ϕ i = r 2 iφ i .(7) Note that, according to eqn. (5), one has E 1 = E 2 /Z and l 1 = l 2 . The Lagrangian of the freely-falling star obeys the constancy L = −1/2 when the affine parameter τ is chosen as the proper time. Since the star falls radially, its angular momentum l i = 0. Due to the traversability of the wormhole, we consider two scenarios with distinct trajectories of the star. In the scenario I, the star with energy E 1 = 1/Z (E 2 = 1) has a nonzero initial velocity at spatial infinity of M 1 . So, the star can pass through the throat and travel towards spatial infinity of M 2 . With the relation (7), the four-velocities of the star in M 1 and M 2 are given by v µ 1 e (r 1 ) = 1 1 − 2r −1 1 R − 2 R − 2k , − 2k − 2 R − 2k + 2 r 1 , 0, 0 , v µ 2 e (r 2 ) = 1 1 − 2kr −1 2 , 2k r 2 , 0, 0 .(8) In the scenario II, the star with energy E 1 = 1 is initially at rest at spatial infinity of M 1 . At first, the star falls freely in M 1 , passes through the throat and reaches a turning point in M 2 . Then, it moves towards the throat in M 2 , returns to M 1 and comes to rest at spatial infinity of M 1 . Similarly, the four-velocities of the star in M 1 and M 2 are v µ 1 e (r 1 ) = 1 1 − 2r −1 1 , ∓ 2M r 1 , 0, 0 , v µ 2 e (r 2 ) = 1 1 − 2kr −1 2 R − 2k R − 2 , ± −2k + 2 R − 2 + 2k r 2 , 0, 0 ,(9) where plus and minus signs represent outward and inward moving, respectively. Moreover, null geodesics on the equatorial plane are also governed by the Lagrangian (6) with L = 0, which rewrites the radial component of the null geodesic equations aṡ r i 2 L 2 i = 1 b 2 i − V i,eff (r i ) ,(10)where b i ≡ l i /E i is the impact parameter, and V i,eff (r i ) = f i (r i )r −2 i is the effective potential. Note that the impact parameters of a null geodesic in M 1 and M 2 , namely b 1 and b 2 , are related by b 1 = Zb 2 . A photon sphere in M i is constituted of unstable circular null geodesics, whose radius r ph i is determined by where b ph i is the corresponding impact parameter. Photons with b i ≈ b ph i are temporarily trapped at the photon sphere and can determine late-time observational appearances of the wormhole. V i,eff (r ph i ) = 1 (b ph i ) 2 , V i,eff (r ph i ) = 0, V i,eff (r ph i ) < 0,(11)Inward Outward Scenario I M 1 R−2 R−2k − cos(α) 2k−2 R−2k + 2 re / M 2 / R−2 R−2k + cos(α) R−2 R−2k 2k re Scenario II M 1 1 − cos(α) 2 re 1 + cos(α) 2 re M 2 1 − cos(α) R−2 R−2k −2k+2 R−2 + 2k re 1 + cos(α) R−2 R−2k −2k+2 R−2 + 2k re If the throat radius satisfies max{2, 2k} < R < min{3, 3k}, the asymmetric thin-shell wormhole can be free of the event horizon and possess two photon spheres, which are located at r ph 1 = 3 and r ph 2 = 3k in M 1 and M 2 , respectively. In this paper, we consider the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6, whose observational appearance of an accretion disk has been discussed in [65]. We assume that the emitted photons are collected by distant observers distributed on a celestial sphere located at r 1 = r o in M 1 . To trace light rays emitting from the star to a distant observer, one needs to supply initial conditions. For a photon of four-momentum p µ i , the momentum measured in the rest frame of the star with four-velocity v µ i e at r i = r e is pt = −v t i e (r e )p t i − v r i e (r e )p r i , pr = − v t i e (r e ) 2 − f −1 i (r e )p t i ± [v r i e (r e )] 2 + f i (r e )p r i ,(12)pθ = 0, pφ = p ϕ i r e , where plus and minus signs correspond to negative and positive v r i e , respectively. The emission angle α is defined as cos α = pr pt ,(13) which is the angle between the propagation direction of the photon and the radial direction in the rest frame of the star. In the rest frame, the photon is emitted with proper frequency ω e = − (v µ i e p µ i ) e = pt. For a distant static observer with four-velocity v µ 1 o = (1, 0, 0, 0), the photon is (12) and (13), we express the normalized frequency ω o /ω e as a function of the star position r e and the emission angle α for two scenarios in Table. I. Furthermore, the luminosity of photons is given by observed with frequency ω o = − (v µ 1 o p µ 1 ) o = p t 1 . With eqns. (5),L k = dE k /dτ k , where E k is the total energy, τ k is the proper time, and k = e and o denote quantities corresponding to the emitter and the observer, respectively. Similar to the normalized frequency, one can define the normalized luminosity L o L e = dE o /dτ o dE e /dτ e ≈ ω o dn o ω e dn e dt o dτ e −1 ,(14) where n o and n e are the observed and emitted photon numbers, respectively, and we replaced dτ o by dt o since they are almost the same for distant observers. III. NUMERICAL RESULTS In this section, we numerically study observational appearances of a star freely falling radially in the asymmetric thin-shell wormhole in the scenarios I and II. During the free fall of the star, photons are emitted isotropically in the rest frame of the star. Specifically, we assume that the star starts emitting photons at t 1 = t 2 = 0 and r 1 = 30.65 in M 1 , and emits 3200 photons, which are uniformly distributed in the emission angle α, every proper time interval δτ e = 0.002. It is worth emphasizing that observational appearances of the freely-falling star, especially late-time appearances, are rather insensitive to the initial position where the star starts emitting. Here, for better comparison with the Schwarzschild black hole case, we simply choose the initial position as r 1 = 30.65, which is in agreement with that of [76]. Here, observational appearances of the star are studied for two kinds of observers in M 1 . The first kind is observers distributed on a celestial sphere at the radius r o = 100, which refers to collecting photons in the whole sky at fixed radial coordinate r o = 100 in M 1 . The measurement by the observers on the celestial sphere would give the frequency distribution and the total luminosity of photons that reach the celestial sphere. The second kind is a specific observer, who is located 1 Since ϕ1 = ϕ2 at the throat, the subscript of ϕ is omitted for simplicity. is strikingly different from the black hole case, where the total luminosity has been found to decay exponentially at late times [76,77]. at ϕ o = 0b 1 =3.579 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 x y b 1 =3.664 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 x y b 1 =4.923 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 x y b 1 =5.238 -8 -6 -4 -2 0 2 4 6 8 -6 For a specific observer located at ϕ o = 0 and θ o = π/2 on the celestial sphere at r o = 100 in M 1 , the angular coordinate change ∆ϕ of light rays connecting the star with the observer is ∆ϕ = 2nπ,(15) where n = 0, 1, 2 · · · is the number of orbits that the light rays complete around the wormhole. To simulate observational appearances of the star seen by the observer, we select photons with M 1 (i.e., the purple region) and those with b 2 b ph 2 , the normalized frequency has high-frequency and low-frequency branches, corresponding to the star falling away from and towards the observer, respectively. If photons are emitted inside the photon sphere in M 1 with b 1 b ph 1 , the highfrequency (low-frequency) branch denotes ingoing and outgoing (outgoing and ingoing) emissions from the star falling away from and towards the observer, respectively. For the high-frequency branches, strong gravitational lensing around the photon spheres can cause blueshifts of nearcritical photons emitted inward at a large r e in M 1 . In particular, the normalized frequency with [76], which is in consistency with the frequency observation. Afterwards, the received blueshifted photons with a small impact parameter dominate the total luminosity, resulting in a peak at t o 220. At late times, the total luminosity is maintained around one since most emitted photons can be collected by the observers. Subsequently, photons emitted in the yellow region determine the luminosity observation again and produce a peak around t o 220. At late times, the star travels towards the observer at a large r e in M 1 , and hence radially emitted photons would make a dominant contribution to the total luminosity. In particular, the late-time luminosity remains fairly constant, which is greatly different from the black hole case. IV. CONCLUSIONS In this paper, we investigated observational appearances of a point-like freely-falling star, which emits photons isotropically in its rest frame, in an asymmetric thin-shell wormhole connecting two spacetimes, M 1 and M 2 . Specifically, two scenarios with different initial velocities of the star were considered. In the scenario I, the star starts with a nonzero velocity at spatial infinity of M 1 and moves towards spatial infinity of M 2 . In the scenario II, the star falls at rest from spatial infinity of M 1 , reaches a turning point in M 2 and returns to M 1 . For the two scenarios, the frequency distribution and luminosity of the star measured by all observers and a specific observer on a celestial sphere were obtained by numerically tracing emitted light rays. Interestingly, it was found that the absence of the event horizon and the presence of two photon spheres play a pivotal role in frequency and luminosity observations. In [76] and [77], observational appearances of a star freely falling in black holes with one or two photon spheres were investigated. To compare the wormhole case with the black hole one, we briefly summarize the main findings of [76,77] and this paper as follows. • Black holes with a single photon sphere: The total luminosity of the star fades out with an exponentially decaying tail, which is determined by quasinormal modes at the photon sphere. At late times, the specific observer sees a series of flashes indexed by the orbit number, whose luminosity decreases exponentially with the orbit number. Moreover, the frequency content of received photons contains a discrete spectrum of frequency lines indexed by the orbit number, which decay sharply at late limes. • Black holes with double photon spheres: At late times, the total luminosity first rises to a peak and then decreases with an exponentially decaying tail. The sub-long-lived quasinormal modes at the outer photon sphere are responsible for the slowly decaying exponential tail, and the leakage of photons trapped between the inner and outer photon spheres results in the luminosity peak. The specific observer sees two series of flashes, which are mainly determined by photons orbiting outside the outer and inner photon spheres, respectively. Moreover, the specific observer detects a discrete spectrum of frequency lines indexed by the orbit number and the photon sphere that received photons orbit around, which fall steeply at late limes. • Wormhole: At late times, the total luminosity first rises to a peak and then gradually decays with time (scenario I) or remains roughly constant (scenario II). The luminosity peak is caused by photons travelling between the two photon spheres (scenario I) or those emitted in M 2 nearly along the radial direction (scenario II). Due to the absence of the event horizon, a considerable number of photons can still reach observers at late times, and hence an exponentially decaying tail would not appear. Similarly, the late-time luminosity measured by the specific observer can be sizable, and therefore he only sees a bright flash and a faint one (scenario I) or two bright flashes (scenario II) due to strong background luminance. Moreover, the specific observer detects frequency lines indexed by the orbit number and the photon sphere that received photons orbit around. The frequency lines produced by photons orbiting around the photon sphere in M 1 decline sharply (scenario I) or grow steadily (scenario II) at late limes; those produced by photons orbiting around the photon sphere in M 2 gradually increase at late limes. In short, we showed that the absence of the event horizon in wormholes gives rise to significantly different optical appearances of a luminous star accreted onto wormholes at late times. Therefore, these findings can provide us a novel tool to distinguish wormholes from black holes in future observations. FIG. 1 .M 2 12on the equator of the celestial sphere. Among all photons collected on the celestial sphere, we select photons with cos ϕ > 0.99 to mimic photons detected by the specific observer. To calculate observed luminosities, the collected photons are grouped into packets of 50 (i.e., dn o = 50) according to their arrival time. As shown in FIG. 1, the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6 has a doublepeak effective potential, corresponding to one photon sphere in M 1 and one in M 2 . Specifically, the photon sphere in M 1 is located at r ph 1 = 3 with the critical impact parameter b The effective potential of null geodesics in the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6. The potential has two peaks at r ph 1 = 3 (solid vertical blue line) and r ph 2 = 3.6 (dashed vertical blue line), corresponding to a photon sphere with b , respectively. The vertical red line denotes the throat at r 1 = r 2 = R. Photons emitted in the pink, brown, orange and purple regions have impact parameters close to the impact parameters of the photon spheres, and hence can be temporarily trapped around the photon spheres. In particular, when photons are emitted towards the throat at r 2 > r ph 2 in the pink region or at r 1 > r ph 1 in the brown, orange and purple regions, they usually orbit the wormhole with ∆ϕ ≥ 2π. that in M 2 is located at r ph 2 = 3.6 with the critical impact parameter b ph 2 = 3.6 √ 3. To discuss how photons with different impact parameters contribute to the observations of the star, we classify received photons into seven categories according to their impact parameter b 1 in M 1 , • b 1 < 3.579. Yellow region in FIG. 1 and yellow dots in FIGs. 4, 5, 7 and 8. • 3 . 3579 ≤ b 1 < Zb ph 2 . Pink region in FIG. 1 and pink dots in FIGs. 4, 5, 7 and 8. In this category, photons emitted inward outside the photon sphere in M 2 can circle around the photon sphere more than once before reaching a distant observer in M 1 . For example, a light ray with b 1 = 3.579, which has ∆ϕ = 2π 1 , is displayed in the upper-left panel of FIG. 2. • Zb ph 2 < b 1 ≤ 3.664. Brown region in FIG. 1 and brown dots in FIGs. 4, 5, 7 and 8. In this category, photons emitted inward would circle around the photon sphere in M 2 roughly with ∆ϕ ≥ 2π before escaping to the celestial sphere in M 1 . For example, a light ray with b 1 = 3.664, which has ∆ϕ = 2π, is displayed in the upper-right panel of FIG. 2. FIG. 2 . 2Photon trajectories in the asymmetric thin-shell wormhole with k = 1.2 and R = 2.6. The red points and circles denote the star and the throat, respectively. The blue solid and dashed circles represent the photon spheres in M 1 and M 2 , respectively. The upper-left panel shows a photon emitted at r e = 5 in M 2 with b 1 = 3.579, and the light ray has ∆ϕ = 2π. Other panels show photons emitted at r e = 5 in M 1 with b 1 = 3.664, 4.923 and 5.238, and the light rays all have ∆ϕ = 2π. The solid and dashed segments of the light rays correspond to the segments in M 1 and M 2 , respectively. • 3 . 3664 < b 1 ≤ 4.923. Blue region in FIG. 1 and blue dots in FIGs. 4, 5, 7 and 8. • 4 . 4923 < b 1 < b ph 1 . Orange region in FIG. 1 and orange dots in FIGs. 4, 5, 7 and 8. In this category, if photons are emitted inward outside the photon sphere in M 1 , they would linger for some time around the photon sphere by orbiting it approximately with ∆ϕ ≥ 2π. For example, a light ray with b 1 = 4.923, which has ∆ϕ = 2π, is displayed in the lower-left panel of FIG. 2. • b ph 1 < b 1 ≤ 5.238. Purple region in FIG. 1 and purple dots in FIGs. 4, 5, 7 and 8. In this category, photons emitted inward outside the photon sphere in M 1 usually circle around the photon sphere more than once. For example, a light ray with b 1 = 5.238, which has ∆ϕ = 2π, is displayed in the lower-right panel of FIG. 2. • b 1 > 5.238. Green region in FIG. 1 and green dots in FIGs. 4, 5, 7 and 8.In short, we use the orbit number of light rays emitted at r 1 = 5 in M 1 or r 2 = 5 in M 2 to determine the threshold impact parameters separating the seven categories. To sum up, light rays emitted inward at r 2 = 5 in the yellow/pink category would circle around the wormhole less/more than once before being received; light rays emitted inward at r 1 = 5 would circle around the wormhole less than once before being received in the blue and green categories, or more than once in the brown, orange and purple categories. Note that the orbit number of light rays with a given impact parameter depends slightly on the emitting position. So, light rays connecting the star and the observers circle around the wormhole approximately more than once in the pink, brown, orange and purple categories, and less than once in the yellow, blue and green categories. In other words, photons in the pink, brown, orange and purple categories can be temporarily trapped near the photon spheres.A. Scenario I In the scenario I, the star with energy E 1 = 1/Z = √ 3 would travel through the throat and move towards spatial infinity of M 2 . For near-critical photons emitted with the impact parameter very close to those of the photon spheres in M 1 (i.e., b 1 b ph 1 ) and M 2 (i.e., b 2 b ph 2 ), their normalized frequencies ω o /ω e measured by observers on the celestial sphere are plotted against the emitted position r e in FIG. 3. The colors of the lines in FIG. 3 match those of the corresponding emitted regions in FIG. 1. Moreover, photons with b 1 b ph 1 and b 2 b ph 2 are denoted by solid and dashed lines, respectively. It is worth emphasizing that the observed frequency of a photon is determined by the gravitational redshift and the Doppler effect, which are controlled by the position and the velocity of the photon when it is emitted, respectively. For photons of b 2 b ph 2 , the normalized frequency can noticeably exceed 1 at a large r e in M 1 since the Doppler effect plays a more important role than the gravitational redshift. As the star falls towards the throat, the normalized frequency decreases due to stronger gravitational redshift, and blueshift becomes redshift at r e = 4.063 in M 1 , where the normalized frequency is 1. When emitted at the throat, the normalized frequency reaches the minimum. After the star enters M 2 , the normalized frequency increases as r e grows, and observed photons become bluershifted when r e > 12.281. For photons of b 1 b ph 1 , the behavior of the normalized frequency is quite similar to FIG. 3. The normalized frequency ω o /ω e as a function of the emitted position r e for photons in the scenario I, whose impact parameter is very close to these of the photon spheres in M 1 (solid lines) and M 2 (dashed lines). The observers are distributed on the celestial sphere at r o = 100 in M 1 . For a large r e in M 1 , inward-emitted and near-critical photons can be blueshifted since the Doppler effect dominates over the gravitational redshift. Due to the relation (5) at the throat, near-critical photons can also be blueshifted when r e is large in M 2 . Photons emitted inward and outward between the two photon spheres can both reach a distant observer after orbiting the photon sphere in M 1 , which gives two branches of the orange line in the inset. Moreover, the normalized frequency reaches the minimum at the throat, which is located at r e = 2.6. those of b 2 b ph 2 when they are emitted outside the photon sphere in M 1 . When the star emits photons between the two photon spheres, inward-emitted and outward-emitted photons can both be captured by a distant observer after they circle around the photon sphere in M 1 , thus leading to two branches as shown in the inset. The upper and lower branches correspond to photons emitted away from and towards the observer, respectively. In the left panel of FIG. 4, we display the normalized frequency distribution of photons, which are emitted from the freely-falling star in the scenario I and collected by observers distributed on the celestial sphere at r o = 100 in M 1 . At early times, received photons are dominated by those emitted in the green region of FIG. 1, among which inward-emitted photons contribute to the high-frequency observation. When t o > 160, photons emitted towards the photon sphere in M 2 in the blue and brown regions start reaching the observers after orbiting around the photon sphere. Subsequently, the observers receive photons emitted towards the photon sphere in M 1 in the purple and orange regions. Since time moves faster in M 2 roughly by a factor of 1/Z = √ 3 relative to in M 1 , photons circling around the photon sphere in M 2 arrive earlier. Moreover, the maximum frequency of photons emitted in the blue and brown regions is higher than that of FIG. 4. The normalized frequency distribution and the total luminosity of the freely-falling star in the scenario I, measured by observers on a celestial sphere at r o = 100 in M 1 . Left: The observers receive photons with a wide range of frequencies. At the early stage, photons emitted in the green region of FIG. 1 give rise to the frequency observation. Afterwards, photons emitted in the brown, blue, orange and purple regions are observed. In particular, photons with a near-critical impact parameter produce high frequency observations. The late-time frequency observation is determined by photons in the yellow and pink regions, which are emitted at a large r e in M 2 . Right: The luminosity is calculated by grouping received photons into packets of 50. An increase of the observed luminosity is caused by photons emitted inward in the blue region, leading to a peak at t o 168. At late times, the total luminosity gradually decays with time and is mainly controlled by photons, which are emitted at a large r e in M 2 and travel through the throat to reach the observers. photons emitted in the green, purple and orange regions. This is expected from FIG. 3, which shows that near-critical photons with b 2 b ph 2 have higher normalized frequency than these with b 1 b ph 1 . Afterwards, the frequency observations are dominated by photons emitted in the orange region, which are trapped at the photon sphere in M 1 for a longer time. At late times, the observers mostly receive photons in the yellow and pink regions, which are emitted towards the throat in M 2 with a small impact parameter. The normalized total luminosity of the freely-falling star is displayed in the right panel of FIG. 4, where a dot corresponds to a packet of 50 photons, and the color of the dot is that having most photons in the packet. The luminosity gradually increases until reaching a peak around t o 145, and is dominated by photons emitted in the green region roughly before t o = 150, which is in agreement with the frequency observation. After t o 160, photons emitted in the blue region give rise to a noticeable increase of the total luminosity. As the star moves towards spatial infinity of M 2 , emitted photons can still propagate to the observers in M 1 through the throat, and a slight decrease of the total luminosity is displayed at late times. Interestingly, this late-time observation FIG. 5. The normalized frequency and the luminosity of the freely-falling star in the scenario I, measured by a distant observer at r o = 100, θ o = π/2 and φ o = 0 in M 1 . The colored dots denote photons emitted in the regions with the same color in FIG. 1. Left: Received photons form several frequency lines indexed by the orbiting number n. The inset displays three frequency lines caused by n = 1 photons with b 1 b ph 1 , b 1 b ph 1 and b 2 b ph 2 . The time delay between the adjacent n ≥ 1 lines formed by photons orbiting around the photon sphere in M 1 and M 2 is roughly the period of circular null geodesics at the photon sphere, respectively. Right: At early times, the luminosity is dominated by photons with a small impact parameter, and decreases first and then increases after the star goes through the throat. Subsequently, blueshifted n = 1 photons start to reach the observer and become the most dominant contribution, which produces a luminous flash at t o 170. Later, the luminosity is mainly contributed by the n = 0 photons emitted in the yellow region of M 2 and almost declines gradually at late times. In addition, a faint flash, which results from the n = 2 photons emitted in the orange region, is observed at t o 200. cos ϕ > 0.99 from all photons received on the celestial sphere. The frequency observation is presented in the left panel of FIG. 5, which shows a discrete spectrum separated by the received time. The yellow line is formed by photons with n = 0, which radially propagate to the observer. At early times, the observed frequency of the n = 0 photons decreases with the received time as the star falls towards the throat. After the star passes through the throat, the observed frequency of the n = 0 photons increases since the gravitational redshift becomes weaker as the star moves further away from the throat, which results in the dip at t o 150. Owing to the existence of two photon spheres, the n = 1 photons with impact parameters b form three frequency lines, which are highlighted in the inset of FIG. 5. As the star falls towards the throat, the three frequency lines decrease rapidly due to strong gravitational redshift near the throat. After the star passes through the throat, the frequency line with b 2 b ph 2 gradually increases. For n = 2, the frequency lines with b 1 b ph 1 and b 1 b ph 1 move closer and are hardly distinguishable from each other. On the other hand, the frequency line with b 2 b ph 2 becomes more separate from them since photons spend more time orbiting around the photon sphere in M 1 . Indeed, it takes ∆T 1 2πb ph 1 33 to orbit around the photon sphere in M 1 one time, and ∆T 2 2πZb ph 2 23 to orbit around that in M 2 2 . Therefore, for b 1 b ph 1 and b 1 b ph 1 (b 2 b ph 2 ), the time delay between the n = 1 and 2 frequency lines roughly equals to ∆T 1 (∆T 2 ). For n = 3, because of the finite number of photons in our numerical simulation, only the frequency line with b 2 b ph 2 can be found and is shown by orange dots around t o 230. The left panel of FIG. 5 shows the observed normalized luminosity as a function of the time, which exhibits a decline before the star reaches the throat. After the star moves through the throat, the luminosity starts to increase since the frequency of received photons grows, which causes a dip at t o 150. Around t o 160, blueshifted photons with n = 1 start to play a dominant role, leading to a luminous flash around t o 170. Afterwards, the luminosity is mainly dominated by photons emitted in the yellow region of M 2 , and slowly decreases except a faint flash at t o 200 caused by the arrival of n = 2 photons. The flashes of photons with n ≥ 3 are much fainter and barely visible in the background of the dominant photons emitted in the yellow region. In contrast,for a black hole with two photon spheres, a series of flashes with decreasing luminosity are observed at late times due to photons orbiting around the hairy black hole different times[77].B. Scenario IIIn the scenario II, the star starts falling from spatial infinity of M 1 and returns to the infinity after going through the throat twice. Similarly, the normalized frequency ω o /ω e for near-criticalphotons is plotted inFIG. 6. Specifically, we focus on photons with b 1 b ph 1 emitted in the purple and orange regions and those with b 2 b ph 2 emitted in the brown region, which are denoted by solid and dashed lines, respectively. For photons with b 1 b ph 1 emitted outside the photon sphere in II, whose impact parameter b 1 is very close to b ph 1 (solid lines) or b 2 is very close to b ph 2 (dashed lines). The observers are distributed on the celestial sphere at r o = 100 in M 1 . The normalized frequency of near-critical photons emitted in the purple and brown regions has two branches. Specifically, the high-frequency (lowfrequency) branch corresponds to photons emitted from the star falling away (towards) from the observer.Similar to the scenario I, the high-frequency branch can be blueshifted for a large r e in M 1 . The normalized frequency reaches the global minimum ω o /ω e 0.139 at the throat for the low-frequency branch. reaches the maximum ω o /ω e = 4/3 (ω o /ω e = 1.392) at r e = 12 (r e = 8.679), becomes one at r e = 5.196 (r e = 3.6), and reaches the minimum ω o /ω e = 0.306 (ω o /ω e = 0.139) at the throat. In M 2 , the normalized frequency with b 2 b ph 2 reaches the maximum ω o /ω e = 1 at r e = 3.6, where the star returns. The normalized frequency distribution of photons received by observers distributed on the celestial sphere is presented in the left panel of FIG. 7. When t o 200, a wide range of frequencies is observed for photons emitted in the green region. After near-critical photons emitted in the purple, orange, blue and brown regions start arriving at the observers around t o 150, they come to dominate the high-frequency part of the frequency distribution. This early-stage frequency distribution bears a resemblance to the Schwarzschild black hole case, in which a star falls from FIG. 7. The normalized frequency distribution (Left) and the total luminosity (Right) of the freely-falling star in the scenario II, measured by observers on the celestial sphere at r o = 100 in M 1 . Similar to the scenario I, photons emitted in the green region of FIG. 1 dominate the frequency and luminosity observations in the early stage. After the star enters M 2 , photons emitted in the yellow region, which propagates to the observers nearly in the radial direction, produce frequency and luminosity peaks around t o 220. Later, near-critical photons with a wide range of frequencies are observed. At late times, the emitted position r e is in M 1 and large, and therefore the observers would collect most of emitted photons, which leads to a nearly constant total luminosity. spatial infinity at rest [76]. Similar to the scenario I, the maximum frequency of photons emitted inward in the blue and brown regions is greater than that of photons emitted inward in the green, purple and orange regions. After the star enters M 2 , the observed frequency of photons emitted in the yellow region starts to increase and reaches a maximum around t o 220, which is associated with the star returning to the throat. Subsequently, photons emitted in the brown and purple regions are observed to have a wide range of frequencies after they circle around the photon sphere in M 1 and reach the observers. At late times, the star comes back to M 1 and moves towards the observer, and thus the low-frequency distribution is dominated by photons emitted towards the throat with b 1 b ph 1 and b 2 b ph 2 . On the other hand, photons emitted towards the observers with a small impact parameter produce the high-frequency observation. The normalized total luminosity of the freely-falling star in the scenario II is displayed in the right panel of FIG. 7. Before t o 200, the total luminosity behaves similarly to the Schwarzschild black hole case studied in FIG. 8 . 8The normalized frequency and the luminosity of the freely-falling star in the scenario II, measured by a distant observer at r o = 100, θ = π/2 and φ = 0 in M 1 . Left: The yellow line denotes radially emitted photons with n = 0 and has a dip (peak) near t o 200 (t o 220), corresponding to emission from the star at the throat. The n = 1 frequency lines with b 1 b ph 1 , b 1 b ph 1 and b 2 b ph 2 steadily increase to a peak followed by a sharp decrease when t o 230, and gradually increase when t o 240. For 230 t o 240, the n = 1 frequency line with b 2 b ph 2 rises to another high point. Right: Similar to the scenario I, the luminosity is dominated by n = 0 photons and gradually decreases before t o 160. Later, blueshifted n = 1 photons start to reach the observer and then become the most dominant contribution, which results in a luminosity peak around t o 180. Afterwards, due to the increasing frequency of n = 0 photons emitted in M 2 , the luminosity rises and reaches a peak around t o 220. At late times, received n = 0 photons emitted in M 1 enable the luminosity to stay roughly constant. In the left panel of FIG. 8, we exhibit the normalized frequency of photons received by an observer located at ϕ = 0 and θ = π/2 on the celestial sphere in M 1 for the scenario II. The observed frequency of radially emitted photons with n = 0 is represented by the yellow line, which displays three periods. In the first and last periods, the photons are emitted when the star moves towards and away from the throat in M 1 , respectively, and the n = 0 frequency line both decreases with the received time; in the intermediate period, the star emits the photons in M 2 , and the n = 0 frequency line increases. There appears a peak and a dip of the n = 0 frequency line, which correspond to the star going through the throat the first time and the second time, respectively. Similar to the scenario I, the n = 1 frequency lines consist of three lines with b 1 b ph 1 , b 1 b ph 1 and b 2 b ph 2 , respectively. The n = 1 frequency line with b 2 b ph 2 increases slowly until the maximum and then decreases rapidly in the first period, rises to a peak followed by a steep decline in the intermediate period, and gradually increases in the last period. For the n = 1 frequency lines with b 1 b ph 1 and b 1 b ph 1 , there is a sharp drop after reaching a peak when the star moves away from the observer, and a steady increase when the star moves towards the observer. For n = 2, only two frequency lines are visible, namely the b 1 b ph 1 (orange dots) and b 2 b ph 2 (brown dots) lines. Note that the n = 2 frequency lines are quite similar to the n = 1 counterparts. In addition, only the frequency line with b 1 b ph 1 is visible for n = 3. The normalized luminosity of the star in the scenario II measured by the observer is displayed in the right panel of FIG. 8. Similar to the scenario I, the luminosity decreases slowly before t o 170, which is dominated by radially emitted photons in the yellow region. Afterwards, photons emitted in the blue region come to control the luminosity observation and lead to a flash around t o 180. TABLE I . IThe normalized frequency ω o /ω e as a function of the star position r e and the emission angle αin the scenarios I and II. Inward and outward correspond to travelling towards and away from the throat, respectively. Eqn.(7) leads to dt/dφ| r ph = b −1 V −1 eff (r ph ) = b ph , which gives ∆T 2πb ph . We are grateful to Guangzhou Guo and Qingyu Gan for useful discussions and valuable comments. This work is supported in part by NSFC (Grant No. 11875196, 11947225, 12105191, 12275183 and 12275184). Scholars of Sichuan UniversityHouwen Wu is supported by the International Visiting Program for Excellent YoungWe are grateful to Guangzhou Guo and Qingyu Gan for useful discussions and valuable com- ments. This work is supported in part by NSFC (Grant No. 11875196, 11947225, 12105191, 12275183 and 12275184). Houwen Wu is supported by the International Visiting Program for Excellent Young Scholars of Sichuan University. First M87 Event Horizon Telescope Results. I. 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Thermodynamics and phase structure of an Einstein-Maxwell-scalar model in extended phase space. Guangzhou Guo, Peng Wang, Houwen Wu, Haitang Yang, 10.1103/PhysRevD.105.064069arXiv:2107.04467Phys. Rev. D. 10562022Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Thermodynamics and phase structure of an Einstein-Maxwell-scalar model in extended phase space. Phys. Rev. D, 105(6):064069, 2022. arXiv:2107.04467, doi:10.1103/PhysRevD.105.064069. I Photon ring and observational appearance of a hairy black hole. Qingyu Gan, Peng Wang, Houwen Wu, Haitang Yang, 10.1103/PhysRevD.104.044049arXiv:2105.11770Phys. Rev. D. 10442021Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon ring and observational appearance of a hairy black hole. Phys. Rev. D, 104(4):044049, 2021. arXiv:2105.11770, doi:10.1103/PhysRevD. 104.044049. I Gravitational Lensing by Black Holes with Multiple Photon Spheres. 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{'abstract': 'It has been recently reported that, at late times, the total luminosity of a star freely falling in black holes decays exponentially with time, and one or two series of flashes with decreasing intensity are seen by a specific observer, depending on the number of photon spheres. In this paper, we examine observational appearances of an infalling star in a reflection-asymmetric wormhole, which has two photon spheres, one on each side of the wormhole. We find that the late-time total luminosity measured by distant observers gradually decays with time or remains roughly constant due to the absence of the event horizon. Moreover, a specific observer would detect a couple of light flashes in a bright background at late times. These observations would offer a new tool to distinguish wormholes from black holes, even those with multiple photon spheres.', 'arxivid': '2210.10948', 'author': ['Yiqian Chen *[email protected]†[email protected]‡[email protected]§[email protected] \nCenter for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina\n', 'Peng Wang \nCenter for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina\n', 'Houwen Wu \nCenter for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina\n\nDepartment of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUK\n', 'Haitang Yang \nCenter for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina\n'], 'authoraffiliation': ['Center for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina', 'Center for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina', 'Center for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina', 'Department of Applied Mathematics and Theoretical Physics\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUK', 'Center for Theoretical Physics\nCollege of Physics\nSichuan University\n610064ChengduChina'], 'corpusid': 253018679, 'doi': '10.1140/epjc/s10052-023-11486-y', 'github_urls': [], 'n_tokens_mistral': 28731, 'n_tokens_neox': 21936, 'n_words': 11834, 'pdfsha': '5c815475ba1df5d972969d110fa27e7760b5b2c0', 'pdfurls': ['https://export.arxiv.org/pdf/2210.10948v1.pdf'], 'title': ['Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole', 'Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole'], 'venue': []}

arxiv

Dynamical Gibbs-non-Gibbs transitions in Widom-Rowlinson models on trees February 14, 2023 12 Feb 2023 Sebastian Bergmann [email protected] Fakultät für Mathematik Ruhr-Universität Bochum Universitätsstraße 15044780BochumGermany Sascha Kissel [email protected] Fakultät für Mathematik Ruhr-Universität Bochum Universitätsstraße 15044780BochumGermany Christof Külske [email protected] Fakultät für Mathematik Ruhr-Universität Bochum Universitätsstraße 15044780BochumGermany Dynamical Gibbs-non-Gibbs transitions in Widom-Rowlinson models on trees February 14, 2023 12 Feb 20231AMS 2020 subject classification: 82B2082C2060K35 Keywords: Widom-Rowlinson modelGibbs measuresNon-GibbsiannessStochastic dynamicsDynamical Gibbs-non-Gibbs transitionsPhase transitionsCayley treePercolationZero-one law * We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order d (which has d + 1 nearest neighbours), depending on repulsion strength β between particles of different signs and on an activity parameter λ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength β.We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d ≥ 2. In our second and main result, we also show that for large β and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity λ increases, assuming that d ≥ 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity. Introduction Dynamical Gibbs-non-Gibbs transitions for spin models can be analysed on various types of graphs. A prototypical example is the low-temperature Ising model on the integer lattice in two or more dimensions, under stochastic independent spin-flip dynamics [8]. The authors showed in particular that the time-evolved model, started from the initial plus Gibbs measure in zero external field, fails to be Gibbsian for all large enough finite times, while the Gibbs property is preserved for small times. This study of the Ising model has been extended to mean-field and Kac-models, where the appropriate notion of sequential Gibbsianness has been employed [10,11,18,26]. For studies of Gibbsian properties of the Potts model under time evolution and also other transformations, see [14,15,16,27]. A famous model in the world of point particles taking positions in R d is the Euclidean Widom-Rowlinson model. In its original form the particles carry one of the possible signs plus or minus and are subjected to hardcore pair interactions, which forbids inter-particle distances smaller than a fixed radius R > 0 when they carry different signs. In equilibrium the continuum model shows a ferromagnetic transition, see [4,5], for dynamical Gibbs-non-Gibbs transitions, see [20], for generalities on Gibbsian point processes see [6]. In this note we turn to the soft-core Widom-Rowlinson model on trees under a spin-flip time evolution. The model has the local state space {−1, 0, 1}, where the spin value 0 stands for an empty site. It has two parameters λ > 0, the activity of particles, and β > 0 describing a soft-core repulsion between neighbouring particles of different signs. Static and dynamical soft-core Widom-Rowlinson models have been studied on the lattice and in mean field, see [17,23,24,25]. The static behaviour of the soft-core Widom-Rowlinson model on a Cayley tree of order d ≥ 2 is known, see [22], where d-dependent non-uniqueness regions in the space of the parameters β, λ are described. In these regions there exist at least three different tree-indexed Markov chain Gibbs measures (splitting Gibbs measures) which are tree-automorphism invariant. Among these there is a unique measure which is also spin-flip invariant, the so-called intermediate measure µ # β,λ . In this note we focus on the corresponding time-evolved intermediate measure µ # β,λ,t which is obtained by drawing the initial condition w.r.t. the infinite-volume measure µ # β,λ and applying independent stochastic spin-flips to the spins at the occupied sites while keeping the holes fixed. Previous study has shown that in the time-evolved Ising model in zero external field a quite unusual behaviour (compared to lattices) occurs on regular trees [7]. One feature was that at large β and sufficiently large times all configurations become bad for the time-evolved intermediate measure µ # Ising;β,t . Bad configurations are non-removable discontinuity points of finite-volume conditional probabilities in the sense of Definition 2.5. If there exists at least one bad point, a Gibbsian representation with a well-behaved specification is impossible. If almost all (or even all) configurations are bad, this signals a particularly strong internal non-locality of the system. As for the Widom-Rowlinson model, the intermediate Ising measure can be defined to be the unique spin-flip invariant tree-automorphism invariant Gibbs measure which is also a tree-indexed Markov chain. The full measure badness was seen only in the time-evolved intermediate Ising measure, for different Ising measures as starting measure even recovery of the Gibbs property for large times was proved in [7]. On the other hand, full-measure badness for time-evolved Widom-Rowlinson measures was discovered in models with hard-core interaction, both in the continuum and on the lattice [20,24]. The mechanisms responsible for this in both models were based on the hard-core interactions and were much different from the mechanism responsible for the result for the Ising model on a tree. All studies of the time-evolved Widom-Rowlinson model with soft-core interactions on discrete graphs have shown non-Gibbsian behaviour at large times, but did not reveal full measure bad configurations. How much of this all-badness can we expect to survive in the time-evolved intermediate measure, and what is the role of the second parameter, the activity λ? Results and some proof ideas Our results split into criteria for badness of the time-evolved measure given in Subsection 2.2 and criteria for goodness of the time-evolved measure given in Subsection 2.3. Let us explain informally our main badness result of Theorem 2.6 which we consider to be the most interesting part. It states that on Cayley trees of order d ≥ 4 for large enough repulsion strength β, and large activity λ, at large enough times, the time-evolved measure has bad configurations of full measure. Bad configurations are by definition discontinuity points of conditional probabilities of the time-evolved measure in the sense of Definition 2.5. To appreciate this result note that our starting measure is provably non-extremal in a regime of large β and large λ by the Kesten-Stigum criterion [21], see Proposition 3.2. While bad configurations generally form a tail-event, our measure of interest will not be extremal and therefore not tail-trivial, therefore there is a priori no reason why their probabilities should be restricted to zero or one. However, we are able to formulate a different zero-one law in Theorem 2.7 which holds in all parameter regimes, and does not assume tail-triviality, but from which the desired statement follows. It applies more generally not only to the intermediate measure, but to all homogeneous tree-indexed Markov chain Gibbs measures of our model. The proof we give uses a representation of configurations of the measure in terms of the (finite or infinite) connected components by means of a renewal construction on the tree, starting from an arbitrary root, see Subsection 3.1. Using our zero-one law of Theorem 2.7, we notice that for the proof of the full-measure badness in Theorem 2.6, it suffices to find positive measure sets of bad configurations. It turns out that sets with this property can be given in terms of the subtree percolation condition of Theorem 2.8. This condition on infinite-volume configurations of the Widom-Rowlinson model looks only at the occupied sites, i.e. the sites with spin-values not equal to zero, disregarding the signs, and asks for the existence of an infinite occupied subtree of large enough order s. A glance at the result of Theorem 2.8 then also shows that for any order d ≥ 2 there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, for large β, independently of the particle activity. This is clear, as it suffices to exhibit just one particular bad configuration, and for s = d the theorem shows that the fully occupied configuration is always bad. Let us outline some of the ideas and difficulties of the proof of Theorem 2.8, i.e. explain how to ensure badness of configurations with percolating subtrees. The detailed proof will be given in Subsection 3.2. Starting from a two-layer representation of the conditional probabilities of the time-evolved model one is first led to an inhomogeneous recursion on a subtree of occupied sites which needs to be run on the first layer (spin configurations at time zero). In this recursion the influence of the configuration in the conditioning on the second layer (infinite-volume configurations at time t) appears as an inhomogeneous magnetic-field term, and unoccupied sites act as a dilution on the first layer. Non-removable discontinuities (bad points of the time-evolved measure) come from non-decaying memory on the boundary condition in this recursion, and this is what needs to be proved to ensure badness. There are mainly two possibly counteracting influences in this recursion, and this is what creates new difficulty for the Widom-Rowlinson model as compared to the non-homogeneous recursion for the Ising model [2,7]. First, there are the terms coming from the infinite tree of occupied sites attached at the origin, from we have cut off all finite parts. As for the second part, there are also contributions caused by the non-percolating appendices of the tree. These may work in the opposite direction, depending on the choice of the signs of the configuration. The proof then consists in showing that, not only are the percolating parts able to carry a boundary condition to the origin regardless of the second-layer spins, but they also win against possible counteractions from the finite parts. The proof of the all-badness result of Theorem 2.6 is then completed by ensuring typicality of s-subtree percolation of occupied sites, see Proposition 2.10. In Section 2.3 we give two results on goodness. The first is the small-time Gibbs property of Theorem 2.12, which follows by the Dobrushin method. The second result of Theorem 2.14 asserts that at any possibly large β, the set of bad configurations, while not necessarily empty, has zero measure, if the activity λ is sufficiently small. The proof is carried out again in the two-layer picture, via comparison to extinction of a suitable Galton-Watson tree. Finally, from Theorem 2.6 and Theorem 2.14 we conclude, that for any fixed large enough β and large enough time, a λ-driven transition between a non-Gibbs regime with zero measure bad configurations to a regime with full measure bad configurations occurs. The remainder of the paper is organized as follows. In Subsection 2.1 we define the soft-core Widom-Rowlinson model, review the relation between tree-indexed Markov chains and boundary laws provided by Zachary's Theorem 2.4, and define the time-evolved measure and notion of a bad configuration. Subsection 2.2 contains our results on badness, Subsection 2.3 contains our results on goodness, and Sections 3 and 4 contain the proofs. Model and main results Definitions and notation The soft-core Widom-Rowlinson model on trees A graph (V, E) is a vertex set V in combination with a set of edges E ⊂ V 2 . If two vertices are neighbours in the sense that they are connected through an edge we write i ∼ j. We will denote the set of oriented edges, which is the set of ordered pairs in E, as ⇀ E and its elements by ⇀ ij or in unambiguous cases ij to lighten the notation. Given a subset Λ ⊂ V of the vertex set we write Λ V if it is finite and denote its boundary by ∂Λ, this is the set of vertices in Λ c directly connected to Λ through an edge: ∂Λ := i ∈ Λ c ∃j ∈ Λ, i ∼ j .(1) We sometimes abuse this notation and write ∂i for the neighbours of i ∈ V . A path between two vertices i, j ∈ V will be a finite collection of vertices (k 0 , . . . , k N ) such that k 0 = i, k N = j and k n ∼ k n+1 for all n = 0, . . . , N − 1. We will call this path non-repeating (or equivalently self-avoiding) if k n = k m for all n = m. If at least one such path exists between every two vertices of a subset Λ we will call Λ connected. In particular we consider trees, connected graphs where each vertex has a finite number of neighbours and where only one unique non-repeating path between two vertices i, j ∈ V exists. We denote this path by P(i, j), the length of the path defines a metric on the tree via d(i, j) := N if P(i, j) = (k 0 , . . . , k N ). A tree is called Cayley tree of order d if each vertex has exactly d + 1 ∈ N neighbours. For the Widom-Rowlinson model we introduce a copy of the spin space {−1, 0, 1} on each vertex. Combined they form the configuration space Ω := {−1, 0, 1} V which is endowed with the σ-algebra F := P({−1, 0, 1}) ⊗V . Let Λ ⊂ V be any set, we denote by Ω Λ the set of configurations ω Λ := (ω i ) i∈Λ restricted on Λ, and by σ Λ : Ω → Ω Λ the mapping with σ Λ (ω) = ω Λ for each ω ∈ Ω. We write ω A η B ∈ Ω A∪B for the concatenation of configurations on disjoint sets A, B ⊂ V . The σ-algebra on Ω generated by the projections to a subset Λ ⊂ V is denoted by F Λ . If a function f : Ω → R is F Λmeasurable for Λ V , f is called a local function. A function f is called quasilocal on Ω if there exists a sequence of local functions (f n ) n∈N with lim n→∞ f − f n ∞ = 0. Note that for finite state spaces quasilocality is equivalent to continuity with respect to the product topology. The soft-core Widom-Rowlinson model is a natural extension of the original Widom-Rowlinson model on graphs where neighbouring vertices are not allowed to take the same spin value. For our model this hard-core type restriction is weakened. Here, it is allowed that vertices with +1 and −1 spins are nearest neighbours, however this will be punished by a repulsion parameter β > 0 called the inverse temperature. The interaction of the model can be written as a potential Φ Λ (ω) =      β1 {ω i ω j =−1} if Λ = {i, j} ∈ E −hω i − log(λ)ω 2 i if Λ = {i} 0 else (2) where h ∈ R is an external magnetic field and λ > 0 serves as an activity parameter of occupied sites. Throughout this paper we consider the case of h = 0. To define Gibbs measures we need the notion of specifications. These are a families of probability kernels γ = (γ Λ ) Λ V from F Λ c to F respectively, which satisfy the properness condition γ Λ (A| · ) = 1 A ( · ) for all A ∈ F Λ c , and the consistency condition γ ∆ γ Λ = γ ∆ for all Λ ⊂ ∆ V . A specification is called quasilocal if for each Λ V and each quasilocal function f : Ω → R the function γ Λ (f | · ) := Ω γ Λ (dω| · )f (ω) (3) is quasilocal. We say a measure µ on (Ω, F) is a Gibbs measure, if it satisfies the Dobrushin-Lanford-Ruelle equations for a quasilocal specification, i.e. µ = µγ Λ (4) for each Λ V . This condition is equivalent to µ Λ (f | · ) := µ(f |F Λ c )( · ) = γ Λ (f | · ) µ-almost-surely for each measurable function f : Ω → R. Given a specification γ we write G(γ) for the set of all Gibbs measures admitted by this specification. As G(γ) is a simplex we are particularly interested in its extremal points, the extremal Gibbs measures. For the potential of the soft-core Widom-Rowlinson model we can define a specification through the probability kernels defined by γ Λ (σ Λ = ω Λ |η Λ c ) = 1 Z Λ (η Λ c ) exp − H Λ (ω Λ η Λ c ) ω, η ∈ Ω,(5) with the finite-volume Hamiltonian H Λ (ω) = A∩Λ =∅, A V Φ A (ω) for all Λ V . The function Z Λ is called partition function and is defined such that γ Λ ( · |η Λ c ) is a probability measure for each η in Ω. Tree-indexed Markov chains and boundary laws We briefly review the notion of tree-indexed Markov chains and Zachary's theorem stating a one-to-one correspondence between boundary laws and Markov chain Gibbs measures. We begin with the definition of a Markov specification Definition 2.1. A specification γ = (γ Λ ) Λ V is called a Markov specification if for each region Λ V and all fixed spin configurations ω Λ ∈ Ω Λ the specification density γ Λ (ω Λ | · ) is F ∂Λ -measurable. One can easily see that the specification associated to the Widom-Rowlinson model is Markovian. To define the stronger notion of tree-indexed Markov chains we need a concept of past. We define the set of vertices that lies in the past of an oriented edge ⇀ ij as those vertices whose unique path to i does not pass over the edge ⇀ ij and thereby does not contain j: (−∞, ⇀ ij) := k ∈ V j / ∈ P(i, k) .(6) This leads to the definition of a tree-indexed Markov chain. Definition 2.2. Let (V, E) be a tree. A measure µ is called a tree-indexed Markov chain if µ σ j = ω j F (−∞, ⇀ ij ) = µ σ j = ω j F i µ − a.s. ∀ω j ∈ Ω j ,(7) for each edge ⇀ ij ∈ ⇀ E . There are some connections between Gibbs measures and tree-indexed Markov chains. To explain these relations we need to introduce boundary laws and transfer operators for Markov chains. Q {i,j} (ω i , ω j ) := exp −Φ {i,j} (ω i , ω j ) − Φ {i} (ω i ) |∂i| − Φ {j} (ω j ) |∂j| . (8) A family of vectors (l ij ) ij∈ ⇀ E with each l ij in (0, ∞) Ω i is called a boundary law consistent with the transfer operators (Q {i,j} ) {i,j}∈E , if for each ⇀ ij ∈ ⇀ E there exists a positive constant c ij > 0 such that the consistency equation l ij (ω i ) = c ij k∈∂i\j ω k ∈Ω k Q {k,i} (ω k , ω i )l ki (ω k )(9) holds for every ω i ∈ Ω i . Note that boundary laws (l ij ) ij∈ ⇀ E are uniquely determined up to a constant only. Hence, one can choose one of the entries of the vectors l ij arbitrarily. In our case it is useful to set l ij (0) = 1 for every ⇀ ij ∈ ⇀ E . With the idea of transfer operators the specification of the Widom-Rowlinson can be rewritten as γ Λ (σ Λ = ω Λ |ω Λ c ) = 1 Z Λ (ω Λ c ) {i,j}∈E {i,j}∩Λ =∅ Q {i,j} (ω i , ω j )(10) which leads to the following theorem, firstly proven by Zachary: Theorem 2.4 (Zachary [30]). Let γ = (γ Λ ) Λ V be a Markov specification on (Ω, F) with transfer operators (Q {i,j} ) {i,j}∈E . Then each boundary law (l ij ) ij∈ ⇀ E consistent with these transfer operators defines a unique tree-indexed Markov chain µ ∈ G(γ) via the equation µ(σ Λ∪∂Λ = ω Λ∪∂Λ ) = 1 Z Λ {i,j}∈E {i,j}∩Λ =∅ Q {i,j} (ω i , ω j ) k∈∂Λ l kk Λ (ω k )(11) where Λ V is a connected set and Z Λ ∈ (0, ∞) a suitable normalizing constant. Here k Λ denotes the unique nearest neighbour vertex of k ∈ ∂Λ that lies inside of Λ. Conversely, every tree-indexed Markov chain µ ∈ G(γ) has the above representation through a boundary law (l ij ) ij∈ ⇀ E which is uniquely defined up to a positive factor. In [22] the authors find solutions of the above recursion problem (11) for the soft-core model depending on the values of β and λ. One of the results is that there always exists a solution with the property l ij (1) = l ij (−1) which via Theorem 2.4 defines a tree-indexed Markov chain and Gibbs measure µ # which we call intermediate measure. It is the only Gibbs measure if β is small enough. Time evolution, good and bad configurations As Gibbs measures describe the equilibrium states of physical systems at constant temperatures, to investigate the behaviour of systems for variable temperatures we need to introduce a suitable transformation. In our case we use a Markovian semigroup (π t ) t∈[0,∞) describing a site-independent spin-flip dynamics whose single-site marginals are given by p t (ω, η) = 1 2 (1 + e −2t )1 {ω=η =0} + 1 2 (1 − e −2t )1 {ωη=−1} + 1 {ω=η=0} ∀ω, η ∈ {−1, 0, 1}. (12) This time evolution acts like a heating of the system where the distribution and quantity of occupied sites does not change. For a given initial Gibbs measure µ on (Ω, F) at time t = 0, we then define the time-evolved measure as the action of this semigroup on the initial measure µ t := µπ t . The expectation of a local function f under the time-evolved measure is thus given by µ t (f ) = ω Λ ∈Ω Λ η Λ ∈Ω Λ f (η) i∈Λ p t (ω i , dη i )µ(dω)(13) where Λ is the support of f . We are interested in the properties of this time-evolved measure, specifically to what extent it still admits to a Gibbsian description. By definition µ t would be a Gibbs measure if it is compatible with a quasilocal non-null specification. To contradict this property it therefore suffices to show the existence of a non-removable point of discontinuity for the expected value µ t (f |F Λ c ) of one local function f , as this discontinuity would persist in each specification admitting the measure µ t , thereby showing the non-quasilocality of all compatible specifications. Such points of non-removable discontinuity will be called bad configurations, configurations which are not bad are called good configurations. A configuration η is a bad configuration if the measure is essentially-discontinuous at η (cp. [9]): Definition 2.5. The measure µ t is essentially-discontinuous at η ∈ Ω if there exists a local function f and a region Λ 0 V such that lim sup Λ V sup ξ 1 ,ξ 2 ∈Ω ∆:Λ⊂∆ V µ t (f |η Λ\Λ 0 ξ 1 ∆\Λ ) − µ t (f |η Λ\Λ 0 ξ 2 ∆\Λ ) > 0.(14) If η is an essential discontinuity in the sense of Definition 2.5, it must be a discontinuity point in the product topology for every specification γ for which µ is a compatible measure, which can be quickly seen as follows: Take an arbitrary compatible specification γ. Then, one may write the terms with finite-volume conditionings of the form µ t (f |η ∆\Λ 0 ) appearing in Definition 2.5 as an integral of γ Λ 0 (f |η ∆\Λ 0 ζ ∆ c ) over the variables ζ ∆ c with respect to some conditional measure which is not important for the argument. Using uniform upper and lower bounds on ζ ∆ c , this shows that the left hand side in (14) is a lower bound for the analogous expression involving the kernel γ Λ 0 , namely lim sup Λ V sup ξ 1 ,ξ 2 ∈Ω γ Λ 0 (f |η Λ\Λ 0 ξ 1 V \Λ ) − γ Λ 0 (f |η Λ\Λ 0 ξ 2 V \Λ ) ,(15) which therefore is strictly positive, too. Results: Badness Theorem 2.6. Let (V, E) be the Cayley tree of order d ≥ 4. Then there exist finite positive constants β b (d) > 0, λ b (d) > 0 such that for all β > β b (d) there exists a finite time t b (β, d) so that the set of bad configurations for µ # β,λ,t has full measure for all t ≥ t b (β, d) and all λ ≥ λ b (d). The proof of the theorem relies on a general zero-one law, Theorem 2.7, together with two ingredients about the set of a bad configurations which are interesting in themselves. A subtree condition for bad configurations, Theorem 2.8, and the typicality of this condition at large activities, Proposition 2.10. Zero-one law First we give a general zero-one theorem for the set of bad configurations on trees, for possibly non-extremal measures. Note that for extremal Gibbs measures on any countable graph, it is well known that the set of bad configurations has probability zero or one. This is clear, as the set of bad configurations form a tail-event, and extremality of a Gibbs measure implies its triviality on the tail-sigma algebra. In our case however, the intermediate measure under consideration is provably non-extremal in the interesting regime of large repulsion β, and large activity λ, as we show in Lemma 3.2, and so the following theorem is necessary: Theorem 2.7 (Zero-one law). Assume that µ is a tree-indexed Markov chain on the Cayley tree with state space {−1, 0, 1} V , which is invariant under tree-automorphisms, and whose transition matrix P has strictly positive matrix elements. Denote by µ t the corresponding time-evolved measure with starting measure µ, obtained under the spin-flip dynamics (12). Then, for each time t, the set of bad configurations B t for the time-evolved measure µ t satisfies the zero-one law µ t (B t ) ∈ {0, 1}. Subtree condition on badness The following theorem gives a sufficient condition for bad configurations of the timeevolved measure, uniformly in the choice of signs. s > d + 1 2 .(16) Then there exists a critical repulsion strength β c (d, s) ∈ (0, ∞) so that for all β > β c (d, s) there exists a time t c (β, d, s) ∈ (0, ∞) such that the time-evolved intermediate measure µ # β,λ,t is essentially-discontinuous at η for all times t ≥ t c (β, Subtree percolation When is the set of provably bad configurations from Theorem 2.8 typical, i.e. when is subtree-percolation ensured? The measure of occupied sites drawn from µ # β,λ,t turns out to be a tree-indexed Markov chain again, see Lemma 3.3, with an explicit transition matrix depending on β, λ but independent of t -which should not be expected from the other Gibbs measures. Hence the connected clusters of occupied sites of the intermediate measure (growing away from the origin, see proof of zero-one law) form Galton-Watson processes that do not depend on t. Let p s (β, λ, d) denote the time-independent probability that a fixed occupied site on the Cayley tree of order d, whose sites are occupied according to the time-evolved intermediate measure µ # β,λ,t , is the root of an outward growing occupied subtree where each vertex has at least s children. In [28] probabilities for occupied subtrees have already been studied, also [1] investigates the probability for the existence of so-called k-forts which implies the existence of an occupied subtree of order d − k. However, both of these more general works do not immediately give the bounds for our special case, thus we give a self-contained proof in Subsection 3.3. Remark 2.11. One may ask if the full-measure badness of Theorem 2.6 could persist on the ternary (d = 3) or even on a binary tree (d = 2), and try to improve the badness condition in Theorem 2.8. A more refined approach for the recursion might allow better estimates by using typical influences of spins on the occupied subtree and the non-percolating branches, instead of estimating uniformly by the worst possible cases (cp. Subsection 3.2). Furthermore we might obtain badness results for occupied subtrees closer to full occupation than the subtree of order d − 1, yet still typical in a region of very large activity, thereby improving the current proof of Theorem 2.6. This remains an open problem which needs a finer analysis. Results: Goodness Almost sure goodness for small density via extinction In contrast to Theorem 2.6 the set of bad configurations has zero measure, since it is empty, for small times and every activity. In the following theorem we handle the case for small λ. Here we prove only an almost-sure result, i.e. the set of bad configurations has zero measure. The idea to rewrite the model to use extinction probabilities for Galton-Watson trees. Proofs: Badness Renewal subtree construction and zero-one law for bad configurations We now give the proof of the zero-one law for bad configurations. Figure 1: The vertices of a tree can be enumerated along a spiral emanating from an arbitrary root index. The occupied vertices, here coloured in grey and black for plus and minus spins, can be seen as active sites growing an outward pointing (away from the root) tree with stopping symbol 0 provided they are not the child of another occupied vertex. Enumerating these active sites along the spiral we obtain an almost surely infinite sequence of active sites (vi) i∈N . The partial trees growing from these active sites are i.i.d.. connected components of occupied sites, along with their signs, anchored at the site on the cluster closest to an origin. Using only the tree-indexed Markov chain property of µ, the clusters will be grown by means of a recursive algorithm described below. Here the anchoring sites of the clusters will appear as so-called active sites. They will be determined depending on the clusters which have grown in the previous steps. These anchored clusters can be viewed as geometric generalizations to the excursions of a stationary Markov chain. More precisely, we start by enumerating the vertices of the tree by N ∪ {0}: Choose an arbitrary root of the vertex set of the tree and label it by 0. Then label the d + 1 sites at distance 1 to the root by the integers 1, . . . , d + 1, in otherwise arbitrary order. Next label the sites at distance 2 to the root by the (d + 1)d next integers, in otherwise arbitrary order. Next label the sites at distance 3 to the root by the next integers, in otherwise arbitrary order. Proceed in this way for all finite distances. Spiralling renewal We construct a configuration of µ according to the following spiralling renewal-construction, based on the tree-indexed Markov chain property: Step 0 -Initialisation: Choose the value σ 0 at the root according to the single-site distribution ρ, which is the invariant distribution for the transition matrix P of the Markov-chain Gibbs measure, i.e. it satisfies ρ = ρP . Note that our assumption on P implies strict positivity of the entries of ρ. In the case σ 0 = s where s ∈ {−1, 1}, choose the active site as v = 0 and turn to the next step, Step 1. In the case σ 0 = 0 choose the value of σ 1 = s 1 with probability P (0, s 1 ) > 0. In the case σ 1 = 0, call 1 the active site and turn to the next step. Otherwise carry on this procedure, i.e. continue according to the enumeration of sites in this way until the first vertex v is reached for which σ v = 0. Call this vertex v the active site for Step 1. Step 1 -Grow partial trees from an active site v, with stopping symbol 0: In the case σ v = s where s ∈ {−1, 1}, grow random partial configurations σ s Vv on the random subset V v , which contains v and points to the outside, in the following way. Here V v by definition should contain the active site v as its smallest site according to the enumeration. Apply the transition matrix P , from inside to outside, away from the root, starting with initial condition σ v = s. Stop to grow the branches to the outside when a first zero appears in the branch, and keep these first zeros together with the configurations of zeros and signs obtained so far. This has filled a (finite or infinite) part V v of the tree emerging from site v, where the value 0 plays the role of a stopping symbol which marks the boundary of the connected component. We note that for v = 0 the distribution of σ s Vv is tree-invariant, and describes the connected component attached at the site v. For the particular case v = 0, maximally d + 1 such components come together to form the component at the origin. Step 2 -Filling more zeros to get to the next active site: Determine the smallest site z according to the spiralling enumeration whose spin value has not yet been determined. By construction the spin value of the parent site of z on the tree was already determined to be zero. Choose the value σ z = s z with probability according to P (0, s z ). In the case σ z = 0 repeat Step 2. In the case σ z = s for s ∈ {−1, 1} go to Step 1 with new active site z. This procedure produces a sequence of active sites (v i ) i∈N , as shown in Figure 1, along with their signs s i . As the transition matrix P has strictly positive matrix elements, this sequence is almost surely infinite. We then obtain a configuration σ V on the full vertex set of the tree which is distributed according to µ as the concatenation of (σ s i Vv i ) i∈N with the empty configuration 0 on the remaining sites ( i∈N V v i ) c . We note that (up to the component of the root) the components σ s i Vv i are (tree-isomorphic to) i.i.d. random objects. Now we come to the proof of the zero-one property of bad configurations. A configuration σ V on the full tree is a bad configuration if and only if there is at least one connected component of its occupied sites which acts as a bad configuration (when the configuration is continued by zero outside). This follows as the first-layer model decouples over the connected components of occupied sites in the conditioning. We can therefore check badness of the time-evolved measure on the connected components grown from v i = 0 with sign s ∈ {−1, 1}. We note for this purpose that the probabilities p bad (s) :=μ(σ s Vv i 0 (Vv i ) c is bad at time t | s i = s)(18) do not depend on i for i ≥ 2, where the measureμ is obtained as the representation of the measure µ via the above renewal construction, together with the semi-group of the time evolution. This is clear by the invariance of the construction of V v i , and the tree-automorphism invariance of the property to be a bad configuration. Case 1: p bad (s ) > 0 for at least one spin value s ∈ {−1, 1}. As the sequence of active sites takes the value v i = s for infinitely many i ≥ 2 with probability one, we conclude that µ-almost surely there are even infinitely many connected components labelled by i ≥ 2 which are bad. This follows by the Borel-Cantelli Lemma applied to the situation of independently many trials with positive probability. In particular we have µ t (B t ) = 1 for the set of bad configurations B t . Case 2: p bad (1) = p bad (−1) = 0. Then all the components for i ≥ 2 carry good configurations almost surely. We remark that also for the first component we have thatμ(σ s Vv 1 0 (Vv 1 ) c is bad at time t|s 1 = s) = 0, for both values of s, if all connected components away from the origin are good. To see this, condition on the event that v 1 = 0 is in fact the origin, and employ a small Gibbsian computation which shows that glueing of finitely many good configurations on connected components preserves the property to be a good configuration. Hence it follows µ t (B t ) = 0. Subtree condition on badness First, we introduce a sufficient criterion for essential discontinuity of the time-evolved intermediate measure. We treat the time evolution kernels as additional fields on the first-layer model, the percolation of information in the time-evolved model can then be expressed as a recursion depending on these fields. Lemma 3.1. The measure µ # β,λ,t is essentially-discontinuous at configuration η ∈ Ω if there exists a vertex 0 ∈ V , an > 0, and cofinal sequences ( Λ n ) n∈N V , (∆ n ) n∈N V with 0 ∈ Λ n ⊂ ∆ n V such that lim n→∞ f k0 [η Λn\0 + ∆n\Λn ] − f k0 [η Λn\0 − ∆n\Λn ] ≥ ,(19) for at least one occupied vertex k ∈ ∂0. The f ij [ξ A\0 ] are boundary fields depending on the configuration ξ in A \ 0 calculated for edges pointing towards 0 through the recursion (22) and homogeneous starting values f ij [ξ A\0 ] = 0 at the boundary of A, i.e. for all edges ⇀ ij pointing towards 0 with i ∈ ∂A, independent of the configuration ξ. f ij [ξ A\0 ] = k∈∂i\j |ξ k |=1 ϕ β/2 (f ki [ξ A\0 ] + h t ξ k ) (20) with ϕ β (x) := 1 2 log cosh (x + β) cosh (x − β) (21) h t := 1 2 log 1 + e −2t 1 − e −2t Using the definition of essential discontinuity (14), this lemma will be proved by representing the single-site probabilities of the time-evolved measure µ # β,λ,t conditioned on a finite neighbourhood as a sum over compatible first-layer configurations, i.e. those configurations at time t = 0 which could eventually evolve to the second-layer configuration in the conditioning. These first-layer configurations have a closed representation through boundary laws, as the initial measure µ # β,λ,0 at time t = 0 is a tree-indexed Markov chain. Each summand is weighted by a modified Hamiltonian that includes field-like terms originating from the time evolution. Executing the sum then leads to the recursion relation above and a representation of the conditioned single-site probabilities through a first-layer Hamiltonian with the additional boundary fields that globally depend on the second-layer configuration. Therefore a discontinuity in these fields translates to essential discontinuity of the measure. Proof of Lemma 3.1. First note that we can naturally write the conditional probability of the time-evolved intermediate measure µ # β,λ,t at a fixed but arbitrary root index 0 conditioned on a second-layer spin configuration η in a finite neighbourhood Λ \ 0 as µ # β,λ,t (σ 0 = η 0 |η Λ\0 ) = μ # β,λ,t [η Λ\0 ](dω 0 )p t (ω 0 , η 0 ),(23) whereμ # β,λ,t is the probability of the first-layer spin value at 0 conditioned on the given second-layer configuration η, which by (13) has the representation µ # β,λ,t [η Λ\0 ](ω 0 ) := 1 Z β,λ,t [η Λ\0 ] ω Λ\0 ∈Ω Λ\0 µ # β,λ,0 σ Λ = ω 0 ω Λ\0 i∈Λ\0 p t (ω i , η i ).(24) Here µ # β,λ,0 denotes the intermediate measure at time t = 0 and Z β,λ,t [η Λ\0 ] is a suitable normalisation -we will use this notation for normalisations of different expressions without further apology. Since p t interpreted as a matrix is bijective, from Equation (23) follows, that we can infer essential discontinuity of µ # β,λ,t by proving that the family of measures μ # β,λ,t [η Λ\0 ] η∈Ω,Λ Ω fulfils the condition lim n→∞ μ # β,λ,t [η Λn\0 + ∆n\Λn ](ω 0 ) −μ # β,λ,t [η Λn\0 − ∆n\Λn ](ω 0 ) ≥(25) for a fixed > 0 and cofinal sequences (Λ n ) n∈N V , (∆ n ) n∈N V with 0 ∈ Λ n ⊂ ∆ n V and some ω 0 ∈ S. Here we have chosen 1 {ω 0 } as local function and +, − denote the fixed configurations that are plus and minus everywhere on the whole tree respectively. To rewrite µ # β,λ,t in terms of boundary fields we first note that the single-site time evolution (12) can be rewritten in an exponential form p t (ω i , η i ) = c t (ω i , η i ) exp h t ω i η i(26) using h t defined in (22) and c t (ω i , η i ) =      1 2 1 − e −4t 1 2 if |ω i | = |η i | = 1 1 if |ω i | = |η i | = 0 0 else .(27) This time evolution prohibits first-layer configurations in the sum of Equation (24) whose set of occupied sites differs from that of the prescribed second-layer configuration η Λ\0 . We therefore introduce for each second-layer configuration η the space of its compatible configurations in the first-layer Ω η by Ω η := ω ∈ Ω |ω i | = |η i | ∀i ∈ V(28) and can restrict the sum to finite-volume configurations of Ω η . Replacing the single-site kernels through their exponential notation then leads tô µ # β,λ,t [η Λ\0 ](ω 0 ) = 1 Z β,λ,t [η Λ\0 ] ω Λ\0 ∈Ω η Λ\0 µ # β,λ,0 σ Λ = ω 0 ω Λ\0 i∈Λ\0 exp h t ω i η i .(29) Here we were allowed to include the factors c t of the time evolution in the normalisation as they are identical for all compatible configurations in Ω η Λ\0 . Since the initial measure µ # β,λ,0 is a tree-indexed Markov chain, it has a representation through boundary laws (cp. Theorem 2.4) µ # β,λ,0 (σ Λ∪∂Λ = ω Λ∪∂Λ ) = 1 Z Λ;β,λ {i,j}∈E {i,j}∩Λ =∅ Q {i,j} (ω i , ω j ) k∈∂Λ l kk Λ (ω k ).(30) Replacing µ # β,λ,0 in (29) through its boundary law representation (30) yields an expression that just consists of functions for spin-values of at most two neighbouring vertices. As trees do not contain loops this means that the sum over configurations on Λ can be split into multiple sums over configurations on disjoint connected components of Λ, which just communicate through one unique path. If this path contains a vertex with a fixed spin-value, as is the case for every compatible first-layer configuration if the second-layer configuration η has spin-value zero at one of the paths sites, these components are fully independent. In particular this implies that Equation (29) only depends on spin-values of sites which have a direct connection to the root through a path that is occupied in the second-layer configuration η. We therefore define the set of vertices connected to the root with regard to occupation in η C η (A) := i ∈ A |η j | = 1 ∀j ∈ P(0, i) ,(31) and can restrict Equation (29) to configurations on this connected component µ # β,λ,t [η Λ\0 ](ω 0 ) = 1 Z β,λ,t [η Λ\0 ] ω C η (Λ\0) ∈Ω C η (Λ\0) {i,j}∈E {i,j}⊂C η (Λ) Q {i,j} (ω i , ω j ) × k∈C η (∂Λ) l kk Λ (ω k ) l∈C η (Λ\0) exp h t ω l η l .(32) Except for the root, all sites in Equation (32) have spins that are guaranteed to be non-zero. As the boundary laws of the intermediate measure µ # β,λ,0 are identical for occupied vertices they can be included in the normalisation resulting in µ # β,λ,t [η Λ\0 ](ω 0 ) = 1 Z β,λ,t [η Λ\0 ] × ω C η (Λ\0) ∈{−1,1} C η (Λ\0) {i,j}∈E {i,j}⊂C η (Λ) Q {i,j} (ω i , ω j ) exp k∈C η (Λ\0) h t η k ω k .(33) Executing the summation over the first-layer spin values in (33) can be done successively for each site, beginning at the boundary and working inwards to the root. By defining boundary fields via the recursion f ij [η Λ\0 ] := k∈∂i\j |η k |=1 ϕ β/2 (f ki [η Λ\0 ] + h t η k ),(34) where ϕ β has already been defined in (21), with homogeneous starting values f ij [η Λ\0 ] = 0 for all i ∈ ∂Λ, we eventually get the representation µ # β,λ,t [η Λ\0 ](ω 0 ) = λ |ω 0 | Z β,λ,t [η Λ\0 ] × ω C η (∂0) ∈{−1,1} C η (∂0) exp k∈∂0 |η k |=1 β1 {ω k ω 0 =−1} + h t η k ω k + f k0 [η Λ\0 ]ω k .(35) We note that the activity λ plays no part in the recursion process, as all spins at vertices on C η (Λ \ 0) are occupied, which allows us to incorporate the λ-dependent parts of the transfer operators in the normalisation. The only components of Equation (35) which are dependent on the global behaviour of η are the boundary fields f k0 [η Λ\0 ] calculated through the η-dependent recursion (34). To show that Inequality (25) holds, it suffices to look at the difference between the boundary fields for both expressions and show lim n→∞ k∈∂0 |η k |=1 f k0 [η Λn\0 + ∆n\Λn ] − k∈∂0 |η k |=1 f k0 [η Λn\0 − ∆n\Λn ] ≥(36) for some > 0. This can be seen by an additional recursion step which results in µ # β,λ,t [η Λ\0 ](ω 0 ) = λ |ω 0 | Z β,λ,t [η Λ\0 ] exp ω 0 |C η (∂0)| k∈∂0 |η k |=1 f 0k [η Λ\0 ](37) with new boundary fields f 0k pointing away from the origin that due to the strict monotonicity of (34) retain condition (36). The benefit of this representation lies in the fact that the influence of the boundary fields is more immediate. To see that jumps in the boundary fields carry over to jumps in the probabilities compare now the fraction µ # β,λ,t [η Λn\0 + ∆n\Λn ](+1) µ # β,λ,t [η Λn\0 + ∆n\Λn ](−1)(38) to the corresponding fraction for η Λn\0 − ∆n\Λn . As the recursion (34) also preserves monotonicity in the configurations we get f ij [η Λn\0 + ∆n\Λn ] ≥ f ij [η Λn\0 − ∆n\Λn ] for each edge ⇀ ij ∈ ⇀ E . To show (36) it is therefore sufficient to find just a single occupied vertex k ∈ ∂0 such that f k0 [η Λn\0 + ∆n\Λn ] > f k0 [η Λn\0 − ∆n\Λn ] . This concludes the proof of the lemma. Having developed the essential discontinuity criterion above, let us now show that it is fulfilled for large repulsion β at large times, if the set of occupied sites O(η) = {i ∈ V | |η i | = 1} of the configuration η contains a rooted subtree S with s children where s satisfies s > d + 1 2 .(39) The main idea of the following proof is depicted in Figure 2. Proof of Theorem 2.8. Using Lemma 3.1 we choose 0 to be the root of the occupied subtree S and take Λ n = D n , ∆ n = D n+m , where D k := {i ∈ V |d(i, 0) ≤ k} is the disc with radius k around 0. The value of m ∈ N will be chosen later in the proof. Using the recursion (20) we can then proceed to calculate the boundary fields on the annuli R k := ∂D k−1 for k ∈ {1, . . . , n + m + 1} (beginning at R n+m+1 working inwards toward ∂0 = R 1 ). We will do so for the plus configuration on ∆\Λ, the proof for the minus condition works analogously. Since + ∆\Λ is homogeneous, the boundary fields on each ring R k ⊂ ∆ \ Λ are also homogeneous. We denote these homogeneous values by F k and it holds From the boundary field recursion (20) we get f ij [η Λ\0 + ∆\Λ ] = F k ∀ ⇀ ij with i ∈ R k , j ∈ R k−1 .(40)F k = dϕ β/2 (F k+1 + h t ) ∀k ∈ {n + 1, . . . , n + m}(41) with starting value F n+m+1 = 0. The function x → dϕ β/2 (x + h t ),(42) always has exactly one positive attractive fixed point F > 0 and we can ensure that F n+1 is arbitrarily close to F by choosing an appropriately large m ∈ N. F n+1 then provides the initial condition for the recursion on Λ. As the recursion on Λ is η-dependent, we will estimate a lower bound for the boundary fields on each ring R k ⊂ Λ for the edges that are part of the subtree S F k := min ij∈ ⇀ E , i,j∈S i∈R k , j∈R k−1 f ij [η Λ\0 + ∆\Λ ] ∀k ∈ {1, . . . , n},(43) and show that this lower bound stays positive up to the root vertex 0. Applying the boundary field recursion (20) to the f ij in F k gives F k = min ij∈ ⇀ E , i,j∈S i∈R k , j∈R k−1 l∈∂i\j |η l |=1 ϕ β/2 f li [η Λ\0 + ∆\Λ ] + h t η l .(44) We can split the sum in terms where the vertex l ∈ R k+1 is part of the subtree S and those where l is not contained in S. The boundary fields of the former terms, of which there are at least s, can be lower-bounded by F k+1 . Estimating η l ∈ {−1, 1} by −1 and using the monotonicity of ϕ then gives a lower bound for the terms on the subtree. The terms for l / ∈ S, of which there are at most (d − s) can be lower-bounded by −β/2 as |ϕ β/2 | is bounded by β/2. Thus we get the estimation F k ≥ min ij∈ ⇀ E , i,j∈S i∈R k , j∈R k−1 l∈∂i\j |η l |=1, l∈S ϕ β/2 F k+1 − h t − l∈∂i\j |η l |=1, l / ∈S β 2 ≥sϕ β/2 F k+1 − h t − (d − s) β 2 ,(45) which holds for all k ∈ {1, . . . , n}. A positive fixed point solution F + > 0 of the function x → sϕ β/2 (x − h t ) − (d − s) β 2 ,(46) would have to be smaller than the fixed point F > 0 of the outer recursion, which can be seen through direct comparison of the functions (42) and (46). Provided such a solution exists, we could fix m to be large enough such that F + < F n+1 . This would inductively imply for all k ∈ {1, . . . , n}. Here we used the monotonicity of ϕ β/2 in the second step. In particular we would get f j0 [η Λ\0 + ∆\Λ ] ≥ F + > 0 for all j ∈ ∂0 ∩ S and arbitrarily large n ∈ N. It remains to show the existence of the fixed point F + > 0. From the analysis of the Ising model on Cayley trees in [13] we know that the function F k (45) ≥ sϕ β/2 (F k+1 − h t ) − (d − s) β 2 ≥ sϕ β/2 (F + − h t ) − (d − s) β 2 = F + > 0 (47) -2 2 -4 -2 2 4 sϕ β/2 (x) − (d − s) β 2 sϕ βc/2 (x) − (d − s) βc 2 F + β F + βc (s − d 2 )β (s − d 2 )βc -2 2 -4 -2 2 4 sϕ β/2 (x − h t ) − (d − s) β 2 sϕ β/2 (x − h tc ) − (d − s) β 2 F + t F + tc (s − d 2 )β −(d − s) β 2 h t h tcx → sϕ β/2 (x − h)(48) has a positive fixed point if β > β (s) and h ≤ h (β, s) where h (β, s) = s − 1 2 β + s − 1 2 log(s − 1) − s 2 log(s) + O β (1).(49) Therefore (46) has a positive solution for parameters β > β (s) and t > 0 that fulfil the condition h t + (d − s) β 2 ≤ h (β, s).(50) For s > (d + 1)/2 from (49) it follows that there exists a finite β c (d, s) > β (s) such that (50) is a strict inequality for the limiting case h t = 0 for each β > β c (d, s), as depicted in the left part of Figure 3. Choosing t c (β, d, s) subsequently such that (50) is an equality for h tc guarantees that it is fulfilled for all t ≥ t c (β, d, s), as h t is a decreasing function in t. Therefore a positive fixed point F + > 0 exists for all β > β c (d, s) and t ≥ t c (β, d, s). An analogous argument shows that in case of the minus configuration on ∆ \ Λ the boundary fields can be upper bounded by f j0 [η Λ\0 − ∆\Λ ] ≤ F − = −F + < 0 for all j ∈ ∂0 ∩ S and all n ∈ N for identical critical values. Setting = 2F + concludes the proof. Subtree percolation In this section we show that a fixed occupied site of the Cayley tree of order d whose spins are distributed by the time-evolved intermediate measure has a positive probability of growing an occupied rooted subtree with s or more children for each s ≤ d − 1 at any repulsion strength β > 0, if the activity is greater than a critical value λ b (d). This leads to the typicality of bad configurations in the regime of Theorem 2.8 for large activities. Note that the spin-flip dynamics (12) Proof. This result follows from the Kesten-Stigum criterion [21] as the in modulus second largest eigenvalue u 2 of the transition matrix for the intermediate measure fulfils the condition u 2 > 1 √ d(51) for large repulsion β and large activity λ if d ≥ 2 as we will show. To calculate the transition matrix, we use Zachary's theorem (Theorem 2.4) and by applying the consistency condition for boundary laws (9) we get an expression for the spin distribution of two vertices along an arbitrary edge {i, j} ∈ E: µ # β,λ (σ i = x, σ j = y) = 1 Z l β,λ (x)Q {i,j} (x, y)l β,λ (y) ∀x, y ∈ {−1, 0, 1}.(52) Here the transfer operator is defined by the Widom-Rowlinson potential through Equation (8). The boundary law for the intermediate measure is the homogeneous solution of the boundary law recursion (9) with l β,λ (−1) = l β,λ (+1) and has the representation l β,λ (−1), l β,λ (0), l β,λ (1) = ξ β,λ λ − 1 d+1 , 1, ξ β,λ λ − 1 d+1(53) where ξ β,λ is the unique positive solution to the equation x = λ 1 + (1 + e −β )x 1 + 2x d ,(54) see [22]. Note, that while the exact value of ξ β,λ is dependent on β it is always contained in the interval (2 −d λ, λ) and can therefore be controlled by the activity λ. Using the distribution of neighbouring spins (52) with an appropriate normalisation the homogeneous matrix of transition probabilities for the intermediate measure µ # β,λ takes the form P # β,λ = 1 1 + (1 + e −β )ξ β,λ    ξ β,λ 1 e −β ξ β,λ α β,λ ξ β,λ α β,λ α β,λ ξ β,λ e −β ξ β,λ 1 ξ β,λ   (55) where α β,λ : = 1 + (1 + e −β )ξ β,λ 1 + 2ξ β,λ .(56) The eigenvalues of P # β,λ ordered by absolute value are: u 1 = 1 u 2 = (1 − e −β )ξ β,λ 1 + (1 + e −β )ξ β,λ u 3 = − u 2 1 + 2ξ β,λ(57) As ξ β,λ can be solely controlled by λ we get lim λ→∞ u 2 = 1 − e −β 1 + e −β = tanh(β/2) ∀β > 0.(58) Therefore for any repulsion strength β with tanh(β/2) > 1 √ d(59) and sufficently large λ the Kesten-Stigum criterion is fulfilled and the measure is nonextremal. The criterion (59) yields the critical value at which the intermediate measure for the Ising model on the Cayley tree of order d with parameter β/2 transitions from extremal to non-extremal, for more information on this transition of the intermediate measure for the Ising model see [3,12,19,29]. This is consistent with the observation that the Widom-Rowlinson model of repulsion strength β conditioned on full occupation yields the Ising model with parameter β/2. Next we briefly look at the occupation measure describing the distribution of occupied sites. For the intermediate measure µ # β,λ this measure proves to be a tree-indexed Markov chain: We define the mapping τ : Ω → {0, 1} V with τ (ω) := (|ω i |) i∈V . Proof. For any S ⊂ V we define F oc S := σ(|σ i |, i ∈ S) the σ-algebra generated by the occupation numbers in S. Using first the tower property for conditional probabilities and then the Markov-chain property of the intermediate measure we get µ # β,λ (|σ j | = x|F oc (−∞,ij) ) = µ # β,λ µ # β,λ (|σ j | = x|F (−∞,ij) ) F oc (−∞,ij) = µ # β,λ µ # β,λ (|σ j | = x|F i ) F oc (−∞,ij) .(60) Proof of Theorem 2.6 Combining all previous results we get a proof for the main result: Proof. First, by Theorem 2.8, for d ≥ 4 with the choice s = d − 1 we get finite critical values β b (d) > 0, t b (β, d) > 0 such that all configurations containing an occupied rooted subtree of order d − 1 are bad for β > β b (d) and all times t ≥ t b (β, d). By Proposition 2.10 there exists a critical activity λ b (d) such that these bad configurations have positive probability for all λ ≥ λ b (d). Therefore the set of all bad configurations has positive probability and using the zero-one law 2.7 we see that the set of bad configurations is an almost sure event. Proofs: Goodness To prove almost-sure Gibbsianness we will use the transition probability u β,λ := µ # β,λ (|σ y | = 1| |σ x | = 1). and Galton-Watson trees. A Galton-Watson tree is a random rooted tree constructed in the following way. Starting at the root one chooses a random number of children according to a known distribution. Then for every child one independently chooses according to the same distribution again the number of children. This procedure will be repeated for every new child. Proof of Theorem 2.14. It is known that a Galton-Watson tree has almost surely no infinite connected component if the expected number of children is smaller than one. In our case the offspring distribution is given by a binomial random variable with probability of success less than u β,λ . If u β,λ < 1/d there almost surely cannot exist an infinite connected component of occupied sites inside the tree. Since lim λ→0 sup β>0 u β,λ = 0 by Lemma 3.4 there exists a critical λ g (β, d) ∈ (0, ∞) such that u β,λ < 1/d for all λ < λ g (β, d). Only configurations with an infinite cluster of occupied sites can be bad which implies that for all λ < λ g (β, d) the measure µ # β,λ,t is almost surely Gibbs. Proof of Theorem 2.12. In [24] the authors have investigated the soft-core Widom-Rowlinson models on the lattice V = Z d . They used Dobrushin-uniqueness theory to show that the time-evolved measure is Gibbs for small times. A quasilocal specification γ satisfies the Dobrushin condition if sup i∈V j∈V sup η,ζ∈Ω: η V \j =ζ V \j d T V,i (γ {i} ( · |η V \{i} ), γ {i} ( · |ζ V \{i} )) < 1(76) where d T V,i is the total variation distance for measures on (Ω {i} , F {i} ). In other words the Dobrushin condition is satisfied if changing only one site in the boundary condition has not a big effect. This condition can be used to prove that there exists a unique Gibbs measure for the specification. Furthermore, one can use the Dobrushin comparison Theorem which is an important ingredient to prove short-time Gibbsianness. The Theorem 2.8 in [24] states that if the absolute value of the external magnetic field h and λ are big enough the soft-core Widom-Rowlinson satisfies the Dobrushin condition. This not only shown for the lattice but also for general locally finite graphs. As Cayley trees are in fact locally finite graphs one can repeat the steps in [24] to prove that the time-evolved measure is Gibbs. Moreover, this result does not need that the starting measure is µ # β,λ . It holds for every starting Gibbs measure for the soft-core Widom-Rowlinson model. Remark 4.1. In [24] it is proven that the soft-core Widom-Rowlinson model satisfies the Dobrushin condition if β(d + 1) < 2. Hence for small enough β the time-evolved model is Gibbs for all times t>0. Definition 2. 3 . 3We define the family of transfer operators (Q {i,j} ) {i,j}∈E of the Widom-Rowlinson model for each {i, j} ∈ E by Theorem 2. 8 . 8Let (V, E) be the Cayley tree of order d and η ∈ Ω any configuration such that the set of occupied sites O(η) := {i ∈ V | |η i | = 1} contains a rooted tree of order s, where s satisfies Proposition 2. 10 . 10For d ≥ 2 there exists a critical activity λ b (d) ∈ (0, ∞) such that p s (β, λ, d) > 0(17)holds for all λ ≥ λ b (d) uniformly for all β > 0 and any s ≤ d − 1. Theorem 2. 14 . 14Let β > 0. Then there exists λ g (β, d) ∈ (0, ∞) such that for every λ < λ g (β, d) the time-evolved measure µ # β,λ,t is almost surely Gibbs for every time t > 0.Remark 2.15. From Theorems 2.6 and 2.14 we get an λ-dependent transition for the set of bad configurations of the time-evolved intermediate measure on Cayley trees of order d ≥ 4 from measure zero to measure one. For β > β b (d) and t ≥ t b (β, d) the set of bad configurations has measure zero for activities λ < λ g (β, d) and measure one for large activities λ ≥ λ b (d). Proof of Theorem 2. 7 . 7The main idea is to describe a configuration drawn from the tree-indexed Markov chain µ in terms of i.i.d. building blocks which are given by the Figure 2 : 2Simulation of the boundary field recursion on the Cayley tree of order d = 4 at repulsion strength β = 2.0 and time t = 0.2. Vertices are coloured according to their second-layer spin values, here the configuration on Λ is such that the centre vertex is the root of an occupied subtree with s = 3 children. Each inward pointing edge ij is coloured according to the value of the boundary field fij plus the time-dependent field h t ηi. The boundary field value at ∂Λ is the fixed point solution F > 0 of the recursion for an all plus configuration on ∆ \ Λ. The boundary field values on the inner rings are subsequently calculated from the recursion (20) with boundary fields from unoccupied sites set to zero as they do not influence the recursion. The positive starting value, combined with a large number of occupied spins due to the subtree structure, guarantees that the boundary field values along the edges of the subtree stay positive until they reach the centre vertex. This is true even for configurations where the subtree is completely occupied by minus spins. For the Ising-like case where a sub-Cayley-tree is occupied, s = 2 children suffice for this result, regardless of the size of the main tree. In the Widom-Rowlinson model however, we might get spin clusters outside of this infinite subtree that are disconnected from the positive influences at ∂Λ, like those depicted in the lower half, which can develop negative boundary field values. These need to be compensated through an in comparison sufficiently large structure of occupied spins percolating the positive boundary fields, leading to condition (39) for subtree percolation. Figure 3 : 3Solutions for the recursion (46) at β = 1.1 s = 7, d = 8 and t = 0.5. The left side depicts the limiting case h t = 0. Increasing the repulsion β lowers the interception point of the function with the y-axis while increasing its slope and maximal value. If s fulfils the condition (39) for fixed d there exists a critical value βc(d, s) such that for all β > βc(d, s) there are exactly two positive fixed points. In particular this allows a positive fixed point to exist for small h t . As h t is decreasing in t there exists a lower bound tc(β, d, s) for the time such that at least one positive fixed point exists for all t ≥ tc(β, d, s) and thereby h t ≤ h tc . Lemma 3. 3 . 3The occupation measure for the intermediate measure defined by µ # β,λ • τ −1 on ({0, 1} V , P({0, 1}) ⊗V ) is a tree-indexed Markov chain. d, s) and all activities λ > 0. Remark 2.9. Applying Theorem 2.8 for d = s ≥ 2 we immediately obtain that the time-evolved intermediate measure is non-Gibbs at large β for all sufficiently large times for any d ≥ 2. This is clear as we are provided with the bad configurations constructed from s-subtrees (which in general may however have zero measure). General result: short-time goodness via DobrushinAs we have seen the time-evolved measure µ # β,λ,t is not Gibbs for large enough times t and activities λ. However, there are regimes of parameters where µ # β,λ,t is Gibbs or at least almost surely Gibbs. First we state that the intermediate dynamical measure satisfies the so-called short-time Gibbs property, i.e. µ # β,λ,t is Gibbs for small times t.Theorem 2.12. For every β > 0 and λ > 0 there exists a time t g (β, λ, d) ∈ (0, ∞] such that for all t < t g (β, λ, d) the time-evolved measure µ # β,λ,t is Gibbs. Remark 2.13. By this theorem and Remark 2.9 we found a Gibbs-non-Gibbs transition for the time-evolved intermediate measure on the Cayley tree of order d at large repulsion strength β. For all activities λ the measure is Gibbs for small times and non-Gibbs for large times. does not change the distribution of occupied sites, therefore it suffices to investigate the intermediate measure without time-evolution. First we present a result regarding the non-extremality of this intermediate measure, which shows the necessity of the general zero-one law 2.7 for possibly non-extremal measures for our results.Proposition 3.2. For d ≥ 2, large repulsion β and large activity λ the intermediate measure µ # β,λ is non-extremal. Due to the symmetries of the transition matrix (55) of the intermediate measure we havefor each edge {i, j} ∈ E, which shows that µ # β,λ (|σ j | = x|F i ) is F oc i -measurable. Therefore the last line of (60) is F oc i -measurable which yieldsThis holds for arbitrary x ∈ {0, 1} and all edges {i, j} ∈ E, proving the Markov chain property for the occupation measure.From the explicit form of the transition matrix we can estimate the transition probabilities of the occupation measure for large and small activities respectively. Lemma 3.4. The transition probability between neighbouring occupied sites of the µ # β,λmeasure can be controlled by the activity λ, to be more precisefor all edges {i, j} ∈ E and for arbitrary fixed d ∈ N.Proof. The transition probabilities are given by the transition matrix P # β,λ (55) from the proof of Lemma 3.2. Due to the matrix symmetry we getAlso from the previous proof we know that ξ β,λ is bounded by (2 −d λ, λ), therefore we can estimate the transition probability in the following way:Taking the limit for large and small activities respectively completes the proof.Finally, using the estimations for the transition probabilities in Lemma 3.4 we can prove that an active site in the sense of the zero-one law 2.7 creates an occupied rooted tree of sufficient size with positive probability.Proof of Proposition 2.10. Let p s (β, λ, d) denote the probability that a fixed occupied site on the Cayley tree of order d whose sites are occupied according to the time-evolved intermediate measure µ # β,λ,t is the root of an occupied subtree where each vertex has at least s children. Note that the distribution of occupied sites does not depend on t, as the spin-flip dynamics(12)does not affect the positions of occupied sites. It is therefore β,λ . We show the existence of a critical activityDenote by p (n) s (β, λ, d) the probability that an occupied site on the Cayley tree of order d is the root of a finite subtree of n generations where each vertex has at least s occupied children, then we get limThe root site is chosen to be occupied, therefore we have p (0) s (β, λ, d) = 1 and as the growing of the tree follows a binomial distribution we get the β-independent inequalitywith u λ := inf β>0 µ oc β,λ (|σ y | = 1| |σ x | = 1). It suffices to show the positivity of p d−1 (β, λ, d), as a subtree of size d − 1 already contains an s subtree for every s ≤ d − 1.A fixed point p λ ∈ (0, 1) for the mappingwould be a lower bound for pfor all β > 0 and all n ∈ N. We show that such a fixed point exists for large activities λ:The functionis continuous and takes the values g(0) = 0, g(1) = 1 as well as g (0) = g (1) = 0 for each d ≥ 2. By the intermediate value theorem the set of fixed points of g on (0, 1) is non-empty and as g is a polynomial it is finite. Choosing x c ∈ (0, 1) to be the largest of these fixed points guarantees that g(x) > x for all x ∈ (x c , 1) as g (1) = 0. By Lemma 3.4 we can choose a critical activity λ b (d) such that for any λ ≥ λ b (d) the transition probability u λ is close enough to 1 so that there exists a fixed point x λ ∈ (x c , u λ ) forSubstituting p λ := x λ /u λ ∈ (0, 1) in (72) shows that p λ is a fixed point for (70). Therefore we have the positive lower boundfor p Bootstrap percolation on infinite trees and non-amenable groups. J Balogh, Y Peres, G Pete, Combin. Probab. Comput. 155J. Balogh, Y. Peres, and G. Pete. "Bootstrap percolation on infinite trees and non-amenable groups". In: Combin. Probab. Comput. 15.5 (2006), pp. 715-730. MR2248323. Stability of the phase transition of critical-field Ising model on Cayley trees under inhomogeneous external fields. 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De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 2011. xiv+545. MR2807681. Is the fuzzy Potts model Gibbsian?. O Häggström, In: Ann. Inst. Henri Poincaré Probab. Stat. 39O. Häggström. "Is the fuzzy Potts model Gibbsian?" In: Ann. Inst. Henri Poincaré Probab. Stat. 39.5 (2003), pp. 891-917. MR1997217. Gibbs properties of the fuzzy Potts model on trees and in mean field. O Häggström, C Külske, Markov Process. Related Fields. 10O. Häggström and C. Külske. "Gibbs properties of the fuzzy Potts model on trees and in mean field". In: Markov Process. Related Fields 10.3 (2004), pp. 477-506. MR2097868. Gibbs-non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: closing the Ising gap. F Henning, R C Kraaij, C Külske, In: Bernoulli. 25F. Henning, R. C. Kraaij, and C. Külske. "Gibbs-non-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction: closing the Ising gap". In: Bernoulli 25.3 (2019), pp. 2051-2074. MR3961240. 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"Dynamical Gibbs-non-Gibbs transitions in Curie-Weiss Widom- Rowlinson models". In: Markov Process. Related Fields 25.3 (2019), pp. 379-413. MR3966346. Dynamical Gibbs-non-Gibbs transitions in lattice Widom-Rowlinson models with hard-core and soft-core interactions. S Kissel, C Külske, J. Stat. Phys. 1783S. Kissel and C. Külske. "Dynamical Gibbs-non-Gibbs transitions in lattice Widom- Rowlinson models with hard-core and soft-core interactions". In: J. Stat. Phys. 178.3 (2020), pp. 725-762. MR4059958. A phase transition in a Widom-Rowlinson model with Curie-Weiss interaction. Y Kozitsky, M Kozlovskii, 073202. MR3845477J. Stat. Mech. Theory Exp. Y. Kozitsky and M. Kozlovskii. "A phase transition in a Widom-Rowlinson model with Curie-Weiss interaction". In: J. Stat. Mech. Theory Exp. (2018). 073202. MR3845477. Spin-flip dynamics of the Curie-Weiss model: loss of Gibbsianness with possibly broken symmetry. C Külske, A Le Ny, Comm. Math. Phys. 271C. Külske and A. Le Ny. "Spin-flip dynamics of the Curie-Weiss model: loss of Gibbsianness with possibly broken symmetry". In: Comm. Math. Phys. 271.2 (2007), pp. 431-454. MR2287911. Dynamical Gibbs-non-Gibbs Transitions in the Curie-Weiss Potts Model in the Regime β < 3. C Külske, D Meißner, J. Stat. Phys. 18415C. Külske and D. Meißner. "Dynamical Gibbs-non-Gibbs Transitions in the Curie-Weiss Potts Model in the Regime β < 3". In: J. Stat. Phys. 184.2 (2021), p. 15. MR4292483. On family trees and subtrees of simple branching processes. A G Pakes, F M Dekking, J. Theoret. Probab4A. G. Pakes and F. M. Dekking. "On family trees and subtrees of simple branching processes". In: J. Theoret. Probab. 4.2 (1991), pp. 353-369. MR1100239. The critical Ising model on trees, concave recursions and nonlinear capacity. R Pemantle, Y Peres, Ann. Probab. 38R. Pemantle and Y. Peres. "The critical Ising model on trees, concave recursions and nonlinear capacity". In: Ann. Probab. 38.1 (2010), pp. 184-206. MR2599197. 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{'abstract': 'We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order d (which has d + 1 nearest neighbours), depending on repulsion strength β between particles of different signs and on an activity parameter λ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength β.We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d ≥ 2. In our second and main result, we also show that for large β and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity λ increases, assuming that d ≥ 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.', 'arxivid': '2012.09718', 'author': ['Sebastian Bergmann [email protected] \nFakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany\n', 'Sascha Kissel [email protected] \nFakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany\n', 'Christof Külske [email protected] \nFakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany\n', 'Sebastian Bergmann [email protected] \nFakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany\n', 'Sascha Kissel [email protected] \nFakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany\n', 'Christof Külske [email protected] \nFakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany\n'], 'authoraffiliation': ['Fakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany', 'Fakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany', 'Fakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany', 'Fakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany', 'Fakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany', 'Fakultät für Mathematik\nRuhr-Universität Bochum\nUniversitätsstraße 15044780BochumGermany'], 'corpusid': 229297776, 'doi': '10.1214/22-aihp1242', 'github_urls': [], 'n_tokens_mistral': 23694, 'n_tokens_neox': 21078, 'n_words': 13044, 'pdfsha': '5e8039a481ab4da117c8bb0fb395ef22b2d836d1', 'pdfurls': ['https://export.arxiv.org/pdf/2012.09718v2.pdf'], 'title': ['Dynamical Gibbs-non-Gibbs transitions in Widom-Rowlinson models on trees', 'Dynamical Gibbs-non-Gibbs transitions in Widom-Rowlinson models on trees', 'Dynamical Gibbs-non-Gibbs transitions in Widom-Rowlinson models on trees', 'Dynamical Gibbs-non-Gibbs transitions in Widom-Rowlinson models on trees'], 'venue': []}

arxiv

Reconfigurable Intelligent Surface-Empowered Self-Interference Cancellation for 6G Full-Duplex MIMO Communication Systems 29 Mar 2023 Chia-Jou Ku Department of Electronics and Electrical Engineering National Yang Ming Chiao Tung University HsinchuTaiwan Li-Hsiang Shen Department of Electronics and Electrical Engineering National Yang Ming Chiao Tung University HsinchuTaiwan Kai-Ten Feng [email protected] Department of Electronics and Electrical Engineering National Yang Ming Chiao Tung University HsinchuTaiwan Reconfigurable Intelligent Surface-Empowered Self-Interference Cancellation for 6G Full-Duplex MIMO Communication Systems 29 Mar 2023arXiv:2112.07833v2 [cs.IT]Index Terms-6Greconfigurable intelligent surfacefull- duplexmulti-input-multi-outputquality-of-serviceinterference mitigation Substantially increasing wireless traffic and extending serving coverage is required with the advent of sixthgeneration (6G) wireless communication networks. Reconfigurable intelligent surface (RIS) is widely considered as a promising technique which is capable of improving the system sum rate and energy efficiency. Moreover, full-duplex (FD) multiinput-multi-output (MIMO) transmission provides simultaneous transmit and received signals, which theoretically provides twice of spectrum efficiency. However, the self-interference (SI) in FD system is a challenging task requiring high-overhead cancellation, which can be resolved by configuring appropriate phase shifts of RIS. This paper has proposed an RIS-empowered full-duplex interference cancellation (RFIC) scheme in order to alleviate the severe interference in an RIS-FD system. We consider the interference minimization of RIS-FD MIMO while guaranteeing quality-of-service (QoS) of whole system. The closed-form solution of RIS phase shifts is theoretically derived with the discussion of different numbers of RIS elements and receiving antennas. Simulation results reveal that the proposed RFIC scheme outperforms existing benchmarks with more than 50% of performance gain of sum rate.Index Terms-6G, reconfigurable intelligent surface, fullduplex, multi-input-multi-output, quality-of-service, interference mitigation. I. INTRODUCTION The requirement of wireless data demands is increasingly high as the sixth-generation (6G) technology evolves. To meet the compellingly high traffic demands, there are tremendous emerging techniques for 6G wireless communications. Reconfigurable intelligent surface (RIS) is widely considered as one of the promising 6G techniques for extending coverage area, reducing power consumption, and enhancing system performance. The RIS is composed of numerous ultra-thin metamaterial-based elements, which can reflect the received signals on the surface without additional signal processing. Moreover, as a benefit of cost-effective RIS, it can be readily employed by reconfiguring its phase shifts to alternate the original channel between transmitters and receivers in order to increase the service coverage and spectrum-energy efficiency [1]- [4]. RIS-assisted 6G wireless communications have been widely discussed due to its cost-effective and unsophisticated deployment which can be utilized in various systems, e.g., signal quality is improved by RIS under non-orthogonal multiple access system [5] and multi-input-multi-output (MIMO) networks [6]. Another arising critical problem of RIS is the joint design of precoding at transmitters and phase shifts of RIS elements [7]- [9]. In [7], the authors aim for minimizing the symbol error rate by deploying RIS, whilst authors in [8] maximize the energy efficiency of the reflected signals. By adjusting RIS configuration, the authors in [9] tend to minimize total base station (BS) power consumption. In addition, enhanced security can be reached by converting the original intruded wireless paths with RIS deployment [10], [11]. In [12], [13], theoretical analysis of service coverage probability have been conducted with RIS deployment. Moreover, RIS can be utilized for interference management, which is able to suppress the strong interference and to enhance the desired signal strength by adjusting RIS phase shifts in different transmission directions [14]. In order to further increase the spectrum efficiency, fullduplex (FD) communication can simultaneously transmit and receive signals at a single operating frequency, which theoretically provides twice of spectrum efficiency compared to the half-duplex technique. However, the main challenging issue is that FD possesses strong self-interference (SI) degrading the signal quality, which results in lowered sum rate. Therefore, existing literatures have researched on SI mitigation via signal processing methods [15]- [19]. In [16], they minimize severe SI power by deriving the optimal eigen-based beamforming through spatial-domain elimination. SI mitigation with oblique projection is applied in an FD system [17]. To maximize the vehicular-to-infrastructure networks under the FD system, a resource allocation method is proposed in [18]. Moreover, in [19], a resource management problem based FD networks with consideration of user equipment's (UE's) queue backlog and time-varying channels is discussed. However, the abovementioned works require additional overhead of channel estimation and signal processing for FD SI cancellation. Therefore, it becomes compellingly imperative to design efficient RIS to alternate wireless channels in order to alleviate interference in an RIS-FD system. In this paper, we conceive an FD MIMO transmission which is empowered by RIS and aim for maximizing total system sum rate in an RIS-FD network while guaranteeing both downlink/uplink (DL/UL) quality-of-service (QoS) and RIS phase constraint. The contributions of this work are summarized as follows. • II. SYSTEM MODEL AND PROBLEM FORMULATION We consider an RIS-assisted FD MIMO system as illustrated in Fig. 1. An RIS-FD MIMO BS consists of N t transmit antennas and N r receiving antennas. Note that we deploy a single RIS to empower the data transmission which is equipped with K reflecting elements. There are N UEs intending UL transmission and M UEs desired for DL reception operating at the identical frequency bands, where each UE is assumed to be equipped with a single antenna. DL and UL transmission paths are categorized by direct path and reflected paths. For UL, U ∈ C Nr×N is the direct path from UL UEs to BS receiving antennas, while the reflected paths from UL UEs to RIS and from RIS to BS are respectively defined as U 1 ∈ C K×N and U 2 ∈ C Nr×K . For DL, D ∈ C M×Nt is the direct path from BS transmit antennas to DL UEs, and the reflected paths from BS transmit antennas to RIS and from RIS to DL UEs are respectively defined as D 1 ∈ C K×Nt and D 2 ∈ C M×K . For this system, the self-interference channel induced by FD MIMO from BS transmit to received antennas is denoted as S ∈ C Nr ×Nt and the co-channel interference from UL UEs to DL UEs is denoted as V ∈ C M×N . As for RIS, it configures the phase shifters for converting transmission directions, which requires zero power without circuit power amplifier. We define Θ = diag(αe jθ1 , ..., αe jθi , ..., αe jθK ) as the phase shift diagonal matrix of RIS, where diag(·) is the diagonal operation of a matrix, α is a constant related to reflection decaying efficiency, and θ i ∈ [0, 2π] is the phase of the i-th element of RIS. The received DL signal at DL UE and received UL signal at BS receiving antennas can be expressed y D = √ p D (D+D 2 ΘD 1 )x D + √ p U (V + D 2 ΘU 1 )x U + n 1 ,(2)y U = √ p U (U+U 2 ΘU 1 )x U + √ p D (S + U 2 ΘD 1 )x D + n 2 ,(3) where p U and p D are respectively the UL and DL transmit power of UEs and BS. x U = [x U,1 , x U,2 , · · · , x U,N ] T ∈ C N and x D = [x D,1 , x D,2 , · · · , x D,Nt ] T ∈ C Nt denote the signals from UL UEs and from DL transmit antennas at BS, respectively, and [·] T is the transpose operation. The notations of n 1 , n 2 ∼ CN (0, N 0 Be Nr ) are defined as the complex white Gaussian noises, where 0 is the zero-mean vector, e Nr denotes the unit vector, N 0 is noise power spectral density and B is the system bandwidth. Note that the signal channel includes both the received signal from direct link and reflected signals through the RIS, so does the interference channel. By deploying the RIS, we can potentially enhance the quality of signal by dynamically adjusting the RIS phase shifter Θ while guaranteeing the performance of DL/UL system and phase constraint of [0, 2π]. The considered problem for sum rate maximization on both UL and DL can be formulated as max Θ R U + R D (4a) s.t. θ i ∈ [0, 2π], ∀ 1 ≤ i ≤ K, (4b) R U ≥ γ thr,U , (4c) R D ≥ γ thr,D ,(4d) where RU = log 2 1 + √ pU (U + U2ΘU1)xU 2 √ pD(S + U2ΘD1)xD 2 + n1 2 ,(5)RD = log 2 1 + √ pD(D + D2ΘD1)xD 2 √ pU (V + D2ΘU1) x U 2 + n2 2 . (6) The notations R U ∈ R and R D ∈ R represents the UL and DL sum rate, respectively. The constraint (4c) represents that the UL sum rate R U is required to be larger than a predefined threshold γ thr,U for guaranteeing UL QoS, whilst (4d) indicates that for DL QoS satisfaction. The proposed optimization problem is transformed as min Θ,µ √ p D (S + U 2 ΘD 1 )x D 2 + µ √ p U (V + D 2 ΘU 1 )x U 2 (7a) s.t. θ i ∈ [0, 2π], ∀ 1 ≤ i ≤ K, (7b) √ p U (U + U 2 ΘU 1 )x U 2 ≥ t thr,U , (7c) √ p D (D + D 2 ΘD 1 )x D 2 ≥ t thr,D , (7d) µ ∈ R. (7e) However, the problem objective and constraints of (4) are nonconvex and non-linear. Note that the sum rates R U and R D increase monotonically with the signal-to-noise-ratio (SINR) term because log 2 is a monotone increasing function, i.e., sum rate = log 2 (1+SINR). While the received signal power is guaranteed to be above a certain value, maximizing the SINR of received signal is approximately equivalent to minimizing the interference power. The objective function can be transformed into minimizing the power of UL SI and DL cochannel interference and guaranteeing the power of DL/UL received signal. The parameter µ represents the importance ratio between DL co-channel interference and UL SI. Note that t thr,U and t thr,D in (7c) and (7d) respectively indicate UL/DL signal quality satisfaction, which implies asymptotic meaning of QoS satisfaction with those in (4c) and (4d). In the following, we will discuss two different cases with and without QoS consideration, i.e., t thr,U = 0 and t thr,D = 0 for QoS-aware scheme and t thr,U = t thr,D = 0 for non-QoS scenario design. III. PROPOSED RIS-EMPOWERED FD INTERFERENCE CANCELLATION (RFIC) SCHEME In the conventional FD MIMO system without RIS deployment, the existing works utilize signal processing methods via either precoding or postcoding to suppress the interference, which requires unaffordable computational overhead in practical implementation with increasing number of transmit or receiving antennas. However, under RIS deployment, we can generate the artificial channel to automatically and adjustably mitigate strong interferences. Our proposed RFIC scheme can optimally determine the RIS phase to substantially mitigate FD interferences. As observed from (7), we can infer that both the objective and corresponding constraints possess convexity property indicating that the optimal solution can be derived via celebrated Lagrange optimization [20]. By applying Lagrangian method, we can obtain the augmented Lagrangian expression with parameters of θ t and λ t as J(θ t , λ t ) = P (θ t , λ ′ t ) − λ 2K+1 ( √ p U (U + U 2 ΘU 1 )x U 2 −t thr,U ) − λ 2K+2 ( √ p D (D + D 2 ΘD 1 )x D 2 −t thr,D ),(8)where P (θ t , λ ′ t ) = √ p D (S + U 2 ΘD 1 )x D 2 + µ √ p U (V + D 2 ΘU 1 )x U 2 − K−1 i=0 λ 2i+1 θ i + λ 2i+2 (θ i − 2π) .(9) The parameter θ t = {θ 1 , θ 2 , ..., θ K } is a set containing K designed RIS variables. λ t = {λ 1 , λ 2 , ..., λ 2K+1 , λ 2K+2 } and λ ′ t = {λ 1 , λ 2 , . .., λ 2K } are the set of 2K + 2 and 2K Lagrangian multipliers. The first-order derivative of the Lagrangian expression of (8) can be given in (10) shown at the top of next page, where z i,j = U 2i,j d 1j , d 1 = [ Nt i=1 D 11,i , Nt i=1 D 12,i , ..., Nt i=1 D 1K,i ] T ∈ C K is the sum of the columns of D 1 , a i,j = D 2i,j d 1j , where U 2i,j , D 1i,j and D 2i,j are the elements in the i-th row and the j-th column of U 2 , D 1 and D 2 , and d 1j is the j-th element of d 1 . The notation b i,j = U 2i,j u 1j is defined with u 1 = [ N i=1 U 11,i , N i=1 U 12,i , ..., N i=1 U 1K,i ] T ∈ C K denoted as the sum of the columns of U 1 , and y i,j = D 2i,j u 1j , where U 1i,j is the element in the i-th row and the j-th column of U 1 , and u 1j is the j-th element of u 1 . The notation s = [ Nt i=1 S 1,i , Nt i=1 S 2,i , ..., Nt i=1 S Nr,i ] T ∈ C Nr is the sum of the columns of S, v = [ N i=1 V 1,i , N i=1 V 2,i , ..., N i=1 V M,i ] T ∈ C M is, u = [ N i=1 U 1,i , N i=1 U 2,i , ..., N i=1 U Nr,i ] T ∈ C Nr is the sum of the columns of U and d = [ Nr i=1 D 1,i , Nt i=1 D 2,i , ..., Nt i=1 D M,i ] T ∈ C M is the sum of the columns of D, where U i,j and D i,j are the elements in the i-th row and the j-th column of U and D, respectively. s n , v n , u n and d n respectively represent the n-th element of s, v, u and d. However, we can observe from (10) that the augmented objective function is unsolvable due to mutually-coupled exponential terms, which provokes no existing closed-forms. Therefore, we adopt the heuristic scheme for acquiring the sub-optimal solution of QoS-aware RIS adjustment. We iteratively optimize a single RIS element via quantized exhausted search with the remaining elements fixed, and then take turns performing the same procedure until convergence. Furthermore, for obtaining the closed-form solution, we relax the problem by considering the case without the QoS constraints, i.e., λ 2K+1 = λ 2K+2 = 0. As explained previously, we can observe from problem (7) that the objective function (7a) is convex due to its quadratic form. Similarly, we adopt the celebrated Lagrange optimization to acquire the optimal solution. The parameter λ ′ t represents a set regarding the RIS constraint in (7a). The augmented objective function with θ t and λ ′ t can be given by J(θ t , λ ′ t ) = P (θ t , λ ′ t ),(12) which represents that the part of the augmented Lagrangian expression in (8) without considering QoS constraints. Then, based on the theory of linear algebra [21], we can analytically derive three potential cases on the condition of N r + M to be ∂J(θt, λt) ∂θi =Ni(θt, λ ′ t ) − λ2K+1 pU x 2 U 2jbn,ie jθ i Nr n=1   bn,i + Nr p =i bn,pe jθp + un   − λ2K+2 pDx 2 D 2jam,ie jθ i M m=1   am,i + M q =i am,qe jθq + dn   , ∀ 1 ≤ i ≤ K,(10) where smaller, equal to, or greater than the number of RIS elements K in the following theorem, where N r + M denotes the total receiving antennas in entire system. Theorem 1. Consider interference mitigation for RISempowered FD MIMO transmission, three potential solutions can be obtained as follows. (a) First, when the number of whole system receiving antennas is equal to RIS elements N r + M = K, the closed-form solution of RIS phase shifts can be derived as Ni(θt, λ ′ t ) = pDx 2 D 2jzn,ie jθ i Nr n=1   zn,i + Nr p =i zn,pe jθp + sn   + µ pU x 2 U 2jym,ie jθ i M m=1   ym,i + M q =i ym,qe jθq + vn   − λ2i−1 − λ2i, ∀ 1 ≤ i ≤ K.(11)e jθi = − det(W c (i) ) det(W c ) , ∀ 1 ≤ i ≤ K,(13) where          W c = U c D c ∈ C Nr×K , W c (i) = U c (i) D c (i) ∈ C M×K .(14) Note that det(·) is the determinant of a matrix, U c = U 2 diag(d 1 ) ∈ C Nr×K and D c = D 2 diag(u 1 ) ∈ C M×K . We define U c u 2n (d 1 • p) + s n = 0, ∀ 1 ≤ n ≤ N r , d 2n (u 1 • p) + v n = 0, ∀ 1 ≤ n ≤ M,(15) where u 2n and d 2n are respectively the n-th channel row of U 2 and D 2 . The symbol • denotes the Hadamard product, and phase shifts p = [e jθ1 , e jθ2 , ..., e jθK ] T . (c) As for the case of N r + M > K, no optimal solution of phase shift for our considered problem can be acquired. Proof. We have N r + M receiving antennas in our considered system including N r receiving antennas at BS for UL and M antennas for DL UEs, which means that N r + M equations are derived based on the first-order derivatives on Lagrangian expression as the total interference received. Accordingly, there exist three conditions as follows: N r + M = K, N r + M < K, and N r + M > K. Based on (12), we can have K first-order derivatives which can be given by ∂J(θ t , λ ′ t ) ∂θ i = N i (θ t , λ ′ t ), ∀ 1 ≤ i ≤ K.(16) In the case that N r + M = K, there are N r + M equations and K variables, and the degree of freedom of variables is equal to the number of equations. Therefore, we can obtain a unique solution of phase shift to null all interference over each channel path. Notice that the optimal solution for phase shifts acquired from (13) does not contain the ratio µ since all interference can be eliminated. Accordingly, we can obtain the following equivalent equation set from (15) which is given by              K i=1 u 2n,i d 1i e θi + s n = 0, ∀ 1 ≤ n ≤ N r , K i=1 d 2n,i u 1i e θi + v n = 0, ∀ 1 ≤ n ≤ M.(17) Therefore, the phase shift in (13) can be acquired by solving (17) via substitution among variables of θ i , ∀1 ≤ i ≤ K. Furthermore, consider the second case with N r + M < K, the variable's degree of freedom is higher than the number of equations, which results in infinite solutions. Accordingly, we can derive a decent form of (15) based on (17). Conversely, if N r + M > K, we have insufficient degrees of freedom to solve these equations which cannot guarantee to reach an optimal solution for phase shift of RIS. This completes the proof. Based on Theorem 1, we can observe that perfect interference cancellation cannot be performed by RIS if N r +M > K. Note that the considered equation set in (16) will still contain the ratio µ. When the value of µ is large, it implies that we focus on eliminating DL co-channel interference instead of UL SI; conversely, the effect will be opposite. On the other hand, in the case that the number of receiving antenna is equal to or less than that of RIS elements, i.e., N r + M ≤ K, the adjustment of RIS phase shifts can effectively cancel all the UL SI and DL co-channel interferences. Meanwhile, when N r + M = K, a closed-form solution in (13) can be acquired indicating an optimal RIS phase shift is obtained to perfectly cancel all considered interferences in an RIS-FD MIMO system. IV. PERFORMANCE EVALUATIONS The performance results of proposed RFIC scheme for RIS-FD MIMO transmission are evaluated via simulations. The channels comprise distance-aware large-scale and small-scale Rayleigh fading. The large-scale pathloss of 3.5 GHz frequency spectrum is defined as P L LoS = 38.88 + 22 log 10 (d 0 ) dB for line-of-sight path [22], where d 0 is the distance between the transmitter and receiver. The Rayleigh fading is considered as an exponential distribution with expectation of one. The main system parameters of our RIS-FD MIMO system are listed in Table I. The transmit antennas of UL UEs and BS are assigned with equal power and the total power is respectively denoted by P U,max = N · p U and P D,max = N t · p D . The simulated scenario is shown in Fig. 2, where d, In Fig. 3, we evaluate sum rate of proposed RFIC for non-QoS case with t thr,D = t thr,U = 0 versus different transmit power P D,max of BS and P U,max of UL UE for N r = 2, N t = 2 and N = 2. In the left plot, as P D,max increases, it can be observed from the cases without RIS that the DL sum rate is intuitively increased; while that for UL is decreased since it becomes SI power for UL. On the other hand, the UL sum rate remains unchanged since our proposed RFIC can null the SI and DL sum rate due to larger signal power P D,max . In the right plot, with similar reason, the RFIC scheme results in enhanced and unchanged sum rate for UL and DL sum rate respectively with an increase in total transmit power of UL UE P U,max . With the increment of RIS elements from K = 4 to 8, higher sum rate can be achieved due to more reflected desired signals. Compared with the cases without RIS, it can be seen that those with RFIC achieve much higher sum rates with its capability of interference cancellation via the adjustment of RIS phase shifts. Fig. 4 depicts the sum rate performance of our proposed RFIC considering t thr,D = t thr,U = 0 versus different horizontal distances from BS to RIS under N r = 2, N t = 2 and N = 2. We can observe from Fig. 4 that there exist three performance trends: d < 60 m, d = 60 m and d > 60 m. If d is smaller than 60 m, the benefit of RIS deployment is significantly high, which can reflect more desired signal power with cancellation of severe interference. Moreover, it performs a convex curve when d < 60 m since the reflected desired signals are the weakest when d = 30 m, which is caused by the longest transmit distance through RIS. If d is equal to 60 m, the highest performance can be reached due to the shortest distance between RIS and UEs. As d becomes larger than 60 m, sum rates are decreasing due to significant effects of higher pathloss. In Fig. 5, we demonstrate sum rate of RFIC scheme with t thr,D = 0 and t thr,U = 0 for non-QoS case and t thr,D = 70 and t thr,U = 70 for QoS-aware case versus different numbers of UL UEs. It reveals that UL sum rate increases as the number of UL UEs increases due to more signal power reflected, whilst the DL sum rate decreases because it causes more co-channel interference for DL. Furthermore, it can be observed that additionally deployed transmit antennas N t will cause performance degradation for UL but result in increased DL sum rate since transmit signals from the BS will induce not only SI but also DL transmit signal. Moreover, the results show that proposed RFIC scheme with QoS-aware cases outperforms those with non-QoS due to joint consideration of guaranteed received signal quality and mitigation of interference. Although RFIC cannot perfectly cancel total interference under the case with QoS restriction, it alleviates most of interference in order to sustain desired signal quality. In Fig. 6, we compare the total sum rate performance of our proposed RFIC with existing benchmarks versus different transmit power P D,max of BS with N t = 2, N r = 2 and N = 2. We consider three benchmarks including conventional deployment without RIS, random RIS configuration, and nullsteering method [12]. Note that random RIS method means arbitrary configuration of phase shifts under the constraints of [0, 2π]. Moreover, null-steering is regarded as a well-known postcoder method which minimizes interference after received superposed signals. We can observe from Fig. 6 that the performance of random RIS setting is similar to that without RIS deployment since it does not provide adequate capability for interference mitigation. Furthermore, null-steering has higher performance than random phase shifts and non-RIS cases. This is because null-steering is capable of mitigating the interference but still with abundant residual interference affecting the desired signal. As a result, the proposed RFIC can perfectly mitigate interference, outperforming the other exiting schemes in open literature, e.g., more than 50% of performance gain of sum rate is achieved by RFIC with K = 8 having 22 bps/Hz of sum rate compared to RIS null-steering with 14 bps/Hz under P D,max = 15 mWatt. V. CONCLUSIONS In this paper, we conceive an RFIC scheme for RISempowered interference mitigation for 6G FD MIMO transmissions considering QoS requirement of desired signal. The proposed RFIC scheme is theoretically proved to posses a closed-form solution of RIS phase shift adjustment when the number of RIS elements and receiving antennas are equivalent. There exist infinite solutions if the number of RIS elements is higher than receiving antennas; conversely, no optimal outcome can be obtained. Simulation results have shown that the proposed RFIC scheme effectively mitigates the SI of FD MIMO system with more equipped RIS elements compared to that without RIS deployment. Moreover, the highest sum rate can be attained when the UEs are located nearest to RIS. Benefited by substantial interference cancellation, the proposed RFIC outperforms the other existing methods including deployment without RIS, random configuration and null-steering method in terms of the highest performance. Fig. 1 . 1RIS-assisted FD MIMO system for 6G wireless communications. the sum of the columns of V, where S i,j and V i,j are the elements in the i-th row and the j-th column of S and V, respectively. Similarly Fig. 2 . 2The illustration of deployment regarding the relative distance between BS, RIS and DL/UL UE. (i) = [U c1 , ..., U ci−1 , s, U ci+1 , ..., U cK ] with U ci ∈ C Nr is the i-th column vector of U c , and D c (i) = [D c1 , ..., D ci−1 , v, D ci+1 , ..., D cK ] with D ci ∈ C Mas the i-th column vector of D c . (b) When N r + M < K, infinite solutions of RIS phase shifts are obtained as Fig. 3 . 3Sum rate of our proposed RFIC versus different total transmit power P D,max and P U,max for Nr = 2 receiving antennas, Nt = 2 transmit antennas, N = 2 UL UEs and M = 2 DL UEs. Fig. 4 .Fig. 5 . 45Sum rate of proposed RFIC versus different distances between BS and RIS with Nt = 2 transmit antennas, Nr = 2 receiving antennas, K = {4, 8} RIS elements, N = 2 UL UEs, M = 2 DL UEs and P D,max = P U,Sum rate versus different numbers of UL UEs with Nt = {2, 4} transmit antennas, Nr = 2 receiving antennas, RIS elements of K = {4, 8}, M = 2 DL UEs and P D,max = P U,max = 1 mWatt. Fig. 6 . 6The comparison of total sum rate performance of proposed RFIC scheme with existing benchmarks of random RIS configuration, null-steering method, deployment without RIS by considering different transmit power of BS, Nt = 2 transmit antennas, Nr = 2 receiving antennas, M = 2 DL UEs, N = 2 UL UEs and P U,max = 1 mWatt. We have proposed to formulate and derive the solution of RIS phase shifts for maximizing system sum rate with the guarantee of DL/UL signal quality in the RIS-FD MIMO systems, considering both DL SI and UE's UL co-channel interference.• The proposed RIS-empowered full-duplex interference cancellation (RFIC) scheme can entirely remove the DL SI and UL UE's co-channel interference when the number of RIS elements is equal to that of BS transmit antennas and DL UEs. The corresponding optimal phase shift of RIS is theoretically proved in a closed-form manner. • Simulations have revealed that the proposed RFIC scheme can largely enhance the sum rate performance by cancelling SI and co-channel interference under proposed RIS-FD system in terms of different configurations. The comparison also demonstrates that our RFIC scheme outperforms the other existing interference cancellation methods in open literatures with more than 50% of performance gain of sum rate. TABLE I PARAMETERS IOF RIS-FD MIMO SYSTEMSystem Parameters Value Carrier frequency 3.5 GHz Noise power spectral density −174 dBm/Hz Bandwidth 20 MHz Total transmit power of UL UEs [1, 15] mWatt Total transmit power of BS [1, 15] mWatt RIS reflection decaying efficiency 0.95 Number of transmit/receiving antennas {2, 4} Number of RIS elements {0, 4, 8} Number of UL UEs [1, 5] Number of DL UEs 2 d B,U and d B,D denote the horizontal distance of BS to RIS, BS to UL UEs and BS to DL UEs, respectively; while d B,R and d UE respectively indicates the distance between BS and UL UEs and between DL UEs to UL UEs. We set d = d B,U = d B,D = 60 m, d B,R = 70 m, and d UE = 60 m. Note that we evaluate received DL/UL sum rate considering different transmit power, horizontal distance between RIS and BS, and the numbers of UL UEs, transmit antennas and RIS elements. The proposed RFIC scheme is also compared to existing benchmarks in open literature. 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{'abstract': 'Substantially increasing wireless traffic and extending serving coverage is required with the advent of sixthgeneration (6G) wireless communication networks. Reconfigurable intelligent surface (RIS) is widely considered as a promising technique which is capable of improving the system sum rate and energy efficiency. Moreover, full-duplex (FD) multiinput-multi-output (MIMO) transmission provides simultaneous transmit and received signals, which theoretically provides twice of spectrum efficiency. However, the self-interference (SI) in FD system is a challenging task requiring high-overhead cancellation, which can be resolved by configuring appropriate phase shifts of RIS. This paper has proposed an RIS-empowered full-duplex interference cancellation (RFIC) scheme in order to alleviate the severe interference in an RIS-FD system. We consider the interference minimization of RIS-FD MIMO while guaranteeing quality-of-service (QoS) of whole system. The closed-form solution of RIS phase shifts is theoretically derived with the discussion of different numbers of RIS elements and receiving antennas. Simulation results reveal that the proposed RFIC scheme outperforms existing benchmarks with more than 50% of performance gain of sum rate.Index Terms-6G, reconfigurable intelligent surface, fullduplex, multi-input-multi-output, quality-of-service, interference mitigation.', 'arxivid': '2112.07833', 'author': ['Chia-Jou Ku \nDepartment of Electronics and Electrical Engineering\nNational Yang Ming Chiao Tung University\nHsinchuTaiwan\n', 'Li-Hsiang Shen \nDepartment of Electronics and Electrical Engineering\nNational Yang Ming Chiao Tung University\nHsinchuTaiwan\n', 'Kai-Ten Feng [email protected] \nDepartment of Electronics and Electrical Engineering\nNational Yang Ming Chiao Tung University\nHsinchuTaiwan\n'], 'authoraffiliation': ['Department of Electronics and Electrical Engineering\nNational Yang Ming Chiao Tung University\nHsinchuTaiwan', 'Department of Electronics and Electrical Engineering\nNational Yang Ming Chiao Tung University\nHsinchuTaiwan', 'Department of Electronics and Electrical Engineering\nNational Yang Ming Chiao Tung University\nHsinchuTaiwan'], 'corpusid': 245144502, 'doi': '10.1109/pimrc54779.2022.9978133', 'github_urls': [], 'n_tokens_mistral': 11366, 'n_tokens_neox': 9948, 'n_words': 6043, 'pdfsha': '31d1f726fe3a0ac3cc5c25f8ef2cfa9a3dd57086', 'pdfurls': ['https://export.arxiv.org/pdf/2112.07833v2.pdf'], 'title': ['Reconfigurable Intelligent Surface-Empowered Self-Interference Cancellation for 6G Full-Duplex MIMO Communication Systems', 'Reconfigurable Intelligent Surface-Empowered Self-Interference Cancellation for 6G Full-Duplex MIMO Communication Systems'], 'venue': []}

arxiv

ON THE RECOVERY OF TRAVELING WATER WAVES WITH VORTICITY FROM THE PRESSURE AT THE BED 8 Oct 2015 Vera Mikyoung Hur And Michael R Livesay ON THE RECOVERY OF TRAVELING WATER WAVES WITH VORTICITY FROM THE PRESSURE AT THE BED 8 Oct 2015 We propose higher-order approximation formulae recovering the surface elevation from the pressure at the bed and the background shear flow for small-amplitude Stokes and solitary water waves. They offer improvements over the pressure transfer function and the hydrostatic approximation. The formulae compare reasonably well with asymptotic approximations of the exact relation between the pressure at the bed and the surface wave in the zero vorticity case, but they incorporate the effects of vorticity through solutions of the Rayleigh equation. Several examples are discussed.HUR AND LIVESAYHere and throughout,stands for the Fourier transform of the function x → f (x). Note that (1.2) becomes (1.1) in the limit as h 0 → 0.Laboratory experiments in[BD87], for instance, support that (1.2) satisfactorily predicts the wave height. Furthermore one can derive it consistently in the regime of small-amplitude Stokes waves in the case of zero vorticity; see [ES08], for instance. On the other hand, numerous studies raised doubts about the adequacy of using the linear theory; see [HmHK66,Cav80,Bie82,LW84,KC94], for instance. Note that the effects of nonlinearity and current are not negligible in shallow water or in the surf zone; see [LW84], for instance.Remarkably, exact relations were derived in [OVDH12,Con12,CC13] between the trace of the pressure at the horizontal bed and the surface elevation for Stokes and solitary water waves. In particular, the formulae apply to large amplitude waves. They are implicit but, nevertheless, easily implemented in numerical computations, and the results agree to varying degrees with laboratory experiments; see [DOV12], for instance.The arguments strongly use that in the case of zero vorticity, one is to solve the Cauchy problem for the Laplace equation. Unfortunately they cannot accommodate underlying shear flows and other physical aspects. We pause to remark that real flows are hardly irrotational. Rather vorticity is generated, for instance, by density stratification, the shear force of the wind, currents or tidal forces, and the effects of bathymetry. At present, no exact relations are available between the pressure at the bed and the surface wave in rotational flows. Furthermore numerical schemes approximating the exact formulae do not converge, because the Cauchy problem for an elliptic PDE is ill-posed.Recently in [CHW15], one of the authors elaborated (1.2) and (1.1) to permit vorticity and density stratification. Specifically, the pressure transfer function and the hydrostatic approximation were consistently derived for small-amplitude surface and interface waves in an arbitrary shear flow. Unfortunately they do not capture the effects of nonlinearity. Furthermore the hydrostatic approximation does not sense the effects of vorticity.Here we take matters further and propose higher-order approximation formulae recovering the surface elevation from the pressure at the bed for small-amplitude Stokes and solitary water waves in an arbitrary shear flow. Specifically, we compute higher-order correction terms to the pressure transfer function and the hydrostatic approximation in[CHW15]. To the best of the authors' knowledge, these are new. We carry out higher-order perturbations of the governing equations, rather than relying on a less empirical approach of higher-order Stokes expansion. We sacrifice the ability to accommodate large amplitude waves, but we are able to work to an arbitrary, albeit finite, degree of accuracy, when exact formulae relating the pressure at the bed and the surface wave are unavailable.The formulae incorporate the effect of vorticity through solutions to the Rayleigh equation, which one must in general investigate numerically. But we make an effort to discuss some examples. In the case of zero vorticity, in particular, we demonstrate that our results compare reasonably well with asymptotic approximations of the exact formulae in [OVDH12], for instance; see Example 3.1 and Example 4.1. The Introduction A basic problem in oceanography is to determine wave parameters -significant wave height, significant wave period, spectral peaks, etc. -from ocean measurements. For instance, the task of tracking the genesis and propagation of tsunamis is of obvious importance. One main source of data extensively used for the purpose is pressure transducers seeded throughout the Pacific and Indian Ocean. They collect pressure readings at various water depths and transmit to monitoring stations. This motivates an interesting mathematical question. Suppose that a wave runs in a channel of water over a long distance practically at a constant velocity without change of form, and that the value of the pressure at the bed is given, and perhaps some other information about the upstream and downstream flow. From such scant data, can one recover the wave height? Incidentally traveling waves may be used as a means to understand more general wave motions; see [OVDH12,DOV12], for instance. Under the assumption that the fluid in the bulk is irrotational, a simple approach, which is in practice, for instance, in tsunami detection, is to take up the hydrostatic approximation (see [DD91,KC10], for instance): (1.1) η(x) = 1 g p(x). Here x denotes the spatial variable in the direction of wave propagation, η is the surface displacement from the undisturbed fluid depth h 0 , say, and p is the dynamic pressure, measuring the departure from the hydrostatic pressure; g is the constant due to gravitational acceleration. Another is the pressure transfer function (see [DD91,KC10], for instance): (1.2) F (η)(k) = 1 g cosh(kh 0 )F (p)(k). Date: October 9, 2015. 1 upshot of the present treatment is highly computationally manageable solutions, which may develop into an easy and effective numerical scheme. The practical use of the results, including numerical and experimental studies, is of future investigation. Preliminaries The water wave problem, in the simplest form, concerns the wave motion at the surface of an incompressible inviscid fluid, below a body of air and acted upon by gravity. For definiteness, we take Cartesian coordinates (x, y), where the x-axis points in the direction of wave propagation and the y-axis vertically upward. In other words, the motion is constant in one horizontal direction. The fluid at time t occupies a region in R 2 , bounded above by the free surface and below by the fixed horizontal bottom y = 0, say. Let y = h(x; t) describe the fluid surface at time t, and we assume that h is a single-valued, non-negative and smooth function. In the bulk of the fluid, the velocity (u(x, y; t), v(x, y; t)) and the pressure P (x, y; t) satisfy the Euler equations for an incompressible fluid:      u t + uu x + vu y = −P x , v t + uv x + vv y = −P y − g in 0 < y < h(x; t), u x + v y = 0. Here and throughout, a subscript denotes partial differentiation. Although an incompressible fluid such as water may have variable density, we assume for simplicity that the density = 1. The flow is allowed to be rotational and characterized by the vorticity: ω = v x − u y . The kinematic and dynamic conditions at the fluid surface v = h t + uh x and P = P atm at y = h(x; t) state, respectively, that each water particle at the surface remains so for all time and that the pressure at the surface equals the atmospheric pressure P atm ; we assume that the air is quiescent and we neglect the effects of surface tension. The flow is required to be tangential to the bottom: v = 0 at y = 0. It is a matter of experience that waves which are commonly seen in the ocean or a lake propagate over a long distance practically at a constant velocity without change of form, namely traveling waves. In other words, u, v and P are functions of (x−ct, y) and h is a function of x−ct for some c > 0, the speed of wave propagation. Under this assumption, we will go to a moving coordinate frame, changing x − ct to x, whereby the time t completely disappears. The result becomes: (2.1)                (u − c)u x + vu y = −P x , (u − c)v x + vv y = −P y − g in 0 < y < h(x), u x + v y = 0, v = (u − c)h x and P = P atm at y = h(x), v = 0 at y = 0. Note that (2.2) h ≡ h 0 , (u, v) = (U (y), 0) and P = P atm + g(h 0 − y) make a solution of (2.1) for arbitrary c > 0, h 0 > 0 and an arbitrary U ∈ C 1 ([0, h 0 ]). Physically, it represents a shear flow, for which the velocity and the fluid surface are horizontal and the pressure is hydrostatic. The present interest is waves propagating in the x-direction over a prescribed shear flow of the form. In what follows, therefore, h 0 and U are held fixed. Note that the vorticity of (2.2) is −U ′ (y). Here and throughout, the prime denotes ordinary differentiation. Scaling of variables. In order to systematically characterize various approximations, we introduce (2.3) δ = the long wavelength parameter and ǫ = the amplitude parameter, and we define the set of scaled variables. Rather than introducing a new notation for the variables, we choose, wherever convenient, to write, for instance, x → x/δ. This is to be read that x is replaced by x/δ, so that hereafter the symbol x will denote a scaled variable. With this understanding, let (2.4) x → x/δ and (2.5) u → U + ǫu 1 + ǫ 2 u 2 + ǫ 3 u 3 + · · · and v → δ(ǫv 1 + ǫ 2 v 2 + ǫ 3 v 3 + · · · ). Moreover, we write (2.6) h = h 0 + ǫη 1 + ǫ 2 η 2 + ǫ 3 η 3 + · · · and (2.7) P = P atm + g(h 0 − y) + ǫp 1 + ǫ 2 p 2 + ǫ 3 p 3 + · · · . Physically, h − h 0 means the surface displacement from the undisturbed fluid depth and P − P atm − g(h 0 − y) is the dynamic pressure, measuring the departure from the hydrostatic pressure; see [Joh97, Section 1.3.2 and Section 3.4.1] for the detail. Substituting (2.4) through (2.7) into (2.1) and restricting the result to the undisturbed fluid domain 0 < y < h 0 , we arrive at that: (2.8)                            (U − c)u 1x + U ′ v 1 + ǫ((U − c)u 2x + U ′ v 2 + u 1 u 1x + v 1 u 1y ) + ǫ 2 ((U − c)u 3x + U ′ v 3 + (u 1 u 2 ) x + u 2 u 1x + v 1 u 2y + v 2 u 1y ) + · · · = −p 1x − ǫp 2x − ǫ 2 p 3x + · · · , δ 2 ((U − c)v 1x + ǫ((U − c)v 2x + u 1 v 1x + v 1 v 1y ) + ǫ 2 ((U − c)v 3x + u 1 v 2x + u 2 v 1x + v 1 v 2y + v 2 v 1y )) + · · · = −p 1y − ǫp 2y − ǫ 2 p 3y + · · · , u 1x + v 1y + ǫ(u 2x + v 2y ) + ǫ 2 (u 3x + v 3y ) + · · · = 0 in 0 < y < h 0 , and (2.9)      (U − c)η 1x + ǫ((U − c)η 2x + u 1 η 1x ) + ǫ 2 ((U − c)η 3x + u 1 η 2x + u 2 η 1x ) + · · · = v 1 + ǫv 2 + ǫ 2 v 3 + · · · , g(η 1 + ǫη 2 + ǫ 2 η 3 ) + · · · = p 1 + ǫp 2 + ǫ 2 p 3 + · · · at y = h 0 and (2.10) v 1 + ǫv 2 + ǫ 2 v 3 + · · · = 0 at y = 0. Note that u = v = p = η = 0 -no disturbances -satisfy (2.8)-(2.10) for arbitrary c > 0, h 0 > 0 and an arbitrary U ∈ C 1 ([0, h 0 ]). The governing equations may be non-dimensionalized (see [Joh97, Section 1.3.1], for instance), but we do not pursue it here. Rather we work in the physical variables as much as possible. Stokes waves By Stokes waves, we mean solutions of (2.1), which are periodic and symmetric in the x-direction (historically, in the case of zero vorticity and practically at rest at great depths). For an arbitrary distribution of vorticity, under some assumptions, incidentally, all periodic solutions of (2.1) are a priori symmetric about their crests; see [Hur07] and [CEW07], for instance. Stokes in his classic memoir in 1847 (see also [Sto80]) made many contributions about waves of the kind, observing, for instance, that crests tend to be sharper and troughs flatter as the amplitude increases and that the crest of a wave of greatest height would be a stagnation point with a 120 • corner. In the case of zero vorticity, the rigorous existence theory of Stokes waves goes back to constructions in [Nek51,LC25] and [Str26] of small amplitude waves, and it includes global bifurcation results in [Kra61,KN78], for instance, and the resolution in [AFT82] of Stokes' conjecture about the wave of greatest height. All these works strongly use the assumption that the flow in the bulk is irrotational, whereby one may reformulate the problem in terms of quantities defined at the fluid surface. We encourage the interested reader to some excellent surveys [Tol96,OS01,BT03,Gro04,Str10]. The zero vorticity assumption may serve as a reasonable approximation in some circumstances. Moreover in the absence of initial vorticity, boundaries or currents, water waves will have zero vorticity for all future time. But rotational effects are significant in many circumstances, for instance, for wind-driven waves, waves riding upon a sheared current, or waves near a ship or pier. In the case of nonzero vorticity, the situation is to look inside the fluid because the velocity potential is no longer viable to use. Consequently it is harder to handle, analytically and numerically, than the zero vorticity case. It was not until recently that Constantin and Strauss [CS04] established the existence, from zero up to (but not including) an "extreme" wave exhibiting a stagnation point. Specifically, for arbitrary wave speed and period and for an arbitrary function relating the vorticity and the stream function, subject to a "bifurcation condition", they constructed a global continuum of Stokes waves with vorticity. This quickly led to a flurry of research activities. It would be impossible to do justice to all the advances in the direction, but we single out a few - [Hur06,Hur11] in the infinite depth, [Wah09,EEW11,CSV14] permitting critical layers, [Var09,VW12] about an extreme wave, and [CS11] permitting discontinuous vorticities. A key idea in [CS04], as in all free boundary problems, is to fix the -a priori unknown -fluid domain. As a matter of fact, the bifurcation condition in [CS04] asks if the linearization of (2.1) about (2.2) admits a nontrivial solution, but in the Dubreil-Jacotin variables, which map the fluid domain of one period to a fixed rectangle. Seeking explicit relations between the pressure at the bed and the surface wave in the physical coordinates (where the underlying shear flow and the fluid depth are fixed), here we carry out all calculations in the physical variables. Below we record the translation of the bifurcation condition in [CS04] to the physical variables, originally derived by one of the authors in [HL08]. The bifurcation condition. For arbitrary h 0 > 0 and U ∈ C 1 ([0, h 0 ]), recall that (2.2) is a solution of (2.1) for all c > 0. We are interested in determining at which values of c > 0 and k > 0, there bifurcates a family of small-amplitude 2π/k-periodic solutions of (2.1). A necessary condition, it turns out, is that (3.1)      (U − c)(φ ′′ − k 2 φ) − U ′′ φ = 0 for 0 < y < h 0 , φ ′ (h 0 ) = g (U (h 0 ) − c) 2 + U ′ (h 0 ) U (h 0 ) − c φ(h 0 ) and φ(0) = 0 admits a nontrivial solution for some c and k. It is in general not a sufficient condition, but in case when c > max 0 y h0 U (y), bifurcation does occur, provided that the kernel of (3.1) is one dimensional. Under this assumption, furthermore, u < c throughout the fluid region. Note that u = c at a stagnation point. In [CS04], the wave speed and the vorticity-stream function relation are fixed whereas the shear flow U and the fluid depth h 0 vary along the branch of solutions. On the contrary, here the shear flow and the fluid depth are fixed and the wave speed is determined upon solving (3.1). The ordinary differential equation in (3.1) goes by the name of the Rayleigh (or inviscid Orr-Sommerfeld) equation. It is not singular, provided that c > max U . In the case of U ≡ 0, namely the zero vorticity, a straightforward calculation reveals that a nontrivial solution to (3.1) exists, provided that (3.2) c 2 = g tanh(kh 0 ) k . This is the well-known dispersion relation of water waves in irrotational flows. In the case of U (y) = γy for some constant γ, namely the constant vorticity −γ, similarly, a straightforward calculation reveals that a nontrivial solution to (3.1) exists, provided that (3.3) c = − γ tanh(kh 0 ) 2k + γ 2 tanh 2 (kh 0 ) 4k 2 + gk tanh(kh 0 ) k . This is the dispersion relation in the case of constant vorticity. (The other solution with the − sign violates c > max U , and hence we discard it.) The bifurcation condition is closely related, but not equivalent, to the dispersion relation. The bifurcation condition is a necessary and sufficient condition for nontrivial solutions to exist whereas the dispersion relation is a necessary condition for plane wave solutions to exist to the associated linear problem. For general non-constant vorticities, one must not expect to solve (3.1) explicitly; see [Kar12], for instance. For a wide range of shear flows, nevertheless, one may be able to verify the bifurcation condition using the ODE theory. If U ∈ C 2 ([0, h 0 ]), U ′′ (h 0 ) < 0 and U (h 0 ) > U (y) for 0 < y < h 0 , for instance, bifurcation takes place for some c > max U for all k > 0; see [HL08, Lemma 2.5]. In the long wave limit as k → 0+, suitable for solitary water waves, the bifurcation condition leads to the Burns condition (3.4) h0 0 dy (U (y) − c) 2 = 1 g ; see [Bur53, Joh97, HL08], for instance. 3.1. The first-order approximation. We set forth the surface reconstruction procedure from the pressure at the fluid bed for small-amplitude Stokes waves with vorticity. We therefore assume that (3.5) δ = 1 and ǫ ≪ 1 and, without loss of generality, u j 's, v j 's, p j 's and η j 's, j = 1, 2, 3, . . . , in (2.5)-(2.7) are 2π-periodic in the x-variable. In other words, the wave number k = 1. We do not assume a priori their symmetry and monotonicity. Under this assumption, (2.8) and (2.9) (2.10) at the leading order become: (3.6)      (U − c)u 1x + U ′ v 1 = −p 1x , (U − c)v 1x = −p 1y in 0 < y < h 0 , u 1x + v 1y = 0, and v 1 = (U − c)η 1x and p 1 = gη 1 at y = h 0 , (3.7) v 1 = 0 at y = 0. (3.8) Stokes waves with vorticity constructed in [CS04], for instance, solve (2.1), and hence (2.8)-(2.10). It follows from bifurcation theory that small-amplitude solutions solve (3.6)-(3.8) with errors in O(ǫ 2 ) in some suitable Hölder space. Thanks to the symmetry of the bifurcation problem, furthermore, the wave speed is approximated by that determined upon solving (3.1) with errors in O(ǫ 3 ). Throughout the section c > 0 is held fixed. Differentiating the first equation in (3.6) in the y-variable and the second equation in the x-variable, and using the third equation, we arrive at that (U − c)∆v 1 − U ′′ v 1 = 0 in 0 < y < h 0 . Suppose that the dynamic pressure (see (2.7)) at the bed is prescribed to the order of ǫ. We assume that it is smooth and 2π-periodic in the x-variable, and thus we write it in the (complex) Fourier series as (3.9) p 1 (x, 0) = ∞ n=−∞ b 1n e inx . The first and third equations in (3.6) restricted at the bed imply that (U (0) − c)v 1y (x, 0) − U ′ (0)v 1 (x, 0) = p 1x (x, 0) = ∞ n=−∞ inb 1n e inx . Note from (3.8) that the second term on the left side vanishes. To recapitulate, in the first-order approximation of the surface elevation as a function of the pressure at the bed for small-amplitude Stokes waves with vorticity, one solves the Cauchy problem for the linear elliptic PDE: (3.10)            (U − c)∆v 1 − U ′′ v 1 = 0 in 0 < y < h 0 , v 1 = 0 at y = 0, v 1y = 1 U (0) − c ∞ n=−∞ inb 1n e inx at y = 0, and one uses equations in (3.6) to determine u 1 and p 1 in terms of p 1 (·, 0) (see (3.9)). One then determines η 1 using the second equation in (3.7) as gη 1 (x) = p 1 (x, h 0 ). Specifically, let's write that v 1 (x, y) = ∞ n=−∞ iφ 1n (y)e inx . Here i is for convenience. For each n, note that (3.10) leads to the Cauchy problem for the Rayleigh equation: (3.11)          (U − c)(φ ′′ 1n − n 2 φ 1n ) − U ′′ φ 1n = 0 for 0 < y < h 0 , φ 1n (0) = 0, φ ′ 1n (0) = nb n U (0) − c . It follows from the ODE theory that (3.11) admits a unique solution. In particular φ 10 ≡ 0. We then infer from the last equation in (3.6) that u 1 (x, y) = u 10 (y) − n =0 φ ′ 1n (y) e inx n for some function u 10 and, similarly, from the first equation in (3.6) that p 1 (x, y) = p 10 (y) + n =0 ((U (y) − c)φ ′ 1n (y) − U ′ (y)φ 1n (y)) e inx n for some function p 10 . Comparing p 1y to the second equation in (3.6), we use (3.11) to deduce that p 10 (y) is a constant. Comparing this to (3.9), moreover, we conclude that p 10 (y) = b 0 . One may not be able to determine u 10 , on the other hand. This is not surprising since adding a function of y to u 1 does not change (3.6)-(3.8). Ultimately the second equation in (3.7) implies that (3.12) gη 1 (x) = b 0 + n =0 ((U (h 0 ) − c)φ ′ 1n (h 0 ) − U ′ (h 0 )φ 1n (h 0 )) e inx n . This furnishes an implicit formula relating η 1 to p 1 (·, 0) (see (3.9)), provided with the background shear flow and the wave speed, subject to the bifurcation condition. It incorporates the effects of vorticity through the solution to the Cauchy problem for the Rayleigh equation. In what follows, we furthermore assume that (3.13) η 1 and u 1 (·, y) for each y ∈ [0, h 0 ] are proportional to cos(x). It follows from local bifurcation theory that the leading part of small-amplitude solutions constructed in [CS04], for instance, satisfy (3.13). As a matter of fact, all solutions in [CS04] satisfy (3.13), after possibly redefining the undisturbed fluid depth h 0 and the background current U . Under this assumption, the pressure at the bed must be prepared as (3.14) p 1 (x, 0) = b cos(x) for some constant b (b = 2b 11 = 2b 1−1 ), and a straightforward calculation reveals that (3.15)      u 1 (x, y) = φ ′ (y) cos(x), v 1 (x, y) = φ(y) sin(x), p 1 (x, y) = ((c − U (y))φ ′ (y) + U ′ (y)φ(y)) cos(x) and (3.16) η 1 (x) = 1 g ((c − U (h 0 ))φ ′ (h 0 ) + U ′ (h 0 )φ(h 0 )) cos(x), where φ is the unique solution of (3.17)          (U − c)(φ ′′ − φ) − U ′′ φ = 0 for 0 < y < h 0 , φ(0) = 0, φ ′ (0) = b c − U (0) . This furnishes an implicit formula relating η 1 and p 1 (·, 0) (see (3.14)), assuming that small-amplitude Stokes waves are sinusoidal to the leading order. Note that (3.13) uniquely determines u 1 . Remark (The pressure transfer function). The last equation in (3.15) is reminiscent of the pressure transfer function found in [CHW15]: (3.18) p(x, y) = ((c − U (y))φ ′ (y) + U ′ (y)φ(y))η(x), but φ in [CHW15] solves the boundary value problem for the Rayleigh equation:      (U − c)(φ ′′ − φ) − U ′′ φ = 0 for 0 < y < h 0 , φ(0) = 0, φ(h 0 ) = c − U (h 0 ). One may use (3.18) to approximate the pressure at the bed as a function of the surface elevation for small-amplitude Stokes waves with vorticity. 3.2. The second-order approximation. We continue to assume (3.5). To proceed, (2.8) and (2.9), (2.10) at the order of ǫ become: (3.19)      (U − c)u 2x + U ′ v 2 + u 1 u 1x + v 1 u 1y = −p 2x , (U − c)v 2x + u 1 v 1x + v 1 v 1y = −p 2y in 0 < y < h 0 , u 2x + v 2y = 0, and v 2 = (U − c)η 2x + u 1 η 1x and p 2 = gη 2 at y = h 0 , (3.20) v 2 = 0 at y = 0. (3.21) Furthermore we assume (3.13), and thus u 1 , v 1 and p 1 are computed using (3.15); otherwise, expressions become quite complicated involving double Fourier series. Suppose that the dynamic pressure at the bed is prescribed to the order of ǫ 2 . We continue to assume that it is smooth and 2π-periodic in the x-variable, and we write that (3.22) p 2 (x, 0) = ∞ n=−∞ b 2n e inx . We then repeat the argument in the previous subsection. To summarize, in the second-order approximation of the surface elevation as a function of the pressure at the bed for small-amplitude Stokes waves with vorticity, one solves the Cauchy problem for the inhomogeneous linear elliptic PDE: (3.23)              (U − c)∆v 2 − U ′′ v 2 = 1 2 (φφ ′′′ − φ ′ φ ′′ ) sin(2x) in 0 < y < h 0 , v 2 = 0 at y = 0, v 2y = 1 U (0) − c 1 2 (φφ ′′ − (φ ′ ) 2 ) sin(2x) + ∞ −∞ inb 2n e inx at y = 0, and uses equations in (3.19) and (3.15) to determine u 2 and p 2 in terms of p 1 (·, 0) and p 2 (·, 0) (see (3.14) and (3.22)). One then determines η 2 using the second equation in (3.20) as gη 2 ( x) = p 2 (x, h 0 ). Let's write that v 2 (x, y) = ∞ n=−∞ iφ 2n (y)e inx . Here i is for convenience. For each n = ±2, note that (3.23) leads to the Cauchy problem for the Rayleigh equation: (3.24)          (U − c)(φ ′′ 2n − n 2 φ 2n ) − U ′′ φ 2n = 0 for 0 < y < h 0 , φ 2n (0) = 0, φ ′ 2n (0) = nb 2n U (0) − c . For n = ±2, similarly, (3.25)            (U − c)(φ ′′ 2±2 − 4φ 2±2 ) − U ′′ φ 2±2 = ∓ 1 4 (φφ ′′′ − φ ′ φ ′′ )e ±2ix for 0 < y < h 0 , φ 2±2 (0) = 0, φ ′ 2±2 (0) = 1 U (0) − c ∓ 1 4 (φφ ′′ − (φ ′ ) 2 )(0) ± 2b 2±2 . It follows from the ODE theory that (3.24) and (3.25) admit unique solutions. In particular φ 20 ≡ 0. We then infer from the last equation in (3.19) that u 2 (x, y) = u 20 (y) − n =0 φ ′ 2n (y) e inx n for some function u 20 and, similarly, from the first equation in (3.19) that p 2 (x, y) = 1 4 (φφ ′′ −(φ ′ ) 2 )(y) cos(2x)+p 20 (y)+ n =0 ((U (y)−c)φ ′ 2n (y)−U ′ (y)φ 2n (y)) e inx n for some function p 20 . Comparing p 2y to the second equation in (3.19), we use (3.24) to deduce that p ′ 20 (y) = −(φφ ′ )(y). Comparing this to (3.22), moreover, we conclude that p 20 (y) = − 1 2 φ 2 (y) + b 20 . One may not able to determine u 20 , on the other hand. This is not surprising since adding a function of y to u 2 does not change (3.19)-(3.21). Ultimately the second equation in (3.20) implies that gη 2 (x) = − 1 2 φ 2 (h 0 ) + 1 4 (φφ ′′ − (φ ′ ) 2 )(h 0 ) cos(2x) + b 20 + n =0 ((U (h 0 ) − c)φ ′ 2n (h 0 ) − U ′ (h 0 )φ 2n (h 0 )) e inx n , (3.26) where φ solves (3.17) and φ 2n solves (3.24) or (3.25). This furnishes an implicit formula relating η 2 to p 1 (·, 0) and p 2 (·, 0) (see (3.14) and (3.22)), provided with the background shear flow and the wave speed, subject to the bifurcation condition. It incorporates the effects of vorticity through solutions of the Cauchy problems for the Rayleigh equations. Note that (3.26) depends linearly on the second-order pressure data and nonlinearly on the first-order pressure data. We may repeat the above argument and continue to higher-order approximations. Expressions become quite complicated, however, and hence we do not pursue here. One shortcoming of our method is that we only determine the horizontal velocities u j 's, j = 1, 2, 3, . . . , up to functions of y. We may continue to assume that (see (3.13)) 2π 0 u j (x, y) dx = 0 for each y ∈ [0, h 0 ], j = 1, 2, 3, . . . , after possibly redefining the shear flow, to uniquely determine them. In the case of zero vorticity, in [OVDH12], for instance, likewise, one determines the velocity potential up to a constant, but the reconstruction formula does not require knowledge of the velocity potential itself; see Appendix A for the detail. Another shortcoming is that the wave speed agrees with the bifurcation speed up to the order of ǫ 2 in the regime of small amplitude waves, even in the case of zero vorticity. Moreover it is in practice difficult to measure. We may continue to assume the bifurcation speed for higher-order approximations. As a matter of fact, numerical computations in [DOV12], for instance, indicate that the maximum wave height does not suffer much from the small amplitude limit of the wave speed. 3.3. Examples. Formulae in Section 3.1 and Section 3.2 must in general be investigated numerically. In some cases, nevertheless, analytical solutions are available, as we discuss below. Throughout the subsection, we assume for simplicity that (3.27) p 1 (x, 0) = b cos(x) and p 2 (x, 0) = 0. Example 3.1 (Zero vorticity). In the case of U ≡ 0, namely the zero vorticity, a straightforward calculation reveals that the unique solution of (3.17) is φ(y) = b c sinh(y), whence (3.15) and (3.16) become This agrees with the pressure transfer function in (1.2) in the case when the wave number k = 1.            u 1 (x, y) = b c cosh(y) cos(x), v 1 (x, y) = b c sinh(y) sin(x), To proceed, note that φφ ′′ − (φ ′ ) 2 = −(b/c) 2 . A straightforward calculation reveals that the unique solution of (3.24) or (3.25) is φ 2n (y) ≡ 0 if n = ±2 and φ 22 (y) = −φ 2−2 (y) = − b 2 8c 3 sinh(2y). Therefore (3.26) becomes (3.29) gη 2 (x) = 1 2 b c 2 sinh 2 (h 0 )(−1 + cos(2x)). We compare (3.28) and (3.29) to asymptotic approximations of the exact formula in [OVDH12], for instance, but in the physical variables. We include the detail of the algebra in Appendix A for completeness. The first-order approximation of the result in [OVDH12] agrees with (3.28); see (A.9). The second-order approximation of the result in [OVDH12] may be written, abusing notation, as (1 + 4 sinh 2 (h 0 ) cos(2x)). This shares with (3.29) that the second-harmonic correction becomes negligible as h 0 → 0. But in (3.29) the second-order depth correction decreases as h 0 → 0 whereas in (3.30) it increases with small depths. Note from (3.2) that c → √ gh 0 as h 0 → 0. A potential reason for disagreements between (3.29) and (3.30) is that equations in (2.9), and hence (3.20), do not take into account of the nonlinear effects of the boundary conditions at the free surface. If we were to include boundary variations in the derivation of (2.9) (see [HL08], for instance) then perhaps we would be able to find a better agreement. Furthermore we compare the results to the small amplitude asymptotics of a true solution in [Whi74, Section 13.13], for instance. The first-order approximation agrees with (3.28) and (A.9). The second-order approximation of the result in [Whi74] may be written, abusing notation, as (3.31) gη 2 (x) = 1 4 b c 2 − 1 cosh 2 (h 0 ) + 2 + 3 sinh 2 (h 0 ) cos(2x) . Observe that the nonlinear corrections become significant as h 0 → 0. Experimental studies in [LW84], for instance, bear out this. (Note that the result in [Whi74] assumes that the integral of η j over one period is zero for all j = 1, 2, 3, . . . . One may think of this as a consequence of absorbing a constant of integration in the hydrostatic pressure. It is straightforward to keep track of nonzero mean values in η j 's, however. We omit the detail.) A potential reason for disagreements between (3.29), (3.30) and (3.31) is that (3.27) may not hold for the pressure distribution of a true solution. If we were to prescribe a more physically realistic pressure input, then perhaps we would be able to find a better agreement. It is interesting to find a higher-order approximation formula, for which the nonlinear effects become significant for small depths. Example 3.2 (Constant vorticity). Let U (y) = γy, 0 y h 0 , for some constant γ. (More generally, one may take U (y) = γy + U 0 for some constant U 0 , but U 0 may be absorbed into the wave speed.) This models the constant vorticity −γ. A straightforward calculation reveals that the unique solution of (3.17) is φ(y) = b c sinh(y), whence (3.15) and (3.16) become              u 1 (x, y) = b c cosh(y) cos(x), v 1 (x, y) = b c sinh(y) sin(x), p 1 (x, y) = c − γy c cosh(y) + γ c sinh(y) b cos(x) and gη 1 (x) = c − γh 0 c cosh(h 0 ) + γ c sinh(h 0 ) b cos(x). To proceed, note that φφ ′′ − (φ ′ ) 2 = −(b/c) 2 . A straightforward calculation reveals that the unique solution of (3.24) or (3.25) is φ 2n (y) ≡ 0 if n = ±2 and φ 22 (y) = −φ 2−2 (y) = − b 2 8c 3 sinh(2y). Therefore (3.26) becomes gη 2 (x) = 1 4 b c 2 − 2 sinh 2 (h 0 ) + c − γh 0 c cosh(2h 0 ) + γ 2c sinh(2h 0 ) − 1 cos(2x) . In the case of γ = 0, these formulae reduce to those in Example 3.1. Solitary water waves By solitary water waves, we mean solutions of (2.1), for which instead of the periodic boundary condition, h(x) tends to a constant and v(x, y) → 0 uniformly for y as x → ±∞. Historically they have stimulated a considerable part of developments in the theory of wave motion, from Russell's famous horseback observations to the elucidation of the Korteweg-de Vries (KdV) solitons. Solitary water waves may formally be viewed as the limit of Stokes waves as the period tends to infinity. As a matter of fact, small-amplitude solitary water waves, if exist, emanate near the critical speed determined upon solving (3.4). They are a genuinely nonlinear phenomenon, however, and classical bifurcation theory fail to yield the existence. In the case of zero vorticity, their rigorous theory goes back to constructions in [FH54,Bea77] of small amplitude waves and it includes a large amplitude result in [AT81]. Recently, these results have been extended in the case of non-zero vorticity in [Hur08a,GW08] and [Whe13]. Specifically, for an arbitrary non-zero vorticity, one of the authors employed the generalized implicit function theorem of Nash-Moser type to construct a family of small-amplitude solitary water waves with super-critical wave speed near the KdV soliton. Moreover they are unique. 4.1. The first-order approximation. We follow the same line of argument as in the previous section and develop the surface reconstruction procedure from the pressure at the fluid bed, for small-amplitude solitary water wave with vorticity near the KdV soliton. We therefore assume that (see [Hur08a], for instance) (4.1) δ = √ ǫ ≪ 1 and u j 's, v j 's, p j 's and η j 's, j = 1, 2, 3, . . . , in (2.5)-(2.7) decay to zero as x → ±∞. For an arbitrary distribution of vorticity, incidentally, all solitary water waves decay to zero exponentially fast at infinity; see [Hur08b], for instance. Under this assumption, (2.8) and (2.9), (2.10) at the leading order become: Small-amplitude solitary water waves with vorticity constructed in [Hur08a], for instance, solve (2.1), and hence (2.8)-(2.10). It follows from the Lyapunov-Schmidt reduction in [Hur08a] that as δ 2 = ǫ → 0, they are approximated by the KdV soliton with errors in O(ǫ 2 ) in the real analytic function space and the wave speed is approximated by that determined upon solving (3.4) with errors in O(ǫ 2 ). For an arbitrary distribution of vorticity in a Hölder space, incidentally, one of the authors in [Hur12] proved that a solitary water wave in the corresponding Hölder space is real analytic. Throughout the section, c means the critical wave speed. Numerical computations in [DOV12], for instance, indicate that the maximum wave height does not suffer much from assuming the critical wave speed. (4.2)      (U − c)u 1x + U ′ v 1 = −p 1x , p 1y = 0 in 0 < y < h 0 , u 1x + v 1y = 0, Suppose that the dynamic pressure (see (2.7)) at the bed is prescribed to the order of ǫ. We write that (4.5) p 1 (x, 0) = b 1 (x), and assume that b 1 is smooth and decays to zero as x → ±∞. Since the second equation in (4.2) implies that the dynamic pressure to the leading order does not vary with the depth, it follows from the second equation in (4.3) the hydrostatic approximation (see (1.1)) (4.6) gη 1 (x) = b 1 (x). This furnishes an explicit formula relating η 1 and p 1 (·, 0). It is independent of the underlying shear flow, and hence it is likely to be a poor approximation. In the following subsection, we shall compute higher-order correction terms to (4.6), which do incorporate the effects of vorticity. For future usefulness, we infer from the first and the last equations in (4.2) that (4.7) v 1 U − c y = b ′ 1 (U − c) 2 . We solve it by quadrature to arrive at that v 1 (x, y) = b ′ 1 (x)(U (y) − c)f r(y), where (4.8) f r(y) = y 0 dz (U (z) − c) 2 . As a matter of fact, one can solve (4.7) analytically for all U , unlike the Rayleigh equations in the previous section. Note in passing that f r(h 0 ) is the inverse square of the Froude number. We then determine u 1 upon integrating the last equation in (4.2) and using that u 1 vanishes at infinity. To summarize, (4.9)          u 1 (x, y) = −b 1 (x) U ′ (y)f r(y) + 1 U (y) − c , v 1 (x, y) = b ′ 1 (x)(U (y) − c)f r(y), p 1 (x, y) = b 1 (x), where f r is defined in (4.8). 4.2. Higher-order approximations. We continue to assume (4.1). To proceed, (2.8) and (2.9), (2.10) at the order of ǫ become: (4.10)      (U − c)u 2x + U ′ v 2 + u 1 u 1x + v 1 u 1y = −p 2x , (U − c)v 1x = −p 2y in 0 < y < h 0 , u 2x + v 2y = 0 and v 2 = (U − c)η 2x + u 1 η 1x and gη 2 = p 2 at y = h 0 , (4.11) v 2 = 0 at y = 0. (4.12) Suppose that the dynamic pressure at the fluid bed is prescribed to the order of ǫ 2 . We write that (4.13) p 2 (x, 0) = b 2 (x), and we continue to assume that b 2 is smooth and decays to zero as x → ±∞. Integrating the second equation in (4.10), we use (4.9) and (4.13) to arrive at that p 2 (x, y) = −b ′′ 1 (x) y 0 (U (z) − c) 2 f r(z) dz + b 2 (x), where f r is defined in (4.8). It then follows from the second equation in (4.11) that (4.14) gη 2 (x) = −b ′′ 1 (x) h0 0 (U (y) − c) 2 f r(y) dz + b 2 (x). This furnishes an explicit formula relating η 2 and p 1 (·, 0) (see (4.5)) and p 2 (·, 0), provided with the background shear flow and the wave speed satisfying (3.4). It incorporates the effects of the vorticity through the solution of the Rayleigh equation (4.7). We may repeat the argument in the previous subsection to find u 2 and v 2 . Specifically, we infer from the first and the last equations in (4.10) that v 2 U − c y = p 2x + u 1 u 1x + v 1 u 1y (U − c) 2 , where p 2 is determined above and u 1 and v 1 are in (4.9). We determine v 2 upon integrating this. We then determine u 2 upon integrating the last equation in (4.10) and using that u 2 vanishes at infinity. To summarize, (4.15)                          u 2 (x, y) = −U ′ (y) y 0 x −∞ p 2x + u 1 u 1x + v 1 u 1y (U − c) 2 (w, z) dwdz − x −∞ p 2x + u 1 u 1x + v 1 u 1y U − c (w, y) dw, v 2 (x, y) = (U (y) − c) y 0 p 2x + u 1 u 1x + v 1 u 1y (U − c) 2 (x, z) dz, p 2 (x, y) = −b ′′ 1 (x) y 0 (U (z) − c) 2 f r(z) dz + b 2 (x), where f r is in (4.8) and u 1 and u 2 are in (4.9). Continuing, (2.8) and (2.9), (2.10) at the order of ǫ 2 become: (4.16)      (U − c)u 3x + U ′ v 3 + u 1 u 2x + u 2 u 1x + v 1 u 2y + v 2 u 1y = −p 3x , (U − c)v 2x + u 1 v 1x + v 1 v 1y = −p 3y , u 3x + v 3y = 0 in 0 < y < h 0 and v 3 = (U − c)η 3x + u 1 η 2x + u 2 η 1x and gη 3 = p 3 at y = h 0 , (4.17) v 3 = 0 at y = 0. Suppose that the dynamic pressure at the fluid bed is prescribed to the order of ǫ 3 : (4.18) p 3 (x, 0) = b 3 (x), and we continue to assume that b 3 is smooth and decays to zero as x → ±∞. Integrating the second equation in (4.16), we use (4.18) to arrive at that p 3 (x, y) = − y 0 ((U − c)v 2x + u 1 v 1x + v 1 v 1y )(x, z) dz + b 3 (x), where u 1 , v 1 and v 2 are in (4.9) and (4.15). Therefore, (4.19) gη 3 (x) = − h0 0 ((U − c)v 2x + u 1 v 1x + v 1 v 1y )(x, y) dy + b 3 (x). This furnishes an explicit formula relating η 3 and p j (·, 0), j = 1, 2, 3, provided with the background shear flow and the wave speed satisfying (3.4). It incorporates the effects of vorticity through the solution of the Rayleigh equation (4.7). We may repeat the above argument and continue to higher-order approximations. As a matter of fact, one is able to derive explicit formulae relating η j as functions of p j ′ (·, 0) for each j and for all j ′ = 1, 2, . . . , j. Expressions become quite complicated, however, and hence we do not pursue here. Examples. We illustrate the results in Section 4.1 and Section 4.2 by discussing some examples. Throughout the subsection, we assume for simplicity that b 2 (x, 0) = b 3 (x, 0) = 0. Example 4.1 (Zero vorticity). In the case of U ≡ 0, namely the zero vorticity, a straightforward calculation reveals that f r(y) = y/c 2 , whence (4.9) and (4.6) becomes            u 1 (x, y) = b 1 (x) c , v 1 (x, y) = − b ′ 1 (x) c y, p 1 (x, y) = b 1 (x), and gη 1 (x) = b 1 (x). Of course, this represents the hydrostatic approximation (1.1). To proceed, (4.15) and (4.14) become              u 2 (x, y) = − 1 2 b ′′ 1 (x) c y 2 + 1 2 b 2 1 (x) c 3 v 2 (x, y) = 1 6 b ′′′ 1 (x) c y 3 − b 1 (x)b ′ 1 (x) c 3 y p 2 (x, y) = − 1 2 b ′′ 1 (x)y 2 and (4.20) gη 2 (x) = − 1 2 h 2 0 b ′′ 1 (x). Continuing, (4.19) becomes (4.21) gη 3 (x) = 1 24 h 4 0 b (4) 1 (x) − h 0 b ′ 1 (x) 2 c . We compare the results to asymptotic approximations of the exact formula in [OVDH12], for instance. The results in [OVDH12, Section 4.2] may be written, in the dimensionless variables, abusing notation, as            η 1 (x) = b 1 (x), η 2 (x) = − 1 2 b ′′ 1 (x), η 3 (x) = 1 24 b (4) 1 (x) − b 1 (x)b ′′ 1 (x) − 1 2 b ′ 1 (x) 2 c 2 + 1 c 2 . The second equation agrees with (4.20) up to physical constants and the last equation agrees with (4.21) up to physical constants except the middle term. Note that c ≈ 1 in the non-dimensionalization. Example 4.2 (Constant vorticity). Let U (y) = γy, 0 y h 0 , for some constant γ. (More generally, one may take U (y) = γy + U 0 for some constant U 0 , but U 0 may be absorbed into the wave speed.) This models the constant vorticity −γ. A straightforward calculation reveals that f r(y) = 1 c y c − γy , whence (4.9) and (4.6) become            u 1 (x, y) = b 1 (x) c , v 1 (x, y) = − b ′ 1 (x) c y, p 1 (x, y) = b 1 (x), and gη 1 = b 1 (x). To proceed, (4.14) becomes gη 2 (x) = b ′′ 1 (x) c 1 3 γh 3 0 − 1 2 ch 2 0 . Other solutions may be computed explicitly using (4.15) and (4.19). Expressions are quite complicated, however, and we do not record them here. and evaluate (4.14) to find that gη 2 (x) = − b ′′ 1 (x) 30(c − h 2 0 ) 3/2 h 0 (15c 2 − 20ch 2 0 + 8h 4 0 ) tan −1 h 0 c − h 2 0 + c − h 2 0 h 2 0 (4c − h 2 0 ) + 4(c − h 2 0 ) 2 log c − h 2 0 c . Other solutions may be computed explicitly. Expressions are quite complicated and we do not record them here. Appendix A. The zero vorticity case, revisited We discuss the exact formula in [OVDH12] relating the Stokes wave with zero vorticity and the pressure at the bed. Under the assumption that the flow is irrotational, one may rewrite the governing equations of the water wave problem in terms of the velocity potential φ(x, y; t) and the surface elevation η(x; t) from the undisturbed depth: (A.1)              ∆φ = 0 in 0 < y < h 0 + η(x; t), φ t + 1 2 (φ 2 x + φ 2 y ) = p in 0 < y < h 0 + η(x; t), φ y = η t + φ x η x and p = gη at y = h 0 + η(x; t), φ y = 0 at y = 0. Recall that p is the dynamic pressure, measuring the departure from the hydrostatic pressure. Let q(x; t) = φ(x, h 0 + η(x; t); t) represent the trace of the velocity potential at the fluid surface y = η(x; t). We appeal to the chain rule and use the former of the third equations in (A.1) to show that φ x = q x − η x η t 1 + η 2 x and φ y = η t + η x q x 1 + η 2 x at y = h 0 + η(x; t). Similarly, φ t = q t − η t (η t + η x q x ) 1 + η 2 x at y = h 0 + η(x; t). In the moving coordinate frame, where q and η are functions of x − ct, φ x = q x + cη 2 x 1 + η 2 x , φ y = (q x − c)η x 1 + η 2 x and φ t = −cq x + cη 2 x (q x − c) 1 + η 2 x at y = h 0 + η(x). After substitution, the second equation and the latter of the third equations in (A.1) imply that q 2 x − 2cq x − c 2 η 2 x + 2gη(1 + η 2 x ) = 0. Therefore q x = c ± (c − 2gη)(1 + η 2 x ). We choose the − sign to guarantee that u − c < 0 throughout the fluid region. Consequently (A.2) φ x (x, h 0 + η(x)) = c − c 2 − 2gη(x) 1 + η 2 x (x) . Restricting the second equation in (A.1) at y = 0, moreover, we arrive in the moving coordinates frame at that −cφ x + 1 2 φ 2 x = −p at y = 0. Therefore φ x (x, 0) = c ± c 2 − 2p(x, 0). We choose the − sign to guarantee that u − c < 0 throughout the fluid region. To recapitulate, in the surface reconstruction from the pressure at the bed for Stokes waves with zero vorticity, one solves the boundary value problem for the Laplace equation: q n cosh(n(h 0 + η(x)))e inx = n =0 2π 0 (c − c 2 − 2p(x, 0))e inx dx cosh(n(h 0 + η(x)))e inx . (A.4) This furnishes an implicit formula relating η and p(·, 0), provided with a suitable wave speed. It agrees with (22) in [OVDH12], for which the sum on the right side ranges over all integers, since q 0 = 0. Nevertheless, we emphasize that the zeroth Fourier mode is excluded, which is important below in the derivation of asymptotic approximations. To proceed, we assume that ǫ ≪ 1 and that η = ǫη 1 + ǫ 2 η 2 + · · · and p(·, 0) = ǫp 1 . Substituting these, we expand the left side of (A.4) in the Taylor fashion to write that c − c 2 − 2gη 1 + η 2 x =c − (c 2 − 2gǫη 1 − 2gǫ 2 η 2 − · · · )(1 − ǫ 2 η 2 1x + · · · ) =c − c 2 − 2gǫη 1 − 2gǫ 2 η 2 − ǫ 2 c 2 η 2 1x + · · · =c − c 1 − 2 g c 2 ǫη 1 + g c 2 ǫ 2 η 2 + ǫ 2 η 2 1x + · · · =c − c 1 − g c 2 ǫη 1 + g c 2 ǫ 2 η 2 + ǫ 2 η 2 1x − 1 2 g 2 c 4 ǫ 2 η 2 1 + · · · = g c ǫη 1 + g c ǫ 2 η 2 + 1 2 cǫ 2 η 2 1x + g 2 2c 3 ǫ 2 η 2 1 + · · · . (A.5) Similarly, cosh(n(h 0 + η)) = cosh(nh 0 + nǫη 1 + nǫ 2 η 2 + · · · ) = cosh(nh 0 ) + sinh(nh 0 )(nǫη 1 + nǫ 2 η 2 + ...) + 1 2 cosh(n 2 ǫ 2 η 2 1 + · · · ). (A.6) and c − c 2 − 2b =c − c 2 − 2ǫp 1 + · · · =c − c 1 − 2 1 c 2 ǫp 1 + · · · = 1 c ǫp 1 + 1 2c 3 ǫ 2 p 2 1 + · · · . (A.7) Let's write that p 1 (x) = ǫ ∞ n=−∞ b n e inx . Substituting (A.5)-(A.7) into (A.4), at the order of ǫ, we gather that gη 1 (x) = n =0 cosh(nh 0 )b n e inx . This agrees with (51) in [OVDH12] if it is written in the dimensional variables. Continuing, at the order of ǫ 2 , we gather that (A.8) g c η 2 + c 2 η 2 1x + g 2 c 3 η 2 1 = n =0 cosh(nh 0 ) 1 2 p 2 1 c 3 + η 1 n =0 n sinh(nh 0 ) p 1 c . This is similar to (52) in [OVDH12] in the dimensional variables, but the convolution sums must ranges over nonzero integers in the Stokes wave setting. To illustrate and to compare the results with those in Example 3.1, we furthermore assume that p 1 (x) = b cos(x). We then find, instead, that (A.9) gη 1 (x) = cosh(h 0 )b cos(x). p 1 1(x, y) = b cosh(y) cos(x), and (3.28) gη 1 (x) = b cosh(h 0 ) cos(x). and v 1 = (U − c)η 1x and gη 1 = p 1 at y = h 0 , Example 4. 3 ( 3Poiseuille flows). Let U (y) = h 2 0 − y 2 , 0 y h 0 . This models Poiseuille flows. Note that c > max 0 y h0 U (y) = h 2 0 to guarantee that solitary water waves exist.One may evaluate (4.8) using Mathematica to find that = c − c 2 − 2p(x, 0) and φ y = 0 at y = 0; one then uses (A.2) to determine η in terms of p(x, (y)e inx and c − c 2 − 2p(− c 2 − 2p(x, 0))e inx dx.For each n = 0, note that (A.3) leads to the Cauchy problem for the linear secondorder constant-coefficient ODE: φ ′′ n − n 2 φ n = 0 for y > 0, φ ′ n (0) = 0 and inφ n (0) = q n , whence φ(x, y) = n =0 −iq n cosh(ny) e inx n up to addition by a constant. Note that q 0 = 0 and φ 0 is an arbitrary constant. It then follows from (A. Acknowledgements. VMH is supported by the National Science Foundation grant CAREER DMS-1352597, an Alfred P. Sloan research fellowship, and an Arnold O. Beckman research award RB14100, the Center for Advanced Study Beckman fellowship at the University of Illinois at Urbana-Champaign. MRL is supported through an Arnold O. Beckman research award RB14100 at the University of Illinois at Urbana-Champaign.To proceed, (A.8) becomesThe second and third equalities use (3.2), where k = 1. The constant term on the right side becomes zero if the sum ranges over all integers. On the Stokes conjecture for the wave of extreme form. C J Amick, L E Fraenkel, J F Toland, 193-214. MR 666110Acta Math. 14835147C. J. Amick, L. E. Fraenkel, and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193-214. MR 666110 (83m:35147) On solitary water-waves of finite amplitude. J Charles, John F Amick, Toland, 9-95. MR 629699Arch. Rational Mech. Anal. 76176017Charles J. Amick and John F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9-95. MR 629699 (83b:76017) Measuring waves with pressure transducers. 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{'abstract': "We propose higher-order approximation formulae recovering the surface elevation from the pressure at the bed and the background shear flow for small-amplitude Stokes and solitary water waves. They offer improvements over the pressure transfer function and the hydrostatic approximation. The formulae compare reasonably well with asymptotic approximations of the exact relation between the pressure at the bed and the surface wave in the zero vorticity case, but they incorporate the effects of vorticity through solutions of the Rayleigh equation. Several examples are discussed.HUR AND LIVESAYHere and throughout,stands for the Fourier transform of the function x → f (x). Note that (1.2) becomes (1.1) in the limit as h 0 → 0.Laboratory experiments in[BD87], for instance, support that (1.2) satisfactorily predicts the wave height. Furthermore one can derive it consistently in the regime of small-amplitude Stokes waves in the case of zero vorticity; see [ES08], for instance. On the other hand, numerous studies raised doubts about the adequacy of using the linear theory; see [HmHK66,Cav80,Bie82,LW84,KC94], for instance. Note that the effects of nonlinearity and current are not negligible in shallow water or in the surf zone; see [LW84], for instance.Remarkably, exact relations were derived in [OVDH12,Con12,CC13] between the trace of the pressure at the horizontal bed and the surface elevation for Stokes and solitary water waves. In particular, the formulae apply to large amplitude waves. They are implicit but, nevertheless, easily implemented in numerical computations, and the results agree to varying degrees with laboratory experiments; see [DOV12], for instance.The arguments strongly use that in the case of zero vorticity, one is to solve the Cauchy problem for the Laplace equation. Unfortunately they cannot accommodate underlying shear flows and other physical aspects. We pause to remark that real flows are hardly irrotational. Rather vorticity is generated, for instance, by density stratification, the shear force of the wind, currents or tidal forces, and the effects of bathymetry. At present, no exact relations are available between the pressure at the bed and the surface wave in rotational flows. Furthermore numerical schemes approximating the exact formulae do not converge, because the Cauchy problem for an elliptic PDE is ill-posed.Recently in [CHW15], one of the authors elaborated (1.2) and (1.1) to permit vorticity and density stratification. Specifically, the pressure transfer function and the hydrostatic approximation were consistently derived for small-amplitude surface and interface waves in an arbitrary shear flow. Unfortunately they do not capture the effects of nonlinearity. Furthermore the hydrostatic approximation does not sense the effects of vorticity.Here we take matters further and propose higher-order approximation formulae recovering the surface elevation from the pressure at the bed for small-amplitude Stokes and solitary water waves in an arbitrary shear flow. Specifically, we compute higher-order correction terms to the pressure transfer function and the hydrostatic approximation in[CHW15]. To the best of the authors' knowledge, these are new. We carry out higher-order perturbations of the governing equations, rather than relying on a less empirical approach of higher-order Stokes expansion. We sacrifice the ability to accommodate large amplitude waves, but we are able to work to an arbitrary, albeit finite, degree of accuracy, when exact formulae relating the pressure at the bed and the surface wave are unavailable.The formulae incorporate the effect of vorticity through solutions to the Rayleigh equation, which one must in general investigate numerically. But we make an effort to discuss some examples. In the case of zero vorticity, in particular, we demonstrate that our results compare reasonably well with asymptotic approximations of the exact formulae in [OVDH12], for instance; see Example 3.1 and Example 4.1. The", 'arxivid': '1510.02396', 'author': ['Vera Mikyoung ', 'Hur And ', 'Michael R Livesay '], 'authoraffiliation': [], 'corpusid': 119702383, 'doi': '10.1016/j.euromechflu.2016.08.003', 'github_urls': [], 'n_tokens_mistral': 23797, 'n_tokens_neox': 19845, 'n_words': 11444, 'pdfsha': 'bc548a95c1b4e65eedefd0060b123e363112091b', 'pdfurls': ['https://arxiv.org/pdf/1510.02396v1.pdf'], 'title': ['ON THE RECOVERY OF TRAVELING WATER WAVES WITH VORTICITY FROM THE PRESSURE AT THE BED', 'ON THE RECOVERY OF TRAVELING WATER WAVES WITH VORTICITY FROM THE PRESSURE AT THE BED'], 'venue': []}

arxiv

SOME NECESSARY AND SOME SUFFICIENT CONDITIONS FOR THE COMPACTNESS OF THE EMBEDDING OF WEIGHTED SOBOLEV SPACES 30 Jan 2003 Francesca Antoci SOME NECESSARY AND SOME SUFFICIENT CONDITIONS FOR THE COMPACTNESS OF THE EMBEDDING OF WEIGHTED SOBOLEV SPACES 30 Jan 2003 We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaceswhere w is some weight on a domain Ω ⊂ R n . Introduction In the present paper, we give some necessary and sufficient conditions for the compactness of the embedding of Sobolev spaces (1.1) W 1,p (Ω, w) −→ L p (Ω, w). Our investigations were originated by a recent paper appeared in the Annals of Statistics ( [17]), in which a new definition of Nonlinear Principal Components is introduced as follows: if X is an absolutely continuous random vector on an open connected set Ω ⊆ R n , with density function f X , zero expectation and finite variance, the j th Nonlinear Principal Component of X is defined as a solution ϕ j of the maximization problem (1.2) max ψ(X)∈W 1,2 (Ω,f X ) E(ψ(X) 2 ) E(|∇ψ(X)| 2 ) subject to the conditions E(ψ(X), ϕ s (X)) = 0 for s = 1, 2, ..., j − 1, j > 1. Moreover, it is required that ϕ j has zero expectation. Here E(ψ, ϕ) denotes the usual scalar product in the Hilbert space L 2 (Ω, f X ) of square integrable functions on Ω with respect to the measure f X dx, and W 1,2 (Ω, f X ) is the weighted Sobolev space W 1,2 (Ω, f X ) = u ∈ L 2 (Ω, f X ) | ∀i = 1, ..., n, ∂ i u ∈ L 2 (Ω, f X ) . It is well-known that the compactness of the embedding (1.3) W 1,2 (Ω, f X ) −→ L 2 (Ω, f X ) turns out to be essential in order to prove the existence of an orthonormal set of Nonlinear Principal Components. The problem of the compactness of the embedding (1.1) for weighted Sobolev spaces has been studied by many authors (see e.g. [9], [10], [11]). For a rich bibliography on this kind of problems we refer also to [5], [14], [16], [19]. Nevertheless, the attention has mainly focused on those classes of weights which arise in the study of partial differential equations, such as polynomial weights in unbounded domains, or, in bounded domains, weights depending on the distance from the boundary and, as concerns the Poincaré-Wirtinger inequality, weights in A p classes and derivatives of quasi-conformal mappings. Under this point of view an enormous amount of work has been done in the last years and a complete list of contributions is not possible here. Just to have an idea, see for example [6], [7], [8], [13], [18] and the references therein. On the contrary, in the maximization problem (1.2) we have to deal with general density functions f X . Moreover, most of the criteria for the compactness of the embedding (1.3) which can be found in mathematical literature are not simple to be handled. In view of possible applications of Nonlinear Principal Components, it is important, instead, to have simple, easily applicable necessary and sufficient conditions on the density function f X and on the set Ω for the compactness of (1.3). In the present paper, we give some necessary conditions and sufficient conditions for the compactness of the embedding (1.1). In section 2, we recall some basic facts about weighted Sobolev spaces. In section 3, we prove the weighted versions of some necessary conditions for compactness due to Adams ([1]). In section 4, we prove a sufficient condition for the compactness of (1.1), showing that, under suitable hypothesis on the weight w, if the subgraph Ω w := {(x, y) ∈ R n × R | x ∈ Ω, 0 < y < w(x)} of the weight is such that the embedding W 1,p (Ω w ) −→ L p (Ω w ) is compact, then the embedding (1.1) is compact. In section 5, applying a sufficient condition for compactness in the non-weighted case due to Adams ([1]), we show some examples to which our sufficient condition is applicable. Preliminary facts Let Ω be an open domain in R n . We will denote by W (Ω) the set of all realvalued, measurable, a.e. in Ω positive and finite functions w(x). Elements in W (Ω) are called weight functions. For any Lebesgue-measurable set U ⊂ R n and for w ∈ W (Ω), we will denote by µ w (U ) the Borel measure defined by µ w (U ) = U w(x) dx. As usual, we will denote by C ∞ c (Ω) the set of all the smooth, compactly supported functions in Ω. Moreover, we will denote by L p (Ω, w), for 1 ≤ p < ∞, the set of measurable functions u = u(x) such that (2.1) u L p (Ω,w) = Ω |u(x)| p w(x) dx 1 p < +∞. It is a well-known fact that the space L p (Ω, w), endowed with the norm (2.1) is a Banach space. The following Proposition holds Proposition 2.1. Let 1 ≤ p < ∞. If the weight w(x) is such that (2.2) w(x) − 1 p−1 ∈ L 1 loc (Ω) (in the case p > 1) (2.3) ess sup x∈B 1 w(x) < +∞ (in the case p=1), for every ball B ⊂ Ω, then (2.4) L p (Ω, w) ⊆ L 1 loc (Ω). For a proof, see [15], [19]. As a consequence, under condition (2.2) ((2.3)), convergence in L p (Ω, w) implies convergence in L 1 loc (Ω). Moreover, every function in L p (Ω, w) has distributional derivatives. Definition 2.2. Let Ω be an open set in R n , let 1 ≤ p < ∞, and let w(x) be a weight function satisfying condition (2.2) (resp. (2.3)). We define the Sobolev space W m,p (Ω, w) as the set of those functions u ∈ L p (Ω, w) such that their distributional derivatives D α u, for |α| ≤ m, belong to L p (Ω, w). It is a well-known fact that Theorem 2.3. If w(x) fulfills condition (2.2) (resp.(2.3)), W m,p (Ω, w) is a Banach space. For a proof, we refer to [15]. Throughout the paper, we will always assume that w satisfy condition (2.2) (resp. (2.3)). Necessary conditions for compactness In this section we derive some necessary conditions for the compactness of the embedding (3.1) W 1,p (Ω, w) −→ L p (Ω, w), which are generalizations to the weighted case of analogous results obtained by Adams ([1], [3]) for non-weighted Sobolev spaces. Let w(x) ∈ W (Ω) be a weight on an open set Ω. Let T be a tesselation of R n with n-cubes of edge h. If H ∈ T , we will denote by N (H), as in [1], the cube of side 3h concentric with H and having faces parallel to those of H, and by F (H) the fringe of H, defined by F (H) := N (H) \ H. The natural extension to the weighted case of the concept of λ-fatness employed by Adams is given by the following Definition 3.1. Let λ > 0. A cube H ∈ T is called (λ, w)-fat (with respect to Ω) if (3.2) µ w (H ∩ Ω) > λ µ w (F (H) ∩ Ω). If H is not (λ, w)-fat, it is called (λ, w)-thin. As in the non-weighted case, the following property holds: Theorem 3.2. Let 1 ≤ p < ∞. If the embedding (3.1) is compact, then for every λ > 0 and for every tesselation T of fixed edge h, T has only finitely many (λ, w)-fat cubes. Proof. The thesis follows from an easy extension of the proof of Theorem 6.33 in [1], where the Lebesgue measure µ is replaced by µ w and λ-fat cubes are replaced by (λ, w)-fat cubes. Remark Theorem 3.2 implies that if w is a weight on R n which has the doubling property, the embedding (3.1) cannot be compact. In particular, the embedding (3.1) for A p weights on R n cannot be compact. A consequence of Theorem 3.2 is the following result, which is the extension to the weighted case of Theorem 6.37 in [1]. The boundedness of the weight plays a crucial role and cannot be removed; a counterexample is given by the weight w(x) = x α on I = (0, 1) ⊂ R for α ≤ −1, for which the embedding is compact (see [10]) even if I w(t) dt = +∞. Theorem 3.3. If w(x) is bounded and the embedding (3.1) is compact, then necessarily (3.3) Ω w(x) dx < +∞. Proof. The proof essentially follows the argument in [1]; we will give some details in order to show where the boundedness of the weight is essential. Let T be a tesselation of R n by cubes of unitary edge, and let λ = (2(3 n − 1)) −1 . If P is the union of the finitely many (λ, w)-fat cubes of T , then µ w (P ∩Ω) ≤ µ w (P ) < +∞. If H is a (λ, w)-thin cube of T , thanks to the choice of λ we can choose H 1 ∈ F (H) such that µ w (H ∩ Ω) ≤ 1 2 µ w (H 1 ∩ Ω). Analogously, if also H 1 is (λ, w)-thin, we can choose H 2 ∈ F (H 1 ) such that µ w (H 1 ∩Ω) ≤ 1 2 µ w (H 2 ∩Ω). If going on in this way we can construct an infinite chain {H, H 1 , H 2 , ...} of (λ, w)-thin cubes, then for every j ∈ N µ w (H ∩ Ω) ≤ 1 2 j µ w (H j ∩ Ω), whence, thanks to the boundedness of w, µ w (H ∩ Ω) ≤ C 2 j for some positive constant C for every j ∈ N; thus, µ w (H ∩ Ω) = 0. As a consequence, if we denote by P ∞ the union of all the (λ, w)-thin cubes in T for which it is possible to construct such an infinite chain, µ w (P ∞ ∩ Ω) = 0. Let P j denote the union of all (λ, w)-thin cubes H ∈ T such that any chain of this type stops at the j-th step (that is, such that H j is (λ, w)-fat). Following the proof of [1], we get µ w (P j ∩ Ω) ≤ (2j + 1) n 2 −j µ w (P ∩ Ω), whence +∞ j=1 µ w (P j ∩ Ω) ≤ µ w (P ∩ Ω) +∞ j=1 (2j + 1) n 2 −j < +∞. Since R n = P ∪ P ∞ ∪ P 1 ∪ P 2 ∪ ..., the thesis follows. Some stronger necessary conditions for compactness are given in the following Theorem, which is a generalization to the weighted case of Theorem 6.40 in [1]: Theorem 3.4. Let w(x) be a continuous, bounded weight. For every r > 0, let Ω r , S r be defined as Ω r := {x ∈ Ω | |x| > r} S r := {x ∈ Ω | |x| = r} . Moreover, let us denote by A r the surface area, with respect to the weight w, of S r . Then if the embedding (3.1) is compact, (1) for every ǫ > 0, δ > 0, there exists R > 0 such that if r ≥ R µ w (Ω r ) ≤ δ µ w ({x ∈ Ω | r − ǫ ≤ |x| ≤ r}); (2) if A r is positive and ultimately decreasing as r → +∞ then for every ǫ > 0 lim r→+∞ A r+ǫ A r = 0. Proof. The thesis follows from an easy extension of the argument in [1], where the Lebesgue measure µ is replaced by µ w , λ-fat cubes are replaced by (λ, w)-fat cubes and A r is computed with respect to the weight w. A consequence of Theorem 3.4 is the following Corollary, whose proof is a simple generalization to the weighted case of the proof of Corollary 6.41 in [1], and is therefore omitted. A sufficient condition Let w ∈ W (Ω) be a lower semicontinuous weight defined on an open set Ω ⊆ R n . Let us suppose that w vanishes only on a closed subset Ω 0 ⊂ Ω. Moreover, let Ω ∞ := {x ∈ Ω | w(x) = +∞} , and suppose that Ω ∞ is closed. Both Ω 0 and Ω ∞ have (Lebesgue) measure equal to zero. Moreover, we suppose that w is bounded from above and from below by positive constants on any compact set K ⊂ Ω \ (Ω 0 ∪ Ω ∞ ). Let us denote by Ω w the subgraph of the weight w(x), that is, the open set (4.1) Ω w := {(x, y) ∈ R n × R | x ∈ Ω, 0 < y < w(x) } and consider the map J : W 1,p (Ω, w) −→ W 1,p (Ω w ) defined by (Ju)(x, y) = u(x) a.e.. J is well-defined. It is not difficult to see that if u ∈ J(W 1,p (Ω, w)), then the distributional derivative in the y-direction ∇ y u is equal to zero. J is an isometry of W 1,p (Ω, w) onto J(W 1,p (Ω, w)) since for every u ∈ W 1,p (Ω, w) Ju p W 1,p (Ωw) = Ωw |Ju(x, y)| p dxdy + Ωw |∇ x (Ju)(x, y)| p dx dy = Ω w(x) 0 |u(x)| p dy dx + Ω w(x) 0 |∇ x u(x)| p dy dx = = Ω |u(x)| p w(x) dx + Ω |∇ x u(x)| p w(x) dx. We will denote by W 1,p y (Ω w ) the set J(W 1,p (Ω, w)). Moreover, we will denote by L p y (Ω w ) the completion of W 1,p y (Ω w ) with respect to the norm of L p (Ω w ). Lemma 4.1. If w(x) satisfies the conditions above, then C ∞ c (Ω \(Ω 0 ∪ Ω ∞ )) is dense in L p (Ω, w), for 1 ≤ p < +∞. Proof. It suffices to show that, given f ∈ L p (Ω, w), for every ǫ > 0 there exists g ∈ C ∞ c (Ω \ (Ω 0 ∪ Ω ∞ )) such that f − g L p (Ω,w) < ǫ. Let {Ω n }, n ∈ N, be an exhaustion of Ω, defined by Ω n := x ∈ Ω | min {d(x, Ω 0 ), d(x, Ω ∞ ), d(x, ∂Ω)} > 1 n . Let f ∈ L p (Ω, w); for every ǫ > 0, there exists n such that ( Ω\Ω n |f (x)| p w(x) dx) 1 p < ǫ 2 . Since w is bounded from above and from below by positive constants on Ω n , u ∈ L p (Ω n , w) if and only if u ∈ L p (Ω n ) and there exist C 1 , C 2 > 0 such that for every u ∈ L p (Ω n , w) C 2 u L p (Ω n ) ≤ u L p (Ω n ,w) ≤ C 1 u L p (Ω n ) . Hence f |Ω n ∈ L p (Ω n ). As a consequence, there exists a function g ∈ C ∞ c (Ω n ) ⊂ C ∞ c (Ω \ (Ω 0 ∪ Ω ∞ )) such that g − f |Ω n L p (Ω n ) < (2C 1 ) −1 ǫ. Hence, g − f |Ω n L p (Ω n ,w) < ǫ 2 . This implies g − f L p (Ω,w) = g − f L p (Ω n ,w) + f L p (Ω\Ω n ,w) < ǫ. Hence, since C ∞ c (Ω \ (Ω 0 ∪ Ω ∞ )) ⊂ W 1,p (Ω, w), W 1,p (Ω, w) is dense in L p (Ω, w). Since for every u ∈ W 1,p (Ω, w) Ω |u(x)| p w(x) dx = Ωw |Ju(x, y)| p dxdy, we get Lemma 4.2. If w(x) satisfies the above conditions, J can be extended to an isometry J : L p (Ω, w) −→ L p y (Ω w ). As a consequence: Theorem 4.3. Let w(x) be a weight, satisfying the above conditions. If the subgraph Ω w of w(x) is such that the embedding I Ωw : W 1,p (Ω w ) −→ L p (Ω w ) is compact, then the embedding (3.1) is compact. Proof. The embedding (3.1) is compact if and only if the embedding I y : W 1,p y (Ω w ) −→ L p y (Ω w ) is compact. But I y coincides with P y • I Ωw • I, where I is the immersion I : W 1,p y (Ω w ) −→ W 1,p (Ω w ) and P y denotes the "projection" P y : L p (Ω w ) −→ L p y (Ω w ), defined for a.e. x ∈ Ω by (P y u)(x) := 1 w(x) w(x) 0 u(x, y) dy. Since P y and I are continuous, the thesis follows. We remark that the sufficient condition of Theorem 4.3 is not a necessary condition; as a matter of fact, whilst the embedding I Ωw can be compact only if we assume a certain regularity for the weight w, the embedding (3.1) can be compact even if the weight w is extremely irregular, once provided that it is controlled from above and below by a "regular" weight Φ . Indeed, the following Proposition holds 4.2) W 1,p (Ω, Φ) −→ L p (Ω, Φ) is compact. Let w(x) be a weight such that there exist α, β > 0 such that a.e. in Ω αΦ(x) ≤ w(x) ≤ βΦ(x). Then the embedding W 1,p (Ω, w) −→ L p (Ω, w) is compact. Proof. It is immediate that u ∈ L p (Ω, Φ) if and only if u ∈ L p (Ω, w) and α u L p (Ω,Φ) ≤ u L p (Ω,w) ≤ β u L p (Ω,Φ) . Analogously, u ∈ W 1,p (Ω, Φ) if and only if u ∈ W 1,p (Ω, w) and α u W 1,p (Ω,Φ) ≤ u W 1,p (Ω,w) ≤ β u W 1,p (Ω,Φ) . Hence, if {u n } is a bounded sequence in W 1,p (Ω, w), it is bounded also in W 1,p (Ω, Φ). Due to the compactness of the embedding (4.2), there exists a subsequence {u n k } such that u n k converges in L p (Ω, Φ). Hence, u n k converges also in L p (Ω, w). Some applications Let us now state some simple applications of Theorem 4.3. In the nonweighted case the following sufficient condition for compactness holds : (2) there exist a flow Φ : U → Ω and a constant c > 0 such that if Ω N = Ω \ Ω * N then (a) Ω N × [0, c] ⊂ U for every N ; (b) Φ t is one-to-one, for every t; (c) there exists M > 0 such that for every (x, t) ∈ U |∂ t Φ(x, t)| ≤ M ; (3) the functions d N (t) = sup x∈Ω N | det JΦ t (x)| −1 satisfy (a) lim N →∞ d N (c) = 0; (b) lim N →∞ c 0 d N (t) dt = 0, then the embedding W 1,p (Ω) −→ L p (Ω) is compact. We recall that a flow on Ω is a continuously differentiable map Φ : U → Ω, where U is an open set in Ω × R containing Ω × {0}, with Φ(x, 0) = x for every x ∈ Ω. Moreover, we denote by Φ t the map Φ t : x −→ Φ(x, t), and by JΦ t the Jacobian matrix of Φ t . Theorem 5.1, together with Theorem 4.3, can be used to get compactness results for weighted Sobolev spaces. Our first result is connected with Example 6.49 in [1]: Lemma 5.2. Let Ω be a bounded domain in R n , with C ∞ boundary, and let w(x) ∈ C 1 (Ω) be a weight on Ω, positive in every compact set K ⊂ Ω, such that, if we denote by r(x) the distance r(x) = dist(x, ∂Ω), near to the boundary w(x) can be expressed as w(x) = f (r), where f ∈ C 1 is positive, nondecreasing, has bounded derivative f ′ and satisfies lim r→0 + f (r) = 0; then the embedding (3.1) is compact. Proof. Since the boundary is regular, there exist an open neighbourhood V of ∂Ω in Ω, a constant a > 0 and a diffeomorphism ∂Ω × [0, a) −→ V, (ξ, r) −→ x(ξ, r) such that x(ξ, r) ∈ ∂Ω if and only if r = 0. We can suppose that r is equal to the distance of x(ξ, r) from the boundary. Since w is strictly positive in Ω \ V , the embedding (3.1) is compact if and only if W 1,p (V, w) is compactly embedded in L p (V, w). Let us consider the subgraph V w of w |V , V w = {(ξ, r, y) ∈ ∂Ω × [0, a) × R | r > 0, 0 < y < f (r)} , and, for N ∈ N, the sets (V w ) N := (ξ, r, y) ∈ V w | 0 < r ≤ 1 N . For N ∈ N, the open sets (V w ) * N := V w \ (V w ) N are such that (V w ) * N ⊆ (V w ) * N +1 ; moreover, they satisfy the cone property, hence the embedding W 1,p ((V w ) * N ) −→ L p ((V w ) * N ) is compact for every N ∈ N. Let us consider the flow Φ(ξ, r, y, t) := ξ, r + t, f (r + t) f (r) y defined on the set U = {(ξ, r, y, t) ∈ ∂Ω × [0, a) × R × R | (ξ, r, y) ∈ V w , −r < t < a − r} . We simply have to check that the sets (V w ) * N and the flow Φ satisfy the conditions of Theorem 5.1. It is easy to see that (V w ) N × [0, a 2 ] ⊂ U for every N ∈ N. Φ t is one-to-one for every t, since det(JΦ t ) = f (r + t) f (r) > 0. Moreover, |∂ t Φ(ξ, r, y, t)| = |(0, 1, f ′ (r + t) f (r) y)| ≤ M for some M > 0 since f ′ is bounded and | y f (r) | < 1 on V w . Further, d N (t) := sup (Vw) N | det(JΦ t )| −1 = sup (Vw) N | f (r) f (r + t) | satisfies lim N →+∞ d N a 2 = lim r→0 + f (r) f (r + a 2 ) = 0. Analogously, for every t > 0 Hence, by Theorem 5.1, the embedding I Vw is compact, and Theorem 4.3 implies that the embedding (3.1) is compact. Remark In particular, for f (r) = r α , α ≥ 1, we find compact embedding as in [10]. This holds also for α = p − 1. In the case of a radial weight on R n , combining Theorems 5.1, 4.3 and 3.4 we can even get a necessary and sufficient condition. The following result is a sort of "radial" version of Example 6.48 in [1]: Lemma 5.3. Let Ω = R n , and w(x) be a radial function w(x) = g(r) where r = |x| and g ∈ C 1 ([0, +∞)) is positive, nonincreasing, with bounded derivative g ′ ; then the embedding (3.1) is compact if and only if (5.1) lim s→+∞ g(s + ǫ) g(s) = 0 for every ǫ > 0. Proof. Suppose, first, that (5.1) holds for every ǫ > 0. Let us consider, on R n , polar coordinates (r, θ). Then the subgraph of w can be described by Ω w = {(r, θ, y) | 0 < y < g(r)} . For every N ∈ N, let us consider the set (Ω w ) N := {(r, θ, y) ∈ Ω w | r ≥ N } . Then (Ω w ) * N := Ω w \ (Ω w ) N is bounded and has the cone property; hence, the embedding W 1,p ((Ω w ) * N ) −→ L p ((Ω w ) * N ) is compact for every N ∈ N. Moreover (Ω w ) * N ⊂ (Ω w ) * N +1 for every N ∈ N. An easy computation shows that the flow Φ(r, θ, y, t) := r − t, θ, g(r − t) g(r) y defined on the set U := {(r, θ, y, t) | 0 < t < r} , satisfies the conditions of Theorem 5.1 (with c = 1). As a consequence, the embedding I Ωw is compact, and Theorem 4.3 yields the thesis. Conversely, suppose that the embedding (3.1) is compact. Then by Theorem 3.4 A r = |x|=r g(r)r n−1 drdθ = C(n)r n−1 g(r) must fulfill the condition lim r→+∞ A r+ǫ A r = 0. As a consequence, lim r→+∞ g(r + ǫ) g(r) = lim r→+∞ C(n)(r + ǫ) n−1 g(r + ǫ) C(n)r n−1 g(r) = 0. Remark In particular, for g(r) = e αr and for g(r) ∼ r α as r → +∞ (α < 0) we get that there is no compactness, as stated in [11]. Theorem 5.1, together with Theorem 4.3, can be used also to deal with weights which are not bounded from above. Lemma 5.4. Let Ω be a bounded domain in R n , with C ∞ boundary, and let w(x) ∈ C 1 (Ω) be a weight on Ω, positive in every compact set K ⊂ Ω; moreover, we suppose that, if we denote by r(x) the distance r(x) = dist(x, ∂Ω), near to the boundary w(x) can be expressed as w(x) = f (r), where f (r) → +∞ as r → 0 + , f is strictly decreasing on 0 < r < δ for some δ > 0, |f ′ (r)| ≥ 1 C and (5.2) lim y→+∞ f −1 (y + ǫ) f −1 (y) = 0 for every ǫ > 0. Then the embedding (3.1) is compact. Proof. As in Lemma 5.2, it suffices to show that W 1,p (V, w) is compactly embedded in L p (V, w), where V is a tubular neighbourhood of ∂Ω in Ω. To this end, consider the subgraph V w of w |V , and, for N ∈ N, the sets (V w ) N := {(ξ, r, y) ∈ V w | n ≤ y < f (r)} . The open sets (V w ) * N := V w \ (V w ) N and the flow Φ(ξ, r, y, t) := ξ, f −1 (y − t) f −1 (y) r, y − t , defined for 0 < t < y < f (r), fulfill the conditions of Theorem 5.1. Hence, Theorem 4.3 implies that the embedding (3.1) is compact. Remark The previous Lemma does not apply to the weight w(x) = r α when α < 0. In this case, the compactness of the embedding (3.1) has been proved in [10]. Via a similar proof it can be shown that Lemma 5.5. Let Ω be an open domain in R n such that 0 ∈ Ω, and let w ∈ C 1 (Ω \ {0}) be a weight of the type w(x) = f (|x|), where f (s) → +∞ as s → 0 + , f is strictly decreasing on 0 < s < δ for some δ > 0, |f ′ (s)| ≥ 1 Corollary 3. 5 . 5Let w(x) be a continuous, upper bounded weight. If the embedding (3.1) is compact, then for every k ∈ Z lim r→+∞ e kr µ w (Ω r ) = 0. Proposition 4 . 4 . 44Let Φ be a weight for which the embedding ( Theorem 5.1. ([1]) Let Ω be an open set in R n . If (1) there exists a sequence {Ω * N } ∞ N =1 of open subsets of Ω such that Ω * N ⊆ Ω * N +1 and for every N the embedding W 1,p (Ω * N ) −→ L p (Ω * N ) is compact; d N (t) dt = 0. every ǫ > 0.Then the embedding (3.1) is compact. Finally, we show an example of weight not belonging to the A p class on a bounded domain:Lemma 5.6. Let us consider the set Ω := (− 1 2 , 1 2 ) and the weightThen the embedding (3.1) is compact.Proof. Consider the subgraph Ω w . Since the sethas the cone property, in order to prove the compactness of the embedding (3.1), it suffices to show that the embedding I Ω + w is compact, where Ω + w := (x, y) ∈ R 2 | x ∈ Ω, x > 0, 0 < y < w(x) . Hence in this case a Poincaré-Wirtinger inequality holds even if the weight does not belong to the A p class.To this purpose, it is not difficult to check that the subsets (Ω Sobolev spaces. R Adams, Academic PressNew YorkR. 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Statist., Vol. 26, No. 2, (1998), 596-616; Weighted inequalities and degenerate elliptic partial differential equations. E W Stredulinski, Lecture Notes in Mathematics. 1074Springer VerlagE. W. Stredulinski, Weighted inequalities and degenerate elliptic partial differential equations, Lecture Notes in Mathematics 1074, Springer Verlag, (1984), Berlin; B O Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces. Springer1736B.O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics 1736, Springer, 2000. Duca degli Abruzzi 24, 10129, Torino E-mail address: antoci@calvino. Politecnico Dipartimento Di Matematica, C Di Torino, polito.itDipartimento di Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129, Torino E-mail address: [email protected]

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{'abstract': 'We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaceswhere w is some weight on a domain Ω ⊂ R n .', 'arxivid': 'math/0301352', 'author': ['Francesca Antoci '], 'authoraffiliation': [], 'corpusid': 2902456, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10581, 'n_tokens_neox': 9302, 'n_words': 5321, 'pdfsha': 'b6e5b628dd68abccbe63370961bb3af2c79e53ad', 'pdfurls': ['https://arxiv.org/pdf/math/0301352v1.pdf'], 'title': ['SOME NECESSARY AND SOME SUFFICIENT CONDITIONS FOR THE COMPACTNESS OF THE EMBEDDING OF WEIGHTED SOBOLEV SPACES', 'SOME NECESSARY AND SOME SUFFICIENT CONDITIONS FOR THE COMPACTNESS OF THE EMBEDDING OF WEIGHTED SOBOLEV SPACES'], 'venue': []}

arxiv

BRST symmetry and W -algebra in higher derivative models 4 Sep 2013 Rabin Banerjee National Centre for Basic Sciences JD Block Sector III700 098Salt Lake City, KolkataIndia Biswajit Paul National Centre for Basic Sciences JD Block Sector III700 098Salt Lake City, KolkataIndia Sudhaker Upadhyay National Centre for Basic Sciences JD Block Sector III700 098Salt Lake City, KolkataIndia S N Bose National Centre for Basic Sciences JD Block Sector III700 098Salt Lake City, KolkataIndia BRST symmetry and W -algebra in higher derivative models 4 Sep 2013 In this paper we discuss the (anti-)BRST symmetries and W -algebra of higher derivative theories of relativistic particles satisfying general gauge conditions. Using this formalism, the connection between the (anti-)BRST symmetries and W -algebra for the massless particle with rigidity is established. Incidentally, the full W -algebra emerges only when the anti-BRST transformations are considered in tandem with the BRST ones. Further, the BRST symmetry is made finite and coordinate-dependent. We show that such finite coordinate-dependent BRST symmetry changes the BRST invariant gauge-fixing fermion within the functional integration. This is exploited to connect two different arbitrary gauge conditions. Introduction It is usual to consider theories where the Lagrangian has only single time derivative of the fields. But in some cases we need to consider terms where higher time derivative of the fields appear. Such theories are known as higher derivative (HD) theories. The concept of introduction of HD field is not new and has been considered by many authors and applied in diverse fields like electrodynamics [1,2], supersymmetry [3,4], noncommutativive theory [5,6], cosmology [7,8], extended Maxwell-Chern-Simon theory [9,10], theory of anyons [11][12][13], relativistic particle with torsion [14], membrane model of the electron [15,16] etc. In gravity theories HD terms were added to ensure renormalizability [17]. There are various models of gravity where HD corrections are added to the Einstein-Hilbert action [18][19][20][21]. HD terms also frequently appear in the context of string theory [22,23]. The importance of HD terms, therefore, cannot be overemphasised. A useful and interesting HD model to be considered is the relativistic particle model with curvature. In this case the curvature term, which is higher derivative in nature, is added to the action of the usual massive relativistic particle. This model was introduced long ago by Pisarski [24] and still continues to be under active consideration [25][26][27][28][29][30][31]. Interestingly, the model has only one gauge symmetry identified as diffeomorphism symmetry although there are two independent primary first class constraint present in the theory [31], which is unusual. The presence of the extra primary first class constraint was successfully explained as an effect of the higher derivative nature. The massless version, known as massless particle model with rigidity, is shown to describe bosons and fermions [28]. This model has two gauge symmetries viz. diffeomorphism and W-symmetry alongwith two primary first class constraints [29][30][31]. One can also add a torsion term to the relativistic particle model and the theory emerges with very interesting results. When quantised, the relativistic particle model with torsion leads to Majorana equations [14]. The massive sector contains infinite number of states when quantised in the Minkowski space and finite number of states in euclidean space [11]. Higher derivative models also shows Majorana equations [13] with the 2+1 dimensional analogous models lead to anyons. Another interesting line where the higher derivative models appear corresponds to finite-gap (algebra-geometric) systems [32]. There, the higher derivative models are given by the so called Novikov equation, with a space variable playing a role of the evolution parameter. On the other hand, BRST symmetry is a very powerful tool to quantize a theory with gauge invariance which also helps in the proof of the renormalizability and unitarity of gauge theories [33][34][35][36]. This transformation, which is characterized by an infinitesimal, global and anticommuting parameter leaves the effective action as well as path integral of the effective theory invariant. In gauge field theories the usual BRST symmetry has been generalized to make it finite and field-dependent [37]. This finite field dependent BRST (FFBRST) transformations have found many applications in various contexts [38][39][40][41][42][43][44][45]. The implementation of BRST symmetries for HD theories is quite nontrivial and poses problems. In this context, therefore, a natural question arises regarding the application of BRST formalism to relativistic particle models. Indeed it is not surprising that in spite of a considerable volume of research on relativistic particle models, this aspect remains unstudied. A basic motivation of this paper is to bridge this gap. In this paper, we consider a relativistic particle model with curvature as a HD theory possessing a gauge symmetry. The constraint analysis of this model and its massless analogue is discussed. Further, the gauge symmetry transformations in the case of massive relativistic particle model with curvature are identified with diffeomorphism invariance. However, for the massless particle model with rigidity it corresponds to W -symmetry in addition to diffeomorphism invariace. We construct the BRST symmetry and anti-BRST symmetry for these particle models. The difficulties of applying BRST transformations to HD theories are bypassed by working in the first order formalism developed in [26,28,31] instead of the conventional Ostrogradski approach [46]. It is shown that the (anti-)BRST symmetry transformations for all variables reproduce the diffeomorphism symmetry of the massive relativistic particle model including curvature. Furthermore, we also show that the massless particle model with rigidity yields both the diffeomorphism and W -invariances. We explicitly demonstrate the W 3 -algebra. For BRST transformations this algebra is shown for all variables, excluding the anti-ghost. Exactly the same features are revealed, but now excluding the ghost variable instead of the anti-ghost, for anti-BRST transformations. To get the complete picture, therefore, both BRST and anti-BRST transformations have to be considered. Further, we implement the concept of FFBRST transformation [37] in the quantum mechanical relativistic particle model. The quantum mechanical version of FFBRST transformation [37] is called as finite coordinate-dependent BRST (FCBRST) transformation. We see that FCBRST transformation for the relativistic particle model is a symmetry of the action only, but not of the generating functional. Analogous to FFBRST the FCBRST transformation changes the Jacobian of path integral measure non-trivially. For an appropriate choice of finite coordinate dependent parameter FCBRST connects two different gauge-fixed action within functional integration. The plan of the paper is as follows. In sections 2 and 3 we introduce the various relativistic particle models and discuss their gauge symmetries. The nilpotent BRST and anti-BRST transformation with emergence of W 3 -algebra is demonstrated in section 4. In section 5, we construct the FCBRST transformation. Two arbitrary gauges are connected with the help of this FCBRST within a path integral formalism in section 6. We draw concluding remarks in the last section. 2 Massive relativistic particle model with curvature The massive relativistic point particle theory with curvature has the action 1 S = −m √ẋ 2 dτ + α (ẋẍ) 2 −ẋ 2ẍ2 1 2 x 2 dτ.(1) The model is meaningful for α < 0 andẋ 2 > 0 . The energy spectrum for this model turns out to be the energy spectrum of the Majorana equation [26,27]. By direct substitution we may verify that (1) is invariant under reparametrisation, τ −→ τ + Λ(τ ),(2) where Λ is an infinitesimal reparametrisation parameter. Under this reparametrisation x µ transforms as, δx µ = x µ (τ − Λ) − x µ (τ ) = −Λẋ µ .(3) Since this is a higher derivative model, the usual Hamiltonian formalism does not apply. There is, however, the well established Ostrogradski method where the momenta are defined in some non-trivial way [46]. Other than this, a first order formalism exists in the literature where the time derivatives of the coordinates are considered as independent variables to convert the theory into a first order one. There are different variants [26,28,31] of this formalism and we adopt the one that was developed by two of us in a collaborative work [31]. To convert the theory into a first order one we introduce the new coordinates q µ 1 = x µ q µ 2 =ẋ µ .(4) The Lagrangian in these coordinates has a first order form given by L = −m q 2 2 + α (q 2q2 ) 2 − q 2 2q 2 2 1 2 q 2 2 + q µ 0 (q 1µ − q 2µ ),(5) where q µ 0 are the Lagrange multipliers that enforce the constraintṡ q 1µ − q 2µ = 0. Let p 0µ , p 1µ and p 2µ be the canonical momenta conjugate to q 0µ , q 1µ and q 2µ respectively, p 0µ = ∂L ∂q µ 0 = 0, p 1µ = q 0µ , p 2µ = α((q 2q2 )q 2µ − q 2 2q 2µ ) q 2 2 (q 2q2 ) 2 − q 2 2q 2 2 .(7) The primary constraints thus obtained are listed below [27,31], Φ 0µ = p 0µ ≈ 0, Φ 1µ = p 1µ − q 0µ ≈ 0, Φ 1 = p 2 q 2 ≈ 0, Φ 2 = p 2 2 q 2 2 + α 2 ≈ 0.(8) Conservation of the constraints (8) yield the following secondary constraints, ω 1 = q 0 q 2 + m q 2 2 ≈ 0, ω 2 = q 0 p 2 ≈ 0.(9) We define Φ ′ 2 as a combination of constraints, Φ ′ 2 = q 2 0 − m 2 Φ 2 − 2p 2 2 (q 0 q 2 ) ω 1 .(10) Φ 1 and Φ ′ 2 form the first class constraint set which are primary in nature. The second class constraints Φ 0µ , Φ 1µ , ω 1 and ω 2 are eliminated by replacing all the Poisson brackets by Dirac brackets. The nonzero Dirac brackets between the phase space variables are listed below 2 {q 1µ , q 1ν } D = 1 p 2 1 − m 2 (p 2µ q 2ν − q 2µ p 2ν ) , {q 1µ , q 2ν } D = − q 2µ p 1ν p 2 1 − m 2 , {q 1µ , p 2ν } D = 1 p 2 1 − m 2   m q 2 2 p 2µ q 2ν + p 2µ p 1ν   , {q 1µ , p 1ν } D = η µν , {q 2µ , p 2ν } D = η µν − 1 p 2 1 − m 2   p 1µ p 1ν + m q 2 2 p 1µ q 2ν   .(11) Now the generator G of the gauge symmetry is given by a combination of the first class constraints, G = ǫ 1 Φ 1 + ǫ 2 (p 2 1 − m 2 )Φ 2 .(12) However, we have shown in [31] that there is only one independent gauge parameter which is a consequence of the higher derivative theory. The generator is then given by, G = q 2µ q 2 2 d dτ 2mp 2 2 q 2 2 ǫ 2 q µ 2 Φ 1 + ǫ 2 (p 2 1 − m 2 )Φ 2 .(13) It may be noted that in the expression of the gauge generator there appears time derivative of the fields as well gauge parameter ǫ 2 . The apparent problem of taking gauge variation of the derivative of the phase space variables does not arise since they are multiplied by constraints which are set to zero after computing the brackets. Now the gauge transformation of the variables are given by δq µ 1 = {q µ 1 , G} D = 2ǫ 2 p 2 2 m q 2 2 q µ 2 ,(14) and δq µ 2 = {q µ 2 , G} D = q 2ν q 2 2 d dτ 2mp 2 2 q 2 2 ǫ 2 q ν 2 q µ 2 + 2ǫ 2 q 2 2 (p 2 1 − m 2 ) p µ 2 .(15) In terms of the reparametrization parameter, Λ = −2ǫ 2 p 2 2 m q 2 2 , the transformation for q µ 1 may be expressed as, δq µ 1 = −Λq µ 2 .(16) Using the identification (4) the relation (16) reproduces the reparametrisation symmetry (3). Thus the gauge symmetry gets identified with the reparametriasation symmetry. 3 The model of massless particle with rigidity The massless version of the model (1) is known as the model of massless particle with rigidity. The massless version is not obtained simply by putting m = 0 in (1) due to reasons of internal consistency [28]. This requires a modification in the curvature term and the model is given by, S = α ẋ 2ẍ2 − (ẋẍ) 2 1 2 x 2 dτ.(17) The action of this model thus is proportional to the curvature and also its classical equation of motion is compatible only for super-relativistic motion of the particle. The importance of the model lies in the fact that it corresponds to massless modes of either integer or half-integer helicity states. The model finds its relevance whenẋ 2 < 0 [28]. Once again we adopt the first order formalism developed in [31]. We introduce the new coordinates q µ 1 = x µ , q µ 2 =ẋ µ .(18) The Lagrangian in these coordinates has a first order form given by L = α q 2 2q 2 2 − (q 2q2 ) 2 1 2 q 2 2 + q µ 0 (q 1µ − q 2µ ),(19) where q µ 0 are the Lagrange multipliers that enforce the same constraints mentioned in equation (6). Let p 0µ , p 1µ and p 2µ be the canonical momenta conjugate to q 0µ , q 1µ and q 2µ respectively, having same expressions as that of (7). Consequently, we obtain the following primary constraints Φ 0µ = p 0µ ≈ 0, Φ 1µ = p 1µ − q 0µ ≈ 0, Φ 1 = p 2 q 2 ≈ 0, Φ 2 = p 2 2 q 2 2 − α 2 ≈ 0.(20) The secondary set of constraints obtained by time conserving the primary ones are, ω 1 = q 0 q 2 ≈ 0, ω 2 = q 0 p 2 ≈ 0.(21) Finally, by conserving ω 2 the tertiary constraint is obtained as ω 3 = q 2 0 ≈ 0.(22) This completes the chain of constraints. In the above constraint structure the first class set is { Φ 1 ,Φ 2 , ω 1 , ω 2 , ω 3 } and all others are second class in nature. Once again we remove all second class constraints as described in the previous section. Fortunately the Dirac bracket comes out to be same as the Poisson brackets between the phase space variables. After removing the nondynamical variables q 0µ and p 0µ by solving the second class constraints Φ 0µ and Φ 1µ , the final set of first class constraints are Ω 1 = Φ 1 = p 2 q 2 ≈ 0, Ω 2 = Φ 1 = p 2 2 q 2 2 − α 2 ≈ 0, Ω 3 = ω 1 = p 1 q 2 ≈ 0, Ω 4 = ω 2 = p 1 p 2 ≈ 0, Ω 5 = ω 3 = p 2 1 ≈ 0.(23) Note that the original conditionẋ 2 < 0 translates into q 2 2 < 0. This ensures the reality of α, as may be easily seen from the constraint Ω 2 ≈ 0. The reality of α is connected to the helicity states of the particles as discussed [28]. As done previously, the gauge generator is written as a combination of all the first class constraints, G = 5 a=1 ǫ a Ω a .(24) However, due to the presence of secondary first-class constraints, the parameters of gauge transformation(ǫ a ) are not independent [35,47]. It is found that only two out of the five gauge parameters are independent.For our convenience we take ǫ 3 and ǫ 5 as independent. So, the expression for the gauge generator becomes [31], G = ǫ 3 + q 2q2 q 2 2 ǫ 3 − α √ g q 4 2ǫ 5 Ω 1 + 1 2q 2 2 ǫ 5 − q 2q2 q 2 2ǫ 5 + α √ g p 2 2 q 2 2 ǫ 3 Ω 2 +ǫ 3 Ω 3 +ǫ 5 Ω 4 + ǫ 5 Ω 5 ,(25) which contains only two independent parameters (here g = q 2 2q 2 2 − (q 2q2 ) 2 ). We now calculate the gauge variations of the dynamical variables, defined as δq = {q, G} D . These are given by, δq µ 1 = ǫ 3 q µ 2 +ǫ 5 p µ 2 + 2ǫ 5 p µ 1 , δq µ 2 = ǫ 3 + q 2q2 q 2 2 ǫ 3 − α √ g q 4 2ǫ 5 q µ 2 + ǫ 5 − q 2q2 q 2 2ǫ 5 + α √ g p 2 2 q 2 2 ǫ 3 p µ 2 +ǫ 5 p µ 1 , δp µ 1 = 0, δp µ 2 = − ǫ 3 + q 2q2 q 2 2 ǫ 3 − α √ g q 4 2ǫ 5 p µ 2 − p 2 2 q 2 2 ǫ 5 − q 2q2 q 2 2ǫ 5 + α √ g p 2 2 q 2 2 ǫ 3 q µ 2 − ǫ 3 p µ 1 .(26) The above transformations can be identified as diffeomorphism (D) and W -symmetry by putting ǫ 5 = 0 and ǫ 3 = 0 respectively, as done in [31]. Detailed calculations on all the phase-space variables show that [29][30][31] δ D ǫ 3 1 , δ D ǫ 3 2 = δ D ǫ 3 ; with ǫ 3 =ǫ 3 1 ǫ 3 2 − ǫ 3 1ǫ 3 2 δ D ǫ 3 , δ W ǫ 5 = δ W ǫ ′5 ; with ǫ ′5 = −ǫ 3ǫ5 δ W ǫ 5 1 , δ W ǫ 5 2 = δ W ǫ 5 ; with ǫ 5 = p 2 2 q 2 2 ǫ 5 2 ǫ 5 1 − ǫ 5 2ǫ 5 1 . This reproduces the usual W 3 -algebra. (Anti-)BRST symmetries and W 3 -algebra In this section we construct the nilpotent BRST and anti-BRST symmetries for the theory. For this purpose we need to fix a gauge before the quantization of the theory as the theory is gauge invariant and therefore has some redundant degrees of freedom. The general gauge condition in this case is chosen as: F 1 [f (q)] = 0,(27) where f (q) is a general function of all the generic variables q. Some explicit examples of gauge conditions corresponding to relativistic particle models are [28]. q 0 1 − τ = 0, q 0 2 − 1 = 0, p 0 2 = 0, q 2 2 = 0.(28) The general gauge condition (27) can be incorporated at a quantum level by adding the appropriate gauge-fixing term to classical action. The linearised gauge-fixing term using Nakanishi-Lautrup auxiliary variable B(q) is given by S gf = dτ 1 2 B 2 + BF 1 [f (q)] .(29) To complete the effective theory we need a further Faddeev-Popov ghost term in the action. The ghost term in this case is constructed as S gh = dτ [csF 1 [f (q)]] , = − dτ [csF 1 [f (q)]] ,(30) where c andc are ghost and anti-ghost variables. Now the effective action can be written as S ef f = S + S gf + S gh .(31) The source free generating functional for this theory is defined as Z[0] = Dq e iS ef f ,(32) where Dq is the path integral measure. The nilpotent BRST symmetry of the effective action in the case of relativistic particle model with curvature is defined by replacing the infinitesimal reparametrisation parameter (Λ) to ghost variable c in the gauge transformation given in equation (16) as s D q µ 1 = −cq µ 2 , s D q µ 2 = −ċq µ 2 − cq µ 2 , s D c = 0, s Dc = B, s D B = 0,(33) where c,c and B are ghost, anti-ghost and auxiliary variables respectively for relativistic particle model with curvature. This BRST transformation, corresponding to gauge symmetry identified with the diffeomorphism invariance, leaves both the effective action as well as generating functional, invariant. Similarly, we construct the anti-BRST symmetry transformation, where the roles of ghost and antighosts are interchanged with some coefficients, as s D q µ 1 = −cq µ 2 ,s D q µ 2 = −ċq µ 2 −cq µ 2 , s Dc = 0,s D c = −B,s D B = 0.(34) These transformations are nilpotent and absolutely anticommuting in nature i.e. (s D ) 2 = 0, (s D ) 2 = 0, s DsD +s D s D = 0.(35) The above (anti-)BRST transformations are valid for both the models. On the other hand, the nilpotent BRST and anti-BRST symmetry transformations, identified with W -symmetry (with ǫ 3 = 0) in (26), for relativistic massless particle model with rigidity only, are constructed as s W q µ 1 =ηp µ 2 + 2ηp µ 1 , s W q µ 2 = − α √ g q 4 2η q µ 2 + η − q 2q2 q 2 2η p µ 2 +ηp µ 1 , s W p µ 1 = 0, s W p µ 2 = α √ g q 4 2η p µ 2 − p 2 2 q 2 2 η − q 2q2 q 2 2η q µ 2 , s W η = 0, s Wη = B, s W B = 0,(36)ands W q µ 1 =ηp µ 2 + 2ηp µ 1 , s W q µ 2 = − α √ g q 4 2η q µ 2 + η − q 2q2 q 2 2η p µ 2 +ηp µ 1 , s W p µ 1 = 0,s W p µ 2 = α √ g q 4 2η p µ 2 − p 2 2 q 2 2 η − q 2q2 q 2 2η q µ 2 , s Wη = 0,s W η = −B,s W B = 0,(37) where η,η and B are ghost, anti-ghost and auxiliary variables, respectively, for relativistic massless particle model with rigidity. Here we observe interestingly that the BRST symmetry transformations of all variables (excluding the anti-ghost variable) given in equations (33) and (36) also satisfy the W 3 -algebra as s D c 1 , s D c 2 = s D c 3 ; with c 3 = c 2ċ1 −ċ 2 c 1 s D c , s W η = s W η ′ ; with η ′ =ηc s W η 1 , s W η 2 = s W η 3 ; with η 3 = p 2 2 q 2 2 (η 2η1 − η 1η2 ) , and the anti-BRST symmetry transformations of all variables (excluding ghost variable) given in equations (34) and (37) also satisfy the W 3 -algebra as s D c 1 ,s D c 2 =s D c 3 ; withc 3 =c 2ċ1 −ċ 2c1 s D c ,s W η =s W η ′ ; withη ′ =ηc s W η 1 ;s W η 2 =s W η 3 ; withη 3 = p 2 2 q 2 2 (η 2η1 −η 1η2 ) . This completes our analysis of the connection between the (anti-)BRST symmetries and W 3 -algebra. FCBRST formulation for higher derivative theory In this section we investigate the finite coordinate-dependent BRST (FCBRST) formulation for general higher derivative theory. To do so, we first define the infinitesimal BRST symmetry transformation with Grassmannian constant parameter δρ as δ b q = sq δρ,(38) where sq is the BRST variation of generic variables q in the HD theories. The properties of the usual BRST transformation in equation (38) do not depend on whether the parameter δρ is (i) finite or infinitesimal, (ii) variable-dependent or not, as long as it is anticommuting and global in nature. These observations give us a freedom to generalize the BRST transformation by making the parameter δρ finite and coordinate-dependent without affecting its properties. We call such generalized BRST transformation in quantum mechanical systems as FCBRST transformation. In the field theory such generalization is known as FFBRST transformation [37]. Here we adopt a similar technique to generalize the BRST transformation in quantum mechanical theory. We start by making the infinitesimal parameter coordinate-dependent with introduction of an arbitrary parameter κ (0 ≤ κ ≤ 1). We allow the generalized coordinates, q(κ), to depend on κ in such a way that q(κ = 0) = q and q(κ = 1) = q ′ , the transformed coordinate. The usual infinitesimal BRST transformation, thus can be written generically as dq(κ) = s[q]Θ ′ [q(κ)]dκ,(39) where the Θ ′ [q(κ)]dκ is the infinitesimal but coordinate-dependent parameter. The FCBRST transformation with the finite coordinate-dependent parameter then can be constructed by integrating such infinitesimal transformation from κ = 0 to κ = 1, to obtain [37] q ′ ≡ q(κ = 1) = q(κ = 0) + s(q)Θ[q],(40) where Θ[q] = 1 0 dκ ′ Θ ′ [q(κ ′ )],(41) is the finite coordinate-dependent parameter. Such a generalized BRST transformation with finite coordinate-dependent parameter is the symmetry of the effective action in equation (31). However, the path integral measure in equation (32) is not invariant under such transformation as the BRST parameter is finite in nature. The Jacobian of the path integral measure for such transformations is then evaluated for some particular choices of the finite coordinate-dependent parameter, Θ[q(x)], as Dq ′ = J(κ)Dq(κ).(42) The Jacobian, J(κ) can be replaced (within the functional integral) as J(κ) → exp[iS 1 [q(κ)]],(43) iff the following condition is satisfied [37] Dq 1 J dJ dκ − i dS 1 [q(x, κ)] dκ exp [i(S ef f + S 1 )] = 0,(44) where S 1 [q] is local functional of variables such that at κ = 0 it must vanish. The infinitesimal change in the J(κ) is written as [37], 1 J dJ dκ = − dτ ±sq(κ) ∂Θ ′ [q(κ)] ∂q(κ) ,(45) where ± sign refers to whether q is a bosonic or a fermionic variable. Thus, the FCBRST transformation with appropriate Θ, changes the effective action S ef f to a new effective action S ef f + S 1 (κ = 1) within the functional integration. 6 Connecting different gauges in relativistic particle models Here we will exploit the general FCBRST formulation developed in the previous section to connect the path integral of relativistic particle models with different gauge conditions. The FCBRST transformations (f b ) for the relativistic particle model with curvature are constructed as follows: f b q µ 1 = −cq µ 2 Θ[q], f b q µ 2 = (−ċq µ 2 − cq µ 2 )Θ[q], f b c = 0, f bc = BΘ[q], f b B = 0,(46) where Θ[q] is an arbitrary finite coordinate-dependent parameter. Now, we show how two different gauges (say F 1 (q) = 0 and F 2 (q) = 0) in the relativistic particle model may be connected by such transformations. For this purpose, let us choose the following infinitesimal coordinate dependent parameter (through equation (41)) Θ ′ [q] = −i dτc(F 1 − F 2 ).(47) Let us first calculate the infinitesimal change in the Jacobian J(κ) for above Θ ′ [q] using the relation (45) as 1 J dJ dκ = i dτ [−B(F 1 − F 2 ) + s(F 1 − F 2 )c], = −i dτ [B(F 1 − F 2 ) +c s(F 1 − F 2 )].(48) To express the Jacobian as e iS 1 [37], we take the ansatz, S 1 [κ] = dτ [ζ 1 (κ)BF 1 + ζ 2 (κ)BF 2 + ζ 3 (κ)c sF 1 + ζ 4 (κ)c sF 2 ],(49) where ζ i (κ)(i = 1, ...4) are constant parameters satisfying the boundary conditions ζ i (κ = 0) = 0.(50) To satisfy the crucial condition (44), we calculate the infinitesimal change in S 1 with respect to κ using the relation (39) as dS 1 [q, κ] dκ = dτ [ζ ′ 1 BF 1 + ζ ′ 2 BF 2 + ζ ′ 3c sF 1 + ζ ′ 4c sF 2 + (ζ 1 − ζ 2 )B(sF 1 )Θ ′ + (ζ 2 − ζ 4 )B(sF 2 )Θ ′ ],(51) where prime denotes the differentiation with respect to κ. Exploiting equations (48) and (51), the condition (44) simplifies to, Dq (ζ ′ 1 + 1)BF 1 + (ζ ′ 2 − 1)BF 2 + (ζ ′ 3 + 1)c sF 1 + (ζ ′ 4 − 1)c sF 2 + (ζ 1 − ζ 3 )B(sF 1 )Θ ′ + (ζ 2 − ζ 4 )B(sF 2 )Θ ′ e i(S ef f +S 1 ) = 0.(52) The comparison of coefficients from the terms of the above equation gives the following constraints on the parameters ζ i ζ ′ 1 + 1 = 0, ζ ′ 2 − 1 = 0, ζ ′ 3 + 1 = 0, ζ ′ 4 − 1 = 0, ζ 1 − ζ 3 = 0, ζ 2 − ζ 4 = 0.(53) The solutions of the above equations satisfying the boundary conditions (50) are ζ 1 = −κ, ζ 2 = κ, ζ 3 = −κ, ζ 4 = κ.(54) With these values of ζ i the expression of S 1 [κ] given in equation (49) becomes S 1 [κ] = dτ [−κBF 1 + κBF 2 − κc sF 1 + κc sF 2 ],(55) which vanishes at κ = 0. Now, by adding S 1 (κ = 1) to the effective action (S ef f ) given in equation (31) we get S ef f + S 1 (κ = 1) = S + dτ 1 2 B 2 + BF 2 [f (q)] +csF 2 [f (q)] ,(56) which is nothing but the effective action for relativistic particle models satisfying the different gauge condition dτ e iS ef f F CBRST − − −− −→ dτ e i[S ef f +S 1 (κ=1)] .(57) We end this section by noting that the FCBRST transformation with appropriate finite coordinatedependent parameter is able to connect two different (arbitrary) gauges of the relativistic particle model. Conclusions The relativistic particle models have always been an interesting area of research as it led to the Polyakov action of string theory [22]. When a curvature term is added to the action of the relativistic particle model it becomes a higher derivative (HD) theory. Due to HD nature, it shows an inconsistency in counting the independent gauge degrees of freedom. The apparent mismatch is due to the interrelation between the variables with higher derivatives. Whereas, if we consider the mass term to be zero (with proper condition on the particle velocities as in [28]) the mismatch vanishes and the number of gauge degrees of freedom and number of independent primary first class constraints are same [31], as happens for all standard theories [47,48]. So, it would be interesting to study the BRST symmetries of both these models. But here we are faced an obstacle. For HD theories there is no well defined prescription for analysing BRST symmetry. In the present case this is avoided by working in the first order formalism developed in [31]. In this paper, we have analysed the different constraint structures of the models of relativistic particle with curvature and of massless relativistic particle with rigidity. The relativistic particle model with curvature is shown to have the diffeomorphism symmetry whereas the gauge symmetries of the model of relativistic massless particle with rigidity contain both diffeomorphism and W -morphisms. The nilpotent BRST and anti-BRST symmetries for these model have also been investigated. A remarkable feature for such symmetries is the manifestation of W 3 -algebra. The BRST symmetries for all variables (excluding for anti-ghost variable) corresponding to diffeomorphism and W -morphism satisfy the W 3 -algebra. Likewise, apart from the ghost variable, the anti-BRST symmetry transformations for all other variables also satisfy the same W 3 -algebra. Thus the full W 3 -algebra for all variables is obtained by taking into account both BRST and anti-BRST transformations. The finite coordinate-dependent BRST (FCBRST) symmetry, which is quantum mechanical analog of finite field-dependent BRST (FFBRST), has also been analysed in full generality for higher derivative particle models. It has been shown that although such a transformation is a symmetry of the effective action, it breaks the invariance of the generating functional of the path integral. The Jacobian of path integral measure changes non-trivially for FCBRST symmetry transformation. We have shown that FCBRST transformation with a suitable coordinate-dependent parameter changes the effective action from one gauge to another within a functional integral. Thus, FCBRST formulation is very useful to connect two different Greens functions for models of relativistic particles. The results were explicitly presented for the massive case. For the massless version, all results go over trivially in the appropriate limit. Finally, we feel that although our analysis was done for relativistic particle models, it is general enough to include other higher derivative models. F 2 [f (q)] = 0 . 20Thus, under FCBRST transformation, the generating functional of HD models changes from one gauge condition (F 1 [f (q)] = 0) to another gauge (F 2 [f (q)] = 0) as contractions are abbreviated as A µ Bµ = AB, A µ Aµ = A 2 . We consider the model in 3 + 1 dimensions. So µ assumes the values 0, 1, 2, 3[26,27,31]. Dirac brackets are denoted by {, }D. We consider p 2 1 − m 2 = 0, else it is a singular case. Explicit constraint structure and Dirac brackets of the singular case can be found in[31]. AcknowledgementOne of the authors (BP) gratefully acknowledges the Council of Scientific and Industrial Research (CSIR), Government of India, for financial assistance. . B Podolsky, Phys. Rev. 6268B. Podolsky, Phys. Rev. 62 (1942) 68. . B Podolsky, C Kikuchi, Phys. Rev. 65228B. Podolsky, C. Kikuchi, Phys. Rev. 65 (1944) 228. . 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{'abstract': 'In this paper we discuss the (anti-)BRST symmetries and W -algebra of higher derivative theories of relativistic particles satisfying general gauge conditions. Using this formalism, the connection between the (anti-)BRST symmetries and W -algebra for the massless particle with rigidity is established. Incidentally, the full W -algebra emerges only when the anti-BRST transformations are considered in tandem with the BRST ones. Further, the BRST symmetry is made finite and coordinate-dependent. We show that such finite coordinate-dependent BRST symmetry changes the BRST invariant gauge-fixing fermion within the functional integration. This is exploited to connect two different arbitrary gauge conditions.', 'arxivid': '1306.0744', 'author': ['Rabin Banerjee \nNational Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia\n', 'Biswajit Paul \nNational Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia\n', 'Sudhaker Upadhyay \nNational Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia\n', 'S N Bose \nNational Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia\n'], 'authoraffiliation': ['National Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia', 'National Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia', 'National Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia', 'National Centre for Basic Sciences\nJD Block\nSector III700 098Salt Lake City, KolkataIndia'], 'corpusid': 118560860, 'doi': '10.1103/physrevd.88.065019', 'github_urls': [], 'n_tokens_mistral': 13117, 'n_tokens_neox': 10811, 'n_words': 6482, 'pdfsha': '2f2c698137983fcec13da68b0fba21f841ff72dd', 'pdfurls': ['https://arxiv.org/pdf/1306.0744v2.pdf'], 'title': ['BRST symmetry and W -algebra in higher derivative models', 'BRST symmetry and W -algebra in higher derivative models'], 'venue': []}

arxiv

Light-Front Nuclear Physics: Mean Field Theory for Finite Nuclei Jun 1999 P G Blunden Canada M Burkardt G A Miller Department of Physics and Astronomy Department of Physics New Department of Physics University of Manitoba Winnipeg Mexico State University Las Cruces Box 351560R3T 2N2, 88003-0001MB, NMU.S.A University of Washington Seattle 98195-1560WAU.S.A Light-Front Nuclear Physics: Mean Field Theory for Finite Nuclei Jun 1999(January 16, 2022)arXiv:nucl-th/9906012v1 4 A light-front treatment for finite nuclei is developed from a relativistic effective Lagrangian (QHD1) involving nucleons, scalar mesons and vector mesons. We show that the necessary variational principle is a constrained one which fixes the expectation value of the total momentum operator P + to be the same as that for P − . This is the same as minimizing the sum of the total momentum operators: P − + P + . We obtain a new light-front version of the equation that defines the single nucleon modes. The solutions of this equation are approximately a non-trivial phase factor times certain solutions of the usual equal-time Dirac equation. The ground state wave function is treated as a meson-nucleon Fock state, and the meson fields are treated as expectation values of field operators in that ground state. The resulting equations for these expectation values are shown to be closely related to the usual meson field equations. A new numerical technique to solve the self-consistent field equations is introduced and applied to 16 O and 40 Ca. The computed binding energies are essentially the same as for the usual equaltime theory. The nucleon plus momentum distribution (probability for a nucleon to have a given value of p + ) is obtained, and peaks for values of p + about seventy percent of the nucleon mass. The mesonic component of the ground state wave function is used to determine the scalar and vector meson momentum distribution functions, with a result that the vector mesons carry about thirty percent of the nuclear plus-momentum. The vector meson momentum distribution becomes more concentrated at p + = 0 as A increases. I. INTRODUCTION The purpose of this paper is to derive the light-front formalism necessary to compute the properties of finite nuclei. Nuclear properties are very well handled within the existing conventional nuclear theory, so it behooves us to explain why we are embarking on this project. Our motivation is that understanding experiments involving high energy nuclear reactions seems to require that light-front dynamics and light cone variables be used. Consider the EMC experiment [1], which showed that there is a significant difference between the parton distributions of free nucleons and nucleons in a nucleus. This difference can interpreted as a shift in the momentum distribution of valence quarks towards smaller values of the Bjorken variable x Bj . The Bjorken variable is a ratio of the plus-momentum k + = k 0 + k 3 of a quark to that of the target. Thus light cone variables are relevant. If one uses k + as a momentum variable the corresponding canonical spatial variable is x − = x 0 − x 3 and the time variable is x + = x 0 + x 3 . It is important to realize that the use of light-front dynamics is not limited to quarks within the nucleon -it also applies to nucleons within the nucleus. This formalism is useful whenever the momentum of initial or final state nucleons is large compared to their mass [2]. In particular, it can be used for (e, e ′ p) and (p, 2p) reactions. If one uses lightfront variables for nucleons in a nucleus, it is also necessary to maintain consistency with the information derived previously using conventional nuclear dynamics. This provides the technical challenge which we address in the present manuscript. The conventional equal-time approach to nuclear structure physics provides an excellent framework, so it is worthwhile to introduce the light-front variables and describe the expected advantages in a general way. The use of the light-cone variables can be obtained using a simple argument based on kinematics [2]. Suppose the virtual photon that is absorbed by a fermion at a space-time point (z 1 , t 1 ). The fermion then starts to move at high momentum and nearly the speed of light and emits the photon at another space time point (z 2 , t 2 ). In between the two times, the wave function of the entire system has undergone a time evolution given by the complicated operator e −iH(t 2 −t 1 ) . But we have z 1 + ct 1 = z 2 + ct 2 , if the z−axis is opposite to the direction of the virtual photon. The two scattering events occur at different times, but at the same value of x + = z + ct. Thus if we use x + as a time variable, no time evolution factor appears. The net result is that the cross section involves light-like correlation functions which involve field operators evaluated at the same light-front time: x + = 0 (see for example the reviews [3,4]). Thus it is a specific and general feature of the light-front wave approach that knowing only the ground state wave function is sufficient for computing the distribution functions. Let us review the salient features of the basic idea that using the light-front approach leads to a simplified treatment. To be specific, consider high energy electron scattering from nucleons in nuclei. The key ingredient in the light-front simplification is to realize the main difference between the two formalisms. In the equal-time formalism, sums over intermediate states are taken over eigenstates of the Hamiltonian, P 0 . The usual three momentum is conserved, but energy is not conserved in intermediate states. In the lightfront approach one sums over eigenstates of the minus component of the total momentum operator. The value of P − is not conserved in intermediate state sums, and the values of P + and P ⊥ are conserved. This is especially convenient for high energy reactions, in which the plus-component is the largest component of the momentum for each projectile or ejectile. The advantage of using P − as an "energy" variable can be easily described. Let the four-momentum q of the exchanged virtual photon be given by ν, 0, 0, − √ Q 2 + ν 2 , with Q 2 = −q 2 , and Q 2 and ν 2 are both very large but Q 2 /ν is finite (the Bjorken limit). In this case it is worthwhile to use the light-cone variables q ± = q 0 ±q 3 in which q + ≈ Q 2 /2ν = Mx, q − ≈ 2ν − Q 2 /2ν, so that q − ≫ q + . Here M is the mass of a nucleon and x is the Bjorken variable. We shall neglect q − in comparison to q + , noting that corrections to this can be handled in a systematic fashion. Then the schematic form of the scattering cross section for e + A → e ′ + (A − 1) f + p, where f represents the final nuclear eigenstate of P − , and p the four-momentum of the final proton, is given by dσ ∼ f d 3 p f E f d 4 p δ(p 2 − M 2 )δ (4) (q + p i − p f − p)| p, f | J(q) | i | 2 . (1.1) Here the operator J(q) is a schematic representation of the electromagnetic current. Performing the four-dimensional integral over p leads to the expression dσ ∼ f d 2 p f dp + f p + f δ (p i − p f + q) 2 − m 2 | p, f | J(q) | i | 2 . (1.2) The argument of the delta function ( p i − p f + q) 2 − M 2 ≈ −Q 2 + 2q − (p i − p f ) + . Thus we see that p − f does not appear in the argument of the delta function, or anywhere else, so that we can replace the sum over intermediate states by unity. In the usual equal-time representation, one finds the argument of the delta function to be −Q 2 + 2ν(E i − E f ). The energy of the final state appears, and one can not do the sum. To proceed further in this schematic approach we take J(q) = d 3 k b † k+q b k , (1.3) where b is a nucleon destruction operator and V ≡ (V + , V ⊥ ). It is useful to define p B ≡ p i − p f because p + B = Q 2 /2ν ≡ Mx,(1.dσ ∼ d 2 p B ⊥ i | b † p B b p B | i = d 2 p B ⊥ n(Mx, p B ⊥ ), (1.5) where n(Mx, p B ⊥ ) is the probability for a nucleon in the ground state to have a momentum (Mx, p B ⊥ ). Integration in Eq. (1.5) leads to σ ∼ d 2 p ⊥ n(Mx, p ⊥ ) ≡ f (Mx),(1.6) with f (Mx) as the probability for a nucleon in the ground state to have a plus momentum of Mx. The quantity f (Mx) has been a widely used prescription UP for handling the light-front in a simple way. The variable Mx is replaced by M − ε α + k 3 , in which the label α denotes a shell-model orbital φ α of binding energy ε α . Then f U P = α n α d 2 p ⊥ dp 3 | φ α (p 3 , p ⊥ ) | 2 δ(M − ε α + p 3 − Mx), (1.7) in which n α is an occupation probability. The validity of this prescription, which rests on a reasonable assumption, is rather suspect because the variable p + = Mx is a kinematic variable, unrelated to discrete eigenvalues of a wave equation. One of the main purposes of the present paper is to see if anything like this prescription emerges from our calculations. We shall see that Eq. (1.7) is not obtained, if a vector potential is a significant part of the nuclear mean field. It is useful to discuss the relation with y-scaling [5]. The arguments that the cross section depends on a plus-momentum distribution are well known when used for quarks in a nucleon, but they also apply to nucleons in a nucleus [2]. Ji and Filippone [6] showed that the y scaling function F (y) extracted in quasi-elastic electron scattering on nuclei is actually the light cone plus-momentum distribution function for nucleons in the nucleus. It is useful to use a relativistic form of the variable y [7] in which y = −q 3 + ν + E s + M, (1.8) as both q 3 and ν are large in magnitude, and E s is the single nucleon separation energy. But x = (q 3 ) 2 −ν 2 2M ν ≈ 1 + Es−y M , so that Mx = M + E s − y ≡ M A y A . (1.9) Here y A is a new y-scaling variable. This means that according to Eq. (1.4) p + B = M A y A /M ≈ Ay A ,(1.10) so that a measurement of σ determines the probability that the struck nucleon has a plusmomentum of Ay A . This probability also enters in convolution model calculations of nuclear deep inelastic scattering. The use of light-front dynamics to compute nuclear wave functions should allow us to compute F (y) from first principles. Furthermore, we claim that using light-front dynamics incorporates the experimentally relevant kinematics from the beginning, and therefore is the most efficient way to compute the cross sections for nuclear deep inelastic scattering and nuclear quasi-elastic scattering. It is worthwhile to review some of the features of the EMC effect [1,4]. The key experimental result is the suppression of the structure function for x ∼ 0.5. This means that the valence quarks of bound nucleons carry less plus-momentum than those of free nucleons. One way to understand this result is to postulate that mesons carry a larger fraction of the plus-momentum in the nucleus than in free space. While such a model explains the shift in the valence distribution, one obtains at the same time a meson (i.e. anti-quark) distribution in the nucleus, which is strongly enhanced compared to free nucleons and which should be observable in Drell-Yan experiments [8]. However, no such enhancement has been observed experimentally [9], and the implications are analyzed in Ref. [10]. The use of light-front dynamics allows us to compute the necessary nuclear meson distribution functions using variables which are experimentally relevant. The need for a computation of such functions in a manner consistent with generally known properties of nuclei led one of us to attempt to construct a light front treatment of nuclear physics [11]. These calculations, using a Lagrangian in which Dirac nucleons are coupled to massive scalar and vector mesons [12], treated the example of infinite nuclear matter within the mean field approximation. In this case, the meson fields are constants in both space and time and the momentum distribution has support only at k + = 0. Such a distribution would not be accessible experimentally, so that the suppression of the plus-momentum of valence quarks would not imply the existence of a corresponding testable enhancement of anti-quarks. However, it is necessary to ask if the result is only a artifact of the infinite nuclear size and of the mean field approximation. The present paper is an attempt to handle finite-sized nuclei using light-front dynamics. A. Recovery of rotational invariance It is worthwhile to discuss, in a general way, how it is that we are able to find spectra which have the correct number of degenerate states. Let us imagine that we try to determine eigenstates of a LF Hamiltonian by means of a variational calculation. Simply minimizing the LF energy obviously leads to nonsensical results since the LF energy scales like the inverse of the LF momentum. Even if one has only a poor ansatz for the intrinsic wave function, one can easily reach zero energy by letting the overall momentum scale to infinity! However, this problem is avoided by performing a constrained variation, in which the total LF momentum is fixed by including a Lagrange multiplier term proportional to the total momentum in the LF Hamiltonian. Note that this is not a problem if one is able to use a Fock space basis in which the total plus and ⊥ momentum of each component are fixed. In calculations involving many particles, the Fock state approach cannot be used in practical calculations -instead one uses a mean field in which each particle moves in an "external" potential. In this case the total momentum is not fixed, and a Lagrange multiplier term needs to be included in order to avoid solutions with infinite LF momentum. In order to fix this potential problem with "runaway solutions" (P + → ∞) to variational calculations for LF Hamiltonians, any term proportional to P + would suffice. However, by setting the coefficient for the term proportional to P + equal to one, i.e. minimizing P − +P + , one automatically guarantees that P + = P − (or P 3 = 0). The reason is that, using covariance, P − has eigenvalues of the form P − n = M 2 n +P 2 ⊥ P + , i.e. it scales like 1/P + . Therefore, when one minimizes P − +P + with respect to P + , the minimum occurs for P + = M 2 n + P 2 ⊥ , which yields P − = M 2 n +P ⊥ 2 P + = M 2 n + P 2 ⊥ as well. This "equipartition" between P + and P − thus arises since the two operators scale in exactly opposite ways under longitudinal boosts. Note that this is quite analogous to the nonrelativistic harmonic oscillator where, under scale transformations, potential and kinetic energy scale in opposite ways, resulting in the equipartition between potential and kinetic energy. The net result is that we minimize the sum of P + + P − . The need to include the plusmomentum can also be seen in a simple example. Consider a nucleus of A nucleons of momentum P + A = M A , P A⊥ = 0, which consists of a nucleon of momentum (p + , p ⊥ ), and a residual (A − 1) nucleon system which must have momentum (P + A − p + , −p ⊥ ). The kinetic energy K is given by the expression K = p 2 ⊥ + M 2 p + + p 2 ⊥ + M 2 A−1 P + A − p + . (1.11) In the second expression, one is tempted to neglect the term p + in comparison with P + A ≈ M A . This would be a mistake. Instead make the expansion K ≈ p 2 ⊥ + M 2 p + + M 2 A−1 P + A 1 + p + P + A ≈ p 2 ⊥ + M 2 p + + p + + M A−1 ,(1.12) because for large A, M 2 A−1 /P 2 A ≈ 1. For free particles, of ordinary three momentum p one has E 2 (p) = p 2 + m 2 and p + = E(p) + p 3 , so that K ≈ (E 2 (p) − (p 3 ) 2 ) E(p) + p 3 + E(p) + p 3 + M A−1 = 2E(p) + M A−1 . (1.13) We see that K depends only on the magnitude of a three-momentum and rotational invariance is restored. The physical mechanism of this restoration is the inclusion of the recoil kinetic energy of the residual nucleus. B. Outline The organization of the paper is as follows. The light-front quantization for our chosen Lagrangian is is presented in Sec. II. This quantization is applied, along with a constrained minimization of the expectation value of P − , to derive a light-front version of mean field theory in Sec. III. We obtain a new light version of the equation that defines the single nucleon modes. The solutions of this equation are approximately a non-trivial phase factor times the solutions of the usual equal-time ET Dirac equation. The consequences of this phase factor are discussed. The meson fields are treated as expectation values of operators. The equations for these expectation values are closely related to the meson field equations appearing in the usual treatment of the Walecka model. However, the mesonic Fock space is accessible in our formalism. Our nucleon mode equation is simplified by the use of a two-component spinor formalism [13], and by an angular momentum reduction in Sec. IV. The numerical aspects are discussed in App. A. The binding energies, nucleon and meson distributions for 16 O and 40 Ca are presented in Sec. V. A concluding discussion appears in Sec. VI. Numerical details of how we evaluate the momentum distributions are given in App. B. A brief discussion of some of the results can be found in Ref. [14]. A related set of solutions of some toy model problems and a heuristic derivation of our nucleon mode equation will appear in a separate paper [15]. II. LIGHT-FRONT QUANTIZATION We start with a model in which the nuclear constituents are nucleons ψ (or ψ ′ ), scalar mesons φ and vector mesons V µ . The Lagrangian L is given by L = 1 2 (∂ µ φ∂ µ φ − m 2 s φ 2 ) − 1 4 V µν V µν + 1 2 m 2 v V µ V µ +ψ ′ γ µ ( i 2 ↔ ∂ µ −g v V µ ) − M − g s φ ψ ′ , (2.1) where the bare masses of the nucleon, scalar and vector mesons are given by M, m s , m v , and V µν = ∂ µ V ν − ∂ ν V µ . We ignore pions here. The field equations are given by: γ · (i∂ − g v V )ψ ′ = (M + g s φ)ψ ′ , (2.2) ∂ µ V µν + m 2 v V ν = g vψ ′ γ ν ψ ′ (2.3) ∂ µ ∂ µ φ + m 2 s φ = −g sψ ′ ψ ′ . (2.4) The next step is obtain the light-front Hamiltonian (P − ) [16] as a sum of a free, noninteracting and a set of terms containing all of the interactions. This is accomplished by separating the independent and dependent degrees of freedom in the usual way [17,3] and then using the energy momentum tensor. Consider the nucleons. Although described by four-component spinors, these fields have only two independent degrees of freedom. The light-front formalism allows a convenient separation of dependent and independent variables via the projection operators Λ ± ≡ 1 2 γ 0 γ ± [18,19], with ψ ′ ± ≡ Λ ± ψ ′ . The independent fermion degree of freedom is chosen to be ψ ′ + , and one finds (i∂ − − g v V − )ψ ′ + = (α ⊥ · (p ⊥ − g v V ⊥ ) + β(M + g s φ))ψ ′ − (i∂ + − g v V + )ψ ′ − = (α ⊥ · (p ⊥ − g v V ⊥ ) + β(M + g s φ))ψ ′ + . (2.5) The relation between ψ ′ − and ψ ′ + is very complicated unless one may set the plus component of the vector field to zero [17]. This is a matter of a choice of gauge for QED and QCD, but the non-zero mass of the vector meson prevents such a choice here. Instead, one simplifies the equation for ψ ′ − by [18,20] transforming the fermion field according to ψ ′ = e −igvΛ(x) ψ, ∂ + Λ = V + . (2.6) This transformation leads to the replacement of Eq. (2.5) by (i∂ − − g vV − )ψ + = (α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ))ψ − i∂ + ψ − = (α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ))ψ + ,(2.7) where ∂ +V µ = ∂ + V µ − ∂ µ V + . (2.8) Note that while it isV µ that enters in the nucleon field equations, it is V µ that enters in the meson field equations. The scalar field can be expressed in terms of creation and destruction operators: φ(x) = d 2 k ⊥ dk + θ(k + ) (2π) 3/2 √ 2k + a(k)e −ik·x + a † (k)e ik·x , (2.9) where k · x = 1 2 (k + x − ) − k ⊥ · x ⊥ ≡ k · x,(2.10) and the fields and their derivatives with respect to x + are evaluated at x + = 0. This notation is used through out this work. The consequence is that the energy momentum tensor T µν does not depend on x + . In the above expansion (and in the expansions for any of our fields) the particles are on the mass-shell. Here k − = k 2 ⊥ +m 2 s k + . The theta function restricts k + to positive values. The commutation relations are [a(k), a † (k ′ )] = δ(k ⊥ − k ′ ⊥ )δ(k + − k ′+ ), (2.11) with [a(k), a(k ′ )] = 0. It is useful to define δ (2,+) (k − k ′ ) ≡ δ(k ⊥ − k ′ ⊥ )δ(k + − k ′+ ). (2.12) The expression for the vector meson field operator is V µ (x) = d 2 k ⊥ dk + θ(k + ) (2π) 3/2 √ 2k + ω=1,3 ǫ µ (k, ω) a(k, ω)e −ik·x + a † (k, ω)e ik·x ,(2.13) where the polarization vectors are the usual ones: k µ ǫ µ (k, ω) = 0, ǫ µ (k, ω)ǫ µ (k, ω ′ ) = −δ ωω ′ , ω=1,3 ǫ µ (k, ω)ǫ ν (k, ω) = −(g µν − k µ k ν m 2 v ). (2.14) Once again the four momenta are on-shell, with k − = k 2 ⊥ +m 2 v k + . The commutation relations are [a(k, ω), a † (k ′ , ω ′ )] = δ ωω ′ δ (2,+) (k − k ′ ), (2.15) with [a(k, ω), a(k ′ , ω ′ )] = 0, and lead to commutation relations amongst the field operators that are the same as in Ref. [20]. We also need the eigenmode expansion forV µ . This is given bȳ V µ (x) = d 2 k ⊥ dk + θ(k + ) (2π) 3/2 √ 2k + ω=1,3ǭ µ (k, ω) a(k, ω)e −ik·x + a † (k, ω)e ik·x ,(2.16) where, using Eqs.(2.8) and (2.13), the polarization vectorsǭ µ (k, ω) arē ǫ µ (k, ω) = ǫ µ (k, ω) − k µ k + ǫ + (k, ω). (2.17) Note that ω=1,3ǭ µ (k, ω)ǭ ν (k, ω) = −(g µν − g +µ k ν k + − g +ν k µ k + ). (2.18) Then we may construct the total four-momentum operator from P µ = 1 2 dx − d 2 x ⊥ T +µ (x + = 0, x − , x ⊥ ), (2.19) with (as usual) 20) in which the degrees of freedom are labelled by φ r . We need T ++ and T +− , which are T µν = −g µν L + r ∂L ∂(∂ µ φ r ) ∂ ν φ r ,(2.T ++ = ∂ + φ∂ + φ + V ik V ik + m 2 v V + V + + 2ψ † + i∂ + ψ + ,(2.21) and T +− = ∇ ⊥ φ · ∇ ⊥ φ + m 2 φ φ 2 + 1 4 (V +− ) 2 + 1 2 V kl V kl + m 2 v V k V k +ψ γ ⊥ · (p ⊥ − g vV − ) + M + g s φ ψ. (2.22) This form is still not useful for calculations because the constrained field ψ − contains interactions. We follow Refs. [18,21] in expressing ψ − as a sum of terms, one ξ − whose relation with ψ + is free of interactions, the other η − containing the interactions. That is, rewrite the second of Eq. (2.7) as [13] ξ − = 1 i∂ + (α ⊥ · p ⊥ + βM)ψ + η − = 1 i∂ + (−α ⊥ · g vV ⊥ + βg s φ)ψ + . (2.23) Furthermore, define ξ + (x) ≡ ψ + (x), so that ψ(x) = ξ(x) + η − (x), (2.24) where ξ(x) ≡ ξ − (x) + ξ + (x) . This separates the dependent and independent parts of ψ. One needs to make a similar treatment for the vector meson fields. The operator V +− , is determined by V −+ = 2 ∂ + g v J + − m 2 v V + − ∂ i V i+ . (2.25) Part of this operator is determined by a constraint equation, because the independent variables are V + and V i+ . To see this examine Eq (2.25), and make a definition V +− = v +− + ω +− ,(2.26) where ω +− = − 2 ∂ + J + . (2.27) The sum of the last term of Eq˙(2.22) and the terms involving ω +− is the interaction density. 28) and the interactions P − 0N = 1 2 d 2 x ⊥ dx −ξ (γ ⊥ · p ⊥ + M) ξ,(2.P − I = v 1 + v 2 + v 3 , (2.29) with v 1 = d 2 x ⊥ dx −ξ g v γ ·V + g s φ ξ, (2.30) v 2 = d 2 x ⊥ dx −ξ −g v γ ·V + g s φ γ + 2i∂ + −g v γ ·V + g s φ ξ,(2.31) and v 3 = g 2 v 8 d 2 x ⊥ dx − dy − 1 ǫ(x − − y − 1 ) ξ † + (y − 1 , x ⊥ )γ + ξ + (y − 1 , x ⊥ ) × dy − 2 ǫ(x − − y − 2 )ξ † + (y − 2 , x ⊥ )γ + ξ + (y − 2 , x ⊥ ), (2.32) where ǫ(x) ≡ θ(x) − θ(−x) . The term v 1 accounts the emission or absorption of a single vector or scalar meson. The term v 2 includes contact terms in which there is propagation of an instantaneous fermion. The term v 3 accounts for the propagation of an instantaneous vector meson. Our variational procedure will involve the independent fields ψ + , so we need to express the interactions P − 0N , v 1,2 in terms of ξ + . A bit of Dirac algebra shows that P − N ≡ P − 0N + v 1 + v 2 = d 2 x ⊥ dx − 2 ξ † + 2g vV − + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) 1 i∂ + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) ξ + . (2.33) It is worthwhile to define the contributions to P ± arising from the mesonic terms as P ± s and P ± v . Then one may use Eqs. (2.22) and (2.21) along with the field expansions to obtain P − s = 1 2 d 2 x ⊥ dx − ∇ ⊥ φ · ∇ ⊥ φ + m 2 s φ 2 = d 2 k ⊥ dk + θ(k + )a † (k)a(k) k 2 ⊥ + m 2 s k + , (2.34) P + s = d 2 k ⊥ dk + θ(k + )a † (k)a(k)k + , (2.35) P − v = ω=1,3 d 2 k ⊥ dk + θ(k + ) k 2 ⊥ + m 2 v k + a † (k, ω)a(k, ω) + v 3 ,(2.36) and P + v = ω=1,3 d 2 k ⊥ dk + θ(k + )k + a † (k, ω)a(k, ω). (2.37) The term v 3 is the vector-meson instantaneous term, and we include it together with the purely mesonic contribution to P − v because it is cancelled by part of that contribution. Thus, our result for the total minus-momentum operator is P − = P − N + P − s + P − v ,(2.38) and for the plus-momentum P + = P + N + P + s + P + v ,(2.39) where from Eq. (2.21) P + N ≡ d 2 x ⊥ dx − 2 2ξ † + i∂ + ξ + . (2.40) III. MEAN FIELD THEORY The light-front Schroedinger equation for the complete nuclear ground-state wave function | Ψ is P − | Ψ = M A | Ψ . (3.1) We choose to work in the nuclear rest frame so that we also need P + | Ψ = M A | Ψ . (3.2) We want to use a variational principle. One might think that one may simply minimize the expectation value of P − , but this makes no sense because P + P − = M 2 A when acting on the wave function. One would get a zero of P − for an infinite value of P + . As explained in the Introduction, one must minimize the expectation value of P − subject to the condition that the expectation value of P + is equal to the expectation value of P − . This is the same as minimizing the average of P − and P + , which is the rest-frame energy of the entire system. To this end we define a light-front Hamiltonian H LF ≡ 1 2 P + + P − . (3.3) We stress that H LF is not usual the Hamiltonian, because the light-front quantization is used to define all of the operators that enter. The wave function | Ψ consists of a Slater determinant of nucleon fields | Φ times a mesonic portion 4) and the mean field approximation is characterized by the replacements | Ψ =| Φ ⊗ | mesons ,(3.φ → Ψ | φ | Ψ V µ → Ψ | V µ | Ψ . (3.5) We shall derive the meson field equations, and then determine the nucleon modes using a variational principle. A. Meson field equations We shall go through the derivation of the equation for the expectation value of φ(x) in a detailed fashion. Consider the quantity H LF a(k) | Ψ , and use commutators to obtain H LF a(k) | Ψ = [H LF , a(k)] | Ψ + M A a(k) | Ψ . (3.6) The operators P ± s of Eqs. (2.34) and (2.35) and the standard commutation relations allow one to obtain [H LF , a(k)] = − k 2 ⊥ + k + 2 + m 2 s 2k + a(k) + J(k) (2π) 3/2 √ 2k + , (3.7) where J(k) (2π) 3/2 √ 2k + is the commutator of the interaction, Eq. (2.29), between the scalar meson and the nucleon: J(k) (2π) 3/2 √ 2k + = 1 2 [P − I , a((k)]. (3.8) We use Eqs. (2.30)-(2.32), and take the commutator of the interactions v i with a(k). Then re-express the results in terms of ξ to obtain J(k) = − 1 2 g s d 2 x ⊥ dx − e ik·xξ (x)ξ(x). (3.9) Take the overlap of Eq. (3.6) with Ψ | to find Ψ | a(k) | Ψ k 2 ⊥ + k + 2 + m 2 s 2k + = Ψ | J(k) | Ψ (2π) 3/2 √ 2k + . (3.10) Multiply the above Eq. (3.10) by a factor √ 2k + (2π) 3/2 e −ik·x . Then add the result of that operation to its complex conjugate. The integral of the resulting equation over all k ⊥ and positive values of k + and using the field expansion (2.9) leads to the result   −∇ 2 ⊥ − 2 ∂ ∂x − 2 + m 2 s   Ψ | φ(x) | Ψ = Ψ | d 2 k ⊥ dk + θ(k + ) (2π) 3 J(k)e −ik·x + J † (k)e +ik·x | Ψ . (3.11) The evaluation of the right-hand-side of Eq. (3.11) proceeds by using Eq. (3.9) and its complex conjugate. The combination of those two terms allows one to remove the factor θ(k + ) and obtain a delta function from the momentum integral. That 1 2 k + appears in the exponential leads to the removal of the factor 1 2 of Eq. (3.9). One can also change variables using z ≡ −x − 2 , x ≡ (z, x ⊥ ). (3.12) The minus sign enters to remove the minus sign between the two terms of the factor k · x in Eq. (2.10). Then one may re-define the operator −∇ 2 ⊥ − 2 ∂ ∂x − 2 appearing in Eq. (3.11) as −∇ 2 . Note that we previously [22] obtained the above relation 3.12 simply by examining the space-time diagram for a static source (independent of x 0 ). The net result is that −∇ 2 + m 2 s Ψ | φ(x) | Ψ = −g s Ψ |ψ(x)ψ(x) | Ψ ,(3.13) which has the same form as the equation in the usual equal-time formulation. Note that the right hand side of Eq. (3.13) should be a function of |x| for the spherical nuclei of our present concern. Our formalism for the nucleon fields uses x ⊥ and x − as independent variables, so that obtaining numerically scalar and vector nucleon densities that depend only x 2 ⊥ +(x − /2) 2 will provide a central, vital test of our procedures and mean field theory. Assuming for the moment that this occurs, the scalar field Ψ | φ(x) | Ψ will depend only |x| according to (3.13). We stress that the use of Eq. (3.12) is merely a convenient way to simplify the calculation -using it allows us to treat the ⊥ and minus spatial variables on the same footing, and to maintain explicit rotational invariance. We will obtain the mesonic plus-momentum distributions from the ground state expectation value of different operators. The procedure of Eqs. (3.6) to (3.13) can also be applied to the vector fields. The appearance of the barred vector potential makes it necessary to display certain steps. The starting point is to consider the expression H LF a(k) | Ψ and the interaction J(k, ω) (2π) 3/2 √ 2k + = 1 2 [P − I , a(k, ω)]. (3.14) Using equations (2.30)-(2.32), taking the commutator of the interactions v i with a(k), and re-expressing the results in terms of ξ, leads to J(k, ω) = − 1 2 g v d 2 x ⊥ dx − e ik·xξ (x)γ ·ǭ(k, ω)ξ(x). (3.15) This, along with the other terms in the expression for H LF a(k, ω) | Ψ , allows us to obtain Ψ | a(k, ω) | Ψ k 2 ⊥ + k + 2 + m 2 v 2k + = Ψ | − 1 2 g v d 2 x ⊥ dx − e ik·xξ (x)γ ·ǭ(k, ω)ξ(x) | Ψ (2π) 3/2 √ 2k + . (3.16) The field equation for the vector mesonsV µ is obtained by multiplying the above byǭ µ (k, ω), summing over ω and performing standard manipulations. We need to know the quantity X µ (k) ≡ α,ω γ αǭ α (k, ω)ǭ µ (k, ω) (3.17) The use of Eqs. (2.18) and (2.17) leads to X µ (k) = −γ µ + 2δ(µ, −) γ · k k + + k µ k + γ + . (3.18) One makes familiar manipulations to obtain the result −∇ 2 + m 2 v Ψ |V µ (x) | Ψ = g v Ψ |ξ(x)γ µ ξ(x) | Ψ + ∆ µ , (3.19) with ∆ µ (x) ≡ − Ψ | d 3 x ′ (2π) 3 e ik·(x−x ′ )ξ (x ′ ) k µ k + γ + + 2δ(µ, −) γ · k k + ξ(x ′ ) | Ψ. (3.20) Note that (as in the derivation given above for φ) the variable k + (confined to positive values) is replaced by the inclusion of the complex conjugate term (a † (k, ω) by a variable k 3 which ranges from −∞ to ∞. We proceed by first assuming that −∇ 2 + m 2 v Ψ | V µ (x) | Ψ = g v Ψ |ξ(x)γ µ ξ(x) | Ψ ≡ J µ ,(3.∂ µ J + = ∂ + ∆ µ . (3.22) For µ = − the above relation is verified by integration by parts in the expression (3.20) for ∆ µ . If µ = −, one may use the definition (2.8) and that Ψ | V + | Ψ does not depend on x + to see that B. Nucleon single-particle wave functions ∂ − Ψ |V − | Ψ = ∂ − Ψ | V − | Ψ . The mesonic field equations are given in the previous subsection. The equation for the nucleon modes are to be found using the procedure of minimizing P − +P + with respect to the nucleon wave function, subject to the condition that the normalization of the independent fields remains fixed. The nucleon field operators enter only in the term P − N + P + N , so that it is useful to define H LF ≡ 1 2 P − N + P + N . (3.24) The specific operator is obtained by using Eq. (2.33) to find H LF = dx − 2 d 2 x ⊥ ξ † + H LF ξ + ,(3.25) where 2H LF ≡ i∂ + + 2g vV − + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) 1 i∂ + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) . (3.26) The potentials appearing in Eq. (3.26) are independent of x + . This implies some simplifications: ∂ +V − = ∂ + V − − ∂ − V + = ∂ + V − , so thatV − = V − , and (for i = 1, 2) ∂ +V i = ∂ + V i − ∂ i V + = −∂ i V + . Using the relation (2.6) we find thatV i = −∂ i Λ. The Slater determinant | Φ is defined by allowing A nucleon states, denoted by the index α to be occupied. For our Slater determinant the constrained minimization is given by the equation δ d 2 x ⊥ dx − 2 α | Λ + H LF − p − α 2 Λ + | α = 0,(3.27) where the quantities p − α are the Lagrange multiplication factors for each occupied orbital. The relation (3.27) leads immediately to our mode equation p − α Λ + | α = i∂ + + 2g vV − Λ − | α + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) 1 i∂ + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) Λ + | α . (3.28) The operators α and β have non-zero values when appearing between Λ + and Λ − , but vanish when appearing between the two identical projection operators. Thus we may obtain Λ − | α as Λ − | α = 1 i∂ + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) Λ + | α ,(3.29) or i∂ + Λ − | α = α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) Λ + | α .(3.p − α Λ + | α = i∂ + + 2g vV − Λ + | α + α ⊥ · (p ⊥ − g vV ⊥ ) + β(M + g s φ) Λ − | α .E s = 1 2 d 2 k ⊥ dk + θ(k + ) k 2 ⊥ + m 2 s k + + k + Ψ | a † (k)a(k) | Ψ . (3.33) In our mean field approximation Ψ | a † (k)a(k) | Ψ = | Ψ | a(k) | Ψ | 2 ,(3.34) with the matrix element already known from Eq. (3.10). Then straightforward calculation leads to the result E s = 1 2 d 3 k (2π) 3 1 k 2 + m 2 s | Ψ | J(k) | Ψ | 2 ,(3.35) where J(k) is given by Eq. (3.9), the replacement (3.12) is used, and as above k + is replaced by k 3 . The above expression is strikingly familiar -it is the result obtained in standard equal-time calculations. The vector meson contribution to the energy E v is defined as the one-half of the sum of the terms of P ± v of Eqs. (2.36) and (2.37). The calculation of P ± v is rather similar to the one just done for the scalar mesons. One uses the results (3.15), (3.16), and X µ (3.18). The effects of the instantaneous term v 3 are cancelled by the non-γ µ term of X µ , so that we find E v = − 1 2 d 3 k (2π) 3 1 k 2 + m 2 v | J v (k) | 2 , (3.36) where J v (k) ≡ g v Ψ | d 3 x e ik·x ψ † (x)ψ(x) | Ψ . (3.37) The nuclear mass M A is then given by M A = occ α p − α 2 + E s + E v ,(3.38) with expressions for each of the contributions given above. D. Relation with the equal-time formulation Our main results obtained using the mean field approximation and including the recoil of the A − 1 nuclear system are embodied in the equations (3.31) and (3.30). We solve these equations below using a mixed momentum-coordinate space procedure in which the wave functions are p + , x ⊥ | α = ψ α (p + , x ⊥ ). The values of p + are greater than zero. Thus the so-called spectrum condition that positive energy particles have only positive plus-momenta is maintained in our mean field approximation. An intermediate step is to make an approximation by using coordinate space techniques. Here one does not maintain the spectrum condition in an exact manner. Then one can show there is a very close relationship (but approximate) between our ψ α (x − , x ⊥ ) and the usual solutions to the Dirac equation obtained from the equal-time ET formulation. To see this, let's first consider the case where there is no vector potential at all (V µ → 0). Then multiply Eq. (3.30) by γ + and Eq. (3.31) by γ − . Use γ ± Λ ∓ | α = γ ± (Λ + + Λ − ) | α = γ ± | α , and then add the two equations. This gives γ 0 p − α − γ 3 (2p + − p − α ) ψ α (x − , x ⊥ ) = 2 γ ⊥ · p ⊥ + M + g s φ(x − , x ⊥ ) ψ α (x − , x ⊥ ). (3.39) Convert this to ordinary coordinates using x − = −2z, so that p + = i∂ + = 2i ∂ ∂x − → −i ∂ ∂z . The operator p + acts as a p 3 operator, and the result (3.39) looks like the Dirac equation of the equal-time formulation, except for the offending term −p − α multiplying the γ 3 . This motivates us to look for a solution of the form ψ α (z, x ⊥ ) = f (z)ψ ET α (z, x ⊥ ) , in which f (z) is chosen to remove to the offending term. The notation ET refers to the usual equal-time solution, because we see that ψ ET α obeys the usual ET Dirac equation γ 0 p − α 2 − γ · p − M − g s φ(z, x ⊥ ) ψ ET α (z, x ⊥ ) = 0, (3.40) provided f (z) = e ip − α z/2 ,(3.41) so that ψ α (z, x ⊥ ) = e ip − α z/2 ψ ET α (z, x ⊥ ). (3.42) The quantity of interest is ψ α (p + , x ⊥ ) which is expressed as ψ α (z, x ⊥ ) ≈ 1 √ 2π ∞ 0 dp + e ip + z ψ α (p + , x ⊥ ). (3.43) The approximation is that the correct version of ψ α (p + , x ⊥ ) will have no support for p + < 0, but the approximation (3.43) does. We can determine this support by examining the inverse Fourier transform. This gives ψ α (p + , x ⊥ ) = 1 √ 2π ∞ −∞ dz e −i(p + −p − α /2)z ψ ET α (z, x ⊥ ), (3.44) which is a Fourier transform of the equal-time Dirac wave function at a z-component of momentum p + − p − α /2. This is not exactly equal to zero when p + is zero or negative, but it is very small because p − α /2 includes the nucleon mass. The relationship between the term p − α /2 and the binding energy of the level denoted by α is p − α /2 = M − ε α . (3.45) Thus the relation (3.44) is just the usual equal-time procedure equal-time prescription, represented by Eq. (1.7), of replacing the kinematic variable p + by the combination of dynamical and kinematic variables M − ε + α + p 3 for the orbital α: p + → M − ε + α + p 3 . (3.46) However, the prescription (3.46) is dramatically changed when the vector potential is included. To see this, multiply Eq. (3.31) by γ − and Eq. (3.30) by γ + . Then Eq. (3.39) becomes (3.47) in which we usedV − = V − = V 0 . We again wish to reduce the coefficient of the γ 3 term to 2p + . This can be done with a new version of the multiplier f (z). We find that the light-front wave function is given by γ 0 (p − α − 2g v V 0 ) − γ 3 (2p + − p − α + 2g v V 0 ) ψ α (x − , x ⊥ ) = 2 (γ ⊥ · p ⊥ + M + g s φ) ψ α (x − , x ⊥ ),ψ α (p + , x ⊥ ) = 1 √ 2π ∞ −∞ dz e −i(p + −p − α /2)z e −igvΛ(z,x ⊥ ) ψ ET α (z, x ⊥ ), (3.48) where ∂ + Λ(x − , x ⊥ ) = V 0 (x − , x ⊥ ), Λ(z, x ⊥ ) = ∞ z dz ′ V 0 (z ′ , x ⊥ ),(3.49) and The nucleon mode equation resulting from the minimization of 1 2 (P + + P − ) is given by the coupled set of equations (3.31) and (3.30). The meson fields φ and V ± obey the equations γ 0 (p − α − g v V 0 )ψ ET α (z, x ⊥ ) = (γ · p + M + g s φ)ψ ET α (z, x ⊥ ).−(∂ + ) 2 − ∂ 2 ⊥ + m 2 s φ(x − , x ⊥ ) = −g s occ αψ α (x − , x ⊥ )ψ α (x − , x ⊥ ), (4.1) −(∂ + ) 2 − ∂ 2 ⊥ + m 2 v V ± (x − , x ⊥ ) = g v occ αψ α (x − , x ⊥ )γ ± ψ α (x − , x ⊥ ), (4.2) in which ψ α (x − , x ⊥ ) ≡ x − , x ⊥ | α . We use the Harindranath-Zhang [13] representation for the Dirac matrices α and β, which allows us to write Eqs. α i = 0 σ i σ i 0 , β = 1 0 0 −1 , (4.3) by the unitary transformation U = 1 √ 2 1 −σ 3 σ 3 1 . (4.4) Hence ψ → Uψ and θ → UθU † , where θ is a Dirac matrix in the standard representation. In our representation, the matrices of interest are: Λ + = 0 0 0 1 , Λ − = 1 0 0 0 , β = 0 σ 3 σ 3 0 , α 3 = −1 0 0 1 , α ⊥ = 0 σ ⊥ σ ⊥ 0 . (4.5) The 4-component wavefunction |ψ α may now be written in the form x − , x ⊥ |ψ α = x − , x ⊥ |ψ − α x − , x ⊥ |ψ + α ,(4.(p − α − 2g v V − − i∂ + )|ψ + α = (σ ⊥ · (p ⊥ − g v ∂ ⊥ Λ) + σ 3 (M + g s φ)) |ψ − α , (4.7a) i∂ + |ψ − α = (σ ⊥ · (p ⊥ − g v ∂ ⊥ Λ) + σ 3 (M + g s φ)) |ψ + α . (4.7b) The scalar and vector densities are defined as ρ s ≡ occ αψ α ψ α = occ α ψ + α σ 3 ψ − α +ψ − α σ 3 ψ + α , (4.8) ρ ± ≡ occ αψ α γ ± ψ α = occ α 2(ψ ± α ) † ψ ± α . (4.9) In the nuclear rest frame, ρ + = ρ − = ρ 0 , where ρ 0 is the usual nucleon density. Hence V ⊥ = 0 and V − = V + in this frame. A. Angular momentum We can write σ ⊥ · p ⊥ = σ (+) p (−) + σ (−) p (+) , (4.10) where σ (±) = 1 2 (σ 1 ± iσ 2 ) and p (±) = p 1 ± ip 2 = −ie ±iφ ∂ ∂r ± 1 r ∂ ∂φ . (4.11) Here r = |x ⊥ | and φ is the azimuthal angle, using cylindrical coordinates. For the nuclear physics problems of interest, we anticipate that there is an axis of azimuthal symmetry. Hence we can expand the 2-component wavefunctions in eigenstates of angular momentum J z , with eigenvalue j z : The equations to be solved are then x − , x ⊥ |ψ ± α = i x − , r|u ± α e i(jz− 1 2 )φ χ1 2 + x − , r|l ± α e i(jz+ 1 2 )φ χ − 1 2 =   iu ± α (x − , r)e i(jz− 1 2 )φ l ± α (x − , r)e i(jz+ 1 2 )φ   ,(4.p − α − 2g v V + − i∂ + u + α = − ∂ ∂r + j z + 1 2 r − ig v ∂Λ ∂r l − α + M * u − α , (4.13a) p − α − 2g v V + − i∂ + l + α = ∂ ∂r − j z − 1 2 r − ig v ∂Λ ∂r u − α − M * l − α , (4.13b) i∂ + u − α = − ∂ ∂r + j z + 1 2 r − ig v ∂Λ ∂r l + α + M * u + α , (4.13c) i∂ + l − α = ∂ ∂r − j z − 1 2 r − ig v ∂Λ ∂r u + α − M * l + α . (4.13d) The wavefunctions u ± α and l ± α , the nucleon effective mass M * = M +g s φ, the vector potential V + , and Λ are all functions of both x − and r. Eqs. (4.13) have a manifest spin degeneracy under j z → −j z . Solutions with the same eigenvalue p − α are obtained with the corresponding replacement u + α l + α → l + α u + α , u − α l − α → − l − α u − α , (4.14) Combined with isospin symmetry, we therefore have a manifest fourfold degeneracy of each single particle state. The numerical solution to Eqs. (4.13) is discussed in App. A. V. NUCLEAR BINDING ENERGIES If these solution to Eqs. (3.31) and (3.30) are to have any relevance at all, they should respect rotational invariance. The success in achieving this is examined in Tables I and II, which give our results for the spectra of 16 O and 40 Ca, respectively. Scalar and vector meson parameters are taken from Horowitz and Serot [24], and we have assumed isospin symmetry. We see that the J z = ±1/2 spectrum contains the eigenvalues of all states, since all states must have a J z = ±1/2 component. Furthermore, the essential feature that the expected degeneracies among states with different values of J z are reproduced numerically. The results shown in Tables I-III are obtained using a basis of 20 splines, a box size of 2L = 24 fm, and 24 Fourier components in the expansion of the wavefunction (see App. A). This value of L is large enough so that our results do not depend on it, and the number of terms in the expression for the density is enough to ensure that the densities are spherically symmetric. Another feature is that the spectrum with p + > 0 has no negative energy states, so that in using the LF method one is working in a basis of positive energy states only. TABLES The values of p − α /2 given in Tables I and II are essentially the same as the single particle energies E α of the ET formalism, to within the expected numerical accuracy of our program. This equality is not mandated by spherical symmetry alone because the solutions in the equal-time framework have non-vanishing components with negative values of p + . Table III gives the contributions to the total P + momentum from the nucleons, scalar mesons, and vector mesons for 16 O, 40 Ca, and 80 Zr, as well as the nuclear matter limit. In the next section we examine in detail the momentum distributions giving rise to these expectation values. VI. PLUS-MOMENTUM DISTRIBUTIONS AND LEPTON-NUCLEUS DEEP INELASTIC SCATTERING We discuss the probability that a nucleon, or meson has a momentum p + . In the lightfront formulation, these distribution functions are determined by the absolute square of the ground state wave function. Each distribution is discussed in turn. A. Nucleon plus-momentum distribution The light-front formulation is very useful for obtaining this observable. The probability that we want, f N (p + ), follows from Eq. (4.9) as f N (p + ) = 2 occ α d 2 x ⊥ | p + , x ⊥ | ψ + α | 2 , (6.1) with A = ∞ 0 dp + f N (p + ),(6.2)P + N = ∞ 0 dp + p + f N (p + ). (6.3) The next step is to define a dimensionless variable y: y ≡ p + A M A ≡ p + M A ,(6.4) and a dimensionless distribution f N (y): f N (y) ≡ f N (p + ) M A . (6.5) The result is shown in Fig. 1 for 16 O, 40 Ca, and 80 Zr. The peaks of the distributions range from y ≈ 0.72 to y ≈ 0.80, whereas the average values y are somewhat lower (see Table III). The distribution is not symmetric about its average value, as it would be if a simple Fermi gas model were used. Both of these effects are caused by the presence of nuclear mesons, which carry the remainder of the plus-momentum. Vector meson plus-momentum distribution yf v (y). In the nuclear matter limit, yf v (y) becomes a delta function. B. Scalar meson distribution The probability we want is given by f s (k + ) = d 2 k ⊥ Ψ | a † (k)a(k) | Ψ . (6.6) Using Eqs. (3.34) and (3.10) this becomes f s (k + ) = d 2 k ⊥ (2π) 3 2k + (k 2 + m 2 s ) 2 | Ψ | J(k) | Ψ | 2 . (6.7) This result which is of the same form as in Ref. [22]. A final step is to define a dimensionless distribution f s (y): f s (y) ≡ f s (k + ) M A . (6.8) The scalar mesons are found to carry less that 1% of the plus-momentum of the nucleus (Table III), which is negligible. C. Vector meson distribution The probability we want is given by f v (k + ) = d 2 k ⊥ ω=1,3 Ψ | a † (k, ω)a(k, ω) | Ψ . (6.9) Using Eqs. (3.16) and the mean field approximation (3.5) this becomes f v (k + ) = d 2 k ⊥ (2π) 3 2k + (k 2 + m 2 v ) 2 ω=1,3 | J(k, ω) | 2 ,(6.10) in which J(k, ω) = d 3 x e ik·x Ψ |ψ(x)γψ(x) ·ǭ(k, ω) | Ψ . (6.11) Using Eq. (2.18) and that only the µ = ν = − term enters, leads to the result that f v (k + ) = d 2 k ⊥ (2π) 3 2k + (k 2 + m 2 v ) 2 k 2 ⊥ + m 2 v k + 2 | J v (k) | 2 ,(6.12) another result which is of the same form as in Ref. [22]. A final step is to define a dimensionless distribution f v (y) f v (y) ≡ f v (k + ) M . (6.13) The vector mesons carry approximately 30% of the nuclear plus-momentum. The technical reason for the difference with the scalar mesons is that the evaluation of a † (k, ω)a(k, ω) counts vector mesons "in the air"and the resulting expression contains polarization vectors that give a factor of 1 k + in Eq. (6.12) which enhances the distribution of vector mesons of low k + . The results for the vector meson distribution are shown in Fig. 2. Clearly as the size of the nucleus increases the enhancement of the distribution at lower values of k + becomes more evident. In the case of nuclear matter the distribution k + f v (k + ) becomes a delta function. D. Lepton-nucleus deep inelastic scattering It is worthwhile to see how the present results are related to lepton-nucleus deep inelastic scattering experiments. We find that the nucleons carry only about 70% of the plus-momentum. The use of our f N in standard convolution formulae lead to a reduction in the nuclear structure function that is far too large (∼95% is needed [4]) to account for the reduction observed [4] in the vicinity of x ∼ 0.5. The reason for this is that the quantity M + g s φ acts as a nucleon effective mass of about 670 MeV, which is very small. A similar difficulty occurs in the (e, e ′ ) reaction [25] when the mean field theory is used for the initial and final states. The use of a small effective mass and a large vector potential enables a simple reproduction of the nuclear spin orbit force [12,24]. However, effects beyond the mean field may lead to a significant effective tensor coupling of the isoscalar vector meson [26] and to an increased value of the effective mass. Such effects are incorporated in Bruckner theory, and a light-front version [27] could be applied to finite nuclei with better success in reproducing the data. VII. SUMMARY AND DISCUSSION The previous Sections present a derivation of a light-front version of mean field theory. The necessary technique is to minimize expectation value of the sum P − + P + . This leads to a new set of coupled equations (3.31) and (3.30) for the single nucleon modes. These depend on the meson fields of Eqs. (3.13) and (3.21). The most qualitatively startling feature emerging from the derivation is that the meson field equations (3.13) and (3.21) are the same as that of the usual theory, except that z of the equal-time theory translates to −x − /2 of the light-front version. This can be understood in a simple manner by noting that light-front quantization occurs at x + = 0. If one then sets z = −t, then x − = t − z = −2z. However, this simple argument is not really justified, because using x ± precludes the use of z and t. A general argument, using the feature that a static source in the usual coordinates corresponds to a source moving with a constant velocity in light front coordinates, will be presented in a separate paper [15]. That paper also contains a number of solutions of toy models. Even though the meson field equations of the light-front and equal-time theories are the same, there are substantial and significant differences between the two theories. In our treatment, the mesonic fields are treated as quantum field operators. The mean field approximation is developed by replacing these operators by their expectation values in the complete ground state nuclear wave function. This means that the ground state wave function contains Fock terms with mesonic degrees of freedom. We can therefore compute expectation values other than that of the field. In particular, we are able to obtain the mesonic momentum distributions (Sec. VI). This feature has been absent in standard approaches. We obtain an approximate solution (3.48) of our nucleon mode equation. Our nucleon mode functions are approximately a phase factor times the usual equal-time mode functions (evaluated at x − = −2z). This shows that the energy eigenvalues of the two theories should have very similar values. But the wave functions are different-the presence of the phase factor explicitly shows that the nucleons give up substantial amounts of plus momentum to the vector mesons. A new numerical technique, discussed in Sec. IV and App. A, is introduced to solve the coupled nucleon and meson field equations. Our results display the expected 2j α + 1 degeneracy of the single nucleons levels, and the resulting binding energies are essentially the same as for the usual equal-time formulation. This indicates that the approximation (3.48) is valid. As discussed in Sec. VI-D, the present results related to lepton-nucleus deep inelastic scattering experiments and (e, e ′ ) reactions are not consistent with experimental findings. This is because, in 40 Ca for example, the nucleons carry only 72% of the plus momentum. This is a result of the quantity M + g s φ, which acts as a nucleon effective mass, is very small, about 670 MeV. The use of a small effective mass and a large vector potential enables a simple reproduction of the nuclear spin orbit force [12,24]. However, effects beyond the mean field may lead to a significant effective tensor coupling of the isoscalar vector meson [26] and to an increased value of the effective mass. Such effects are incorporated in Bruckner theory [27] which, for infinite nuclear matter, results in nucleons having about 80-85% of the nuclear plus-momentum. A light-front version [27] should be applied to finite nuclei with better success in reproducing the data. Another approach could be to use different Lagrangians, with non-linear couplings between scalar mesons and the nucleons [12], or ones in which the coupling is of derivative form [28]:ψγ µ ψ∂ µ φ. These models are known to have significantly smaller magnitudes of the scalar and vector potentials. In particular, in nuclear matter vector mesons carry only about 10-15% of the nuclear-plus momentum. Another interesting possibility would be to obtain a light-front version of the quark-meson coupling model [29], in which confined quarks interact by exchanging mesons with quarks in other nucleons. This model, also has smaller magnitudes of the scalar and vector potentials. In any case, these kinds of nuclear physics calculations can be done in a manner in which modern nuclear dynamics is respected, boost invariance in the z-direction is preserved, and in which the rotational invariance so necessary to understanding the basic features of nuclei is maintained. ACKNOWLEDGMENTS P.G.B. was supported in part by the Natural Sciences and Engineering Research Council of Canada. Meson fields The equations for the meson fields are solved using Green function methods. We illustrate this for the vector field V + , with results for φ following by analogy. Starting with − ∂ 2 ∂z 2 − ∂ 2 ∂r 2 − 1 r ∂ ∂r + m 2 v V + (z, r) = g v ρ + (z, r),(A8) we expand V + (z, r) in the same form as the density ρ + (z, r), Eq. (A4): V + (z, r) = m V + m (r) cos mqz.(A9) The functions V + m (r) satisfy − ∂ 2 ∂r 2 − 1 r ∂ ∂r + m 2 v + m 2 q 2 V + m (r) = g v ρ + m (r),(A10) and their solution may be written as V + m (r) = g v ∞ 0 dr ′ r ′ G(r, r ′ )ρ + m (r ′ ). (A11) The Green function is G(r, r ′ ) = I 0 (m * v r)K 0 (m * v r ′ )θ(r ′ − r) + I 0 (m * v r)K 0 (m * v r ′ )θ(r − r ′ ).(A12) We have introduced the definition m * v ≡ m 2 v + m 2 q 2 , and I 0 and K 0 are modified cylindrical Bessel functions of zeroth order. The meson fields are computed numerically from Eq. (A11) by an outward and an inward integration. Solution of nucleon equation To streamline the notation, we drop the explicit dependence on the single particle label α in this section. Equation (4.13) can be rewritten in the form of a 2 × 2 matrix equation p − z, r|u + z, r|l + =    2g v V + − i ∂ ∂z I + H 1 −i ∂ ∂z I H    z, r|u + z, r|l + ,(A13) with the constrained subsidiary relation z, r|u − z, r|l − = 1 −i ∂ ∂z I H z, r|u + z, r|l + .(A14) Here I is the 2 × 2 identity matrix, and H = M * D 1 + ig v ∂Λ ∂r D 2 − ig v ∂Λ ∂r −M * ,(A15)D 1 = − ∂ ∂r + j z + 1 2 r , (A16) D 2 = ∂ ∂r − j z − 1 2 r . (A17) If we take N Fourier components n = 1, 2, 3, . . . , N in the expansion of Eq. (A2) in z, then u + (z, r) = z, r|u + and l + (z, r) = z, r|l + have the matrix representation       u + 1 (r) u + 2 (r) . . . u + N (r)       ,       l + 1 (r) l + 2 (r) . . . l + N (r)       ,(A18) where u + n (r) = p + n , r|u + and l + n (r) = p + n , r|l + . Equation (A13) becomes a 2N × 2N matrix equation. Matrix elements of the N × N sub-blocks are determined from the integrals V + (r) (nn ′ ) = p + n |V + (z, r)|p + n ′ (A19) = 1 2L L −L dz e i(p + n ′ −p + n )z V + (z, r),(A20)= 1 + δ m,0 2 V + m (r)δ |n−n ′ |,m .(A21) Similarly, [M * (r)] (nn ′ ) = Mδ n,n ′ + 1 + δ m,0 2 g s φ 0 (r)δ |n−n ′ |,m , (A22) −i ∂ ∂z (nn ′ ) = p + n δ n,n ′ ,(A23)[D 1 ] (nn ′ ) = D 1 δ n,n ′ , (A24) [D 2 ] (nn ′ ) = D 2 δ n,n ′ ,(A25)[iΛ(r)] (nn ′ ) = − 1 2 V + m (r) + (−1) m V + 0 (r) p + n − p + n ′ δ |n−n ′ |,m m = 0 .(A26) The last relation comes from the definition − ∂Λ ∂z = V + . Using integration by parts, and a careful treatment of surface terms, gives matrix elements in the form (A26). The problem has now been reduced to an eigenvalue problem involving 2N coupled differential equations in the variable r. To solve this, we make a further expansion of u ± n (r) and l ± n (r) in a finite basis of B-splines of degree k [30,31]: u ± n (r) = N i=1 α ± n,i B (k) i (r),(A27)l ± n (r) = N i=1 β ± n,i B (k) i (r).(A28) The B-splines {B (k) i , i = 1, . . . , N } are polynomials of degree k − 1 spanning a domain of equally spaced knots {t i , i = 1, . . . , N + k} in r. They are smooth "local" functions that are non-zero only on the interval t i < r < t i+k . This basis forms an accurate, but non-orthogonal set. Hence the overlap matrix S ij = ∞ 0 dr r B (k) i (r)B (k) j (r) (A29) is non-diagonal. It is diagonally banded, however, since B We now have a matrix equation with the 2 × 2 block structure    2g v V + − i ∂ ∂z ⊗ I + H 1 −i ∂ ∂z ⊗ I H    α + β + = p − [1] ⊗ I α + β + ,(A30) where ⊗ denotes an outer product of matrices. The last relation follows from using integration by parts. Numerical methods For our numerical calculations we use L = 12 fm and N = 24 Fourier components, and set the number of terms in the cosine expansion of the densities to be N/2, or 12. We choose k = 5 for the degree of the B-splines, N = 20 B-splines in the expansion in r, and take 0 < r < L. The integrals over r are performed using Gaussian integration between knots, which gives exact results for the matrix elements S ij , Eq. (A29). The matrix eigenvalue problem Eq. (A30) is of the form Ax = λBx, where A and B are real, symmetric matrices. In our problem, A and B are diagonally banded, and there are efficient EISPACK routines that take advantage of this [32]. Cholesky decomposition is used to efficiently compute the matrix 1 −i ∂ ∂z ⊗ I H,(A38) which is needed both to determine u − and l − as well as to construct the matrix in Eq. (A30). APPENDIX B: MOMENTUM DISTRIBUTIONS In momentum space the meson field equations become φ(k + , k ⊥ ) = − g s k 2 + m 2 s ρ s (k + , k ⊥ ),(B1)V + (k + , k ⊥ ) = g v k 2 + m 2 s ρ + (k + , k ⊥ ),(B2) with k 2 = (k + ) 2 + k 2 ⊥ , and the convention V + (k + , k ⊥ ) = d 2 x ⊥ dz e −ik ⊥ ·x ⊥ −ik + z V + (z, x ⊥ ).(B3) The scalar meson momentum distribution from Eq. (6.7) can be rewritten as f s (k + ) = 2k + 2π d 2 k ⊥ (2π) 2 g 2 s (k 2 + m 2 s ) 2 |ρ s (k + , k ⊥ )| 2 (B4) = 2k + 2π d 2 k ⊥ (2π) 2 |φ(k + , k ⊥ )| 2 (B5) = 2k + 2π d 2 x ⊥ |φ(k + , x ⊥ )| 2 .(B6) Then P + s = ∞ 0 dk + k + f s (k + ) (B7) = 1 2 ∞ −∞ dk + 2π 2(k + ) 2 d 2 x ⊥ |φ(k + , x ⊥ )| 2 (B8) = d 3 x (∂ + φ(x)) 2 (B9) = T ++ s .(B10) The vector meson momentum distribution is a little more complicated. Starting from Eq. (6.12), f v (k + ) = 2k + 2π d 2 k ⊥ (2π) 2 g 2 v (k 2 + m 2 v ) 2 k 2 ⊥ + m 2 v (k + ) 2 |ρ + (k + , k ⊥ )| 2 (B11) = 2k + 2π d 2 k ⊥ (2π) 2 g v k 2 + m 2 v k 2 + m 2 v − (k + ) 2 (k + ) 2 V + (k + , k ⊥ )ρ + (k + , k ⊥ ) (B12) = 2 2π d 2 k ⊥ (2π) 2 1 k + g v V + (k + , k ⊥ )ρ + (k + , k ⊥ ) − k + |V + (k + , k ⊥ )| 2 .(B13) Then P + v = ∞ 0 dk + k + f v (k + ) (B14) = 1 2 ∞ −∞ dk + 2π 2 d 2 x ⊥ g v V + (k + , x ⊥ )ρ + (k + , x ⊥ ) − (k + ) 2 |V + (k + , x ⊥ )| 2 (B15) = d 3 x g v V + (x)ρ + (x) − (∂ + V + (x)) 2 (B16) = T ++ v . (B17) Clearly f v (k + ) is singular at k + = 0, so we plot k + f v (k + ) instead. Momentum distributions involve integrals over x ⊥ , or equivalently over k ⊥ , so one really only needs to Fourier transform V + (z, x ⊥ ) in z. If we define V + (k + , x ⊥ ) = dz e −ik + z V + (z, x ⊥ ),(B18) then for k + m = mq, q = π/L, it follows from the definition Eq. (A9) that V + (k + m , x ⊥ ) = 2L 1 + δ m,0 2 V + m (r),(B19) with a similar result for ρ + (k + m , x ⊥ ), φ(k + m , x ⊥ ), and ρ s (k + m , x ⊥ ). Hence we calculate the momentum distributions from the expressions The nucleon momentum distribution for p + n = nq is given by f N (p + n ) = 2 q ∞ 0 dr r occ α (u + α,n (r)) 2 + (l + α,n (r)) 2 , so that A = ∞ 0 dp + f N (p + n ) ≈ q n f N (p + n ), and P + N = ∞ 0 dp + p + f N (p + n ) ≈ q n p + n f N (p + n ). We interpolate between the discrete values of k + n and p + n to produce the plots of Figs. 1 and 2. ( 3 . 23 ) 323Thus the validity of Eq. (3.21) is established. and (3.30) are the essential results of this section. We have obtained the light-front version of the Hartree equations. C. Nuclear energy There are contributions to the expectation value of P − + P + from the nucleons, scalar mesons, and vector mesons. The nucleonic term is given from the expectation value of the nucleonic part H LF (3.26). Taking the nuclear expectation value of H LF leads to a sum of matrix elements in the occupied states | α . The use of the wave equation (3.28) leads to the result Ψ | H LF | Ψ the sum is over α includes only occupied states. The contribution from the scalar mesons E s is given from the scalar meson terms of Eqs. (2.34) and (2.35) by (3.48) tells us that the influence of the vector potential is to remove plusmomentum from the nucleons. This removal and enhancement of the nuclear vector meson content is the most dramatic result we have.How accurate is Eq.(3.48)? This can only be addressed by solving the problem in a manner which respects the spectrum condition. The results show an astonishing agreement between the eigenvalues of Eq. (3.31) and those of the equal-time Dirac equation. Thus it should be safe to use Eq. (3.48) for qualitative purposes. of the nucleon and meson field equations are discussed. The reduction of Eqs. (3.31) and (3.30) to a two-dimensional matrix equation is presented here. The new numerical technique involving splines is elaborated in App. A. ( 3 . 331) and(3.30) in 2-component form. This representation can be obtained from the standard representation FIGURESFIG. 1 . 1Nucleon plus-momentum distribution function, f N (y), for 16 O, 40 Ca, and 80 Zr. Here y ≡ p + /(M A /A). FIG. 2 . 2FIG. 2. Vector meson plus-momentum distribution yf v (y). In the nuclear matter limit, yf v (y) becomes a delta function. j (r) are functions that only overlap if |i − j| ≤ k − 1. The property of being diagonally banded also applies to the matrix elements of other operators. jj α + = {α + n,i } and β + = {β + n,i } are column vectors of length (N × N ), and the (N × N ) × (N × N ) sub-blocks have matrix elements (r) [M * (r)] (nn ′ ) , (A32) −i ∂ ∂z (nn ′ ),(ij)= p + n δ n,n ′ S ij , (A33)[1] (nn ′ ),(ij) = δ n,n ′ S ij ,(A34)[D 1 ] (nn ′ ),(ij) = δ n,2 ] (nn ′ ),(ij) = δ n,(r) [iΛ(r)] (nn ′ ) . g v V + m (r)φ + m (r) − (mq) 2 (V + m (r)) 2 . TABLE I . IComparison of the single particle spectra of 16 O in the equal-time (ET) formalism (E α − M ) with the light-front (LF) method (p − α /2 − M ). Units are in MeV.TABLE III. Total plus-momentum per nucleon for 16 O, 40 Ca, 80 Zr, and nuclear matter (NM) in MeV. No Coulomb interaction is included here.ET LF State α E α − M J z = ±1/2 J z = ±3/2 0s 1/2 −41.73 −41.73 0p 3/2 −20.77 −20.79 −20.77 0p 1/2 −12.49 −12.51 TABLE II. Comparison of the ET and LF single particle spectra of 40 Ca. ET LF State α E α − M J z = ±1/2 J z = ±3/2 J z = ±5/2 0s 1/2 −55.40 −55.39 0p 3/2 −38.90 −38.91 −38.90 0p 1/2 −33.18 −33.18 0d 5/2 −22.75 −22.76 −22.75 −22.74 1s 1/2 −14.39 −14.36 0d 3/2 −13.87 −13.88 −13.89 Nucleus P + N /A P + s /A P + v /A P + /A 16 O 704.7 6.4 221.8 932.9 40 Ca 672.6 4.7 253.3 930.6 80 Zr 655.2 3.6 270.2 929.0 NM 569.0 0.0 354.2 923.2 APPENDIX A: NUMERICAL TECHNIQUESThere are many possible numerical approaches to solving Eqs.(4.13). We choose a method that is robust, and emphasizes the physical content of the wavefunctions, at the expense of being computationally intensive. We begin by making a Fourier expansion of the wavefunctions in the variable x − :with a similar expansion for l ± α (x − , r). Boundary conditions are imposed by constraining the system to be in a "box" of a given length in the variable x − . In the nuclear rest frame,with {p + n = nq, n = 1, 2, 3, . . .}, and q = π/L. For bound states, the functions u ± α,n (r) and l ± α,n (r) are real, and so the scalar and vector densities take the formwith m = 0, 1, 2, . . ., and ρ s m (r) = 2 2L 2πρ ± m (r) = 2 2L 2π 2 1 + δ m,0 occ α n u ± α,n (r)u ± α,n+m (r) + l ± α,n (r)l ± α,n+m (r) .The normalization integral for a nucleus with A nucleons is (u + α,n (r)) 2 + (l + α,n (r)) 2 .(A7) . J Aubert, Phys. Lett. 123275J. Aubert et al., Phys. Lett. 123B, 275 (1982); . R G Arnold, Phys. Rev. Lett. 52727R.G. Arnold et al., Phys. Rev. Lett. 52, 727 (1984); . A Bodek, Phys. Rev. Lett. 51534A. Bodek et al., Phys. Rev. Lett. 51, 534 (1983). . L L Frankfurt, M I Strikman, Phys. Rep. 76215L.L. Frankfurt and M.I. Strikman, Phys. Rep. 76, 215 (1981). . M Burkardt, Adv. Nucl. Phys. 231M. Burkardt, Adv. Nucl. Phys. 23, 1 (1996). R L Jaffe, Relativistic Dynamics and Quark-Nuclear Physics. M.B. Johnson and A. PicklesimerNew YorkWileyR.L. Jaffe, in Relativistic Dynamics and Quark-Nuclear Physics, edited by M.B. Johnson and A. 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Rev. Lett. 642479D.M. Alde et al., Phys. Rev. Lett. 64, 2479 (1990). . G F Bertsch, L Frankfurt, M Strikman, Science. 259773G.F. Bertsch, L. Frankfurt, and M. Strikman, Science 259, 773 (1993). . G A Miller, Phys. Rev. C. 562789G.A. Miller, Phys. Rev. C 56, R8 (1997); 56, 2789 (1997). . B D Serot, J D Walecka, Adv. Nucl. Phys. 161B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1 (1986); . Int. J. Mod. Phys. 6515Int. J. Mod. Phys. E6, 515 (1997). . A Harindranath, W.-M Zhang, Phys. Rev. D. 484903A. Harindranath and W.-M. Zhang, Phys. Rev. D 48, 4861 (1993); 48, 4881 (1993); 48, 4903 (1993). . P G Blunden, M Burkardt, G A Miller, nucl- th/9901063Phys. Rev. C. P.G. Blunden, M. Burkardt, and G.A. Miller, Phys. Rev. C (to be published), nucl- th/9901063. Light-Front Nuclear Physics: Toy Models, Static Sources and Tilted Light-Front Coordinates, preprint NT@UW-99-22, to be submitted to. P G Blunden, M Burkardt, G A Miller, Phys. Rev C. P.G. Blunden, M. Burkardt and G.A. Miller, Light-Front Nuclear Physics: Toy Mod- els, Static Sources and Tilted Light-Front Coordinates, preprint NT@UW-99-22, to be submitted to Phys. Rev C. Our notation is that a four vector A µ is defined by the plus, minus and perpendicular components as (A 0 + A 3 , A 0 − A 3 , A ⊥ ) and A · B = 1 2 A + B − + 1. Our notation is that a four vector A µ is defined by the plus, minus and perpendicular components as (A 0 + A 3 , A 0 − A 3 , A ⊥ ) and A · B = 1 2 A + B − + 1 . A − B + − A ⊥ · B ⊥, A − B + − A ⊥ · B ⊥ . . S J Brodsky, H-C Pauli, S S Pinsky, Phys. Rep. 301299S. J. Brodsky, H-C Pauli, S.S. Pinsky, Phys. Rep. 301, 299 (1998); S J Brodsky, G P Lepage, Perturbative Quantum Chromodynamics. A. MuellerSingaporeWorld ScientificS.J. Brodsky and G.P. Lepage, in Perturbative Quantum Chromodynamics, edited by A. Mueller, (World Scientific, Singapore, 1989); . X D Ji, Comm. Nucl. Part. Phys. 21123X.D. Ji, Comm. Nucl. Part. Phys. 21, 123 (1992); . W.-M , W.-M. 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Burkardt and G.A. Miller, Phys. Rev. C 58, 2450 (1998). because the fermion scalar density is independent of x + . Then taking the Fourier transform as needed in Eq. ∂ − Vanishes, 3.20) leads to the equality (3.22) for the case µ = −∂ − vanishes, because the fermion scalar density is independent of x + . Then taking the Fourier transform as needed in Eq. (3.20) leads to the equality (3.22) for the case µ = −. We neglect here the electromagnetic effects as well as those of the ρ meson. C J Horowitz, B D Serot, Nucl. Phys. 368503C.J. Horowitz and B.D. Serot, Nucl. Phys. A368, 503 (1981). We neglect here the electromagnetic effects as well as those of the ρ meson. . H Kim, C J Horowitz, M R Frank, Phys.Rev. C. 51792H. Kim, C.J. Horowitz, and M.R. Frank, Phys.Rev. C 51, 792 (1995). . R J Furnstahl, J J Rusnak, B D Serot, Nucl. Phys. 632607R.J. Furnstahl, J.J. Rusnak, and B.D. Serot, Nucl. Phys. A632, 607 (1998). . G A Miller, R Machleidt, nucl-th/9811050Phys. Lett. B. G.A. Miller and R. Machleidt, Phys. Lett. B (to be published), nucl-th/9811050; Phys. Rev. C. to be published. Infinite nuclear matter on the light-front: Nucleon-nucleon correlations. G A Miller, R Machleidt, nucl-th/9903080G.A. Miller and R. Machleidt, Phys. Rev. C. to be published. Infinite nuclear mat- ter on the light-front: Nucleon-nucleon correlations, nucl-th/9903080. . J Zimanyi, S A Moszkowski, Phys. Rev. C. 421416J. Zimanyi, S.A. Moszkowski, Phys. Rev. C 42, 1416 (1990); . N K Glendenning, F Weber, S A Moszkowski, Phys. Rev. C. 45844N.K. Glendenning, F. Weber, and S.A. Moszkowski, Phys. Rev. C 45, 844 (1992); . S A Moszkowski, private communicationS.A. Moszkowski, (private communication). . K Saito, A W Thomas, Phys. Lett. 3279K. Saito and A.W. Thomas, Phys. Lett. 327B, 9 (1994); . P G Blunden, G A Miller, Phys. Rev. C. 54359P.G. Blunden and G.A. Miller, Phys. Rev. C 54, 359 (1996); . X Jin, B K Jennings, Phys. Rev. C. 541427X. Jin and B.K. Jennings, Phys. Rev. C 54, 1427 (1996). 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{'abstract': 'A light-front treatment for finite nuclei is developed from a relativistic effective Lagrangian (QHD1) involving nucleons, scalar mesons and vector mesons. We show that the necessary variational principle is a constrained one which fixes the expectation value of the total momentum operator P + to be the same as that for P − . This is the same as minimizing the sum of the total momentum operators: P − + P + . We obtain a new light-front version of the equation that defines the single nucleon modes. The solutions of this equation are approximately a non-trivial phase factor times certain solutions of the usual equal-time Dirac equation. The ground state wave function is treated as a meson-nucleon Fock state, and the meson fields are treated as expectation values of field operators in that ground state. The resulting equations for these expectation values are shown to be closely related to the usual meson field equations. A new numerical technique to solve the self-consistent field equations is introduced and applied to 16 O and 40 Ca. The', 'arxivid': 'nucl-th/9906012', 'author': ['P G Blunden ', 'Canada M Burkardt ', 'G A Miller ', '\nDepartment of Physics and Astronomy\nDepartment of Physics New\nDepartment of Physics\nUniversity of Manitoba Winnipeg\nMexico State University Las Cruces\nBox 351560R3T 2N2, 88003-0001MB, NMU.S.A\n', '\nUniversity of Washington Seattle\n98195-1560WAU.S.A\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nDepartment of Physics New\nDepartment of Physics\nUniversity of Manitoba Winnipeg\nMexico State University Las Cruces\nBox 351560R3T 2N2, 88003-0001MB, NMU.S.A', 'University of Washington Seattle\n98195-1560WAU.S.A'], 'corpusid': 119357119, 'doi': '10.1103/physrevc.60.055211', 'github_urls': [], 'n_tokens_mistral': 26585, 'n_tokens_neox': 23275, 'n_words': 13831, 'pdfsha': '3f8cd07b0c7c0ce15613eef5e1c8fbb7c6d67bb3', 'pdfurls': ['https://export.arxiv.org/pdf/nucl-th/9906012v1.pdf'], 'title': ['Light-Front Nuclear Physics: Mean Field Theory for Finite Nuclei', 'Light-Front Nuclear Physics: Mean Field Theory for Finite Nuclei'], 'venue': []}

arxiv

arXiv:hep-lat/9308023v1 31 Aug 1993 DETERMINING THE ORDER OF SU (3) DECONFINING PHASE TRANSITION Zheng-Kun Zhu Department of Physics and Atmospheric Science Center for Theoretical Physics Department of Physics and Atmospheric Science Drexel U niversity † Philadelphia Massachusetts Institute of Technology Cambridge Drexel University Philadelphia 19104-9984, 02139, 19104-9984Pennsyvalnia, Massachusetts, Pennsyvalnia Da Hsuan Feng Department of Physics and Atmospheric Science Center for Theoretical Physics Department of Physics and Atmospheric Science Drexel U niversity † Philadelphia Massachusetts Institute of Technology Cambridge Drexel University Philadelphia 19104-9984, 02139, 19104-9984Pennsyvalnia, Massachusetts, Pennsyvalnia arXiv:hep-lat/9308023v1 31 Aug 1993 DETERMINING THE ORDER OF SU (3) DECONFINING PHASE TRANSITION Physics Division Oak Ridge National Laboratory, Oak Ridge, Tennessee37996 An effective spin model for the finite temperature non-abelian lattice gauge theory is derived. The outcome is the surprising result that only nearest neighbor coupling survives, thus confirming the well known numerical results that the deconfining phase transition of the (3+1)-dimensional SU (3) pure gauge theory is first order. A major success of lattice gauge theory (LGT) is the numerical demonstration of a transition from the low temperature color confining to the high temperature non-confining phase [1]. Determining the order of the transition is a subtle matter. For the (3+1) dimensional SU(3) finite temperature LGT, it was confirmed numerically on the N τ = 4 lattice (N τ is the lattice size in the temporal direction) that the transition is first order [2,3,4]. More recent calculations on the N τ = 6 [4] also suggest that this is the case. Still, the definitive statement about the order of phase transition remains a challenge for the next generation of computers. Apart from the full numerical determination the order of the finite temperature phase transition, there exists in the literature universality arguments, first suggested by Svetitsky and Yaffe in 1982 [5], which relate the field theory to a simpler three dimensional effective spin model. The derivation of the effective spin model is highly complex. Nevertheless, by conjecturing a short range coupling, it was shown that the transition is also first order [3,6,7]. Although the conjecture may not be beyond reproach, these important results certainly lend confidence to the conclusion that the transition could indeed be first order. We therefore feel that there remain the urgent task of deriving an effective spin model. The purpose of this letter is to carry out the derivation. Instead of directly studying the (3+1) dimensional SU(3) finite temperature LGT, we will begin by studying a generic (3 + 1)-dimensional SU(N) LGT. The finite temperature behavior can be described by a partition function defined on a hypercubic lattice of size N 3 s × N τ , Q = e S ,(1) where S is the Wilson action S = β E n µ<ν 1 N ReT r(U µ n U ν n+µ U µ + n+ν U ν + n ),(2) β E is the coupling constant, n and µ, ν represent the space-time coordinate and directions respectively. Finite temperature is introduced by imposing a periodic boundary condition in the time direction, with period N τ . Accordingly, the temperature is 1 Nτ a , where a is the lattice spacing. It is well known that the nature of the deconfining phase transition can straightforwardly be studied by constructing an effective theory (of the Wilson line) from the partition function in eq.(1) with all the link variables integrated out except those for the Wilson lines. Then the effective action S W ef f has the following form [8]: S W ef f = S W ef f ({ 1 N T rW n }).(3) Unfortunately, this process could only be carried out in the strong coupling limit. There are two main obstacles to obtain S W ef f . The first is that in eq.(2) there is no explicit Wilson line variable in S. The second is that it is very complicated to carry out the integration. We will now discuss the removal of these difficulties. It turns out that by taking a thermal gauge choice [9] the first difficulty can easily be removed. U τ n,nτ = 1 (1 ≤ n τ ≤ N τ − 1) The link U τ n,Nτ remains unchanged, and the trace of U τ n,Nτ becomes a Wilson line, relabelled here as W n . By inserting the above gauge choice into eq.(2), we can rewrite the action S of eq.(2) as a sum of S g and S τ , where S g = β E n,i<j Nτ nτ =1 1 N ReT r(U i n,nτ U j n+i,nτ U i + n+j,nτ U j + n,nτ ) +β E n,i Nτ −1 nτ =1 1 N ReT r(U i n,nτ U i + n,nτ +1 )(4) and S τ = β E n,i 1 N ReT r(U i n,Nτ W n+i U i + n,1 W + n ).(5) Straightforward removal of the second difficulty is arduous. To this end, we will instead effectively decouple the partition function into two independent sub-partition functions; one describes the Wilson line field and the other the space-like link field. We will then derive the effective spin model by means of a variational principle. In the following, we will show how to decouple the partition function. First we will formally write the effective action as a sum of two parts: one is an effective spin model, the other has only space-like link variables [10]. S ef f ≡ S W ef f ({W n }) + S U ef f ({U i n }).(6) where S W ef f and S U ef f are S W ef f = ln( {U i n } exp(S g + S τ ) {U i n } exp(S U ef f ) ),(7)S U ef f = ln( {W } exp(S g + S τ ) {W } exp(S W ef f ) ). (8) We can now see that the partition function of the effective action given by eq. (6) is the same as that of the action given by eqs. (4) and (5), U,W e S ef f = U,W e S g +S τ .(9) In this way at least the decoupling of the fields W and U can formally be achieved. We can now see that W and U fields can be described exactly by S W ef f and S U ef f respectively. It should be noted that S ef f cannot describe the correlation between W and U. However, such a correlation is irrelavant for the physics discussed in this letter. Of course, S W ef f can in principle be obtained by intergating U in eq.(7). However as we have discussed earlier, it is highly complex to obtain S W ef f by this approach. Instead we have resorted to the variational principle to derive S W ef f . Since S U ef f does not explicitly connect with the deconfining transition in our scheme, we will not comment on it further. Indeed, there is no loss of generality if we simply assume that it is already known. The action S ef f resembles S except for the coupling between W and U fields. Therefore it is quite reliable to determine S W ef f by a variational method [11] where S ef f is a trial action. To this end, we will first determine its form. We notice that there is a local gauge invariance for W in S of eqs.(4) and (5). Hence, in order to maintain this invariance, S W ef f must only depend on T r(W m ) (where m is an integer) [12]. Then we can describe the action S W ef f as follows: S W ef f = β E α n,i Re( 1 N T rW n+i 1 N T rW + n ) + S W r ({ 1 N T r(W m n )}).(10) where α is a variational parameter. From eq.(5), we note that the first term of eq.(10) is the dominant part of the effective spin model and S W r the residue. What remains is to determine α and the form of S W r . Ignoring the residue part in eq.(10) for the moment, we write the action as: S W 0 = αβ E n,i Re( 1 N T rW n+i 1 N T rW + n ),(11) then the trial action becomes S 0 = S W 0 + S U ef f .(12) To determine the value of α, we calculate the partition function as follows, Q = Q 0 < exp{S g + S τ − S 0 } > 0 ,(13) where < · · · > 0 represents the average in action S 0 and Q 0 is Q 0 = e S 0 .(14) Using Jensen's inequality (< expX >≥ exp < X >), we can obtain F = lnQ ≥ lnQ 0 + < S g + S τ − S 0 > 0(15) and α can then be determined by maximizing F var with respect to α F var = lnQ 0 + < S g + S τ − S 0 > 0 .(16) To accomplish this, we will compute < S g + S τ − S 0 > 0 ( ≡ K): K N 3 s = 3(β E Re(< 1 N T rW + n 1 N T rW n+i > W 0 < 1 N T r(U i n,Nτ U i + n,1 ) > U ) −αβ E Re(< 1 N T rW + n 1 N T rW n+i > W 0 )) + < S g − S U ef f > U N 3 s (17) where < · · · > U and < · · · > W 0 represent the average in S U ef f and S W 0 respectively. We then notice that S W 0 is real and invariant under the transformation W → W + ,(18) hence < 1 N T rW + n 1 N T rW n+i > W 0 is also real. By maximizing F var with respect to α , we obtain α = Re(< 1 N T r(U i n,Nτ U i + n,1 ) > U )(19) We are now ready to determine the form of S W ef f . Let's first write S W r as S W r ≡ ξS W M r (20) where ξ is a variational parameter and S W M r the remaining part of the effective action which is independent of ξ. We notice that the action S of eqs. (4) and (5) is invariant if all the link variables U transform as U → U + (here U means either U or W ). To maintain this invariance in the effective spin model, S W ef f must be invariant under the transformation W → W + . Since S W ef f is real and invariant under W → W + , < 1 N T rW + n 1 N T rW n+i > W (where < · · · > W represents the average in action S W ef f ,) is still real. With this, we will again compute the partition function Q ef f in the action S ef f and K as: K = < S g − S U ef f > U − < S W r > W ,(21) By maximizing F var = lnQ ef f + K with respect to ξ, we obtain < ∂S W r ∂ξ > W − ∂ < S W r > W ∂ξ = 0.(22) From eqs. (20) and (22), we obtain ξ ∂ < S W M r > W ∂ξ = 0.(23) and ∂ < S W M r > W ∂ξ =< (S W M r ) 2 > W − < S W M r > 2 W(24) Clearly according to eq.(24), ∂<S W M r > W ∂ξ does not vanish unless S W M r is a constant, a situation which is not relavant in our discussion. From eq.(23), we see that ξ must vanish. Our derivation is general since S W M r includes any kind of coupling terms. Hence surprisingly, we have obtained an effective spin model with only nearest neighbor coupling terms. It is important to understand why by using the variational principle, the effective spin model appears simple. Actually, it is very complex in the sense that to compute α according to eq.(19) is complicated. The reason is as follows: In order to obtain α, we have to know the form of S U ef f which can be obtained by integrating over W in eq. (8). With this knowledge, α can then be calculated according to eq.(19). Both procedures are nontrivial. In fact, the difficulty of deriving the effective spin model has actually been transferred from the integration of U in eq.(7) to the integration of W in eq.(8) and to calculate α from eq.(19). Therefore the simplicity of the form of the effective spin model is not obtained without a price. What we have done is to "organize" the theory such that the discussion of the order of deconfining phase transition can be vividly studied. Having the effective spin model, we can now focus on studying the deconfining phase transition. To this end, we note that there were numerous investigation made in this direction already. These studies enable us to discuss the order of the deconfining phase transition. For the SU(2) gauge theory, it is well known that the plaquette is continuous with the change of β E . Therefore, according to eq.(19), α is also a continuous function of β E . Hence our study directly shows that the phase transition of the Z(2) spin model possesses the same universality property as the SU(2) gauge theory. This agrees with the recently reached conclusion by the Monte-Carlo real space renormalization study [13,14]. For the SU(3) gauge theory, it was shown via the Monte Carlo study of the effective spin model with a nearest neighbor coupling assumption that the transition is first order [1,7]. Results reported here confirm the nearest neighbor assumption and are consistent with the recent Monte-Carlo results about the transition order [3,6]. Finally, we mention that our derivation appears to provide an analytical approach to study the thermodynamical behavior of LGT [15]. We are very grateful to Lay Nam Chang, Xiangdong Ji, Robert Perry, Hai-Chang Ren and Yong-Shi Wu for illuminating discussions. We also thank N. Christ, M. Fukugita, S. Ohta and C. DeTar for several useful communications and suggestions. One of us (ZKZ) is very grateful for Center for Theoretical Physcis of MIT for its hospitality. This work is supported by the National Science Foundation and the Department of Energy. † Permanent address . A See For Example, Ukawa, Nucl. Phys.,B (Proc. Suppl.). 17118See for example, A. Ukawa, Nucl. Phys.,B (Proc. Suppl.) 17 ,118(1990); . B Petersson, Nucl. Phys. 525237B. Petersson, Nucl. Phys., A525,237(1991). . N Christ, Nucl. Phys. B (Proc. Suppl.). 20129N. Christ, Nucl. Phys. B (Proc. Suppl.) 20, 129(1991). . F R Brown, Phys. Rev. Lett. 612058F. R. Brown et al., Phys. Rev. Lett. 61, 2058(1988); . F R Brown, Nucl. Phys. B (Proc. Suppl.). 17214F. R. Brown, Nucl. Phys. B (Proc. Suppl.) 17,214(1990); . S Cabasino, Nucl. Phys. B (Proc. Suppl.). 17218S. Cabasino et al., Nucl. Phys. B (Proc. Suppl.)17,218(1990); . N A Alves, B A Berg, S Sanielevici, Phys. Rev. Lett. 643107N. A. Alves, B. A. Berg and S. Sanielevici, Phys. Rev. Lett., 64,3107(1990). . M Fukugita, M Okawa, A Ukawa, Phys. Rev. Lett. 631768M. Fukugita, M. Okawa, A. Ukawa, Phys. Rev. Lett., 63,1768(1989); . Nucl. Phys. 337181Nucl. Phys., B337,181(1990). . Y Iwasaki, Phys. Rev. Lett. 673343Y. Iwasaki et al., Phys. Rev. Lett. 67, 3343(1991). . B Svetitsky, L G Yaffe, Nucl. Phys. 210423B. Svetitsky and L. G. Yaffe, Nucl. Phys. B210, 423(1982). . F R Brown, Phys. Lett. 224412F.R.Brown, Phys. Lett. B224,412(1989); . M Fukugita, M Okawa, Phys. Rev. Lett. 6313M. Fukugita, M. Okawa, Phys. Rev. Lett., 63,13(1989). . S Gupta, A Irback, B Pettersson, R V Gavai, F Karsch, Nucl. Phys. B (Proc. Suppl.). 17199S. Gupta, A. Irback, B. Pettersson, R. V. Gavai and F. Karsch, Nucl. Phys. B (Proc. Suppl.)17,199(1990). . B Svetitsky, Phys. Rep. 1321B. Svetitsky, Phys. Rep., 132,1(1986). Alessandrini and ph. Boucaud. M Creutz, Nucl. Phys. 15599Phys. Rev.M. Creutz, Phys. Rev., D15,1128(1977), V. Alessandrini and ph. Boucaud, Nucl. Phys. B235,599(1984). There is some difference between our definition of S W ef f and eq. There is some difference between our definition of S W ef f and eq.(3) from R P Feynman, A R Hibbs, Quantum Mechanics and Path Integrals. New YorkMcGraw-HillR. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Inte- grals, McGraw-Hill, New York, 1965; . J M Drouffe, J B Zuber, Phys. Rep. 1021J. M. Drouffe and J. B. Zuber, Phys. Rep.102,1(1983). For N τ = 1, the idea which S W ef f depends only on T rW was discussed by. M. Ogilvie521369For N τ = 1, the idea which S W ef f depends only on T rW was discussed by M. Ogilvie, Phys. Rev. Lett., 52,1369(1984). . M Okawa, Phys. Rev. Lett. 601805M. Okawa, Phys. Rev. Lett., 60,1805(1988); . K M Bitar, Nucl. Phys. 61B300[FS22K. M. Bitar, Nucl. Phys., B300[FS22],61(1988). There exist non-nearest coupling terms in the Potts model, although the range is exponentially short. Our results are not in contradiction with such results because we have used the SU(N) matrix and not the Z(N) spin in our calculations. There exist non-nearest coupling terms in the Potts model, although the range is exponentially short. Our results are not in contradiction with such results because we have used the SU(N) matrix and not the Z(N) spin in our calculations. . Z K Zhu, D H Feng, Phys. Rev. 48397Z. K. Zhu and D. H. Feng, Phys. Rev. D48, 397(1993).

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{'abstract': 'An effective spin model for the finite temperature non-abelian lattice gauge theory is derived. The outcome is the surprising result that only nearest neighbor coupling survives, thus confirming the well known numerical results that the deconfining phase transition of the (3+1)-dimensional SU (3) pure gauge theory is first order.', 'arxivid': 'hep-lat/9308023', 'author': ['Zheng-Kun Zhu \nDepartment of Physics and Atmospheric Science\nCenter for Theoretical Physics\nDepartment of Physics and Atmospheric Science\nDrexel U niversity † Philadelphia\nMassachusetts Institute of Technology Cambridge\nDrexel University Philadelphia\n19104-9984, 02139, 19104-9984Pennsyvalnia, Massachusetts, Pennsyvalnia\n', 'Da Hsuan Feng \nDepartment of Physics and Atmospheric Science\nCenter for Theoretical Physics\nDepartment of Physics and Atmospheric Science\nDrexel U niversity † Philadelphia\nMassachusetts Institute of Technology Cambridge\nDrexel University Philadelphia\n19104-9984, 02139, 19104-9984Pennsyvalnia, Massachusetts, Pennsyvalnia\n'], 'authoraffiliation': ['Department of Physics and Atmospheric Science\nCenter for Theoretical Physics\nDepartment of Physics and Atmospheric Science\nDrexel U niversity † Philadelphia\nMassachusetts Institute of Technology Cambridge\nDrexel University Philadelphia\n19104-9984, 02139, 19104-9984Pennsyvalnia, Massachusetts, Pennsyvalnia', 'Department of Physics and Atmospheric Science\nCenter for Theoretical Physics\nDepartment of Physics and Atmospheric Science\nDrexel U niversity † Philadelphia\nMassachusetts Institute of Technology Cambridge\nDrexel University Philadelphia\n19104-9984, 02139, 19104-9984Pennsyvalnia, Massachusetts, Pennsyvalnia'], 'corpusid': 16617829, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5212, 'n_tokens_neox': 4524, 'n_words': 2842, 'pdfsha': 'a2b3b21b4d38c95bf507afc76bbf1ea5e59cbdd4', 'pdfurls': ['https://arxiv.org/pdf/hep-lat/9308023v1.pdf'], 'title': ['arXiv:hep-lat/9308023v1 31 Aug 1993 DETERMINING THE ORDER OF SU (3) DECONFINING PHASE TRANSITION', 'arXiv:hep-lat/9308023v1 31 Aug 1993 DETERMINING THE ORDER OF SU (3) DECONFINING PHASE TRANSITION'], 'venue': ['Physics Division']}

arxiv

Refined finite-size analysis of binary-modulation continuous-variable quantum key distribution Takaya Matsuura Department of Applied Physics Graduate School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan School of Science Centre for Quantum Computation & Communication Technology RMIT University Melbourne VIC 3000Australia Shinichiro Yamano Department of Applied Physics Graduate School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan Yui Kuramochi Department of Physics Faculty of Science Kyushu University 744 Motooka, Nishi-kuFukuokaJapan Toshihiko Sasaki Department of Applied Physics Graduate School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan Photon Science Center Graduate School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan Masato Koashi Department of Applied Physics Graduate School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan Photon Science Center Graduate School of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan Refined finite-size analysis of binary-modulation continuous-variable quantum key distribution Recent studies showed the finite-size security of binary-modulation CV-QKD protocols against general attacks. However, they gave poor key-rate scaling against transmission distance. Here, we extend the security proof based on the complementarity, which is used in the discrete-variable QKD, to the previously developed binary-modulation CV-QKD protocols with the reverse reconciliation under the finite-size regime and obtain large improvements in the key rates. Notably, the key rate in the asymptotic limit scales linearly against the attenuation rate, which is known to be optimal scaling but is not achieved in previous finite-size analyses. This refined security approach may offer full-fledged security proofs for other discrete-modulation CV-QKD protocols.Takaya Matsuura: [email protected] 1 arXiv:2301.03171v2 [quant-ph] 26 Jan 2023 U (N )-invariant CV-QKD protocols[33,34]. However, in practice, ideal Gaussian modulation cannot be implemented and should be approximated by a finite number of coherent states. It turns out that an overwhelming number of coherent states is needed to directly approximate the Gaussian ensemble for the security condition to be satisfied[35,36]. If we try to mitigate the required number, additional assumptions are needed, which makes it difficult to apply it in the finite-size regime[15]. The other completely different approach[37,38]is targeted at the discrete-modulation CV QKD from the beginning. Refs.[37,38]show the finite-size security against general attacks for a binary-modulation protocol. It also takes into account the discretization of the signal processing, such as binned homodyne and heterodyne measurements (see also[39]for this topic). Although it has a nice feature, the obtained key rate has very poor scaling against transmission distance. A possible reason for this bad performance is the fact that its security proof is based on the entanglement distillation[40,41]. It is known that the security proof based on the entanglement distillation is too stringent in general for secure key distribution. There are alternative types of security proofs[42][43][44]that can be applied to general cases. In particular, for CV-QKD protocols, the security proof based on the reverse reconciliation often provides better performance than that based on the direct reconciliation [10], which may be unattainable by a security proof based on the entanglement distillation due to its symmetric nature between the sender and the receiver in the security proof.Contributions of this paper. In this article, we aim to develop another approach to carry out the finite-size security proof for the discrete-modulation CV QKD against general attacks. The approach should be able to exploit the benefit of the reverse reconciliation. To do it concretely, we develop refined security proofs based on the reverse reconciliation for the binary-modulation CV-QKD protocols proposed in Refs.[37,38], i.e., the protocol in which the sender Alice performs BPSK-type modulation according to her randomly generated bit and the receiver Bob performs homodyne measurement, heterodyne measurement, and trash randomly [37], or performs heterodyne measurement followed by a random selection of the post-processing of the outcome[38]. We use the same apparatuses and setups as those in Refs.[37,38]but slightly change the protocols. To refine the security proofs, we use an approach based on the complementarity[43,45]under the reverse reconciliation, which is more general than the one based on the entanglement distillation [46] and treats Alice and Bob asymmetrically in the security proof. In these refined security proofs, we have degrees of freedom that did not appear in the previous analyses. By setting these degrees of freedom to be optimal in the pure-loss channel [47], we obtain a significant improvement in the key gain rates; in fact, the asymptotic key rates of the protocols scale linearly with regards to the attenuation rate of the pure-loss channel, which is known to be the optimal scaling for the one-way QKD[47]. This shows that we can exploit the benefit of using the reverse reconciliation in the approach based on complementarity. Although the protocols are still fragile against the excess noise, this approach itself may be a step towards the full-fledged security proofs for discrete-modulation CV QKD.Organization of this paper. The article is organized as follows. In Section 2, we provide the refined security proofs based on the complementarity [43] for protocols that use the same experimental setups as proposed in Refs.[37,38]. The section is further divided into three parts. The first part 2.1 defines the actual protocols, which are almost the same as the ones in Refs.[37,38], and develops virtual protocols for the complementarity approach[43]. In the second part 2.2, we derive an explicit form of the phase error operator defined by the virtual procedure of the previous part. In the third part 2.3, we finish the finite-size security proof by developing operator inequalities. In Section 3, we numerically demonstrate the improved performance of the protocols with our refined security proof. Finally, in Section 4, we wrap up our article by discussing future work and open problems. Introduction Quantum key distribution (QKD) [1] enables two remote parties to share identical secret bits that are secure against arbitrary eavesdropping allowed in the law of quantum mechanics. QKD combined with the one-time pad [2] can thus realize the information-theoretic security of bipartite communication. Nowadays, there is increasing interest in implementing QKD in the real world. Among other things, continuous-variable (CV) QKD [3][4][5][6][7][8][9] has advantages over short-distance highbit-rate QKD due to the low cost of its implementation and the affinity to the wavelength division multiplexing. This is because homodyne and heterodyne detectors used in CV-QKD protocols do not require a low-temperature environment and have good wavelength selectivity. The (single-)photon detectors used in discrete-variable (DV) QKD, on the other hand, typically require a lowtemperature environment for stable operation and a high-quality frequency filter for the wavelength division multiplexing. The main problems of CV-QKD protocols are difficulties in their complete security proofs. Compared to the DV-QKD protocols, most of which have complete security proofs even in the finite-size regime, almost all the CV-QKD protocols only have asymptotic security proofs [10][11][12][13][14][15][16][17][18][19] or security proofs against collective attacks [20][21][22][23][24][25]. There are, however, some results for the composable finite-size security against general attacks. One is for the protocol using the two-mode squeezed vacuum state [26,27], whose security proof is based on the entropic uncertainty relation on the infinite dimension [28]. Unfortunately, this protocol has difficulty in its implementation and poor key-rate scaling against the transmission distance. Another is for the U (N )-symmetric protocol that uses coherent states with their complex amplitudes modulated according to a Gaussian distribution [29,21,30]. The security proof for this type of protocol utilizes the de Finetti reduction theorem [31,32] to the i.i.d. case. This methodology has proved the security of several Failure" with probability 1 − suc 2 − suc − 2 Figure 1: Setups of the protocols. In both protocols, the sender Alice modulates the optical phase of a laser pulse prepared in a coherent state |µ with 0 or π according to her random bitâ = 0 or 1. (a) "Homodyne protocol", which is similar to the protocol proposed in Ref. [37]. In this protocol, the receiver Bob randomly switches three types of measurements according to probability psig, ptest, and p trash , respectively. In "Signal", Bob performs homodyne measurement and obtains the outcomex ∈ R. Then he obtainsb ∈ {0, 1} with probability fsuc (−1)bx , respectively, or announces "Failure" with probability 1 − fsuc(x) − fsuc(−x). In "Test", Bob performs heterodyne measurement and obtains the outcomeω ∈ C. Then he computes Λm,r |ω − (−1)â| 2 with Alice's bitâ announced. In "Trash", Bob discards the received optical pulse and produces no outcome. (b) "Heterodyne protocol", which is similar to the protocol proposed in Ref. [38]. In this protocol, the receiver Bob performs heterodyne measurement, obtains the outcomeω, and randomly switches three types of postprocessings according to probability psig, ptest, and p trash , respectively. In "Signal", Bob definesx = Re[ω] and follows the same procedure of obtaining the bit b or "Failure" as in Homodyne protocol. In "Test", Bob follows the same procedure of computing Λm,r |ω − (−1)â| 2 as in Homodyne protocol. In "Trash", Bob discards the outcomeω. 2 Security proof 2.1 Actual, virtual, and estimation protocols In this section, we define two binary-modulation CV-QKD protocols that are closely related to the ones proposed in Refs. [37,38], and present their security proofs based on the reverse reconciliation. The definition of the (composable) security is the same as that in Ref. [37]. The setups of the protocols are illustrated in Fig. 1. In the following, a random number is denoted with a hat such aŝ ·. For the places where the slash "/" is used, one can adopt either its left-hand side or right-hand side depending on which of "Homodyne protocol" or "Heterodyne protocol" defined in Fig. 1 one chooses. Note that Homodyne protocol is the same as the protocol proposed in Ref. [37] except for the definition of f suc (x) as well as the way of bit error correction, and Heterodyne protocol is the same as the protocol proposed in Ref. [38] except for the additional trash round as well as the way of bit error correction. Prior to the protocol, Alice and Bob determine the number N of total rounds, the acceptance probability function f suc (x) (x ∈ R) of the homodyne/heterodyne measurement satisfying f suc (x) + f suc (−x) ≤ 1, an odd integer m and a real r for the test function Λ m,r (ν) := e −rν (1 + r)L (1) m ((1 + r)ν) with L (1) m being the associated Laguerre polynomial [37], and the protocol parameters (µ, p sig , p test , p trash , β, s, κ, γ) satisfying p sig + p test + p trash = 1 and β < √ µ, where all the parameters are positive. Alice and Bob then run the protocol described in Box 1. Unless aborted, the protocol generates a shared final key of lengtĥ N fin = max N suc − N suc h U (F ,N trash )/N suc − s, 0 ,(1) where · is the ceiling function, the function h(x) is defined as h(x) := −x log 2 (x) − (1 − x) log 2 (1 − x) (x ≤ 1/2) 1 (x > 1/2) ,(2) and the function U (F ,N trash ) will be specified later. 2. For the received pulse C in each round, Bob chooses a label from {signal, test, trash} with probabilities p sig , p test , and p trash , respectively, and announces it. According to the label, Alice and Bob do one of the following procedures. [signal] Bob performs a homodyne/heterodyne measurement on the received optical pulse C and obtains an outcomex ∈ R. (For the heterodyne measurement,x is defined as the real part of the outcomeω ∈ C.) Bob defines a sifted-key bitb asb = 0 with a probability f suc (x) andb = 1 with a probability f suc (−x). When Bob has defined his sifted key bit, he announces "success", and otherwise, he announces "failure". In the case of a success, Alice (resp. Bob) records a bitâ (b). [test] Bob performs a heterodyne measurement on the received optical pulse C and obtains an outcomeω. Alice announces her bit a. Bob calculates the value of Λ m,r (|ω − (−1)âβ| 2 ). [trash] Alice and Bob produce no outcomes. For simplicity, we omitted the bit-error-sampling rounds in the above protocol. To satisfy the required correctness ε cor for the final key, Alice and Bob randomly insert N smp sampling rounds among N rounds in which Bob performs the same measurement as that of the signal round and estimate an upper bound e qber on the bit error rate. LetN suc smp be the number of "success" in N smp sampling rounds, and letÊ obs be the number of discrepancies between Alice's and Bob's bits observed in the "success" sampling rounds. Then, Bob sets e qber to e qber = MN suc +N suc smp ,N suc smp ,εcor/2 (Ê obs ) −Ê obs N suc ,(3) where the functionM N,n, is defined in Eq. (101) in Appendix A. The proof that this definition of e qber upper-bounds the actual bit error rate with probability no smaller than 1 − ε cor /2 is also shown in Appendix A. The required amount H EC of the error syndrome Bob sends to Alice in the bit error correction depends on the error correction method; here we assume H EC =N suc [f h(e qber ) + (1 − f )] ,(4) where f ∈ [0, 1] denotes an error correction efficiency [48,49,[16][17][18]50] for the error correction to succeed with the probability no smaller than 1 − ε cor /2. The net key gainĜ per pulse is thus given byĜ = (N fin − H EC )/(N + N smp ).(5) Here, we do not use verification in the post-processing, unlike Refs. [37,38], due to the subtleties to incorporate it in our security proof. The acceptance probability f suc (x) should be chosen to post-select the rounds with larger values of x, for which the bit error probability is expected to be lower. The definition of f suc (x) in this article follows Ref. [38] and is slightly more general than that of Ref. [37]. (Note that Ref. [37] can also use this definition of f suc (x).) It is ideally a step function with a threshold x th (> 0), but our security proof applies to any form of f suc (x). The test function Λ m,r (ν) is the same as the one defined in Ref. [37] where it is shown to satisfy E ρ [Λ m,r (|ω − β| 2 )] ≤ β| ρ |β(6) for any odd integer m, positive real r, and density operator ρ (see Corollary 1 in Ref. [37]). The parameter β is typically chosen to be √ ηµ with η being a nominal transmissivity of the quantum channel, while the security proof itself holds for any choice of β. The parameter s is related to the overall security parameter in the security proof below. We determine a sufficient amount of the privacy amplification according to the complementarity, or in other words, the phase error correction [43,45], which has been widely used for the DV-QKD protocols. We aim at showing the secrecy of Bob's final key against the adversary Eve. To do so, we consider a virtual protocol in which Bob has a qubit for each success signal round such that the outcome of the Z-basis measurement on it is equivalent to his sifted key bit b. Alice can do arbitrary quantum operations in the virtual protocol as long as all the statistics and available information to the adversary Eve are the same as those in the actual protocol. Then, after Bob's Z-basis measurement on the qubit, the reduced classical-quantum state between Bob and Eve in the virtual protocol is the same as that in the actual protocol. In the following, we explicitly describe the virtual protocol. For Alice, we introduce a qubit A and assume that she entangles it with an optical pulseC in a state |Ψ AC := |0 A | √ µ C + |1 A |− √ µ C √ 2 ,(7) where |ω C with ω ∈ C denotes the coherent state with the amplitude ω, which is defined as |ω C := e − |ω| 2 2 ∞ n=0 ω n √ n! |n C .(8) Then, the optical pulseC emitted by Alice is in the same state as that in the actual protocol. For Bob, we construct a process of probabilistically converting the received optical pulse C to a qubit B, which can be regarded as a coherent version of Bob's signal measurement. For Homodyne protocol, consider a map K hom C→B defined as [37] K hom C→B (x)(ρ C ) := K hom suc (x) ρ C K hom suc (x) † (9) with K hom suc (x) := f suc (x) |0 B x| C + |1 B −x| C ,(10) where x| maps a state vector to the value of its wave function at x; i.e., for a coherent state vector |ω , x| acts as x|ω = 2 π 1 4 exp −(x − ω r ) 2 + 2iω i x − iω r ω i ,(11) where ω = ω r + iω i with ω r , ω i ∈ R. Let Π ev(od) denote a projection operator onto the subspace of even(odd) photon numbers. Since x| (Π ev − Π od ) = −x| holds, we have K hom suc (x) = 2f suc (x) |+ B x| C Π ev + |− B x| C Π od .(12) This defines an instrument I hom C→B for the process of producing the outcomex and leaving C in a post-measurement state; i.e., given a measurable set ∆ ⊆ R, the unnormalized post-measurement state is given by I hom C→B (∆)(ρ C ) = ∆ dx K hom C→B (x)(ρ C )(13) with Tr[I hom C→B (∆)(ρ C )] being a probability of "success" signal event with the outcomex ∈ ∆. Similarly, for Heterodyne protocol, consider a map K het C→B defined as [38] K het C→B (ω)(ρ C ) := K het suc (ω) ρ C K het suc (ω) † (14) with K het suc (ω) := f suc (ω r ) π |0 B ω| C + |1 B −ω| C = 2f suc (ω r ) π (|+ B ω| C Π ev + |− B ω| C Π od ) ,(15) where |ω denotes a coherent state vector and ω = ω r + iω i with ω r , ω i ∈ R. Similarly to Homodyne protocol, we can define an instrument I het C→B composed of the heterodyne outcome and the (unnormalized) post-measurement state, which is given by I het C→B (∆ )(ρ C ) = ∆ dω r dω i K het C→B (ω)(ρ C ),(16) where ∆ ⊆ R 2 is a measurable set. If Bob measures the qubit B on the Z basis after the instrument (13) (resp. (16)), he obtains the same sifted key bit with the same probability as in the actual protocol whenx ∈ ∆ (resp.ω ∈ ∆ ) [37,38]. At this point, one has a degree of freedom to perform quantum operations on the system AB for each outcomex (resp.ω) as long as it does not change the Z-basis value of the qubit B. This is because we aim at showing the secrecy of Bob's final key against the adversary Eve with Alice's system traced out. Thus, after applying the map K hom C→B (resp. K het C→B ), we assume that Alice and Bob perform a controlled isometry V hom B;A→R (x) (resp. V het B;A→R (ω)) of the form V hom B;A→R (x) := |0 0| B ⊗ V (0) A→R (x) + |1 1| B ⊗ V (1) A→R (x) C-X BA (17) V het B;A→R (ω) := |0 0| B ⊗ V (0) A→R (ω) + |1 1| B ⊗ V (1) A→R (ω) C-X BA ,(18) where C- X BA := |0 0| B ⊗ I A + |1 1| B ⊗ X A denotes the Controlled-NOT gate and V (j) A→R (x) (resp. V (j) A→R (ω)) for j = 0, 1 denotes an isometry from the system A to another system R that is no smaller than A 1 . If V (j) A→R (x) (resp. V (j) A→R (ω)) is an identity, then the analysis reduces to the previous results [37,38]. Let V hom B;A→R (x) (resp. V het B;A→R (ω)) be an adjoint action (i.e., a CPTP map) for the isometry V hom B;A→R (x) (resp. V het B;A→R (ω)). The composition of the map V hom B;A→R (x) and the map (9) (resp. the mad V het B;A→R (ω) and the map (14)) with Alice's system traced out at the end defines a quantum operation F hom AC→B (resp. F het AC→B ) that (probabilistically) outputs Bob's qubits for his sifted key as F hom AC→B (ρ AC ) = ∞ −∞ dx K hom AC→B (x)(ρ AC ),(19)F het AC→B (ρ AC ) = ∞ −∞ dω r dω i K het AC→B (x)(ρ AC ),(20) with K hom AC→B (x) (resp. K het AC→B (ω)) given by K hom AC→B (x)(ρ AC ) := Tr R V hom B;A→R (x) • Id A ⊗ K hom C→B (x) (ρ AC ) ,(21)K het AC→B (ω)(ρ AC ) := Tr R V het B;A→R (ω) • Id A ⊗ K het C→B (ω) (ρ AC ) ,(22) 1 Here, a subtlety for using the verification comes in. In order to know whether verification succeeds or not, Alice has to confirm the syndrome bits for the verification. However, this procedure may not commute with the action of V hom B;A→R (x) (resp. V het B;A→R (ω)). We do not currently have a method to evaluate how much the verification affects the secrecy condition. where Id denotes the identity map. Note that the idea of acting the isometry V hom B;A→R (x) or V het B;A→R (ω) is closely related to the twisting operation on the shield system [51][52][53][54][55]. The difference is that in our case it acts on the system A in a way that is incompatible with the Z-basis measurement on A. This is allowed in a security proof based on the complementarity since what we need to prove in the virtual protocol is that the outcome of the Z-basis measurement on B is secret to Eve when the system A is traced out [43]; i.e., the system A works as a shield system. We then introduce a virtual protocol that explicitly incorporates the action of F hom AC→B in Eq. (19) (resp. F het AC→B in Eq. (20)) in Box 2. Box 2: Virtual protocol 1 . Alice prepares a qubit A and an optical pulseC in a state |Ψ AC defined in (7) and sends the pulseC to Bob. She repeats it for N rounds. Bob receives an optical pulse C for each of the N rounds. 2 . For the received pulse C in each round, Bob announces a label in the same way as that at Step 2. Alice and Bob do one of the following procedures according to the label. [signal] Alice and Bob perform the quantum operation on the system A and the received pulse C specified by the map F hom AC→B defined in Eq. (19) (resp. F het AC→B defined in Eq. (20)) to determine success or failure of detection, obtain the qubit B upon success, and perform the controlled isometry given in Eq. (17) (resp. Eq. (18)). Bob announces the success or failure of the detection. [test] Bob performs a heterodyne measurement on the received optical pulse C, and obtains an outcomeω. Alice measures her qubit A on Z basis and announces the outcomeâ ∈ {0, 1}. Bob calculates the value of Λ m,r (|ω − (−1)âβ| 2 ). [trash] Alice measures her qubit A on X basis to obtainâ ∈ {+, −}. 3 .N suc ,N fail ,N test ,N trash , andF are defined in the same way as those at Step 3. Let Q − be the number of rounds withâ = − among theN trash trash rounds. 4 . According to (the upper bound on) the bit error rate e qber , Bob performs H EC bits of encrypted communication consuming a pre-shared secret key to send a dummy message. 5 . Bob computes and announces the final key lengthN fin according to Eq. (1). Bob performs a randomly chosen unitary on his qubits (see the main text), and measures the firstN fin qubits on the Z bases. In the last line of Step 5 , the random choice of a unitary is constructed so that, along with the subsequentN fin -qubit measurement on the Z bases, it is equivalent to the privacy amplification. This is possible because for any n × n linear transformation C on the n-bit sequence, there always exists a corresponding unitary U (C) that satisfies U (C) |z = |Cz on the Z basis. As has already been claimed, if Eve performs the same attacks as those in the actual protocol, the resulting classical-quantum state between Bob and Eve is the same as that in the actual protocol. The complementarity argument [43] in a reverse reconciliation scenario relates the amount of privacy amplification to the so-called phase error patterns of Bob's qubits. Suppose that, just before the Z-basis measurement at Step 5 of the virtual protocol, Bob's quantum state on the firstN fin qubits is arbitrarily close to |+ +| ⊗N fin . Then, the secrecy condition of the final key is satisfied [43,45,56]. For this to be true, the errors on the X bases (i.e., the phase errors) on Bob's qubits should be corrected by the procedure at Step 5 of the virtual protocol. To see the correctability of the phase errors at Step 5 , suppose that Bob measured hisN suc qubits on the X basis {|+ , |− } at the end of Step 3 , and obtained a sequence of + and −. The minuses in the sequence are regarded as phase errors. It has already been known that, if we can find an upper bound on the number of possible phase-error patterns, then we can prove the security [43]. To make the argument more precise, we introduce the estimation protocol in Box 3. Box 3: Estimation protocol 1 . Alice prepares a qubit A and an optical pulseC in a state |Ψ AC defined in (7) and sends the pulseC to Bob. She repeats it for N rounds. Bob receives an optical pulse C for each of the N rounds. 2 . For the received pulse C in the ith round (i = 1, . . . , N ), Bob announces a label in the same way as that at Step 2. Alice and Bob do one of the following procedures according to the label and obtain the values of random variablesN suc (i) ph ,F (i) , andQ (i) − . Unless explicitly written, these random variables are set to be zeros. [signal] Alice and Bob do the same procedure as that at "signal" of Step 2 . Upon "success", Bob performs the X-basis measurement on qubit B and obtainsb ∈ {+, −}. When b = −,N suc (i) ph is set to be unity. [test] Alice and Bob do the same procedure as that at "test" of Step 2 . ThenF (i) is set to be Λ m,r (|ω − (−1)âβ| 2 ). [trash] Alice does the same procedure as that at "trash" of Step 2 . Whenâ = −,Q (i) − is set to be unity. . Same as Steps of the virtual protocol. Note thatF = N i=1F (i) andQ − = N i=1Q (i) − hold. 4 . Regarding + as zero and − as unity for eachb in success signal round, define theN suc -bit sequencex ph . LetN suc ph be the Hamming weight ofx ph , i.e.,N suc ph = N i=1N suc (i) ph . The task of proving the security of the actual protocol is then reduced to constructing a function U (F ,N trash ) that satisfies Pr N suc ph ≤ U (F ,N trash ) ≥ 1 −(23) for any attack in the estimation protocol and setting the final-key length toN fin =N suc − H PA − s, where H PA is defined as H PA := N suc h U (F ,N trash )/N suc .(24) In fact, if the condition (23) is satisfied, then the number of possible phase-error patterns can be bounded from above by 2 H PA [57]. Therefore, by extracting the (H PA +s)-bit error syndrome ofx ph using the universal 2 hash function, Bob could uniquely identifyx ph with a failure probability no smaller than 1 − 2 −s [43,58,46,44]. In the virtual protocol, the quantum operations at Step 5 can be made equivalent to the (N suc −N fin )-bit syndrome extraction via the universal 2 hash function and the error correction on the X bases ofN fin qubits. Since a unitary U (C −1 ) that acts as the matrix C −1 on the Z bases acts as C on the X bases, i.e., U (C −1 ) |x X = |C x X where · X denotes the X basis, this procedure corresponds to the privacy amplification via the dual universal 2 hashing on the Z bases [58,44] (i.e., in the actual protocol). Combining these, the condition (23) implies that the actual protocol with the final key length given in Eq. (1) is sec -secure with a security parameter sec = √ 2 √ + 2 −s + cor [43,45,56]. From now on, we thus focus on the estimation protocol for finding a function U (F ,N trash ) to satisfy Eq. (23). Phase error operator In this section, we explain how our new security analysis can be reduced to the previous analyses carried out in Refs. [37,38] with a tighter operator inequality. The number of phase errors depends on the choice of the controlled isometry V hom B;A→R (x) (resp. V het B;A→R (ω)) in the virtual and the estimation protocol. We here take a suboptimal strategy; fix V hom B;A→R (x) (resp. V het B;A→R (ω)) so that the probability of the phase error eventb = − in the estimation protocol is minimized for an ideal pure-loss channel [47] with transmission η = β 2 /µ. When the state |Ψ AC in Eq. (7) is put into a pure-loss channel with the channel output being |±β C , the resulting state |Φ ACE on systems A, C, and an adversary's system E (i.e., an environment of the pure-loss channel) is given by |Φ ACE = 1 √ 2 |0 A |β C µ − β 2 E + |1 A |−β C − µ − β 2 E .(25) Tracing out the system E, the reduced state Φ AC is given by Φ AC = (1 − q µ,β ) |φ + φ + | AC + q µ,β |φ − φ − | AC ,(26) where |φ + AC := 1 √ 2 (|0 |β + |1 |−β ) = |+ A ⊗ Π ev |β C + |− A ⊗ Π od |β C ,(27)|φ − AC := 1 √ 2 (|0 |β − |1 |−β ) = |+ A ⊗ Π od |β C + |− A ⊗ Π ev |β C = (Z A ⊗ I C ) |φ + AC ,(28) and q µ,β := 1 − e −2(µ−β 2 ) 2 (> 0).(29) For Homodyne protocol, we observe that C-X BA Id A ⊗ K hom C→B (x) (Φ AC ) C-X BA (30) = 2f suc (x) C-X BA (1 − q µ,β )P x| Π ev |β |++ AB + x| Π od |β |−− AB +q µ,βP x| Π od |β |+− AB + x| Π ev |β |−+ AB C-X BA (31) = 2f suc (x) (1 − q µ,β )P x| Π ev |β |+ A + x| Π od |β |− A ⊗ |+ +| B +q µ,βP x| Π od |β |+ A + x| Π ev |β |− A ⊗ |− −| B (32) = f suc (x) (1 − q µ,β )P g β,1/4 (x) |0 A + g −β,1/4 (x) |1 A ⊗ |+ +| B +q µ,βP g β,1/4 (x) |0 A − g −β,1/4 (x) |1 A ⊗ |− −| B ,(33) whereP (ψ) := ψψ † (and thusP (|ψ ) = |ψ ψ|), and g m,V is the normal distribution with the mean m and the variance V , i.e., g m,V (x) := 1 √ 2πV exp − (x − m) 2 2V .(34) We define τ hom AB (x) as τ hom AB (x) :=(1 − q µ,β )P g β,1/4 (x) |0 A + g −β,1/4 (x) |1 A ⊗ |+ +| B + q µ,βP g β,1/4 (x) |0 A − g −β,1/4 (x) |1 A ⊗ |− −| B .(35) From Eqs. (17), (21), (33), and (35), the probability density of an outcome x with occurrence of the phase error is given by Tr |− −| B K hom AC→B (x)(Φ AC ) = f suc (x) 2 Tr 0| B τ hom AB (x) |0 B + 1| B τ hom AB (x) |1 B − V (1) A→R (x) † V (0) A→R (x) 0| B τ hom AB (x) |1 B − 1| B τ hom AB (x) |0 B V (0) A→R (x) † V (1) A→R (x)(36)= f suc (x) 1 2 Tr τ hom AB (x) − Re Tr V (1) A→R (x) † V (0) A→R (x) 0| B τ hom AB (x) |1 B (37) ≥ f suc (x) 1 2 Tr τ hom AB (x) − 0| B τ hom AB (x) |1 B 1 ,(38) where the last inequality follows from the matrix Hölder inequality. If we write the polar decom- position of 0| B τ hom AB (x) |1 B by W hom A (x) 0| B τ hom AB (x) |1 B , the equality in (38) can be achieved by setting V (1) A→R (x) † V (0) A→R = W hom A (x) † .(39) From Eq. (35), 0| B τ hom AB (x) |1 B is given by 0| B τ hom AB (x) |1 B = 1 2 (1 − q µ,β )P g β,1/4 (x) |0 A + g −β,1/4 (x) |1 A −q µ,βP g β,1/4 (x) |0 A − g −β,1/4 (x) |1 A ,(40) which is hermitian with two eigenvalues having opposite signs. Let |u hom + (x) A and |u hom − (x) A be eigenvectors of 0| B τ hom AB (x) |1 B with positive and negative eigenvalues, respectively. Then, W hom A (x) is given by W hom A (x) = |u hom + (x) u hom + (x)| A − |u hom − (x) u hom − (x)| A .(41) The explicit form of |u hom ± (x) A is given in Eq. A→R (x) = I A , V(1)A→R (x) = W hom A (x),(42) which, with Eqs. (17) and (41), leads to V hom B;A→A (x) = |u hom + (x) u hom + (x)| A ⊗ I B + |u hom − (x) u hom − (x)| A ⊗ Z B C-X BA .(43) For Heterodyne protocol, the calculation similar to Eqs. (30)-(35) leads to C-X BA Id A ⊗ K het C→B (ω) (Φ AC ) C-X BA(44)= 2f suc (ω r ) π C-X BA (1 − q µ,β )P ω| Π ev |β |++ AB + ω| Π od |β |−− AB +q µ,βP ω| Π od |β |+− AB + ω| Π ev |β |−+ AB C-X BA (45) = f suc (ω r ) π (1 − q µ,β )P ω|β |0 A + −ω|β |1 A ⊗ |+ +| B +q µ,βP ω|β |0 A − −ω|β |1 A ⊗ |− −| B ,(46) Since ω|β = e − 1 2 [(ωr−β) 2 +ω 2 i +2iωiβ] is not real in general, we insert a θ-rotation around the Z basis R Z A (θ) := exp(−iθZ A /2)(47) in order to have R Z A (2ω i β) † C-X BA Id A ⊗ K het C→B (ω) (Φ AC ) C-X BA R Z A (2ω i β) (48) = e −ω 2 i f suc (ω r ) √ π (1 − q µ,β )P g β,1/2 (ω r ) |0 A + g −β,1/2 (ω r ) |1 A ⊗ |+ +| B +q µ,βP g β,1/2 (ω r ) |0 A − g −β,1/2 (ω r ) |1 A ⊗ |− −| B .(49) We define τ het AB (ω r ) as τ het AB (ω r ) := (1 − q µ,β )P g β,1/2 (ω r ) |0 A + g −β,1/2 (ω r ) |1 A ⊗ |+ +| B + q µ,βP g β,1/2 (ω r ) |0 A − g −β,1/2 (ω r ) |1 A ⊗ |− −| B .(50) Thus, the structure of the matrix τ het AB (ω r ) is essentially the same as τ hom AB (x) of Homodyne protocol. In the same way as Homodyne protocol, the probability density of outcome ω with the occurrence of a phase error is given by Tr |− −| B K het AC→B (ω)(Φ AC ) = e −ω 2 i f suc (ω r ) √ π 1 2 Tr τ het AB (ω r ) (51) −Re Tr V (1) A→R (ω r ) † V (0) A→R (ω r )R Z A (2ω i β) 0| B τ het AB (ω r ) |1 B R Z A (2ω i β) † (52) ≥ e −ω 2 i f suc (ω r ) √ π 1 2 Tr τ het AB (ω r ) − 0| B τ het AB (ω r ) |1 B 1 .(53) If we write the polar decomposition of 0| B τ het AB (ω r ) |1 B by W het A (ω r ) 0| B τ het AB (ω r ) |1 B , then the equality of Eq. (53) can be achieved by setting R Z A (2ω i β) † V (1) A→R (ω) † V (0) A→R (ω)R Z A (2ω i β) = W het A (ω r ) † .(54) From Eq. (50), 0| B τ het AB (ω r ) |1 B is given by 0| B τ het AB (ω r ) |1 B = 1 2 (1 − q µ,β )P g β,1/2 (ω r ) |0 A + g −β,1/2 (ω r ) |1 A −q µ,βP g β,1/2 (ω r ) |0 A − g −β,1/2 (ω r ) |1 A ,(55) which is hermitian. Let |u het + (ω r ) A and |u het − (ω r ) A be eigenvectors of 0| B τ het AB (ω r ) |1 B with positive and negative eigenvalues, respectively. Then, W het A (ω r ) is given by W het A (ω r ) = |u het + (ω r ) u het + (ω r )| A − |u het − (ω r ) u het − (ω r )| A .(56) We can choose V (j) A→R (ω) to satisfy Eq. (54) in the same way as Homodyne protocol. We set R = A and set V (0) A→R (ω) = R Z A (2ω i β) † , V(1)A→R (ω) = W het A (ω r ) R Z A (2ω i β) † ,(57) which, with Eqs. (18) and (56), leads to V het B;A→A (ω) = |u het + (ω r ) u het + (ω r )| A ⊗ I B + |u het − (ω r ) u het − (ω r )| A ⊗ Z B R Z A (2ω i β) † C-X BA .(58) As explained previously, we set V (j) A→R (x) to the one in Eq. (42) (resp. V (j) A→R (ω) to the one in Eq. (57)) also for arbitrary channels, i.e., arbitrary coherent attacks by Eve. This choice is suboptimal for general channels but is expected to be close to optimal for channels that are close to the pure-loss one. Now that the controlled isometry V hom B;A→A (x) (resp. V het B;A→A (ω)) is fixed, we can interpret the event that Bob announces "success" and obtainsb = − (i.e., the phase error) at the signal round of Estimation protocol as the outcome of a generalized measurement on Alice's qubit A and the optical pulse C and define the corresponding POVM element M hom/het ph through Eq. (19) (resp. Eq. (20)) as M hom ph := F hom ‡ AC→B |− −| B = ∞ −∞ dx K hom AC→B (x) ‡ |− −| B ,(59)M het ph := F het ‡ AC→B |− −| B = ∞ −∞ dω r dω i K het AC→B (ω) ‡ |− −| B ,(60) where ‡ denotes the adjoint map. Then, for any density operator ρ on the joint system AC, M hom ph (resp. M het ph ) satisfies E ρ N suc (i) ph = p sig Tr ρ M hom/het ph (61) in Homodyne (resp. Heterodyne) protocol. For Homodyne protocol, by using Eqs. (9), (21), and (43), we have M hom ph = ∞ −∞ dx I A ⊗ K hom suc (x) † V hom B;A→A (x) † I A ⊗ |− −| B V hom B;A→A (x) I A ⊗ K hom suc (x) (62) = ∞ −∞ dx P I A ⊗ K hom suc (x) † C-X BA |u hom + (x) A ⊗ |− B +P I A ⊗ K hom suc (x) † C-X BA |u hom − (x) A ⊗ |+ B ,(63) where we used the fact that the adjoint map of the tracing-out Tr A is taking the tensor product with I A . Using the relation C-X BA = |+ +| A ⊗ I B + |− −| A ⊗ Z B as well as Eq. |u hom + (x) A ⊗ |x C +P Π (−,od),(+,ev) AC |u hom − (x) A ⊗ |x C ,(64) where two orthogonal projections Π := |+ +| A ⊗ Π od + |− −| A ⊗ Π ev ,(65) Π (−,od),(+,ev) AC := |− −| A ⊗ Π od + |+ +| A ⊗ Π ev .(66) A similar relation holds for Heterodyne protocol by replacing K hom suc (x) with K het suc (ω) and V hom B;A→A (x) with V het B;A→A (ω) as well as using Eqs. M het ph = ∞ −∞ dω r dω i P I A ⊗ K het suc (ω) † C-X BA R Z A (2ω i β) |u het + (ω r ) A ⊗ |− B +P I A ⊗ K het suc (ω) † C-X BA R Z A (2ω i β) |u het − (ω r ) A ⊗ |+ B (67) = ∞ −∞ 2f suc (ω r ) π dω r dω i P Π (+,od),(−,ev) AC R Z A (2ω i β) |u het + (ω r ) A ⊗ |ω C +P Π (−,od),(+,ev) AC R Z A (2ω i β) |u het − (ω r ) A ⊗ |ω C .(68) Using Eq. (11), we observe that 1 π dω i exp(±2iω i β) |ω ω| = 1 π dω i dxdx 2 π e ±2iωiβ−(x−ωr) 2 +2iωix−(x −ωr) 2 −2iωix |x x | (69) = 2 dxdx δ(2(x ± β − x )) |x x|ω r ω r |x x | (70) = dx |x x|ω r ω r |x ± β x ± β| .(71) Applying this to Eq. (68) and changing the integration variable appropriately, we have M het ph = ∞ −∞ 2f suc (ω r )dω r dx P Π (+,od),(−,ev) AC O β AC (x) |u het + (ω r ) A ⊗ |ω r C +P Π (−,od),(+,ev) AC O β AC (x) |u het − (ω r ) A ⊗ |ω r C ,(72)where the operator O β AC (x) is defined as O β AC (x) := |0 0| A ⊗ |x x| C + |1 1| A ⊗ |x − β x − β| C .(73) Finite-size analysis Since the phase error operator was defined on systems A and C, we can follow essentially the same analysis as that in Ref. [37]. Let us define the following operators: Π fid := |0 0| A ⊗ |β β| C + |1 1| A ⊗ |−β −β| C (74) = |φ − φ − | AC + |φ + φ + | AC ,(75)Π trash − := |− −| A ⊗ I C ,(76) where |φ ± AC are defined in Eqs. (27) and (28). For any density operator ρ on the joint system AC, these operators satisfy E ρ F (i) ≤ p test Tr ρ Π fid ,(77)E ρ Q (i) − = p trash Tr ρ Π trash − ,(78) where the first inequality follows from Theorem 1 in Ref. [37] as well as the definition ofF (i) . Let M hom/het [κ, γ] for positive numbers κ and γ determined prior to the protocol be defined as M hom/het [κ, γ] := M hom/het ph + κΠ fid − γΠ trash − .(79) In Corollaries 2 and 3 in Appendix B, we show an inequality M hom/het [κ, γ] ≤ B hom/het (κ, γ) I AC (80) with a computable convex function B hom/het (κ, γ). LetT (i) [κ, γ] be a linear combination of random variables at ith round in Estimation protocol given bŷ T (i) [κ, γ] := p −1 sigN suc (i) ph + p −1 test κF (i) − p −1 trash γQ (i) − .(81+ p −1 test κF − p −1 trash γQ − ≤ N B hom/het (κ, γ) + δ 1 ( /2),(82) holds with a probability no smaller than 1 − /2. (See Proposition 1 as well as Eqs. (92)-(105) in Ref. [37].) Here, δ 1 ( ) is defined as [37] δ 1 ( ) := max p −1 sig , p −1 test κ max ν≥0 Λ m,r (ν) − min p −1 test κ min ν≥0 Λ m,r (ν), −p −1 trash γ N 2 ln 1 . (83) SinceQ − is determined solely by Alice's qubits, each in the state TrC(|Φ Φ| AC ) with |Φ AC given in Eq. (7), it follows the same statistics as a tally ofN trash Bernoulli trials with a probability q − := −| A |Ψ AC 2 = (1 − e −2µ )/2. Hence we observe that Q − ≤ q −N trash + δ 2 ( /2;N trash ) (84) holds with a probability no smaller than 1 − /2. (See Eq. (31) in Ref. [37].) Here, δ 2 ( ; n) is defined as [37] D(q − + δ 2 ( ; n)/n q − ) = − 1 n log 2 ( ) ( > q n − ) δ 2 ( ; n) = (1 − q − )n ( ≤ q n − ) ,(85) where D(x y) := x log 2 x y + (1 − x) log 2 1 − x 1 − y (86) is the Kullback-Leibler divergence. Combining Eqs. (81), (82), and (84), by setting U (F ,N trash ) = p sig N B hom/het (κ, γ) + δ 1 ( /2) − p sig p test κF + p sig p trash γ q −N trash + δ 2 ( /2;N trash ) , (87) we observe that Eq. (23) holds from the union bound. Numerical simulations We compute (the lower bound on) the net key gain per pulse (i.e., key rateĜ) against the transmission distance with various values of excess noise at the channel output. In this model, Bob receives Gaussian states ρ (â) model obtained by randomly displacing attenuated coherent states |(−1)â √ ηµ with attenuation rate η to increase their variances via factor of (1 + ξ), i.e., ρ (â) model := 2 πξ C e −2|γ| 2 /ξ |(−1)â √ ηµ + γ (−1)â √ ηµ + γ| d 2 γ.(88) For simplicity, the number N smp of the sampling rounds is set to be N/100, and the bit error correction efficiency f in Eq. (4) is to be 0.95 2 . The acceptance probability f suc (x) is assumed to be a step function Θ(x − x th ) with a threshold x th (> 0), where Θ(x) denotes the Heaviside step function. The expected amplitude of the coherent state β is chosen to be √ ηµ. We set the security parameter sec = 2 −50 , and set cor = sec /2 and = 2 −s = 2 sec /16. We assume that the number of "success" signal roundsN suc is equal to its expectation, i.e., E[N suc ] = p sig N ∞ −∞ (f suc (x) + f suc (−x)) x| 1 2 a∈{0,1} ρ (a) model |x dx (89) = p sig N (P + hom + P − hom ),(90) where Figure 2: Key rates of the Homodyne protocol against transmission distance over an optical fiber. The attenuation rate of the optical fiber is assumed to be 10 −0.02L with transmission distance L km, an error correction efficiency f in Eq. (4) is set to be 0.95, and the number of sampling rounds Nsmp is set to be N/100. a) Key rates when the excess noise ξ at the channel output is zero; that is, the channel is pure loss. The bold solid lines show the key rates with our refined analysis developed here, the broken lines show those with the previous analysis [37], and the black thin line shows the PLOB bound, which is the ultimate limit of the key rate of one-way QKD [47]. One can see that the logarithm of the asymptotic key rate decreases in parallel to the PLOB bound with our refined analysis against the transmission distance ( 1 km) as opposed to the previous results [37]. Improvement in the key rate is sustained in the finite-size case. b) Key rates when N = 10 12 with various values of excess noise parameter ξ. (The detail of the noise model is given in the main text.) The solid lines show the key rates with our refined analysis, and the broken lines show those with the previous results [37]. One can see that, although the key rate significantly improves for the pure-loss channel, the excess noise as high as ξ = 10 −3 -10 −2 degrades the performance to almost the same level as that of the previous results. P ± hom := ∞ −∞ f suc (±x) 2 a∈{0,1} (−1) a x| ρ (a) model |(−1) a x dx (91) = 1 2 erfc (x th ∓ √ ηµ) 2 1 + ξ ,(92) for Homodyne protocol [37]. For Heterodyne protocol [38], it is given by E[N suc ] = p sig N (P + het + P − het ),(93)P ± het := ∞ −∞ f suc (±ω r ) 2π a∈{0,1} (−1) a ω| ρ (a) model |(−1) a ω dω r dω i (94) = 1 2 erfc (x th ∓ √ ηµ) 2 2 + ξ .(95) We also assume that the number of "success" sampling rounds is equal to (P + hom/het +P − hom/het )N smp , the number of test roundsN test is equal to p test N , and the number of trash roundsN trash is equal to p trash N . The test outcomeF is assume to be equal to its expectation given by [37] E[F ] = p test N 1 2 a∈{0,1} E ρ (a) model [Λ m,r (|ω − (−1) a √ ηµ| 2 )](96)= p test N 1 + ξ/2 1 − (−1) m+1 ξ/2 1 + r(1 + ξ/2) m+1 .(97) For the test function Λ m,r in the above, we adopt m = 1 and r = 0.4120, which leads to (max ν≥0 Λ m,r (ν), min ν≥0 Λ m,r (ν)) = (2.824, −0.9932). We assume that the numberÊ obs of bit errors observed in the "success" sampling rounds is equal to its expectationÊ obs = P − hom/het N smp . The upper-bound e qber on the bit error rate is thus given by Eq. (3) with the parametersN suc , N suc smp , andÊ obs given above. Under these assumptions, the remaining parameters to be determined are six parameters (µ, x th , p sig , p test , κ, γ). We determined (κ, γ) via a convex optimization using CVXPY 1. 2.1 and (µ, x th , p sig , p test ) via the Nelder-Mead in the scipy.minimize library in Python, for each transmission distance L with the attenuation rate η assumed to be 10 −0.02L . Figure 2 shows the key rates of Homodyne protocol for the channel model explained above. Figure 3: Key rates of the Heterodyne protocol against transmission distance over an optical fiber. The noise models are the same as those of Homodyne protocol. a) Key rates when the excess noise ξ at the channel output is zero; that is, the channel is pure loss. The bold solid lines show the key rates with our refined analysis developed here, the broken lines show those with the previous analysis [38], and the black thin line shows the PLOB bound, which is the ultimate limit of the key rate of one-way QKD [47]. One can see that the logarithm of the asymptotic key rate is in parallel to the PLOB bound when the transmission distance is large in the same way as that of Homodyne protocol. The key rate is still less (about half) than that of Homodyne protocol. b) Key rates when N = 10 12 with various values of excess noise parameter ξ. The solid lines show the key rates with our refined analysis, and the broken lines show those with the previous result [38]. higher key rates and longer transmission distance than that of the previous results [37] even in the finite-key case. Furthermore, the logarithm of the asymptotic key rate in the pure-loss case (i.e., ξ = 0) is in parallel to the PLOB bound [47] against the transmission distance; that is, it achieves a linear scaling against the channel transmission, which is known to be optimal for one-way QKD in the pure-loss channel. When the excess noise ξ is around 10 −3.0 -10 −2.0 , however, the improvements in our refined analysis are lost. The result of the parameter optimization implies that our refined analysis generates the key with relatively small intensity µ of the input coherent states compared to the previous analyses; e.g., the optimized input intensity µ of Homodyne protocol is ∼ 0.04 in our refined analysis compared to ∼ 0.2 in the previous analysis [37] at η = 0.1 (i.e., 50 km) for the asymptotic pure-loss case. The key rate of Heterodyne protocol has a similar behavior. Figure 3 shows the key rates of Heterodyne protocol with the same noise model as above. Figures show that our refined analysis significantly improves the key rate against the pure-loss channel, but is fragile against excess noise. One can see, however, that, while the key rate of Heterodyne protocol is still low compared to that of Homodyne protocol, the achievable distance (i.e., the distance with a non-zero key rate) now becomes comparable with our refined analysis. This implies that our refined analysis based on the reverse reconciliation is more effective for Heterodyne protocol. Discussion We propose a refined security analysis for the protocol proposed in Ref. [37] based on the reverse reconciliation. The motivating ideas of our refinement come from the facts that the distillability of a secret key from a quantum state is a looser condition than the distillability of an entanglement from it [51-53, 42, 54, 43] and the reverse reconciliation can increase the key rate for CV QKD protocols [10]. To exploit the ideas, we developed the procedure of "twisting" Alice's system with V hom B;A→R (x) (resp. V het B;A→R (ω)) controlled by Bob's qubit, while the similar techniques have already appeared in previous works [51,42,53,54,43,55]. Our finding is that by using the twisting operation that minimizes the phase error probability for the pure-loss channel, the protocol has asymptotically optimal scaling in the key rates both for Homodyne and Heterodyne protocols. This is a clear distinction from the previous results [37,38]; there, the asymptotic key rate nonlinearly decreases against the channel transmission. The improvement in the performance remains in the finite-key case but is lost under the existence of excess noise as high as ξ = 10 −3 -10 −2 at the channel output. This may limit the feasibility of our binary-modulation protocol, but current theoretical progress in CV QKD reveals that the discrete-modulation CV-QKD protocols with four types of modulation have more tolerance against excess noise than those with binary modulation [16][17][18]. What is important is that our security proof can be extended to the four-state protocols with binary outcomes, such as Protocol 2 in Ref. [17] and a protocol in Ref. [18], by replacing the bit-extracting measurements of these protocols with the qubit-extracting maps as shown in Eq. (9) and constructing the corresponding phase error operator. This is, however, much more complicated than the previous analysis, and we leave the problem as future work. There are several remaining questions with our present results. The first and foremost is whether we can obtain higher tolerance against excess noise by extending our analysis to the fourstate protocols. As explained above, our analysis can be extended to the four-state protocols with binary outputs [17,18], i.e., protocols that use homodyne measurement to distinguish signals. With the same type of argument based on the phase error estimation, we can carry out the finitesize security proof for these protocols in principle. However, developing the analyses that preserve the robustness against excess noise for these protocols still has non-triviality. A more challenging problem is to apply our finite-size security proof to the four-state protocols with more than two outputs, such as a protocol in Ref. [16] and Protocol 1 in Ref. [17]. In this case, the definition of phase errors is already non-trivial as opposed to those with binary outputs, and we have to develop more elaborated finite-size security proof. Whether we can extend our techniques to these protocols or protocols with even more constellations [19] is still open. Another important theoretical question is whether the trusted-noise model can be applied to our security analysis. In practice, even the excess noise of ξ = 10 −3 at the channel output is difficult to realize if all the noises are untrusted. Recently, efforts have been made in the field of CV QKD on how to incorporate noises that are intrinsic to apparatuses and thus inaccessible to Eve into the security proof as trusted noises. This effectively eases the requirement on the experimental apparatuses. In the present security analysis as well as ones in Refs. [37,38], the fidelity test measures the fidelity to a pure coherent state, which cannot be naively generalized to the fidelity to a mixed state. Whether we can incorporate trusted noises into the fidelity test may be crucial in this direction. From the viewpoint of the feasibility of the protocol, the total number of 10 12 of rounds to obtain a tolerable finite-size performance may be demanding. The finite-size performance may be improved by applying recently developed refinement [62] of the Azuma's inequality [59] that utilizes unconfirmed knowledge. What is non-trivial for the application of this is that the random variable in our application of Azuma's inequality can not directly be observed even at the end of the protocol. Whether we can apply the refined concentration inequality [62] with the information accessible in our protocol (in a similar fashion to Ref. [63]) may be an interesting problem. Lemma 1 (Tail bound for the hypergeometric distribution [64]). Let X 1 , . . . , X N be a binary sequence, and M be the number of elements with X i = 1, i.e, M := N i=1 X i . LetŶ 1 , . . . ,Ŷ n (n ≤ N ) be randomly sampled from X 1 , . . . , X N without replacement. Letm := n i=1Ŷ i be the number of ones inŶ 1 , . . . ,Ŷ n . Then, for any δ ∈ [0, M/N ], the following inequality holds: Pr m n ≤ M N − δ ≤ 2 −nD( M N −δ M N ) ,(98) where D(· ·) is defined in Eq. (86). Then, the following corollary is essential for the bit error sampling. Pr m n ≤ M N − M N − f (M ) n ≤ 2 −nD f (M ) n M N .(102) We With Corollary 1, we can bound the number of total bit-error events from the sample under the given failure probability ε cor /2 by setting N =N suc +N suc smp , n =N suc smp , and = ε cor /2 for M N,n, . As a result, we have the following statement; the number E of bit errors inN suc -bit sifted key is bounded from above by Pr E ≤MN suc +N suc smp ,N suc smp ,εcor/2 (Ê obs ) −Ê obs ≥ 1 − ε cor /2. Thus, we can define an upper bound e qber of the bit error rate as in Eq. (3), which holds with probability no smaller than 1 − ε cor /2. B Proof of the operator inequality In this section, we prove the inequality (80) used in the security proof in the main text. We first prove the following lemma. Lemma 2. Let Π ± be orthogonal projections that have the rank no smaller than three or infinite. Let M be a self-adjoint operator satisfying M = (Π + +Π − )M (Π + +Π − ) ≤ α(Π + +Π − ), where α is a real constant. Let |ψ be a vector satisfying (Π + +Π − ) |ψ = |ψ and Π ± |ψ = 0. Assume Π ± |ψ are not proportional to eigenvectors of Π ± M Π + (if they have). Define the following quantities with respect to |ψ : C ± := ψ| Π ± |ψ (> 0),(107)λ ±± := C −1 ± ψ| M ±± |ψ ,(108)λ +− := (C + C − ) − 1 2 ψ| M +− |ψ , λ −+ := λ * +− ,(109)σ ±+ := C −1 + M ±+ |ψ 2 − |λ ±+ | 2 1 2 ,(110)σ ±− := σ −1 ±+ (C + C − ) − 1 2 ψ| M +± M ±− |ψ − λ +− λ ±± ,(111)∆ ±− := C −1 − M ±− |ψ 2 − |λ ±− | 2 − |σ ±− | 2 1 2 ,(112) Then, for any real numbers γ ± , we have σ sup (M + |ψ ψ| − γ + Π + − γ − Π − ) ≤ σ sup (M 6d ),(114) where σ sup (X) denotes the supremum of the spectrum of the operator X, and M 6d is given by M 6d :=         α − γ + 0 0 ∆ +− 0 0 0 α − γ + σ ++ σ +− 0 0 0 σ ++ C + + λ ++ − γ + C + C − + λ +− σ −+ 0 ∆ +− σ * +− C + C − + λ −+ C − + λ −− − γ − σ * −− ∆ −− 0 0 σ −+ σ −− α − γ − 0 0 0 0 ∆ −− 0 α − γ −         .(115) Proof. We choose orthonormal vectors {|e (1) ± , |e(2)± , |e(3) ± } in the domains of Π ± , respectively, to satisfy C ± e (1) ± = Π ± |ψ ,(116)+ λ −− e (1) − + σ −− e (2) − + ∆ −− e (3) − ,(118) which is well-defined due to Eqs. where h.c. denotes the hermitian conjugate of the terms in the preceding parenthesis. The last inequality comes from M ≤ α(Π + + Π − ). Using Eq. (126), we have M + |ψ ψ| − γ + Π + − γ − Π − ≤ M 6d ⊕ (α − γ + )Π (≥4) + ⊕ (α − γ − )Π (≥4) − ,(127) where M 6d is given in Eq. (115) with the basis {|e + , |e(2)+ , |e(1) − , |e Proof. We first derive the explicit form of |u hom ± (x) A introduced in Eq. (41). Notice that 1 − 2q µ,β = e −2(µ−β 2 ) , g β,1/4 (x) g −β,1/4 (x) = e 4βx .(148) Let θ(x) be defined to satisfy |θ(x)| < π 2 , tan θ(x) = Tr Z A 0| B τ hom AB (x) |1 B Tr X A 0| B τ hom AB (x) |1 B .(150) ( 151 ) 151in Appendix B. The choice of the isometry V (j) A→R (x) to satisfy Eq. (39) is not unique; one of the reasons is the arbitrariness of the dimension of the system R. Here, we set R = A and set V (0) (15),(58), (65), and (66): Figures show that under the condition of low excess noise, our refined analysis results in significantly Corollary 1 ( 1Estimation by the simple random sampling without replacement). Let X 1 , . . . , X N be a binary sequence with M := N i=1 X i . LetŶ 1 , . . . ,Ŷ n (n ≤ N ) be randomly sampled from X 1 , . . . , X N without replacement, and definem := n i=1Ŷ i . Then, for any ∈ (0, 1), the following inequality holds:Pr M N,n, (m) < M ≤ ,(99)where the functionM N,n, (m) is defined to satisfy m n ≤M N,n, (m) N ≤ 1 (100) and for 0 ≤ m < n, D m/n M N,n, (m)/N = − 1 n log . (101) Proof. Let f (M ) be a function of M satisfying 0 ≤ f (M )/n ≤ M/N . Then, from Lemma 1, we have set the function f (M ) to the restriction of the function f N,n, (M ) of the real numberM that satisfies D f N,n, (M )/n M /N = − 1 n log , (103) forM ∈ [(1 − n √ )N, N ). The function f N,n, (M ) increases monotonically with increasingM in [(1 − n √ )N, N ), and its image lies in [0, n). Thus, from Eq. (102), we have Pr f −1 N,n, (m) ≤ M ≤ (104) for anym ∈ [0, n). We define the functionM N,n, (m) := f −1 N,n, (m) for m ∈ [0, n). To incorporate the casem = n, we use the following weaker condition that trivially follows from Eq. (104): Pr M N,n, (m) < M ≤ , (105) and defineM N,n, (n) = N so that the above holds also form = n. These show that Eq. (99) holds whileM N,n, (m) satisfies Eqs. (100) and (101) by construction in Eq. (103). where M ++ , M −− , M +− , and M −+ are given respectively by M ±± := Π ± M Π ± , M +− := Π + M Π − , M −+ := M † +− . (107)-(113) and M = (Π + + Π − )M (Π + + Π − ). Actually, Eqs. (110)-(112) are derived by taking inner product of appropriate pairs among M ±± |ψ and M ±∓ |ψ . Overall phases of |e so that σ ±+ and ∆ ±− are positive. From (Π + + Π − ) |ψ = |ψ , we have |ψ = C + e sup (M 6d ), the supremum of the spectrum of the right-hand side of Eq. (127) is equal to the maximum eigenvalue of the six-dimensional matrix M 6d . We then obtain Eq. (114).Define the following (real) parameters:C o := β| Π od |β = e −|β| 2 sinh |β| 2 , C e := β| Π ev |β = e −|β| 2 cosh |β| 2 , M hom oo M hom (+,o)(−,e) |β − λ hom oo λ hom (+,o)(−,e) = (σ hom (+,o)(−,e) ) * , (140) σ hom (−,e)(−,e) := (σ hom (−,e)(+,o) ) −1 (C o C e ) Define a convex function B hom (κ, γ) := max{σ sup (M Then, for κ, γ ≥ 0, we have M hom [κ, γ] ≤ B hom (κ, γ)I AC . Define a convex functionB het (κ, γ) := max{σ sup (M (0) 6d ), σ sup (M (1) 6d )}. Box 1: Actual protocol1. Alice generates a random bitâ ∈ {0, 1} and sends an optical pulseC in a coherent state with amplitude (−1)â √ µ to Bob. She repeats it for N rounds. Bob receives an optical pulse C for each of the N rounds. 3 . 3We refer to the numbers of "success" and "failure" signal rounds, test rounds, and trash rounds asN suc ,N fail ,N test , andN trash , respectively. (N =N suc +N fail + N test +N trash holds by definition.) Bob calculates the sum of Λ m,r (|ω − (−1)âβ| 2 ) obtained in theN test test rounds, which is denoted byF .4. For error correction, they use H EC -bits of encrypted communication consuming apre-shared secret key to do the following. According to (the upper bound on) the bit error rate e qber , Bob randomly chooses an error-correcting code and sends it with the H EC -bits syndrome to Alice. Alice reconciles her sifted key accordingly.5. Bob computes and announces the final key lengthN fin according to Eq. (1). Alice and Bob apply privacy amplification to obtain the final key. Currently, this level of efficiency may be too optimistic because the bit error correction in our protocol must succeed with probability no smaller than 1 − εcor/2 without the use of the verification. A Bit error samplingIn this section, we summarize how to determine an upper bound on the bit error rate from the given sample. As explained in the main text, N smp sampling rounds are randomly inserted in the actual protocol in which Alice and Bob announce their bit values if Bob's detection succeeds (in the same way as in the signal round). The number of "success" sampling rounds is denoted bŷ N suc smp , and the observed number of discrepancies between Alice and Bob is denoted byÊ obs . Let us first introduce a Chernoff-type bound for the hypergeometric distribution.As a corollary of this lemma, we obtain the followings. First, we consider Homodyne protocol.Corollary 2.Let |β be a coherent state and θ hom µ,β (x) be defined to satisfyNoticing that TrFrom Eqs.(40), (148), (149), and (150), we can see that θ(x) coincides with θ hom µ,β (x) defined in Eq. (128). We now observe thatwhere |φ − AC is defined in Eq.(28). Since so-defined M only has continuous spectrum, we can apply Lemma 2 and obtainIn the same way, we apply Lemma 2 to Πwhere |φ + AC is defined in Eq.(27). Then, we observeCombining inequalities(161)and(168)completes the proof.Next, we consider Heterodyne protocol.Corollary 3.Let |β be a coherent state and θ hom µ,β (x) be defined to satisfyDefine the following parameters: Then, for κ, γ ≥ 0, we haveProof. In the same way as Homodyne protocol, we have from Eqs.(55)and(56)thatCombining this with Eqs.(72)and(73), we observe thatAs can be seen from Eq. (72) as well as Eqs.where |φ − AC is defined in Eq.(28). 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K Horodecki, M Horodecki, P Horodecki, D Leung, J Oppenheim, 10.1103/PhysRevLett.100.110502Phys. Rev. Lett. 100110502K. Horodecki, M. Horodecki, P. Horodecki, D. Leung, and J. Oppenheim, Phys. Rev. Lett. 100, 110502 (2008). . K Horodecki, M Horodecki, P Horodecki, J Oppenheim, 10.1109/TIT.2008.2009798IEEE Transactions on Information Theory. 551898K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, IEEE Transactions on Infor- mation Theory 55, 1898 (2009). . J E Bourassa, I W Primaatmaja, C C W Lim, H.-K Lo, 10.1103/PhysRevA.102.062607Phys. Rev. A. 10262607J. E. Bourassa, I. W. Primaatmaja, C. C. W. Lim, and H.-K. Lo, Phys. Rev. A 102, 062607 (2020). . T Matsuura, T Sasaki, M Koashi, Physical Review A. 9942303T. Matsuura, T. Sasaki, and M. Koashi, Physical Review A 99, 042303 (2019). T M Cover, J A Thomas, Elements of information theory. John Wiley & SonsT. M. Cover and J. A. Thomas, Elements of information theory (John Wiley & Sons, 2012). . 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{'abstract': 'Recent studies showed the finite-size security of binary-modulation CV-QKD protocols against general attacks. However, they gave poor key-rate scaling against transmission distance. Here, we extend the security proof based on the complementarity, which is used in the discrete-variable QKD, to the previously developed binary-modulation CV-QKD protocols with the reverse reconciliation under the finite-size regime and obtain large improvements in the key rates. Notably, the key rate in the asymptotic limit scales linearly against the attenuation rate, which is known to be optimal scaling but is not achieved in previous finite-size analyses. This refined security approach may offer full-fledged security proofs for other discrete-modulation CV-QKD protocols.Takaya Matsuura: [email protected] 1 arXiv:2301.03171v2 [quant-ph] 26 Jan 2023 U (N )-invariant CV-QKD protocols[33,34]. However, in practice, ideal Gaussian modulation cannot be implemented and should be approximated by a finite number of coherent states. It turns out that an overwhelming number of coherent states is needed to directly approximate the Gaussian ensemble for the security condition to be satisfied[35,36]. If we try to mitigate the required number, additional assumptions are needed, which makes it difficult to apply it in the finite-size regime[15]. The other completely different approach[37,38]is targeted at the discrete-modulation CV QKD from the beginning. Refs.[37,38]show the finite-size security against general attacks for a binary-modulation protocol. It also takes into account the discretization of the signal processing, such as binned homodyne and heterodyne measurements (see also[39]for this topic). Although it has a nice feature, the obtained key rate has very poor scaling against transmission distance. A possible reason for this bad performance is the fact that its security proof is based on the entanglement distillation[40,41]. It is known that the security proof based on the entanglement distillation is too stringent in general for secure key distribution. There are alternative types of security proofs[42][43][44]that can be applied to general cases. In particular, for CV-QKD protocols, the security proof based on the reverse reconciliation often provides better performance than that based on the direct reconciliation [10], which may be unattainable by a security proof based on the entanglement distillation due to its symmetric nature between the sender and the receiver in the security proof.Contributions of this paper. In this article, we aim to develop another approach to carry out the finite-size security proof for the discrete-modulation CV QKD against general attacks. The approach should be able to exploit the benefit of the reverse reconciliation. To do it concretely, we develop refined security proofs based on the reverse reconciliation for the binary-modulation CV-QKD protocols proposed in Refs.[37,38], i.e., the protocol in which the sender Alice performs BPSK-type modulation according to her randomly generated bit and the receiver Bob performs homodyne measurement, heterodyne measurement, and trash randomly [37], or performs heterodyne measurement followed by a random selection of the post-processing of the outcome[38]. We use the same apparatuses and setups as those in Refs.[37,38]but slightly change the protocols. To refine the security proofs, we use an approach based on the complementarity[43,45]under the reverse reconciliation, which is more general than the one based on the entanglement distillation [46] and treats Alice and Bob asymmetrically in the security proof. In these refined security proofs, we have degrees of freedom that did not appear in the previous analyses. By setting these degrees of freedom to be optimal in the pure-loss channel [47], we obtain a significant improvement in the key gain rates; in fact, the asymptotic key rates of the protocols scale linearly with regards to the attenuation rate of the pure-loss channel, which is known to be the optimal scaling for the one-way QKD[47]. This shows that we can exploit the benefit of using the reverse reconciliation in the approach based on complementarity. Although the protocols are still fragile against the excess noise, this approach itself may be a step towards the full-fledged security proofs for discrete-modulation CV QKD.Organization of this paper. The article is organized as follows. In Section 2, we provide the refined security proofs based on the complementarity [43] for protocols that use the same experimental setups as proposed in Refs.[37,38]. The section is further divided into three parts. The first part 2.1 defines the actual protocols, which are almost the same as the ones in Refs.[37,38], and develops virtual protocols for the complementarity approach[43]. In the second part 2.2, we derive an explicit form of the phase error operator defined by the virtual procedure of the previous part. In the third part 2.3, we finish the finite-size security proof by developing operator inequalities. In Section 3, we numerically demonstrate the improved performance of the protocols with our refined security proof. Finally, in Section 4, we wrap up our article by discussing future work and open problems.', 'arxivid': '2301.03171', 'author': ['Takaya Matsuura \nDepartment of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan\n\nSchool of Science\nCentre for Quantum Computation & Communication Technology\nRMIT University\nMelbourne VIC 3000Australia\n', 'Shinichiro Yamano \nDepartment of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan\n', 'Yui Kuramochi \nDepartment of Physics\nFaculty of Science\nKyushu University\n744 Motooka, Nishi-kuFukuokaJapan\n', 'Toshihiko Sasaki \nDepartment of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan\n\nPhoton Science Center\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan\n', 'Masato Koashi \nDepartment of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan\n\nPhoton Science Center\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan\n'], 'authoraffiliation': ['Department of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan', 'School of Science\nCentre for Quantum Computation & Communication Technology\nRMIT University\nMelbourne VIC 3000Australia', 'Department of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan', 'Department of Physics\nFaculty of Science\nKyushu University\n744 Motooka, Nishi-kuFukuokaJapan', 'Department of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan', 'Photon Science Center\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan', 'Department of Applied Physics\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan', 'Photon Science Center\nGraduate School of Engineering\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-8656TokyoJapan'], 'corpusid': 255546298, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 25940, 'n_tokens_neox': 22620, 'n_words': 13063, 'pdfsha': '7536bf3e923b44e55209efcb7bfe83cc4aa84449', 'pdfurls': ['https://export.arxiv.org/pdf/2301.03171v2.pdf'], 'title': ['Refined finite-size analysis of binary-modulation continuous-variable quantum key distribution', 'Refined finite-size analysis of binary-modulation continuous-variable quantum key distribution'], 'venue': []}

arxiv

Snowmass White Paper: The Quest to Define QFT 10 Jan 2023 Mykola Dedushenko Simons Center for Geometry and Physics Stony Brook University 11794-3636Stony BrookNYUSA Snowmass White Paper: The Quest to Define QFT 10 Jan 2023 This article provides a review of the literature on rigorous definitions and constructions in Quantum Field Theory, spanning the period of seven decades. Comparing with the ideas and constructions found in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians. arXiv:2203.08053v2 [hep-th] 10 Jan 2023 2 Existing axiomatics 2.1 Correlator-focused approaches Wightman axioms. One of the older axiom systems that remains relevant to date is that due to Wightman [20-23] (see also books [25, 28, 29]). The axioms view fields as operator-valued tempered distributions and formalize the notion of expectation values of their products ("Wightman functions"). One starts with the assumption (W0) of relativistic invariance (as in Wigner's classification [30] 1 , see also [31-34]): physical Hilbert space is a unitary representation of the Poincare group. This assumption is supplemented by the spectral condition (the energy-momentum spectrum lies in the closed upper light-cone) and the uniqueness and Poincare-invariance of the vacuum state Ψ 0 ∈ H. The axiom W1 states that there is a set of fields ϕ i [f ] given by tempered distributions 2 valued in the operators defined on (and preserving) a dense subset D (which includes the vacuum) of the Hilbert space H. The subset D is assumed to be Poincare-invariant. Then W2 states covariance of fields with respect to the Poincare group, and W3 requires locality (also called microcausality) in the form of (anti)commutativity of spacelike-separated fields. A quantum field theory is said to satisfy W0-W3, and in addition obey cyclicity of the vacuum: The span of vactors of the form ϕ 1 [f 1 ] . . . ϕ n [f n ]Ψ 0 (for all possible n and f i ) is dense in H. The latter condition guarantees that there are enough fields in the theory. Introduction The subject of Quantum Field Theory is nearing the centennial, with its inception dating back to the papers [1,2], followed by [3,4] and many others. Growing mostly out of the need to reconcile special relativity with quantum mechanics, both young subjects at that time, it led to the development of the early version of perturbative QFT over the following two decades. An interesting historical account of those early years can be found in the first chapter of Weinberg's excellent textbook [5]. One of the biggest challenges that had to be overcome were the UV divergences, which eventually led to the development of renormalization techniques by the end of forties in the works of Dyson, Feynman, Schwinger, and Tomonaga, [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. A new field, with all its strange renormalization machinery, desperately needed a clear set of rules, or axioms, from which everything else would follow in a logical manner. Such rules would "distill" the subject into a mathematical subfield, but they were also necessary due to the limitations of the perturbative Lagrangian techniques. Thus starting from the fifties, various axiomatics for QFT began to appear. For the purpose of this paper, we will consider Wightman's axioms [20][21][22][23] as the start of that process, which will inevitably miss some earlier attempts, such as the S-matrix program of Heisenberg [24], a closely related extended (off-shell) S-matrix approach of Bogolyubov-Medvedev-Polivanov [25], or axioms of Lehmann-Symanzik-Zimmermann [26,27] (the LSZ reduction formula, however, became part of the standard QFT formalism). Some of the other approaches are still being developed nowadays. Here we would like to give a brief overview of these issues, and discuss the range of applicability of various axiom systems. A point that we want to make is that none of the existing definitions covers the full range of notions of Quantum Field Theory that appears in the physics literature. Having a rigorous system of axioms for a physics subfield has two philosophical motivations. On the one hand, it provides a starting point for mathematical investigations, sort of extracting the abstract truth from the messy reality. On the other, it indicates that the physical understanding of the subject has matured enough. Indeed, most other physics subfield (all the "non-quantum" physics and the nonrelativistic quantum mechanics) have already undergone this process. The fact that QFT does not have (as we will see) one clear and universal set of axioms likely shows that the physical understanding is still lacking. Hence, we argue, it is a challenge both for physicists and for mathematicians to define QFT. Below we will provide a brief overview of the existing approaches. Announcement: comments are highly appreciated! Axioms of Euclidean QFT. The Osterwalder-Schrader (OS) axioms [65][66][67], as well as their modifications by Glimm-Jaffe (GJ) [68] and the axioms of Nelson [69] provide, roughly, the Euclidean version 3 of Wightman axioms. OS axioms are also based on formalizing the notion of correlation functions, known as Schwinger functions S n (x 1 , . . . , x n ) = φ 1 (x 1 ) . . . φ n (x n ) in the Euclidean case. They include: OS0 temperedness of S n as distributions; OS1 Euclidean covariance; OS2 Reflection positivity; OS3 (anti)symmetry under permutations ("anti" in the fermionic case); OS4 cluster decomposition. Under a subtle additional property of linear growth condition, the OS theorem [66] (sometimes called the OS reconstruction theorem) states that the Schwinger functions can be analytically continued to the Minkowski signature to obey Wightman axioms there (see also modification by Zinoviev [77]). The Glimm-Jaffe axioms similarly formalize the generating functional S[f ] = e φ[f ] ≡ e φ[f ] dµ, where dµ is a measure on the space of distributions φ. They demand its analyticity, regularity (in the form of a certain growth bound), Euclidean covariance, reflection positivity, and ergodicity of the time translations. These axioms imply OS with the growth condition, and thus also Wightman axioms. Finally, Nelson axioms [69,69,[78][79][80] similarly take the measure-theoretic approach seriously and, importantly, require Markov property, which essentially captures locality and implies that the state of a system in some region is assigned to the boundary of this region (and in particular, the Hilbert space is assigned to the boundary). He also requires ergodicity and proves that the Wightman axioms follow upon analytic continuation to the Minkowsi space ("Nelson's reconstruction theorem", see book [81] by B.Simon for the review of this and other topics; see also [82]). Also note the result of [83] extending the OS reconstruction to equilibrium statistical mechanics (see also [84]), and the result of [85] studying the reconstruction of representations of the Poincare group in the same context of Euclidean QFT. A recent work [86] also provides generalizations of W and OS axioms (and reconstruction theorems) that are supposed to be suitable for gauge theories. Constructive Field Theory. The axioms discussed so far are completely non-constructive, one has to do extra work to provide examples. At first, only free fields (including generalized free fields defined in [87],) and various solvable models related to free fields were known to satisfy the Wightman axioms (and, naturally, other axiom systems). This led to the subject of Constructive Field Theory (CQFT) emerging in the 1960's [88][89][90][91][92], whose main goal was to provide rigorous interacting examples of QFTs. An extensive review (as of 1987) can be found in the book [68] (it focuses on the Euclidean path integral approach, see also other books on the subject: [93][94][95][96]), also see [97][98][99][100][101]. An early vision of the field can be found in [102,103], as well as a slightly later review [104]; a review summarizing some successes of CQFT as of 2000 is in [105], as well as a slightly more detailed review in [106]. More recent reviews include [107][108][109] and a talk [110]. Through a lot of work starting from the late sixties, success has been achieved in rigorously constructing and studying 2d scalar theories with arbitrary polynomial interaction (the so-called P(φ) 2 theories) [91,[111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128][129][130] (see also book [81], recent papers [131,132], and [133][134][135][136] on other potentials), threedimensional λφ 4 theory [137][138][139][140][141][142][143][144][145] (see also more recent [146][147][148][149][150]), Gross-Neveu theory [151][152][153][154][155][156], Thirring model [157][158][159][160][161] (in particular, all these theories were shown to obey Wightman axioms); other theories with fermions include 2D and 3D Yukawa models [65, (see also [187]) and some supersymmetric models [188][189][190][191][192][193][194]. Random walks representations of Euclidean theories were introduced [73,75] and developed later [195][196][197][198], resulting in various applications [145,199], most prominently the proof of triviality of the φ 4 theory in d ≥ 5 spacetime dimensions [200][201][202] (see book [203]). The four-dimensional case turned out to be much more subtle [204][205][206][207][208][209][210][211][212][213] and has been resolved only recently in [214] (see also lecture notes [215]) confirming triviality of the φ 4 4 model. 4 Lattice regularization has played role, especially in gauge theories [217][218][219], see for example [220] and many references therein, especially works [221][222][223][224][225][226][227][228][229][230][231][232][233][234][235][236][237][238][239][240] of Balaban, see also [241][242][243] that revisits Balaban's approach to the renormalization group (illustrated with the φ 4 theory) and [244][245][246]. The results of [232,234] and [247] (using different methods) provide a significant progress towards solving the Millennium problem on the four-dimensional Yang-Mills, 5 see [248,249] for discussions. Algebraic QFT Haag-Kastler axioms. Algebraic QFT (AQFT) is another approach to axiomatizing QFT that de-emphasizes the notion of fields, and instead formalizes the algebra of observables without referring to any Hilbert space at first. This subject was initiated by the formulation of Haag-Kastler (HK) axioms in [250] (with some elements appearing in earlier works, such as [39,40,53,[251][252][253][254], see also reference [2] in [250]). There exists a number of books and monographs on AQFT [255][256][257][258][259][260][261] and on operator algebras [262][263][264][265][266][267][268][269][270][271][272][273][274][275][276][277][278][279][280], which should be consulted for details. Among the more recent literature, we mention a collection [281], a concise review [282], books and monographs [283][284][285] and a related book [286]. The key points and references are also summarized in [287]. The HK axioms (sometimes called Araki-Haag-Kastler axioms) are about relativistic local unitary QFTs in flat Minkowski space-time. For every causally closed 6 subset U of observables one assigns a C * -algebra 7 of observables A(U ). Under an inclusion U 1 ⊂ U 2 , one has an inclusion A(U 1 ) ⊂ A(U 2 ) 4 "Trivial" means "free" or "Gaussian", and the statements are about UV-complete models (i.e., with cutoffs removed) in precisely integral dimensions. Of course nothing prevents models with cut-offs from being nontrivial effective field theories, and furthermore, d = 4 − [216] is not covered by such statements. 5 Curiously, after two decades of rapid progress in the 70s and 80s, the field of Constructive Field Theory has gone so far away from the mainstream that, even though it hosts one of the most famous problems in theoretical physics, many young people nowadays do not even know that this field exists. In authors opinion, this state of affairs will change in the future, as more mathematicians are starting to think about QFT nowadays again. 6 See discussion of causal closedness in [288] 7 C * -algebras were introduced in the works of Gelfand and Naimark. See above block of references and [256] on C * algebras. In short, a C * -algebra is characterized by the following data: a C-algebra with an involution * obeying natural properties (this is a * -algebra); norm A obeying AB ≤ A B and A * = A ; completeness with respect to the topology induced by · (Banach * -algebra, or B * -algebra); the C * -property A * A = A A * . of C * -algebras (this property is called isotony), which is functorial, i.e., respects compositions (this data is often called a local net of algebras; we could say that A(·) is an isotonic pre-cosheaf, except it is defined on causally closed subsets rather than opens). The requirement of causal locality says that A(U 1 ) and A(U 2 ) commute with each other (inside A(U ), where U i ⊂ U ) if U 1 and U 2 are spacelike separated. Furthermore, one usually imposes Poincare covariance in the form of a morphism α(p) : A(U ) → A(pU ) for any Poincare transformation p. Another requirement, due to the existence of linear dynamics, known as a time slice axiom, states that if U 1 ⊂ U 2 and U 1 contains a Cauchy surface of U 2 , then A(U 1 ) → A(U 2 ) is an isomorphism. One also often requires positivity of the energy spectrum (i.e., of the operator of time translations). In the algebraic formulation, quantum states are understood as linear maps ω : A → C satisfying the positivity condition ω(A * A) ≥ 0 for all A ∈ A, where ω(A) is the "expectation value of A". One can consider faithful representations π ω of algebras A by bounded operators π ω (A(U )) ⊂ B(H) on the Hilbert space H obtained via the GNS construction from the state ω [289,290]. The algebra B(H) has two useful notions of closed * -subalgebras: the above-mentioned C * -algebras (closed in the norm topology), and von Neumann algebras [291] (which are closed in either one of the three topologies: strong operator, weak * , and weak operator), see for example [255] or any other textbook referred above. Sometimes A(U ) is taken to be a C * -algebra, but quite often one focuses on R(U ) = (A(U )) , which is the minimal von Neumann algebra containing A(U ), where (·) denotes the commutant 8 inside B(H). One often talks about the net R(U ) forming a vacuum representation, see, e.g., a review [292]. This then connects to the rich theory of von Neumann algebras [293][294][295][296][297][298][299] (see collection [300] and textbooks cited previously), in particular such topics as: Decomposition [299] into factors of Type I, II, III [293] depending on whether the spectrum of dimensions of projectors on the invariant subspaces in H is, respectively, discrete containing all integers in an interval, continuous, or consisting of just 0 and ∞, and a deep result that in QFT we deal with the Type III factors [301][302][303]; modular or Tomita-Takesaki theory (introduced by Tomita [304] and clarified by Takesaki [305], see also [275] and expositions by Borchers [306] and Summers [307], or the book [255],) which provides the structural theory of the Type III factors, and has connections to other topics, such as KMS states (see [308][309][310]). There has been a lot of interest in the modular theory recently due to its connection to the entanglement properties in QFT (see [311] as the entrance point into this portion of literature). Similar to other approaches, it is possible to study general structural properties within the AQFT system of axioms, such as existence of scattering states [40,252], superselection sectors 9 [315,316] (see also, e.g., [317][318][319][320]), spin-statistics and CPT theorems (see [321][322][323], also see the 8 In the subject of operator algebras, the "commutant" of X means everything that commutes with X, while in the rest of math this notion would be called a centralizer. 9 The concept of superselection sectors [312,313], as it is apparent from [250], was from the beginning important in the development of AQFT, see also review [314]. The idea was that different superselection sectors arise from the inequivalent representations of one algebraic structure. topological version in [324]), the Reeh-Schlieder theorem [56] (it was already mentioned earlier, but traditionally this theorem is viewed as part of the AQFT machinery), Goldstone theorem [325,326]. One difference from the Wightman axioms should be clear: while they did not require boundedness of the operators (e.g., the momentum operator has unbounded spectrum), this was sort of an idealization. Any realistic experiment involves devices with finite ranges of possible values, thus any outcome should be predictable with arbitrary precision by a theory dealing with bounded operators only, like in the AQFT framework. 10 In the discussion of connection to the Wightman axioms, one asks two questions: whether, starting with a Wightman field smeared out with a compactly supported test function, one can find a self-adjoint bounded operator, and whether, starting with a net of algebras of bounded operators, one can obtain Wightman fields by a limiting process shrinking the regions to points (such questions were studied, e.g., in [327][328][329][330][331][332][333]). Perturbative AQFT. Requirements of bounded, C * or von Neumann, are dropped in perturbative AQFT, where instead one deals with formal series star-algebras. Reviews include [284,334,335], see also a book [336] and expositions [337,338]. A few references on causal perturbation theory relevant in this context are: [339][340][341][342][343][344][345], books [346][347][348] and a review [349]. A more recent block of papers on the formalism of perturbative AQFT is [350][351][352][353][354][355][356][357][358][359][360][361][362] (including [353] on the 1/N expansion) and [363] (see also comments in [364]). See also [365][366][367][368][369] and [284,Chapter 7] on the role of Batalin-Vilkovisky formalism [370,371], and [372][373][374][375][376] on the relation to deformation quantization. AQFT in curved space. Quantum field theory on curved space pushes the limits of applicability of the QFT machinery. 11 It comes with new physical phenomena (such as particle production effects [377]; Hawking effect [378], with earlier precursors [379][380][381]; Fulling-Davies-Unruh effect [382][383][384]). They generally follow from the absence of Poincare invariance, and, as a result, absence of the distinguished vacuum, no particle interpretation, no momentum space representation, etc. Continuation from Mikowskian to Euclidean signature is also not generically available, and, relatedly, there is no unique choice of the Feynman propagator. All these subtleties put traditional particle-based techniques in danger, and it was recognized early on 12 that the AQFT framework extended to general curved backgrounds must be the right way to proceed. 13 By the 80s, some version of such an approach was available [386][387][388][389][390][391], but it had shortcomings: it could only describe free fields; there were also problems in subtracting singularities 14 when renormalizing composite operators, such as the stress-energy tensor [394] or general Wick polynomials, where the answer depended on the choice of a reference quasi-free Hadamard state. This prevented both analyzing backreaction and building consistent perturbation theory in the interacting case. Imposing locality and covariance [394] would eventually help to fix these issues. Real progress, however, began in the 90s when it became clear that the microlocal analysis gave a more refined control over the singularities of distributions and allowed to overcome these issues in a more systematic way [395][396][397]. The works [398][399][400][401][402][403][404][405][406][407] studied the microlocal aspects, in particular: Formulated the microlocal spectral condition, developed a proper (local and covariant) notion of Wick polynomials, including constructions of the covariantly conserved stress tensor, and reduced the renormalization ambiguity to that generated by local gravity counterterms. The gravity counterterms (and more general background counterterms) are generic in the discussion of QFT on classical backgrounds, they lead to fundamental ambiguities and regularization scheme-dependencies that will be mentioned later. QFT in curved space has been a subject of books, monographs and reviews [394,[408][409][410][411][412][413][414][415], in particular see a recent accessible introduction [416] (and the follow-up [417]). In this context one usually talks about globally hyperbolic spacetimes 15 (see [418] for a departure from this condition). The AQFT axioms on globally hyperbolic spacetimes were formulated in [400,419,420], see also reviews [414,[421][422][423][424][425], in particular [422,423] for the history briefly outlined above, and the review [424] and the collection [414] for more technical details. These axioms are often referred to as locally covariant quantum field theory (LCQFT). They are similar to the Haag-Kastler axioms, yet have important differences. Again, there is a net of C * -algebras, but not just on a single globally hyperbolic M and its causal globally hyperbolic subsets; instead, it is defined on all globally-hyperbolic d-manifolds simultaneously, with a natural local covariance property with respect to isometric embeddings. 16 Clearly, the Poincare covariance is dropped, and other conditions present in the Haag-Kastler system are replaced by their locally covariant analogs. In [322,429,430] superselection sectors and the spin-statistics on curved spaces were considered, for further aspects of the theory see: [431][432][433][434][435][436][437][438][439]. An alternative approach to AQFT formalizing the OPE on curved spacetime is presented in [440]. Constructions of concrete interacting models proceed via the perturbation theory and renormalization, see the original papers and the reviews [284,334,335,363,399,401,405,406,441]. Note also a construction of quantum Yang-Mills (YM) as the perturbative AQFT in [442], see also [366]. Other references on gauge theories include [443][444][445][446][447][448]. Dynamical C * algebras. A novel C * -algebraic approach to QFT is being developed in the recent series of papers [449][450][451][452][453][454][455][456]. It is based on the Lagrangian formulation of field theory, and could probably be called constructive AQFT. Indeed, given a Lagrangian L, this approach produces a concrete C * -algebra A L called the dynamical C * -algebra in this context. The output obeys the 15 A pseudo-Riemannian spacetime (of Lorentzian signature) is globally hyperbolic if it has no closed causal curves, and for any two points, the intersection of the causal past of one with the causal future of the other is compact. 16 Further readings on the principle of "same physics in all spacetimes" (SPASs) include [425][426][427][428]. Haag-Kastler axioms and at the same time incorporates ideas from perturbative QFT. Homotopical AQFT. Perturbative gauge theories live in the topologically trivial sector. Inclusion of the topological effects like instantons, however, breaks some the axioms of LCQFT: The isotony is violated, as well as the ability to reconstruct global algebras from the local ones. 17 This fact is explained, for example, in the talks [457][458][459], see also [445,460]. One way to address this problem replaces the space of gauge orbits (configuration space) by a stack given by the corresponding gauge groupoid (a category, whose objects are the bundle-connection pairs and whose morphisms are gauge equivalences). Correspondingly, the "quantized algebra of functions on fields" typical to the usual approach is now replaced by some appropriate homotopy 18 dg-algebra. In such a generalized approach (called by practitioners the homotopical LCQFT ) one obtains, instead of locally covariant nets of C * -algebras, their homotopic dg-versions. Such structures are currently under investigation, see reviews [461,462], the monograph [463] and the papers [464][465][466][467][468][469][470][471][472][473][474][475][476] in which the subject is being developed (see also [477]). Haag duality and DHR. The global symmetries and their role in AQFT (in particular, superselection sectors) were studied by Doplicher, Haag, and Roberts (DHR) [315,316,478,479]. To include gauge theories, a modification of the local QFT rules was proposed [259,480,481], suggesting to consider, in addition to the bounded regions in Minkowski space, infinite cones. Another approach is being developed in [482][483][484], where the violation of Haag duality A(U ) = A(U ) [255,485] is at the heart of issue (these authors also consider generalized symmetries and associated extended operators that are responsible for the breakdown of Haag duality, see also [320,486]). Factorization algebras and Euclidean perturbative AQFT. Another approach to Euclidean perturbative QFT, which spiritually fits into the AQFT philosophy, is that of factorization algebras (FA). The notion of FA goes back to [487,488]. The perturbative renormalization in QFT and formulation via factorization algebras was developed in [489][490][491] (see also [492]). The idea of FA looks superficially similar to the nets of algebras in AQFT, and indeed [493] made comparison between the FA approach and the perturbative AQFT, concluding that the two are closely related. At least for free theories, they show them to be equivalent. In a later paper [494] the same authors relate observables in the perturbative AQFT and the FA. A general result of [495] (where FA are considered on Lorentzian oriented time-oriented globally hyperbolic spaces) abstractly shows their equivalence, modulo natural hypotheses. Therefore, the likely status of the FA approach is that it provides an alternative viewpoint, and technically quite a different approach to constructing concrete models. Some papers that use this framework include [496][497][498][499][500][501][502][503][504][505][506][507][508][509][510][511], also [512][513][514][515]. It is also suggested by [516] that this approach has close relations to [517]. Atiah-Segal-like approach, or Functorial Field Theory Following the Atiyah-Segal's axiomatization of TQFT [518] (and its many successes, e.g., classification of fully extended TQFTs [488]), as well as earlier ideas from the work on CFT [519,520], G.Segal, in a series of lectures [521], proposed another set of axioms that are supposed to define a general Euclidean QFT. Similar axioms have been used by Stolz and Teichner [522,523], and, apparently, were also considered by Kontsevich (unpublished). These are sometimes referred to as Functorial Field Theory (FFT) [524], though the name is slightly abused, since locally covariant AQFTs discussed earlier are also defined as functors between the appropriate categories (of globally hyperbolic spaces and C * -algebras). We will nevertheless use the term FFT here for concreteness, but we should mention that some authors [525][526][527] call it geometric field theory because it depends on some geometric data on the spacetime, such as the metric. These latter authors seem to have seriously undertaken the task of developing the geometric FFT ideas, and claim to have a definition and even classification (as in the cobordism hypothesis of [488]) of the fully extended geometric FFTs [526,527]. Recently, the FFT framework was used by Segal and Kontsevich [528], where the definition of QFT on Riemannian manifolds was extended to "allowable" complex metrics (a notion serving as a bridge between the Euclidean and the Lorentzian cases). In general, the FFT philosophy for non-topological QFTs has been gaining momentum in the past decade, 19 even though the number of papers devoted to non-topological FFTs is still relatively small. 20 We should note that an approach to FFTs on CW-complexes (serving as a discretization of spacetime) was developed in [530][531][532][533]. The main idea of FFT is that the field theory is a functor from the category of geometric bordisms (i.e., decorated by some geometric structure) to the category of topological vector spaces (see [528]). The functoriality here encodes the gluing axiom that follows from the locality and means that spacetime can be glued from pieces, and these pieces talk to each other only through the boundaries. Namely, FFT on each piece produces a state (co)vector in the tensor product of vector spaces assigned to its boundary components, and gluing (at least in the absence of corners) is dove via composing vectors and covectors. In essence, this is the very same Markov property of Euclidean path integrals that was noticed by Nelson in the 70s [69], as we discussed earlier (see also [534]). The relation between FFT and AQFT and how the former implies the latter is proposed in [535]. See also a discussion of physics and formal properties of the gluing axiom in [536]. Conformal Field Theory (CFT) CFTs form a special subclass of QFTs as they correspond to fixed points of the RG flows. Due to the constraints of conformal symmetry, the operator product expansion (OPE) of their local observables becomes much more concrete and tangible than in generic QFTs. Axiomatically, we could start with any of the approaches reviewed above and specialize them to CFTs. Wightman's and Osterwalder-Schrader axioms in the presence of conformal invariance (really, scale-invariance is enough) are supplied with the OPE relations under the correlators. Historically, this has been the most popular approach to CFT, see recent works [537,538] reviewing, among other things, Euclidean CFT axioms and their relation to (W) and (OS) axioms. Functorial QFT in the presence of conformal symmetry in two dimensions leads to the definition of conformal field theory by Segal [519,520], which actually predates the FQFT axioms. Finally, AQFT axioms supplied by conformal invariance in 2D lead to the notion of conformal nets [539][540][541][542][543][544][545][546] (they are known to be related to Vertex Operator Algebras (VOAs), see [546][547][548][549], the last two of which also mention the relation to FQFT), see also series of works [550][551][552][553][554][555], [556] and [557]. Most results on CFTs appear in the physics literature, but they are often mathematically rigorous (or there are no conceptual obstacles to making them rigorous). The CFTs are usually characterized by the spectrum of local observables and their OPE data, and in two-dimensional spacetime, the enhanced Virasoro symmetry often affords exact solutions [558], connecting to the theory of Vertex Operator Algebras (VOAs). Higher-dimensional CFTs are a subject of an active subfield reviewed in a separate Snowmass paper [559], and there is another one reviewing some aspects of the VOAs [560]. Here, we only briefly scratched the surface of the subject, mostly because the CFT literature is really vast and cannot be given any justice in this review. Discussion As is apparent from this review and an inevitably incomplete yet huge list of references, the amount of intellectual resources invested into understanding QFT is enormous. Despite that, it is also clear that we are still lacking a single satisfactory unifying viewpoint on the subject. To some extent, the LCQFT axioms of Brunetti-Fredenhagen-Verch-Fewster and the FFT axioms of the last section present the most general and advanced attempts to axiomatize QFT, but even the oldest Wightman axioms still play role in the modern literature (see, for example, [537,538]). There are, however, some obvious issues with these axioms: • The fact that LCQFT faces difficulties in gauge theories and has to be replaced by homotopic AQFT teaches us something. Over the past decades we have learned about dualities in field theories, and understood that "being a gauge theory" is not an intrinsic property of a QFT but merely a construction. Indeed, there are known cases when a gauge theory admits a dual non-gauge formulation. Therefore, a model-independent formalism such as AQFT should not treat gauge theories separately. In fact, topological effects occur not only in gauge theories. This suggests that perhaps homotopic AQFT is the right arena for general AQFT machinery, not only gauge theories (it goes in line with the derived mathematics playing more and more role in physics, starting, perhaps, with the Batalin-Vilkovisky formalism). On the other hand, some progress on the issue of gauge theories is being made in [482,483]. • One can ask a few obvious questions about the FFT approach. It implants the notion of spaces of states, essentially, into the axioms, while the AQFT paradigm emphasizes that the Hilbert space of states is a secondary object not part of the axioms. Furthermore, in case the spatial slices are non-compact, as was emphasized recently in [416], the Hilbert space does not even have to exist. Of course one may overcome this in FFT by only allowing compact spatial slices, but the situation seems a bit uncomfortable. • Another issue that does not seem to be addressed in the FFT framework are ambiguities. As we mentioned in the text, QFTs on generic backgrounds have ambiguities due to the background counterterms that render partition functions regularization-dependent. In some cases, in the presence of extra symmetries, such ambiguities make partition functions valued in bundles, like the S 2 partition function in 2d (2, 2) SCFTs, which is valued in the Kähler bundle [561] over the moduli space. In more generic cases (like 2d theories with (1, 1) SUSY or less), such an interpretation is lost and the partition function appears completely ambiguous. Thus it might be too naive to assume that the FFT functor always produces a unique answer, in particular always assigns a complex number to a closed spacetime. However, this might be just a normalization issue. • Currently available axiomatic approaches to non-topological QFT do not take extended operators and defects very seriously. That is not say it is impossible: One can include extended operators in the nets of local observables, and it is possible to include defects by modifying the algebras assigned to regions that intersect the defect. 21 One can also incorporate all sorts of (extended or not) observables in the FFT formalism by excising a tubular neighborhood of the observable and assigning the corresponding state to the boundary. However, these issues do not appear to be particularly explored. Even less understood and more mysterious is the case of corners (i.e., extending the theory to codimension ≥ 2 in the non-topological case). Additionally, we should note that there exists a philosophy typical to the condensed matter literature that QFT describes small perturbations around a critical point of some lattice, many-body or other finite system. We did not include this in the main text as it does not provide a system 21 We thank O. Gwilliam for a discussion on this point. of axioms. It is not very clear how to relate such a philosophy to any of the axiomatic approaches we have, especially to the AQFT. For example, a lattice system usually comes with a well-defined unique Hilbert space, while the QFT that should emerge from it must, somehow, lose this property. These are of course old questions, some of them have been partially answered in the Constructive Field Theory program for concrete models. We also note a recent increased interest in the lattice approach to QFTs, in particular papers [562][563][564], where a specially designed continuum limit is supposed to address the above questions. Besides the issues mentioned before, a number of QFTs studied in the modern literature do not fit into any of the axiom systems currently available. QFT was originally introduced to marry quantum mechanics with special relativity, but today we know that this was more of a historic accident. For instance, QFTs exist outside the Lorentz-invariant setting, examples including Lifshitz field theories (see a review [565] and references therein), Horava gravity [566] (which, however, is a gravitational theory,) and many others. Surely, when placed on curved spaces, such theories are expected to obey some modified version of local covariance, if any at all. Hence they do not fit into any of the axiom systems described above. More generally, it has been recently appreciated in the hep-th community [567] that our understanding of QFT is incomplete. The standard techniques are very limited, and a number of physically acceptable theories do not fit the old profile of QFT. Such theories include field theories on non-commutative spaces, little string theories, and various exotic theories such as those from [568][569][570][571] and references therein. Combining everything we said, there is a clear problem: we do not have the general definition of QFT. While it is not known how to generalize the notion of QFT yet, one idea is worth mentioning. In [572] A.Losev and S.Hu made a bold proposal that one should modify the geometry on which the QFT is defined. Instead of working on ordinary manifolds, Riemannian or Lorentzian, one should consider a certain generalization that captures the algebraic operations that are used in constructing QFTs. The authors of [572] coined the name "Feynmann geometry," and suggested that it should be described by an A ∞ -algebra with trace-class operations (such a definition covers many UV regulators: momentum cut-offs, lattices, non-commutativity). In this respect, one should also mention the work of Kontsevich-Soibelman on the A ∞ approach to non-commutative geometry [573], which perhaps can be of use. It is also possible that the correct notion of "Feynman geometry" should be even more general to cover all the instances of exotic QFTs, if this is the right approach. See[70][71][72][73][74][75] as well as references in[76] for origins of the Euclidean QFT. We did not describe the latter property in this review, but it is discussed in most of the references.18 In this context, the word "homotopy" means that various relations like commutativity or associativity hold up to higher homotopies. 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R Brunetti, M Dütsch, K Fredenhagen, K Rejzner, arXiv:2108.13336The unitary Master Ward Identity: Time slice axiom, Noether's Theorem and Anomalies. math-phR. Brunetti, M. Dütsch, K. Fredenhagen, and K. Rejzner, "The unitary Master Ward Identity: Time slice axiom, Noether's Theorem and Anomalies," arXiv:2108.13336 [math-ph]. Unitary, anomalous Master Ward Identity and its connections to the Wess-Zumino condition. R Brunetti, M Dütsch, K Fredenhagen, K Rejzner, arXiv:2210.05908BV formalism and L ∞ -algebras. math-phR. Brunetti, M. Dütsch, K. Fredenhagen, and K. Rejzner, "Unitary, anomalous Master Ward Identity and its connections to the Wess-Zumino condition, BV formalism and L ∞ -algebras," arXiv:2210.05908 [math-ph]. Locally covariant approach to effective quantum gravity. R Brunetti, K Fredenhagen, K Rejzner, arXiv:2212.07800gr-qcR. Brunetti, K. Fredenhagen, and K. Rejzner, "Locally covariant approach to effective quantum gravity," arXiv:2212.07800 [gr-qc]. 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M Benini, C Dappiaggi, A Schenkel, 10.1007/s00220-014-1917-0arXiv:1303.2515Commun. Math. Phys. 330math-phM. Benini, C. Dappiaggi, and A. Schenkel, "Quantized Abelian principal connections on Lorentzian manifolds," Commun. Math. Phys. 330 (2014) 123-152, arXiv:1303.2515 [math-ph]. Homotopy colimits and global observables in Abelian gauge theory. M Benini, A Schenkel, R J Szabo, 10.1007/s11005-015-0765-yarXiv:1503.08839Lett. Math. Phys. 1059math-phM. Benini, A. Schenkel, and R. J. Szabo, "Homotopy colimits and global observables in Abelian gauge theory," Lett. Math. Phys. 105 no. 9, (2015) 1193-1222, arXiv:1503.08839 [math-ph]. Quantum field theories on categories fibered in groupoids. M Benini, A Schenkel, 10.1007/s00220-017-2986-7arXiv:1610.06071Commun. Math. Phys. 3561math-phM. Benini and A. Schenkel, "Quantum field theories on categories fibered in groupoids," Commun. Math. Phys. 356 no. 1, (2017) 19-64, arXiv:1610.06071 [math-ph]. M Benini, A Schenkel, U Schreiber, 10.1007/s00220-018-3120-1arXiv:1704.01378The Stack of Yang-Mills Fields on Lorentzian Manifolds. 359math-phM. Benini, A. Schenkel, and U. Schreiber, "The Stack of Yang-Mills Fields on Lorentzian Manifolds," Commun. Math. Phys. 359 no. 2, (2018) 765-820, arXiv:1704.01378 [math-ph]. Operads for algebraic quantum field theory. M Benini, A Schenkel, L Woike, 10.1142/S0219199720500078arXiv:1709.08657Commun. Contemp. Math. 23022050007math-phM. Benini, A. Schenkel, and L. Woike, "Operads for algebraic quantum field theory," Commun. Contemp. Math. 23 no. 02, (2021) 2050007, arXiv:1709.08657 [math-ph]. Homotopy theory of algebraic quantum field theories. M Benini, A Schenkel, L Woike, 10.1007/s11005-018-01151-xarXiv:1805.08795Lett. Math. Phys. 1097math-phM. Benini, A. Schenkel, and L. Woike, "Homotopy theory of algebraic quantum field theories," Lett. Math. Phys. 109 no. 7, (2019) 1487-1532, arXiv:1805.08795 [math-ph]. Linear Yang-Mills Theory as a Homotopy AQFT. M Benini, S Bruinsma, A Schenkel, 10.1007/s00220-019-03640-zarXiv:1906.00999Commun. Math. Phys. 3781math-phM. Benini, S. Bruinsma, and A. Schenkel, "Linear Yang-Mills Theory as a Homotopy AQFT," Commun. Math. Phys. 378 no. 1, (2019) 185-218, arXiv:1906.00999 [math-ph]. Categorification of algebraic quantum field theories. M Benini, M Perin, A Schenkel, L Woike, 10.1007/s11005-021-01371-8arXiv:2003.13713Lett. Math. Phys. 11135math-phM. Benini, M. Perin, A. Schenkel, and L. Woike, "Categorification of algebraic quantum field theories," Lett. Math. Phys. 111 (2021) 35, arXiv:2003.13713 [math-ph]. Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories. M Benini, A Schenkel, B Vicedo, 10.1007/s00220-021-04304-7arXiv:2008.01829Commun. Math. Phys. 3893hep-thM. Benini, A. Schenkel, and B. Vicedo, "Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories," Commun. Math. Phys. 389 no. 3, (2022) 1417-1443, arXiv:2008.01829 [hep-th]. Algebraic Quantum Field Theories: a homotopical view. V Carmona, arXiv:2107.141762021math-phV. Carmona, "Algebraic Quantum Field Theories: a homotopical view," 2021. arXiv:2107.14176 [math-ph]. Relative Cauchy evolution for linear homotopy AQFTs. S Bruinsma, C J Fewster, A Schenkel, arXiv:2108.10592math-phS. Bruinsma, C. J. Fewster, and A. Schenkel, "Relative Cauchy evolution for linear homotopy AQFTs," arXiv:2108.10592 [math-ph]. A Anastopoulos, M Benini, arXiv:2201.06464Homotopy theory of net representations. math-phA. Anastopoulos and M. Benini, "Homotopy theory of net representations," arXiv:2201.06464 [math-ph]. Spacetimes categories and disjointness for algebraic quantum field theory. A Grant-Stuart, arXiv:2201.09166math-phA. Grant-Stuart, "Spacetimes categories and disjointness for algebraic quantum field theory," arXiv:2201.09166 [math-ph]. Homotopical algebraic quantum field theory. 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{'abstract': 'This article provides a review of the literature on rigorous definitions and constructions in Quantum Field Theory, spanning the period of seven decades. Comparing with the ideas and constructions found in the modern physics literature, we conclude that none of the existing systems of QFT axioms can cover all the physical situations. Therefore, it is still an outstanding open problem to formulate a complete definition of QFT. We argue that the question is of relevance for both physicists and mathematicians. arXiv:2203.08053v2 [hep-th] 10 Jan 2023 2 Existing axiomatics 2.1 Correlator-focused approaches Wightman axioms. One of the older axiom systems that remains relevant to date is that due to Wightman [20-23] (see also books [25, 28, 29]). The axioms view fields as operator-valued tempered distributions and formalize the notion of expectation values of their products ("Wightman functions"). One starts with the assumption (W0) of relativistic invariance (as in Wigner\'s classification [30] 1 , see also [31-34]): physical Hilbert space is a unitary representation of the Poincare group. This assumption is supplemented by the spectral condition (the energy-momentum spectrum lies in the closed upper light-cone) and the uniqueness and Poincare-invariance of the vacuum state Ψ 0 ∈ H. The axiom W1 states that there is a set of fields ϕ i [f ] given by tempered distributions 2 valued in the operators defined on (and preserving) a dense subset D (which includes the vacuum) of the Hilbert space H. The subset D is assumed to be Poincare-invariant. Then W2 states covariance of fields with respect to the Poincare group, and W3 requires locality (also called microcausality) in the form of (anti)commutativity of spacelike-separated fields. A quantum field theory is said to satisfy W0-W3, and in addition obey cyclicity of the vacuum: The span of vactors of the form ϕ 1 [f 1 ] . . . ϕ n [f n ]Ψ 0 (for all possible n and f i ) is dense in H. The latter condition guarantees that there are enough fields in the theory.', 'arxivid': '2203.08053', 'author': ['Mykola Dedushenko \nSimons Center for Geometry and Physics\nStony Brook University\n11794-3636Stony BrookNYUSA\n'], 'authoraffiliation': ['Simons Center for Geometry and Physics\nStony Brook University\n11794-3636Stony BrookNYUSA'], 'corpusid': 247450696, 'doi': '10.1142/s0217751x23300028', 'github_urls': [], 'n_tokens_mistral': 76243, 'n_tokens_neox': 57787, 'n_words': 26222, 'pdfsha': '7f70610031d6ff1f0cb1f7e8b95274d5db0747df', 'pdfurls': ['https://export.arxiv.org/pdf/2203.08053v2.pdf'], 'title': ['Snowmass White Paper: The Quest to Define QFT', 'Snowmass White Paper: The Quest to Define QFT'], 'venue': []}

arxiv

ANALYTIC TORSION AND R-TORSION OF WITT REPRESENTATIONS ON MANIFOLDS WITH CUSPS Pierre Albin Frédéric Rochon David Sher ANALYTIC TORSION AND R-TORSION OF WITT REPRESENTATIONS ON MANIFOLDS WITH CUSPS We establish a Cheeger-Müller theorem for orthogonal representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps. 1045119. The authors are happy to acknowledge useful conversations with or R-torsion, is a combinatorial invariant of simplicial complexes while the analytic torsion is a smooth invariant defined via the spectrum of the Hodge Laplacian. This connection is behind many applications in topology, number theory, and mathematical physics. One particularly interesting aspect of this theorem is that it allows us to use analysis to study the size of the torsion in homology. For example, in [Che79, Example 1.3] Cheeger points out that if F 0 d 0 −−→ F 1 d 1 −−→ . . . −→ F n d n − −− → 0 is a complex of free Abelian groups and K i = F i ⊗ R then the Reidemeister torsion (after some canonical choices) is given by R-torsion = |H 2k+1 (F • ) torsion | |H 2k (F • ) torsion | . A recent paper of Bergeron-Venkatesh [BV13] exploits this relationship to study the growth of torsion in group homology by studying the analytic torsion of locally symmetric spaces. This paper has initiated a lively discussion [BV13,Mül12,MM13,MP13c,MP13b,MP14,BMZ11,BSV14,MFP14], which has recently expanded to non-compact hyperbolic spaces. Analytic torsion in this context was first studied by Park [Par09] who proved that a relation discovered by Fried [Fri86] between analytic torsion and dynamical zeta functions continues to hold on noncompact hyperbolic spaces. Recently there has been an impressive sequence of papers by Müller and Pfaff [MP12,Pfa14b,MP13a,Pfa12,Pfa14a,Pfa13] in which the Selberg trace formula is used to great effect in analyzing analytic torsion. The methods in the papers cited above are closely tied to the algebraic structure of locally symmetric spaces. In [ARS14] and the present paper we use the geometric microlocal analysis methods of Melrose to extend the study of analytic torsion to a larger class of spaces devoid of this algebraic structure. Indeed, in [ARS14] we have established a Cheeger-Müller theorem on non-compact manifolds with fibered cusp ends, a class of manifolds including many locally symmetric spaces of rank one. Specifically, let N be the interior of a manifold with boundary N and assume that the boundary participates in a fiber bundle of closed manifolds Z -∂N φ − − → Y. Let x be a 'boundary defining function' for ∂N, that is, a smooth function on N that vanishes precisely at ∂N and with non-vanishing differential there. A metric g d on N is a fibered cusp metric, or d-metric, if it is asymptotically of the form g d ∼ dx 2 x 2 + x 2 g Z + φ * g Y where g Z + φ * g Y is a submersion metric on ∂M, and an 'even' d-metric if g Z and g y are functions of x 2 , see [Vai01], [ARS14,Definition 7.3] for more details. Let F −→ N be vector bundle with flat connection ∇ F and g F a compatible bundle metric smooth all the way down to ∂N. In fact we assume that g F is even, meaning that it extends smoothly to the double of N across ∂N. The analytic torsion of (N, g d , F, g F ) is defined in [ARS14,§9] following [Mel93] by means of the renormalized trace of the heat kernel, since the usual operator trace is infinite. We prove that this analytic torsion can be expressed in terms of the Reidemeister torsion of N relative to ∂N and in terms of the Reidemeister torsion of Dar [Dar87], Iτ m of the singular space N obtained from N by 'coning-off' the fibers of the fiber bundle φ. Theorem 1 ( [ARS14, Corollary 11.2]). Let (N, g d ) be an odd-dimensional Riemannian manifold with fibered cusp ends and an even d-metric. Let α : π 1 (N ) −→ GL(k, R) be a unimodular representation meaning that | det α| = 1. Endow the associated flat vector bundle F −→ M with an even bundle metric g F and assume that it is strongly acyclic in that Iτ m (C φ ∂N, α) = log τ (N , ∂N , α) + 1 2 log τ (∂N, α). where C φ ∂N is the mapping cylinder of ∂N φ − − → Y. In this paper we specialize from fibered cusps to non-fibered cusps (i.e., we assume that Y is a point), and we replace the strong acyclicity condition on α with a much weaker 'Witt condition'. We can describe exactly how we will manage this extension by recalling the proof of Theorem 1 in the case Y = pt . Let M be a smooth closed manifold obtained by doubling N across ∂N = Z, and F a flat bundle over M. We consider a family of metrics ε → g ε,hc that, in a tubular neighborhood of Z has the form g ε,hc = dx 2 x 2 + ε 2 + (x 2 + ε 2 )g Z . This can be visualized as stretching the manifold M in the direction normal to the hypersurface Z until it has two infinite cusp ends in place of the hypersurface. In fact, while for ε > 0 the metrics g ε,hc are smooth Riemannian metrics on M, as ε → 0 the metric degenerates along Z, and has two non-singular limits: a hyperbolic cusp metric on M \Z and a complete metric with cylindrical ends on the normal bundle of Z, N M Z ∼ = Z ×R. The de Rham operator ð dR = d + δ associated to g ε,hc has a model operator corresponding to each of these limiting spaces. On M \ Z, we obtain ð dR,hc , the de Rham operator of the limiting fibered hyperbolic cusp metric. The other model operator is actually on R. It is the de Rham operator ð dR,b of a metric with cylindrical ends, but twisted by the 'vertical cohomology bundle', H * (Z; F ) −→ R. In [ARS14] we carried out a careful analysis of the spectrum of ð dR,hc as ε → 0 by describing the precise asymptotics of the Schwartz kernels of the resolvent and heat kernel. In particular we proved that there are finitely many eigenvalues of ð dR,hc that converge to zero as ε → 0. We call these the 'small eigenvalues' and denote the product of the non-zero small eigenvalues by det(ð dR ) Bsm . Theorem 2 ( [ARS14, Theorem 10.2]). If log det(ð dR ) Bsm is polyhomogeneous in ε, the metric g ε,hc is of 'product-type' and the flat bundle is Witt in that H dim Z/2 (Z; F ) = 0 then the determinant of the Laplacian satisfies (1) FP ε=0 log det ð 2 dR,hc = log det ð 2 dR,hc + log det ð 2 dR,b − FP ε=0 log det(ð 2 dR ) Bsm . The acyclicity condition we imposed in the fibered cusp setting implies that the last two terms in (1) do not contribute to the analytic torsion. To remove the acyclicity condition and compute the limit of analytic torsion as ε → 0 we will establish that log det(ð dR ) Bsm is polyhomogeneous in ε and compute its finite part as ε → 0 (Corollary 3.4) and we will compute the determinants of the model operators ð dR,b ( §2.2 especially (2.15)). Another consequence of removing the acyclicity condition is that the analytic torsion should be thought of not as a number, but as a function that assigns a number to each basis of the cohomology H * (M ; F ), so we will also compute the behavior of a basis of harmonic forms as ε → 0 ((3.30)). These pieces together determine the limit of analytic torsion as ε → 0. In Theorem 5.4 we determine how the Reidemeister torsion of M is related to the Reidemeister torsion of N. All together we establish the following theorem in Corollary 6.2. 2 −1 q=0 (−1) q dim H q (Z; F ) [(m − 1 − 2q) log(m − 1 − 2q)] where CZ is the cone over Z. In the literature cited above, a theorem close to ours is an interesting Cheeger-Müller theorem due to Pfaff [Pfa13]. This theorem applies to noncompact hyperbolic manifolds with cusps N of odd dimension m and flat vector bundles F induced by the irreducible representations of SO 0 (m, 1) or Spin(m, 1) that are not invariant with respect to the Cartan involution. Pfaff uses constructions of Harder [Har75] to define a canonical Reidemeister torsion τ Eis (N ; F ) (a similar construction is used by Calegari and Venkatesh [CV12] in three dimensions). Let C be a neighborhood of the cusps Pfaff uses the renormalized trace of Melrose to define analytic torsion and is then able to compute the difference log τ Eis (N ; F ) − log AT (N ; F ) AT (C, ∂C; F ) in terms of the rank of F, the Betti numbers and volume of ∂C and some weights associated to the holonomy representation of F. Notice that in this setting, the Witt condition is never satisfied. A very interesting preprint, using different methods, has recently been posted by Vertman [Ver14] that includes a Cheeger-Müller theorem for flat unitary bundles over three dimensional manifolds with product-type cusps satisfying the Witt condition. For hyperbolic surfaces cusp formation corresponds to converging to the boundary of Teichmüller space and so has been the subject of much study. For example Seeley and Singer [SS88] studied the ∂ operator as a cusp is formed. In Propositions 7.1 and 7.2 we recover results of Wolpert and Burger [Wol87,Wol90,Wol10,Bur88] on the asymptotics of small eigenvalues and the blow-up of the determinant. This paper is organized as follows: Section 1 recalls our conventions for cusp metrics and analytic torsion. After this section we work in the context of a closed manifold M with hypersurface Z along which the metric is degenerating to form cusp ends. In §2 we analyze the model operator D b on R and compute its contribution to the asymptotics of analytic torsion. Then section 3 is devoted to the study of the small eigenvalues including their polyhomogeneity in ε and culminating in the computation of the corresponding determinant. This section also includes an analysis of the asymptotics of an appropriately chosen basis of harmonic forms. These results are collected in §4 and yield the asymptotics of analytic torsion along degeneration to a manifold with cusp ends. Section 5 contains our study of Reidemeister torsion, particularly of how the R-torsion of the closed manifold M relates to the R-torsion of N, the manifold with cusp ends. Then in §6 we combine this study with our analysis of analytic torsion to obtain our Cheeger-Müller theorem. In the final section, §7, we specialize to dimension two and explain the relevance of our results to families of hyperbolic metrics approaching the boundary of Teichmüller space. Cusp metrics and analytic torsion In this section we recall the definition of cusp metrics and a very useful replacement for the tangent bundle that is adapted to the geometry. We also recall the definition of analytic torsion on closed manifolds and manifolds with asymptotic cusps. 1.1. Analytic torsion. On a closed Riemannian manifold (M, g) of dimension m, the heat kernel of any Laplace-type operator satisfies Tr(e −t∆ ) ∼ t −m/2 k≥0 a k t k as t → 0, Tr(e −t∆ ) − dim ker ∆ = O(e −tλ 1 ) as t → ∞ with λ 1 > 0. Hence its zeta function ζ(s) = ζ(s; ∆) = 1 Γ(s) ∞ 0 t s Tr(e −t∆ − P ker ∆ ) dt t extends from a holomorphic function on Re s > m/2 to a meromorphic function on all of C which has at worst simple poles and is regular at the origin. If F −→ M is a flat vector bundle with flat metric, and ∆ q is the Hodge Laplacian on F -valued differential forms of degree q, then LAT(M, g; F ) = 1 2 q (−1) q qζ (0; ∆ q ) is the logarithm of the analytic torsion of (M, g; F ). If F is acyclic, that is if H * (M ; F ) = 0, then the analytic torsion is independent of the choice of Riemannian metric. Otherwise, we choose a basis {µ q j } of each H q (M ; F ) and let ω ε be an orthonormal basis of harmonic representatives with respect to the metric g ε,hc ; then we define LAT(M, {µ q j }, F ) = LAT(M, g ε,d ; F ) − log Π n q=0 [µ q |ω q ε ] (−1) q , where [µ q |ω q ε ] = | det W q | with W q the matrix such that µ q i = j W q ij ω q j . It is this quantity that is independent of the choice of metric. 1.2. Cusp metrics. Let L be a smooth manifold with boundary Z. Let x be a smooth, non-negative function on L that vanishes precisely on Z and such that dx does not vanish anywhere on Z. We call such a function a 'boundary defining function' for Z, or 'bdf' for short. We fix a choice of bdf, and our constructions will depend (mildly) on this choice. Let us single out a subset of the vector fields on L, V φ (L) = V ∈ C ∞ (L; T L) : V is tangent to Z, and V x ∈ O(x 2 ) and point out that there is a vector bundle over L whose space of sections is V φ (L). We denote this bundle φ T L −→ L and refer to it as the 'φ-tangent bundle' of L. (The φ more generally denotes a fibration on the boundary of L; in our present context the fibration is Z -Z −→ pt .) The φ-tangent bundle is isomorphic to the usual tangent bundle of L. This isomorphism is canonical over L • , but not over all of L. The dual bundle φ T * L −→ L is called the 'φ-cotangent bundle' of L. Note that dx x 2 is a section of φ T * L that is nondegenerate at Z = {x = 0}. We can use x to rescale the φ-tangent bundle at Z (see [Mel93,Chapter 8]), and we refer to the bundle hc T L = 1 x φ T L as the hc-tangent bundle or 'hyperbolic cusp tangent bundle'. Its dual bundle hc T * L −→ L is the hc-cotangent bundle of L, and we point out that the one form dx x , as a section of hc T * L, is non-degenerate at Z. Similarly if z is a local coordinate on Z then xdz, as a local section of hc T * L, is non-vanishing at x = 0. An hc-metric is a bundle metric on the hc-tangent bundle. The simplest hc-metrics are those that in some collar neighborhood of Z of the form [0, 1] x × Z take the form g hc,pt = dx 2 x 2 + x 2 g Z with g Z a metric on Z independent of x. We refer to such metrics as product-type hcmetrics. An hc-metric g hc is product-type to order if there is a product-type metric g hc,pt such that g hc − g hc,pt ∈ x C ∞ (L; S 2 ( hc T * L)) where S 2 ( hc T * L) denotes the bundle of symmetric bilinear forms on hc T * L. In this paper our results will hold for hc-metrics that are product-type to order 2. The heat kernel of a Laplace-type operator associated to an hc-metric is not as wellbehaved as the corresponding object on a closed manifold ( [Vai01], [ARS14,§7]). First, the heat kernel is possibly not trace class. Fortunately it is well-behaved enough that we can make sense of its renormalized trace R Tr e −t∆ = FP z=0 Tr(x z e −t∆ ). Moreover, from [ARS14, § 7] and the appendix of [AR13], the asymptotics of the renormalized trace of the heat kernel are more complicated as t → 0 : R Tr e −t∆ ∼ t −m/2 k≥0 a k/2 t k/2 + t −1/2 k≥0 b k/2 t k/2 log t. And finally, one does not always have exponential converge of R Tr e −t∆ to dim ker ∆ as t → ∞. We will deal with these differences by adding appropriate additional assumptions. Let us say that a flat bundle F is Witt if, upon restricting to Z, we have H v/2 (Z; F ) = 0 where v = dim Z = m − 1. If ∆ is a Hodge Laplacian associated to a Witt bundle, then we know from [ARS14] that R Tr e −t∆ − dim ker ∆ = O(e −tλ 1 ) as t → ∞ for some λ 1 > 0. If g hc is product-type to order two and m is odd, then a m/2 = b 1/2 = 0. Again, if ∆ is a Hodge Laplacian associated to a Witt bundle, the zeta function ζ(s; ∆) = 1 Γ(s) ∞ 0 t s R Tr e −t∆ − P ker ∆ dt t is a holomorphic function on Re s > m/2 that extends to a meromorphic function on C, with at worst double poles but regular at the origin. Thus for flat Witt bundles we may define analytic torsion for a cusp manifold just as for a closed manifold. 1.3. Cusp degeneration. We say that a closed Riemannian manifold (M, g) with a twosided hypersurface Z is undergoing cusp degeneration if the metric is degenerating from a smooth metric to a cusp metric on M \ Z. We will carry out these degenerations in a controlled fashion by studying 'cusp surgery metrics'. Let us start by performing the 'radial blow-up' of Z × {0} in M × [0, 1] ε . Recall that this is a smooth manifold with corners, X s = [M × [0, 1] ε , Z × {0}], obtained by replacing Z ×{0} with its inward pointing spherical normal bundle (see [Mel93]). Figure 1 represents the space X s . There is a natural map, known as the blown-down map, The manifold X s has three boundary hypersurfaces. One, β −1 ({ε = 1}), will not be relevant to our studies and will be cheerfully ignored. The other two are β −1 (Z × {0}), known as the surgery boundary and denoted B sb , and B sm = β −1 (M × {0} \ Z × {0}), where the m in the subscript recalls that this is where most of M × {0} ended up. Given any blow-down map, the 'interior lift of a set' is equal to the closure of the lift of that set minus the set being blown-up; thus B sm is the interior lift of M × {0}, which we denote B sm = β (M × {0}). There is a natural choice of boundary defining function for B sb , which we fix once and for all: ρ sb = √ x 2 + ε 2 . When there is no possibility of confusion, we will denote this simply as ρ. The interior of B sb can be identified with the normal bundle to Z in M ; B sb corresponds to its fiberwise compactification. The normal bundle to Z is trivial by assumption, and so we have B sb ∼ = Z × [−π/2, π/2]. (Of course any closed interval would serve, but our usual choice of coordinates will correspond to [−π/2, π/2], so we use this interval throughout.) We endow B sb with a trivial fibration Z -B sb φ + − −− → [−π/2, π/2]. Analogously to the hc-tangent bundle, we will define a 'cusp surgery tangent bundle' or ε, hc-bundle. First let π ε : X s −→ [0, 1] ε be the composition of β with the obvious projection and define ε T X s = ker π ε * ⊆ T X s . Next let V ε,φ = {V ∈ C ∞ (X s ; ε T X s ) : V B sb tangent to fibers of φ + and V ρ ∈ O(ρ 2 )} and define ε,φ T X s so that V ε,φ is its space of sections. Finally, let ε,hc T X s = 1 ρ ε,φ T X s , and let ε,hc T * X s denote the dual bundle. The one-forms dx ρ , ρ dz, where z denotes a coordinate along Z, lift from the interior of X s to a spanning set of sections of ε,hc T * X s . Note that, as sections of ε,hc T * X s , these do not degenerate at Z. A cusp surgery metric is a bundle metric on ε,hc T X s . We say that an ε, hc-metric is of product type if there is a tubular neighborhood Tub(Z) ∼ = [−1, 1] x × Z ⊆ M around Z in which the metric takes the form g ε,hc,pt = dx 2 x 2 + ε 2 + (x 2 + ε 2 )g Z where g Z is a metric on Z that is independent of both x and ε. We say that an ε, hc-metric g ε,hc is of product type to order if g ε,hc − g ε,hc,pt ∈ ρ C ∞ (X s ; S 2 ( ε,hc T * X s )) for some product type metric g ε,hc,pt , where S 2 ( ε,hc T * X s ) denotes the bundle of symmetric two-tensors on ε,hc T * X s . Let F −→ X s be a flat Euclidean vector bundle and let ð dR = d + δ be the corresponding de Rham operator. We will consider this as an operator on the bundle E = Λ * ε,hc T * X s ⊗ F. One of the advantages to using the ε, hc-cotangent bundle, as opposed to the usual cotangent bundle of X s , is that the leading order behavior of ð dR will be described by tractable model operators, discussed below. We are interested in the action of ð dR as an unbounded operator on L ε,hc (M ; E), the natural L 2 space associated to an ε, hc-metric g ε,hc and the bundle metric on F. However, for some constructions it will be easier to work with (1.1) L 2 ε,b (M ; E) = ρ v/2 L 2 ε,hc (M ; E), where v = dim Z = m − 1. Thus our main object of interest is the operator D dR = ρ v/2 ð dR ρ −v/2 acting as an unbounded operator on L 2 ε,b (M ; E). If g ε,hc is of product-type to order two, then we have simple expressions for the model operators of D dR . First let us write Λ ε,hc T * Tub(Z) ∼ = ρ Λ T * Z ⊕ dx ρ ∧ ρ −1 Λ −1 T * Z ; this splitting distinguishes between forms with a dx and forms without a dx. With respect to this splitting, a direct computation tells us that D dR is given by (1.2) D dR = 1 ρ ð Z dR −ρ∂ x + (N Z − 1 2 v) x ρ ρ∂ x + (N Z − 1 2 v) x ρ − 1 ρ ð Z dR , near B sb up to higher order terms in ρ as ε, hc differential operators. Here N Z is the number operator on Z that multiplies a differential form by its degree. The first model operator, known as the vertical operator, is (1.3) D v = ρD dR B sb = ð Z dR 0 0 −ð Z dR . Its null space forms a vector bundle over B sb which is just the space of scaled harmonic forms on Z, thought of as a trivial vector bundle over [−π/2, π/2] and then pulled-back along φ + . We will denote this bundle by ρ N H * (Z; F ) −→ B sb . The second model operator, known as the horizontal operator, is defined by D b u = Π h D b u, where Π h denotes the projection onto Z-harmonic forms, u is a section of ker D v and u is any choice of extension off B sb . In terms of (1.2), the operator D b is given by (1.4) D b u = Π h 0 −ρ∂ x + (N Z − 1 2 v) x ρ ρ∂ x + (N Z − 1 2 v) x ρ 0 u, where Π h denotes the projection onto Z-harmonic forms. In projective coordinates near B sb , X = x ε , z, ε, in which B sb = {ε = 0}, we let X = √ 1 + X 2 so that D b = 0 − X ∂ X + (N Z − 1 2 v) X X X ∂ X + (N Z − 1 2 v) X X 0 , as an operator acting on C ∞ (R X ; ρ N Z H * (Z; F ) ⊕ dX X ∧ ρ N Z H * (Z; F )), where the restriction of ρ k H k (Z; F ) to B sb is well defined as a section of Λ k ( ε,hc T * X s ). Thus, D b is a b-operator in the sense of Melrose [Mel93]. If is the twisted de Rham operator corresponding to the hc-metric g 0 = g ε,hc Bsm and the flat bundle F Bsm . Analysis of the model operator Let M be a closed manifold of odd dimension m, Z a two-sided hypersurface with fixed boundary defining function x and g ε,hc a cusp surgery metric of product-type to second order. If F −→ X s is a flat vector bundle of Witt type, then we have seen that there is a model b-operator D b = 0 − X ∂ X + (N Z − 1 2 v) X X X ∂ X + (N Z − 1 2 v) X X 0 , X = √ 1 + X 2 acting on C ∞ (R X ; ρ N H * (Z; F ) ⊕ dX X ∧ ρ N H * (Z; F )) . In this section we study this operator and its contribution to the asymptotics of analytic torsion. 2.1. Null space of the horizontal operator. First let us compute its null space, and that of its square. Note that if 0 − X ∂ X + (N Z − 1 2 v) X X X ∂ X + (N Z − 1 2 v) X X 0 f (X) h(X) = 0 0 then the projections of f and h onto the spaces of forms of fixed vertical degree k (that is, having degree k in Z), which we denote f k and h k respectively, are also in the null space of D b . Now since P (a) = X ∂ X + a X X = X −a ( X ∂ X ) X a we see that X −(k−v/2) ( X ∂ X ( X (k−v/2) f k (X))) = 0 =⇒ f k (X) = C X v/2−k and that − X (k−v/2) ( X ∂ X ( X −(k−v/2) h k (X))) = 0 =⇒ h k (X) = C X k−v/2 . Thus we have found ker D b = span u X v/2−k v X k−v/2 : u, v ∈ ρ k H k (Z; F ), k ∈ N 0 . We are interested in D b as an unbounded operator on L 2 b , i.e., with the measure dX X on R. With respect to this measure, X a is in L 2 iff a < 0, and hence the L 2 kernel of D b is (2.1) ker L 2 D b = span u X v/2−k v X k−v/2 : u ∈ ρ k H k (Z; F ), k > v/2, v ∈ ρ k H k (Z; F ), k < v/2 . Next consider D 2 b = −P ( 1 2 v − N Z )P (N Z − 1 2 v) 0 0 −P (N Z − 1 2 v)P ( 1 2 v − N Z ) and note that, for j ∈ C ∞ (R), P (−a)P (a)j = 0 =⇒ P (a)j = C X a =⇒ j = C a (X) + C X −a , with a (X) = X −a X 0 s 2a−1 ds. Notice that as X → ±∞, | a (X)| is of order |X| |a| for a = 0, while for a = 0, 0 (X) = sinh −1 (X), so a is never in L 2 with respect to the density dX X . Hence the null space of D 2 b acting on smooth sections of X N H * (Z) is (2.2) ker D 2 b = span u X v/2−k + u k−v/2 (X) v X k−v/2 + v v/2−k (X) : u, u , v, v ∈ X k H k (Z; F ), k ∈ N 0 . Since the function a (X) is never in L 2 b , ker L 2 D 2 b = ker L 2 D b , which could also have been deduced from the formal self-adjointness of D b . Let us emphasize in particular that D 2 b has no L 2 -kernel on forms of total degree (i.e., degree in dX X plus degree in Z) equal to zero or m. Analytic torsion contribution of the horizontal operator. Let (D 2 b ) j,k = D 2 b Λ j R X ∧ X k H k (Z;F ) , j ∈ {0, 1}, k ∈ {0, . . . , v} where v = dim Z = m − 1. Each of these is a Laplace-type operator on R X and we denote the corresponding zeta function by ζ j,k (s). From [ARS14, Theorem 8.1], we know that the contribution of the horizontal operator D b to the asymptotics of analytic torsion is through (2.3) 1 2 (−1) j+k (j + k)ζ j,k (0). In this subsection we will compute this contribution. From the previous subsection we see that the heat kernel of D b satisfies e −tD 2 b ρ q H q (Z;F )⊕ dX X ∧ρ q−1 H q−1 (Z;F ) = e tP ( v−2q 2 )P ( 2q−v 2 ) 0 0 e tP ( 2(q−1)−v 2 )P ( v−2(q−1) 2 ) . We introduce the abbreviation F R a = R Tr(e tP (a)P (−a) ) and note that R Tr(e −tD 2 b ) ρ q H q (Z;F )⊕ dX X ∧ρ q−1 H q−1 (Z;F ) = b q F R (v−2q)/2 + b q−1 F R (2(q−1)−v)/2 , where b q = dim H q (Z; F ). So we can write (2.4) (−1) j+k (j + k) R Tr e −t(D 2 b ) j,k = q (−1) q q b q F R (v−2q)/2 + b q−1 F R (2(q−1)−v)/2 . Let us use Poincaré duality on Z and the corresponding identification of H q (Z; F ) with H v−q (Z; F ) to group together all of the terms acting on vertical harmonic forms of degree q and v − q, (2.5) (−1) q b q qF R (v−2q)/2 − (q + 1)F R (2q−v)/2 + (−1) v (v − q)F R (v−2(v−q))/2 − (−1) v (v − q + 1)F R (2(v−q)−v)/2 = (−1) q b q (q − (−1) v (v − q + 1))F R (v−2q)/2 + (−q − 1 + (−1) v (v − q)))F R (2q−v)/2 . Summing the expression (2.5) over q < v 2 yields (2.4), so overall, (2.3) is equal to (2.6) 1 2 v q=0 (−1) q b q (q − (−1) v (v − q + 1)) − log det −P ( v−2q 2 )P ( 2q−v 2 ) +((−q − 1 + (−1) v (v − q))) − log det −P ( 2q−v 2 )P ( v−2q 2 ) . It thus suffices to compute the determinant of −P (−a)P (a) on R (endowed with the metric dX 2 / X 2 and bdf ρ = X −1 ), det (−P (−a)P (a)) = e −ζ −P (−a)P (a) (0) . Our strategy will be to compute the variation in a of the renormalized trace (see (2.10)) and use this to compute the determinant (see (2.13)). Once we have computed these onedimensional determinants, we return to (2.6) in (2.14). Let us start with the two cases we can compute directly. Lemma 2.1. When a = 0, we have e tP (0) 2 (X, X ) = 1 √ 4πt exp − | sinh −1 (X) − sinh −1 (X )| 2 4t , R Tr e tP (0) 2 = log 2 √ πt , ζ −P (0) 2 (s) = 0, log det −P (0) 2 = 0. When a = −1, we have e tP (1)P (−1) (X, X ) = e −t e tP (0) 2 (X, X ), R Tr e tP (1)P (−1) = e −t log 2 √ πt , ∞ 0 t s R Tr e tP (1)P (−1) dt t = Γ(s − 1/2) log 2 √ π , log det (−P (1)P (−1)) = 2 log 2. Proof. In the coordinate u = sinh −1 (X), P (a) = ∂ u + a tanh u, −P (−a)P (a) = −∂ 2 u + a 2 − (a 2 + a)sech 2 u. Hence −P (0) 2 is the Euclidean Laplacian and −P (1)P (−1) is the Euclidean Laplacian plus one. In the former case the restriction to the diagonal is (4πt) −1/2 and in the latter e −t (4πt) −1/2 ; as these are independent of u, the renormalized trace is computed by multiplying these expressions by the renormalized volume. The renormalized volume is, for our choice of measure dX X and of boundary defining function = X −1 , R R dX X = 2 R R + dX X = 2 R 1 0 d 1 − 2 = 2 FP ε=0 1 ε d 1 − 2 = 2 FP ε=0 log( √ 1 − ε 2 + 1) − log ε = 2 FP ε=0 log 2 − log ε + O(ε 2 ) = 2 log 2. This proves that R Tr e tP (0) 2 = log 2 √ πt . It is easy to see that the renormalized Mellin transform over R + t of a power of t is equal to zero; indeed, let us define, for any function f (t) with an asymptotic expansion in t as t → 0 and t −1 as t → ∞ : (2.7) M 0 (f, s) = 1 0 t s f (t) dt t , M ∞ (f, s) = ∞ 1 t s f (t) dt t . Each of these extends to a meromorphic function on C, which we denote by the same symbol, and the renormalized Mellin transform of f is M(f, s) = M 0 (f, s) + M ∞ (f, s). If f (t) = t ν , then M 0 (f, s) = 1 s+ν and M ∞ (f, s) = − 1 s+ν , and so M(f, s) = 0. Hence we see that the zeta function of −P (0) 2 is identically zero. For a = −1 we have, for any s > 1 2 , ∞ 0 t s Tr(e tP (1)P (−1) ) dt t = log 2 √ π ∞ 0 t s−1/2 e −t dt t = Γ(s − 1/2) log 2 √ π ∼ −2 log 2 + O(s); hence log det −P (1)P (−1) = − ∂ ∂s s=0 ζ(s) = − ∂ ∂s s=0 Γ(s − 1/2) log 2 Γ(s) √ π = 2 log 2. Next let us compute the variation of the renormalized trace for arbitrary a. First note that ∂ a P (a) = ∂ a X −a ( X ∂ X ) X a = − log X P (a) + P (a) log X = [P (a), α] where α denotes log X . Similarly Next, by Duhamel's formula, we have ∂ ∂a R Tr(e tP (−a)P (a) ) = − R Tr t 0 e τ P (−a)P (a) ∂ a (−P (−a)P (a))e (t−τ )P (−a)P (a) dτ = T 1 + R 1 where T 1 = −t R Tr(e tP (−a)P (a) ∂ a (−P (−a)P (a))), R 1 = − t 0 R Tr e τ P (−a)P (a) ∂ a (−P (−a)P (a)), e (t−τ )P (−a)P (a) dτ. Note that since the renormalized trace does not vanish on commutators, R 1 is not automatically zero. Let us focus first on T 1 . We can rewrite it as In turn let us write this as T 2 +R 2 where R 2 consists of the summands involving commutators. From the uniqueness of the solution to the heat equation, we note that P (−a)e P (a)P (−a) = e P (−a)P (a) P (−a), and so we can write T 2 as T 1 = −t R Tr(e tP (−T 2 = −2t R Tr(P (a)P (−a)e tP (a)P (−a) α) + 2t R Tr(P (−a)P (a)e tP (−a)P (a) α) = 2t∂ t R Tr(e tP (−a)P (a) α) − R Tr(e tP (a)P (−a) α) which we can rewrite as 2t∂ t R Str(e t P 2 (a) α), where P (a) = 0 P (−a) P (a) 0 and R Str A B C D = R Tr(A) − R Tr(D). Thus we have (2.8) ∂ ∂a R Tr(e tP (a)P (−a) ) = 2t∂ t R Str(e t P 2 (a) α) + R 1 + R 2 where R 1 and R 2 involve renormalized traces of commutators. To address these terms we will make use of appropriate trace defect formulae. We will use the same conventions as in [Mel93] regarding Mellin transform and indicial operators, namely: M(u)(λ) = ∞ 0 x −iλ u(x) dx x , (M −1 v)(x) = 1 2π Im λ=η x iλ v(λ) dλ I(A, λ) = x −iλ Ax iλ ∂ . The latter means that I(A, λ) acts on a section u over the boundary by choosing an extension u off of the boundary, applying x −iλ Ax iλ to u, and then restricting back to the boundary; the result is independent of the choice of extension. R Tr([A, B]) = i 2π R Tr ∂ (∂ λ I(A, λ)I(B, λ)) dλ, where the trace in the integrand is the trace Tr ∂ : Ψ −∞ (∂M ) −→ C. b) With A and B as above, R Tr([A, B log ]) = − 1 4π R Tr ∂ (∂ 2 λ I(A, λ)I(B, λ)) dλ. Proof. Following [MN96] (cf. [MR04,Alb09]), let us use Riesz renormalization to define the renormalized trace, R Tr(B) = FP z=0 Tr( z B) for any operator B such that z B is trace-class for large enough Re(z). We have R Tr([A, B]) = FP z=0 Tr( z [A, B]) = FP z=0 Tr([ z , A]B) = FP z=0 Tr(z z A(z)B) = Res z=0 Tr( z A(z)B) where A(z) = A− −z A z z . Note that this is a holomorphic function of z and has indicial operator I( A(z), λ) = I(A, λ) − I(A, λ + 1 i z) z with Taylor expansion at z = 0 given by − k≥1 z k−1 k!i k ∂ k λ I(A, λ). In particular, we point out that (2.9) I( A(0), λ) = i∂ λ I(A, λ) = −I([A, log ], λ). To compute the residue Res z=0 Tr( z A(z)B) note that δ 0 z ( k ) d = δ z+k z + k so only the term with k = 0 contributes to the residue at z = 0. Hence Tr( z A(z)B) extends to a meromorphic function with a simple pole at z = 0 and residue equal to Tr ∂ ( A(0)B =0 ) = 1 2π R Tr ∂ (I( A(0)B, λ)) dλ = i 2π R Tr ∂ (∂ λ I(A, λ)I(B, λ)) dλ as required. Now replace B with B log . As before, since δ 0 z ( k log ) d = δ z+k log δ z + k − δ (z+k) (z + k) 2 , only the terms with k = 0 contribute to the residue. Furthermore, for small z we have δ z = 1 + z log δ + O(z 2 ), and so δ z log δ z − δ z z 2 ∼ log δ z − 1 + z log δ z 2 + O(1) = − 1 z 2 + O(1) . We see that to compute the residue we will only need (minus one times) the O(z) term in Tr( A(z)B). Since the O(z) term in the expansion of I( A(z), λ) is z 2 ∂ 2 λ I(A, λ), we find Res z=0 Tr( z A(z)B log ) = − 1 4π R Tr ∂ (∂ 2 λ I(A, λ)I(B, λ)) dλ. To compute the indicial operator of P (a) let us recall that our bdf is = X −1 , so that X = sign(X) −2 − 1 , X ∂ X = − sign(X) 1 − 2 ∂ and hence P (a) = a − sign(X) 1 − 2 ∂ −a , = − sign(X) 1 − 2 ( ∂ − a) I(P (a), λ) = −iλ + a at X → +∞, iλ − a at X → −∞. It follows that I(−P (−a)P (a), λ) = λ 2 + a 2 and I(e tP (−a)P (a) , λ) = e −t(λ 2 +a 2 ) at both ends of R. We can use this observation and Lemma 2.2 to compute R 1 and R 2 . Indeed, note from (2.9) and the fact that α = − log that I(∂ a (−P (−a)P (a)), λ) = I(−[P (−a), log ]P (a) + P (−a)[P (a), log ]), λ) = −( 1 i ∂ λ I(P (−a), λ))I(P (a), λ) + P (−a)( 1 i ∂ λ I(P (a), λ)) = 2a (the same at both ends of R). Hence R 1 = − t 0 R Tr e τ P (−a)P (a) ∂ a (−P (−a)P (a)), e (t−τ )P (−a)P (a) dτ = −2 t 0 i 2π R ∂ λ e −τ (a 2 +λ 2 ) 2a e −(t−τ )(a 2 +λ 2 ) dλdτ where we multiply by two in applying the trace defect formula since we have the same contribution from each end of R. Since the integrand is odd in λ, we see that R 1 = 0. Next for R 2 we have (again multiplying by two to take into account both ends of R) R 2 = −t R Tr P (−a)P (a), e tP (−a)P (a) α − 2 P (a), e tP (−a)P (a) P (−a)α = t 2π R −(∂ 2 λ I(P (−a)P (a), λ))e −t(a 2 +λ 2 ) + 2(∂ 2 λ I(P (a), λ))e −t(a 2 +λ 2 ) I(P (−a), λ) dλ = t π R e −t(a 2 +λ 2 ) dλ = t π e −ta 2 , and so altogether (2.10) ∂ ∂a R Tr(e tP (−a)P (a) ) = t π e −ta 2 − 2t ∂ ∂t R Str(e t P 2 (a) log ). Lemma 2.3. When a = 1, we have log det (−P (−1)P (1)) = 0. Proof. Note that since R Str e t P 2 (a) log is an odd function of a for each fixed t, as are its derivatives in t, we have ∂ ∂a R Tr(e tP (−a)P (a) ) − R Tr(e tP (a)P (−a) ) . = 2 t π e −ta 2 Since R Str e t P 2 (0) = 0, integrating from 0 to a yields (2.11) R Tr(e tP (−a)P (a) ) − R Tr(e tP (a)P (−a) ) = 2 a 0 t π e −tb 2 db. Hence from Lemma 2.1 we have R Tr e tP (−1)P (1) = R Tr e tP (1)P (−1) + 2 1 0 t π e −tb 2 db = e −t log 2 √ πt + 2 1 0 t π e −tb 2 db. Consider this integral as a function of t. Writing it alternately as 2 1 0 t π e −tb 2 db = 2 √ π √ t 0 e −v 2 dv we see that it is O(t 1/2 ) as t → 0 and 1 + O(t −1/2 e −t ) as t → ∞. It follows that the integral 2 ∞ 0 t s 1 0 t π e −tb 2 db dt t exists for all s with real part in (−1/2, 0). For these s we can use Fubini's theorem to find 2 ∞ 0 t s 1 0 t π e −tb 2 db dt t = 2 √ π 1 0 ∞ 0 t s+1/2 e −tb 2 dt t db = − Γ(s + 1/2) s √ π . We also know that M( R Tr e tP (1)P (−1) , s) = Γ(s−1/2) log 2 √ π , so altogether it follows that the zeta function of −P (−1)P (1) is equal to ζ 1 (s) = 1 Γ(s) M R Tr e tP (−1)P (1) , s = Γ(s − 1/2) log 2 √ π Γ(s) − Γ(s + 1/2) s √ π Γ(s) ∼ −1 + O(s 2 ) and hence log det −P (−1)P (1) = −∂ s s=0 ζ 1 (s) = 0. Next, using the notation of (2.7), we point out that ∂ ∂a M 0 R Tr e tP (−a)P (a) , s = M 0 ∂ ∂a R Tr e tP (−a)P (a) , s , ∂ ∂a M ∞ R Tr e tP (−a)P (a) , s = M ∞ ∂ ∂a R Tr e tP (−a)P (a) , s , since this interchange is justified in the particular regions of s ∈ C where these functions are holomorphic. Hence, for a = 0, writing ζ a (s) = 1 Γ(s) M R Tr e tP (−a)P (a) , s , we have ∂ ∂a ζ a (s) = 1 Γ(s) M t π e −ta 2 − 2t∂ t R Str e t P 2 (a) log =: ζ a (s) + ζ a (s). We now examine ζ a (s) and ζ a (s) in a neighborhood of s = 0. For ζ a (s), note that ∞ 0 t s+1/2 e −ta 2 dt t = a −2s−1 ∞ 0 y s+1/2 e −y dy y = a −2s−1 Γ(s + 1/2) and hence, ζ a (s) = a −2s−1 Γ(s + 1/2) √ π Γ(s) ∼ 1 a s + O(s 2 ) as s → 0. For ζ a (s), we start by integrating by parts to find ζ a (s) = 2s Γ(s) M R Str e t P 2 (a) log , s . R a = − Tr(Π ker L 2 P (a) log ) − Tr(Π ker L 2 P (−a) log ) , where Π ker L 2 P (b) is the orthogonal projection onto the L 2 -null space of P (b). We will need a more explicit formula for this residue. We have shown that ker L 2 P (a) = span{ X −a } if a > 0 {0} if a ≤ 0. Let us write (2.12) c k = X −k 2 L 2 = R X −2k dX X = 2 1 0 2k d 1 − 2 = 1 0 r k−1 (1 − r) 1/2−1 dr = B(k, 1/2) = Γ(k)Γ(1/2) Γ(k + 1/2) . So, for b > 0, the Schwartz kernel of the projection onto ker P (b) is given by K Π (X, X ) = 1 c b X −b X −b dX X , and hence Tr(Π ker P (b) log ) = 1 c b R X −2b log 1 X dX X = 1 2c b ∂ b (c b ). Thus, for a = 0, the residue equals R a = − Tr(Π ker P (a) log ) − Tr(Π ker P (−a) log ) = − 1 2ca ∂ a c a if a > 0 − 1 2c |a| ∂ a c |a| if a < 0, which determines the behavior of ζ a (s) near s = 0. Indeed, since s Γ(s) = s 2 + O(s 3 ) and M( R Str e tP 2 (a) log , s) = 1 s R a + O(1), we have ζ a (s) = 2 R a s + O(s 2 ). Thus altogether we have ∂ ∂a ζ a (s) = ζ a (s) + ζ a (s) ∼ 1 a + 2 R a s + O(s 2 ), and, interchanging ∂ a and −∂ s s=0 , this shows that ∂ ∂a log det (−P (−a)P (a)) = − 1 a − 2 R a = ∂ ∂a (− log a + log c a ) if a > 0 ∂ ∂a log |a| + log c |a| if a < 0. Hence there exist constants C ± such that log det (−P (−a)P (a)) = C + − log a + log c a if a > 0 C − + log |a| + log c |a| if a < 0 . Note that c 1 = 2, so from Lemmas 2.1 and 2.3 we have log det (−P (1)P (−1)) = 2 log 2 = C − + log 2 log det (−P (−1)P (1)) = 0 = C + + log 2, which shows that C − = log 2 and C + = − log 2. Thus altogether we have shown that (2.13) log det (−P (−a)P (a)) =      log(2|a|c |a| ) if a < 0 0 if a = 0 log ca 2a if a > 0 Finally, let us compute the contribution to the analytic torsion we have been looking for. From (2.6) and (2.13), we get 1 2 q< v 2 (−1) q b q (q − (−1) v (v − q + 1)) − log det −P ( v−2q 2 )P ( 2q−v 2 ) +(−q − 1 + (−1) v (v − q))) − log det −P ( 2q−v 2 )P ( v−2q 2 ) = − 1 2 q< v 2 (−1) q b q (q − (−1) v (v − q + 1)) log((v − 2q)c v/2−q ) +((−q − 1 + (−1) v (v − q)) log( 1 v−2q c v/2−q ) , so that (2.14) LAT([−π/2, π/2], D b , H * (Z; F )) = 1 2 q< v 2 (−1) q b q (1 + (−1) v ) log c v/2−q + ((−1) v (2v − 2q + 1) − (2q + 1)) log(v − 2q) . In particular, when v is even, we obtain (2.15) LAT([−π/2, π/2], D b , H * (Z; F )) = q< v 2 (−1) q b q log c v/2−q + (v − 2q) log(v − 2q) . Cusp degeneration and small eigenvalues Let g ε,hc be an ε, hc metric of product-type to order two and F −→ X s a flat Euclidean vector bundle. In [ARS14, § §4-5] we showed that, as long as F | Z is Witt, there exists a δ > 0 and ε 0 > 0 such that Spec(D dR ) ∩ S δ (0) = ∅ for all ε < ε 0 , and such that every eigenvalue in B δ (0) converges to zero as ε → 0. We call the eigenvalues of D dR in B δ (0) the small eigenvalues and the sum of their eigenspaces the small eigenforms, Ω k small (M ; ε); note that the space of harmonic forms ker D 2 dR = ker D dR is a subspace of Ω k small (M ; ε). Write d ε = ρ v/2 dρ −v/2 , δ ε = ρ v/2 δ ε ρ −v/2 , so that ð dR = d + δ ε and D dR = d ε + δ ε ; the ε subscripts are used to remind us that all of the operators except d depend on ε. Note that the small eigenvalues of D dR and of ð dR are the same, and the eigenforms differ by a factor of ρ v/2 , so we may usually speak without ambiguity. As we will see in Corollary 3.4 below, the product of the positive eigenvalues in a given degree are polyhomogeneous in ε. The quantity log τ small (∆ q ) = FP ε=0 log λ∈Spec small (∆q)\{0} repeated with multiplicity λ is thus well-defined, where ∆ q denotes the action of D 2 dR on forms of degree q. In this section we will compute the contribution to the analytic torsion from these small eigenvalues, which by [ARS14, Theorem 10.2] is given by H k (2) (M 0 ; F ) ∼ = IH k m ( M 0 ; F ) ∼ =      H k (M 0 ; F ) k ≤ m−1 2 Im H k (M 0 , ∂M 0 ; F ) −→ H k (M 0 ; F ) k = m 2 H k (M 0 , ∂M 0 ; F ) k > m−1 2 where H k (2) (M 0 ; F ) denotes the k th L 2 -cohomology group of (M 0 , g 0 ) with coefficients in F. On the other hand, from the computation of the L 2 -null space of the horizontal model operator in §2.1, we see that for k ≤ m−1 2 , dim Ω k small = dim H k (M 0 ; F ) + dim H k−1 (Z; F ). For positive ε, a subspace of dimension dim ker ∆ ε,hc = dim ker D dR of these eigenforms will correspond to the eigenvalue zero, and the rest will correspond to positive small eigenvalues. This suggests that, to understand the small eigenforms corresponding to non-zero small eigenvalues, we look for a long exact sequence linking H k (M 0 ; F ), H k (M ; F ) and H k−1 (Z; F ). Observe first that M is homeomorphic to the union of M 0 with Z × (−1, 1), with an overlap region homotopic to Z Z. The associated Mayer-Vietoris sequence is (3.2) . . . −→ H k−1 (Z; F ) ⊕ H k−1 (Z; F ) ∂ k−1 − −−− → H k (M ; F ) −→ H k (M 0 ; F ) ⊕ H k (Z; F ) j k −−→ H k (Z; F ) ⊕ H k (Z; F ) −→ . . . , where the map j k is given by j k (µ, λ) = (ι * + µ−λ, ι * − µ−λ) and ι ± : ∂ ± M 0 → M 0 is the natural inclusion. Note that j k is injective when restricted to H k (Z; F ). By using the identification H k (Z, F ) ⊕ H k (Z, F ) / j k ({0} × H k (Z, F )) → H k (Z, F ) [λ 1 , λ 2 ] → λ 1 − λ 2 , we obtain from (3.2) a new long exact sequence (3.3) . . . −→ H k−1 (Z; F ) ∂ k−1 − −−− → H k (M ; F ) i k −−→ H k (M 0 ; F ) j k −−→ H k (Z; F ) −→ . . . with j k (µ) = ι * + µ − ι * − µ. Moreover, for k ≤ m−1 2 , we can replace H k (M 0 ; F ) with H k (2) (M 0 ; F ). Replacing singular cohomology with Hodge cohomology (notationally by replacing H with H) endows these spaces with inner products, so for k ≤ m−1 2 , we have short exact sequences 0 −→ (ker ∂ k−1 ) ⊥ −→ H k (M ; F ) −→ ker j k −→ 0, and hence the dimension of the space of eigenforms corresponding to positive small eigenvalues is equal to dim Ω k small (M ; F ) − dim H k (M ; F ) = dim ker ∂ k−1 + dim(ker j k ) ⊥ . This in turn suggests that we relate the vector spaces H k + (Z; F ) := ker ∂ k , H k + (M 0 ; F ) := (ker j k ) ⊥ to the space of small eigenforms with positive eigenvalue, and that we relate their respective orthocomplements, which we denote H k H (Z; F ) and H k H (M 0 ; F ), to the space of harmonic forms on M . Indeed, as the notation suggests, we will see in the next subsection that these vector spaces are restrictions of the subspaces of small eigenforms of D 2 dR with positive eigenvalue, or with zero eigenvalue, to the two boundary faces of X s . Finally, from (3.3), we deduce the decomposition (3.4) H k (M ; F ) = L 2 H k H (M 0 ; F ) ⊕ L 2 H k−1 H (Z; F ) for k ≤ m − 1 2 . For k > m−1 2 , we need to consider the dual of the long exact sequence (3.3) under Poincaré duality. Notice that the long exact sequence (3.3) could have alternatively been obtained by looking at the long exact sequence associated to the pair (M, M 0 ) and by using the Thom isomorphism H k+1 c (Z × (−1, 1); F ) ∼ = H k (Z; F ) along with the identification H k (2) (M 0 ; F ) ∼ = H k (M 0 ; F ) for k ≤ m−1 2 . This means that the long exact sequence dual to (3.3) can be obtained by looking the long exact sequence associated to the pair (M, Z): (3.5) · · · / / H k c (M 0 ; F ) / / H k (M ; F ) / / H k (Z; F ) / / H k+1 c (M 0 ; F ) / / · · · . Indeed, it is obtained from (3.5) by using the identification H k c (M 0 ; F ) ∼ = H k (2) (M 0 ; F ) for k > m−1 2 . We get (3.6) · · · H k (2) (M 0 ; F ) / / i k / / H k (M ; F ) ∂ k / / H k (Z; F ) j k / / H k+1 (2) (M 0 ; F ) / / · · · where i k , j k and ∂ k are Poincaré duals of the maps i m−k , j m−1−k and ∂ m−1−k in (3.3). Using the Hodge * -operators of g Z and g 0 , we can define for k > m 2 the space of harmonic forms (3.7) H k + (Z; F ) := * Z H m−1−k + (Z; F ), H k H (Z; F ) := * Z H m−1−k H (Z; F ), L 2 H k + (M 0 ; F ) := * g 0 L 2 H m−k + (M 0 ; F ), L 2 H k H (M 0 ; F ) := * g 0 L 2 H m−k H (M 0 ; F ), so that for k > m 2 we have the decompositions (3.8) H k (Z; F ) = H k + (Z; F ) ⊕ H k H (Z; F ), H k (2) (M 0 ; F ) = L 2 H k + (M 0 ; F ) ⊕ L 2 H k H (M 0 ; F ), H k (M ; F ) = L 2 H k H (M 0 ; F ) ⊕ H k H (Z; F ). This last decomposition is of course related to the long exact sequence (3.6) via the natural identifications 3.2. Positive small eigenvalues. As discussed in the previous subsection, the surgery long exact sequences (3.3) and (3.5) suggest that the vector spaces H k + (Z; F ) and H k + (M 0 ; F ) correspond to restrictions to the boundary faces of X s of the eigenforms corresponding to positive small eigenvalues. In this subsection, we will both see that this is indeed the case and compute the rate at which these eigenvalues approach zero as ε → 0. We first make the following simple observation: Lemma 3.1. If λ ε is a positive small eigenvalue with eigenform u ε = 0, D 2 dR u ε = λ ε u ε , then d ε u ε and δ ε u ε are also eigenforms with the same small eigenvalue λ ε . Furthermore, at least one of these two eigenforms is non-zero. Proof. The first statement is immediate since both d ε and δ ε commute with D dR . To see that du ε and δ ε u ε cannot be both zero, it suffices to notice that for ε > 0, 0 = λ ε u ε = d ε ( δ ε u ε ) + δ ε ( d ε u ε ). The basis for the computation of the decay rates of the small eigenvalues is the following lemma. Lemma 3.2. Suppose u ε is a section of Λ k ( ε,hc T * X s ), with k ≤ m−1 2 , such that • Π small u ε = u ε and u ε L 2 b = 1; • u ε is polyhomogeneous on X s ; • u ε | B sb = 0 and u 0 := u ε | Bsm is such that u 0 L 2 b = 1; • δ ε u ε = 0; • j k ([u 0 ]) = 0, where [u 0 ] ∈ H k (2) (M 0 ; F ) is the cohomology class associated to u 0 . Then d ε u ε 2 L 2 b = 1 4c (m−1)/2−k ||j k ([u 0 ])|| 2 L 2 ε m−1−2k + o(ε m−1−2k ), where c k = B(k, 1/2) is the constant from (2.12) and j k is understood as a map into the harmonic forms on Z (identified with the cohomology by the de Rham isomorphism). Moreover, the (k + 1)-form v ε = dεuε dεuε L 2 b is well-defined and • Π small v ε = v ε and v ε L 2 b = 1; • v ε is polyhomogeneous on X s ; • v ε | Bsm = 0 and v b := v ε | B sb is such that v b L 2 b = 1; • d ε v ε = 0; • v b ∈ ker ∂ k ⊂ H k (Z; F ) ∼ = ker L 2 D k+1 b is a non-zero multiple of j k ([u 0 ]) ∈ H k + (Z; F ). In particular, if u ε is an eigenform associated to a small eigenvalue λ ε , then λ ε = d ε u ε 2 L 2 b = 1 c (m−1)/2−k ||j k ([u 0 ])|| 2 L 2 ε m−1−2k + o(ε m−1−2k ), and v ε is also an eigenform for the small eigenvalue λ ε . Proof. Since j (m−1)/2 ([u 0 ]) is always 0 by the Witt condition, the theorem statement is empty for k = (m − 1)/2, so we may assume that k < (m − 1)/2. With this understood, let u 0 be the restriction of u ε to B sm . Since Π small u ε = u ε , we know from [ARS14] that u 0 is in the L 2 -kernel of D dR | B Bsm . This is helpful in describing the expansion of u 0 near B sm ∩ B sb , which has two components, which we write as ∂ + and ∂ − . Near ∂ ± , using the splitting into tangential and normal parts, we see from (1.2) that u 0 = |x| u ± |x| v ± + o(|x| ) o(|x| ) , for some u ± , v ± ∈ H k (Z; F ) and , ∈ R. Indeed, clearly, the coefficients u ± and v ± must be in H k (Z; F ) for u 0 to be in the kernel of D dR | B Bsm . Then the powers and are obtained by solving the equation 0 −|x|∂ |x| + (k − 1 2 v) |x|∂ |x| + (k − 1 2 v) 0 o(|x| ) o(|x| ) = 0, which gives = v 2 − k, = k − v 2 . Since < 0 and u 0 is in L 2 b (M 0 ; E) = |x| v/2 L 2 hc (M 0 ; E), this means v ± = 0, so that in fact we have that (3.11) u 0 = |x| m−1 2 −k u ± 0 + o(|x| m−1 2 −k ), for some u ± ∈ H k (Z; F ). Now, since j k [u 0 ] = u + − u − and j k [u 0 ] = 0, we must have u + = u − , so at least one must be nonzero. Therefore the expansion of u ε at B sb must have a nonzero term of order ε m−1 2 −k . Moreover, since u + = u − , the coefficient of this term cannot be of the form (3.12) and hence cannot be in the L 2 -kernel of d b = P (k − m−1 2 ). A priori, u ε could have lower order terms at B sb . Notice however that u ε cannot have terms of order less than ε m−1 2 −k at B sb that are in the kernel of d b . Indeed, if there were such a term of order ε α (log ε) q with α < m−1 2 − k, or with α = m−1 2 − k and q > 0, the coefficient u α,q of this term would be in the kernel of d b , so of the form Let ρ sm = ε ρ ; it is a boundary defining function for B sm . Since √ 1 + X 2 = ρ ε , near B sm , ε α (log ε) q u α,q would be of order ρ α− m−1 2 +k sm (log ρ sm ) q . Since u ε is bounded, this forces u α,q = 0. Notice then that exterior differential d does not depend on ε, and that d ε = ρ m−1 2 dρ − m−1 2 . Therefore, to understand u ε , we must examine the first term in the expansion of u ε at B sm (resp. B sb ) which is not in the kernel of d 0 (resp. d b ); the discussion above indicates that such a term exists. Let us denote it by u lead and suppose for contradiction that u lead occurs at order ε α (log ε) q with either α < m−1 2 − k or α = m−1 2 − k and q > 0, and with (α, q) minimal. Then consider d ε u ε ; it is polyhomogeneous on X s with a nontrivial term of order either ε α−1 (log ε) q or ε α (log ε) q (the loss of an order happens if and only if if the term is at B sb and is not a section of ker D v ). Let w ε be d ε u ε divided by its leading-order coefficient in ε, so that w ε has expansions at B sb and B sm with restriction w 0 to B sm and/or restriction w b to B sb being nontrivial. Since d ε Π small = Π small d ε , w ε is also in the range of Π small , and so w 0 and w b are in |x| (m−1)/2 H k L 2 (M 0 ; F ) and ker L 2 D b respectively. In particular, we immediately see that w b must be a section of ker D v . Suppose that u lead is at B sb but is not a section of the ker D v = ρ N H * (Z; F ) −→ B sb ; then the Hodge decomposition on Z would imply that w b is not a section of ker D v , which is a contradiction. So u lead must either be at B sm or a section of ker D v at B sb . Next suppose it is at B sm at order (α, q); then by the Hodge decomposition on B sm = M 0 , the coefficient of u lead at B sm cannot be in L 2 b (M 0 ; E), so must have a term of order zero or smaller at B sb ∩ B sm , which would contradict our assumption that u ε is bounded with u ε | B sb = 0. Thus w 0 = 0, w b = 0, and u lead occurs at B sb and is a section of ker D v . Furthermore, the same logic as in the previous paragraph tells us that no term at B sb within one order of u lead , inclusive, is not a section of ker D v ; if it were, w b would have a contribution from d ε applied to that term and would not be a section of ker D v either. We conclude that u lead occurs at B sb , is equal to u α,q ε α (log ε) q , and that d b u α,q = w b . Since w b ∈ ker L 2 (d b + δ b ) and d 2 b = 0, we must have u α,q ∈ ker(δ b d b ) but is not contained in ker d b . By examining (2.2), we see that there must be a nontrivial term of order X m−1 2 −k in the expansion of the first component of u α,q at the boundary of B sb . Since X = ρ ε , this means that u α,q ε α (log ε) q is of order ρ m−1 2 −k ε α−( m−1 2 −k) (log ε) q at B B sb ∩ B Bsm . However, unless α ≥ (m − 1)/2 − k and q = 0 if α = m−1 2 − k, this would contradict the assumption that u ε is bounded at B sm . And since there is in fact a nontrivial term of order (m − 1)/2 − k at B sb , we must in fact have α = (m − 1)/2 − k and q = 0. Thus let u b = u (m−1)/2−k,0 be the coefficient of the term of order m−1 2 − k of u ε at B sb . By the argument above, u b must be an element of the form (2.2). To be consistent with (3.11), this means that (3.13) u b = u + − u − 0 (X 2 + 1) m−1 4 − k 2 2 f k (X) + u + + u − 0 (X 2 + 1) m−1 4 − k 2 2 + ν with ν ∈ ker L 2 b D b and where f k (X) = 2 cm−1 2 −k X 0 s 2k−m ds is such that lim X→±∞ f (X) = ±1. Now a simple computation using the explicit form of d b shows that the leading order term w b is precisely the vector with zero in the first factor and (3.14) 1 c (m−1)/2−k (u + − u − )(X 2 + 1) k 2 − m−1 4 in the second factor. Squaring and then integrating with respect to b-surgery densities, this finally gives (3.15) d ε u ε 2 L 2 b = 1 c (m−1)/2−k u + − u − 2 L 2 ε m−1−2k + o(ε m−1−2k ). From there, the properties of v ε are easily obtained. The only slightly tricky part is to show that v ε is polyhomogeneous. However, we may write v ε = ε −(m−1)/2+k d ε u ε /||ε −(m−1)/2+k d ε u ε || L 2 b . Since ||ε −(m−1)/2+k d ε u ε || 2 L 2 b , is bounded and has limit a positive constant as ε approaches zero, it follows from a direct series expansion construction that its inverse is also polyhomogeneous in ε with limit a positive constant as ε approaches zero. Since the product of a polyhomogeneous function of ε and a polyhomogeneous function on X s is polyhomogeneous on X s , the result follows. We can now consider the projections. In the following theorem shows that their leadingorder behavior at B sm and B sb is what we expect. In the statement of the result, we will tacilty make the following natural identification following from (2.1), ker L 2 D 2 b Λ k ( ε,hc T * Xs)| B sb ∼ = H k−1 (Z; F ), k < m−1 2 , H k (Z; F ), k ≥ m−1 2 . Theorem 3.3. Recall from [ARS14] that for some index family K ≥ 0, Π small ∈ Ψ −∞,K b,s (X s ; E) is the projection onto eigenforms of D dR corresponding to small eigenvalues, where E = Λ * ( ε,d T * X s ) ⊗ F and where Ψ −∞,K b,s is the space of operators whose kernels are smooth and polyhomogeneous on X 2 s with index family K. Then let Π k small be the subspace of Π small consisting of forms of pure degree k. We have: (i) Π k small = Π k H + Π k + , where Π k H , Π k + ∈ Ψ −∞,K b,s (X s ; E) are projections onto the harmonic forms and onto the eigenforms associated to positive small eigenvalues respectively. (ii) Each of the projections Π k H and Π k + can itself be written as a sum of two projections in Ψ −∞,K b,s (X s ; E), Π k H = Π k H,Bsm + Π k H,b , Π k + = Π k +,Bsm + Π k +,b , where Π k H,(iii) For k ≤ m−1 2 , δ ε Π k +,Bsm = 0, D 2 dR Π k +,Bsm = O(ε m−1−2k ), d ε Π k +,b = 0, and D 2 dR Π k +,b = O(ε m+1−2k ) , and the maps (3.16) ε − m−1 2 +k−1 d ε : Im Π k−1 +,Bsm → Im Π k +,b , ε − m−1 2 +k−1 δ ε : Im Π k +,b → Im Π k−1 +,Bsm are isomorphisms. (iv) For k > m−1 2 , d ε Π k +,Bsm = 0, D 2 dR Π k +,Bsm = O(ε 2k−1−m ), δ ε Π k +,b = 0, and D 2 dR Π k +,b = O(ε 2k+1−m ) , and the maps (3.17) ε m+1 2 −k−1 δ ε : Im Π k+1 +,Bsm → Im Π k +,b , ε m+1 2 −k−1 d ε : Im Π k +,b → Im Π k+1 +,+,b = ε m−1 2 d ε Im Π m 2 −1 +,Bsm ⊕ ε m−1 2 δ ε : Im Π m 2 +1 +,Bsm . Proof. In degree k = 0, Π 0 H is just the projection onto ker D 2 dR | k=0 , which is ρ m−1 2 Γ flat (M ; F ), where Γ flat (M ; F ) is the space of flat sections of F. Since Γ flat (M ; F ) does not depend on ε, we clearly see that Π 0 H ∈ Ψ −∞,K b,s (X s ; E) and that its range admits a basis of polyhomogeneous sections of F on X s . Since ker L 2 (d b + δ b ) is trivial in degree zero, Π 0 +,b = Π 0 H,b = 0 and Π 0 H = Π 0 H,Bsm . Thus, we can take Π 0 +,Bsm = (Π 0 H,Bsm ) ⊥ ⊂ Π 0 small ; this projection is polyhomogeneous on X 2 b,s and its image restricts to the image of the projection onto (ker j 0 ) ⊥ at B sm and to 0 at B sb . We claim the image has a basis which is polyhomogeneous on X s and restricts to a basis of harmonic forms corresponding to (ker j 0 ) ⊥ at B sm and to 0 at B sb . Indeed, the argument is standard and proceeds as follows. Take a basis of harmonic forms corresponding to (ker j 0 ) ⊥ at B sm ⊂ X s ; each is polyhomogeneous on B sm , so we can extend each basis element to a polyhomogeneous form on X s . Then applying the projection Π 0 +,Bsm to each element yields a polyhomogeneous basis for ε small. (This argument to go from polyhomogeneity of the projection to polyhomogeneity of a basis works for any of the projections involved in this proof, as all are polyhomogeneous on X 2 b,s and have restrictions at B mf and B φbf to projections which have polyhomogeneous bases on B sm and B sb respectively.) By Lemma 3.2, we see that for u ε and v ε in the range of Π 0 +,Bsm , v ε , D 2 dR u ε L 2 b = d ε v ε , d ε u ε L 2 b = O(ε m−1 ) . Since D dR and Π small commute, this means that D 2 dR Π 0 +,Bsm = O(ε m−1 ). From here, we now proceed inductively to prove the rest of the theorem for k ≤ m−1 2 . Suppose it is true for degree k − 1; we must show it is true for degree k. First, we take Π k +,b to be the projection onto the range of ε − m−1 2 +k−1 d ε Π k−1 +,Bsm . We claim that it has all the required properties. Indeed, it is clear that d ε Π k +,b = 0. If {u i } is a polyhomogeneous basis of Im Π k−1 +,Bsm , then by Lemma 3.2, v i = ε − m−1 2 +k−1 dd ε u i is a polyhomogeneous basis of Im Π k +,b = H k−1 + (Z; F ). Next we construct Π k H,b . If (ker ∂ k ) ⊥ = 0, then Π k H,b = 0. If not, let ω ∈ (ker ∂ k ) ⊥ be a non-zero element and let u b = (1 + X 2 ) k 2 − m−1 4 0 ω be the corresponding element in ker L 2 D b . Let χ ∈ C ∞ c (R) be a cutoff function taking values between 0 and 1, with χ(t) = 1 for |t| < δ and χ(t) = 0 for |t| > 2δ, where δ > 0 is chosen small enough, and consider u ε := χ(x)u b (x/ε). By direct computation, d ε u ε = 0, u ε | Bsm = 0, and δ ε u ε is polyhomogeneous and bounded on X s , vanishing at both B sm and B sb . First set v ε := u ε − Π k +,b u ε . Since d ε Π +,b = 0, we still have that d ε v ε = 0. By the properties of Π k +,b , we also have that v ε | B sb = u ε | B sb = u b and v ε | sm = 0. By construction, Π k +,b v ε = 0, which implies by the inductive step that Π k−1 +,Bsm δ ε v ε = 0. However, by the Hodge decomposition δ ε v ε is orthogonal to the harmonic forms, and since δ ε δ ε v ε = 0, the inductive hypothesis implies that δ ε v ε is also orthogonal to the image of Π k−1 +,b . Therefore Π k−1 small δ ε v ε = 0. Finally, note that Π k +,b u ε is also polyhomogeneous and bounded on X s , vanishing to positive order at both B sm and B sb . Therefore δ ε Π k +,b u ε is polyhomogeneous on X s , and (3.16) implies that it is in ε m−1 2 −k+1 L 2 and thus bounded, and in fact vanishing at both B sm and B sb . The same is therefore true for δ ε v ε . Now set µ ε = −( d ε + δ ε − Π small ) −1 ( δ ε v ε ). By the Hodge decomposition, µ ε is a pure form of degree k with d ε µ ε = 0 and Π k small µ ε = 0. By the resolvent construction and mapping properties of surgery operators from [ARS14], µ ε is polyhomogeneous on X s , bounded, and vanishes to positive order at B sm and B sb . Finally, set w ε = v ε + µ ε ; we have that D 2 dR w ε = d ε δ ε v ε + d ε δ ε µ ε = d ε δ ε v ε + d ε ( d ε + δ ε − Π k small )µ ε = d ε δ ε v ε − d ε δ ε v ε = 0. So w ε is harmonic. And since w ε = u ε − Π k +,b u ε + µ ε , w ε has the same restrictions at B sm and B sb as u ε , namely 0 and u b . Thus, taking a basis ω 1 , . . . , ω p of (ker ∂ k−1 ) ⊥ , we can find harmonic forms w 1 , . . . , w p on X s , polyhomogeneous and bounded on X s , such that w i | sm = 0, w i | B sb = (1 + X 2 ) k 2 − m−1 4 0 ω i and with [w i | ε=c ] ∈ H k (M ; F ) a positive multiple of ∂ k−1 [ω i ] for c > 0. We can thus define Π k H,b to be the projection on the span of w 1 , . . . , w p . To construct Π k H,Bsm we can proceed in a similar fashion. If ker j k = {0}, then we can just pick Π k H,Bsm = 0. Otherwise, take a class µ ∈ ker j k and choose a class τ ∈ H k (M ; F ) such that i k (τ ) = µ. Represent τ by a smooth form v of degree k on M ; without loss of generality, we can assume that in a tubular neighborhood Z ⊂ M, v is of the form v| Z×(−1,1)x = ω with ω ∈ H k (Z; F ) independent of x. In particular, if k = m−1 2 , this means by the Witt condition that ω = 0 so that v| Z×(−1,1)x = 0. If instead k < m−1 2 , then the norm of v ∈ L 2 Λ k (M ε ; F ) is uniformly bounded as ε approaches zero. Thus, for any k ≤ m−1 2 , we have that the form v ε := ρ m−1 2 v on X s is in L 2 b . Since dv = 0, it is such that d ε v ε = 0. Moreover, we have that v ε | B sb = 0, while v ε | sm represents the class µ ∈ ker j k ⊂ H 2 (2) (M 0 ; F ). Consider then u ε := v ε − (Π k H,b + Π k +,b )v ε . Then we still have that d ε u ε = 0, u ε | B sb = 0 and u ε | Bsm represents the class µ. We also have that Π k−1 small δ ε u ε = 0, so the form µ ε = −( d ε + δ ε − Π small ) −1 δ ε u ε is a well-defined k-form with Π k small µ ε = 0. As in the construction of Π k H,b , the form w ε = u ε + µ ε is harmonic with w ε | B sb = 0 and with w ε | Bsm the harmonic representative of the class µ. Thus, starting with with a basis µ 1 , . . . , µ of ker j k , we can construct harmonic forms w 1 , . . . , w such that w i | B sb = 0, w i | sm represents µ i and Π k H,b w i = 0. Then we define Π k H,Bsm to be the projection on the range of w 1 , . . . , w . Finally, we set Π k +,Bsm = (Π k +,b ⊕ Π k H,b ⊕ Π k H,Bsm ) ⊥ ⊂ Π k small ; as before, it is polyhomogeneous with a basis which is polyhomogeneous on X s ; since we understand the restrictions of every space on the right-hand side, we conclude that it has the appropriate restrictions at B sm and B sb . Let u ε ∈ Π k +,Bsm ; then consider ε − m−1 2 +k δ ε u ε , which we claim is equal to zero. Suppose not. Then it is certainly in the image of Π k−1 small , and as it is in the image of δ ε , it must be in the image of one of the + projections. By duality and the fact that d ε Π k−1 +,b = 0, it is orthogonal to everything in the image of Π k−1 +,b , so it must be in the image of Π k−1 +,Bsm . However, Π k +,b u ε = 0; therefore, by the inductive hypothesis, Π k−1 +,Bsm δ ε u ε = 0. Hence δ ε Π k +,Bsm = 0 as required. Then Lemma 3.2 immediately gives the required estimates for D 2 dR Π k +,Bsm . This completes the inductive step, and the proof for k ≤ m−1 2 . For k ≥ m+1 2 , we obtain the corresponding results by applying Poincaré duality. Finally, if m is even, then, applying Lemma 3.2 as in the case k = 0 to Π Corollary 3.4. In every degree, the product of positive small eigenvalues is polyhomogeneous in ε > 0. Furthermore, for k ≤ (m − 1)/2, the product of all positive small eigenvalues of D 2 dR in degree k is asymptotic to ( 1 c v/2−k ε m−1−2k ) dim H k + (Z;F ) | det((j k ) ⊥ )| 2 · ( 1 c v/2−(k−1) ε m−1−2(k−1) ) dim H k−1 + (Z;F ) | det((j k−1 ) ⊥ )| 2 as ε 0. For k ≥ (m + 1)/2, the product is the same as the product for m − k. If m is even and k = m 2 , then the product is asymptotic to ( 1 c 1/2 ε) dim H m 2 −1 + (Z;F ) | det((j m 2 −1 ) ⊥ )| 2 2 as ε 0. Proof. By Theorem 3.3, the product of positive small eigenvalues in degree k is polyhomogeneous in ε since it is given by det(Π k + D 2 dR Π k + ).In the asymptotic behavior of this product for k ≤ m−1 2 , the first term comes from the eigenvalues corresponding to Π k +,Bsm and is computed using (3.15). Here, we are relying on the fact that the map (j k ) ⊥ maps an orthonormal basis by harmonic representatives of H k + (M 0 ; F ) to an orthogonal basis of harmonic representatives of H k + (Z; F ). Indeed, if u 0 and v 0 are orthogonal harmonic representatives of classes in H k + (M 0 ; F ) and u ε and v ε are extensions in the range of Π small , then similarly to (3.15), we have that o(ε m−1−2k ) = v ε , ∆ ε u ε L 2 = d ε v ε , d ε u ε L 2 = 1 c v/2−k v + − v − , u + − u − L 2 ε m−1−2k + o(ε m−1−2k ), from which we see that j k (u 0 ) = u + −u − and j k (v 0 ) = v + −v − are orthogonal as claimed. The second term comes from the small eigenvalues corresponding to Π k +,b , which by Lemma 3.1 are the same as those corresponding to Π k−1 +,Bsm . The second statement in the theorem follows immediately by Poincaré duality. For the last statement, it suffices to notice that from part (v) of Theorem 3.3, the positive small eigenvalues in degree m 2 are the same as the small eigenvalues coming from Π We can now compute (3.1). First note that, for q ≤ (m − 1)/2, with ∆ q = ð 2 dR in degree q, taking the finite part in ε gives log τ small (∆ q ) = − dim H q + (Z; F ) log(c (m−1)/2−q ) + 2 log | det(j q ) ⊥ | − dim H q−1 + (Z; F ) log(c (m−1)/2−(q−1) ) + 2 log | det(j q−1 ) ⊥ | =: a q + a q−1 with the convention that a m−1 2 = 0 since dim H (m−1)/2 + (Z; F ) = 0 and (j (m−1)/2 ) ⊥ = 0. Next, since log τ small (∆ q ) = log τ small (∆ m−q ) and a (m−1)/2 = 0, we have that for m odd, (3.18) − 1 2 m q=0 (−1) q q log τ small (∆ q ) = − 1 2 (m−1)/2 q=0 (−1) q q + (−1) m−q (m − q) (log τ small (∆ q )) = − 1 2 (m−1)/2 q=0 (−1) q (2q − m)(log τ small (∆ q )) = − 1 2 (m−1)/2 q=0 (−1) q (2q − m)(a q + a q−1 ) = (m−1)/2 q=0 (−1) q a q = (m−1)/2 q=0 (−1) q (− dim H q + (Z; F ) log(c v/2−q ) + 2 log | det(j q ) ⊥ |). When m is even, then using the fact that log τ small (∆ m 2 ) = −2 dim H m 2 −1 + (Z; F ) log(c 1 2 ) + 4 log | det((j m 2 −1 ) ⊥ |, a similar computation shows that (3.19) − 1 2 m q=0 (−1) q q log τ small (∆ q ) = 0. 3.3. Harmonic bases. The second consequence of the analysis in [ARS14] that we will use is about the asymptotics of harmonic forms as ε → 0. Recall from §1.1 that the analytically defined metric invariant quantity is LAT(M, {µ q j }, F ) = LAT(M, g ε,d ; F ) − log Π n q=0 [µ q |ω q ε ] (−1) q where µ = {µ q j } is a fixed basis of H * (M ; F ) , ω ε is an orthonormal basis of harmonic representatives with respect to the metric g ε,hc and where [µ q |ω q ε ] = | det W q | with W q the matrix such that µ q i = j W q ij ω q j . In this section we compute the asymptotic expansion of log Π n q=0 [µ q |ω q ε ] (−1) q . We are interested in the coefficient of ε 0 , as terms dependent on ε will cancel out with those in the expansion of LAT(M, g ε,d ; F ). To compute this contribution, we will make a specific choice for the basis µ. Namely, we let µ k M 0 and µ k Z be bases of orthonormal harmonic representatives for M 0 and for Z with respect to the metrics g 0 and g Z respectively; by an orthogonal transformation we can also assume without loss of generality that they are compatible with the decompositions IH k m (M 0 ; F ) = L 2 H k H (M 0 ; F ) ⊕ L 2 H k + (M 0 ; F ), (3.20) H k (Z; F ) = H k H (Z; F ) ⊕ H k + (Z; F ). (3.21) Then take µ k to be a subset of (µ k M 0 , µ k−1 Z ) which is a basis compatible with the canonical decomposition (3.22) H k (M ; F ) = (ker j k ) ⊕ (ker ∂ k−1 ) ⊥ =: H k H (M 0 ; F ) ⊕ H k−1 H (Z; F ) for k ≤ m − 1 2 . Similarly, for k > m−1 2 , we take µ k to be a subset of (µ k M 0 , µ k Z ) compatible with the canonical decomposition (3.23) H k (M ) = H k H (M 0 ; F ) ⊕ H k H (Z; F ), k > m − 1 2 . With these choices, the constant term in the asymptotic expansion of [µ k |ω k ε ] is coming from µ k−1 Z if k ≤ m−1 2 and from µ k Z otherwise. Precisely, let α be a harmonic form on Z with [α] ∈ H k−1 H (Z; F ) and α L 2 = 1 and let β ∈ H k (Z; F * ) be a harmonic form Poincaré dual to α, so that On the other hand, by Theorem 3.3, the harmonic form ω ε with respect to g ε,hc of L 2 -norm equal to 1 representing a positive multiple of the class as [ν] in H k (M ; F ) is asymptotically of the form ω ε ∼ 1 c m−1 2 −(k−1) X k−1− m−1 2 ρ − m−1 2 dx ρ ∧ ρ k−1 α = ε k−1− m−1 2 c m−1 2 −(k−1) X 2k−2−(m−1) dX X ∧ α when ε approaches 0. Thus, from (3.25), we see that asymptotically as ε tends to zero, [ν] ∼ γ ε [ω ε ] with γ −1 ε = ∞ −∞ ε k−1− m−1 2 cm−1 2 −(k−1) X 2k−2−(m−1) dX X = ε k−1− m−1 2 c m−1 2 −(k−1) . This implies that for k ≤ m−1 2 , (3.26) log[µ k |ω k ] = − dim H k−1 H (Z; F ) 1 2 log cm−1 2 −(k−1) + (k − 1 − m − 1 2 ) log ε + o(1) as ε tends to zero. When k > m−1 2 , the form ν representing the cohomology class [ν] corresponding to the form α under the decomposition (3.23) is such that its restriction to Z is [α], in other words, (3.27) Z ν ∧ β = 1. Another application of Theorem 3.3 shows that the harmonic form ω ε with respect g ε of L 2 -norm equal to 1 representing a positive multiple of the class as [ν] in H k (M ; F ) is asymptotically of the form (3.28) ω ε ∼ 1 c k− m−1 2 X m−1 2 −k ρ − m−1 2 ρ k α = ε k− m−1 2 c k− m−1 2 α as ε tends to zero. From (3.27), we thus see that [ν] ∼ c k− m−1 2 ε k− m−1 2 [ω ε ] as ε tends to zero. Taking the logarithm, we obtain that for k > m−1 2 , (3.29) log[µ k |ω k ] = dim H k H (Z; F ) 1 2 log c k− m−1 2 + ( m − 1 2 − k) log ε + o(1) Using Poincaré duality on Z and the fact m is odd, we see from (3.26) and (3.29) that the constant term in the expansion of − log[Π n q=0 [µ q |ω q ] (−1) q ] is given by Cusp degeneration and analytic torsion Let M be a closed manifold with a two-sided hypersurface Z. We endow X s with a flat Euclidean bundle F −→ X s and an ε, hc metric g ε,hc and in this section we determine the limit as ε → 0 of analytic torsion. In [ARS14,Theorem 8.1] we have computed the constant term in the expansion as ε → 0 of the logarithm of analytic torsion: + LAT([−π/2, π/2], D b , H * (Z; F )) − 1 2 (−1) q q log τ small (∆ q ). We have computed the last two terms, which leads to the following theorem. Proof. When m is odd, the term LAT([−π/2, π/2], D b , H * (Z; F )) is computed in (2.15) and equals + m−1 2 −1 q=0 (−1) q 2 log | det((j q ) ⊥ )| + dim H q (Z; F ) ((m − 1 − 2q) log(m − 1 − 2q)) .v/2−1 q=0 (−1) q dim H q (Z; F ) log c v/2−q + (v − 2q) log(v − 2q) . The term − 1 2 (−1) q q log τ small (∆ q ) is computed in (3.18) and equals (m−1)/2 q=0 (−1) q (− dim H q + (Z; F ) log(c v/2−q ) + 2 log | det(j q ) ⊥ |). Finally, recall from §3.3 that LAT(M, µ, F ) = LAT(M, g ε,hc ; F ) − log Π n q=0 [µ q |ω q ε ] (−1) q and from (3.30) that the constant term in the expansion of − log[Π n q=0 [µ q |ω q ] (−1) q ] as ε → 0 is given by m−1 2 q=0 (−1) q (− dim H q H (Z; F ) log c v/2−q ). The formula when m is odd then follows immediately from the formulas above and the fact that dim H q (Z; F ) = dim H q + (Z; F ) + dim H q H (Z; F ), which yields cancellation of all the terms involving the constants c k . When m is even, then LAT(M, µ, F ) = 0 by the argument of [RS71]. We also computed that − 1 2 (−1) q q log τ small (∆ q ) = 0, so the result follows from (4.1) and (2.14) with v odd. Applying this result to the case where M 0 = N 0 N 0 is the disjoint union of two copies of (N 0 , g 0 ), an even-dimensional manifold with cusp with link Z, and flat Euclidean vector bundle F satisfying the Witt condition, we obtain the following. Corollary 4.2. The analytic torsion of an m-dimensional manifold (m even) with cusp (N 0 , g 0 ) with link Z, equipped with a flat vector bundle F satisfying the Witt condition, is given by LAT([M ; Z], µ 0 ; F ) = m 2 q< m−1 2 (−1) q dim H q (Z; F ) log(m − 1 − 2q), where µ 0 is a basis of L 2 -harmonic forms on N 0 with values in F . Cusp degeneration and Reidemeister torsion We assume as in the previous section that F → X s is a flat Euclidean vector bundle such that H m−1 2 (Z; F ) = {0} (the Witt condition). Moreover, we will now assume that m is odd. To study the change of the R-torsion under a pinching surgery, we will make use of the long exact sequence (3.3). As a complex, we will denote this long exact sequence by H 1 . Let (5.1) M = B sm (CZ CZ), B sm (CZ CZ) = Z Z, be the singular space associated to B sm , where CZ is the disjoint union of the cones of each connected components of Z. To relate the R-torsion of M with an appropriate intersection R-torsion on M , we will need another exact sequence, namely the Mayer-Vietoris sequence obtained by writing M as the union of B sm with CZ CZ. The pseudomanifold M has a natural stratification of depth one, with singular stratum given by a disjoint union of points. Let T be a choice of triangulation on M compatible with this stratification and the decomposition (5.1). Recall from [ARS14] that we can then use T and its first barycentric subdivision T to defined the complex of cochains R * m (M , α) where α is the orthogonal representation induced by the holonomy of F. This complex has natural restrictions to CZ CZ and Z Z, so there is an induced Mayer-Vietoris short exact sequence of finite dimensional complexes (5.2) 0 / / R * m (M ; F ) / / C * T (M 0 ; F ) ⊕ R * m (CZ CZ; F ) / / C * T (Z Z; F ) / / 0. Here, C * T (M 0 ; F ) = C * T ( B sm ) ⊗ Zπ 1 (Bsm) R q , where R q is seen as a Zπ 1 (B sm )-module via the representation α : π 1 (M ) → O(q) given by the holonomy of F, T is the lift of T to the universal cover B sm of B sm , and C * On the other hand, identifying the span of ( ω 1 √ 2 , ω 1 √ 2 ), . . . ( ω k √ 2 , ω k √ 2 ) isometrically with H k + (Z; F ), we see that the map √ 2 ∂ k : H k + (Z; F ) → L 2 H k+1 + (M 0 ; F ) corresponds to j k , hence to the adjoint of j m−k−1 : L 2 H m−k−1 + (M 0 ; F ) → H m−k−1 + (Z; F ). Thus, the contribution of ∂ k : H k + (Z; F ) → L 2 H k+1 + (M 0 ; F ) to τ (H 1 ) −1 τ (H 2 ) is given by (5.15) k< m−1 2 ( √ 2 − dim H k + (Z;F ) | det((j k ) ⊥ |) (−1) k . Combining (5.14) and (5.15) with Lemma 5.2 and using the fact that dim H k (Z; F ) = dim H k H (Z; F ) + dim H k + (Z; F ), along with Poincaré duality, gives the result. Remark 1. Even though we have made a specific choice of basis for H k (M 0 ; F ) to prove Theorem 5.4, the final result is independent of such a choice. Theorem 6.1. Let M be an odd-dimensional manifold, Z ⊆ M a two sided hypersurface, g ε,hc a cusp surgery metric which is product-type to order two, and F −→ X s a flat Euclidean vector bundle satisfying the Witt condition. Let µ k M 0 and µ k Z be bases of IH k m (M 0 ; F ) ∼ = L 2 g 0 H k (M 0 ; F ) and H k (Z; F ), consisting of harmonic forms which are orthonormal with respect to the metrics g 0 and g Z . Choose the basis µ CZ for IH k m (CZ; F ) induced by (5.9). Using these bases to define the corresponding R-torsions and analytic torsion, we have the following formula: In particular, applying this result to the case where M 0 = N 0 N 0 is the disjoint union of two copies of (N 0 , g 0 ), a manifold with cusp with link Z, where g 0 is product-type to order two and E is a flat vector bundle on N 0 satisfying the Witt condition, we obtain the following: (6.1) LAT([M ; H], µ M 0 , F ) = log Iτ m (M 0 , µ M 0 , F ) Iτ m (CZ, µ CZ , F ) 2 − χ(Z; F ) 4 log 2 − m−1 2 −1 q=0 (−1) q dim H q (Z; F ) [(m − 1 − 2q) log(m − 1 − 2q)] . Corollary 6.2. Let µ N 0 and µ Z be bases of IH k m (N 0 ; E) ∼ = L 2 g 0 H k (N 0 ; E) and H k (Z; E) respectively, consisting of harmonic forms orthonormal with respect to g 0 and g Z . The canonical identification (5.9) gives a basis µ CZ for IH k m (CZ). Using these bases to define the corresponding R-torsions and analytic torsion, we have the following formula: (6.2) LAT(N 0 , µ N 0 , E) = log Iτ m (N 0 , µ N 0 , E) Iτ m (CZ, µ CZ , F ) − χ(Z; E) 8 log 2 − 1 2 m−1 2 −1 q=0 (−1) q dim H q (Z; E) [(m − 1 − 2q) log(m − 1 − 2q)] . Note that if E is a flat vector bundle on N 0 , it extends to a flat vector bundle on the double M 0 which can then be pulled back to obtain a flat vector bundle on X s , allowing us to apply the main theorem. Cusp degeneration and the boundary of Teichmüller space In addition to analyzing the analytic torsion, our results can also be used to analyze the behavior of families of hyperbolic metrics on surfaces which approach the boundary of Teichmüller space, giving a new perspective on results of Wolpert and of Burger [Wol87,Wol90,Wol10,Bur88]. In particular, we analyze the so-called 'plumbing construction'. First we describe this construction: let R 0 be a hyperbolic surface with nodes p 1 , . . . , p m ; each node represents a pair of cusps at punctures a i and b i of R 0 \ {p 1 , . . . , p m }. Let U j i , j = 1, 2, be neighborhoods of a i and b i respectively. Suppose without loss of generality that each U j i is a disk of radius γ 0 for some fixed γ 0 > 0 and that we have local coordinates z i : U 1 i → U and w i : U 2 i → U with z i (a i ) = 0, w i (b i ) = 0 and such that the hyperbolic metric g 0 takes the forms (7.1) |dz i | |z i | log |z i | 2 and |dw i | |w i | log |w i | 2 in terms of these coordinates. Then, for a sufficiently small choice of γ 0 , there exists an open set V disjoint from each U j i and a set of Beltrami differentials ν i , i = 1, . . . , 3g − 3 − m, which are supported in V and which span the tangent space of the boundary of Teichmüller space at R 0 . For s sufficiently close to the origin in C 3g−3−m , we can solve the Beltrami equation for ν(s) = m i=1 s i ν i and obtain a family of hyperbolic surfaces (with cusps) which we denote R ν(s) . The hyperbolic metrics g s on these surfaces are smooth in s and are conformal to the hyperbolic metric g 0 on R 0 in each U j i [Wol90,Wol10]. We now introduce a degeneration parameter σ = (σ 1 , . . . , σ m ) ∈ C m describing the opening of the nodes. For each σ sufficiently close to the origin, and each i ∈ [1, m], we remove a pair of disks D j σ i of radius |σ i | around the ith node and identify the annuli U 1 i \ D 1 σ i and U 2 i \ D 2 σ i , in complex coordinates (z, w), by zw = σ i . This produces new Riemann surface R σ,s spanning a neighborhood of R 0 in Teichmüller space [Wol87]. Each surface R σ,s can be equipped with a unique hyperbolic metric g σ,s in the conformal class specified by the complex structure. To describe the behavior of g σ,s as σ → 0, notice that the local model describing the opening of the node, the degeration fixture P = {(z, w, τ ) : zw = τ, |z|, |w|, |τ | < 1}, is a complex manifold fibering over the disk D = {|τ | < 1} with fiber above τ naturally identified with the annulus |τ | < |z| < 1 for τ = 0. On each fibre, the unique complete hyperbolic metric in the conformal class specified by the complex structure is given by g P,τ = π log |τ | csc π log|z| log |τ | dz z 2 = π log |τ | csc π log|w| log |τ | dw w 2 . At τ = 0 this model degenerate to give two cusps as in (7.1). In fact, making the change of variables x = cot π log|z| log |τ | , θ = arg z, we obtain (7.2) g P,τ = dx 2 x 2 + ε 2 + (x 2 + ε 2 )dθ 2 , with ε = −π log |τ | , which is precisely the degeneration model considered in the present paper. For each sufficiently small σ, we can use this model and construct an approximate hyperbolic metric h σ,s by gluing. More precisely, let g P,σ be the metric which on each U j i \ D 1 σ i is given by the metric g P,σ i . Then let η be a cutoff function on R 0 which is zero within a distance γ 0 /2 of each node and identically 1 outside a distance 2γ 0 of each node, and whose gradient has support in a union of annuli with inner radius γ 0 /2 and outer radius 2γ 0 about each node. Finally, as in [Wol10], define a new metric h σ,s on R 0 by h σ,s = g η 0 g 1−η P,σ . This family of metrics is smooth in (σ, s) away from the nodes [Wol10]. The metrics h σ,s are not necessarily exactly hyperbolic; their curvature may not be identically −1 on the annuli where ∇η may be nonzero. However, their curvature may be computed directly; call it K σ,s . Then we must have g σ,s = e 2ϕσ,s h σ,s , where ϕ σ,s is the solution of the prescribed-curvature equation where ∆ hσ,s is the Laplacian with non-negative spectrum corresponding to the metric h σ,s . As shown in [Wol87, p.293], the metric h σ,s is a good approximation of g σ,s in the sense that (7.4) lim (σ,s)→0 g σ,s h σ,s = 1, see also [Wol90] for an expansion at (σ, s) = 0. In [Wol87], Wolpert also obtains estimates for the small eigenvalues and for the determinants of the Laplacian of the metrics g σ,s , see also [Bur88] for a sharpening of the eigenvalue asymptotics. Since the metrics h σ,s are exactly of the form (7.2) in a fixed neighborhood of the nodes, our work may be used to recover and extend these results in many cases. Indeed, we may apply directly our results to h e −π/ε τ,s , so using (7.4) together with the prescribed curvature equation (7.3) and its solution ϕ e −π/ε τ,s , we can derive corresponding results for g e −π/ε τ,s . 7.1. Small eigenvalues. First we analyze the behavior of the small eigenvalues of ∆ gσ,s . The curvature K e −π/ε τ,s and the corresponding solution of the prescribed curvature equation are analyzed carefully in [Wol90]. From [Wol90, Section 3], we conclude that ||K e −π/ε τ,s +1|| C 0 → 0 as ε → 0, uniformly in s. (In fact, ||K e −π/ε τ,s + 1|| C 0 = Cε 2 + O(ε 4 ))). As a consequence, it is proved in [Wol90, Section 4] that for any δ there exists an ε 0 such that if ε < ε 0 , then for all sufficiently small s and all points on the surface, (1 − δ)h e −π/ε τ,s ≤ g e −π/ε τ,s ≤ (1 + δ)h e −π/ε τ,s . By the well-known result of Dodziuk [Dod82,Prop. 3.3], the quotients of the nonzero eigenvales of the Laplacians for (R, g e −π/ε τ,s ) and (R, h e −π/ε τ,s ) therefore approach 1 as ε → 0, which allows us to apply our analysis of the small eigenvalues in section 3 to conclude: Proposition 7.1. As ε goes to zero, the positive small eigenvalues λ ε,s of ∆ g e −π/ε τ,s satisfy λ ε,s ∼ cε + o(ε), where c can be computed explicitly for each small eigenvalue using the methods of section 3. As a particular case, consider the situation with i = 1. From the long exact sequence (3.3), there is one small eigenvalue if the manifold R ε becomes disconnected in the limit and zero if it does not. If it becomes disconnected and the volumes of the two connected components of the limit are V 1 and V 2 , we conclude from Lemma 3.2 that the leading asymptotic of the single small eigenvalue is V 1 +V 2 πV 1 V 2 ε. This agrees with the result of Burger [Bur88], who computed these eigenvalue asymptotics, to the same accuracy and with specific values of c, using methods involving a comparison with the graph Laplacian. 7.2. Determinant. We can also analyze the determinant of the Laplacian ∆ g e −π/ε τ,s . Observe that it is easy to show, by applying the maximum principle to the prescribed curvature equation, that whenever ε is small enough so that ||K e −π/ε τ,s + 1|| C 0 ≤ 1/2, then there is a constant C such that ||ϕ e −π/ε τ,s || C 0 ≤ C. By substituting these bounds into the prescribed curvature equation, we also get a C 0 bound for ∆ϕ e −π/ε τ,s . Therefore, for sufficiently small ε, there is a universal constant C such that ||ϕ e −π/ε τ,s || C 0 + ||∆ϕ e −π/ε τ,s || C 0 ≤ C. We may now apply the Polyakov formula in the form from [OPS98]: log det ∆ g e −π/ε τ,s = log det ∆ h e −π/ε τ,s − 1 12π R (|∇ϕ e −π/ε τ,s | 2 + 2K e −π/ε τ,s )dh e −π/ε τ,s . Using integration by parts and the bounds we have just proven, it is straightforward to show that for sufficiently small ε, there is a universal constant C (possibly different from the one above) such that (7.5) | log det ∆ g e −π/ε τ,s − log det ∆ h e −π/ε τ,s | ≤ C. We now claim: Proposition 7.2. As ε → 0, log det ∆ h e −π/ε τ,s → −∞. Proof. Since h e −π/ε τ,s is a family of cusp surgery metrics, the log determinant in question has a polyhomogeneous expansion as ε → 0. We need to investigate all the divergent terms in this expansion, so we need to be specific about the leading orders. Recall from [ARS14, Sec. 7] that the polyhomogeneous expansion of the log determinant is obtained by a pair of renormalized pushforwards: first an integration in the spatial variables to get the renormalized trace, then an integration in time to get the log determinant. From [ARS14,Eq. (7.18)], the first integral shows that the renormalized trace is an element of A −2,(−2+2N 0 )∪(−1+2N 0 ),0∪0 (E T ); that is, it has order −2 at B tf , −2 at B tf f , and order 0∪0 at B af . At B tf f , the term of order −2 comes from B ε,τ and the term of order −1 comes from B tf f ⊂ ∆ HX . The second pushforward sends both B tf f and B af to ε = 0, so the log determinant has index set equal to −2∪(0∪0), which would normally yield a term of order ε −2 coming from B tf f as the leading term. However, the term of order ε −2 is actually zero. To see why, note that it comes from the leading-order term in the expansion of the diagonal heat kernel at B ε,τ , which is a constant mutiple of τ −2 . Performing two renormalized pushforwards, first in x and then in τ , on such a term yields zero. Therefore the true leading term is of order ε −1 and comes from B tf f ⊂ ∆ HX . We conclude that there exist constants C 1 , C 2 , C 3 , and C 4 such that, as ε → 0, (7.6) log det ∆ h e −π/ε τ,s ∼ C 1 ε −1 + C 2 log log ε + C 3 log ε + C 4 . If C 1 < 0, or if C 1 = 0 and C 2 > 0, et cetera, we are done. However, from the definition of the determinant and the renormalized pushforward theorem, we know that C 1 = − 1 2 ∞ R 0 A tf f dσ σ , where A tf f is the coefficient of the ε −1 term in the expansion of the renormalized trace at B tf f . From [ARS14, Sec. 7], A tf f = N tf f (A) = e −σ 2 ∆ S 1 1 4πσ exp − | · | 2 ε,d N B tf 2σ 2 µε,φ N B sb × Y H/H . Therefore, restricting to the diagonal and integrating yields C 1 = − 1 8π ∞ R 0 Tr(e −σ 2 ∆ S 1 )σ −2 dσ = − 1 16π ∞ R 0 Tr(e −t∆ S 1 )t −3/2 dt. This integral may be evaluated and gives C 1 = − 1 16π Γ(−1/2)ζ Riem (−1/2), which is negative. This completes the calculation. Combining this lemma with (7.5), we obtain log det ∆ g e −π/ε τ,s → −∞ as ε → 0, which agrees with the result of Wolpert [Wol87, Theorem 5.3] when we use the description of the determinant in terms of the Selberg Zeta function. H * (H/Y ; F ) = 0, H * L 2 (M ; F ) = 0. The logarithm of analytic torsion satisfies LAT(N, g d , F ) = log Iτ m ( N , α)τ (∂N, Theorem 3 ( 3A Cheeger-Müller theorem for manifolds with cusps). Let F −→ N be a flat Euclidan bundle satisfying the Witt condition. For compatible choices of orthonormal bases of cohomology, µ N of IH * m (N ; F ) ∼ = H * L 2 (N ; F ) and µ CZ of IH * m (CZ) LAT(N, µ N , F ) = log Iτ m (N, µ N , E) Iτ m (CZ, µ CZ , F ) β : X s −→ M × [0, 1] ε obtained by collapsing the new boundary hypersurface of X s back to Z × {0}. Figure 1 . 1The single surgery space X s . F is a Witt bundle, then D b is Fredholm [ARS14, Lemma 2.1]. Finally, D dR induces an operator on B sm . This face is the manifold with boundary M 0 = [M ; Z] and D d = D dR Bsm ∂ a P (−a) = [α, P (−a)], ∂ a (−P (−a)P (a)) = −αP (−a)P (a)+2P (−a)αP (a)−P (−a)P (a)α. a)P (a) (−αP (−a)P (a) + 2P (−a)αP (a) − P (−a)P (a)α)) = −t R Tr −P (−a)P (a)e tP (−a)P (a) α + 2P (a)e tP (−a)P (a) P (−a)α − e tP (−a)P (a) P (−a)P (a)α + P (−a)P (a), e tP (−a)P (a) α − 2 P (a), e tP (−a)P (a) P (−a)α . Lemma 2 . 2 . 22On any manifold with boundary M with a fixed choice of bdf , a) [Mel93, Lemma 5.10] If A is a b-pseudodifferential operator and B is a smoothing b-pseudodifferential operator, then Indeed, the integration by parts is justified for each of M 0 and M ∞ in the region where it is holomorphic, and the resulting boundary terms cancel out when we add together the meromorphically continued M 0 and M ∞ . As above, M 0 R Str e t P 2 (a) log , s extends meromorphically from Re s > 1/2 to the complex plane with simple poles at a subset of { 1 2 − N}. For a = 0, M ∞ R Str e t P 2 (a) log , s extends meromorphically from Re s < 0 to the complex plane with a single, simple pole at s = 0 and residue ( −1) q q log τ small (∆ q ).3.1. Surgery long exact sequence. From[ARS14, §5], the dimension of the space of small eigenforms is equal todimΩ * small = dim ker D d + dim ker D b .We can express these dimensions directly in terms of the topology of M, Z, and M 0 = B sm = [M ; Z]. Let us write M 0 for the singular space obtained from M 0 by coning off ∂M 0 , which is two copies of Z; so we have M 0 = M 0 ∪ CZ ∪ CZ. It follows from work of Hausel, Hunsicker, Mazzeo [HHM04] and Lemma 8.19 of [ARS14] that ( 3 . 9 ) 39H k H (Z; F ) ∼ = ker j k and L 2 H k+ (M 0 ; F ) ∼ = ker i k for k > m 2 .If m is even, then we also have to discuss the case k = m 2 , where we have top and bottom rows come from the long exact sequences (3.5) and (3.3). Alternatively, we could proceed more analytically and deduce the decompositions (3.10) from Theorem 3.3 below. ( 3 . 312) u α,q = ω(1 + X 2 ) some ω ∈ H k (Z; F ). Bsm . Using the Hodge * -operator we get the other part of Π m 2 +,b . Since each eigenspace of positve small eigenvalues is even dimensional and formed of pairs of eigenfunctions by Lemma 3.1, we see from the statement of the theorem when k = m 2 that we get all of Π the element [ν] ∈ H k (M ; F ) corresponding to the form α in the decomposition (3.22) can be represented by a form ν with support in a tubular neighborhood ( (( −1) q+1 dim H q−1 H (Z; F ) log c m−1 2 −(q−1) = − −1) q dim H q H (Z; F ) log cm−1 2 −q . M, g ε,hc ; F ) = LAT([M ; Z], g 0 ; F ) Theorem 4 . 1 . 41Let M be a closed manifold, Z a two-sided hypersurface in M, and F −→ X s a flat Euclidean vector bundle. Let g ε,hc be an ε, hc metric, product-type to order two, and let µ k M 0 and µ k Z be orthonormal bases of harmonic representatives with respect to the metrics g 0 and g Z ; then let µ be a choice of basis for H * (M ; F ) compatible with the decompositions(3.22) and (3.23). If m is odd, then(4.2) LAT(M, µ; F ) = LAT([M ; Z], µ 0 ; F ) If instead m is even, then LAT(M, µ; F ) = 0 and LAT([M ; Z], µ 0 ; F ) = m q dim H q (Z; F ) log(m − 1 − 2q). 6.A Cheeger-Müller theorem for Witt representations on manifolds with cusps Let (M, g) be a closed Riemannian manifold with a flat bundle F −→ M corresponding to an orthogonal representation of the fundamental group of M. The classical Cheeger-Müller theorem gives an equality of analytic torsion and R-torsion for every choice of basis µ of the homology groups: LAT(M, µ, F ) = log τ (M, µ, F ). We have analyzed the behavior of both sides of this equation under analytic cusp surgerythe left-hand side in Theorem 4.1 and the right-hand side in Theorem 5.4. In this section we use these results to conclude a Cheeger-Müller theorem for Witt representations on manifolds with cusps. Proof. Assume (by an orthogonal change of basis, if necessary) that µ M 0 and µ Z respect the decompositions (3.20), then pick the basis µ for H k (M ; F ) induced by (3.22) and (3.23); the theorem then follows immediately from Theorems 4.1 and 5.4 together with the Cheeger-Müller theorem on M . (7. 3 ) 3− ∆ hσ,s ϕ σ,s − K σ,s = e 2ϕσ,s , Bsm restricts to the projection onto L 2 H k H (M 0 ; F ) on B sm and restricts to zero on B sb , while Π k H,b vanishes on B sm and on B sb restricts to the projection onto ; F ) on B sm and vanishes on B sb , while Π k +,b vanishes on B sm and on B sb restricts to the projection onto H k−1 Furthermore, the image of each of these projections admits a basis of polyhomogeneous forms on X s .H k−1 H (Z; F ) for k ≤ m−1 2 and onto H k H (Z; F ) for k ≥ m+1 2 . Similarly, Π k +,Bsm restricts to the projection onto L 2 H k + (M 0 + (Z; F ) for k ≤ m−1 2 and onto H k + (Z; F ) for k ≥ m+1 2 . Acknowledgements. P. A. was supported by NSF grant DMS-1104533 and Simons Foundation grant #317883. F. R. was supported by a Canada Research Chair, NSERC and FRQNT. D. S. was supported by a CRM postdoctoral fellowship and by NSF EMSW21-RTGT ( B sm ) is the group of cochains associated to the triangulation T . Similarly, we have thatwhere i labels the connected components of Z. By[ARS14,Proposition 8.14], the short exact sequence (5.2) induces a Mayer-Vietoris long exact sequence involving intersection cohomology (5.3)where IH k m (M ; F ) = IH k m (M , α) and IH k m (CZ CZ; F ) = IH k m (CZ CZ, α). We will denote the long exact sequence (5.3) by H 2 .Proof. By the formula of Milnor[Mil66], we have that. Combining these two relations gives the result. Note that the direct sum assumption is used to write, for example, τ (Z Z; F ) = (τ (Z; F )) 2 .We now make a particular choice of bases for these spaces that allows a direct comparison with (4.2) and also makes some of the terms in (5.4) more explicit, in particular τ (H 2 )τ (H 1 ) −1 . Recall from section 3.3 the decompositions:We use the same bases as in section 3.3 for IH k m (M ; F ), H k (Z; F ), and H k (M ; F ); namely, orthonormal bases µ k M 0 , µ k Z , and µ k compatible with the decompositions (5.5), (5.7) and (5.8). We also need to make a choice of basis of IH k m (CZ; F ); we will take the one induced by our choice of basis for H k (Z; F ) and the canonical identificationNote immediately that since Z is even-dimensional, we have τ (Z; F ) = 1 by [Che79, Proposition 1.19]. It remains to make a choice of basis for H k (M 0 ; F ) when k > m−1 2 , but at the moment we can at least make a partial computation of τ (H 2 )τ (H 1 ) −1 .Lemma 5.2. With the choice of bases made above, the contribution to τ (H 2 )τ (H 1 ) −1 coming from cohomology classes of degree k ≤ m−1 2 is given byProof. First we compute the contribution to τ (H 2 ). To do this, it suffices to notice that the restriction of j k to the second factor induces the identity map j k : IH k m (CZ CZ; F ) → H k (Z Z; F ) with respect to our choice of bases, while i k composed with the projection on the first factor gives the canonical identification IH k m (M ;As for τ (H 1 ), the decompositions (5.5) and (5.7)are such that in the long exact sequence H 1 ,With our choice of bases, this means that the contribution to τ (H 1 ) coming from cohomology classes of degree k ≤ m−1 2 is given by In degree k > m−1 2 , we will take advantage of some cancellations occurring between τ (H 1 ) and τ (H 2 ) to compute τ (H 2 )τ (H 1 ) −1 directly. First, notice that for k > m−1 2 , the long exact seqence H 2 corresponds to the relative long exact sequence associated to the pair (B sm , ∂B sm ),under the canonical identification H k c (M 0 ; F ) = IH k m (M ; F ). This leads to the following commutative diagram between the long exact sequences H 1 and H 2 when k ≥ m+1 2 : (5.11)where α k : H k c (M 0 ; F ) → H k (M ; F ) is the standard push-forward map and the map β k is given by. This definition suggests that we take a different orthonormal basis of harmonic forms on H k (Z; F )⊕H k (Z; F ). Namely, if ν 1 , . . . , ν i k is our chosen basis for H k (Z; F ), then we take the basisSince this change of basis is orthogonal, it has no effect on the torsion of Z Z, and in particular (5.4) still holds if we compute the torsion of Z Z with respect to this new basis. We can now make the following simple observations.is a basis of H k+1 + (M 0 ; F ). This basis is, however, not necessarily orthonormal;Proof. The proof of (1) and (2) follows from the identification of H k2 . For (3), it follows by noticing that, still under the identification H kis the diagonal inclusion. For (4), this is by exactness of the bottom sequence in (5.11), sinceFinally, (5) follows from (3.9) and the definition of the map i k .Therefore, removing the span of ( ω 1, we obtain from (5.11) the following commutative diagram of long exact sequences:In this diagram, the contribution of the top row to τ (H 1 ) −1 τ (H 2 ) is cancelled by the contribution from the bottom row. Therefore, in degree k ≥ m+1 2 , the only contributions to τ (H 1 ) −1 τ (H 2 ) come from• ∂ k when restricted to the span of ( ω 1To reach this conclusion, we have tacitly assumed that we have chosen a basis of H k (M 0 ; F ) for k ≥ m+1 2 which includes a basis of i k (H k H (Z; F )). We can go one step further and choose this basis so that the corresponding basis of i k (H k H (Z; F )) is the image under i k of the chosen basis on H k H (Z; F ). With this choice, i k does not contribute to τ (H 1 ) −1 τ (H 2 ) and we are left with the contributions of ∂ k and j k . We are now ready to state the refinement of Theorem 5.1.Theorem 5.4. Let µ k M 0 and µ k Z be bases of IH k m (M ; F ) and H k (Z; F ), orthonormal with respect to the metrics g 0 and g Z respectively, and compatible with the decompositions (5.5). Let µ k be a basis of H k (M ; F ) compatible with (5.7) and (5.8) and choose the basis for IH k m (CZ; F ) induced by (5.9). Using these bases to define the corresponding R-torsions, we have the relation Proof. By [Che79, Proposition 1.19], we know that τ (Z; F ) = 1. By Theorem 5.1, it remains therefore to compute τ (H 1 ) −1 τ (H 2 ). By Lemma 5.2 and the discussion above, it remains to compute the contributions coming from j k and ∂ k in degree k ≥ m+1 2 . 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{'abstract': 'We establish a Cheeger-Müller theorem for orthogonal representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps. 1045119. The authors are happy to acknowledge useful conversations with', 'arxivid': '1411.1105', 'author': ['Pierre Albin ', 'Frédéric Rochon ', 'David Sher '], 'authoraffiliation': [], 'corpusid': 118690500, 'doi': '10.1215/00127094-2018-0009', 'github_urls': [], 'n_tokens_mistral': 41033, 'n_tokens_neox': 35743, 'n_words': 22311, 'pdfsha': 'db3055a09614049e6b5f26c824265ca171fb4334', 'pdfurls': ['https://arxiv.org/pdf/1411.1105v3.pdf'], 'title': ['ANALYTIC TORSION AND R-TORSION OF WITT REPRESENTATIONS ON MANIFOLDS WITH CUSPS', 'ANALYTIC TORSION AND R-TORSION OF WITT REPRESENTATIONS ON MANIFOLDS WITH CUSPS'], 'venue': []}

arxiv

Performance Analysis of MIMO-HARQ Assisted V2V Communications With Keyhole Effect Sep 2022 Huan Zhang Zhengtao Liao s:[email protected] Zheng Shi [email protected] Guanghua Yang [email protected] Qingping Dou [email protected] Shaodan Ma [email protected]. Huan Zhang Shaodan Ma Zhengtao Liao Zheng Shi Guanghua Yang Qingping Dou Department of Electrical and Computer Engineering School of Intelligent Systems Science and Engineering are with the State Key Laboratory of Internet of Things for Smart City University of Macau 999078MacaoChina Jinan University 519070ZhuhaiChina Performance Analysis of MIMO-HARQ Assisted V2V Communications With Keyhole Effect Sep 20221 (Corresponding author: Zheng Shi.) Vehicle-to-vehicle (V2V) communications under dense urban environments usually experience severe keyhole fading effect especially for multi-input multi-output (MIMO) channels, which degrades the capacity and outage performance due to the rank deficiency. To avoid these, the integration of MIMO and hybrid automatic repeat request (HARQ) is proposed to assist V2V communications in this paper.By using the methods of integral transforms, the outage probabilities are derived in closed-form for different HARQ-assisted schemes, including Type I-HARQ, HARQ with chase combining (HARQ-CC), and HARQ with incremental redundancy (HARQ-IR). With the results, meaningful insights are gained by conducting the asymptotic outage analysis. Specifically, it is revealed that full time diversity order can be achieved, while full spatial diversity order is unreachable as compared to MIMO-HARQ systems without keyhole effect. Moreover, we prove that the asymptotic outage probability is a monotonically increasing and convex function of the transmission rate. More importantly, although HARQ-IR performs better than HARQ-CC owing to its higher coding complexity, this advantage becomes negligible in the large-scale array regime. Finally, the numerical results are verified by Monte-Carlo simulations along with some in-depth discussions.Index TermsHuan Zhang and Zhengtao Liao are co-first authors. ). 2 hybrid automatic repeat request (HARQ), keyhole effect, MIMO, outage probability, V2V communications. I. INTRODUCTION With the rapid development of intelligent transportation systems, vehicle-to-vehicle (V2V) communications have been widely studied in recent years. Specifically, V2V communications only allow the exchange of information between adjacent vehicles with short distance, which enhances the transmission reliability, supports delay-sensitive applications, and improves traffic safety [1]. However, V2V communications considerably differ from mobile cellular communications. On one hand, both transmit and receive vehicles are in motion, which results in more significant Doppler effects and more rapid channel dynamics than cellular communications. Thus, V2V communications usually undergo time-varying fading channels, which lead to frequent estimations of channel state information (CSI). On the other hand, the transceiver antennas are mounted at almost the same height, where the local scatters in the surrounding environment incur more than one multiplicative small-scale fading processes, i.e., cascaded fading [2]. The local scattering objects include buildings, vehicles, street corners, tunnels, etc., which obstruct the direct link between two vehicles and lead to non-line-of-sight (NLOS) propagation channel condition [3] (cf. Fig. 1). To characterize cascaded fading in V2V communications, the authors in [4]- [7] proposed to use double-bounce/multiple scattering distributions, such as double-Rayleigh, double-Nakagami-m, double-Weibull, and double-generalized Gamma distributions. Moreover, a decode-and-forward relaying scheme was developed in [8] for V2V communications by considering double-Nakagami fading, where both the exact and asymptotic outage probabilities were derived. It was demonstrated experimentally in [3] that the double-bounce scattering distributions can provide an accurate statistical fit for channel modeling of V2V communications. Both experimental and theoretical results verified that the error performance of the multiple scattering model is worse than that of the traditional Rayleigh channel model for cellular communication systems. Therefore, V2V communications often suffer from a more severe fading leading to lower spectral efficiency and reception reliability than cellular communications. To boost the spectral efficiency and reliability, multi-input multi-output (MIMO) aided V2V communications have drawn an ever-increasing attention, because multiple antennas can be easily placed on vehicles with large surface [9]- [11]. In contrast to single-input single-output (SISO) systems, MIMO systems equipping with multiple antennas are capable of reaping the benefit of spatial multiplexing gain. Nevertheless, in realistic propagation environments, the performance of MIMO systems is also susceptible to multiple scattering propagation. Particularly for MIMO assisted V2V (MIMO-V2V) communications, multiplicative fading processes encountered in multiple scattering condition are inevitable due to the mobility of the vehicles and low elevation height of the transceiver antennas. In dense urban environments, all the MIMO-V2V propagation paths travel through the same narrow pipe, which results in the so-called keyhole effect [9]. In practice, the keyhole effect is an important and non-negligible characteristic of MIMO-V2V channels that jeopardizes the diversity gain [12], [13]. The keyhole effect brings about the cascaded fading for MIMO-V2V channels. In particular, the coefficient matrix of the keyhole MIMO channel is expressed as a product of those of the multiple-input single-output (MISO) channel from transmitter to keyhole and the single-input multiple-output (SIMO) channel from keyhole to receiver. This introduces the spatial correlation among MIMO-V2V channels and rank deficiency issue, which degrade the capacity and outage performance of MIMO communications [14]. To investigate the keyhole effect, the ergodic capacity of MIMO systems and the average symbol error rate (SER) of space-time block codes (STBC) were derived in closed-form in [15], [16], respectively. In [17], the system performance for antenna selection under MIMO-keyhole channel was studied. In [18], a statistical analysis of signal-to-noise ratio (SNR) was carried out for MIMO keyhole channels by considering double-Rayleigh and double-Nakagami-Rice fadings. In [19], the authors derived the deterministic approximation for the ergodic rate of a large scale MISO system over keyhole fading channel, where the maximum ratio transmission (MRT) precoding was adopted. Furthermore, the impact of the keyhole channels on satellite communications has been studied in [20]. Apart from keyhole fading channels in outdoor environment, the authors in [21] found that furniture, windows and doors can also incur the keyhole effect. Moreover, it was found in [22] that the keyhole effect also appears in dual-hop reconfigurable intelligent surface (RIS) aided wireless systems. All the relevant works, i.e., [3], [13], [15]- [17], [19]- [24], have demonstrated that keyhole channels not only offset the advantage of spatial diversity of MIMO, but also degrade the spatial multiplexing gain. Since the keyhole effect negatively impacts the reliability of MIMO-V2V communications, it is of necessity to remedy the performance loss for MIMO-V2V communications. To address the above issue, hybrid automatic repeat request (HARQ) is a promising technique to support reliable communications [25]- [29]. Specifically, the essence of HARQ is utilizing both forward error control and automatic repeat request [30]. As opposed to the adaptive modulation and coding (AMC) scheme that requires perfect instantaneous CSI at the transmitter (CSIT), HARQ only needs the statistical/outdated CSIT by relying on the acknowledgement feedback. As aforementioned, the feature of rapid channel dynamics emerged in V2V communications entails excessive CSIT acquisition overhead. By comparing to AMC, the adoption of HARQ in V2V communications is favorable for overcoming channel uncertainties. Moreover, HARQ facilitates the implementation of MIMO in V2V communications thanks to its neglected signaling overhead. Based on whether the erroneously received packets are discarded or not, and what types of coding&decoding strategies are used, HARQ can be further divided into the following three basic schemes, namely, Type-I HARQ, HARQ with chase combining (HARQ-CC) and HARQ with incremental redundancy (HARQ-IR). In particular, Type-I HARQ performs the decoding based on the currently received packet without storing the failed packets. In contrast to Type-I, both HARQ-CC and HARQ-IR schemes save the failed packets and carry out the joint decoding with the subsequent packets by using maximum ratio combining (MRC) and code combining, respectively. Thanks to the outstanding potential of HARQ schemes, they have been widely adopted to assist MIMO communications. For example, in [31], by considering both HARQ-CC and HARQ-IR schemes, the fundamental performance limits and linear dispersion code design for the MIMO systems were studied. Additionally, by aiming at the maximization of energy efficiency, the HARQ-IR assisted massive MIMO systems were investigated in [32]. Nevertheless, the performance of HARQ schemes over keyhole fading channels was seldom reported in the literature except for [4] and [33]. In [4] and [33], the outage probability, throughput and delay analyses were conducted for the SISO-HARQ-CC and SISO-HARQ-IR schemes over double-Rayleigh fading channels, respectively. It is worth mentioning that keyhole MIMO fading encompasses double-Rayleigh fading as a special case by assuming only a single antenna at the transceiver. In order to extend the application of MIMO to V2V communications, this paper focuses on the performance investigation of MIMO-HARQ systems over keyhole fading channels. Since the outage probability is the key performance matric, this paper thoroughly investigates the outage performance of MIMO-HARQ assisted V2V communications with keyhole effect, where three different HARQ schemes are considered. However, the cascaded property of fading channels and the complexity of HARQ schemes significantly challenge the outage analysis. To the best of our knowledge, this is the first treatise that touches upon MIMO-HARQ communications with keyhole effect, and many helpful physical insights will be extracted by conducting asymptotic outage analysis in the high SNR and large-scale array regimes. Specifically, the main contributions of this paper can be summarized as follows. 1) Closed-form expressions are derived for the outage probabilities of MIMO-HARQ assisted V2V communications with keyhole effect by using integral transforms, e.g., moment generating function (MGF) and Mellin transform. Besides, the outage expressions of MIMO-HARQ-CC and MIMO-HARQ-IR schemes in our work collapse to those of SISO-HARQ schemes obtained by [4] and [33], respectively. 2) In order to reveal more insights, the asymptotic outage analyses are conducted in this paper. With asymptotic results, it is concluded that full spatial diversity order is unreachable with MIMO, while full time diversity order can be achieved from using HARQ. This obviously justifies the effectiveness of using HARQ to conquer keyhole effect. More specifically, the spatial diversity order is determined by the minimum of the numbers of transmit and receive antennas. Moreover, it is proved that the asymptotic outage probability is a monotonically increasing and convex function with respect to (w.r.t.) transmission rate. This property facilitates the optimal rate selection for practical system design. 3) More interestedly, it is found that the MIMO-HARQ-CC assisted V2V communications are able to achieve a comparable performance as the MIMO-HARQ-IR assisted ones in the large-scale array regime. This indicates that HARQ-CC is more effective than HARQ-IR for massive MIMO systems due to its lower computational complexity and hardware requirement. The remainder of this paper is organized as follows. Section II introduces the system model of MIMO-HARQ assisted V2V communications with keyhole effect. In Section III, the outage analysis are conducted to obtain the exact and asymptotic expressions of the outage probabilities. Additionally, with the asymptotic results, some profound discussions are undertaken in Section IV. In Section V, numerical results are presented for verifications. Finally, Section VI concludes this paper. Notation: The following notations will be used throughout this paper. Bold uppercase and lowercase letters are used to denote matrices and vectors, respectively. X H , X −1 , det(X) and tr(X) stand for the conjugate transpose, the inverse, the determinant and the trace of matrix X, respectively. I represents an identity matrix. · denotes the Euclidean norm of a vector. antennas, respectively. We assume that there are a number of obstacles between the transmitter and receiver, and the transmitted signal propagates through electromagnetically small apertures (or keyholes) among obstacles. By following the keyhole channel model in [16], the channel matrix H for MIMO-HARQ V2V communications is given by H = uv H =        u 1 v * 1 u 1 v * 2 · · · u 1 v * N T u 2 v * 1 u 2 v * 2 · · · u 2 v * N T . . . . . . . . . . . . u N R v * 1 u N R v * 2 · · · u N R v * N T        ,(1) where each entry of v and u follows independent and identically distributed ( i.i.d.) complex normal distribution, i.e., v ∼ CN (0, I N T ) and u ∼ CN (0, I N R ). Clearly from (1), H is a rank-one matrix in the presence of the keyhole effect. The rank-one deficiency significantly reduces the spatial multiplexing gain and degrades the MIMO capacity. To enhance the reception reliability, three types of HARQ schemes, i.e., Type-I HARQ, HARQ-CC and HARQ-IR, are employed in this paper. Therefore, the accumulated mutual information obtained by the three types of HARQ schemes after K HARQ rounds can be expressed as [34] I =            max k=1,··· ,K log 2 det I N R + γ k N T H k H k H , Type-I, log 2 det I KN R + H CC H CC H , HARQ-CC, K k=1 log 2 det I N R + γ k N T H k H k H , HARQ-IR,(2) where H k , k ∈ {1, · · · , K} represents the keyhole channel matrix for the k-th HARQ transmission round and is modelled according to (2), γ k stands for the average transmit SNR at the transmitter for the k-th round, and H CC = [ γ 1 /N T H 1 H , · · · , γ K /N T H K H ] H . Besides, H 1 , · · · , H K are assumed to be independent random matrices. Clearly from (1), since each entry of the channel matrix is expressed as a product of two complex Gaussian random variables, this complex form hinders the following outage analyses. Besides, the accumulated mutual information for the three types of HARQ schemes involves many matrix operations, such as determinant operations, product of block matrix, which further complicates the analysis. To the best of our knowledge, there is no available results for the outage performance of MIMO-HARQ V2V communications in the literature. III. ANALYSIS OF OUTAGE PROBABILITY To investigate the performance of MIMO-HARQ assisted V2V communication systems by considering the three different HARQ schemes, the outage probability is the most significant and essential performance metric. More specifically, the outage probability is defined as the probability of the event that the accumulated mutual information is less than the preset transmission rate R. Accordingly, the outage probability of MIMO-HARQ assisted V2V communication systems can be written as P out = Pr (I < R) .(3) By substituting (2) into (3), the outage analyses for the three types of MIMO-HARQ schemes will be undertaken individually in the following. Furthermore, other performance metrics such as throughput, delay, and ergodic capacity can be expressed in terms of the outage probability. For instance, the throughput of HARQ can be obtained by using [33,Eq. (5)]. In addition, the ergodic capacity is defined as the average mutual information [35]. The ergodic capacity can be expressed in terms of the MGF of the probability density function (PDF) of the mutual information I, which is determined by the distribution of I in (3) [36]. We omit the detailed discussions due to the page limitation. A. MIMO-Type-I HARQ For Type-I HARQ scheme, the erroneously received packets are discarded and negative acknowledgement is sent back to request the retransmission of the message until the maximum number of transmissions, i.e., K. By substituting the accumulated mutual information of Type I-HARQ scheme (2) into (3) and combining with MIMO keyhole channel model (1), the outage probability of MIMO-Type-I HARQ scheme can be rewritten as P T ype−I out = Pr max k=1,...,K log 2 det I N R + γ k N T H k H k H < R = K k=1 Pr log 2 1 + γ k N T u k 2 v k 2 < R = K k=1 F X k N T γ k 2 R − 1 ,(4) where the second step holds by using det(I + AB) = det(I + BA), and F X k (x) denotes the (4), the outage analysis of MIMO-Type-I HARQ scheme boils down to determining the distribution of the equivalent channel gain u k 2 v k 2 . Since u k 2 and v k 2 are central chi-square random variables with 2N R and 2N T degrees of freedom, respectively. The PDF of X k is given as [15] f cumulative distribution function (CDF) of X k = u k 2 v k 2 . FromX k (x) = 2x (N T +N R )/2−1 K τ (2 √ x) Γ (N T ) Γ (N R ) ,(5)f X k (x) = G 2,0 0,2   − N T , N R x   xΓ (N T ) Γ (N R ) .(6) Then, the CDF of X k can be derived by using [38,Eq. (26)] as 9 Finally, substituting (7) into (4) yields the closed-form expression of outage probability for the MIMO-Type-I HARQ scheme as F X k (x) = x 0 f X k (y) dy = G 2,1 1,3   1 N T , N R , 0 x   Γ (N T ) Γ (N R ) .(7)P T ype−I out = 1 (Γ (N T ) Γ (N R )) K K k=1 G 2,1 1,3   1 N T , N R , 0 N T γ k 2 R − 1   .(8) However, although the Meijer G-function in (8) is a built-in function in many popular mathematical software packages, such as MATLAB, the integral form of Meijer G-function are still complex which hampers the extraction of useful insights, such as diversity order and coding gain. To overcome this issue, an asymptotic expression of the asymptotic outage probability is obtained in the high SNR regime, as shown in the following theorem. Theorem 1. In the high SNR regime, i.e., γ k → ∞, the asymptotic outage probability of MIMO-Type-I HARQ scheme is given by P T ype−I out ≃          K k=1 N T γ k (2 R −1) N T ln γ k N T (Γ(N T )) 2 , N T − N R = 0, K k=1 Γ(τ ) Γ(N T )Γ(N R ) N T γ k (2 R −1) (N T +N R )/2−τ/2 (N T +N R )/2−τ /2 , N T − N R = 0. (9) Proof. Please see Appendix A. B. MIMO-HARQ-CC With regard to the MIMO-HARQ-CC scheme, the same packet is retransmitted, all the previously failed packets are stored for the subsequent decoding. Based on (2) together with the definition of H CC , the accumulated mutual information for the MIMO-HARQ-CC scheme can be rewritten as I CC = log 2 det I N T + K k=1 γ k N T u k 2 v k v k H .(10) Clearly, since u k 2 is a central chi-square random variable and v k v k H is a complex central Wishart matrix with one degree of freedom, the complex form of (10) makes the derivation of the distribution of I CC fairly intractable. Unfortunately, the integral transform based approaches developed in [11], [16], [39], [40] are inapplicable herein. Hence, we resort to a lower bound to approximate I CC , as shown in the following Lemma. Lemma 1. The accumulated mutual information of the MIMO-HARQ-CC scheme is lower bounded as I CC ≥ log 2 1 + K k=1 γ k N T u k 2 v k 2 . (11) Proof. By defining A n = I N T + n k=1 γ k N T u k 2 v k v k H , n ∈ {0, · · · , K − 1}, it follows that I CC = log 2 det A K−1 1 + γ K N T u K 2 v K H A −1 K−1 v K = log 2 det (A K−1 ) 1 + γ K N T u K 2 v K H A −1 K−1 v K ≥ log 2 det (A K−1 ) + γ K N T u K 2 v K 2 ≥ · · · ≥ log 2 det (A n ) + K k=n+1 γ k N T u k 2 v k 2 ,(12) where the inequality holds because A n is a positive definite matrix with all eigenvalues no less than 1. Thus we have A −1 n ≻ I N T /det (A n ). By repeatedly applying this property, (11) can be finally obtained. By substituting (11) into (3), the upper bound of outage probability for the MIMO-HARQ-CC scheme is given by P CC out ≤ Pr log 2 1 + K k=1 γ k N T u k 2 v k 2 < R = F X CC 2 R − 1 ,(13) where X CC = K k=1 γ k /N T u k 2 v k 2 = K k=1 γ k /N T X k . To determine the CDF of X CC , it is essential to obtain the distribution of the sum of multiple random variables X 1 , · · · , X K , which can be tackled by using the method of MGF [39]. As a result, the CDF of X CC is given by the following theorem. Theorem 2. The CDF of X CC for the cases of N T ≥ N R and N T < N R are given by F X CC (x) =            K k=1 N T γ k N T 1 2πi a+i∞ a−i∞ K k=1 Ψ N T ,τ +1; N T γ k s −1 s KN T +1 e sx ds, N T ≥ N R K k=1 N T γ k N R 1 2πi a+i∞ a−i∞ K k=1 Ψ N R ,τ +1; N T γ k s −1 s KN R +1 e sx ds, N T < N R ,(14) where Ψ (α, γ; z) denotes the Tricomi confluent hypergeometric function [37, eq. (9.211.4)]. It is worth mentioning that (14) can be evaluated via numerical inversion of Laplace transform [41]. Proof. Please see Appendix B. Accordingly, substituting x = 2 R − 1 into (14) gives rise to the upper bound of the outage probability for the MIMO-HARQ-CC scheme. Similarly to Theorem 1, the asymptotic outage analysis in the high SNR regime is performed for the MIMO-HARQ-CC scheme, as given by the following theorem. Theorem 3. Under high SNR, i.e., γ k → ∞, the CDF of X CC asymptotically behaves as F X CC (x) ≃                1 (KN R )! K k=1 Γ(τ ) Γ(N T ) N T γ k x N R , N T > N R , 1 (KN T )! K k=1 Γ(τ ) Γ(N R ) N T γ k x N T , N T < N R , 1 (KN T )! K k=1 ln γ k Γ(N T ) N T γ k x N T , N T = N R ,(15) and the asymptotic outage probability for the MIMO-HARQ-CC scheme can be directly obtained by replacing x in (15) with 2 R − 1. Proof. Please see Appendix C. C. MIMO-HARQ-IR Different from HARQ-CC, HARQ-IR employs code combining for joint decoding with the erroneously received packets. Besides, the redundant information is incrementally transmitted in each HARQ round. By substituting the accumulated mutual information for MIMO-HARQ-IR into (3), the corresponding outage probability is rewritten as P IR out = Pr K k=1 log 2 det I N R + γ k N T H k H k H < R = Pr K k=1 1 + γ k N T X k < 2 R ∆ = F X IR 2 R ,(16) where F X IR (x) represents the CDF of X IR = K k=1 (1 + γ k /N T X k ). It is easily found that X IR is the product of multiple independent random variables. Hence, we recourse to Mellin transform which is widely adopted to derive the distribution of the product of multiple independent random variables [11], [40]. Specifically, the Mellin transform is a kind of integral transform like the Laplace and Fourier transforms. The Mellin transform has the property that the Mellin transform of the PDF of the product of multiple independent random variables is equal to the product of their Mellin transforms. With this property, the CDF of X IR is obtained in the following theorem. Theorem 4. The CDF of X IR is written in terms of an inverse Mellin transform as F X IR (x) = 1 2πi −c+i∞ −c−i∞ x s s K k=1 G 3,1 1,3   1 s, N T , N R N T γ k   Γ (N T ) Γ (N R ) Γ (s) ds.(17) Proof. Please see Appendix D. Therefore, the outage probability of the MIMO-HARQ-IR scheme can be obtained by substituting x = 2 R into (17). Therefore, the outage probability of the MIMO-HARQ-IR scheme can also be expressed as an inverse Laplace transform, which can be numerically evaluated. However, due to the involvement of Meijer G-function, it is intractable to extract meaningful insights from (17). To address this issue, the asymptotic outage probability of the MIMO-HARQ-IR scheme in the high SNR region is derived in the following theorem. Theorem 5. Under high SNR, i.e., γ k → ∞, the outage probability of the MIMO-HARQ-IR scheme is asymptotically equal to P IR out ≃                            K k=1 Γ(τ ) N T γ k N R Γ(N T ) G 0,K+1 K+1,K+1   1, N R + 1, · · · , N R + 1 1, · · · , 1, 0 2 R   , N T > N R , K k=1 Γ(τ ) N T γ k N T Γ(N R ) G 0,K+1 K+1,K+1   1, N T + 1, · · · , N T + 1 1, · · · , 1, 0 2 R   , N T < N R , K k=1 N T γ k N R ln γ k Γ(N T ) G 0,K+1 K+1,K+1   1, N R + 1, · · · , N R + 1 1, · · · , 1, 0 2 R   , N T = N R .(18) Proof. Please see Appendix E. Comparing with the exact outage expressions for the three MIMO-HARQ schemes, the corresponding asymptotic expressions, i.e., (9), (15), and (18), are more concise and useful, which will be detailed in the next section. IV. DISCUSSIONS OF ASYMPTOTIC RESULTS To facilitate our discussion, we assume equal transmit SNRs, i.e., γ 1 = · · · = γ K = γ. By identifying the asymptotic results (9), (15), and (18) P out = (C(R)γ) −d (ln γ) KI(Nt−Nr) + o(γ −d (ln γ) KI(Nt−Nr) ),(19) where I(N t − N r ) denotes the indicator function, C(R) represents the modulation and coding gain, d stands for the diversity order, and o(γ −d ) denotes the higher order terms. In what follows, the diversity order d, and the modulation and coding gain C(R) of the three types of MIMO-HARQ schemes are individually discussed. Moreover, the asymptotic property of MIMO-HARQ schemes in the large-scale array regime is also examined. A. Diversity Order The diversity order measures the degrees of freedom of communication systems, which is defined as the declining slope of the outage probability with regard to the transmit SNR on a log-log scale in the high SNR regime as [33] d = − lim γ→∞ log (P out ) log (γ) . By substituting the asymptotic outage expressions (i.e., (9), (15), and (18)) into (20), we can obtain the diversity orders for three MIMO-HARQ schemes are equal and are given by d = Kmin (N T , N R ), where K and min(N T , N R ) represent the achievable time and spatial diversity order, respectively. As proved in [11], the diversity order of MIMO systems without keyhole effect is N T N R . However, the spatial diversity order of the MIMO-HARQ with keyhole effect is the minimum of the numbers of transmit and receive antennas. Thus, full spatial diversity order is unreachable due to the rank deficiency of the keyhole effect. This result is also consistent with [17]. Besides, as substantiated in [40], the time diversity order that can be achieved by HARQ is determined by the maximum number of transmissions, i.e., K. Clearly, full time diversity order can be achieved from using HARQ for the proposed scheme. This confirmed the validity of using HARQ to combat keyhole effect. B. Modulation and Coding Gain The modulation and coding gain C(R) quantifies how much transmit power can be reduced by using a certain modulation and coding scheme (MCS) to achieve the same outage performance. In other words, C(R) characterizes how much gain can be benefited from the adopted MCS 1 . With the asymptotic results, the explicit expression of C(R) can be obtained for different relationships between N T and N R . To simplify our discussion, we only consider the case of N T = N R . By plugging (9), (15), and (18) into (19), we can obtain the modulation and coding gain for each MIMO-HARQ scheme in the case of N T = N R as C(R) =          1 N T (2 R −1) (N T (Γ (N T )) 2 ) 1 N T , Type-I, 1 N T (2 R −1) ((KN T )!) 1 KN T (Γ (N T )) 1 N T , HARQ-CC, 1 N T g(R) − 1 KN T (Γ (N T )) 1 N T , HARQ-IR,(21) where g(R) = G 0,K+1 K+1,K+1   1, N R + 1, · · · , N R + 1 1, · · · , 1, 0 2 R   . The convexity and increasing monotonicity of g(R) w.r.t. R has been proved in [40,Lemma 4]. It is evident from (21) that the modulation and coding gain is a decreasing function w.r.t. the transmission rate R. Moreover, the coding gains of the three MIMO-HARQ schemes follow the relationship as C IR (R) ≥ C CC (R) ≥ C I (R), where C I (R), C CC (R) and C IR (R) denote the coding gains of Type-I HARQ, HARQ-CC and HARQ-IR, respectively. The relationship between C CC (R) and C I (R) directly follows from (21). Besides, C IR (R) is the maximum among them, which can be proved by using the integral form of g(R) in [40, eq. (26)]. The similar results also apply to the cases of N T > N R and N T < N R . The details are omitted here to avoid redundancy. In summary, the performance of the MIMO-HARQ schemes commonly depends on their implementation complexity. More specifically, the MIMO-HARQ-IR scheme performs the best in terms of the modulation and coding gain at the cost of its highest coding complexity, while the MIMO-Type-I HARQ scheme exhibits the worst performance due to its simple coding mechanism. Nevertheless, the MIMO-Type-I HARQ 1 For example, if there are two candidate MCSs for a MIMO-HARQ assisted V2V communication system, each having the modulation and coding gain Ci(R), where i is the index of the MCS and i = 1, 2. To achieve the same outage target Pout = ε, the average transmit SNR for each MCS is required to be ε = (C1(R)γi) −d , where the outage probability is approximated by using Pout ≈ (C(R)γ) −d according to (19) and the term ln γ is ignored due to the dominance of γ by comparing to ln γ in the high SNR regime. Thus, if we change the adopted MCS from 1 to 2 while guaranteeing the same outage target, the SNR reduction for MCS 2 is given by 10log 10 γ2 − 10log 10 γ1 ≈ 10log 10 C1(R) − 10log 10 C2(R). This result is consistent with the observation in Figs. 6 and 7. scheme has the lowest hardware requirement by comparing to the other two schemes, because it does not require extra buffer to store the failed packets. Furthermore, it is worth highlighting that the MIMO-HARQ-CC scheme can attain a balanced tradeoff between the complexity and the performance. C. Large-Scale Array Regime It is well known that the HARQ-IR scheme outperforms the HARQ-CC scheme in terms of the spectral efficiency, albeit at the cost of very high coding complexity. It is worth mentioning that the HARQ-CC scheme can achieve a balanced tradeoff between the spectral efficiency and the coding complexity. Fortunately, for large-scale antenna systems, it can be proved that the MIMO-HARQ-CC scheme can achieve a comparable performance as the MIMO-HARQ-IR scheme, as shown in the following theorem. Proof. Please see Appendix F. With Theorem 6, the MIMO-HARQ-CC scheme is more appealing for practical V2V communications due to its lower computational complexity and hardware requirement by comparing to the MIMO-HARQ-IR scheme. V. NUMERICAL ANALYSIS In this section, numerical results and Monte-Carlo simulations are presented for verifications and discussions. Unless otherwise specified, the system parameters are set as R = 3 bps/Hz, N T = N R = 2 and K = 3. Moreover, we assume that the transmit SNR are identical across all HARQ rounds, i.e., γ 1 = · · · = γ K = γ in the sequel, and the labels "Sim.", "Exa." and "Asy." represent the simulated, the exact and the asymptotic outage probabilities, respectively. should be properly chosen in practical V2V communication systems. Fortunately, the increasing monotonicity and convexity of the asymptotic outage probability with respect to R can greatly ease the optimal rate selection. Additionally, Fig. 6 shows the impacts of the transmission rate R on the modulation and coding insights. In particular, we have proved that the three MIMO-HARQ schemes can achieve the same diversity order. However, it has been shown that only full time diversity order can be achieved, while full spatial diversity order is unreachable as compared to MIMO-HARQ systems without It is obviously that the asymptotic behavior of (8) K τ (x) ≃    1 2 Γ (τ ) 1 2 x −τ , τ > 0, − ln (x), τ = 0.(23) By substituting the asymptotic expression (23) into (5), the asymptotic outage probability for MIMO-Type-I HARQ scheme can be written as P T ype−I out ≃            K k=1 N T γ k (2 R −1) 0 2x (N T +N R )/2−1 1 2 Γ(τ )( 1 2 2 √ x) −τ Γ(N T )Γ(N R ) dx, τ > 0, (−1) K K k=1 N T γ k (2 R −1) 0 2x N T −1 ln(2 √ x) Γ 2 (N T ) dx, τ = 0.(24) Accordingly, the following two cases are treated individually. A. τ > 0 For the case of τ > 0, we have P T ype−I out ≃ K k=1 Γ (τ ) N T γ k 2 R − 1 (N T +N R )/2−τ /2 Γ (N T ) Γ (N R ) ((N T + N R ) /2 − τ /2) .(25) B. τ = 0 For the case of τ = 0, one has P T ype−I out ≃ (−1) K K k=1 2 N T γ k 2 R − 1 N T Γ 2 (N T ) 1 0 x N T −1 ln 2 N T γ k (2 R − 1) x dx (a) ≃ K k=1 N T γ k 2 R − 1 N T ln (γ k ) Γ 2 (N T ) 1 0 x N T −1 dx = K k=1 N T γ k 2 R − 1 N T ln (γ k ) N T Γ 2 (N T ) ,(26) where step (a) holds by using −2 ln 2 N T /γ k (2 R − 1) x ≃ ln (γ k ). As a consequence, the asymptotic outage probability for the MIMO-Type-I HARQ scheme is derived as (9). APPENDIX B PROOF OF THEOREM 2 It is obviously found that X CC is the sum of independent random variables, which motivate us to use the method of MGF. More specifically, the MGF of the PDF of X CC = K k=1 γ k /N T X k can be derived as a product of the MGFs of the PDFs of the random variables X 1 , · · · , X K as M X CC (s) = K k=1 ∞ 0 e γ k N T xs f X k (x) dx = 2 Γ (N T ) Γ (N R ) K K k=1 ∞ 0 x (N T +N R )/2−1 e γ k N T xs K τ 2 √ x dx.(27)M X CC (s) =            K k=1 ∞ 0 t N T −1 e −t 1− γ k N T st −N R dt Γ(N T ) , N T ≥ N R , K k=1 ∞ 0 t N R −1 e −t 1− γ k N T st −N T dt Γ(N R ) , N T < N R .(28)M X CC (s) =        K k=1 − N T γ k s N T Ψ N T , τ + 1; − N T γ k s −1 , N T ≥ N R , K k=1 − N T γ k s N R Ψ N R , τ + 1; − N T γ k s −1 , N T < N R ,(29) where Ψ (α, γ; z) denotes the confluent hypergeometric function [37, eq. (9.211.4)]. Finally, by applying the inverse Laplace transform, the CDF of X CC can be obtained as F X CC (x) = L −1 {M X CC (−s)} = 1 2πi a+i∞ a−i∞ M X CC (−s) s e sx ds.(30) By substituting (29) into (30), the CDF of X CC can be obtained as (14). We thus complete the proof. (29) are asymptotically equal to Ψ N T , τ + 1; N T γ k s −1 ≃    Γ(τ ) Γ(N T ) N T γ k s −1 −τ , N T > N R , 1 Γ(N T ) ln (γ k ) , N T = N R ,(31) Ψ N R , τ + 1; N T γ k s −1 ≃ Γ(τ ) Γ(N R ) N T γ k s −1 −τ , N T < N R .(32) Then, by putting (31) and (32) into (30) along with the help of inverse Laplace transform, we can easily arrive at (15). APPENDIX D PROOF OF THEOREM 4 The Mellin transform of the PDF of X IR can be written as a product of the Mellin transforms of the PDFs of 1 + γ 1 /N T X 1 , · · · , 1 + γ K /N T X K . With this property, the Mellin transform w.r.t. X IR can be rewritten as {Mf X IR } (s) = ∞ 0 x s−1 f X IR (x) dx = K k=1 ∞ 0 1 + γ k N T x s−1 f X k (x) dx ∆ = ϕ (s) .(33) By putting (5) Γ (N T ) Γ (N R ) ∞ 0 x −1 x + N T γ k s−1 G 2,0 0,2   − N T , N R x   dx = K k=1 G 3,1 1,3   1 1 − s, N T , N R N T γ k   Γ (N T ) Γ (N R ) Γ (1 − s) .(34) According to [11,Eq. (6)], the CDF of X IR can be derived by invoking inverse Mellin transform as F G (x) = M −1 − 1 s ϕ (s + 1) (x) = 1 2πi c+i∞ c−i∞ x −s −s ϕ (s + 1) ds.(35) By substituting (34) into (35), the CDF of X IR can finally be obtained as (17). APPENDIX E PROOF OF THEOREM 5 In order to derive the asymptotic expression of the MIMO-HARQ-IR scheme, by following [37,Eq. 9.303], the Meijer G-function of (17) can be expanded as + Γ (s − N R ) Γ (N T − N R ) Γ (N R ) N T γ k N R 1 F 2 N R ; 1 + N R − s, 1 + N R − N T ; (−1) −3 N T γ k ,(36) where 1 F 2 (·; ·, ·; x) is a hypergeometric function as [37, Eq. 9.14.1]. By using the representation of the series expansion of the hypergeometric function and ignoring the higher order terms relative to γ −1 k , the asymptotic expression of (36) for N T < N R and N T > N R can be obtained as Moreover, for the case of N T = N R , by utilizing the residue theorem, the Meijer G-function in (17) can be rewritten as G 3,1 1,3   1 s, N T , N R N T γ k   = 1 2πi L Γ (s − t) Γ 2 (N R − t) Γ (t) N T γ k t dt ≃ − Res Γ (s − t) Γ 2 (N R − t) Γ (t) N T γ k t , t = N R = − d dt (t − N R ) 2 Γ (s − t) Γ 2 (N R − t) Γ (t) N T γ k t t=N R ,(38) where the first step holds by using the definition of Meijer G-function [37, Eq. (9.301)] and ignoring the higher order terms. Thus we can obtain the asymptotic expression for the case of N T = N R as G 3,1 1,3   1 s, N T , N R N T γ k   = − d dt Γ (s − t) Γ 2 (N R − t + 1) Γ (t) t=N R N T γ k N R − Γ (s − N R ) Γ 2 (N R − N R + 1) Γ (N R ) N T γ k N R ln N T γ k ≃Γ (s − N R ) Γ (N R ) N T γ k N R ln γ k .(39) Finally, by substituting (37) and (39) into (17) together with [37, Eq. (9.301)], we can obtain the asymptotic outage probability for the MIMO-HARQ-IR scheme as (18). APPENDIX F PROOF OF THEOREM 6 To prove (22), it suffices to show that det I N T + K k=1 α k v k v k H a.s. − −−− → N T →∞ K k=1 1 + α k v k 2 ,(40) where a.s. − − → denotes "almost sure convergence" and α k = γ k /N T u k 2 , k ∈ {1, · · · , K}. To proceed with the proof, we reuse the definition A n = I N T + n k=1 α k v k v k H , where n ∈ [0, K] CN ( 0 , 0I) represents the complex Gaussian vector with zero mean vector and identity covariance matrix. A ≻ B means that A − B is a positive definite matrix. i = √ −1 denotes the imaginary Fig. 1 : 1An example for HARQ-MIMO assisted V2V communications with keyhole effect. unit. The symbol "≃" denotes "asymptotically equal to". The definitions of any other notations are deferred to the place where they arise. II. SYSTEM MODEL As shown in Fig. 1, we consider a MIMO-HARQ V2V communication system under urban environments where the transmit vehicle and the receive vehicle are equipped with N T and N R Theorem 6 . 6For a large number of antenna arrays at the transmitter, i.e., N T → ∞, the gap between the accumulated mutual informations of the MIMO-HARQ-CC and the MIMO-HARQ-IR schemes converges almost surely to zero, namely, Pr lim N T →∞ (I IR − I CC ) = 0 = 1. (22) This result indicates that the MIMO-HARQ-CC scheme can provide a comparable capacity/outage performance as the MIMO-HARQ-IR scheme in the large-scale array regime. Fig. 2 : 2the three MIMO-HARQ schemes versus the transmit SNR under different relationships between the numbers of transmit and receive antennas. It is observed from Figs. 2 -4 that the exact results match well with the simulated ones for the MIMO-Type-I and the MIMO-HARQ-IR schemes. Moreover, it can be seen from Figs. 2 -4 that the approximate error between the exact and simulated outage probabilities of the MIMO-HARQ-CC scheme is acceptable. In addition, the asymptotic outage curves of MIMO-HARQ schemes are parallel to each other under high SNR. This observation is consistent with the results of the diversity order, which reflects the decreasing slope of the outage curve. Although the three MIMO-HARQ schemes have the same diversity order, the HARQ-IR scheme achieves the best outage performance among them because of its highest coding and modulation gain. Furthermore, by comparing to the MIMO-V2V communication scheme without using HARQ (labeled as "No-HARQ" in Figs. 2 -4), the proposed HARQ-assisted schemes exhibit a superior performance. The outage probability P out versus the transmit SNR γ with N T = N R = 2. Figs. 5 Fig. 3 : 53depicts the outage probability of the three MIMO-HARQ schemes versus the transmission rate R. As observed fromFig. 5, the exact outage curves coincide with the simulated outage ones except for the MIMO-HARQ-CC scheme, this is because the outage probability of the MIMO-HARQ-CC scheme is derived by using an upper bound. Besides, it can be seen that the outage probability is an increasing function of the transmission rate R, this is essentially due to the tradeoff between the throughput and reliability. Hence, the transmission rate The outage probability P out versus the transmit SNR γ with N T = 3 and N R = 2. Fig. 4 : 4The outage probability P out versus the transmit SNR γ with N T = 2 and N R = 3. Fig. 5 :Fig. 6 : 56gain C(R) for the three MIMO-HARQ schemes. As expected, the modulation and coding gain decreases with the increase of the transmission rate. Besides, one can observe that the HARQ-IR scheme obtains the maximum modulation and coding gain among the three MIMO-HARQ schemes, and the MIMO-HARQ-CC scheme performs better than the MIMO-Type-I HARQ The outage probability P out versus the transmission rate R by setting parameters as γ The modulation and coding gain C(R) versus the transmission rate R by setting parameters as N T = N R = 2 and K = 3.scheme. The observations in Fig. 6 are consistent with the asymptotic analysis in Section IV-B. To investigate the impact of the number of antennas on the outage performance, Fig. 7 plots the outage probability against the number of transmit antennas N T . It is shown that the increase of N T would improve the outage performance. Meanwhile, as the number of transmit antennas increases, there exists an outage floor for MIMO-HARQ schemes, which is only determined by the number of receive antennas. This result can be proved by applying the deterministic (40), k = 1, · · · , K. Besides, it is found that the MIMO-Type-I HARQ scheme exhibits the worst outage performance among the three MIMO-HARQ schemes for γ = 5 dB. By increasing the transmit SNR of the MIMO-Type-I HARQ scheme from 5 dB to 11.5 dB, it can reach almost the same performance as the MIMO-HARQ-IR scheme at 5 dB. Although the MIMO-HARQ-IR scheme has the lowest outage probability, it relies on larger buffer size and higher computational complexity compared to the MIMO-HARQ-CC scheme. Furthermore, as the number of transmit antennas N T increases, one can observe from Fig. 7 that the MIMO-HARQ-CC scheme achieves almost the same performance as the MIMO-HARQ-IR scheme. This result is consistent with our analysis in Section IV-C. Therefore, in MIMO-HARQ systems with large-scale antenna arrays (i.e., massive MIMO), the MIMO-HARQ-CC scheme could be the best choice due to its lower coding complexity and hardware requirement without significantly degrading the outage performance. Additionally, to understand the physical meaning of the modulation and coding gain, the case of N T = N R = 2, K = 3 and R = 3 bps/Hz is taken as an example. From Fig. 6, compared to the MIMO-Type-I HARQ scheme, the MIMO-HARQ-IR scheme improves the modulation and coding gain at R = 3 bps/Hz by roughly 6.5 dB. This observation agrees well with Fig. 7, where the MIMO-HARQ-IR scheme reduces the required SNR by 11.5 − 5 = 6.5 dB to ensure the same outage as the MIMO-Type-I HARQ scheme. Nonetheless, the obtained modulation and coding gain for the MIMO-HARQ-CC scheme (i.e., (21)) is a lower bound for its actual value, because the result is established by relying on the upper bound of the outage probability. VI. CONCLUSIONS In this paper, we have investigated the outage performance for MIMO-HARQ assisted V2V communications with keyhole effect. To compensate the performance degradation caused by the rank-deficiency of keyhole effect, three types of HARQ, i.e., Type-I, HARQ-CC, and HARQ-IR, have been employed to boost the transmission reliability. With the help of MGF and Mellin transform, we have derived the exact outage expressions for the MIMO-Type-I and MIMO-HARQ-IR schemes, and an outage upper bound has been derived for the MIMO-Type-CC scheme. Moreover, the asymptotic outage analyses have been conducted to obtain meaningful Fig. 7 : 7The outage probability P out versus the number of transmission antenna N T with N R = 2 and K = 3.keyhole effect. Furthermore, in large-scale antenna systems, we have proved that the MIMO-HARQ-CC scheme can achieve almost the same performance as the MIMO-HARQ-IR scheme.In the end, the Monte Carlo simulations have validated the analytical outcomes. 1 2 s; 1 + s − N T , 1 + s − N R ; (−1) −3 N T γ k + Γ (s − N T ) Γ (N R − N T ) Γ (N T ) F 2 N T ; N T + 1 − s, 1 + N T − N R ; (−1) −3 N T γ k s − N T ) Γ (τ ) Γ (N T ) N T γ k N T , N T < N R , Γ (s − N R ) Γ (τ ) Γ (N R ) N T γ k N R , N T > N R . where Γ(·) is the gamma function[37, Eq. (8.310)], K τ (·) represents the modified Bessel function of the τ -th order [37, Eq. (8.432.6)], and τ = |N T − N R |. In analogy to[38], it is suggested to invoke Meijer G-function to generalize our analytical results. By utilizing [37, Eq.(9.34.3)], the PDF of X k can be expressed in the form of Meijer G-function as [37, Eq. (9.301)] under high SNR, i.e., γ k → ∞, corresponds to the behavior of the PDF (5) at small x, i.e., x → 0. Based on [44, Eq. (10.30.2), Eq. (10.30.3)], the modified Bessel function for x → 0 is asymptotic to By using[37, Eq. (6.643.3), Eq. (9.222.2)], the MGF of the PDF of X CC can be expressed as With the help of [37, Eq. (9.211.4)], we can obtain the closed-form expression for the MGF of the PDF of X CC as APPENDIX C CPROOF OF THEOREM 3 By using [44, Eq. (13.2.16), Eq. (13.2.19)] and ignoring the higher order terms, the confluent hypergeometric functions in into(33) and using [37, Eq. (7.811.5)], we can obtain the Mellin transform of the PDF of X IR as ϕ (s) =K k=1 γ k N T s−1 By repeatedly applying the steps (41)-(46) to det (A K−1 ), the deterministic equivalent of det (A K−1 ) can be obtained as the recursive relationship in (46). By recursively using (46), we finally complete the proof. and we stipulate A 0 = I N T . 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Beming, 3G Evolution: HSPA and LTE for Mobile Broadband. Academic press, 2010. Hybrid ARQ in multiple-antenna slow fading channels: Performance limits and optimal linear dispersion code design. C Shen, M P Fitz, IEEE Trans. Inf. Theory. 579C. Shen and M. P. Fitz, "Hybrid ARQ in multiple-antenna slow fading channels: Performance limits and optimal linear dispersion code design," IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 5863-5883, Sept. 2011. Energy-efficient HARQ-IR for massive MIMO systems. S Kim, H Yu, IEEE Trans. Commun. 669S. Kim and H. Yu, "Energy-efficient HARQ-IR for massive MIMO systems," IEEE Trans. Commun., vol. 66, no. 9, pp. 3892-3901, Sept. 2018. Performance and delay analysis of hybrid ARQ with incremental redundancy over double Rayleigh fading channels. A Chelli, E Zedini, M Alouini, J R Barry, M Pätzold, IEEE Trans. Wireless Commun. 1311A. Chelli, E. Zedini, M. Alouini, J. R. Barry, and M. 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Communications (ICC'07), Glasgow, UK, Jun. 2007, pp. 750-755. Outage information rate of doubly correlated multi-cluster scattering MIMO channels. G Taricco, G Alfano, IEEE Wireless Commun. Lett. 76G. Taricco and G. Alfano, "Outage information rate of doubly correlated multi-cluster scattering MIMO channels," IEEE Wireless Commun. Lett., vol. 7, no. 6, pp. 1042-1045, Dec. 2018. A Jeffrey, D Zwillinger, Table of Integrals, Series, and Products. ElsevierA. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products. Elsevier, 2007. The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce system. V Adamchik, O Marichev, Proc. IEEE ISSAC. IEEE ISSACTokyo, JapanV. Adamchik and O. Marichev, "The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce system," in Proc. IEEE ISSAC, Tokyo, Japan, Aug. 1990, pp. 212-224. Achievable diversity order of HARQ-aided downlink NOMA systems. Z Shi, C Zhang, Y Fu, H Wang, G Yang, S Ma, IEEE Trans. Veh. Technol. 691Z. Shi, C. Zhang, Y. Fu, H. Wang, G. Yang, and S. Ma, "Achievable diversity order of HARQ-aided downlink NOMA systems," IEEE Trans. Veh. Technol., vol. 69, no. 1, pp. 471-487, Jan. 2020. Asymptotic outage analysis of HARQ-IR over time-correlated Nakagami-m fading channels. Z Shi, S Ma, G Yang, K Tam, M Xia, IEEE Trans. Wireless Commun. 169Z. Shi, S. Ma, G. Yang, K. Tam, and M. Xia, "Asymptotic outage analysis of HARQ-IR over time-correlated Nakagami-m fading channels," IEEE Trans. Wireless Commun., vol. 16, no. 9, pp. 6119-6134, Sept. 2017. Numerical inversion of Laplace transforms of probability distributions. J Abate, W Whitt, ORSA Journal on Computing. 71J. Abate and W. Whitt, "Numerical inversion of Laplace transforms of probability distributions," ORSA Journal on Computing, vol. 7, no. 1, pp. 36-43, Feb. 1995. D Tse, Fundamentals of Wireless Communication. Cambridge university pressD. Tse, Fundamentals of Wireless Communication. Cambridge university press, 2005. A simple and general parameterization quantifying performance in fading channels. Z Wang, G B Giannakis, IEEE Trans. Commun. 518Z. Wang and G. B. Giannakis, "A simple and general parameterization quantifying performance in fading channels," IEEE Trans. Commun., vol. 51, no. 8, pp. 1389-1398, Aug. 2003. Nist digital library of mathematical functions. D W Lozier, Annals of Mathematics and Artificial Intelligence. 381D. W. Lozier, "Nist digital library of mathematical functions," Annals of Mathematics and Artificial Intelligence, vol. 38, no. 1, pp. 105-119, 2003. R Couillet, M Debbah, Random Matrix Methods for Wireless Communications. Cambridge University PressR. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications. Cambridge University Press, 2011.

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{'abstract': 'Vehicle-to-vehicle (V2V) communications under dense urban environments usually experience severe keyhole fading effect especially for multi-input multi-output (MIMO) channels, which degrades the capacity and outage performance due to the rank deficiency. To avoid these, the integration of MIMO and hybrid automatic repeat request (HARQ) is proposed to assist V2V communications in this paper.By using the methods of integral transforms, the outage probabilities are derived in closed-form for different HARQ-assisted schemes, including Type I-HARQ, HARQ with chase combining (HARQ-CC), and HARQ with incremental redundancy (HARQ-IR). With the results, meaningful insights are gained by conducting the asymptotic outage analysis. Specifically, it is revealed that full time diversity order can be achieved, while full spatial diversity order is unreachable as compared to MIMO-HARQ systems without keyhole effect. Moreover, we prove that the asymptotic outage probability is a monotonically increasing and convex function of the transmission rate. More importantly, although HARQ-IR performs better than HARQ-CC owing to its higher coding complexity, this advantage becomes negligible in the large-scale array regime. Finally, the numerical results are verified by Monte-Carlo simulations along with some in-depth discussions.Index TermsHuan Zhang and Zhengtao Liao are co-first authors. ). 2 hybrid automatic repeat request (HARQ), keyhole effect, MIMO, outage probability, V2V communications.', 'arxivid': '2209.12783', 'author': ['Huan Zhang ', 'Zhengtao Liao s:[email protected] ', 'Zheng Shi [email protected] ', 'Guanghua Yang [email protected] ', 'Qingping Dou [email protected] ', 'Shaodan Ma [email protected]. ', 'Huan Zhang ', 'Shaodan Ma ', 'Zhengtao Liao ', 'Zheng Shi ', 'Guanghua Yang ', 'Qingping Dou ', '\nDepartment of Electrical and Computer Engineering\nSchool of Intelligent Systems Science and Engineering\nare with the State Key Laboratory of Internet of Things for Smart City\nUniversity of Macau\n999078MacaoChina\n', '\nJinan University\n519070ZhuhaiChina\n'], 'authoraffiliation': ['Department of Electrical and Computer Engineering\nSchool of Intelligent Systems Science and Engineering\nare with the State Key Laboratory of Internet of Things for Smart City\nUniversity of Macau\n999078MacaoChina', 'Jinan University\n519070ZhuhaiChina'], 'corpusid': 247905944, 'doi': '10.1109/tcomm.2022.3163779', 'github_urls': [], 'n_tokens_mistral': 20900, 'n_tokens_neox': 18152, 'n_words': 10780, 'pdfsha': 'd048961c0153bcf5fe8046b89cb5453a22d0dcc7', 'pdfurls': ['https://export.arxiv.org/pdf/2209.12783v1.pdf'], 'title': ['Performance Analysis of MIMO-HARQ Assisted V2V Communications With Keyhole Effect', 'Performance Analysis of MIMO-HARQ Assisted V2V Communications With Keyhole Effect'], 'venue': []}

arxiv

VOCAL SIGNAL DIGITAL PROCESSING. INSTRUMENT FOR ANALOG TO DIGITAL CONVERSION STUDY Ovidiu-Andrei Schipor [email protected] Universitatea "Ştefan cel Mare" Suceava Str. Universităţii nr.9720225SuceavaRomania Felicia-Florentina Gîză Universitatea "Ştefan cel Mare" Suceava Str. Universităţii nr.9720225SuceavaRomania VOCAL SIGNAL DIGITAL PROCESSING. INSTRUMENT FOR ANALOG TO DIGITAL CONVERSION STUDY PCM methodvocal signalssignal representation The goal of this article is to present interactive didactic software for analog to digital conversion using PCM method. After a short introduction regarding vocal signal processing we present some method for analog to digital conversion. The didactic software is an applet that can be direct accessed by any interested person. 1.Prelucrarea numerică a semnalelor Procesarea numerică de semnal (PNS) se referă la o varietate de tehnici de îmbunătăţire a acurateţei şi siguranţei comunicaţiilor digitale. Teoria care stă la baza prelucrării numerice de semnal este suficient de complexă, însă la baza acestei tehnici de procesare se găseşte stabilirea şi standardizarea nivelelor şi a stărilor unui semnal digital. Semnalele analogice reprezintă o mărime care se modifică in mod continuu (de exemplu intensitatea sonoră a unui ton). În schimb, semnalele digitale pot prelua numai anumite valori (discrete), de exemplu valorile 0 şi 1 respectiv "sub tensiune" sau "fără tensiune". Toate circuitele de comunicaţie conţin zgomot. Acest lucru este valabil indiferent dacă semnalele sunt analogice sau digitale şi indiferent de tipul de informaţie transmis. Zgomotul este principala sursă de neplăceri pentru inginerii de comunicaţie care încearcă mereu să găsească noi metode de îmbunătăţire a raportului semnal/zgomot (S/N) în sistemele de comunicaţie. Metodele tradiţionale de optimizare a raportului semnal/zgomot includ creşterea puterii semnalului transmis şi creşterea sensibilităţii receptorului de semnal. Principalul avantaj al utilizării semnalelor digitalizate este că orice prelucrare ulterioară a acestora este în principiu lipsită de pierderi de informaţie (numerele nu sunt afectate de zgomot iar precizia calculelor poate fi matematic controlată). Un sistem de prelucrarea numerică a semnalelor îndeplineşte în esenţă un ansamblu de operaţii şi anume: -conversia semnalului analogic în semnal numeric prelucrarea semnalului numeric obţinut -conversia semnalului numeric prelucrat în semnal analogic. Dacă un semnal de intrare este analog, acesta este mai întâi convertit într-o formă digitală de un convertor analog-digital (CAD). Semnalul rezultat are două sau mai multe nivele. Ideal, aceste nivele sunt cunoscute exact, reprezentând curenţi sau tensiuni. Totuşi, deoarece semnalul de intrare conţine zgomot, nivelele nu sunt întotdeauna egale cu valorile standard. Circuitele de prelucrare a semnalului ajustează aceste nivele astfel încât să reprezinte valorile corecte. De fapt acestea elimină zgomotul. 2.Tehnici de conversie analog-numerică Conversia analogică digitală este un proces electronic în care un semnal variabil continuu în timp şi amplitudine (analog) este transformat (cu o eroare controlată), într-un semnal digital. Intrarea unui convertor analogic digital (CAD) constă într-un nivel de tensiune care variază într-un interval infinit de valori. Exemple sunt: formele de undă sinusoidale, formele de undă care reprezintă vorbirea umană, semnalele de la o cameră video etc. Semnalele digitale sunt propagate mult mai eficient decât semnalele analogice, în mare parte datorită faptului că impulsurile digitale, care sunt foarte bine definite şi ordonate, sunt mult mai uşor de deosebit de zgomot, care este haotic. Acesta este avantajul principal al modurilor de comunicare digitală. 2.1.Modulaţia în cod a impulsurilor (PCM) Marele progres în fabricarea componentelor electronice, în special a circuitelor înalt integrate şi cel mai înalt integrate, îl reprezintă utilizarea frecventă a tehnicii numită Pulse Code Modulation (PCM -modulaţia impulsurilor în cod) în tehnica informaţiilor şi în electronică, precum şi în electronica de divertisment, ca de exemplu la video-disc şi la discul digital (compact-disc). PCM este unul din cele mai vechi şi conceptual cele mai simple procese de conversie de la analogic la digital utilizate în semnalele vocale şi video. În cazul modulării în cod a impulsurilor (PCM), semnalele care se prelucrează (de exemplu tonuri sonore) nu se prezintă sub forma oscilaţiilor, ci sub forma numerelor binare formate din mai mulţi biţi care au fost obţinute ca rezultat al eşantionării şi cuantizării cursului oscilaţiilor. Eşantionarea are loc de cele mai multe ori cu ajutorul unui aşa-numit circuit sample-and-hold (eşantionare-şi-menţinere) care înregistrează mărimile continue de intrare (de exemplu valorile tensiunii unei oscilaţii electrice) sub formă de semnale periodice de foarte scurtă durată. Semnalul format prin eşantionare este transformat apoi într-o mărime digitală de ieşire prin intermediul unităţilor de cuantizare şi codare. Cu ajutorul modulării în cod a impulsurilor (PCM) este posibilă recunoaşterea, acoperirea sau corectarea transmiterii erorilor in cazul transmiterii semnalului. Valorile răspunzătoare de erori pot fi eliminate prin prelucrarea semnalului şi înlocuire prin media valorilor corecte învecinate. 2.2.Modulaţia numerică diferenţială În cadrul modulaţiei numerice diferenţiale (DNUM), în locul informaţiei despre un anumit eşantion se transmite o informaţie despre diferenţa dintre acesta şi un eşantion determinat prin predicţie. Cele mai cunoscute tehnici de modulaţie numerică diferenţială sunt: -modulaţia diferenţială a impulsurilor în cod (DPCM) la care sunt generate doar diferenţele dintre amplitudini consecutive (în acest mod se diminuează rata de transfer necesară); -modulaţia delta (M) la care se generează un singur bit de diferenţă între amplitudini succesive (acest bit indică creşterea / descreşterea eşantionului curent faţă de eşantionul precedent). 2.2.Modulaţia numerică adaptativă În cadrul modulaţiei numerice adaptive (ADPCM) se realizează corespondenţa de la eşantion la alfabet funcţie de istoria semnalului. În acest scop se constituie o stare S n a sistemului în intervalul nT, iar corespondenţa între valoarea eşantionului x(nT) şi valoarea discretă y k se va face ţinând cont şi de această stare. Acest procedeu de modulaţie este cel mai eficient. 3.Conversia analog-numerică prin metoda PCM 3.1.Eşantionarea Fie x a (t) un semnal analogic (continuu în timp) şi {t n } n o mulţime numărabilă de valori reale distincte ordonate (t n < t m dacă n < m). Eşantionarea este transformarea semnalului x a (t) în semnalul discret x[n] definit prin relaţia: x[n] = x a (t n ) (1) Eşantionarea uniformă este dată de relaţia: x[n] = x a (nT) (2) unde T>0 este perioada de eşantionare, iar t n = n*T. Figura 1. Eşantionarea (semnal sinusoidal, doua perioade, 20 eşantioane) Prin eşantionare reţinem numai valorile continue în amplitudine şi discrete în timp. În mod ideal, eşantionarea nu are ca rezultat o pierdere de informaţie şi nici nu introduce distorsiuni în semnal dacă sunt respectate condiţiile teoremei eşantionării. Un semnal analogic poate fi refăcut din eşantioanele sale dacă a fost eşantionat la o frecvenţă de cel puţin două ori mai mare decât lărgimea de bandă a semnalului analogic (lărgimea de bandă = frecvenţa superioară -frecvenţa inferioară). De exemplu, pentru muzică, al cărui domeniu de frecvenţe se situează între 20 si 20 000 Hz (limitele analizatorului auditiv uman), sunt necesare deci cel puţin 40 000 de eşantioane pe secundă. Pentru compact-disc, semnalul se eşantionează, de exemplu, cu 44 100 de eşantioane pe secundă si astfel sunt reţinute tot atât de multe valori pe secundă. Vocea umană, poate fi redată optim prin sunete cu frecvenţe cuprinse între 100 şi 8.000 Hz.(limitele aparatului fonoarticulator). Acesta este motivul pentru care sistemele de telefonie au o gamă de frecvenţe de răspuns relativ îngustă, eliminând sunetele de înaltă frecvenţă. Drept rezultat, sunetul înregistrat de un sistem de recunoaştere a vorbirii poate fi eşantionat la o rată minimă de numai 8kHz, cu toate că 16kHz ar putea oferi rezultate mai bune, dacă sistemul dispune de suficientă putere de procesare şi de stocare de date. La ieşirea dispozitivului de eşantionare se obţine o secvenţă de impulsuri. Amplitudinea fiecărui impuls este proporţională cu amplitudinea semnalului de intrare analogic în momentul eşantionării. Din acest motiv, acest pas se numeşte modulaţie de amplitudine a impulsurilor. 3.1.Cuantizarea Cuantizarea reprezintă transformarea semnalului continuu în amplitudine şi discret în timp într-un semnal discret în timp şi în amplitudine. Este un proces ireversibil care transformă amplitudinile notate cu x[n] sau x a [nT] în valori y k dintr-un set finit de valori. Fie D domeniul semnalului de intrare. Acesta este împărţit în L intervale: I k = {x k < x[n] <= x k+1 }, k = 1, 2 … L (3) Având în vedere acest lucru, se obţin nivelele de cuantizare notate cu y 1 , y 2 ...y k astfel: x q [n] = Q(x[n]) = y[n] = y k , pentru x[n]  I k. , iar Q(x[n]) reprezentând intervalul I k în care se găseşte x[n]. numărului de nivele de cuantizare necesită o rată de transfer mai mare. Gama dinamică (diferenţa de volum dintre cel mai slab şi cel mai puternic sunet) pentru vocea umană este mult mai mică decât pentru muzica de înaltă calitate. În cele mai multe cazuri este nevoie de doar 8 biţi pe eşantion cu toate că rezultatele sunt mult mai bune în cazul folosirii a 16 biţi pe eşantion, care este şi numărul de biţi caracteristic CD-urilor audio. Diferenţele dintre ratele de eşantionare şi numărul de biţi pot avea un impact major asupra cantităţii de date pe care calculatorul trebuie să o proceseze. O secundă de sunet digital la 8 kHz şi 8 biţi pe eşantion înseamnă doar 8.000 octeţi de date. Aceeaşi secundă de sunet digital la 16 kHz şi 16 biţi înseamnă de patru ori mai multe date: 32.000 octeţi. Standardul pentru CD-uri audio, de 44 kHz şi 16 biţi, înseamnă că o secundă de sunet necesită un spaţiu de stocare de 88.000 octeţi. Pentru rezultate superioare, cuantizarea nu ar trebui efectuată uniform. Unele semnale sunt de amplitudini joase, iar altele sunt de amplitudini ridicate. În practică, cuantizarea este neuniformă, existând mai multe nivele de cuantizare pentru amplitudinile care predomină în cadrul semnalului. Pentru o transmisie a semnalelor inteligibilă şi de o calitate acceptabilă a comunicaţiei se poate realiza reducerea vitezei de kbit/s cu algoritmi de codare şi de cuantizare vectorială. Scopul acestor algoritmi este de a transmite, memora şi sintetiza semnalul vocal de o calitate dată, utilizând mai puţini biţi. Această reducere este realizată eliminând redundanţa din semnalul vocal 4. Instrument didactic pentru studiul conversiei analog-numerice prin metoda PCM Apletul a fost realizat utilizând limbajul Java, versiunea Sun SDK 1.4. O parte din clasele utilizate au fost realizate de către autori în cadrul bursei de studiu ERASMUS-SOCRATES EUDIL-Lille Franta. Interfaţa este realizată utilizând clasele din pachetul Abstract Window Toolkit.Diferenţa dintre x[n] şi x q [n] se numeşte eroare de cuantizare sau zgomot de cuantizare: e q = x[n] -x q [n] (5) Eroarea de cuantizare nu poate depăşi o jumătate din pasul de cuantizare: 2   ≤ e q ≤ 2  (6) unde  = pas de cuantizare. În cazul în care eroarea de cuantizare depăşeşte limitele admise, trebuie mărit numărul de nivele de cuantizare. Figura 2. Cuantizarea (8 nivele de cuantizare) 3.3.Codarea Codarea este procesul prin care fiecărei valori discrete x q [n] i se atribuie o secvenţă egală cu b biţi. Pentru codificarea celor k nivele de cuantizare posibile sunt necesari log 2 k biţi. Apropierea necesară faţă de procesul oscilatoriu reclamă o gradare fină a valorilor măsurătorilor rezultate prin eşantionare, rezultând deci şiruri relativ lungi de biţi (lungimi de cuvinte). Astfel, printr-o lungime de cuvânt de 4 biţi, pot fi redate numai 7% din procesele oscilatorii. Figura 3. Codificarea (în cazul folosirii a 16 nivele de cuantizare) Pentru un semnal cu frecvenţa de eşantionare de f Hz şi k nivele de cuantizare este necesară o rată de transfer: RT = f * log 2 k bps (7) De exemplu pentru transmiterea unui semnal eşantionat cu rata de eşantionare de 8 KHz şi codat pe 8 biţi (adică are 256 nivele de cuantizare) este necesară o rată de transfer de 64 Kbps. Se observă că o creştere a Prin conversia analog-numerică, semnalul continuu este eşantionat şi cuantizat, fiecare nivel obţinut fiind reprezentat printr-un cuvânt binar care se aplică la intrarea sistemului de prelucrare numerică. Această primă fază a prelucrării numerice este realizată cu convertoare specializate analognumerice şi este deosebit de importantă, deoarece prin aproximările pe care le efectuează, contribuie la precizia de calcul şi la raportul semnal zgomot final. Apletul permite generarea unui semnal periodic nesinusoidal pe baza primelor 6 armonici. Ulterior, acest semnal este convertit numeric, pas cu pas, evidenţiindu-se aspectele esenţiale. Apletul se utilizează în modul următor. Apletul permite generarea unui semnal periodic nesinusoidal pe baza primelor 6 armonici. Ulterior, acest semnal este convertit numeric, pas cu pas, evidenţiindu-se aspectele esenţiale. Apletul se utilizează în modul următor: Se introduc coeficienţii A1…A6 (sin) şi B1…B6 (cos) ai primelor 6 armonici ale semnalului analogic. Se introduc coeficienţii A1…A6 (sin) şi B1…B6 (cos) ai primelor 6 armonici ale semnalului analogic. Se introduce frecvenţa armonicii fundamentale (F1) în format mantisă -ordin de marime. Se introduce frecvenţa armonicii fundamentale (F1) în format mantisă -ordin de marime Se introduce numărul de perioade (Perioade) ale armonicii fundamentale ce se doresc vizualizate. Se introduce numărul de perioade (Perioade) ale armonicii fundamentale ce se doresc vizualizate. Se introduce componenta constantă (U_ctn). Se introduce componenta constantă (U_ctn). Se introduce numărul eşantioanelor de timp (Eşantioane). Se introduce numărul eşantioanelor de timp (Eşantioane). Se introduce numărul de biţi utilizaţi pentru cuantizare (Biţi). Se introduce numărul de biţi utilizaţi pentru cuantizare (Biţi). Se apasă butonul <OK> pentru a vizualiza semnalul analogic. Se apasă butonul <OK> pentru a vizualiza semnalul analogic. Butoane de ajutor: <Autorii> şi <Ajutor>. Opt butoane permit o mai bună vizualizare a graficelor. ^ , V , &gt; , &lt; -Pentru Deplasarea Graficului, Se apasă butoanele <înainte> şi <înapoi> pentru vizualizarea succesivă a etapelorSe apasă butoanele <înainte> şi <înapoi> pentru vizualizarea succesivă a etapelor. Butonul <RST> permite aducerea apletului în starea iniţială. Butoane de ajutor: <Autorii> şi <Ajutor>. Opt butoane permit o mai bună vizualizare a graficelor: -^ , v , > , < -pentru deplasarea graficului; pentru zoom pozitiv şi negativ pe axa OX (a timpului) şi OY (a valorii semnalului. -X+ , X- , Y+ , Y- , -X+ , X-, Y+ , Y-pentru zoom pozitiv şi negativ pe axa OX (a timpului) şi OY (a valorii semnalului). Pentru o utilizare rapidă (fără introducere de date) se poate selecta un exemplu: -sinusoidă -un semnal sinusoidal pe 2 perioade; -triunghiular -un semnal triunghiular pe 2 perioade; -dreptunghiular -un semnal dreptunghiular pe 4 perioade. Pentru o utilizare rapidă (fără introducere de date) se poate selecta un exemplu: -sinusoidă -un semnal sinusoidal pe 2 perioade; -triunghiular -un semnal triunghiular pe 2 perioade; -dreptunghiular -un semnal dreptunghiular pe 4 perioade; -o perioadă -un semnal oarecare pe o perioadă. -o perioadă -un semnal oarecare pe o perioadă. Comunicaţii digitale avansate. Kamilo Dr, Feher, Editura Tehnică. Dr. Kamilo Feher (1993) -Comunicaţii digitale avansate, Editura Tehnică, Bucureşti, Romania Constantin Ioan, Marghescu Ion, Transmisiuni analogice şi digitale. Bucureşti, RomaniaConstantin Ioan, Marghescu Ion (1995) -Transmisiuni analogice şi digitale, Editura Tehnică, Bucureşti, Romania Măsurări electrice şi electronice. Cornelia Marcuta, Mihai Creţu, Editura Tehnică-Info. Cornelia Marcuta, Mihai Creţu (2002) -Măsurări electrice şi electronice, Editura Tehnică-Info, Chişinău, Republica Moldova Error rate characteristics of oversampled analog to digital conversion. Zoran Cvetkovic, Martin Vetterli, IEEE Transactions on Information Theory. 445Zoran Cvetkovic, Martin Vetterli (2004) -Error rate characteristics of oversampled analog to digital conversion, IEEE Transactions on Information Theory vol.44, nr. 5 Current techniques of measurement, acquisition and processing test data. A L Wicks, Department of Mechanical Engineering Virginia Tech site-uri webA.L. Wicks (2003) -Current techniques of measurement, acquisition and processing test data, 2003, Department of Mechanical Engineering Virginia Tech site-uri web

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{'abstract': 'The goal of this article is to present interactive didactic software for analog to digital conversion using PCM method. After a short introduction regarding vocal signal processing we present some method for analog to digital conversion. The didactic software is an applet that can be direct accessed by any interested person.', 'arxivid': '1405.7866', 'author': ['Ovidiu-Andrei Schipor [email protected] \nUniversitatea "Ştefan cel Mare"\nSuceava Str. Universităţii nr.9720225SuceavaRomania\n', 'Felicia-Florentina Gîză \nUniversitatea "Ştefan cel Mare"\nSuceava Str. Universităţii nr.9720225SuceavaRomania\n'], 'authoraffiliation': ['Universitatea "Ştefan cel Mare"\nSuceava Str. Universităţii nr.9720225SuceavaRomania', 'Universitatea "Ştefan cel Mare"\nSuceava Str. Universităţii nr.9720225SuceavaRomania'], 'corpusid': 351994, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 6354, 'n_tokens_neox': 5976, 'n_words': 2475, 'pdfsha': '37835c6f951a6140055aad51c8a07d56025ddcf5', 'pdfurls': ['https://arxiv.org/pdf/1405.7866v1.pdf'], 'title': ['VOCAL SIGNAL DIGITAL PROCESSING. INSTRUMENT FOR ANALOG TO DIGITAL CONVERSION STUDY', 'VOCAL SIGNAL DIGITAL PROCESSING. INSTRUMENT FOR ANALOG TO DIGITAL CONVERSION STUDY'], 'venue': []}

arxiv

MORSE POSITION OF KNOTS AND CLOSED INCOMPRESSIBLE SURFACES 13 Nov 2006 Makoto Ozawa MORSE POSITION OF KNOTS AND CLOSED INCOMPRESSIBLE SURFACES 13 Nov 2006 In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have following results.• Smallness of Montesinos tangles with length two and Kinoshita's theta curve• Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions 1991 Mathematics Subject Classification. Primary 57M25, 57M50; Secondary 57Q35, 57Q37. Introduction Morse theory had a tremendous impact on low-dimensional topology. Bridge positions of knots that developed by Schubert ([7]) and Heegaard splittings of 3manifolds that developed by Heegaard ([3]) are independent of Morse theory, but those can be explained integrally by Morse theory. A bridge position of a knot is defined by using a Morse function from the 3sphere to the 1-dimensional Euclidean space with two critical points so that any maximal critical point of the knot is situated above any minimal critical point. This general idea was expanded to a notion of thin position by Gabai, and he settled the Property R conjecture ( [1]). Thin position also brought a celebrated solution of the knot complement conjecture by Gordon and Luecke ( [2]) In this paper, we show that we can isotope a closed incompressible surface to a related Morse position in the complement of a knot placed in a bridge or thin position. Then as an applications, we show smallness of Montesinos tangles with length two and Kinoshita's theta curve, and classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions. Result Let K be a knot in the 3-sphere S 3 and h : S 3 → R be the composition of the inclusion S 3 ⊂ R 4 and the height function R 4 → R defined by (x, y, z, w) → w. We say that K is in a Morse position with respect to h if the restriction of h to K is a Morse function. Let a 0 , . . . , a n be the critical points of K which are labeled so that the corresponding critical values t i = h(a i ) satisfy t i−1 < t i for all i. For a regular value s i ∈ (t i−1 , t i ), P i = h −1 (s i ) is called a level sphere between a i−1 and a i . The width of K is w(K) = n i=1 |P i ∩ K|. A knot K is in thin position if w(K) is minimal up to isotopy of K. A level sphere P i between a i−1 and a i is said to be thin if a i−1 is a local maximum and a i a local minimum. Similarly, a level sphere P i between a i−1 and a i is said to be thick if a i−1 is a local minimum and a i a local maximum. A knot K is in bridge position if there is only one thick level sphere P n+1 2 . Then, the bridge number of K is b(K) = |P n+1 2 ∩ K|/2. A knot K is in minimal bridge position if b(K) is minimal up to isotopy of K. For each thick level sphere P i between a i−1 and a i , we define the thick region of K as h −1 ([t i−1 + ǫ, t i − ǫ]) for a sufficiently small positive real number ǫ. Then, no thick region contains a maximum or minimum of K, and the rest of all thick regions in S 3 contain all maxima and minima of K, each of which is called a thin region of K. Any thick region contains only one thick level sphere, and any thin region, except for the top 3-ball and the bottom 3-ball, contains only one thin level sphere. Let M thin be the disjoint union of all thin regions of K, and M thick the disjoint union of all thick regions of K. Let F be a closed surface in S 3 disjoint from K or intersecting K transversely. We say that F is in a Morse position with respect to h if the restriction of h to F is a Morse function. The critical points of K and F are assumed to be mutually disjoint. Moreover, we say that F is in a Morse position related to K if the following conditions are satisfied. (1) F is in a Morse position with respect to h, (2) F and K are in general position and F ∩ K is contained in M thick , (3) all maxima and minima of F are contained in M thin , and all saddles of F are contained in M thick . These three conditions are easily satisfied for any closed surface F . We prepare definitions to describe the next condition (4). Let M be a 3-manifold, T a 1-manifold properly embedded in M , and F a surface properly embedded in M such that F is disjoint from T or intersects T transversely in the interior of F . An isotopy φ t of F in (M, T ) should be assumed that φ t (F ) and T are in general position for all t ∈ [0, 1]. We recall the definitions about incompressible surfaces and meridionally incompressible surfaces. First, suppose that there exists a disk D in M −T such that D ∩F = ∂D and ∂D is essential in F − T . Then by cutting F along ∂D and pasting two parallel copies of D, we have a new surface F ′ . We say that F ′ is obtained from F by compressing F along D, and D is called a compressing disk for F . A surface F is incompressible in (M, T ) if for any disk D in M − T with D ∩ F = ∂D, there exists a disk D ′ in F − T such that ∂D ′ = ∂D. In the case that F is a disk disjoint from T , F is said to be incompressible in (M, T ) if for any disk D in ∂M − ∂T with ∂D = ∂F , F ∪ D does not bound a 3-ball in M − T . In the case that F is a 2-sphere disjoint from T , F is said to be incompressible in (M, T ) if F does not bound a 3-ball in M − T . Next, suppose that there exists a disk D in M such that D ∩ F = ∂D and D intersects T in one point of the interior of D. Then by cutting F along ∂D and pasting two parallel copies of D, we have a new surface F ′ . If F is incompressible in (M, T ), then F ′ is also incompressible in (M, T ). We say that F is meridionally incompressible in (M, T ) if for any disk D in M such that D ∩ F = ∂D and D intersects T in one point of the interior of D, there exists a disk D ′ in F such that ∂D ′ = ∂D and D ∪ D ′ bounds a trivial (3-ball, arc)-pair in (M, T ). In the case that F is a disk intersecting T in one point in the interior of F , we say that F is meridionally incompressible in (M, T ) if for any disk D in ∂M with ∂D = ∂F , F ∪ D does not bound a trivial (3-ball, arc)-pair in (M,T). In the case that F is a 2-sphere intersecting T in two points, we say that F is meridionally incompressible in (M, T ) if F does not bound a trivial (3-ball, arc)-pair in (M, T ). We note that this definition is slightly different from a usual one as a matter of convenience. Suppose that K ⊂ S 3 is a knot in a Morse position with respect to h and F ⊂ S 3 is a closed surface in a Morse position related to K. Furthermore, we say that F is in an essential Morse position related to K if the following condition is satisfied. (4) each component of F ∩ M thin and F ∩ M thick is incompressible and meridionally incompressible in (M thin , K ∩ M thin ) and (M thick , K ∩ M thick ) respectively. The following main theorem has a philosophy that makes thin regions M thin simple, but thick regions M thick essential. Theorem 2.1. Let K ⊂ S 3 be a knot in a Morse position with respect to h and F ⊂ S 3 be a closed surface disjoint from K or intersecting K transversely such that F is incompressible and meridionally incompressible in (S 3 , K). Then F can be isotoped so that; (1) F is a thin level sphere, or (2) F is in an essential Morse position related to K. Example 2.2. Let K be a composite knot of two trefoils, and F be the decomposing sphere for K. If we put K in a bridge position, then by Theorem 2.1, F can be isotoped so that F is in an essential Morse position related to K. See Figure 1. E(K) = S 3 − intN (K) is small, a tangle (B, T ) is small if B − intN (T ) is small, and a graph G ⊂ S 3 is small if S 3 − intN (G) is small. A Montesinos tangle T (r 1 , . . . , r n ) is defined as a series of n rational tangles of slope r i (i = 1, . . . , n). See Figure 3. We note that if r i = ∞ for some i, then T (r 1 , . . . , r n ) is splittable by a properly embedded disk, and that if r i is an integer, then the length n of T (r 1 , . . . , r n ) can be taken short. For this reason, it is reasonable to assume that r i = ∞ and r i is not an integer for all i. Example 2.5. As a corollary of Theorem 2.3, we can show that Kinoshita's theta curve is small. Kinoshita's theta curve is known as a most popular theta curve, which is normally illustrated as Figure 4. It is known to be minimally knotted and hyperbolic. Therefore, the exterior is ∂-irreducible. We shorten an edge and obtain an another form of Kinoshita's theta curve as in Figure 5. Then, the 2-string tangle on the left-hand side is a Montesinos tangle with length two and the right-hand side is a trivial H-shaped graph tangle. Hence, the exterior of Kinoshita's theta curve is homeomorphic to the exterior of the Montesinos tangle. This fact was also observed by Wu ([8, Example 3]). Therefore, Kinoshita's theta curve is small by Theorem 2.3. Let G be a theta curve or handcuff graph embedded in S 3 . We say that G is in a Morse position with respect to h if the restriction of h to each edge of G is a Morse function. In this paper, we assume that near each vertex v of G, three edges either go up or go down from v. We say that a theta curve or handcuff graph G in a Morse position is 2-bridge if there exists a level 2-sphere P which intersects G in 5 points other than vertices and critical points, and edges of G have exactly two critical points. Let F ⊂ S 3 be a closed surface disjoint from G or intersecting edges of G transversely. Concerning the definition for F to be meridionally incompressible, we say that F is meridionally incompressible in (S 3 , G) if F is not parallel to ∂N (v) for a vertex v of G, together with the definition for knot case. The next theorem asserts there is no incompressible and meridionally incompressible surface of positive genus in the complement of 2-bridge theta curves. Theorem 2.6. Let G be a 2-bridge theta curve or handcuff graph in S 3 , and F be an incompressible and meridionally incompressible surface. Then F is isotopic to a sphere with one maximum and one minimum illustrated in Figure 6. As a corollary of Theorem 2.6, we obtain the Motohashi's result ( [5]) about composite 2-bridge theta curves. A theta curve G is composite if there exists a 2-sphere which intersects each edge of G in one point and decompose G into two non-trivial theta curve. By Theorem 2.6, the decomposing sphere is isotopic to a sphere with one maximum and minimum as Figure 7. Hence, G is decomposed into two rational theta curves, that is, a theta curve obtained by pasting a rational tangle and a trivial H-shaped graph tangle. Let L(x 1 , x 2 ) be the set of links which are obtained by gluing two Hopf tangles (B 1 , T 1 ) and (B 2 , T 2 ) so that x 1 coincides with x 2 , where x i represents either the equator e i of (B i , T i ) or the meridian m i of (B i , T i ) for i = 1, 2. We note that if a link L in S 3 admits a Hopf tangle decomposition (S 3 , L) = (B 1 , T 1 ) ∪ (B 2 , T 2 ), then the decomposing sphere S = ∂B 1 = ∂B 2 is incompressible and meridionally incompressible in (S 3 , L). Remark 2.7. We remark that L(m 1 , m 2 )∩(L(m 1 , e 2 )∪L(e 1 , m 2 )) = ∅ and (L(m 1 , e 2 )∪ L(e 1 , m 2 )) ∩ L(e 1 , e 2 ) = ∅, and that L(m 1 , m 2 ) ∩ L(e 1 , e 2 ) consists of one link which is obtained by the orientation reversing identity map between two Hopf tangles, namely, the Borromean rings. . . since it can be deformed as Figure 10. By Remark 2.7, it does not belong to L(m 1 , m 2 ) and L(e 1 , e 2 ). Theorem 2.8 shows that the Borromean rings has exactly three Hopf tangle decomposing spheres F 1 , F 2 and F 3 , and that it is small. Matsuda ([4]) also proved that the Borromean rings is small by using clasp disks bounded by the Borromean rings. Theorem 2.8. Let L be a link in S 3 which admits a Hopf tangle decomposition (S 3 , L) = (B 1 , T 1 ) ∪ (B 2 , T 2 ). Then,(1)L(m 1 , m 2 ) L(m 1 , e 2 ) ∪ L(e 1 , m 2 ) L(e 1 , e 2 )× I such that F is disjoint from T or intersects T transversely in the interior of F . An isotopy φ t of S × I is called a horizontal isotopy if h • φ t = h for all t ∈ [0, 1]. A surface F is said to be horizontally ∂-parallel in S × I if F is ∂-parallel in S × I and there exists a horizontal isotopy φ t of S × I such that p • φ 1 is a homeomorphism on F . We say that F is T -compressible in (S × I, T ) if there exists a disk D in S × I such that D ∩ T = ∂D ∩ T = α is a subarc of T and D ∩ F = ∂D − intα. A surface F is T -incompressible in (S × I, T ) if it is not T -compressible. An n-dimensional submanifold V in S × I is vertical if p(V ) is an (n − 1)- dimensional manifold. A submanifold H in S × I is horizontal if H is contained in a level 2-sphere S × {t} for some t ∈ I. Lemma 3.1. Let S be a 2-sphere, X a union of points in S, and F ⊂ (S ×I, X ×I) an (X × I)-incompressible surface. If F is horizontally ∂-parallel in S × I, then F is ∂-parallel in (S × I, X × I). Proof. After a suitable horizontal isotopy of S × I, we may assume that p is a homeomorphism on F . Moreover, by a suitable horizontal isotopy of S × I fixing F , we may assume that p is a homeomorphism on (X × I) ∩ V since X × I is monotone in S × I, where V is a 3-manifold bounded by F and p(F ) in S × I. Then p((X × I) ∩ V ) consists of disjoint arcs embedded in p(F ). See Figure 11. ¿ À ¿ À p Figure 11. F is α-compressible Suppose that there exists an arc α of (X × I) ∩ V such that ∂α is contained in F . We note that the vertical disk p −1 (p(α)) ∩ V is disjoint from ((X × I) ∩ V ) − α. Then, there exists a disk D in p −1 (p(α)) ∩ V such that D ∩ α = ∂D ∩ α = α and ∂D − intα ⊂ F . This shows that F is an (X × I)-compressible in (S × I, X × I), a contradiction. Hence, any arc of (X × I) ∩ V connects F and p(F ) monotonously, and F is ∂-parallel in (S × I, X × I). Lemma 3.2. Let S be a 2-sphere, X a union of points in S, and F ⊂ (S ×I, X ×I) an incompressible and meridionally incompressible surface in a Morse position with respect to h. Suppose that the number of critical points of F is minimal up to isotopy of F in (S × I, X × I). If F has a maximal critical point, then F is ∂-parallel in (S × I, X × I). Proof. If F is horizontally ∂-parallel in S × I, then by Lemma 3.1, F is ∂-parallel in (S × I, X × I). So, hereafter we assume that F is not horizontally ∂-parallel in S × I. Let a 0 , . . . , a n be the critical points of F which are labeled so that the corresponding critical values t i = h(a i ) satisfy t i−1 < t i for all i, and let a m be the lowest maximal critical point of F . If m = 0, then F is a disk with only one maximal critical point a 0 , and hence horizontally ∂-parallel in S × I. This contradicts the assumpution. Hence, we have m ≥ 1. Let F max be the maximal horizontally ∂-parallel subsurface of F containing a m , that is, a component of F ∩ h −1 ([t m−j + ǫ, t m + ǫ]) which is horizontally ∂-parallel in h −1 ([t m−j + ǫ, t m + ǫ]) , for an integer j (≤ m) as large as possible and a fixed sufficiently small positive real number ǫ. See Figure 12. a a m F max F' max m-j V Figure 12. maximal horizontally ∂-parallel subsurface F max Since F max is horizontally ∂-parallel in S × [t m−j + ǫ, t m + ǫ] and (X × [t m−j + ǫ, t m + ǫ])-incompressible in (S × [t m−j + ǫ, t m + ǫ], X × [t m−j + ǫ, t m + ǫ]), by Lemma 3.1, it is a ∂-parallel planar surface in (S × [t m−j + ǫ, t m + ǫ], X × [t m−j + ǫ, t m + ǫ]). Therefore, by a suitable horizontal isotopy of S × I, we may assume that p is a homeomorphism on F max . Then, F max is isotopic to F ′ max = p −1 (p(F max )) ∩ (S × {t m−j +ǫ}) in (S×[t m−j +ǫ, t m +ǫ], X ×[t m−j +ǫ, t m +ǫ]), and F max and F ′ max bound a submanifold V in S × [t m−j + ǫ, t m + ǫ] such that p(F max ) = p(F ′ max ) = p(V ). We note that intV ∩ F = ∅ sinceF max ∪ B. Hence, a component of F ∩ h −1 ([t m−j − ǫ, t m + ǫ]) containing a m is horizontally ∂-parallel in h −1 ([t m−j − ǫ, t m + ǫ]) . This contradicts the maximality of F max . In Case 2, by an isotopy of F max to F ′ max and pushing out it through F ′ max , the maximal critical point a m and the saddle point a m−j are mutually canceled without increasing the number of critical points of F . This contradicts the minimality of the number of critical points of F . In Case 3, there exists a vertical compressing disk D for In Case 4, there exists a vertical compressing disk D for F max ∪ B, as same as Case 3. Actually, this disk D gives a compressing disk for F in (S × I, X × I) since ∂D is essential in F . This contradicts the incompressibility of F in (S × I, X × I). F max ∪B in (S × [t m−j − ǫ, t m + ǫ], X × [t m−j − ǫ, t m + ǫ]) such that D ∩ (X × I) = ∅, D ∩ B = ∂D ∩ B = α In Case 5, first we consider the case that F max is a disk. If p(F max ) contains p(E), then F = F max ∪ E is a 2-sphere which bounds a 3-ball V ′ in S × I. We note that V ′ ∩ (X × I) = ∅ since E ∩ (X × I) = ∅ and F max is isotopic to E in (V ′ , (X ×I)∩V ′ ). Hence F bounds a 3-ball V ′ in S ×I −X ×I, and this contradicts the incompressibility of F in (S × I, X × I). Otherwise, F max ∪ E is isotopic to a level 2-sphere h −1 (t m−j ) in (S × I, X × I), but this contradicts the assumpution that F is not horizontally ∂-parallel in S × I. Next, we consider the case that F max is not a disk. In this case, p(F max ) does not contain p(E). Hence, F max ∪E is isotopic to a planar surface in h −1 (t m−j ), and there exists an isotopy of F max ∪ E so that it has only one maximal point a m and (|∂F max | − 2)-saddle points since a saddle point of F max and a minimal point a m−j are mutually canceled. This contradicts the minimality of the number of critical points of F . Hence, any case does not occur, the lemma is proved. The next lemma is a special case of Lemma 3.2. In the case that the number of strings is less than or equal to 5, incompressible and meridionally incompressible surfaces are very restricted. Lemma 3.3. Let S be a 2-sphere, X a union of points in S less than or equal to 5, and F ⊂ (S × I, X × I) an incompressible and meridionally incompressible surface in a Morse position with respect to h. Suppose that the number of critical points of F is minimal up to isotopy of F in (S × I, X × I). Then, one of the following holds. (1) F has no critical point and any loop of F ∩ (S × {t}) does not bound a disk D in S × {t} such that |D ∩ (X × I)| ≤ 1. (2) F is ∂-parallel in (S × I, X × I). Proof. If F has a maximal or minimal critical point, then by Lemma 3.2, F is ∂-parallel in (S × I, X × I) and we have a conclusion 2 of Lemma 3.3. Hereafter, we assume that F has neither a maximal nor minimal critical point, and suppose that F has a saddle point a s . Let F s be a component of F ∩h −1 ([h(a s )− ǫ, h(a s )+ǫ]) containing a s . Then, at least one of boundary components of F s bounds a disk in S ×{h(a s )±ǫ} which is disjoint from X ×I or intersects X ×I in one point. Therefore, there exists a horizontal compressing disk or horizontal meridionally (1) F is ∂-parallel in (B, T ). compressing disk D for F in (S × [h(a s ) − ǫ, h(a s ) + ǫ], X × [h(a s ) − ǫ, h(a s ) + ǫ]). Since (2) F is a disk separating two strings of T . Proof. Let B be the north hemisphere {(x, y, z, w) ∈ R 4 |x 2 + y 2 + z 2 + w 2 = 1, w ≥ 0}, and h : B → R be the height function defined by (x, y, z, w) → w. Since (B, T ) is a rational tangle, T can be isotoped so that the restriction of h to T is a Morse function with two maximal critical points. Let In the latter case, if F ∩ M thick is ∂-parallel into S, then F can be isotoped so that it is entirely contained in M thin . Since (M thin , T ∩ M thin ) is a trivial tangle, F is compressible or meridionally compressible in (M thin , T ∩ M thin ). This contradicts the incompressibility or meridionally incompressibility of F in (B, T ∩ B). Otherwise, F ∩ M thick is ∂-parallel into ∂B, and we obtain the conclusion 1 of Lemma 3.4. In the formar case, F ∩ M thick has no critical point and for any level 2-sphere P in M thick , any loop of F ∩ P separates four points of T ∩ P into two points and two points in P . It follows that F ∩ M thick is a vertical annulus disjoint from T . Hence, F is a disk separating two strings of T , and we have the conclusion 2 of Lemma 3.4. Proof . Put K ∩ M ± i = α ± i ∪ β ± i , where α ± i is a union of arcs each of which has only one critical point, and β ± i is a union of vertical arcs. See Figure 14. Figure 14. S i M i _ i _ ¿ M i + + i ¿ À + i À i _M ± i , S i , α ± i , β ± i± i , K ∩ M ± i ) disjoint from K. Moreover, if F is meridionally incompressible in (S 3 , K), then we may assume that each component of F ∩ M ± i is meridionally incompressible in (M ± i , K ∩ M ± i ). Proof. (of Claim 4.1) By an incompressibility of F in (S 3 , K), we may assume that F ∩ S i consists of essential loops in S i − K. Let T ± i ⊂ M ± i be a union of mutually disjoint monotone arcs connecting a minimal/maximal critical point of K ∩ M ± i to a point in S i −(F ∩S i ). We may assume that F and T ± i are in general position. Put Figure 15. Moreover, suppose that F is meridionally incompressible in (S 3 , K). If there exists a component of F ∩M ± i which is meridionally compressible in (M ± i , K ∩M ± i ), then it is a vertical annulus. We choose an innermost vertical annlus A and a meridionally compressing disk D for A in (M ± i , K ∩ M ± i ). Then, there exists a disk D ′ in F such that ∂D ′ = ∂D and D ∪ D ′ bounds a trivial (3-ball, arc)-pair in (S 3 , K). An isotopy of F bringing D ′ to D in (S 3 , K) turns the vertical annulus A to a meridionally compressing disk A ′ in (M ± i , K ∩ M ± i ), and it can be pushed out from M ± i . Here, if A ′ enters M ∓ i , then it can be pushed out from M i entirely. Hence, N ± i = N (S i ∪T ± i ; M ± i ). Then, each component of F ∩N ± i is either an incompressible disk in (N ± i , K ∩ N ± i ) or an incompressible annulus in (N ± i , K ∩ N ± i ) disjoint from K. Since (M ± i −intN ± i , K ∩(M ± i −intN ± i )) admits a product structure, there exists an isotopy of ∂N ± i − S i into ∂M ± i − S i in (M ± i , K ∩ M ± i ) keeping such conditions. Hence, we have that each component of F ∩ M ± i is either an incompressible disk in (M ± i , K ∩ M ± i ) with only one critical point or an incompressible vertical annulus in (M ± i , K ∩ M ± i ) disjoint from K. Seeeach component of F ∩ M ± i is meridionally incompressible in (M ± i , K ∩ M ± i ). S i N i T i + _ + _ M i + _ Figure 15. M ± i , S i , T ± i and N ± i Hence, F ∩ M i consists of incompressible disks with only one minimal/maximal critical point or vertical incompressible annuli in (M i , K ∩ M i ). Moreover, if F is meridionally incompressible in (S 3 , K), then F ∩ M i is also meridionally incompressible in (M i , K ∩ M i ). Hereafter, we suppose that |F ∩ M thin | is minimal up to isotopy of F in (S 3 , K) under such conditions, and moreover that the number of critical points of F ∩ M thick is minimal among all Morse positions of F . We note that |F ∩ M thin | = 0 since if F ⊂ M thick , then by Lemma 3.2, F is parallel to a thick level sphere and it is compressible. F ∩ M thick in (M thick , K ∩ M thick ). Since F is incompressible in (S 3 , K), there exists a disk D ′ in F such that ∂D ′ = ∂D. Then, an isotopy of F from D ′ to D decreases |F ∩ M thin | since D ′ contains at least one component of F ∩ M thin . This is a contradiction to the minimality of |F ∩ M thin |. Finally, in the case that F is meridionally incompressible in (S 3 , K), the proof is similar to the case that F is incompressible. By the above, the conditions (1), (2) and (4) in Section 1 are satisfied. Moreover, if F ∩ M thick does not contain a maximal or minimal critical point, then the condition (3) is satisfied. Hence F is in an essential Morse position related to K, and we have the conclusion 2 of Theorem 2.1. Hereafter, we suppose that F ∩ M thick has a maximal or minimal critical point, and we will show that F is isotopic to a thin level sphere, which is the conclusion 1 of Theorem 2.1. Without loss of generality, let a m be the lowest maximal critical point of F ∩ M thick in a thick region. Then, by Lemma 3.2, a component F max of F ∩ M thick containing a m is ∂-parallel in (M thick , K ∩ M thick ). If F max is parallel into ∂M k , then F is isotoped so that it is entirely contained in the trivial tangle (M k , K ∩ M k ), and it is compressible in the trivial tangle. Therefore, without loss of generality, we may assume that F max is isotopic to a planar surface contained in ∂M + i − S i for a thin region M i . Put F ∩ M ± i = D ± i ∪ A ± i , where D ± i is a union of incompressible disks with only one critical point and A ± i is a union of vertical annuli in (M ± i , K ∩ M ± i ). Since each arc of α ± i has only one critical point, there exists a union ∆ ± i of vertical disks in M ± i for α ± i such that ∆ ± i ∩ α ± i = ∂∆ ± i ∩ α ± i = α ± i and γ ± i = ∂∆ ± i − intα ± i ⊂ ∂M ± i − S i . Moreover, by a horizontal isotopy, we can take ∆ ± i so that ∆ ± i ∩ (D ± i ∪ A ± i ) = ∅ since each component of D ± i has only one critical point and A ± i is vertical. See Figure 16. Now, we isotope F max so that it is contained in ∂M + i − S i . If F max contains a component of γ + i , then F max is α + i -compressible and hence F is K-compressible. This contradicts the incompressibility of F in (S 3 , K) and we have F max ∩ ∆ + i = ∅. Let F ′ max be a component of F max ∪ D + i ∪ A + i ∪ A − i containing F max and push it into M i slightly. Since ∆ ± i ∩ (F max ∪ D ± i ∪ A ± i ) = ∅, we may assume that F ′ max is contained in M ′ i = M i − intN (∆ ± i ; M i ). We note that (M ′ i , K ∩ M ′ i ) admits a product structure (M ′ i , β + i ∪ β − i ) . There are following two cases for F ′ max . Case 1: ∂F max ∩ A + i = ∅ Case 2: ∂F max ∩ A + i = ∅ In Case 1, F is a 2-sphere which is entirely contained in (M ′ i , β + i ∪ β − i ). By Lemma 3.2, F is ∂-parallel in (M ′ i , β + i ∪ β − i ) . It follows that F is isotopic to a thin level sphere S i , and we have the conclusion 1 of Theorem 2.1. By Claim 4.3, F ′ is a ∂-parallel 2-sphere in (B, T ). Now, we recover F from F ′ by tubing F ′ along T . Then F is ∂-parallel in B − intN (T ), hence the Montesinos tangle T (r 1 , r 2 ) is small. S i i + _ M i _ ¿ i M i + ¿ ¢ + i i _ ¢ Á i _ Á + i D + i À + i À i _ D i _ A A + i i _ Figure 16. M ± i , S i , α ± i , β ± i , γ ± i , D ± i and A ± i In Case 2, F ′ max is a horizontally ∂-parallel planar surface in (M ′ i , β + i ∪ β − i ). By Lemma 3.1, F ′ max is ∂-parallel in (M ′ i , β + i ∪ β − i ). We isotope F ′ max in (M ′ i , β + i ∪ β − i ) so that it is contained in M − i . Then, Remark 4.4. In general, Montesinos tangles are not small. Tsutsumi informed me that there are Montesinos tangles with length more than two that are not small. Actually, any closed incompressible and meridionally incompressible surface in a Montesinos tangle T (r 1 , . . . , r n ) are parallel to ∂(B i ∪ · · · ∪ B j ), where B i , . . . , B j are successive two or above rational tangles. Then, we perform a meridional tubing along T i ∪ · · · ∪ T j and obtain a closed incompressible surface which is not ∂-parallel in the exterior of T (r 1 , . . . , r n ). In the latter case, F can be isotoped so that it is entirely contained in M thin . Then, F is compressible or meridionally compressible in (M thin , G ∩ M thin ) since G∩M thin is contained in a disk properly embedded in M thin . This is a contradiction. In the formar case, F ∩ M thick has no critical point and for any thick level 2sphere P for G, any loop of F ∩ P separates five points of G ∩ P into two points and three points in P . It follows that F ∩ M thick is a vertical annulus, and F is a sphere with one maximum and one minimum. Hence, F is a 2-sphere illustrated in Figure 6. Proof. (of Theorem 2.8) Let L be a link in S 3 which admits a Hopf tangle decomposition (S 3 , L) = (B 1 , T 1 ) ∪ (B 2 , T 2 ), and F be an incompressible and meridionally incompressible surface in (S 3 , L). For each i = 1, 2, the trivial loop C i ⊂ T i bounds a disk D i in B i which intersects each string of T i in only one point. Suppose that |F ∩ (D 1 ∪ D 2 )| is minimal up to isotopy of F . By the incompressibility and meridional incompressibility of F , each component of F ∩ D i is either an arc separating two points of T i ∩ D i in D i or a loop parallel to ∂D i in D i − T i . Put H i = N (D i ). Then, (H i , T i ∩ H i ) is also a Hopf tangle and each component of F ∩ H i is either a "meridian" disk bounded by the meridian of (H i , T i ∩ H i ) or a "vertical" annulus whose boundary component is parallel to the equator of (H i , Figure 17. We remark about F ∩ H i that a meridian disk is not compatible with a vertical annulus. Figure 17. a "meridian" disk and a "vertical" annulus T i ∩ H i ) in ∂H i − ∂(T i ∩ H i ). See The remainder of these two small Hopf tangles S 3 −int(H 1 ∪H 2 ) admits a product structure (S × I, X × I), where S is the decomposing sphere for (S 3 , L) and X is four points of L ∩ S. Under the minimal condition of |F ∩ (D 1 ∪ D 2 )|, we put F ∩ (S × I) in a Morse position with respect to the height function h : S × I → I and suppose that the number of critical points of F ∩ (S × I) is minimal among all Morse positions of F ∩ (S × I). In the same way as Claim 4.2, each component of F ∩ (S × I) is incompressible and meridionally incompressible in (S × I, X × I). It follows from Lemma 3.3 that each component of F ∩ (S × I) is either a vertical annulus in (S×I, X ×I) which separates four strings of X ×I into two and two, or ∂parallel in (S × I, X × I). In the latter case, F is parallel to the decomposing sphere S when F ∩ (D 1 ∪ D 2 ) = ∅, or it contradicts the minimality of |F ∩ (D 1 ∪ D 2 )| or the incompressibility and meridional incompressibility of F ∩ (S × I). In the former case, L belongs to L(m 1 , m 2 ) ∪ L(m 1 , e 2 ) ∪ L(e 1 , m 2 ) ∪ L(e 1 , e 2 ) since any loop of F ∩ ∂H i is either the meridian or the equator of (H i , T i ∩ H i ) and each component of F ∩ (S × I) is a vertical annulus disjoint from X × I. Thus the conclusion 1 of Theorem 2.8 holds. Next, if L ∈ L(m 1 , m 2 )∪L(m 1 , e 2 )∪L(e 1 , m 2 ), then there exists an another Hopf tangle decomposing sphere which is made of meridian disks and vertical annuli in H i and vertical annuli in S × I as Figure 9. Coversely, if there exists a Hopf tangle decomposing sphere different from S, then it is an incompressible and meridionally incompressible closed surface in (S 3 , L). Hence, by the conclusion 1 of Theorem 2.8, we have L ∈ L(m 1 , m 2 ) ∪ L(m 1 , e 2 ) ∪ L(e 1 , m 2 ) ∪ L(e 1 , e 2 ). But for a link which belongs to L(e 1 , e 2 )− L(m 1 , m 2 ), there does not exist an another Hopf tangle decomposing sphere, therefore we have L ∈ L(m 1 , m 2 )∪L(m 1 , e 2 )∪L(e 1 , m 2 ). Thus the first sentence of conclusion 2 of Theorem 2.8 holds. Moreover, since any Hopf tangle decomposing sphere different from S is made of meridian disks and vertical annuli in H i and vertical annuli in S × I as Figure 9, L ∈ L(m 1 , m 2 ) ∪ L(m 1 , e 2 ) ∪ L(e 1 , m 2 ) has exactly two Hopf tangle decompositions, except the Borromean rings has exactly three Hopf tangle decompositions as Example 2.9. Thus the conclusion 2 of Theorem 2.8 holds. Finally, suppose that L is not small and let Q be an incompressible and not ∂parallel closed surface in E(L). Then, by meridional compressing Q as possible, we get an incompressible and meridionally incompressible closed surface Q ′ in (S 3 , L). By the conclusion 1 of Theorem 2.8, we have L ∈ L(m 1 , m 2 )∪L(m 1 , e 2 )∪L(e 1 , m 2 )∪ L(e 1 , e 2 ). But if L ∈ L(e 1 , e 2 ), then any meridional tubing Q ′ cannot be performed since Q ′ must be a Hopf tangle decomposing sphere. Hence, L ∈ L(e 1 , e 2 ) and Q is an essential torus illustrated in Figure 9. Thus the conclusion 3 of Theorem 2.8 holds. The following exercise can be proved by the similar method as Lemma 3.4. So, we omit the proof. Exercise 4.5. Let (B, T ) be a Hopf tangle and F a surface properly embedded in B which is disjoint from T or intersects T transversely. If F is incompressible and meridionally incompressible in (B, T ), then one of the following holds. (1) F is a meridian disk for (B, T ). (2) F is a vertical annulus for (B, T ). (3) F is ∂-parallel in (B, T ). Figure 1 . 1Simple example of type 2 in Theorem 2.1If we put K in a thin position, then by Theorem 2.1, F can be isotoped so that F is in a thin level sphere. SeeFigure 2. Figure 2 . 2Simple example of type 1 in Theorem 2.1 A 3-manifold M is said to be small if any closed incompressible surface properly embedded in M is ∂-parallel. Analogously, a knot K ⊂ S 3 is small if the exterior Figure 3 . 3Montesinos tangle T (r 1 , . . . , r n ) Theorem 2.3. A Montesinos tangle T (r 1 , r 2 ) is small if r i = ∞ for i = 1, 2. Remark 2. 4 . 4The referee suggested the possibility to show that a tangle is small by showing the double branched cover is small. Indeed, it seems true and Theorem 2.3 follows the smallness of the double branched cover over a Montesinos tangle T (r 1 , r 2 ). Theorem 2.3 gives an another direct proof without branched covering spaces or orbifolds ([6, Theorem 1]). Figure 4 . 4standard form of Kinoshita's theta curve Figure 5 . 5another form of Kinoshita's theta curve Figure 6 . 6incompressible and meridionally incompressible surface in 2-bridge theta curve or handcuff graph complements Figure 7 . 7composite 2-bridge theta curve and the decomposing sphere Next, we consider closed incompressible surfaces in the complements of links that admit Hopf tangle decompositions. The Hopf tangle (B, T ) is a 2-string trivial tangle (B, t 1 ∪ t 2 ) with a trivial loop C as illustrated as Figure 8. The equator of the Hopf tangle (B, T ) is defined as a loop e in ∂B which cobounds an annulus in B − (t 1 ∪ t 2 ) with the trivial loop C, and the meridian is defined as a loop m in ∂B which bounds a disk D in B − (t 1 ∪ t 2 ) separating two strings of T . Figure 8 . 8Hopf tangle (B, T ) There exists an incompressible and meridionally incompressible closed surface in (S 3 , L) different from the decomposing sphere if and only if L ∈ L(m 1 , m 2 ) ∪ L(m 1 , e 2 ) ∪ L(e 1 , m 2 ) ∪ L(e 1 , e 2 ). (2) L admits a unique Hopf tangle decomposition if and only if L ∈ L(m 1 , m 2 )∪ L(m 1 , e 2 ) ∪ L(e 1 , m 2 ). If L ∈ L(m 1 , m 2 ) ∪ L(m 1 , e 2 ) ∪ L(e 1 , m 2 ) except the Borromean rings, then L admits exactly two Hopf tangle decompositions. The Borromean rings admits exactly three Hopf tangle decompositions. (3) L is small if and only if L ∈ L(e 1 , e 2 ). If L ∈ L(e 1 , e 2 ), then there exists an essential torus in S 3 − intN(L).SeeFigure 9for the types of link classes and associated closed incompressible and meridionally incompressible surfaces, where dotted lines represent monotone strings (which may have some twists). . Figure 9 . 9three types of link classes and closed incompressible and meridionally incompressible surfaces Example 2.9. The Borromean rings is the only element of L(m 1 , e 2 ) ∩ L(e 1 , m 2 ) Figure 10 . 10the Borromean rings and three Hopf tangle decomposing spheres 3. Preliminary Let S be a 2-sphere and T a 1-manifold properly embedded in S × I. In this section, we consider incompressible surfaces embedded in (S × I, T ), with the projection p : S × I → S × {0} and the height function h : S × I → I. Let F be a surface properly embedded in S a m is the lowest maximal critical point of F . Moreover, by the minimality of the number of critical points of F , F max has only one maximal point a m and (|∂F max | − 1)-saddle points.Hereafter, we consider the critical point a m−j . If a m−j is a saddle point, then the saddle transformation at a m−j can be regarded as attaching a horizontal rectangular band B to F ∩ (S × [t m−j + ǫ, t m + ǫ]) in a usual way. Otherwise, since a m is the lowest maximal point, a m−j is a minimal point and can be regarded as attaching a horizontal disk E to F ∩ (S × [t m−j + ǫ, t m + ǫ]). Then, there are the following possibility for the critical point a m−j . Case 1 : 1The band B is outside of F ′ max and connects a same boundary component of F max . Case 2: The band B is outside of F ′ max and connects a boundary component of F max and a component of F ∩ (S × [t m−j + ǫ, t m + ǫ]) other than F max . is an arc through B, and p(D) = p(α). By the incompressibility of F in (S ×I, X ×I), there exists a disk D ′ in F such that ∂D ′ = ∂D. Then, by an isotopy of D ′ to D and a suitable isotopy of F , the saddle point a m−j is eliminated without increasing the number of critical points of F . This contradicts the minimality of the number of critical points of F . F is incompressible and meridionally incompressible in (S × I, X × I), there exists a disk D ′ in F such that ∂D ′ = ∂D and D∪D ′ bounds a 3-ball in S ×I −X ×I or a trivial (3-ball, arc)-pair in (S × I, X × I). Moreover, since D is horizontal, D ′ contains at least one maximal or minimal critical point. This contradicts the assumption and hence F has no critical point. Similarly, if there exists a loop of F ∩ (S × {t}) which bounds a disk D in S × {t} such that |D ∩ (X × I)| ≤ 1, then it contradicts the assumption. Therefore, we have a conclusion 1 of Lemma 3.3. Lemma 3 . 4 . 34Let (B, T ) be a rational tangle, F a surface properly embedded in B which is disjoint from T or intersects T transversely. If F is incompressible and meridionally incompressible in (B, T ), then one of the following holds. Figure 13 . 13S a level 2-sphere below two maximal points of T , and put M thin = B ∩ h −1 ([h(S), 1]) and M thick = B ∩ h −1 ([0, h(S)]). M thin -M thick decomposition of (B, T ) We isotope F so that F is a Morse position with respect to h. In the same way as Claim 4.1, we may assume that F ∩ M thin consists of incompressible disks which separate two strings of T ∩ M thin . Hereafter, we suppose that |F ∩ M thin | and the critical points of F are minimal up to isotopy of F in (B, T ). Then each component of F ∩ M thin is an incompressible disk with only one maximal critical point, and by the proof of Claim 4.2, each component of F ∩M thick is incompressible and meridionally incompressible in (M thick , T ∩ M thick ). Hence, by Lemma 3.3, F ∩ M thick has no critical point or F is ∂-parallel in (M thick , T ∩ M thick ). Proof. (of Theorem 2.1) First, we focus the thin regionsM thin = M 1 ∪ M 2 ∪ · · · ∪ M k−1 ∪M k , whereM 1 is the top 3-ball, M 2 , . . . , M k−1 are thin regions which contain thin level spheres S 2 , . . . , S k−1 respectively, and M k is the bottom 3-ball, where h(S i ) > h(S i+1 ) for i = 2, . . . , k − 2. Put S 1 = h −1 (1 − ǫ) and S k = h −1 (−1 + ǫ) for a sufficiently small positive real number ǫ. Then, for each i, S i separates M i into two submanifolds M + i and M − i , where M + i is the upper part and M − i is the lower part Claim 4. 2 . 2Each component of F ∩ M thick is incompressible and meridionally incompressible in (M thick , K ∩ M thick ). Proof. (of Claim 4.2) Suppose that there exists a component of F ∩ M thick which is compressible in (M thick , K ∩ M thick ). Then, there exists a compressing disk D for there exists a union δ of vertical ∂-compressing disks for F ′ max in (M − i , K ∩ M − i ) such that ∂δ cuts F ′ max into a single disk, andδ ∩ (D − i ∪ ∆ − i ) = ∅.By a ∂-compression of F ′ max along δ and a suitable isotopy, F ′ max turns into a disk with only one maximal critical point in M − i . During the transformation above, |F ∩ M i | does not increase, but at least one maximal critical point a m of F ∩M thick is eliminated. This contradicts the supposition of the number of critical points of F ∩ M thick . Hence, Theorem 2.1 is proved.Proof. (of Theorem 2.3) Let (B, T ) be a Montesinos tangle T (r 1 , r 2 ) with r i = ∞ for i = 1, 2, and D a disk dividing (B, T ) into two rational tangles (B 1 , T 1 ) and (B 2 , T 2 ) so that D = B 1 ∩ B 2 = ∂B 1 ∩ ∂B 2 and (B i , T i ) has a slope r i . Let F be a closed incompressible surface properly embedded in B − intN (T ). If F is meridionally compressible in (B, T ), then we perform meridionally compressions as possible and obtain a disjoint union F ′ of closed incompressible and meridionally incompressible surfaces. Conversely, F is obtained from F ′ by meridional tubings along T . Note that F ′ ⊂ intB since F is a closed surface properly embedded in B − intN (T ). Claim 4 . 3 . 43Let S be a closed incompressible and meridionally incompressible surface in (B, T ) disjoint from T or intersecting T transversely. Then S is ∂-parallel in (B, T ). Proof. (of Claim 4.3) Suppose that |S ∩ D| is minimal up to isotopy of S. Then, by the incompressibility and meridionally incompressibility of S in (B, T ), each component of S ∩ B i is incompressible and meridionally incompressible in (B i , T i ).We note that S ∩ D = ∅ since a rational tangle contains no closed incompressible and meridionally incompressible surface, and that any loop of S ∩ D is parallel to ∂D in D − T . By Lemma 3.4, each component P of S ∩ B i is ∂-parallel in (B i , T i ) or a disk separating two strings of T i . If there exists a component P of S ∩ B i of the latter type, then r i = ∞ and this contradicts the hypothesis for r i . Hence, by the minimality of |S ∩ D|, both of S ∩ B 1 and S ∩ B 2 consist of ∂-parallel disks that are not parallel into D. By connecting these components of S ∩ B i , we have a 2-sphere S which is ∂-parallel in (B, T ). Proof. (of Theorem 2.6) Let G be a 2-bridge theta curve or handcuff graph, F an incompressible and meridionally incompressible closed surface in (S 3 , G). In the same way as the case of knots, we decomposeS 3 into M thin = M 1 ∪ M 2 and M thick , where M 1 is the top 3-ball and M 2 is the bottom 3-ball. Then, one vertex and the maximal critical point of G are contained in M 1 , and another vertex and the minimal critical point of G are contained in M 2 . Moreover, by the proof of Theorem 2.1, we may assume that F ∩M 1 are consists of incompressible disks in (M 1 , G∩M 1 ) each of which has only one maximal point and disjoint from G, F ∩ M thick consists of incompressible and meridionally incompressible surfaces each of which has no minimal/maximal critical point, F ∩ M 2 are consists of incompressible disks in (M 2 , G ∩ M 2 ) each of which has only one minimal point and disjoint from G. Suppose that |F ∩ M thin | and the number of critical points of F ∩ M thick is minimal up to isotopy of F in (S 3 , G). By Lemma 3.3, F ∩ M thick has no critical point or F is ∂-parallel in (M thick , G ∩ M thick ). Case 3 : 3The band B is inside of F ′ max and connects a same boundary component of F max . Case 4: The band B is inside of F ′ max and connects two different boundary components of F max . Case 5: The disk E caps F max along a boundary component of F max . In Case 1, p is a homeomorphism on Acknowledgement. The author would like to thank Yukihiro Tsutsumi for informing Remark 4.4, and the referee for pointing out the third Hopf tangle decomposing sphere for the Borromean rings. Foliations and the topology of 3-manifolds III. D Gabai, J. Diff. Geom. 26D. Gabai, Foliations and the topology of 3-manifolds III, J. Diff. Geom. 26 (1987), 479-536. Knots are determined by their complements. C Mca, J Gordon, Luecke, J. Amer. Math. Soc. 2C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. Sur'l Analysis Situs. M P Heegaard, Bull. Soc. Math. France. 44M. P. Heegaard, Sur'l Analysis Situs, Bull. Soc. Math. France 44 (1916), 161-242. H Matsuda, Small knots and links. preprintH. Matsuda, Small knots and links, preprint. 2-bridge θ-curves in S 3. T Motohashi, Topology Appl. 108T. Motohashi, 2-bridge θ-curves in S 3 , Topology Appl. 108 (2000), 267-276. Closed incompressible surfaces in complements of star links. U Oertel, Pacific J. Math. 111U. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984), 209-230. Über eine numerische Knoteninvariante Math. H Schubert, Z. 61H. Schubert,Über eine numerische Knoteninvariante Math. Z. 61 (1954), 245-288. Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa. Y.-Q Wu, Math. Ann. 304The classification of nonsimple algebraic tangles. Japan E-mail address: [email protected]. Wu, The classification of nonsimple algebraic tangles, Math. Ann. 304 (1996), 457-480. Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa Univer- sity, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan E-mail address: [email protected]

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{'abstract': 'In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have following results.• Smallness of Montesinos tangles with length two and Kinoshita\'s theta curve• Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions 1991 Mathematics Subject Classification. Primary 57M25, 57M50; Secondary 57Q35, 57Q37.', 'arxivid': 'math/0503375', 'author': ['Makoto Ozawa '], 'authoraffiliation': [], 'corpusid': 8496818, 'doi': '10.1142/s021821650800618x', 'github_urls': [], 'n_tokens_mistral': 15484, 'n_tokens_neox': 13964, 'n_words': 9354, 'pdfsha': '6df2f9d1149e619d2248c2c3df8ba6429ec70491', 'pdfurls': ['https://export.arxiv.org/pdf/math/0503375v6.pdf'], 'title': ['MORSE POSITION OF KNOTS AND CLOSED INCOMPRESSIBLE SURFACES', 'MORSE POSITION OF KNOTS AND CLOSED INCOMPRESSIBLE SURFACES'], 'venue': []}

arxiv

A Primer on Rate-Splitting Multiple Access: Tutorial, Myths, and Frequently Asked Questions 10 Jan 2023 Fellow, IEEEBruno Clerckx [email protected] Member, IEEEYijie Mao Fellow, IEEEEduard A Jorswieck Fellow, IEEEJinhong Yuan [email protected]. Fellow, IEEEDavid J Love Fellow, IEEEElza Erkip Fellow, IEEEDusit Niyato [email protected] Department of Electrical and Electronic Engineering School of Information Science and Tech-nology and with Silicon Austria Labs (SAL) Imperial College London SW7 2AZ, A-8010London, GrazUK, Austria School of Electrical Engineering and Telecommunica-tions ShanghaiTech University 201210ShanghaiChina School of Computer Engineering University of New South Wales 2052SydneyNSWAustralia Nanyang Techno-logical University Singapore A Primer on Rate-Splitting Multiple Access: Tutorial, Myths, and Frequently Asked Questions 10 Jan 2023Manuscript received August 25, 2022; accepted Dec 15, 2022.arXiv:2209.00491v2 [cs.IT] 1 (Invited Paper) This work has been supported in part by the National Nature Science Foundation of China under Grant 62201347. (Corresponding author: Yijie Mao.) B. Clerckx is with the A. Jorswieck is with the Institute for Communications Technology at TU Braunschweig, Brunswick, Germany (email: [email protected]). J. Yuan is with the D. J. Love is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA (email: [email protected]). E. Erkip is with the Electrical and Computer Engineering Department, New York University Tandon School of Engineering, Brooklyn, NY 11201, USA (email: [email protected]). D. Niyato is with theIndex Terms-Rate-SplittingRate-Splitting Multiple AccessNext Generation Multiple AccessNon Orthogonal Multiple AccessSpace Division Multiple AccessMulti-user MIMOIn- terference Management6G Rate-Splitting Multiple Access (RSMA) has emerged as a powerful multiple access, interference management, and multi-user strategy for next generation communication systems. In this tutorial, we depart from the orthogonal multiple access (OMA) versus non-orthogonal multiple access (NOMA) discussion held in 5G, and the conventional multi-user linear precoding approach used in space-division multiple access (SDMA), multiuser and massive MIMO in 4G and 5G, and show how multi-user communications and multiple access design for 6G and beyond should be intimately related to the fundamental problem of interference management. We start from foundational principles of interference management and rate-splitting, and progressively delineate RSMA frameworks for downlink, uplink, and multicell networks. We show that, in contrast to past generations of multiple access techniques (OMA, NOMA, SDMA), RSMA offers numerous benefits: 1) enhanced spectral, energy and computation efficiency; 2) universality by unifying and generalizing OMA, SDMA, NOMA, physical-layer multicasting, multi-user MIMO under a single framework that holds for any number of antennas at each node (SISO, SIMO, MISO, and MIMO settings); 3) flexibility by coping with any interference levels (from very weak to very strong), network loads (underloaded, overloaded), services (unicast, multicast), traffic, user deployments (channel directions and strengths); 4) robustness to inaccurate channel state information (CSI) and resilience to mixed-critical quality of service; 5) reliability under short channel codes and low latency. We then discuss how those benefits translate into numerous opportunities for RSMA in over forty different applications and scenarios of 6G, e.g., multi-user MIMO with statistical/quantized CSI, FDD/TDD/cell-free massive MIMO, millimeter wave and terahertz, cooperative relaying, physical layer security, reconfigurable intelligent surfaces, cloud-radio access network, internetof-things, massive access, joint communication and jamming, non-orthogonal unicast and multicast, multigroup multicast, satellite, space-air-ground integrated networks, unmanned aerial vehicles, integrated sensing and communications, grant-free access, network slicing, cognitive radio, optical/visible light communications, mobile edge computing, machine/federated learning, etc. We finally address common myths and answer frequently asked questions, opening the discussions to interesting future research avenues. Supported by the numerous benefits and applications, the tutorial concludes on the underpinning role played by RSMA in next generation networks, which should inspire future research, development, and standardization of RSMA-aided communication for 6G.2 The Degrees-of-Freedom (DoF), or multiplexing gain, is a first-order approximation of the rate at high signal-to-noise ratio (SNR). It can be viewed as the pre-log factor of the rate at high SNR and be interpreted as the number or fraction of interference-free stream(s) that can be simultaneously communicated to a user (or multiple users). The DoF achieved depends on the communication strategy used. The larger the DoF, the faster the rate increases with the SNR. Hence, ideally a communication strategy should achieve the highest DoF possible. Readers are referred to[7]for more details on various definitions used to assess the DoF performance of multiple access schemes.3 Perfect CSIT is obtained by setting the channel estimation error in the imperfect CSIT model to zero. I. INTRODUCTION C Ommunication systems are inherently multi-user systems. Multiple access (MA) techniques play the crucial role of deciding how to make use of the resources (e.g., time, frequency, power, antenna, and code) to serve those multiple users. Next generation communications systems, e.g., 6G and beyond, will have to cope with increasing demands for high throughput, reliability, heterogeneity of quality of service (QoS), and massive connectivity to satisfy the requirements of further-enhanced mobile broadband (FeMBB), extremely ultra reliable and low-latency communication (eURLLC), ultra massive machine type communication (umMTC), mixture thereof, and of new services such as integrated sensing and communications (ISAC), integrated satellite-terrestrial, and extended reality. To that end, it is critical to understand how next generation MA can fulfill those demands and requirements by going beyond the conventional orthogonal versus nonorthogonal discussion held in 5G. Table I details the main abbreviations used throughout this work. A. Beyond Orthogonal versus Non-Orthogonal In the past decade, MA schemes have often been classified into two categories, namely orthogonal (serving a single user per resource) versus non-orthogonal (serving multiple users per resource). This classification has triggered the question Is a non-orthogonal approach to MA better than an orthogonal approach? and has led to the emergence of the rich literature on non-orthogonal multiple access (NOMA) versus orthogonal multiple access (OMA) [1], [2]. This question was motivated by the claim that prior generations of cellular communication networks are based on OMA serving multiple users on orthogonal resources using time division multiple access (TDMA), frequency division multiple access (FDMA), code division multiple access (CDMA), or orthogonal frequency division multiple access (OFDMA) [3]. Classifying MA schemes into non-orthogonal vs orthogonal is however over-simplistic as it does not fully reflect modern communication designs. Indeed, those systems are equipped with multiple antennas, and spatial domain processing in the form of multi-user linear precoding (MU-LP), space division multiple access (SDMA), multi-user multiple-input multipleoutput (MU-MIMO), massive MIMO, is an integral part of 4G and 5G for twenty years. Importantly, MU-LP/SDMA/MU-MIMO serves users in a non-orthogonal manner since multiple users are allocated different precoders/beams in the same timefrequency grid and interfere with each other in the same cell. Hence both 4G and 5G already use a combination of orthogonal (in the time-frequency domains) and non-orthogonal (in the spatial domain) approaches in the form of OFDMA combined with SDMA/MU-MIMO/massive MIMO. It is well documented that non-orthogonality can be beneficial in both single and multi-antenna settings [4]- [6]. A major drawback of such classification is that it tends to amalgamate many different MA schemes under the nonorthogonal umbrella without contrasting them or truly understanding the essence of those schemes. This has caused unnecessary confusions and misunderstandings in the past few years [7]. For instance, SDMA and power-domain NOMA 1 are two different non-orthogonal approaches to MA but are fundamentally different. Indeed, SDMA and NOMA can be 1 In the sequel, we simply use NOMA to refer to power-domain NOMA. seen as two extreme interference management strategies where the former treats interference as noise and the latter fully decodes interference [8]. An alternative interpretation of this difference is to note that SDMA (and other forms of linearly precoded and non-linearly precoded MU-MIMO schemes) relies on a transmit-side interference cancellation strategy while NOMA can be seen as a receive-side interference cancellation strategy [9]. Such major differences have however not been captured and not been addressed when answering the aforementioned question, though they lead to drastic performance and complexity gaps between the two schemes [7]. Hence, instead of contrasting orthogonal vs non-orthogonal, a different classification should be considered in next generation wireless networks. In this paper, we will show that the fundamental question behind MA design should instead be how to manage multi-user interference? Answering this question will shed the light on the differences between non-orthogonal approaches to MA designs and on a new classification of MA schemes based on how the interference is managed. Even more importantly, this exercise will bring to light the powerful and newly emerging Rate-Splitting Multiple Access (RSMA) for downlink and uplink communications. B. Toward Rate-Splitting Multiple Access RSMA refers to a broad class of multi-user schemes whose commonality is to rely on the rate-splitting (RS) principle [10]- [12]. RS consists in splitting a user message (e.g., information bits) into two or multiple parts such that each of those parts can be decoded flexibly at one or multiple receivers. A receiver would have to retrieve each part to reconstruct the original message. A key benefit of RS and its message splitting capability is to flexibly manage inter-user interference, as it will appear clear throughout this paper. Though the RS principle first appeared in the information theoretic literature in the late 70's and early 80's, RS (and consequently the emerging RSMA) has received a renewed interest in the past decade in the broader communication community. What triggered this renewed interest is a different line of research, seemingly unrelated to the MA literature at first, on understanding the fundamental limits of robust interference management, i.e., how to manage interference in a multiuser multi-antenna communications system in the presence of imperfect channel state information at the transmitter (CSIT) [13]. Conventional multi-user multi-antenna approaches such as SDMA/MU-MIMO heavily rely on timely and highly-accurate CSIT. Unfortunately, in practice, CSIT is always imperfect due to pilot reuse, channel estimation errors, pilot contamination, limited and quantized feedback accuracy, delay/latency, mobility (due to ever-increasing speeds of vehicle/trains/satellite/flying objects and emerging applications as Vehicle-to-Everything), radio frequency (RF) impairments (e.g., phase noise), inaccurate calibrations of RF chains, subband level estimation [14]. Consequently SDMA/MU-MIMO are inherently non-robust. The classical approach to dealing with this practical limitation takes a "robustification" stancetechniques that have been developed under the assumption of perfect CSIT are tweaked to account for imperfect CSIT [15], [16]. The challenge is that any CSIT inaccuracy results in a residual multi-user interference that needs proper management instead of "robustification". Despite its significance in realistic wireless deployments, the fundamental limits of multi-user multi-antenna communications with imperfect CSIT are still an open problem, i.e., the capacity and capacity achieving schemes remain to be found. Consequently, this lack of understanding of how to design robust communication schemes has led to modern systems like 4G and 5G designs to be fundamentally designed for perfect CSIT (using SDMA and treat interference as noise) instead of being designed from scratch to be truly robust to imperfect CSIT [9], [14]. Interestingly, we are now in a much better position to design truly robust MIMO wireless networks accounting for imperfect CSIT and its resulting multi-user interference [9], [13], [14]. It is indeed known that to benefit from imperfect CSIT and tackle the multi-user interference, the transmitter should take an RS approach that splits the messages into common and private parts, encodes the common parts into a common stream, and private parts into private streams and superposes in a non-orthogonal manner the common stream on top of all private streams [14]. The common stream is decodable by all receivers, while the private streams are to be decoded by their corresponding receivers only. Such an approach is optimal from an information theoretic perspective (Degrees-of-Freedom, DoF 2 ) for downlink multi-user mutliple-input singleoutput (MISO) and MIMO transmissions with imperfect CSIT [17]- [22]. This literature opens various doors that have a major impact on MA designs. First, since imperfect CSIT is more general than perfect CSIT 3 , finding efficient schemes for imperfect CSIT leads to discovering a broader and general class of communication strategies that would subsume perfect CSIT strategies as particular instances. Second, it gives communication engineers clear and fundamentally grounded guidelines on how to design robust schemes. Third, it provides refreshing and new thoughts about low complexity non-orthogonal scheme that are applicable and beneficial even in perfect CSIT settings. Fourth, it brings RS, originally developed for the twouser single-antenna interference channel [10], [11], into MU-MIMO, which was never investigated despite the rich literature on MU-MIMO schemes in the past two decades [6]. Fifth, RS serves users in a non-orthogonal manner by partially treating interference as noise and partially decoding interference. This highlights the usefulness to depart from the extremes of (fully) treat interference as noise (as in SDMA) and (fully) decode interference (as in NOMA) in multi-antenna networks, but also the usefulness to bridge and unify those two extremes [8]. Sixth, RS can be seen as a smart combination of transmit-side and receive-side interference cancellation strategy where the contribution of the common stream is adjusted to benefit from the best balance between transmit and receive cancellation. This departs from the transmit-side only and receive-side only interference cancellation strategies of SDMA (and the rich literature on MU-MIMO) and NOMA, respectively [9]. The above design makes RS a fundamental building block of a powerful MA framework for downlink and uplink communications, namely, RSMA [8], [23]. In the downlink, RSMA uses linearly or non-linearly precoded RS at the transmitter to split each user message into one or multiple common messages and a private message. The common messages are combined and encoded into common streams for the intended users. Successive interference cancellation (SIC) -or any other forms of joint decoding -is used at each user to sequentially decode the intended common streams (and therefore decode part of the interference). Such linearly or non-linearly precoded generalized RSMA has been demonstrated to be a powerful framework to bridge, reconcile, and generalize SDMA, NOMA, OMA, and physical-layer multicasting and further boost system spectral and energy efficiencies for downlink transmissions with both perfect and partial CSIT [8], [24], [25]. In the uplink, users split their message into multiple streams and allocate proper transmit power to each stream. The re-ceiver at the base station performs SIC to retrieve each stream and reconstruct the original messages. By splitting messages, inter-user interference can be dynamically managed without utilizing time sharing among users to achieve the capacity. Here again, NOMA is a subset of RSMA and is obtained when user messages are not split. This message splitting property can enable core services where users have intermittent access behaviour such as URLLC and mMTC, as well as enable different combinations of core services to serve users with heterogeneous profiles [23], [26]. C. Challenges and Opportunities for RSMA In our view, MA schemes will have to address new challenges and requirements in next generation networks: • Efficient: Time, frequency, power, spatial, and codedomain resources should be used to serve users in the most efficient way, from both spectral efficiency and energy efficiency perspectives, while enhancing QoS and fairness across devices and services (e.g., unicast, multicast) and making the best use of the available computational resources (transmitter and receiver complexity). This calls for performance evaluations of MA schemes beyond OMA vs NOMA [7]. • Universal: Simpler is better, i.e., a single unified and general MA scheme would be easier to implement and optimize than a combination of multiple MA schemes, each optimized for specific conditions. This calls for a deeper understanding of non-orthogonality in MA designs and of how non-orthogonal approaches to MA schemes relate to each other [7], [24]. • Flexible: Wireless networks are dynamic and MA schemes should be flexible or versatile enough to cope with various interference levels, network density and load (underloaded, overloaded), topology, services, user distribution, channel directions and strengths in general deployments. This calls for MA schemes that depart from those two extreme interference management strategies, namely fully treat interference as noise and fully decode interference [8]. • Robust and Resilient: MA designs need to depart from the conventional aforementioned "robustification" stance and adopt a true robustness to practical scenarios subject to imperfect CSIT [14]. Aside being robust, MA schemes need to also become increasingly resilient to emerging services involving mixed-criticality, i.e., services with different priorities. This is particularly relevant for applications in safety-critical contexts such as driverless traffic [27]. • Low Latency and Reliable: Providing reliable and lowlatency communications for intelligent transportation and industrial automation is key in next generation communications. Critical sources of latency relate to link establishment, packet re-transmissions, and data blocklength. Hence, an MA scheme needs to be able to perform reliably under short channel codes and decrease the number of re-transmissions [28], [29]. RSMA uniquely appeared in recent years to fulfill all these requirements [25], thanks to its inherent message splitting capability, which is not featured in any other MA schemes. This split capability provides several benefits at once as it allows to: 1) partially decode interference and partially treat interference as noise (hence its efficiency, flexibility, reliability, and resilience), 2) reconcile the two extreme strategies of interference management and multiple MA schemes into a single framework (hence its generality/universality), 3) achieve the optimal DoF in practical scenarios subject to imperfect CSIT (hence its robustness). This contrasts with conventional approaches like OMA, SDMA, and NOMA, that achieve only some of the aforementioned features [7]. Indeed, SDMA has low complexity and works well in perfect CSIT conditions especially in underloaded regimes, while downlink multi-antenna NOMA is inefficient in general as it incurs a severe DoF loss despite an increased receiver complexity due to an inefficient use of SIC receivers [7]. OMA, SDMA, and NOMA are not general as they are suited for particular propagation conditions [24]. Similarly they are not as flexible as RSMA since they rely on eliminating interference by orthogonalization (OMA), treat interference as noise (SDMA), and decode interference (NOMA), which are suboptimal strategies in general settings [8], [30]. Though NOMA can be made more robust than SDMA in the presence of imperfect CSIT, the DoF of NOMA and SDMA are both suboptimal, therefore incurring a rate loss compared to RSMA [7]. Similarly SDMA has also been shown to be less resilient than RSMA [31]. Finally, SDMA and NOMA are less reliable with finite block code lengths [32]- [34]. D. Objectives and Contributions It is worth to point out (again) that SDMA, NOMA, and RSMA are all non-orthogonal approaches to MA design. However, classifying them all as non-orthogonal does not give us a clue as to where the difference in efficiency, universality, flexibility, robustness and resilience, reliability and low latency comes from. Answering the fundamental question how to manage multi-user interference? is crucial because it helps us to design efficient MA schemes, to reveal how MA schemes manage interference, and identify and predict under what conditions a given MA scheme may be efficient, universal, flexible, robust, resilient, reliable and have low latency. In this paper, we provide a tutorial on RSMA and address common myths and frequently asked questions. To that end, we make the following contributions: • We start this tutorial by departing from the orthogonal vs non-orthogonal classification and rather classify schemes as a function of how they manage interference. We go back to the basics of RS proposed for the two-user interference channel, and show step by step, how to build downlink, uplink, and multi-cell RSMA and its various schemes. We also show how interference management and MA designs are closely related and how OMA, SDMA, NOMA, and RSMA schemes differ in terms of how to manage interference. We then explain that by understanding the interference management capability of a scheme, we can understand the drawbacks and benefits of the MA schemes. • We generalize the downlink, uplink, and multi-cell RSMA frameworks to MIMO settings with multiple antennas at all nodes. This builds upon recent efforts to design MIMO RSMA [35], contrast with recent survey and works [7], [25] that were primarily limited to MISO settings, and confirm that the generality and universality of RSMA are not limited to single-input single-output (SISO), MISO, or single-input multiple-output (SIMO) settings, but hold for general MIMO settings. Consequently, recent MIMO NOMA schemes [36], [37] Those myths and frequently asked questions about RSMA nicely complement the tutorial part. This paper also differs from the recent survey paper on RSMA [25] by providing a tutorial flavour instead of survey, by diving further into the key role of interference management, by addressing myths and frequently asked questions, and by expanding the framework to general MIMO settings. It also differs from [26], [38], [39] that give an overview of RSMA and focus on the specific interplay between RSMA and ISAC and reconfigurable intelligent surfaces (RIS). E. Organization and Notations The remainder of this paper is organized as follows. Section II introduces fundamentals about how to manage interference, delineates the RS strategy for the 2-user interference channel, relates the problem of MA design with interference management, and draws important observations. Section III builds upon important lessons drawn from previous section and introduces various architectures of 2-user RSMA for both downlink and uplink for general MIMO settings at all nodes. Important lessons about the relationships between existing MA schemes and RSMA are summarized. Section IV extends the RSMA design to general K-user settings for downlink, uplink and multi-cell deployments. Section V shows how the RSMA architectures find applications in 40 different scenarios relevant in 6G. The breadth and depth of those applications and scenarios showcase how powerful and underpinning RSMA is for 6G. Section VI debunks some myths that have appeared in the RSMA literature. Section VII presents answers to several common questions that are often asked about RSMA. Section VIII concludes this tutorial overview. Notations: Bold upper and lower case letters denote matrices and column vectors, respectively. (·) T , (·) H , | · |, · , E{·}, and tr(·) represent the transpose, Hermitian, absolute value, Euclidean norm, expectation, and trace operators, respectively. CN (0, σ 2 ) denotes the circularly symmetric complex Gaussian (CSCG) distribution with zero mean and variance σ 2 . II. KEY QUESTION BEHIND MULTIPLE ACCESS DESIGN: HOW TO MANAGE INTERFERENCE? In this section, we present fundamentals about how to manage interference. To that end, we focus on an example of a symmetric two-user interference channel, which is the simplest scenario that enables to capture the essence of interference management. We then use that simple example to draw major lessons for MA design. A. Interference Channel Let us consider a two-user symmetric Gaussian interference channel (IC) of Fig. 1. We have four nodes, two transmitters (Tx) and two receivers (Rx), each equipped with a single antenna. Tx-1 (resp. Tx-2) wants to transmit a message W 1 (resp. W 2 ) to Rx-1 (resp. Rx-2). To simplify the number of parameters and obtain some insight into the role of interference, we assume some symmetry in that the direct channel from a Tx to its intended Rx (Tx-1 → Rx-1, Tx-2 → Rx-2) is h d and the cross channel (Tx-1 → Rx-2, Tx-2 → Rx-1) is h c . Hence, the stronger h c compared to h d , the stronger the interference from Tx-k to Rx-j, j = k. A few strategies come to mind to manage this interference: • Orthogonalize: Tx-1 and Tx-2 transmit on orthogonal resources (e.g., time or frequency) so that that they do not interfere with each other at all. This is suboptimal since this strategy does not care about how weak the interference is. • Treat Interference as (additive white Gaussian) Noise: This is a natural approach whenever h c is very weak compared to h d since the interference created by Tx-k would be buried in the noise of Rx-j, j = k. The drawback of this strategy is that the interference actually carries information and has a structure that could potentially be exploited in mitigating its effect [30]. 2. RS for two-user SISO IC (HK scheme) [11]. • Decode Interference: Fully decode the interference at the receivers is especially relevant and actually optimal when the interference is strong enough that h c is large compared to h d [40]. In that case, Rx-k has a better reception of Tx-j's signal than the intended receiver Rxj, k = j. However, we can do more than those seemingly three different strategies by adopting a rate-splitting (RS) approach [10], [11], [30]. RS splits the transmitted information into two parts, namely a common part to be decoded by both receivers and a private part to be decoded only by the intended receiver. The key benefit of RS is that by enabling a receiver to decode the common part, part of the interference is cancelled off, while the remaining private part from the other transmitter is treated as noise. This enables RS (by properly adjusting the power to the common and private parts) to partially decode interference and partially treat the remaining interference as noise, therefore bridging, unifying, and actually outperforming the two extremes of treat interference as noise and decode interference, as it will appear clearer in the sequel. B. Rate-Splitting for Two-User Interference Channel We adopt an RS strategy 4 to split each transmitter message W k into two parts denoted as a common part W c,k and a private part W p,k as per Fig. 2 [10], [11], [30]. On the one hand, a codebook shared between both transmitters 5 /5G New Radio (NR) system, all codebooks are shared since all users use the same family of modulation and coding schemes (MCS) specified in the standard. is used to encode the common parts W c,k , k = 1, 2, into the common stream s c,k , k = 1, 2, such that they are decodable by both receivers. The private parts W p,k , k = 1, 2 are on the other hand encoded into private streams s k , k = 1, 2, constructed from independent codebooks. Tx-k then transmits the superposition of its common stream s c,k and private stream s k with proper power allocation to both streams as x k = P c s c,k + P k s k .(1) By defining s k = [s c,k , s k ] T and assuming that E[s k s H k ] = I, the average transmit (sum) power constraint at each transmitter is P with P c and P k = P −P c being the power allocated to the common and private stream respectively. Let us also denote 4 In the two-user IC, RS is also known as Han-Kobayashi strategy [11]. 5 This is not an issue in modern systems since, for example, in an 4G Long Term Evolution (LTE) by t = Pc P the fraction of the transmit power allocated to the common stream. The received signals at both receivers can be written as y 1 = h d x 1 + h c x 2 + n 1 ,(2)y 2 = h c x 1 + h d x 2 + n 2 ,(3) where n k ∼ CN (0, σ 2 n,k ) is the additive white Gaussian noise (AWGN). Without loss of generality, we assume the variance σ 2 n,k = 1. We also assume perfect CSIT and perfect CSI at the receivers (CSIR). Each Rx decodes the common streams and its intended private stream as illustrated in Fig. 2. To that end, each Rx jointly decodes the common streams into messages W c,1 and W c,2 by treating the private streams as noise. Then Rx-k cancels the common streams from the received signal and decodes the intended private stream s k into message W p,k by treating the other private stream s j , j = k, as noise. Rx-k finally reconstructs the original message by recombining W p,k and W c,k into W k , which is the same as W k in the absence of any decoding error. By doing so, Rx-k has partially decoded the message of Rx-j, namely W c,j , and therefore cancelled off part of the interference. Because of symmetry, the rates of the common streams s c,1 and s c,2 are the same and are simply denoted as R c . Similarly, the rates of the private streams s 1 and s 2 are also the same and are denoted as R p . Assuming Gaussian signaling and infinite blocklength, R c must satisfy the following inequalities at Rx-1 (and equivalently at Rx-2 due to symmetry) R c ≤ log 2 1 + tP |h d | 2 1 + I ,(4)R c ≤ log 2 1 + tP |h c | 2 1 + I ,(5)2R c ≤ log 2   1 + tP |h d | 2 + |h c | 2 1 + I   ,(6) where I = (1 − t) P |h d | 2 + |h c | 2 . Inequalities (4), (5), and (6) originate from the fact that the common streams are jointly decoded first at each receiver by treating the private streams as noise. Those three inequalities are obtained by writing the rate constraints of a two-user multiple access channel (MAC) formed by two virtual transmitters sending respectively s c,1 and s c,2 to Rx-1 (or 2 due to symmetry) subject to the additional noise power I created by the private streams s 1 and s 2 . Hence, (4) refers to the individual rate constraint of decoding s c,1 at Rx-1 by treating s 1 and s 2 as noise. Similarly, (5) refers to the individual rate constraint of decoding s c,2 at Rx-2 by treating s 1 and s 2 as noise. The sum-rate constraint writes as (6). The rate R p of the private streams in (7) is obtained by noticing that a private stream is decoded after cancelling the common streams while treating the other interfering private stream as noise. We can therefore write R p ≤ log 2 1 + (1 − t) P |h d | 2 1 + (1 − t) P |h c | 2 ,(7) where the presence of (1 − t) P |h c | 2 at the denominator expresses the interference from the private stream of Tx-k owed to Rx-j private stream. Combining (4)-(6) and (7), the achievable (symmetric) rate in this two-user IC is written as the sum of the private and common rates R sym = R p + R c in (8). The terminology ratesplitting appears clearly here where we note that R sym is split into two parts, namely, the private rate R p and the common rate R c . The benefit of this RS architecture is the ability to adjust the power allocation to the common and private streams through parameter t (and therefore the amount of information to be carried by the common and private streams) to maximize R sym depending on the channel strengths of h d and h c . Remark 1: The terminology common and private is taken from the information theory literature. In simple words, common (also sometimes called public) simply refers to a message or stream that is to be decoded by multiple users though the content of the message is not necessarily intended to all those users. Indeed, s c,k , k = 1, 2, are decoded by both receivers though the content of s c,k is only intended to Rx-k. Common also contrasts with a multicast message that is a message decoded by multiple users but that is genuinely intended to all those users. Private on the other hand refers to a message or stream that is only decoded by its intended receiver and is treated as noise at other receivers. Indeed, s k is decoded only by Rx-k and is treated as noise by Rx-j with j = k. Messages W k , k = 1, 2, are unicast messages (each intended to a single user), but they are split into a common and private parts for interference management benefits. C. Interference Regimes We can identify four interference regimes depending on the relative strengths 6 of h d and h c : Very weak (|h c | ≪ |h d |): Whenever |h c | is much smaller than |h d |, the maximization of R sym ends up allocating all the transmit power to the private stream such that t = 0 and R sym = log 2 1 + P |h d | 2 1+P |hc| 2 . In such a regime, message W k is entirely encoded in the private stream s k (i.e., no splitting of the messages occurs) such that any interference from Tx-k to Rx-j is treated as noise. This corresponds to the regime where the Treat Interference as Noise strategy is optimal. Weak (|h c | ≤ |h d |): As the strength of h c increases relatively to h d but as long as |h c | remains smaller than |h d |, one enters the weak interference regime where splitting the messages becomes necessary to maximize R sym . In that regime, inequality (4) is inactive (since r.h.s. of (5) < r.h.s. of (4)) and t > 0 is to be chosen, i.e., a non-zero power is allocated to the common stream. There exists a tradeoff between achieving a large private rate and minimizing the interference caused to the other receiver. A qualitative and insightful way of allocating power to the common stream is to choose t so that the interference level caused by the private stream has roughly the same level as the other receiver's noise level, i.e., from (7) choose t such that (1 − t) P |h c | 2 ≈ 1 or equivalently t ≈ P |hc| 2 −1 P |hc| 2 [30]. 6 In the asymmetric IC, the interference regimes take a more complicated form [30]. By doing so, the interference caused by a private stream has little impact on the other receiver's performance (compared to the impairments already caused by the noise). At the same time, it does not prevent each transmitter from experiencing a relatively large private rate as long as |h c | ≤ |h d |. Strong (|h d | 2 ≤ |h c | 2 ≤ |h d | 2 1 + P |h d | 2 ): As the strength of h c further increases relatively to h d and enters the regime where |h c | is larger than |h d |, each receiver is able to decode both the interfering signal and the desired signal by performing for instance SIC or joint decoding. In other words, R sym is maximized by choosing t = 1 and messages W k , k = 1, 2, are entirely encoded in the common streams s c,k , respectively, so that they are both decoded by both receivers (i.e., there are no private streams in this regime and I = 0). With |h c | ≥ |h d |, inequality (5) is inactive and R sym = 1 2 log 2 1 + P |h d | 2 + |h c | 2 as long as 1 2 log 2 1 + P |h d | 2 + |h c | 2 ≤ log 2 1 + P |h d | 2 , i.e., P |h d | 2 ≤ P |h c | 2 ≤ P |h d | 2 1+P |h d | 2 . In other words, the sum-rate constraint (6) is active and forces each transmitter to transmit at a rate smaller than log 2 1 + P |h d | 2 , i.e., smaller than the rate achievable without any interference. It is worth noting that 1 2 log 2 1+P |h d | 2 +|h c | 2 ≤ log 2 1+P |h d | 2 can equivalently be written as log 2 1+ P |hc| 2 1+P |h d | 2 ≤ log 2 1+ P |h d | 2 , which expresses that when Rx-k performs SIC to decode the interfering signal s c,j (j = k) before decoding s c,k , the decodability of s c,j at Rx-k puts a constraint on the transmission rate of s c,j at Tx-j. Very strong (|h c | 2 ≥ |h d | 2 1 + P |h d | 2 ): In this regime, the interference link is even stronger such that 1 2 log 2 1 + P |h d | 2 + |h c | 2 ≥ log 2 1 + P |h d | 2 and the sumrate constraint (6) becomes inactive. In other words, when performing SIC, the decodability of s c,j at Rx-k does not restrict the transmission rate of s c,j at transmitter j. Instead the rates are only limited by the direct links and each transmitter can transmit at a rate R sym = log 2 1 + P |h d | 2 equal to the one achievable without any interference. The strong and very strong interference regimes correspond to the regimes where the Decode Interference strategy is optimal. What distinguishes the weak interference regime from the strong (and very strong) regime is that the interference in the former is not strong enough so that only a part of the interference (through the common message) can be decoded by a receiver while the interference in the latter is so strong that the receiver can fully decode the interference. Consequently, moving from the very weak interference regime to the strong interference regime, the messages are respectively encoded into private streams only, a mixture of common and private streams, and into common streams only as further summarized in Table II. Though the capacity of the two-user IC remains unknown, RS can achieve a rate that is within a single bit per second per hertz (bit/s/Hz) of the capacity for all values of the channel parameters (even in the non-symmetric case) and is information-theoretically optimal at asymptotic high SNR regimes [30]. Fig. 3 illustrates the symmetric rate of the various interference management strategies using a concrete numerical example. The x-axis is the ratio between the interference-tonoise ratio (INR) and SNR; hence the higher INR/SNR, the stronger the interference. The different interference regimes are clearly visible. We note how RS can softly bridge all strategies and outperform them in the weak interference regime 7 . Rsym = log 2 1 + (1 − t) P |h d | 2 1 + (1 − t) P |hc| 2 + min    log 2 1 + tP |h d | 2 1 + I , log 2 1 + tP |hc| 2 1 + I , 1 2 log 2   1 + tP |h d | 2 + |hc| 2 1 + I     (8) In contrast, the treat interference as noise strategy (resp. decode interference) quickly becomes suboptimal as INR/SNR increases (resp. decreases). Remark 2: If each transmitter is equipped with a directive antenna, each transmitter could increase the direct channel h d and decrease the cross channel h c by suitably pointing the directive antenna. If this operation can be done accurately, the interference level is more likely to shift toward weaker regimes. This suggests that it would make more sense to combine multi-antenna processing (enabling directive beams) with an interference management strategy designed for weaker interference regimes (treat interference as noise and RS) rather than stronger interference regimes (decode interference). D. Lessons Learned • Conventional interference management strategies rely on orthogonalization, treat interference as noise, or decode interference. • Orthogonalize the resources so as to completely eliminate multi-user interference is clearly suboptimal. • Treat interference as noise is an efficient strategy whenever the interference level is very weak but is inefficient whenever the interference level grows to weak, strong or very strong. • Decode interference is an efficient strategy whenever the interference level is strong and very strong but is inefficient whenever the interference level is very weak or weak. • RS splits messages into common and private parts so as to partially decode interference and partially treat interference as noise. This allows RS to bridge, unify, and generalize the two extremes of treat interference as noise and decode interference and being efficient in all four interference regimes. • RS is a superset of treat interference as noise and decode interference strategies, and can specialize to each of them depending on how messages are mapped to streams. Those lessons are summarized in Table III and Fig. 4. III. TWO-USER RATE-SPLITTING MULTIPLE ACCESS Building upon the RS scheme for two-user IC, we can obtain some insight into how to design RSMA for twouser downlink (broadcast channel) and uplink (multiple access channel) in the next two sub-sections. We can also relate the above discussion to MA design and show a direct relationship/analogy between interference management and conventional MA designs such as OMA, SDMA, and NOMA. A. Two-User Downlink Rate-Splitting Multiple Access Let us consider a single transmitter potentially equipped with M antennas serving two users. To that end, suppose that the two transmit antennas in Fig. 1 cooperate by exchanging CSI and messages such that they effectively belong to the same transmitter as per Fig. 5. The two-user IC then becomes a two-user MISO broadcast channel (BC) with M = 2 transmit antennas where the base station transmits to two active receiving users whose channels are given by h 1 and h 2 , respectively. In the example of Fig. 5, h 1 = h d h c and h 2 = h c h d . Considering a transmit signal vector x spanning across the transmit antennas, the received signal at user-k can be written as y k = h k x + n k ,(9) where n k ∼ CN (0, σ 2 n,k ). Such a multi-antenna BC is a basic building block of modern downlink communication systems. Assuming perfect CSIT and CSIR, the capacity of this channel is known and is achieved by Dirty Paper Coding (DPC) [41]- [43]. DPC is nevertheless complex to implement due to its inherent nonlinear encoding/precoding mechanism. RSMA has appeared in the past few years as an appealing strategy to achieve close performance to DPC while maintaining the low complexity of linear precoding [8]. In the presence of imperfect CSIT, the capacity and the capacity-achieving strategy are unknown. It is nevertheless known that RS plays a central role to achieve the optimal DoF (and generalized DoF) [17]- [22], and RSMA can outperform DPC [9]. In the sequel, we delineate progressively the key design principles and schemes of RSMA in the two-user downlink scenario. 1) MISO RSMA: Inspired by the IC, the first MISO RSMA architecture is obtained by splitting the two messages W 1 and W 2 into common and private parts and encoding each part into a corresponding stream such that W c,k → s c,k and W p,k → s p,k , k = 1, 2. Given the presence of multiple antennas, precoding/beamforming can be performed across the two antennas and the transmit signal model can be written as x = k=1,2 p c,k s c,k + p k s k ,(10) where p c,k is the precoder of common stream k and p k is the precoder of private stream k. Defining s = [s c,1 , s 1 , s c,2 , s 2 ] T and assuming that E[ss H ] = I, the average transmit sum power constraint 8 at the transmitter is k=1,2 P c,k + P k ≤ P with P c,k = p c,k 2 and P k = p k 2 the power allocated to common stream k and private stream k, respectively. Uniquely, since s c,1 and s c,2 are transmitted from the same transmitter and are decoded by both users in this downlink setting, one can choose p c,1 = p c,2 = p c so that we only have a single common precoder and x = p c (s c,1 + s c,2 ) + k=1,2 p k s k . User-k can now decode using SIC or joint decoding common streams s c,1 and s c,2 and then private stream s k . Using SIC, this architecture would require each receiver k to be equipped with two SIC layers so as to decode three streams s c,j , s c,k , and s k , j = k. In summary, from two messages, this architecture creates four streams, and two SIC are required to recover the original messages. This RSMA architecture is illustrated 9 in Fig. 6. A second MISO RSMA architecture is obtained by noting that an additional benefit of the downlink is that instead of encoding each common part into a common stream W c,k → s c,k , we can first combine the common parts into a common message W c = {W c,1 , W c,2 } that is then encoded into a single common stream W c → s c such that x = p c s c + k=1,2 p k s k .(11) By defining s = [s c , s 1 , s 2 ] T and assuming that E[ss H ] = I, the average transmit (sum) power constraint at the transmitter is P c + k=1,2 P k ≤ P with P c = p c,k 2 the power allocated to the unique common stream. The received signal at user-k, k = 1, 2, j = k, is written as This is the so-called 1-layer RS architecture of RSMA because it relies on a single common stream and therefore a single SIC layer at each receiver, hence simplifying the encoding complexity (three streams used instead of four) and the decoding complexity (one SIC layer instead of two) compared to (10) and (1). Indeed both users decode the single common stream s c into W c by treating the interference from all private streams as noise. Each user-k then retrieves the estimate W c,k from W c . Using SIC, W c is then re-encoded, precoded, and subtracted from the received signal such that user-k decodes its private stream s k into W p,k by treating the remaining interference from the other private streams as noise. Finally user-k recombines W c,k and W p,k into the message W k which is the same as the original message W k if no decoding error occurs. This architecture is illustrated in Fig. 7. y k = h k p c s c + h k p k s k + h k p j s j + n k .(12) 2) Revisiting the Interference Regimes: It is worth relating (12) to the lessons learned in Section II. In (12), after precoding, the two-user MISO BC can effectively be seen as a two-user IC where user-k decodes a common stream s c and a private stream s k subject to interference from private stream s j , j = k. The interference regime experienced by user-k in this effective two-user IC is therefore determined by the strength of the precoded channels h k p k and h k p j (with h k p k and h k p j taking the role of h d and h c , respectively): • |h k p j | ≪ |h k p k |: very weak interference regime and interference should be treated as noise, i.e., p c = 0 and only private streams are used. Decoding interference would perform badly. This scenario can typically occur when the channels are close to being orthogonal. • |h k p j | ≤ |h k p k |: weak interference regime and nonzero power should be allocated to all common and private streams. This would typically occur whenever the channels are neither orthogonal nor aligned. • |h k p j | ≥ |h k p k |: strong interference regime and interference should be decoded. Treat interference as noise would perform badly. This scenario can typically occur when the channels are aligned. This shows that depending on the propagation conditions in multi-antenna settings (e.g., angle between user channel directions), the interference regime can change from very weak to strong. Consequently, downlink MA schemes therefore need the ability to softly evolve from the extreme of decode interference to treat interference as noise. RSMA has the flexibility to cope with all those interference regimes and propagation conditions. It will now appear clear in the sequel how existing MA schemes such as SDMA and NOMA are specifically tailored for one specific interference management strategy (e.g., treat interference as noise, decode interference), one specific interference regime (e.g., very weak, strong) and one type of channel conditions (e.g., orthogonal, aligned), and how RSMA can unify them all. 3) Unifying OMA, SDMA, NOMA, and Multicasting: In the simple two-user case, OMA, SDMA, NOMA, and physicallayer multicasting are particular instances of RSMA, as illustrated by Fig. 8, the message to stream mapping in Table IV [24], and Fig. 9. SDMA is a special case of RSMA by forcing p c = 0. In this way, W k is directly encoded into s k and the system model writes as x = k=1,2 p k s k . By doing so, each stream is decoded by its intended user by treating any residual interference from the other stream as noise. Recalling Tables II and III, from an interference management strategy, SDMA is reminiscent of and builds upon the Treat Interference as Noise strategy which would be efficient only if the residual multiuser interference |h k p j | is sufficiently weak as in orthogonal channels [24]. s 1 s 2 sc SDMA W 1 W 2 - NOMA W 1 - W 2 OMA W 1 - - Multicasting - - W 1 , W 2 RSMA W p,1 W p,2 W c,1 , W c, NOMA is a special case of RSMA by forcing the encoding of message W 2 entirely into s c (i.e., W c = W 2 ) and W 1 into s 1 while turning off s 2 (P 2 = 0). By doing so, user-1 fully decodes the message of user-2 (and therefore interference created by user-2 stream) and 1-layer RS system model becomes the NOMA system model x = p c s c +p 1 s 1 . Note that NOMA also utilizes a common stream since the message of one of the two users, namely W 2 in this example, is decoded by both users. Connecting back to Tables II and III, from an interference management strategy, NOMA is reminiscent of and builds upon the Decode Interference strategy which would be efficient only if the multi-user interference level is sufficiently strong as in aligned channels [24]. OMA is a special case of RSMA by forcing only one user to be scheduled, e.g., user-1 (i.e., p c 2 = p 2 2 = 0). Physical-layer multicasting is a special case of RSMA obtained when messages W 1 , W 2 are both encoded into s c (i.e., W c = {W 1 , W 2 }) and the private streams are turned off (P 1 = P 2 = 0). Drawing an analogy with Table III, we can conclude that MA schemes operate in some preferred residual multi-user interference regimes as shown in Table V. The above clearly shows how three different non-orthogonal approaches to MA designs, namely SDMA, NOMA, and RSMA, fundamentally differ based on how multi-user interference is managed. NOMA is such that at least one user is forced to fully decode the message(s) of other co-scheduled user. SDMA and RSMA do not follow this approach since they both do not force a user to fully decode the messages of another coscheduled user. SDMA actually treats any residual interference as noise, and RSMA is built upon the principle of splitting the messages so as to partially treat interference as noise and partially decode the remaining interference. Consequently, this difference in managing interference has deep consequences on the universality of MA schemes, with RSMA being a superset of SDMA and NOMA as per Fig. 8, but also on the performance of those MA schemes as a function of the propagation conditions [24]. RSMA can also be seen as a smart combination of transmitside and receive-side interference cancellation strategy where the contribution of the common stream is adjusted according to the level of interference that needs to be canceled by the receiver, therefore departing from the transmit-side only and receive-side only interference cancellation strategies of SDMA (and DPC) and NOMA, respectively. This is further summarized in Table VI. Example 1: To further illustrate the split of the messages and the flexibility of RSMA, let us imagine that the message of user- 1 W 1 = (a 1 a 2 a 3 a 4 ) ∈ W 1 = {0000, 0001, 0010, . . ., 1111}, where |W 1 | = 16. Similarly, the message of user-2 is W 2 = (b 1 b 2 b 3 ) ∈ W 2 = {000, 001, 010, . . . , 111}, where |W 2 | = 8. In SDMA, W 1 would be encoded into s 1 and W 2 into s 2 . Assuming uncoded transmission for simplicity, s 1 and s 2 would then be a 16-QAM symbol and a 8-PSK symbol, respectively. In NOMA, W 1 would be encoded into s 1 and W 2 into s c , also using a 16-QAM symbol and a 8-PSK symbol, respectively. In RS, we split user-1's message in, e.g., W c,1 = (a 1 a 2 ), W p,1 = (a 3 a 4 ), and user-2's message in, e.g., W c,2 = (b 1 ), W p,2 = (b 2 b 3 ). The common message is then constructed as W c = (W c,1 W c,2 ) = (a 1 a 2 b 1 ), which is then encoded into s c using a 8-PSK symbol. W p,1 and W p,2 are encoded into s 1 and s 2 using QPSK symbols, respectively. The interested reader is referred to [25] for more examples. 4) Rate Analysis: Under the assumption of Gaussian signaling and infinite blocklength, and perfect CSIT and CSIR, the instantaneous rates for decoding the common and private streams at user-k are given as R c,k = log 2 1 + |h k p c | 2 |h k p 1 | 2 + |h k p 2 | 2 + 1 , R k = log 2 1 + |h k p k | 2 |h k p j | 2 + 1 ,(13) where the noise variance was normalized σ 2 n,k = 1 without loss of generality. To ensure that s c is successfully decoded by both users, its rate cannot exceed R c = min {R c,1 , R c,2 } .(14) As s c contains sub-messages W c,1 , W c,2 of the two users, the rate distribution of R c among the users is adapted to the amount of sub-messages that each user contributed. Let C k denote the portion of rate R c allocated to user-k for W c,k . Then, we have C 1 + C 2 = R c .(15) The overall achievable rate of user-k is R k,tot = C k + R k .(16) Here again, the terminology rate-splitting appears clearly in (16) where we note that the rate of each user is split into two parts, namely, the rate of s k (i.e., the private rate) and part of the rate of s c (i.e., the common rate). Common metrics to design the systems include 1) weighted sum rate (WSR) u 1 R 1,tot + u 2 R 2,tot where u 1 and u 2 are weights set to account for fairness among users (for instance when conducting proportional fair scheduling); 2) max-min fair (MMF) that aims at maximizing min k=1,2 R k,tot ; 3) energy efficiency (EE) R1,tot+R2,tot 1 η (P1+P2+Pc)+Pcir where η ∈ (0, 1] and P cir are respectively the power amplifier efficiency and the circuit power consumption. All three metrics could also be subject to a QoS constraint R k,tot ≥ R th k with R th k a minimum rate threshold to be achieved by user-k. Remark 3: We can wonder what happens if M = 1, namely downlink SISO. RSMA strategy (11) yields x = P c s c + P 1 s 1 + P 2 s 2 (17) and the common and private rates write as R c,k = log 2 1 + |h k | 2 P c |h k | 2 P 1 + |h k | 2 P 2 + 1 , R k = log 2 1 + |h k | 2 P k |h k | 2 P j + 1 ,(18) for k = 1, 2. Without loss of generality, we consider |h 2 | ≤ |h 1 | so that R c = log 2 1 + |h 2 | 2 P c |h 2 | 2 P 1 + |h 2 | 2 P 2 + 1 .(19) Since the capacity of the Gaussian SISO BC with perfect CSIT and CSIR is obtained by performing superposition coding (SC) with SIC (SISO NOMA), optimizing P 1 , P 2 , P c to maximize the WSR would lead to choosing P 2 = 0 so that the message of the weaker user (user-2) W 2 is entirely encoded in s c and SISO RSMA becomes SISO NOMA x = √ P c s c + √ P 1 s 1 and its achievable rates are R 1 = log 2 1 + |h 1 | 2 P 1 ,(20)R c = log 2 1 + |h 2 | 2 P c |h 2 | 2 P 1 + 1 .(21) This shows how RSMA can act as NOMA in Gaussian SISO downlink. Remark 4: The capacity region of a Gaussian SISO BC is known for both cases with fixed and fading channels when there are perfect CSIT and CSIR. However, the problem is open in general when there is imperfect CSIT and only limited cases are known [44]. The fading SISO BC with only perfect CSIR but imperfect CSIT lacks the degraded structure in general for arbitrary fading distributions. In these cases, SC and SIC (and therefore NOMA) can not achieve the capacity region. In [45, Theorem 1], the sufficient condition that the second channel is stochastically larger than or equal to the first channel, i.e., H 2 ≥ st H 1 10 , is derived for which the capacity region is given by R 1 ≤ I(V ; Y 1 |H 1 ), R 2 ≤ I(X; Y 2 |V, H 2 ),(22) with Markov chain V −X −Y 2 −Y 1 . Even though the capacity expressions in (22) look familiar to the rates achievable with SC and SIC (and NOMA), the optimal coding and decoding strategy to achieve (22) remains unknown because the same marginal property of the BC is applied to transform the joint distribution of the underlying channel to fully coupled. 5) Precoder Design and Power Allocation: Recall from Section II-C that a qualitative and insightful way of allocating power to the common stream is so that the interference level caused by the private stream has roughly the same level as the other receiver's noise level, i.e., (1 − t) P |h c | 2 ≈ 1. In the downlink scenario, looking at (13), we can think in a similar way and choose the power allocation P 1 and P 2 to the two private streams such that |h 2 p 1 | 2 ≈ 1 and |h 1 p 2 | 2 ≈ 1, i.e., instead of allocating all the transmit power P to the private streams (as done in SDMA), we allocate a fraction of the total available power to them such that the signal-tointerference-plus-noise ratios (SINRs) of the private streams are not interference limited. If we were allocating a higher power to the private streams with |h k p j | 2 ≫ 1, the SINR |h k p k | 2 |h k pj| 2 +1 ≈ |h k p k | 2 |h k pj | 2 would saturate and the private rates would not increase further. Hence, the key is for the private streams not to enter the interference limited regime, and allocate any remaining power to the common stream whose SINR will keep increasing linearly as P c increases. Doing so, R 1 + R 2 would be roughly equivalent to the sum-rate achieved by SDMA, but the common rate R c would provide an additional rate increase over what SDMA can offer. The above is particularly insightful when the CSIT is imperfect and the transmitter only has an estimateĥ k of userk channel h =ĥ k +h k withh k the channel acquisition error. In such setting, residual multi-user interference scaling as h k p j 2 would be unavoidable regardless of how the precoder p j is designed. Following such an approach, we can demonstrate that RSMA is information theoretically optimal from the DoF perspective in the presence of imperfect CSIT [18], [19], [46] while SDMA and NOMA are not [7]. Because of its inherent robustness to imperfect CSIT, RSMA can also afford smaller feedback overhead than conventional SDMA [47]. More systematic methods to design the precoder and power allocation can be obtained using either closed form low complexity suboptimal techniques or using optimization techniques detailed in [25]. Extensive results have demonstrated that the power allocation to the common stream vs private stream depends on a number of factors, including the angle between user channels [8], [24], the disparity of channel strengths [8], [24], the objective function to maximize [18], [48], the quality of CSIT [18], [48], the QoS, the network load, etc. 10 Stochastic order [52]. H 2 ≥st H 1 means that P(H 2 ≥ s) ≥ P(H 1 ≥ s) for all s from the support of H 1 , H 2 .s 1 s 2 sc θ SDMA W 1 W 2 - θ = 1 NOMA W 1 - W 2 θ = 1 OMA W 1 - - θ = 1 Multic. - - W 1 , W 2 θ = 1 RSMA W p,1 W p,2 W c,1 , W c,2 θ = 1 C-NOMA W 1 - W 2 θ < 1 DF - - W 2 θ < 1 C-RSMA W p,1 W p,2 W c,1 , W c,2 θ < 1 decoded by its intended user and decoded by treated as noise by the other user both users Notations: θ = 1 refers to non-cooperative schemes as in Table IV. θ < 1 refers to cooperative RSMA schemes. 6) Cooperative RSMA: An interesting extension is achieved by considering that the transmitter can opportunistically ask one of the users to act as a relay. This is known as a cooperative relay BC in information theory [49]- [51]. Specifically, in our setting, the common stream is decoded by both users and the transmitter can ask the relay user to forward the decoded common message/stream to the other user to efficiently cope with a wide range of propagation conditions (disparity of user channel strengths and directions) and compensate for the performance degradation due to deep fading [52], [53], as illustrated in Fig. 10. The parameter 0 < θ ≤ 1 refers to the time split between the direct transmission phase and the relaying phase, i.e., θ = 1 indicates that no time is allocated to relaying, hence boiling down to the conventional downlink transmission of Fig. 5. The user relaying feature of cooperative RSMA enables to enlarge the pool of possible schemes within the RSMA framework as shown by the messages-to-streams mapping in the two-user cooperative MISO downlink of Table VII. We note how the mapping of Table IV has been extended to include cooperation where conventional decode and forward (DF) and cooperative NOMA (C-NOMA) are particular instances of cooperative RSMA (C-RSMA). 7) Space-Time / Space-Frequency RSMA: Most RSMA literature deal with a single channel use (be it in time or frequency). This means the signal model as expressed in (11) is applied in a given time slot or given frequency subband. For instance, in the non-alternating CSIT pattern of Table VIII, (11) would be applied separately on subbands A and B, i.e., each subband determining the split of the message, power allocation and precoders based on the available quality of CSIT in that subband (good on subband A and bad on subband B). One could also apply the same strategy to the alternating CSIT pattern of Table VIII. However, because of the alternating feature, simply doing the spatial domain RSMA transmission (11) would not be as efficient as doing an RSMA transmission across the two subbands. In other words, RSMA can benefit from a multi-channel transmission in space-time (ST) or in space-frequency (SF) depending on the CSIT pattern [47], [54], [55]. Such a multi-channel RSMA transmission is particularly helpful in the presence of alternating CSIT, i.e., whenever the CSIT changes across time or frequency in an alternating user-specific manner. The alternating CSIT pattern of Table VIII is a typical (and practical) scenario where the transmitter wants to serve two users but the CSIT of user-1 (resp. user-2) is better on time/frequency B (resp. A) and worse on time/frequency A (resp. B). In such scenario, an ST/SF RSMA scheme can further increase the DoF over conventional RSMA [54], [55]. Compared to the RSMA scheme of (11), ST/SF-RSMA scheme transmits an additional common stream (obtained from a further split of the messages), i.e., s 0 , across the two channel uses. Specifically, considering the alternating CSIT pattern of Table VIII, the transmitted signals in subbands A and B write can be expressed as follows: x (A) = p (A) 0 s 0 + p (A) c s (A) c + k=1,2 p (A) k s (A) k ,(23)x (B) = p (B) 0 s 0 + p (B) c s (B) c + k=1,2 p (B) k s (B) k ,(24) where the superscript (i) refers to the subband. We note the addition of the new common stream s 0 that has been repeated across the two channel uses (precoded by p (i) 0 in channel use i = 1, 2). If the CSIT quality becomes non-alternating, the common stream s 0 becomes useless (zero power is allocated to s 0 ) and SF-RSMA boils down to (11) in each subband. The receiver at both users is more complicated since two common streams have to be decoded at each user before decoding the respective private stream. The decoding works as follows. Let us focus on user-1 for simplicity. First, user-1 decodes s by treating the other private stream as noise. More details on the CSIT pattern conditions needed for the optimality of separate RSMA in each subband can be found from [56]. Another interesting use of ST-RSMA, where RS is combined with space-time block coding to address the lack of knowledge of the channel phase information at the transmitter, has been developed in [57]. 8) MIMO RSMA: Let us now consider a two-user MIMO downlink with M transmit antennas and N receive antennas at each user. The transmitter wants to transmit two Ndimensional vectors of messages w 1 and w 2 to user-1 and user-2, respectively. To that end, for each user, each of those N messages is split into a common part and a private part. Common parts of both users are combined and encoded into a N -dimensional vector of common streams s c and private parts are encoded into two N -dimensional vectors of private streams s k . M × N precoding are performed on the common and private stream vectors such that the transmit signal is written as x = P c s c + k=1,2 P k s k .(25) Let us consider an example with N = 2 to illustrate the universality of the framework. We have four messages to transmit (two for each user) w 1 = [W (1) 1 , W (2) 1 ] and w 2 = [W (1) 2 , W (2) 2 ], where W (j) i refers to the jth message of user-i, i = 1, 2, j = 1, 2. We split the four messages into common and private parts such that W (1) 1 = W (1) c,1 , W (1) p,1 ,(26)W (2) 1 = W (2) c,1 , W (2) p,1 ,(27)W (1) 2 = W (1) c,2 , W (1) p,2 ,(28)W (2) 2 = W (2) c,2 , W (2) p,2 .(29) Common parts are then combined into W W (1) c = W (1) c,1 , W (1) c,2 → s (1) c ,(30)W (2) c = W (2) c,1 , W (2) c,2 → s (2) c ,(31) so as to form the 2-dimensional common stream vector s c = s (1) c s (2) c T . Private parts are encoded into private streams W (1) p,1 → s (1) 1 , W (2) p,1 → s (2) 1 ,(32)W (1) p,2 → s (1) 2 , W (2) p,2 → s (2) 2 ,(33) so as to create two 2-dimensional private stream vectors s 1 = s (1) 1 s (2) 1 T and s 2 = s (1) 2 s (2) 2 T . At the receivers, both users decode the vector of common streams first using the two receive antennas, perform SIC, and then decode their respective private stream vector while treating the co-scheduled user's private stream vector as noise. Fig. 11 illustrates the MIMO RSMA architecture. The mapping of messages to streams of Table IV is now further extended to account for MIMO RSMA in Table IX. RSMA again is a superset of all schemes. We note that SDMA is replaced by MU-MIMO where a vector of private streams is transmitted to each user. NOMA in the form of having one user (user-1) decoding all the messages of the other user (user-2) is illustrated, along with OMA. Additionally, other subschemes of RSMA are listed for the sake of illustration. For instance, in subscheme 1, both users decode the 2nd messages and, in subscheme 2, both users decode the 1st messages, i.e., each user decodes one message of the other user. Subscheme 3 is when both users decode the 2nd message of user-2. Note that the MIMO NOMA scheme proposed in [36], [37] is also a subscheme of MIMO RSMA, and it switches among MU-MIMO, NOMA, or the three additional subschemes of RSMA when different numbers of transmit antennas and receive antennas are considered. Optimization of MIMO RSMA with both perfect and imperfect CSIT can be performed as in [35]. Similar to the MISO case, RSMA outperforms MU-MIMO and NOMA in the MIMO case. Note that RSMA is information theoretically optimal from a DoF perspective in MIMO BC with imperfect CSIT [21], [22]. In the asymmetric case where the number of receive antennas is not the same at each user, RSMA with a multi-channel transmission (similar to ST/SF RSMA) needs to be used to achieve optimality. The reader is referred to [22] for more details on such a ST RSMA scheme for asymmetric MIMO BC. B. Two-User Uplink Rate-Splitting Multiple Access In the uplink, we consider two single-antenna users simultaneously transmitting their messages to a receiver equipped with M antennas. When M = 1, such two-user MAC can be considered as a special case of the two-user IC in Fig. 1 where the two receivers are colocated and cooperatively decode the messages of the two users. 1) Two-User Architectures: Inspired by the RSMA design in IC, the uplink RSMA architecture has been proposed in [23] by splitting the message of one user into two parts. Without loss of generality, we assume that user-1's message W 1 is split into W 1,1 and W 1,2 . By independently encoding the two parts into s 1,1 , s 1,2 , respectively allocating transmit power P 1,1 , P 1,2 , and superposing the two streams, the transmit signal at user-1 is given by x 1 = P 1,1 s 1,1 + P 1,2 s 1,2 .(34)s 1 s 2 sc s (1) 1 s (2) 1 s (1) 2 s (2) 2 s (1) c s (2) c MU-MIMO W (1) 1 W (2) 1 W (1) 2 W (2) 2 - - NOMA W (1) 1 W (2) 1 - - W (1) 2 W (2) 2 OMA W (1) 1 W (2) 1 - - - - Multicasting - - - - W (1) 1 , W (1) 2 W (2) 1 , W (2) 2 RSMA W (1) p,1 W (2) p,1 W (1) p,2 W (2) p,2 W (1) c,1 , W (1) c,2 W (2) c,1 , W (2) c,2 subscheme 1 W (1) 1 - W (1) 2 - W (2) 1 W (2) 2 subscheme 2 - W (2) 1 - W (2) 2 W (1) 1 W (1) 2 subscheme 3 W (1) 1 W (2) 1 W (1) 2 - - W (2) 2 decoded by its intended user and treated as noise by the other user decoded by both users At user-2, the message W 2 is directly encoded into s 2 . By allocating certain power P 2 , the transmit signal at user-2 is x 2 = √ P 2 s 2 . The signal received at the receiver is y = h 1 x 1 + h 2 x 2 + n = P 1,1 h 1 s 1,1 + P 1,2 h 1 s 1,2 + P 2 h 2 s 2 + n,(35) where h 1 , h 2 ∈ C M×1 are the channel vectors and n ∼ CN (0, I M ) is the AWGN vector. The receiver can employ SIC or joint decoding to decode the three streams s 1,1 , s 1,2 , s 2 . If using SIC, two layers of SIC are required at the receiver. In summary, two-user uplink RSMA creates two virtual users from the user that splits its message into two parts and in total three streams are sent to the receiver. Two SIC layers are required to recover the original messages of the two users. This uplink RSMA architecture is illustrated in Fig. 12. 2) Unifying OMA and NOMA: In the two-user MAC, both OMA and NOMA are subschemes of RSMA, as per Fig. 13. The corresponding messages to streams mapping is illustrated in Table X. User-1 User-2 s 1,1 s 1,2 s 2 NOMA W 1 - W 2 OMA W 1 - - RSMA W 1,1 W 1,2 W 2 NOMA is a subscheme of RSMA by forcing the encoding of message W 1 entirely into s 1,1 and W 2 into s 2 while turning off s 1,2 (P 1,2 = 0). There is no message splitting at user-1 and one layer of SIC is required at the receiver to decode and remove the entire message of one user (i.e., user-1) before decoding the message of the other user (i.e., user-2). Note that, though we here refer to this strategy as (uplink) NOMA, this strategy is nothing else than the traditional SIC at the receiver to achieve the corner points of the MAC capacity region. OMA is also a subscheme of RSMA by forcing one user to be scheduled, i.e., W 1 is directly encoded into s 1,1 while W 2 is turned off. 3) Uplink MIMO RSMA: An extension to the two-user uplink SIMO RSMA in Fig. 12 is the two-user uplink MIMO RSMA with N transmit antennas at each user and M receive antennas at the receiver. In this case, each user- k transmits a N -dimensional vector of messages w k = [W (1) k , . . . , W (N ) k ] to the receiver. At user-1, each of its N messages W 1,2 ] T are linearly precoded by the precoders P 1,1 , P 1,2 ∈ C M×N and superposed such that the transmit signal at user-1 is x 1 = P 1,1 s 1,1 + P 1,2 s 1,2 .(36) At user-2, the message vector w 2 is encoded into a Ndimensional stream vector s 2 and linearly precoded by P 2 . The transmit signal at user-2 is x = P 2 s 2 . An example of the two-user uplink MIMO RSMA when N = 2 is illustrated in Fig. 14. Each message in the message User-1 User-2 s 1,1 s 1,2 s 2 s (1) 1,1 s (2) 1,1 s (1) 1,2 s (2) 1,2 s (1) 2 s (2) 2 NOMA W (1) 1 W (2) 1 - - W (1) 2 W (2) 2 OMA W (1) 1 W (2) 1 - - - - RSMA W (1) 1,1 W (2) 1,1 W (1) 1,2 W (2) 1,2 W (1) 2 W (2) 2 vector w 1 of user-1 is split into two parts as W (1) 1 = W (1) 1,1 , W (1) 1,2 , W (2) 1 = W (2) 1,1 , W (2) 1,2 .(37) The four submessages are respectively encoded into four streams and create two 2-dimensional stream vectors s 1,1 = s 1,1 s and linearly precoded to form the transmit signal x 2 . The mapping of messages to streams for this example is illustrated in Table XI. Again, MIMO NOMA and OMA are subschemes of MIMO RSMA in the uplink. Though we here did not split user-2 messages, other variants of uplink RSMA schemes can also be derived by also splitting w 2 . C. Lessons Learned • OMA, SDMA, and NOMA are respectively based on the orthogonalization, treat interference as noise, and decode interference strategy. This limits the application of each of those MA schemes to a specific interference regime. • RSMA relies on the RS interference strategy and is efficient in all interference regimes. • RSMA bridges, unifies, and generalizes OMA, SDMA and NOMA. • RSMA is a superset of OMA, SDMA and NOMA, and can specialize to each of them depending on how messages are mapped to streams. • RSMA is applicable to both downlink and uplink, to SISO, MISO and MIMO settings. • RSMA can be extended to the cooperative relay setting and to space-time/frequency transmission. IV. K -USER RSMA The schemes developed for two-user in the previous sections can all be extended to K-user and conclusions derived for two-user hold for K-user as well. Nevertheless, the K-user scenario also opens the door to other RSMA schemes with a variable number of SIC layers. In this section, the state-ofthe-art K-user RSMA schemes are delineated in the downlink, uplink, and multi-cell MIMO settings where each node is equipped with multiple antennas. A. Downlink We first consider a downlink symmetric MIMO BC where a transmitter equipped with M antennas serves K users, each equipped with N receive antennas. The users are indexed by K = {1, . . . , K}. Without loss of generality, we assume a Q k -dimensional vector of messages is transmitted to user-k, i.e., w k = [W (1) k , . . . , W (Q k ) k ] T , where Q k ≤ min(M, N ). Depending on the different RSMA schemes adopted, user messages {w k , k ∈ K} are split, combined, and encoded into different stream vectors, which are then mapped to the transmit antennas using linear or non-linear precoders and forms the transmit signal x. The signal is transmitted through a MIMO BC and the receive signal at user-k is given by y k = H H k x + n k , where H k ∈ C M×N is the channel matrix between the base station and user-k, n k ∼ CN (0, σ 2 n,k I N ) is the AWGN vector at user-k. Without loss of generality, the noise variances of all users are assumed to be equal to 1, i.e., σ 2 n,k = 1, ∀k ∈ K. Next, we detail the transceiver architectures for different RSMA schemes in such symmetric MIMO BC. 1) 1-layer RS: The system model of 1-layer RS for MIMO BC is first proposed in [35]. As illustrated in Fig. 15, each message W (i) k in w k for user-k is split into one common sub-message and one private sub-message as W (i) k = {W (i) c,k , W (i) p,k }, ∀i ∈ {1, . . . , Q k },(1) k , . . . , s (Q k ) k ] T to be decoded by user-k only. The K + 1 data stream vectors s c , s 1 , . . . , s K are linearly precoded by the precoders P c , P 1 , . . . , P K and superposed at the transmitter. The resulting transmit signal is x = P c s c + K k=1 P k s k .(38) The signal received at user-k is y = H H k P c s c + K j=1 H H k P j s j + n k .(39) User-k employs SIC or joint decoding to decode two stream vectors s c and s k . Only one single layer of SIC is required at each user if employing SIC to decode the data streams. Assuming Gaussian signalling, the instantaneous rates of decoding the common and private stream vectors based on SIC are given by R c,k = log 2 det I + P H c H k (R c,k ) −1 H H k P c , R k = log 2 det I + P H k H k (R k ) −1 H H k P k ,(40) where R c,k and R k are noise plus interference covariance matrices defined as R c,k = I + K j=1 H H k P j P H j H k , R k = I + K j=1,j =k H H k P j P H j H k .(41) To guarantee the common stream vector is successfully decoded by all users, it is achievable rate is R c = min k {R c,1 , . . . , R c,K }. As s c contains sub-messages of K users, we have k∈K C k = R c , where C k denotes the portion of R c allocated to user-k for the transmission of w c,k . The overall achievable rate of user-k is R k,tot = C k + R k . 2) Hierarchical RS: Hierarchical RS (HRS) is proposed in [58] for MISO BC where the transmitter is equipped with multiple antennas and each receiver is equipped with a single antenna. In this subsection, we extend the system model to MIMO BC where each node has multiple antennas. In HRS, users are clustered into G separate groups indexed by G = {1, . . . , G} according to the similarity of their channel covariance matrices. Each group-g contains a subset of users K g such that g∈G K g = K. In contrast to 1-layer RS where each user message is split into two parts, in HRS, each user message W (i) k , i ∈ {1, . . . , Q k } in w k for user- k (assuming user-k is in group g) is split into three sub- messages as W (i) k = {W (i) K,k , W (i) Kg ,k , W (i) k,k }, resulting in three sub-message vectors, namely, an inter-group common message vector w K k = {W (1) K,k , . . . , W (Q k ) K,k }, an inner-group common message vector w Kg k = {W (1) Kg,k , . . . , W (Q k ) Kg ,k }, and a private message vector w k k = {W (1) k,k , . . . , W (Q k ) k,k }. The inter-group common messages of all users {w K 1 , . . . , w K K } are combined into Q c (Q c ≤ min(M, N )) common messages w K ∈ C Qc×1 and encoded into one common stream vector s K = [s (1) K , . . . , s (Qc) K ] T , which is decoded by all users. For groupg, g ∈ G, the inner-group common message vectors of users in group-g {w Kg k , ∀k ∈ K g } are combined into Q cg (Q cg ≤ min(M, N )) common messages w Kg ∈ C Qcg ×1 and encoded into one common stream vector s Kg = [s (1) Kg , . . . , s (Qcg ) Kg ] T to be decoded by all users in group-g. The private message vector w k k of user-k is independently encoded into the private stream vector s k = [s (1) k , . . . , s (Q k ) k ] T to be decoded by user-k only. The overall K + G + 1 data stream vectors s K , s K1 , . . . , s KG , s 1 , . . . , s K are linearly precoded by the corresponding precoders P K , P K1 , . . . , P KG , P 1 , . . . , P K . The resulting transmit signal is x = P K s K + G g=1 P Kg s Kg + K k=1 P k s k .(42) The signal received at user-k is y = H H k P K s K + G g=1 H H k P Kg s Kg + K j=1 H H k P j s j + n k . (43) In contrast to 1-layer RS where each user only decodes two streams, in HRS, each user-k (k ∈ K g ) employs SICs or joint decoding to decode three stream vectors s K , s Kg , and s k . Two layers of SIC are required at each user if employing SIC. Assuming Gaussian signalling, the instantaneous rates of decoding s K , s Kg , and s k based on SIC are given by R K,k = log 2 det I + P H K H k (R K,k ) −1 H H k P K , R Kg,k = log 2 det I + P H Kg H k R Kg,k −1 H H k P Kg , R k = log 2 det I + P H k H k (R k ) −1 H H k P k ,(44) where the noise plus interference covariance matrices R K,k , R Kg,k , and R k are defined as R K,k = I + G g=1 H H k P Kg P H Kg H k + K j=1 H H k P j P H j H k , R Kg,k = I + G i=1,i =g H H k P Ki P H Ki H k + K j=1 H H k P j P H j H k , R k = I + G i=1,i =g H H k P Ki P H Ki H k + K j=1,j =k H H k P j P H j H k .(45) To guarantee the successful decoding of the inter-group and inner-group common stream vectors, the achievable rates follow R K = min k {R K,k , k ∈ K} and R Kg = min k {R Kg,k , k ∈ K g }. As s K and s Kg contain sub-messages of multiple users, we have k∈K C K k = R K and k∈Kg C Kg k = R Kg , where C K k and C Kg k respectively denote the portion of R K and R Kg allocated to user-k for the transmission of w K k and w Kg k . The overall achievable rate of user-k is R k,tot = C K k + C Kg k + R k . It is easy to observe that HRS is a more general framework than 1-layer RS for MIMO BC. HRS boils down to 1-layer RS when the inner-group common stream vectors {s Kg , ∀g ∈ G} are turned off, i.e., P Kg = 0, ∀g ∈ G. In this case, s K for HRS is equivalent to s c for 1-layer RS. In Fig. 16, a four-user two-group HRS example is illustrated with user-1 and user-2 in group-1, user-3 and user-4 in group-2. The user sets K = {1, 2, 3, 4}, K 1 = {1, 2}, K 2 = {3, 4} are simply denoted as 1234, 12, 34, respectively. User-1 and user-2 are required to decode the inter-group common stream vector s 1234 and the inner-group common stream vector s 12 while user-3 and user-4 are required to decode the s 1234 and s 34 . 3) Generalized RS: Generalized RS (GRS) is proposed in [8] for MISO BC to further enhance the spectral efficiency by splitting and encoding multiple common streams, which are required to be decoded by different subsets of users. It is a generalized transmission framework that embraces 1layer RS, HRS, linearly precoded MU-MIMO, and MIMO NOMA as special cases. We here extend it to MIMO BC. At the transmitter, each user message is split into 2 K−1 parts, i.e., W (i) k = {W (i) A ′ ,k | A ′ ∈ K, k ∈ A ′ }. The message vector w k is therefore split into 2 K−1 sub-message vectors w A ′ k , ∀A ′ ∈ K, k ∈ A ′ } with w A ′ k = {W (1) A ′ ,k , . . . , W (Q k ) A ′ ,k }. The sub-message vectors {w A k | k ∈ A} for all users in a given subset A ∈ K are combined and encoded into one stream vector s A of dimension Q cA × 1 which is only decoded by users in A and treated as noise by the remaining users. Following the concept of stream order introduced in GRS for MISO BC [8], we denote the stream vectors to be decoded by l number of users as l-order stream vectors. All l-order stream vectors {s A ′ | A ′ ∈ K, |A ′ | = l} are wrapped up into a larger dimensional vector s l ∈ C A ′ ∈K,|A ′ |=l Q cA ′ ×1 and linearly precoded by the precoding matrix P l composed by {P A ′ | A ′ ∈ K, |A ′ | = l}. The resulting transmit signal is x = K l=1 P l s l = K l=1 A ′ ⊆K,|A ′ |=l P A ′ s A ′ .(46) At user-k, 2 K−1 − 1 layers of SIC or joint decoding is employed to decode the required streams vectors. The set of l-order stream vectors required to be decoded at user-k is denoted by Following the decoding order from the K-order stream vector down to the 1-order stream vector as well as a certain decoding order π l,k to decode the l-order stream vectors in S l,k (i.e., s π l,k (i) is decoded before s π l,k (j) if i < j), we obtain the rate of decoding l-order stream vector s π l,k (i) at user-k under Gaussian signalling and SIC decoding as S l,k = {s A ′ | A ′ ⊆ K, |A ′ | = l, k ∈ A ′ }.R π l,k (i),k = log 2 det I + P H π l,k (i) H k R π l,k (i),k −1 H H k P π l,k (i) ,(47) where R π l,k (i),k is given as R π l,k (i),k = I + j>i H H k P π l,k (j) P H π l,k (j) H k + l−1 l ′ =1 |S l ′ ,k | j=1 H H k P π l ′ ,k (j) P H π l ′ ,k (j) H k + A ′ ⊆K,k / ∈A ′ H H k P A ′ P H A ′ H k .(48) To guarantee the successful decoding of each stream vector s A , the achievable rates follow R A = min k {R A,k , k ∈ A}. As s A contains sub-messages of users in A, we have k∈A C K k = R A , where C A k denotes the portion of R A allocated to userk for the transmission of w A k . The overall achievable rate of user-k is R k,tot = k∈A ′ C A ′ k + R k,k . GRS is a more general framework than 1-layer RS and HRS for MIMO BC. It boils down to 1-layer RS when only the K-order stream vector s K and the 1-order stream vectors s k , k ∈ K are active. s K for GRS is equivalent to s c for 1-layer RS in such case. It reduces to HRS when only s K , s k , k ∈ K, and s Kg , g ∈ G are active. A toy 3-user GRS example is illustrated in Fig. 17. The user message vector of each user is split into 4 sub-message vectors. For user-1, w 1 is split into {w 123 1 , w 12 1 , w 13 1 , w 1 1 }, each is required to be decoded by different groups of users. {w 123 1 , w 123 2 , w 123 3 } are combined into w 123 , which is then encoded into the 3-order stream vector s 123 . Similarly, {w 12 1 , w 12 2 } are combined into w 12 and encoded into the 2order stream vector s 12 . Following this method, we obtain in total 7 transmit stream vectors. 3 layers of SIC is adopted at each user to decode the intended 4 stream vectors. In this example, user-1 decodes s 123 , s 12 , s 13 , s 1 sequentially with s π2,1(1) = s 12 and s π2,1(2) = s 13 . 4) Dirty paper coded RS: Dirty paper coded RS (DPCRS) is a non-linearly precoded RS framework built upon DPC. It is proposed in [9] for MISO BC and is extended to MIMO BC in this subsection. We first illustrate the transmission architecture of the simplest DPCRS model 1-layer Dirty Paper Coded RS (1-DPCRS), and then briefly discuss its extension to multilayer dirty paper coded rate-splitting (M-DPCRS). Fig. 18 illustrates the proposed 1-layer MIMO DPCRS. It is an extension of 1-layer RS in Section IV-A1 by enabling dirty paper encoded private streams. Following 1-layer RS, the K user messages are split and combined into one common message vector w c and K private message vectors w p,1 , . . . , w p,K . w c is encoded into a common stream vector s c using a codebook shared by all users while w p,1 , . . . , w p,K are encoded and precoded by DPC for a certain encoding order π into the private stream vectors s π(1) , . . . , s π(K) . The resulting transmit signal is x = P c s c + K k=1 P π(k) s π(k) ,(49) where P c , P π(1) , . . . , P π(K) are the precoders for the corresponding streams s c , s π(1) , . . . , s π(K) . Each user-π(k) decodes s c and s π(k) based on SIC or joint decoding. Assuming Gaussian signalling, the instantaneous rates of decoding the common and private stream vectors based on SIC are given by R c,π(k) = log 2 det I + P H c H π(k) R c,π(k) −1 H H π(k) P c , R π(k) = log 2 det I + P H π(k) H π(k) R π(k) −1 H H π(k) P π(k) ,(50) where R c,π(k) and R π(k) are noise plus interference covariance matrices defined as R c,π(k) = I + K j=1 H H π(k) P π(j) P H π(j) H π(k) , R π(k) = I + K j>k H H π(k) P π(j) P H π(j) H π(k) .(51) The calculation of the overall achievable rate for each user follows 1-layer RS, which is not detailed here. Besides 1layer MIMO DPCRS, we can also integrate DPC and different RS schemes to enhance the system performance especially in imperfect CSIT. For instance, Fig. 19 illustrates another DPCRS scheme, namely, multi-layer MIMO DPCRS, which marries DPC and GRS. In contrast to GRS in Fig. 17 which relies on linear precoding to precode all streams, multi-layer MIMO DPCRS considers dirty paper coded private streams. B. Uplink Uplink RSMA is first introduced in [23] for SISO MAC and further extended to SIMO MAC in [25]. The main benefit of uplink RSMA discovered in existing works is that the capacity region of the Gaussian MAC can be achieved by uplink RSMA without time sharing among users. In the following, we introduce a generic uplink RSMA transmission framework for MIMO MAC, which is simply denoted by uplink MIMO RSMA. k,2 , resulting in two sub-message vectors w k,1 and w k,2 . Note that splitting K k=1 Q k − 1 messages is sufficient to achieve the capacity region without time sharing in this setting. For illustration simplicity, we here consider a more general model in which the user messages of all users are split. At user-k, the submessage vectors w k,1 and w k,2 are independently encoded into s k,1 and s k,2 , precoded by P k,1 and P k,2 , and superposed at the transmitter. The resulting transmit signal is x k = P k,1 s k,1 + P k,2 s k,2 , ∀k ∈ K.(52) The signals of all users are transmitted via MIMO MAC, the receive signal is y = k∈K H k x k + n,(53) where H k ∈ C M×N is the channel vector between user-k and the receiver, and n ∼ CN (0, I M ) is the AWGN vector. The receiver then employs the receive filters W k,1 , W k,2 , k ∈ K to detect the stream vectors for all users. Denote the decoding order of the 2K received stream vectors {s k,i | k ∈ {1, . . . , K}, i ∈ {1, 2}} by π such that π k,i < π k ′ ,i ′ if s k,i is decoded before s k ′ ,i ′ . 2K − 1 layers of SIC is needed to decode all stream vectors. Assuming Gaussian signaling and infinite blocklength, the rate of decoding s ki at the receiver is given as follows R k,i = log 2 det I + P H k,i H H k W H k,i (R k,i ) −1 W k,i H k P k,i ,(54)where R k,i is defined as R k,i = I + π k ′ ,i ′ >π k,i W k,i H k ′ P k ′ ,i ′ P H k ′ ,i ′ H H k ′ W H k,i .(55) The proposed uplink MIMO RSMA model is generic. It embraces uplink SISO RSMA in [23] and uplink SIMO RSMA in [25] as two sub-schemes. The study of uplink MIMO RSMA is still in its infancy. Its SE/EE performance and applications in different services such as eMBB, URLLC, and mMTC or their hybrid services are worth more investigations. C. Multi-cell In multi-cell networks, the transmission from multiple transmitters to multiple receivers can be categorized into "coordinated transmission" and "cooperative transmission" depending on whether the data is shared among the transmitters [6]. RSMA has been investigated in both transmissions [59]- [62]. It is shown to be a promising strategy to enhance SE by providing a powerful inter-cell and intra-cell interference management capability. In the following subsections, the transmission models of MIMO RSMA for "coordinated transmission" and "cooperative transmission" are respectively delineated. 1) Coordinated transmission: A K-cell coordinated MIMO is illustrated in Fig.21(a). The message of each user is sent from the serving cell (with one RSMA-enabled transmitter) while the resource allocation and user scheduling are coordinated among cells. At each transmitter-k, the message vector w k for user-k is split into two sub-message vectors w c,k and w p,k , independently encoded into two stream vectors s c,k and s p,k , and linearly precoded by the precoding matrices P c,k and P p,k . The transmit signal at transmitter-k is given as x k = P c,k s c,k + P p,k s p,k .(56) The signal of all transmitters are transmitted to all users. The signal received at user-k is where H H kj ∈ C N ×M is the channel between transmitter-j and user-k and n k ∼ CN (0, I N ) is the AWGN vector at user-k. Each user-k is required to decode all common stream vectors s c,1 , . . . , s c,K and the intended private stream s p,k based on SIC or joint decoding. For a certain decoding order π k such that s c,π k (i) is decoded before s c,π k (j) at user-k if i < j. Assuming Gaussian signaling and infinite blocklength, the rate of decoding s c,π k (i) and s p,k at user-k are obtained as follows y k = K j=1 H H kj (P c,j s c,j + P p,j s p,j ) + n k ,(57)R c k,π k (i) = log 2 det I + P H c,π k (i) H kπ k (i) R c,π k (i) −1 H H kπ k (i) P c,π k (i) , R p k = log 2 det I + P H p,k H kk (R k ) −1 H H kk P p,k .(58) where R c,π k (i) and R k are defined as R c,π k (i) = I + j>i H H kπ k (j) P c,π k (j) P H c,π k (j) H kπ k (j) + K j=1 H H kj P p,j P H p,j H kj , R k = I + j∈K,j =k H H kj P p,j P H p,j H kj .(59) The achievable rate of user-k is R k,tot = min R c 1,k , . . . , R c K,k + R p k .(60) As discussed in [59], The performance of coordinated MIMO RSMA could be further boosted by splitting the message of each user into N > 2 parts. Each part is decoded by a different group of users with the same instantaneous CSIT quality. 2) Cooperative transmission: As discussed in [25], cooperative MIMO requires all transmitters to be connected as a virtual giant transmitter to serve all users in the downlink. All downlink transmission frameworks of RSMA discussed in IV-A therefore can be applied for cooperative MIMO. Compared with single-cell MIMO BC which is subject to a sum-power constraint, multi-cell cooperative MIMO BC requires per-cell transmit power constraint and the fronthaul capacity constraints [62]. D. Numerical Results Next, we show numerically the benefits of RSMA in terms of the significant wireless communication performance met- Readers are referred to [35] for more details on the simulation setting and the rate region optimization algorithm. Fig. 22 illustrates the ergodic rate region of RS, MU-MIMO, NOMA, and DPC in perfect CSIT with equal channel variances or 10 dB channel variance disparity between the users. Note that in the two-user case, HRS and GRS are equivalent to 1-layer RS, which are therefore simply denoted by "RS". We observe that in both subfigures, the rate region of RS is larger than that of MU-MIMO and NOMA, and is much closer to the capacity region of MIMO BC achieved by DPC. When the users have equal channel variances as in Fig. 22(a), NOMA achieves the worst performance as no channel strength disparity can be leveraged to manage interference. Although MU-MIMO is capable of exploiting the full DoF in this setting (i.e., underloaded and perfect CSIT), there is still a rate region gap between RS and MU-MIMO. As MU-MIMO always treats interference as noise, it only works well when user channels are (semi-)orthogonal [8]. When the users have a 10 dB channel variance disparity, the rate region gap between NOMA and RS reduces. Interestingly, the rate region of NOMA and MU-MIMO outperform each other in part while RS always outperforms both. Although NOMA can utilize the channel variance disparity to improve its rate region, it incurs a theoretical DoF loss and therefore rate loss due to its interference management principle of forcing one user to fully decode the streams of all users [7]. Owing to its powerful interference management capability of partially decoding the interference and partially treating the interference as noise, RS enhances the spectral efficiency in various user deployments. In Fig. 23, we further compare SE of different strategies in imperfect CSIT. Consider a downlink MU-MISO with M = 4 and K = 2. The power of the channel error is defined as σ 2 e,k = σ 2 k P α , where P = tr(PP H ) and α is fixed to 0.6. More details of the simulation setting and the rate region optimization algorithm can be found in [9]. Besides linearly precoded RS, the ergodic rate region of 1layer DPCRS is also illustrated. In both subfigures, the ergodic rate region of DPC drops significantly as it is sensitive to CSIT uncertainty. Surprisingly, linearly precoded RS achieves a larger rate region than DPC while maintaining a much lower transceiver complexity. As RS has been proved to achieve the optimal multiplexting gain/DoF of MISO BC with imperfect CSIT [19], such gain is reflected in its spectral efficiency gain over MU-MIMO and NOMA. DPCRS, which marries the advantages of DPC and RS, achieves the largest achievable rate region in both subfigures at the sacrifice of a higher transceiver complexity than DPC and RS. Fig. 24 compares EE of different strategies for two-user MU-MISO with perfect CSIT, M = 2, N = 2. EE is defined as K k=1 R k,tot 1 η tr(PP H )+Pcir [63], [64], where η = 0.35 is the power amplifier efficiency, P cir = M P dyn + P sta is the static circit power consumption with P dyn = 27 dBm and P sta = 1 mW. There is a 1 bit/s/Hz minimum rate threshold for each user. RS achieves a larger EE than NOMA and SDMA in both subfigures. Again, NOMA achieves the EE worst performance when there is no channel strength disparity while it slightly outperforms SDMA when there is a 10 dB channel strength disparity and low transmit power constraint. Fig. 22 downlink for different user deployments and CSIT conditions. In the uplink, the authors in [65] have also shown that RSMA achieves a higher spectral efficiency than NOMA and other OMA strategies, as illustrated in Fig. 25 thanks to its capability of achieving each point at the boundary of the capacity region. In contrast, NOMA can only reach the corner points at the boundary of the capacity region and time sharing is needed to achieve the points along the line segment between the two corner points. 2) General and Unified: As widely acknowledged and discussed in the literature of RSMA [8], [9], [24], [25], [29], RSMA is a general and universal MA scheme, interference management strategy, and non-orthogonal transmission framework that unifies SDMA, NOMA, OMA, physical-layer multicasting and treats them as sub-schemes or special cases. Such universality is reflected in the SE and EE gain over existing MA schemes as per Fig. 22-25, as well as in its operational region illustrated in Fig. 26. In Fig. 26 h1 2 h2 2 , which indicates the channel angle between the two users (ρ = 0 and ρ = 1 corresponding to aligned and orthogonal channel directions, respectively). The y-axis is the channel disparity between the two users in dB, which is given as γ dB = 20 log 10 (γ). The parameters and the precoder design are detailed in [24]. RS automatically reduces to the existing MA schemes in some regimes while it outperforms all existing schemes in the remaining regimes especially when the SNR is high. Therefore, instead of using OMA, NOMA, SDMA and optimize them for each propagation deployment, one can use a single and universal RS scheme in wireless communication networks. 3) Flexible: Fig. 27 illustrates the impact of network load to the system performance for a MISO BC with K = 6 users and imperfect CSIT with α = 0.5. The variances of user channels are randomly selected from [0.1, 1]. Two MISO NOMA schemes are considered as baseline schemes, namely, MISO NOMA with a single user group (G = 1) which is motivated by SC and SIC in SISO BC, and MISO NOMA with three user groups (G = 3) which clusters 6 users into 3 user groups and inner-group users are served by SC and SIC. Readers are referred to [7] for more details on the parameter settings and baseline schemes of MISO NOMA. When the number of transmit antennas is M = 3, the network load is extremely overloaded. In this case, MISO NOMA (G = 1) achieves a better MMF rate than SDMA and MISO NOMA (G = 3). However, as the number of transmit antennas increases, the performance of MISO NOMA (G = 1) decreases due to a significant loss in DoF. MU-LP outperforms the two MISO NOMA when the network is underloaded (M = 6). In all subfigures, 1-layer RS, which only uses a single layer of SIC at each user, achieves the best MMF rate no matter the network is underloaded or overloaded. As per Fig. 22-27, RSMA is a flexible MA scheme which is suited for different user deployments (diverse channel directions and strength), network loads (underloaded and overloaded), CSIT condition (perfect and imperfect CSIT), and SNR regimes (low, medium, and high SNR regimes). The root of such flexibility is the powerful interference management ability of RSMA introduced by the common stream (vectors). By adjusting the sub-messages encapsulated in each common stream as well as the power allocation, RSMA dynamically alters the portion of interference to be pre-canceled at the transmitter and decoded at the receivers. 4) Robust: Robustness is one of the most important benefits of RSMA discovered in recent years [9], [18], [19], [67]. The robustness of RSMA is grounded in deep information theoretic results, where RSMA is shown to achieve the optimal DoF in MISO BC when CSIT is imperfect [19]. Motivated by such discovery, different sources that impair CSIT are studied with RSMA such as pilot contamination [68], channel estimation errors [18], user mobility [66], etc. RSMA is shown a significant SE gain over existing MA baselines for all sources of CSIT impairments. Taking user mobility as an example in Fig. 28, we have shown in [66] that RSMA achieves a significantly higher user speed (i.e., 40 km/h) than SDMA (i.e., 5 km/h) for a given QoS rate constraint (i.e., 8 bps/Hz) thanks to its powerful interference management ability. 5) Reliable and Low Latency: : As shown in [32], [34] for downlink and in [69] for uplink, RSMA is more reliable than SDMA and NOMA under finite blocklength. This is illustrated in Fig. 29 for underloaded and overloaded downlink settings where we note that RSMA can achieve a given MMF rate at a lower blocklength, therefore enabling lower latency communications, compared to other MA schemes. It is important to recall here again that 1-layer RS scheme of RSMA is used in Fig. 29, and therefore requires a single SIC. This is in contract with NOMA that achieves lower performance at the expense of the need for 7 SIC layers at the receivers in Fig. 29(b). E. Lessons Learned • RSMA framework is a superset of OMA, SDMA and NOMA, and can specialize to each of them depending on how messages are mapped to streams. This holds for both uplink and downlink, as well as in SIMO, MISO and MIMO settings. • All instances of RSMA are supersets of SDMA and OMA, but not all, such as 1-layer RS scheme, are supersets of NOMA. This enables RSMA schemes with better performance than NOMA at a much lower receiver complexity (e.g., only one SIC layer). • RSMA can be linearly precoded or non-linearly precoded. • RSMA is applicable to all major settings of a cellular network, namely downlink, uplink, and multicell, for general MIMO deployments (with SISO, SIMO, MISO as special cases). • RSMA is spectrally and energy efficient, general and unified, flexible, robust, reliable and has lower latency. V. NUMEROUS APPLICATIONS FOR RSMA Given its fundamental communication theoretic principles, and its unique features (efficiency, universality, flexibility, robustness and resilience, reliability and low latency), RSMA finds applications in all modern multi-user scenarios. We here provide a description of over forty promising scenarios and applications of RSMA and briefly explain how and why RSMA is beneficial 11 . A subset of those scenarios and applications is illustrated in Fig. 30. All those applications demonstrate the suitability of RSMA for FeMBB, eURLLC, umMTC, and new wireless services in 6G [28], [29]. Importantly, the benefits and applications of RSMA have not only been explored from an academic perspective with the performance superiority over other MA techniques confirmed via stochastic analysis [70]- [73] but have also been confirmed using realistic link-level simulations over 5G compliant channel models [33], [35], [74]- [77]. Interested readers are also encouraged to consult [25], [26], [38], [39] for some more descriptions of interesting applications and future works. 1) Downlink SISO: RSMA can be used for both degraded or non-degraded BC. In the degraded BC, RSMA boils down to NOMA (since NOMA is capacity achieving for degraded BC) but RSMA can be used to decrease the receiver complexity and still come close to the capacity region with a reduced number of SIC layers compared to NOMA [8]. In the nondegraded BC, RSMA achieves a strictly larger rate region than NOMA (see Remark 4). 2) Downlink MISO: RSMA can be used in various MISO settings with perfect and imperfect CSIT. The design and precoder optimization of RSMA would depend on the objective value, e.g., WSR, MMF, EE, and the modeling of the CSIT imperfection, e.g., unbounded vs bounded error [8], [18], [48], [64], [78]- [81]. One challenge with imperfect CSIT is to make sure that the rate optimized are achievable. 1-layer RS is the most common scheme but other schemes based on multi-layer 11 Other promising scenarios and applications are also discussed in Section VII on frequently asked questions. RSMA can also be used [8]. A strong benefit of RSMA is its inherent robustness to imperfect CSIT. 3) Downlink MIMO: RSMA can be used in settings with multiple receive antennas at the users. MIMO RSMA enables to transmit vectors of common and private streams, which requires a special design [22], [35], [82]- [86]. Various CSIT/CSIR assumptions and objective values can also be considered in the MIMO setting. 4) Uplink: Uplink RSMA avoids the need of time sharing to achieve the capacity region, which finds new applications in modern systems subject to latency constraints [23], [65], [69], [87]- [89]. Much attention has been brought to the uplink SISO RSMA, but uplink multi-antenna RSMA is also possible as shown in Section IV-B. 5) Unifying OMA, SDMA, NOMA, Multicasting: RSMA is universal and general in the sense that it encompasses many different schemes (such as OMA, SDMA, NOMA, physical-layer multicasting) as particular instances [7], [8], [24], [90]. It is interesting to study and understand under what propagation conditions, e.g., disparity of channel strengths and angle between user channels, RSMA boils down to each of those schemes [24]. 6) Statistical CSIT: RSMA is helpful in scenarios where only statistical CSIT (i.e., the distribution information of the user channels) is available, This is a particular instance of imperfect CSIT, but important in practice as it leads to low feedback overhead. This can be for instance a channel covariance matrix calculated by averaging over frequency and time resources and reported once in a while to the base station. RSMA has been found robust to scenarios where only statistical CSIT is available [58], [91]- [93] 7) Quantized Feedback: Quantized feedback is another popular CSI feedback mechanism that relies on a codebook to quantize the channel. Due to quantization error, the CSIT is imperfect and RSMA has been found robust to the quantization error [47], [94]. Interestingly the number of feedback bits needed to achieve a given sum-rate performance is decreased with RSMA compared to conventional SDMA/MU-MIMO, therefore enabling an overhead reduction [47], [95]. 8) Imperfect CSIT and CSIR: Aside imperfect CSIT, for which we know RSMA is robust, imperfect CSIR is also an important problem that has received less attention [96]. The rate achievability and the SIC error propagation are two important issues to consider when studying RSMA with imperfect CSIR. 9) Frequency Division Duplex (FDD) Massive MIMO: One popular strategy in FDD massive MIMO is to rely on two-tier precoders where the first tier precoder is based on channel statistics to cancel inter-cluster interference and the second tier based on instantaneous CSI feedback to manage intra-cluster interference [97], [98]. Unfortunately if the space spanned by the covariance matrices of the clusters overlap or if the channel statistics and/or the CSIT is imperfect, the performance of such approaches degrade and RSMA can be used to mitigate those issues [58]. 10) Time Division Duplex (TDD) Massive MIMO: TDD massive MIMO also has its impairments such as pilot contam-ination due to multiple users sharing the same 12 pilot sequence when performing uplink channel sounding and channel aging due to mobility and latency between the channel sounding phase and the data transmission phase. Those two issues effectively lead to imperfect CSIT that need to be tackled. RSMA has been shown robust to mitigate both problems [66], [68], [99] 11) Hardware Impairments: Another impairment in Massive MIMO comes from the hardware, e.g., phase noise, digital-to-analogue (DAC) and analogue-to-digital (ADC) converters. Phase noise can effectively lead to a scenario where the CSIT is imperfect and RSMA has been shown to be robust against hardware impairments such as phase noise [100]. Quantization errors due to finite resolution DAC and ADC also lead to multi-user interference that can be efficiently and flexibly managed using RSMA [101]- [103]. 12) Cell-Free Massive MIMO: Cell-free refers to distributed antennas systems, which is subject to the same pilot contamination problem as conventional TDD massive MIMO. The disparity of path loss nevertheless makes the analysis and performance gap between schemes different. RSMA was found robust to pilot contamination in various cell-free topologies [104]. 13) Millimeter Wave and TeraHertz Systems: Higher frequency bands are subject to their own set of challenges such as high path loss, blocking and expensive RF chains. RSMA can be used and designed in conjunction with hybrid analoguedigital precoding to outperform conventional SDMA strategies at high frequencies [91], [105], [106]. RSMA in combination with cooperative communications can also be used to further combat the path loss and blocking issues [107]. 14) Non-Linear Precoding and DPC: RSMA is commonly designed using linear precoding due to its practical use in real systems. Nevertheless, non-linear precoding could also be used to precode the private streams, using for instance DPC, Tomlinson-Harashima precoding, vector perturbation, etc. Recent works have demonstrated the benefits of non-linear precoded RSMA [9], [108]- [110]. The combination of RSMA and DPC (so-called DPCRS strategy as per Fig. 18 and Fig. 19) is particularly suited to imperfect CSIT settings since we know that DPC is capacity achieving with perfect CSIT. As the CSIT quality improves, the power allocation to the common stream decreases, and DPCRS progressively boils down to conventional DPC for multi-antenna BC. It is of interest to investigate how we could further enhance the performance in imperfect CSIT beyond that achieved by DPCRS strategy [9]. 15) Cooperative and Relaying Systems: Cooperative communications and RSMA, i.e., cooperative RSMA as in Fig. 10, form a happy marriage as shown in [52], [53] since a user can decode the common stream and its private stream and forward the common stream to help improving the decodability of the common stream at a cell edge user. By doing so, RSMA can efficiently cope with a wide range of propagation conditions (disparity of user channel strengths and directions), compensate for the performance degradation due to large path loss, extend the coverage, and outperforms SDMA, NOMA, and cooperative NOMA. Those observations also hold in the imperfect CSIT setting [111]. RSMA can also find other interesting applications in multi-user relaying systems [112]- [115]. It would be worth further exploring how, in the classical relay channel [116], as well as the cooperative communication works [117], [118], RS inherently plays a role in the decode-and-forward protocol and its variations. 16) Full Duplex: Self interference is a major problem in full duplex systems. Thanks to its flexible interference management ability, RSMA can increase the range of selfinterference over which full duplex outperforms half duplex [119]. 17) Physical Layer Security: Security at the physical layer has attracted a lot of attention in the past decade [120]. Two interesting scenarios occur for RSMA: first, the eavesdropper is not one of the users served by RSMA (hence, the eavesdropper is not the intended recipient of any message transmitted by the transmitter, but only intercepts confidential messages sent to other authorized users), and second, the eavesdropper is one of the users and can therefore decode the common stream [121], [122]. In the former case, the common stream can be used to manage the interference between users and therefore increase the sum-rate of those users but also to act as an artificial noise (AN) to confuse the eavesdroppers. In the latter case, an eavesdropper could decode a common stream and could therefore reconstruct the original message if it can decode the corresponding private. This leads to a secrecy constraint for the private stream and a tradeoff between sum-rate and secrecy rate [123]- [125]. RSMA achieves better WSR and is more robust to channel errors than SDMA while ensuring all users' security requirements. In contrast, because the entirety of the message of a user is mapped to the common stream in NOMA, NOMA cannot guarantee all users' secrecy rate constraints [123]. Other interesting analysis of RSMA in the presence of untrusted users has recently appeared in [126]. 18) Energy Efficient Networks: EE is an increasingly important metric. In general, the superiority of RSMA in terms of spectral efficiency translates into an EE gain over conventional MA baselines such as OMA, SDMA and NOMA [63], [127]. Nevertheless, SE maximization and EE maximization are two conflicting objectives in the moderate and high SNR regimes. This calls for the study of the tradeoff between the two criteria. In [128], the performance of RSMA is shown to be superior to or equal to SDMA and NOMA in terms of SE, EE, and their tradeoff. 19) Reconfigurable Intelligent Surfaces (RIS): RIS is equipped with a large number of passive elements placed in the environment that can be adjusted so as to provide a passive beamforming gain and make the channel propagation more favourable. The interplay between RIS and RSMA is very attractive and has been shown very promising by several works [39], [129]- [138]. The advantages of RIS-aided RSMA are higher spectral efficiency, coverage extension and beam control flexibility thanks to the presence of the common stream, robustness to CSI imperfection which is welcome given the channel acquisition challenge in RIS, and lower computational and hardware complexity because RSMA superiority means RSMA can afford operating with less complex RIS architectures while maintaining the same overall performance [39]. 20) Multi-Cell Networks: The concept of RS for the 2-user IC can be extended to more than 2 users, or to multiple cells in cellular network context [139]. Furthermore each node in each cell can be equipped with multiple transmit antennas and precoders jointly with the message split need to be optimized across all cells to maximize the objective function accounting for any potential imperfection in the CSIT [22], [59], [60], [140]- [142]. Of particular interest, [59] showed that in a multicell multi-antenna setting with imperfect CSIT, the transmitters should adopt a so called topological RS (TRS) strategy that consists in a multi-layer structure and in transmitting multiple common messages to be decoded by groups of users rather than all users. Though TRS was studied from a communication and information theoretic point of view, it remains to be studied how TRS could be optimized using optimization tools so as to maximize its performance at finite SNR. 21) Coordinated Multi-Point (CoMP): RSMA can be used for CoMP joint transmission where all base stations collaborate by sharing CSI and data. In such a setting, all antennas at the base station act together to form a giant BC or MAC (depending on whether the focus is on downlink or uplink) and precoders and power across streams can be optimized as a function of the CSI and path loss disparity subject to the percell power constraints. It was shown in [61] that whenever there is little inter-user channel strength disparity but large inter-cell channel disparity, SDMA was a suitable option. On the other hand, whenever there is a large inter-user channel strength disparity but little inter-cell channel disparity, NOMA was a better option. In comparison, RSMA always bridges, generalizes, and outperforms existing SDMA and NOMA strategies. It was shown to be suited to any deployment with any inter-user and inter-cell channel disparities. Applications of RSMA to uplink CoMP would be worth investigating. 22) Cloud and Fog-Radio Access Network (C/F-RAN): C-RAN consists of multiple remote radio heads (RRH) connected through fronthaul links to a baseband unit that performs central processing. CoMP joint transmission can be implemented across all those RRH while accounting for the challenging feature that the fronthaul links have a limited capacity. Various RSMA approaches have been proposed for C-RAN: 1) RSMA transmit signal is compressed before being transmitted over the fronthaul [143]; 2) the common and private streams of RSMA to be transmitted over each fronthaul are wisely selected so as to satisfy the fronthaul constraints [62], [144]- [152]. For a given fronthaul capacity constraint, both approaches have demosntrated that RSMA is more spectrally and energy efficient than SDMA and NOMA. 23) Dynamic Resource Management and Cross-Layer Optimization: In RSMA-aided networks, network resource management has received more and more attention. Many of these works, however, only consider the short-term optimization of system resources without taking into account the long-term network operation constraints and objectives. Furthermore, they do not consider random traffic arrivals. To meet the explosive access demands of mobile devices, the multicast communication of a satellite and aerial-integrated network with RSMA is studied in [153]. Specifically, the unmanned aerial vehicle (UAV) sub-network uses RSMA to support massive access of internet of things (IoT) devices in a content delivery scenario. However, in practice, users may require different types of data traffic, such as high-quality video streaming, voice, video phone, online games, network broadcasting, and so on. This necessitates the cross-layer design of RSMA-aided system to cope with traffic exposure rate and long-term system constraints. It is recommended that joint adaptive source encoding and cross-layer resource management schemes be studied. For example, using a supervised learning-based approach for cross-layer resource allocation in a NOMA-aided system [154], the execution time can be reduced by 98% while ensuring that each user has at least one subcarrier. Given the outstanding characteristics of RSMA, more cross-layer (across physical -PHY, medium access control -MAC, and application -APPL -layers) dynamic resource management solutions and optimizations are expected to bring further improvement to the system performance. Though most of the RSMA optimization works focus on PHY and MAC layers (beamforming and power allocation, time and frequency resource allocation), some works apply RSMA for the wireless streaming of video and consider joint optimization across PHY, MAC, and APPL (hence, with the optimization of the encoding rate adaptation for video) [155]. This is an important research direction for RSMA, especially for mobile internet (where video takes more than 70% of traffic) and emerging applications such as 360 video, autonomous driving, metaverse for 6G and beyond. 24) Wireless Caching: The small storage capacity of edge devices combined with the high energy consumption of active caching shorten the standby period of user devices. By positioning caching devices adjacent to the user terminal, wireless caching networks (WCN) can reduce recurrent file transfers to achieve low power consumption caching. Furthermore, the spectrum efficiency can be increased by combining RSMA with WCN. Two caching policies can be examined, i.e., the most popular content (MPC) and the intelligent coded caching (CC) policy. In MPC policy, multiple popular files can be superimposed in the power domain to form a mixed file based on RSMA, which is broadcasted to cache devices by the base station [156]- [158]. Another approach of cache placement is the coded caching (CC), where partitions of the files are stored instead [159]. Multiple cache-enabled receivers can be served using both caching policies within the help of RSMA [156]- [158]. It is shown that the caching gains can be improved significantly, the mutual benefit can be achieved through collaborative design of caching placements and RSMA [156]- [158]. In a way, RSMA for caching-aided multi-antenna BC expands upon, and inherits the main features of, RSMA used for the classical multi-antenna BC with imperfect CSIT. Additionally, using RSMA, the caching device can provide services to numerous users in the same time-frequency resource block. The quantity of caching files, their level of popularity, and their network resources all affect the key indicator of wireless caching networks, e.g., the system hit probability. 25) Overloaded Cellular Internet of Things and Massive Access: Cellular networks will have to cope with extensive IoT devices, and consequently serve simultaneously a large number of devices with heterogeneous demands and CSIT qualities. In [160], [161], RSMA is used to tackle such a scenario by considering an overloaded MISO downlink with two groups of CSIT qualities, namely, one group of users (representative of high-end devices) for which the transmitter has partial knowledge of the CSI, the other group of users (representative of IoT devices) for which the transmitter only has knowledge of the statistical CSI. RSMA is shown to be DoF-optimal in such a setting with heterogeneous CSIT qualities, more efficient than various MA baselines, robust to CSIT inaccuracy, and flexible to cope with heterogeneous QoS rate constraints of all high-end and low-end users. 26) Joint Communication and Jamming: Thanks to its flexibility and robustness, RSMA can be used to efficiently communicate to information users (IUs) and simultaneously jam adversarial users (AUs) to disrupt their communications [162], [163]. The precoders and power allocation to common and private streams can be optimized based on imperfect CSIT for IUs and statistical CSIT for AUs to maximize the sumrate under jamming power constraints on the pilot subcarriers of AUs (as jamming pilot subcarriers is known to be very effective disruptive method). RSMA is shown to outperform significantly conventional MA schemes. 27) Non-Orthogonal Unicast and Multicast (NOUM) : NOUM refers to mixed traffic services where a multicast (genuinely intended to multiple users, i.e., not user-specific) message is transmitted to multiple users and additionally one unicast (user-specific) message is transmitted per user. Conventionally OMA is used to transmit unicast and multicast services on different resources. This is suboptimal and a better approach is to superpose the multicast message on top of the unicast messages and use one SIC at each receiver to decode the multicast message first and then the intended unicast message. RSMA can do better by making a more efficient use of the SIC. Indeed, by splitting the unicast message into common and private parts and encoding jointly the common parts and the multicast message into a common stream to be decoded by all users, the SIC can be efficiently exploited for the dual purpose of separating multicast from unicast but also better manage interference between unicast streams [127], [164]- [166]. Taking the 2-user architecture of Fig. 7 as an example, RSMA in NOUM is obtained by encoding a multicast message W 0 (genuinely intended to both users) along with common parts W c,1 and W c,2 into a common stream. User-k then decodes the common stream to retrieve W 0 and W c,k , before decoding its private stream. RSMA has been shown to outperform SDMA and NOMA counterparts in NOUM [127], [164], [165]. 28) Multigroup Multicast: This scenario considers K users grouped into G < K groups and a transmitter that delivers on the downlink one multicast message per group, i.e., all users in the same group are interested in the same multicast message. Conventional BC is a subset of that setting where there is only one user per group. Such a scenario can occur in broadcasting services, caching settings, satellite communi-cations, multi-view video, virtual reality (360) video, online gaming, metaverse, etc. The challenge of multigroup multicast is that the number of users is often large compared to the number of transmit antennas, i.e., overloaded network, which creates severe multigroup interference issues. RSMA, thanks to its flexibility, is able to tackle that multigroup interference efficiently and outperform both SDMA and NOMA schemes significantly [67], [167]- [172] and cope with various other multicast scenarios relevant to 6G [173]. 29) Multibeam Satellite Communications: Multibeam satellite is often modeled by a multigroup multicast where each beam can be thought of one group. This comes from the superframe/frame-based precoding assumption of multibeam satellite communication which is that one codeword encodes all users' messages in one beam. This is used in practice to increase the efficiency of the error correcting codes. Other challenges in satellite systems include the frequency reuse across multiple spot beams creating high levels of interference, the per-feed power constraints, the satellite channel subject to line-of-sight and large path loss, the imperfect CSIT and latency in CSI acquisition due to large round trip delay, the high doppler for low earth orbit satellite, and the overloaded settings (many terminals in each beam). RSMA has been found to be quite suited to tackle all those challenges thanks to its efficiency, flexibility, and robustness [168], [174]- [177]. 30) Unmanned Aerial Vehicles (UAV)-Assisted Networks: Due to the great mobility and flexibility, UAVs are able to provide services to users when they are outside of the base station's coverage area or when the user channel conditions are unfavorable. However, a UAV relies on its limited capacity battery to fly, hover and communicate, which leads to a limited endurance. Fortunately, RSMA can reduce the communication energy consumption of UAV. By acting as an aerial base station, the UAV can serve multiple ground users simultaneously using RSMA [178]- [181]. RSMA for UAV leads to new joint cross-layer optimization problem of UAV location, transmit power allocation, bandwidth, and RSMA, accounting for CSI availability and traffic. RSMA and UAV locations can also be designed along with methods used for predicting the cellular traffic according to the analysis of previous data, therefore further saving transmit power consumption [182]. 31) Space-Air-Ground/Satellite-Terrestrial Integrated Networks: In satellite-terrestrial (or in space-air-ground) integrated network, the satellite sub-network shares the same RF band with the terrestrial sub-network. A higher spectrum efficiency and throughput is achieved via dynamic spectrum access sharing to enhance spectrum utilization. However, severe interference in and between the sub-networks is induced by an aggressive frequency reuse, which calls for the use of efficient multi-antenna [183] and multiple access strategies [153], [184], [185]. In this context, RSMA has been shown to exhibit significant performance gains compared with various traditional transmission strategies such as SDMA and NOMA in various settings where only CSI is shared among subnetworks (to enable coordination of precoders) or where CSI and data are shared (to enable cooperation as in CoMP joint transmission across all antennas of both sub-networks) [184]. 32) Constructive Interference Exploitation/Symbol-Level Precoding: RSMA is conventionally studied with Gaussian inputs, but in practice finite constellations need to be used. Constructive interference (CI), also called symbol-level precoding, exploits the finite constellation such the information symbols are used, along with the CSI, in order to exploit the multi-user interference to increase the useful signal received power. In other words, CI designs the transmit precoders such that the resulting interference is constructive to the desired symbol, i.e., the interference signal pushes/moves the received symbols away from the decision thresholds of the constellation towards the direction of the desired symbol. Interestingly, RSMA and CI techniques can be combined to further enhance the sum-rate achieved by RSMA with finite input alphabet [186], [187]. 33) Reliability and Low Latency : There are many reliability and latency-sensitive applications, such as industrial automation, smart grid and intelligent transportation. In order to reduce the transmission latency, shortpackets with finite blocklength codes are typically adopted. This brings a stringent latency requirement to the physical layer. Interestingly, the efficiency, robustness, flexibility benefits of RSMA over SDMA and NOMA in the infinite blocklength regime were also confirmed in the finite blocklength regime for both downlink and uplink, offering therefore the additional reliability and the low latency needed to enable those applications [32], [34], [69]. In other words, RSMA can achieve a given performance requirement with a smaller blocklength than that needed by SDMA and NOMA. The significant performance gains of RSMA over SDMA and NOMA were also confirmed in linklevel evaluations with practical codes and finite blocklengths [33], [75]. 34) Integrated Radar Sensing and Communications: Integrated (radar) sensing and communications (ISAC) merges wireless communications and remote sensing into a single system, where both functionalities are combined via shared use of the spectrum, the hardware platform, and a joint signal processing framework. It also enables sensing capabilities of the network to help communications and inversely. The challenge is that a transmitter not only has to serve multiple users simultaneously but also has to satisfy radar performance requirements, lwhich leads to a tradeoff between communication and radar performance and calls for agile and versatile MA schemes. The flexibility and robustness of RSMA becomes particularly handy in this setting as it was shown that RSMA can provide a better communication-radar tradeoff than SDMA and NOMA schemes for a wide range of propagation conditions and radar metrics [38], [76], [101], [102], [188]- [191]. 35) Grant-based, Grant-Free and Semi-Grant Free Transmission and Massive Random Access: Grant-based transmission has been a conventional approach to access the network when the network load is small. With the emergence of IoT devices and mMTC, hybrid grant-based (GB) and grant-free (GF) transmissions are needed to reduce latency. Grant-free access however leads to collisions that need to be managed. In such scenario, GF users meet opportunities to share wireless resources with GB users. In [192], RSMA was used in this setting where the GF users split their messages to realize distributed contentions and utilize transmit power most effectively for robust transmissions, meanwhile keeping themselves transparent to the GB user. RSMA was shown to significantly decrease outage probability and achieve full multi-user diversity gain without restricting the GB and GF users' target rates. 36) Network Slicing: To guarantee the performance of heterogeneous services involving FeMBB, eURLLC, umMTC in 6G, network slicing is needed to allocate resources to different services. Network slicing can be done in an OMA fashion, which means that different services are isolated and allocated orthogonal (non-interfering) resources. However, as the number of users grows, OMA-based slicing may not be the optimal scheme for all scenarios, and a non-orthogonal scheme may achieve a better performance. Thanks to its message splitting and corresponding flexibility, RSMA has emerged as a superior MA scheme for network slicing, outperforming OMA and NOMA in many scenarios [193], [194]. 37) Cognitive Radio: Cognitive radio may increase spectral efficiency through secondary spectrum sharing / dynamic spectrum access, transmitting under the interference temperature. Various setups can be considered where primary transmitterreceiver links communicate simultaneously with a group of cognitive secondary users that form one of classical multiuser channels such as MAC, IC, or BC [195]. In the BC, RSMA can for instance be employed at the secondary transmitter to communicate with secondary users while limiting the interference to primary users [163], [196]. Since the primary link remains oblivious to the secondary system operation in cognitive radio, another potential benefit of RSMA is whether the primary system does not need to time-share the channel with the secondary users, unlike the conventional spectral-gap filling approaches [195]. 38) Optical and Visible Light Wireless Communications: RSMA is commonly studied for RF communications, but it can also be applied to other communication systems such as optical and visible light, though the constraints from those signals and systems need to be captured in RSMA design and optimization, e.g., visible light signals have peak and average optical power constraints (limited for eye safety and practical illumination requirement), are non-negative and real due to the intensity modulation and direct detection technique [197]- [199]. Similarly to RF systems, RSMA outperforms SDMA and NOMA in optical and visible light communications [197]- [199]. 39) Multi-carrier: Frequency domain using multi-carrier transmission (e.g., OFDM/OFDMA) can be combined with RSMA in the same way SDMA and OFDM/OFDMA work together in 4G and 5G, namely the spectrum is divided into subbands (made of contiguous or distributed subcarriers) and multiple users are paired together on one or multiple subbands using RSMA [162], [163], [171], [200]. Resource allocation, including user pairing per subband and power allocation to common and private streams, needs to be optimized. RSMA-OFDMA inherits the same benefits as SDMA-OFDMA since RSMA builds upon SDMA/MU-MIMO. 40) Wireless Information and Power Transfer (WIPT): WIPT has the similarity with ISAC that both systems need to use the spectrum to deliver two services: either sensing and communications, or power and communications. Interestingly, similarly to ISAC, the common stream in RSMA can be helpful to manage multi-user interference and at the same time boost the performance of the other service, i.e., sensing or power. Consequently, RSMA has been found more efficient than SDMA and NOMA in WIPT [201], [202]. 41) Vortex Wave Communications/Orbital Angular Momentum: RSMA can be combined with orbital angular momentum (OAM) to benefit from the flexibility and robustness of RSMA and the additional degrees of freedom of OAM [203]. 42) Mobile Edge Computing (MEC): Equipped with powerful computing and storage capabilities, MEC can cope with the challenges of providing superior and latency-critical computing by enabling edge users to offload their tasks for nearby processing, therefore reducing the backhaul bottlenecks, network delays, and transmission costs. Since RSMA can achieve the full rate boundary of the MAC, while NOMA can achieve only several separated points on the rate boundary, RSMA can be used more efficiently than NOMA to aid MEC where multiple users can offload their tasks while maintaining the QoS of each user [204]. 43) Mixed Criticality: Thanks to the efficiency of RSMA, QoS is enhanced [8], [25]. Thanks to its flexibility, RSMA can also deliver QoS enhancements in applications where, due to the diversity of users, services and applications in 6G, mixed criticality QoS levels are assigned to those users and services [27], [31]. VI. MYTHS Myth 1: RSMA is a special (power-domain) NOMA It is actually the opposite with (power-domain) NOMA being a special RSMA technique. In the same way as decoding interference is a particular instance of RS (and has been known to be so since the 80s and the seminal works on RSMA [11], [23]), NOMA is a particular instance of RSMA, as illustrated in Fig. 4, 8 and 13. However, one needs to check more carefully at how the schemes have been built as NOMA is not a special case of all RSMA schemes. In uplink, NOMA simply relies on SIC. In other words, there is nothing special to NOMA as it is just an SIC receiver (similarly to the SIC used in spatial multiplexing for point to point MIMO). RSMA on the hand not only relies on SIC but also on splitting of the messages at the users. As shown in Section III-B, by adjusting the split and the power allocation to the resulting streams, uplink RSMA boils down to uplink NOMA. In downlink SISO and MISO, all power-domain NOMA schemes are characterized by having at least one user being forced to fully decode the message(s) of other co-scheduled user(s) [7]. In the two-user, NOMA (as well as SDMA, OMA, and physical-layer multicasting) is a subset of RSMA as shown in Section III-A. In the general K-user MISO case as it would depend on the specific RSMA scheme used. 1-layer RS is a superset of SDMA since by turning off (i.e., allocating no power to) the common stream, 1-layer RS boils down to MU-LP. On the other hand, 1-layer RS is not a superset of NOMA. 1-layer RS and NOMA are particular instances/schemes of the RSMA framework based on the generalized RS relying on multiple layers of SIC at each receiver, as discussed in Section IV and [7], [25]. In downlink MIMO, there is less research on RSMA. Nevertheless, RSMA is shown in [7], [35] to outperform and be a superset of NOMA whenever at least one user is forced to fully decode the multiple streams of other coscheduled users. More generally, as it appeared in Table IX and related discussion, RSMA in a MIMO setting will always be a superset than NOMA because RSMA has the message splitting capability for each message (and therefore the related interference management capability), which does not feature in NOMA schemes. This implies that the optimization space of RSMA will be larger than that of NOMA. The relationship between SDMA, NOMA, 1-layer RS, 2layer (hierarchical) RS (as introduced first in [58] for FDD massive MIMO), and RSMA is further illustrated in Fig. 31. Myth 2: RSMA cannot outperform MU-MIMO MU-MIMO schemes can rely on linear or nonlinear precoding schemes [6]. For both types of precoders, we can design linear precoded or non-linear precoded RSMA that are always a superset of MU-MIMO, and would therefore always achieve at least the same performance as MU-MIMO. Taking K-user 1-layer RS for simplicity (but the discussion holds for other RSMA architectures), by decreasing the amount of power allocated to the common stream, 1-layer RS progressively converges to K-user SDMA/MU-MIMO and in the limit where no power is allocated to the common stream, K-user 1-layer RS swiftly boils down to K-user SDMA/MU-MIMO. Hence, 1-layer RS really builds upon SDMA/MU-MIMO and SDMA/MU-MIMO is a subscheme of 1-layer RS, which provides a guarantee to 1-layer RS that its rate and DoF are always the same or better than those of SDMA/MU-MIMO. The same observation holds for non-linear precoded RSMA as discussed in [108], [109] for Tomlinson-Harashima precoded RSMA and in [9], [165] for dirty paper coded RSMA. Note that since dirty paper coding achieves the capacity of MIMO Gaussian BC with perfect CSIT, applying dirty paper coded RSMA to a perfect CSIT setting would end up allocating zero power to the common streams, however non-zero power would be allocated to common streams to boost the performance in imperfect CSIT settings [9], [165]. Other types of non-linear precoded RSMA schemes are left for further investigations. This is completely different from NOMA. NOMA does not build upon SDMA/MU-MIMO. With G groups, K-user MISO NOMA can boil down to G-user SDMA by turning off the power to the weaker users in each group, but K-user MISO NOMA can mathematically never boil down to K-user SDMA [7]. The rate/DoF of K-user NOMA can therefore be worse than that of K-user SDMA [7]. Myth 3: RSMA is only beneficial for multi-antenna downlink RSMA is definitely beneficial in multi-antenna downlink, but RSMA is also beneficial in single-antenna downlink, single/multi-antenna uplink, in multi-cell, and in relaying. In single-antenna downlink (SISO BC), the benefits of RSMA depends on whether the BC is degraded or nondegraded. In the degraded BC, RSMA boils down to NOMA (since NOMA is capacity achieving for degraded BC) but RSMA can be used to decrease the receiver complexity and still come close to the capacity region with a reduced number of SIC layers compared to NOMA [8]. We illustrate in Fig. 32 a three-user example to compare the sum rate of NOMA and 1-layer RS when users' channel variances are σ 2 1 = 1, σ 2 2 = 0.3, σ 2 3 = 0.1 and users are with and without QoS rate constraints. In the two subfigures, 1-layer RS respectively achieves 99.84% and 97.65% rate of NOMA while only a single layer of SIC is required at each user (instead of 2 for NOMA). In the non-degraded BC, RSMA achieves a strictly larger rate region than NOMA (see Remark 4). Examples for non-degraded SISO BC comprise (but are not limited to) multicell BC or BC with interference [205], BC with imperfect CSIT [45], RIS-assisted BC [206], IC with moderate or low interference [11]. Intuitively the underlying feature of these examples is, that the channels to the receivers cannot be ordered. For all examples above, there exists reasons, why the channels, either because there are some carriers where one channel is better than the other and vice versa, or because on some channel realizations one user receives more or less interference than the other, or because the channel state is not perfectly known at the transmitter and therefore, the encoder cannot determine the optimal pre-coding order. In practice, we do not have perfect CSIT. Therefore, we will always have to operate on non-degraded SISO BCs. In uplink (MAC), RSMA achieves every point at the bound-ary of the capacity region without time sharing [23]. As illustrated in Fig. 25, RSMA can also enhance the spectral efficiency when proportional user fairness is considered [65]. In multi-cell (IC), RSMA outperforms SC and SIC in the weak interference regime. It enables an enhanced interference management of the intra-cell and inter-cell interference as the amount of intra/inter-cell interference to be canceled at the transmitters or decoded at the receivers can be flexibly adjusted. RSMA therefore achieves improved spectral and energy efficiency over the conventional coordinated schemes without RSMA [60], [141]. In relaying and cooperative systems, by enabling the users with strong channel strength to relay and forward the common stream to the users with weaker channel strength, not only can cooperative RSMA improves spectral and energy efficiency [53], [207], but also offering substantial benefits in terms of coverage extension [25]. Myth 4: The common stream in RSMA is a multicast stream required for multiple users A common stream in RSMA is multicasted at the physical layer since it is to be decoded by multiple users. However the content of the common stream is not necessarily intended to those users. This is a difference from multicasting and broadcasting where a message is genuinely intended to multiple users, and therefore decoded by multiple users. In RSMA, the common stream is created for interference management purpose, not because the content of the common stream is intended to multiple users. It is also possible to do multicasting on top of RSMA. This is the case where K users want to receive unicast messages W 1 , . . . , W K (one for each user), but additionally a multicast message W 0 is transmitted and intended to all K users. In that case, RSMA can be used to encode in a common stream the multicast message W 0 along with parts of the unicast messages W 1 , . . . , W K , as discussed in NOUM subsection V-27 [127], [165]. Recall that NOMA also has a common message/stream, though commonly not denoted using such terminology in the NOMA literature. Hence, the common message is not a message that is originally intended for all users. It is required to be decoded by all users but is not necessarily intended for all users. Myth 5: As the common stream needs to be decoded by multiple users, it causes privacy/security issues Note that decoding the common stream at the physical layer does not imply the sharing of data as encryption is commonly implemented at higher layers and decryption is performed using user-specific codes. Same would go for other schemes relying on SIC and interference decoding such as NOMA. There is therefore no privacy/security issues as long as higher layer encryption is performed. However, from a physical layer security/secrecy perspective [208], [209], the problem is different and RSMA can be designed to maximize the secrecy rate as discussed in Subsection V-17. Myth 6: The message of each user is required to be split into one or multiple common parts and a private part In RSMA, the message of each user could be split into one or multiple common parts and a private part, but is not always required to be split. Whether the message of a single user or the messages of multiple users are split depends on the objective function. For instance in Example 1, both messages W 1 and W 2 are split, but it could happen that only W 1 is split or only W 2 is split. Splitting the message of a single user (as in [18]) or more users (as in [8]) at the transmitter has no impact on the performance if the objective is to maximize the WSR or EE (defined by sum rate dividing the sum transmit power) subject to transmit power constraint. In such case, the major question is whether RS is helpful or not, and how much of the total (sum) information should be carried by the common message regardless of how the common message is split. However, if more user fairness is considered, i.e., when the objective is to maximize the minimum rate among users or/and subject to QoS rate constraints for each user, the choice of which users to split the messages at the transmitter will influence the final performance. The best method is to leave the possibility to split the messages of all users so as to provide rooms for allocating the rate of the common stream among users. Myth 7: RSMA has to sacrifice a higher receiver complexity in order to outperform (power-domain) NOMA Because RSMA does not enforce a given stream to be fully decoded or to be fully treated as noise, but rather split one or multiple messages such that a message is partially decoded by another users, RSMA can explore a wider space of communication schemes. This consequently leads to relatively simple schemes like 1-layer RS that relies on a single SIC to outperform multi-SIC NOMA schemes. It was for instance shown in [7] how the DoF, and therefore rate, of 1-layer RS can be significantly larger than that of complicated NOMA schemes. Also RSMA builds upon SDMA/MU-MIMO and the addition of SIC layers comes with a performance enhancement. This contrasts with NOMA where the performance (DoF and rate) can degrade as we increase the number of SIC, as a consequence of the restrictive design philosophy of NOMA [7]. The above discussion is further illustrated in Fig. 27. Myth 8: With more data streams to send from the transmitter, beamforming design and power control become very complicated in RSMA For any MA scheme, the larger the number of users and streams to serve, the higher the complexity for beamforming and power control. Hence this is not an issue specific to RSMA, but would hold for SDMA, NOMA, etc. Nevertheless, with RSMA, low complexity beamforming and power control can be perofrmed. For instance, in 1-layer RS, private streams could be precoded using zero-forcing beamforming (ZFBF) and have the power allocated uniformly (that would be much compliant with the way MU-MIMO is implemented in practice in 4G and 5G). The precoder of the common streams can be designed using low complexity techniques [24], [25], [58], [66]. What remains to be designed is the power allocated to the common stream that could depend on the network load, propagation conditions, and metric [24], [58], [66]. Myth 9: RSMA requires high SNR to achieve performance gain over NOMA or SDMA The gain of RSMA over SDMA and NOMA depends on many parameters, including network load, propagation conditions, objective function and QoS constraints, and CSIT quality. The more we favour fairness (e.g., MMF objective function, WSR, QoS constraints), the higher the gain of RSMA over a wide range of SNR. The lowest gain of RSMA would be experienced in massive MIMO regime with sumrate maximization and accurate CSIT. In such scenario, RSMA would boil down to SDMA or allocate a very small amount of power to the common streams only at high SNR. We here illustrate a MMF rate comparison result when the network is overloaded and the CSIT is imperfect, as per Fig. 33. The relative rate gain of 1-layer RS over MISO NOMA (G = 3) is 35% when SNR is 15 dB, and 21% when SNR is 10 dB, which is still non-negligible. Importantly, recall that this gain is achieved with a single SIC layer, while NOMA (G = 1) and NOMA (G = 3) require 5 and 1 SIC layers, respectively, and achieve much worse MMF rate. Myth 10: RSMA only works when instantaneous channel state information is available at the transmitter RSMA can operate on any form of CSIT, such as instantaneous CSIT with high or low accuracy and various CSI acquisition mechanisms [18], [68], [104], delayed CSIT [66], or statistical CSIT based on for instance second order statistics of the channel, e.g., spatial covariance matrix [58], [91], [92]. A detailed summary of the imperfect CSIT models that have been adopted in the existing works of RSMA can be found in [25]. Myth 11: The gain of RSMA in massive MIMO is marginal The gain of RSMA in massive MIMO with perfect CSIT is negligible or inexistent since the transmitter can form pencil beams that would provide high beamforming gain and simultaneously eliminate multi-user interference. Hence private streams are sufficient and RSMA boils down to conventional SDMA-based massive MIMO. As we depart from perfect CSIT and considers practical imperfection in CSIT acquisition, RSMA starts providing gains. In [66], RSMA was shown to provide significant gains in massive MIMO and to maintain multi-user connectivity in mobility conditions despite the delayed CSIT. In [58], [91], RSMA was applied to resolve the imperfect CSIT problem in FDD massive MIMO and was shown to provide performance gain over conventional massive MIMO based on SDMA processing. In [68], [104], RSMA was investigated in TDD massive MIMO and cell-free massive MIMO and was shown to be robust to pilot contamination and to provide significant gains whenever there is a likelihood of users being allocated the same uplink pilot sequence. This is motivated by mMTC where the probability of multiple users sharing the same pilot is very high. In [100], RSMA was used to enhance the robustness of massive MIMO in the presence of hardware impairments and in particular phase noise. RSMA was shown to outperform conventional massive MIMO in the presence of phase noise. In [119], RSMA was finally shown to be robust in multi-pair massive MIMO relay systems. VII. FREQUENTLY ASKED QUESTIONS We here classify frequently asked questions about RSMA in three different categories, namely principles and benefits, standardization and implementation, and applications and interplay with other technologies. A. Principles and Benefits of RSMA Question 1: What is the design principle of RSMA? In the downlink, the design principle of RSMA is much different to SDMA and NOMA. The NOMA design philosophy is based on having a stream to be fully decoded by another user. For instance, in the MISO case, NOMA forces one user to fully decode all streams in a group, i.e., its intended stream and the co-scheduled streams in the group. This leads to the strong constraint that the entire message of one of the users is mapped onto a common stream, e.g., W 2 mapped to s c decoded by both user-1 and user-2 in Table IV. This is radically different from SDMA design philosophy where messages are independently encoded into private streams and each receiver decodes its intended stream treating any residual multi-user interference as noise (even when the interference level is not weak enough to be treated as noise), as per Table IV. Interference is never decoded at the receivers due to the absence of common stream(s). In RSMA, a message of a given user is not forced to be treated as noise or be decoded by a user. Instead we have full flexibility on how to encode it and can evolve in the grey zone in-between. Hence, similarly to SDMA, in MISO downlink, K private streams are enabled, but in contrast to SDMA, each user can partially decode the message of another user thanks to the presence of the common streams. In contrast to NOMA, no user is forced to fully decode the stream(s) of a co-scheduled user since all private streams are encoded independently and each receiver decodes its intended private stream treating any residual interference from the other private streams as noise. This comes with huge benefits as RSMA builds nonorthogonal transmission strategies upon SDMA (and therefore MU-MIMO) so that the performance benefits of SDMA are guaranteed but extra performance is observed by the use of SIC receivers. Indeed, a performance gain over SDMA is expected from a more complex receiver architecture. To achieve this, one should enable the versatility at the transmitter to encode messages such that parts of them can be decoded by all users using SIC while the remaining parts are decoded by their intended receivers and treated as noise by non-intended receivers. This is uniquely achieved with RSMA by providing flexibility in the message-to-streams mapping, as demonstrated in Tables IV,VII, and IX. Indeed, instead of keeping a rigid mapping of a message into a predefined stream (as in SDMA and NOMA), RSMA allows each user to send part of a message in one or multiple common stream(s) and the rest in one of the K private streams. By adjusting the power levels of the common and private streams, one can adjust the amount of interference that occurs on the private stream, so that its level can be weak enough to be treated as noise. This enables RSMA to manage multi-user interference by partially decoding the interference and treating the remaining interference as noise. In the uplink, the design philosophy of RSMA and NOMA is similar at the receivers since both aim at using SIC to decode all incoming streams. However, for a fixed number of users, the number of streams to decode at the receivers is different because of the specific features in the encoding operation at the transmitters between RSMA and NOMA. NOMA philosophy is to map a message into a stream such that in a two-user system, NOMA decodes user-1 before or after user-2. On the other hand, RSMA splits a message (say of user-1) into sub-messages, encodes them into independent streams, and superposes them. This split enables the receiver to decode part of user-1's message before user-2 and the other part of user-1's message after user-2. Question 2: What are the major benefits of RSMA? Universal: RSMA is a more general multiple access framework that unifies and generalizes (and consequently outperforms) OMA, SDMA (and multi-antenna) NOMA. OMA, SDMA, and NOMA, (and other schemes as physical-layer multicasting) are all particular instances of RSMA [8], [24]. Flexible: RSMA is flexible to cope with all user deployments (with a diversity of channel directions, channel strengths), network loads (underloaded and overloaded regimes), and interference levels (weak, medium, strong). RSMA automatically reduces to other MA techniques according to the channel conditions, i.e., it reduces to SDMA when user channels are orthogonal in the underloaded MISO BC with perfect CSIT. When the channels are aligned with certain channel strength disparities, it reduces to power-domain NOMA. For other channel conditions, RSMA takes advantage of the common streams and more efficiently manages multiuser interference by partially decoding the interference and partially treating the remaining interference as noise [8], [24]. Robust: RSMA is robust to any CSIT inaccuracy [14], [18]. This is very relevant in modern downlink multi-antenna deployments. While OMA, NOMA, SDMA, all incur a DoF loss in the presence of imperfect CSIT, RSMA is DoF-optimal and therefore, less sensitive to CSIT inaccuracy. Spectrally efficient: The spectral efficiency of RSMA is always larger than or equal to that of existing MA techniques. Considering a downlink without QoS constraints, the rate region of RSMA comes much closer to the capacity region (achieved by DPC) than SDMA and NOMA when CSIT is perfect [8]. When CSIT becomes imperfect CSIT, linearly precoded RSMA is able to achieve a larger rate region than complex DPC (and SDMA and NOMA) [9]. As RSMA achieves the optimal DoF in both perfect and imperfect CSIT [19], it optimally exploits the spatial dimensions and the availability of CSIT. This contrasts with SDMA and NOMA that are suboptimal [7]. Considering an uplink, RSMA outperforms NOMA without time sharing [23], [69]. Energy efficient: Thanks to its flexibility and universality, the EE of RSMA is also larger than or equal to that of existing MA techniques (OMA, SDMA, NOMA) in a wide range of user deployments [63], [127], [128]. Enhancing QoS and fairness: RSMA exhibits an even larger performance gain over other MA techniques whenever each user is subject to a QoS rate constraint or whenever a higher weight is allocated to the user with a weaker channel condition [7], [8], [25]. Therefore, the ability of a wireless network architecture to partially decode interference and partially treat interference as noise leads to enhanced QoS and user fairness. Reducing complexity: RSMA has the double benefit of simultaneously boosting the performance and decreasing the complexity (at the transmitter and the receiver) compared with multi-antenna NOMA. Recall indeed that multi-antenna NOMA that requires user grouping, ordering, switching (between NOMA and SDMA) at the transmit scheduler and multiple layers of SIC at the receivers. On the other hand, 1-layer RS without any user ordering, grouping or dynamic switching at the transmit scheduler and with a single layer of SIC at each receiver is capable of achieving significant performance gain over NOMA [7]. In contrast to SDMA that requires user pairing to pair users with semi-orthogonal channels, RSMA is suited to all channel conditions and it does not require complex user scheduling and pairing [58]. Moreover, RSMA is capable of further reducing CSI feedback overhead in the presence of quantized feedback [47]. RSMA nevertheless incurs a higher receiver complexity than SDMA due to the use of SIC. Reducing latency and improving reliability: Reducing the transmit packet size is widely known as one major approach of achieving low-latency communications, also known as, short-packet transmission [210]. RSMA for short packet transmission has been investigated in [32], [34], [69] for both downlink and uplink, and RSMA has been shown to use shorter blocklength and therefore lower latency to achieve equal sum rate and MMF rate with SDMA and NOMA. It therefore has a great potential to enhance URLLC services in 6G [28]. Question 3: What does the common message contain? From Example 1, we note that the common message contains information bits from the original unicast (user-specific) messages of user-1 and user-2. In that example, both messages W 1 and W 2 are split to create the common message; hence the common message carries some bits from user-1 and from user-2. But we could have instances where only one of the users message is split, say W 1 only, and in such case, the common message carries some bits of user-1 only. The common message, whether it carries bits of one user or multiple users, is always decoded by multiple users. Question 4: Why do we combine the common parts of user messages into a single common message? Transmitter and receiver design is greatly simplified by combining the common parts of multiple users into a single common message, as shown with the two RSMA strategies of the 2-user downlink in Section III-A1. Moving to a K-user scenario, if each user's message is split into two parts without combining the messages, the sender would have to encode the 2K message and design 2K precoders. Each user needs more SIC layers and the decoding order needs to be optimized at the transmitter. In contrast, the 1-layer RS transmitter only encodes and precodes K + 1 streams, and only one layer of SIC is needed for each user, hence no worry about decoding order. This greatly reduces the complexity of 1-layer RS. Question 5: How does the optimized rate allocation for the common stream guide the practical message split at the transmitter and the modulation and coding scheme? The reader is invited to consult [25] for detailed examples of practical message splits and modulation and coding schemes. Question 6: In what scenarios can RSMA achieve explicit performance gain over SDMA and NOMA? Let us start with the downlink. From a DoF perspective, RSMA achieves the same or higher DoF than SDMA and than NOMA with a lower number of SIC layers [7]. This means that in the high SNR regime, RSMA will always outperform those two schemes and the gains would be larger as the objective functions accounts for fairness, the system gets more overloaded, or the quality of the CSIT degrades. A DoF gain also translates to a rate gain at finite SNR though the exact gain depends on the disparity of channel strengths among users and the angle between user channels. In the low to medium SNR regime, the gains of RSMA over SDMA and NOMA will depend on the user channel orthogonality and the disparity of channel strength, as shown for a 2-user scenario in Fig. 26. SDMA favors orthogonal channels, accurate CSIT and similar channel strengths among users. NOMA favours aligned channels in each group and a large disparity of channel strengths. As we depart from those extremes, or as the CSIT quality degrades, RSMA provides explicit gains over SDMA and NOMA even more when constraints on fairness, QoS, or minimum rate are imposed by the network [8], [24], [25]. RSMA was shown to reduce to SDMA in the presence of orthogonal channels (or close to orthogonal) and outperform SDMA otherwise. RSMA was shown to reduce to NOMA and achieve the same rate performance whenever: 1) the SNR is low, 2) the channels are closely aligned, 3) there is a sufficiently large disparity of channel gains, and 4) the CSIT is perfect. If all four conditions are met, NOMA, RSMA, and DPC schemes achieve very similar performance (if not the same performance). As we depart from those conditions, NOMA incurs a loss over RSMA. Same observations also hold in more general K-user settings [9]. When it comes to the role played by the channel gain disparity among users, it is important to note that with RSMA, the larger the disparity of channel gains, the larger the benefits of using RSMA schemes with multiple SIC. In other words, given a cell size, the disparity of channel strengths among users could be statistically obtained using the conventional path loss model, and the designer can then decide how many SIC would be worth given the complexity that can be afforded. Nevertheless, 1-layer RSMA already brings significant gain even for realistic channel strength disparities. For instance, in Fig. 27, with 10dB path loss difference and additional Rayleigh fading, 1-layer RS with a single layer of SIC outperforms NOMA with five SIC layers. Further gains could be obtained by using RSMA scheme with say two SIC, but this shows how powerful, RSMA is to efficiently make use of the SIC architecture, and therefore reduce receiver complexity [7]. In the uplink, NOMA is heavily dependent on time sharing to achieve good performance and attain capacity. The gain of RSMA over NOMA is explicit whenever we cannot afford time sharing among users. RSMA achieves the capacity without the use of time sharing, which finds applications in scenarios where communication overhead and stringent synchronization requirements due to the coordination of the transmissions of all users is not possible [23]. This occurs for instance in services requiring grant-free access, which allow collisions to reduce the access latency stemming from the channel grant procedure [25], [88], [192]. Other uplink scenarios where RSMA outperforms NOMA is in uplink with finite blocklength [69] or in network slicing [193]. The reader is also invited to consult the many applications and scenarios discussed in Section V where references have demonstrated that RSMA outperforms SDMA and NOMA. Question 7: What can we learn from information theory about RSMA? How can information theory guide the modern study of RSMA? As re-visited in Section II, the roots of RS can be dated back to [10], [11], where the coding scheme was introduced for the SISO IC in the weak interference regime. At that time, the MAC capacity region was characterized completely by separate random coding and SIC at the receiver. The capacity region of the degraded BC was also known to be achieved by SC and SIC. Initial results on the IC showed that capacity region can be achieved by the same techniques in the case of very strong [40] and strong [211] interference. The basic idea to develop a coding scheme which allows the receives to decode part of the interference, to bridge the two extreme cases treat interference as noise (TIN) and decode interference, led to the development of RS. A very important ingredient in the development of the best achievable rate region for the IC is time sharing [212]. Due to the remaining interference, the achievable rate region is non-convex. Only time sharing between different coding and decoding strategies including TIN, FDMA, and RS can guarantee the best achievable rate region. It must be stressed that there is a gap to the outer bound of the capacity of IC with weak interference. A characterization of the capacity region within a finite number of bits is derived in [30]. The achievable scheme in this work is based on a simplified RS approach. In the BC and IC, the RS approach was introduced in recent works in the context for multi-antenna settings (MISO and MIMO) with imperfect CSIT . Though the Gaussian MIMO BC with perfect CSIT is known and achieved by DPC [42], the capacity and capacity-achieving schemes of those channels with imperfect CSIT are still unknown, but RS is known to play a crucial role in achieving optimality in DoF [18]- [22], [46], generalized DoF [17], [213], [214], and constant gap [215], [216]. In the MAC, the RS approach was first introduced in [23]. The motivation behind this development was to apply singleuser coding without requiring synchronization among users. There the term RSMA was coined. Already in [23], the important cases with fading and interference were considered to bring the proposed coding and decoding scheme to practical applications. In multihop-multiflow communication, e.g., in a 2 × 2 × 2 setting comprised of 2 sources, 2 relays, and 2 destinations, the combination of RS with decode-and-forward and amplifyand-forward schemes plays a crucial role in achieving the fundamental limits [217]. RS is also a crucial ingredient in the theoretical and practical principles of the broadcast approach to communication over state-dependent channels and networks [218]. This is relevant in scenarios where the transmitters have access to only the probabilistic description of the time-varying states while remaining unaware of their instantaneous realizations [218]. Later the term multiple access was used also for scenarios in which several users share a link including BC. The term (power-domain) NOMA was introduced mainly for the downlink transmission corresponding to BC setup. It corresponds to SC SIC [219] and achieves the capacity region in degraded BC [220]. The term RSMA was also applied to the BC in [8] where it can provide significant achievable rate gains. Network information theory [12] provides the solid basis for system modeling, a taxonomy of known results with achievable encoding and decoding schemes with unique taxonomy, and a toolbox of methods and schemes for the modern study of multi-user communication systems. All currently standardized and developed transceiver schemes have their roots in the fundamental information theoretic results -even if the name and the terminology might have changed. RSMA is still very new in 3GPP and has not been discussed by standard bodies yet. The machinery required for RSMA is nevertheless already partially being studied, discussed and developed in 3GPP. Indeed, past 3GPP study/work items such as MU-MIMO, full-dimensional MIMO, coordinated multi-point (CoMP), multi-user superposition transmission (MUST), NOMA, network-assisted interference cancellation and suppression (NAICS), multicast and broadcast services, can be leveraged to design RSMA. However the key novelty of RSMA, namely relying on message split, has not been discussed in standardization bodies. Inversely, RSMA, once introduced in the standard, would address numerous issues and therefore boost the performance of all those work items. B. Standardization and Implementation of RSMA NOMA was heavily investigated in 5G but not so well received at the end. From a theoretical point of view, it is predictible that (power-domain) NOMA would not fly given its deficiencies and the lack of clear gains (and even loss) over MU-MIMO [7], [221]. In contrast, RSMA does not suffer from those issues as it really builds upon SDMA/MU-MIMO. Hence 6G could envision a single transmission mode based on RSMA as a replacement or an enhancement of the MU-MIMO transmission mode used in 4G and 5G, but also play numerous other roles in the entire air interface, such as enabling efficient simultaneous unicast-multicast-broadcast transmissions, with numerous new applications in automotive driving, VR, 360 video, metaverse, etc. Question 9: Does RSMA create more complications for implementation? The 1-layer RS strategy and its benefits in terms of implementation and complexity over SDMA and NOMA have been already discussed above. There are nevertheless other challenges to overcome to make RSMA practical. In most of the RSMA works, its transmitter-side design assumes Gaussian inputs, and it can be tricky to fit a Gaussian-optimized RSMAbased precoder into a real physical layer, where it often deals with the finite blocklength and finite constellations, or with pre-standardized MCS. Interestingly, recent efforts have been made to make the 1-layer RS strategy work with the state-of-the-art channel codes and modulation and significant throughput performance have been observed using realistic link-level simulations [33], [35], [74]- [77], though more work is needed in this area. Moving to other RSMA schemes, like the generalized RS strategy [8], can be complex to implement especially for a large number of users. Nevertheless, as a generalized framework of RSMA, it embraces SDMA, NOMA, physicallayer multicasting, OMA as special cases, and suggests a novel method to softly bridge existing MA techniques without using naive hard switching. Moreover, the generalized RS is applicable to the scenarios with relatively small K and it achieves non-negligible performance gain over existing MA techniques. Hence, the transmitter could schedule a small number of users in each resource block. Another use of the generalized RS is to act as a benchmark to demonstrate the performance of other low-complexity RS strategies such as 1-layer RS and 2-layer HRS. It enables to identify which common streams are effective or ineffective, and consequently trim the generalized RS into a low complexity RS scheme that would rely on a subset of the common streams. From our experience, in many applications, low-complexity RS strategies achieve performance reasonably close to that of generalized RS while their complexities are much lower than the generalized RS and NOMA strategies. This demonstrates that by departing from the extremes of treating interference as noise and fully decoding interference, one can find alternative MA strategies that are spectrally and energy efficient and simultaneously computationally efficient (relatively low complexity and small number of SIC layers). It helps us draw the conclusion that 1-layer RS is a good alternative to the generalized RS in many practical scenarios. Another benefit of generalized RS is the ability to come up with a set of schemes whose performance improve as the number of SIC layers increases. This is helpful to figure out the performance gap vs complexity tradeoff between lowcomplexity RS strategies and generalized RS and understand whether the addition of one or multiple common streams is worth the complexity increase. This contrasts with NOMA where a larger number of SIC layers can lead to a lower performance [7]. Therefore, the generalized RS is a significant strategy in the framework of RSMA. C. Applications and Interplay between RSMA and other Wireless Technologies Question 10: Can RSMA be integrated with other MA techniques such as OFDMA, SDMA, power-domain NOMA, and code-domain NOMA? RSMA can definitely be combined with OFDMA in the same way as it is done with SDMA/MU-MIMO in 4G and 5G, namely a group of users are paired and served using RSMA on a given resource block or subband. Though much remains to be investigated in OFDMA-RSMA design, some research on the user grouping and power allocation optimization has been initiated in [171], [172]. Interplay between SDMA or power-domain NOMA and RSMA would not bring benefits, since SDMA is always part of any RSMA scheme, namely when it comes to the transmission and reception of the private streams, and powerdomain NOMA is part of the generalized RS architecture of RSMA. On the other hand, there is no effort so far on the interplay between code-domain NOMA and RSMA and this is an area of interest to see whether we can further enhance RSMA performance by bringing the code-domain dimension. Codedomain NOMA (CD-NOMA) employs carefully designed interleavers and/or code sequences to multiplex users. The idea was inspired by the traditional CDMA [222] or interleaverdivision multiple access (IDMA) [223]. Some well-known examples of CD-NOMA include sparse code multiple access (SCMA) [224] and non-orthogonal coded access (NOCA) [225]. In SCMA, each user is assigned a sparse codeword according to its message. To exploit the sparsity introduced by the codeword, the receiver adopts message passing algorithm for multi-user detection. In contrast, NOCA assigns each user a dense spreading signature to fully utilize the available time-frequency resources. The receiver exploits the low cross-correlation properties of spreading sequences for interference mitigation/suppression. Nevertheless, it is possible to incorporate the design principle of CD-NOMA into RSMA. Notice that the split common and private messages in RSMA can be seen as virtual users. Each of the virtual user can be assigned a dedicated sequence for enabling code-domain multiplexing. Some interesting research directions on codedomain RSMA can include but not limited to: design of sparse/dense codewords, detection and decoding architectures, resource allocation for optimizing achievable rate and energy efficiency. It should also be noted that the RSMA framework can be expanded in the time or frequency domains to get a spacetime or space-frequency RSMA framework as in [22], [47], as discussed in Section III-A7. This is particularly relevant when the CSIT quality changes across users and time or frequency or when we deal with asymmetric downlink or multi-cell framework where the receivers have a different number of receive antennas. Question 11: Can RSMA be integrated with emerging waveform, e.g., orthogonal time frequency space (OTFS), orthogonal delay-Doppler division multiplexing (ODDM)? OTFS was recently proposed as a new two-dimensional modulation scheme [226] [227], which multiplexes information symbols in the delay-Doppler (DD) plane (or domain) rather than the time-frequency (TF) domain as for conventional multi-carrier modulation or OFDM schemes in the current 4G/5G cellular and WiFi networks. The DD domain symbol multiplexing enables a direct coupling of the transmitted symbols with the channel's delay-Doppler spread function, which has nice properties, such as quasi-static, compact, and sparse, to be exploited to achieve a low channel estimation overhead and full channel diversity with low complexity receivers [228] [229] [230] [231] [232]. OTFS has also stimulated additional research on delay-Doppler plane modulation with orthogonal pulses, such as orthogonal delay-Doppler division multiplexing (ODDM) [233] [234], which can achieve orthogonality with respect to the channels' delay and Doppler resolutions that are generally much smaller than the symbol duration and subcarrier spacing in conventional OFDM. ODDM, or general delay-Doppler plane multi-carrier modulation, is a promising waveform for future wireless systems, particularly on doubly-selective channels. This type of schemes not only provides robust performance in highmobility channels, they can also be a viable choice for the future ISAC, due to that their transmit/receive pulses and the corresponding ambiguity function satisfy the orthogonality property with respect to the delay-Doppler resolutions. Similar to OFDMA, ODDM itself can be employed as an orthogonal multiple access scheme. With this property, it is natural to have RSMA combined with OTFS/ODDM waveform in MISO or MIMO systems. The combination of RSMA and OTFS/ODDM can provide high system design flexibility in terms of its resource allocation and optimization, in the mean time having a potential to offering high spectral efficiency, signal localisation, integrated sensing and communication capabilities. Question 12: Can RSMA use discrete signaling without SIC? To analyze the performance of RSMA, such as achievable rate and energy efficiency, Gaussian signaling is often assumed. As we know, Gaussian signaling is the optimal signaling in many channels, e.g., point-to-point Gaussian channels, Gaussian multiple access channels, and Gaussian broadcast channels. Hence, assuming Gaussian signaling becomes natural in RSMA. Under this assumption, RSMA can handle interference with different strengths effectively. In practical communication systems, discrete signaling, i.e., coded modulation, is used. In addition, for some applications with stringent requirements on decoding complexity and latency, e.g., downlink URLLC services, TIN is more favorable than interference decoding. If Gaussian signaling is assumed, low-complexity TIN is only optimal when the interference is very weak, e.g., see Table III. Indeed, despite being the optimal input distribution for many channels, Gaussian signaling is also the worst noise. On the other hand, discrete signaling can behave differently from Gaussian signaling when being treated as noise. When the interference strength is not very weak, it is possible for discrete signaling with TIN to achieve a strictly larger achievable rate region than that for Gaussian signaling with TIN. Discrete signaling with TIN, i.e., without SIC, has been investigated in recent works, e.g., [235]-[238]. In [235], a lattice partition based NOMA scheme was proposed for the single antenna K-user Gaussian broadcast channel. It was rigorously proved that the scheme based on discrete signaling and TIN is capable of achieving the whole capacity region to within a constant gap independent of the number of users and channel parameters. The same results hold true for the aforementioned channel model with only statistical CSI available at the transmitter side [236]. By further exploiting the algebraic properties of lattices, the full diversity of the broadcast channel with block fading and close-to-perfect SIC error performance can be attained for each user with TIN decoding [237]. Finally, in [238], it was shown in the first time that the capacity region of the Gaussian (asymmetric) interference channel can be achieved to within a constant gap by discrete signaling and TIN. To unleash the full potential of RSMA in practical communication systems, it is important to exploit the properties of discrete inputs with low-complexity TIN in RSMA. In light of the above works [235]- [238], discrete signaling with TIN should also benefit RSMA to achieve a better tradeoff between performance and complexity. Question 13: How can machine/federated learning help RSMA? What is the role of machine/federated learning in RSMA design? Machine learning (ML) and federated learning (FL) can be used on multiple fronts. ML can be used at the receiver of RSMA. In [77], a model-based deep learning (MBDL) method was used to propose new and practical RSMA receiver designs exploiting the conventional SIC receiver and the robustness and model agnosticism of deep learning techniques. Thanks to its ability to generate on demand non-linear symbol detection boundaries in a pure data-driven manner, the MBDL receiver was shown to significantly outperform conventional SIC receiver with imperfect CSIR. ML can be used at the transmitter of RSMA to optimize the resource allocation, power allocation, task offloading (as in MEC), and beamforming design as an alternative to conventional convex optimization methods. In [239], the power allocation for each transmit stream was designed using a deep reinforcement learning algorithm. It was shown that RSMA achieves a significant performance gain over SDMA. Different from the centralized ML method, FL uploads trained model parameters rather than raw data [240]. However, when training involves wireless edge devices at the edge network, communications could become a significant problem. In conventional FL, TDMA is typically used for uplink transmission. However, the central cloud has to wait until it receives information from all the user. RSMA enables multiple users to upload information by sharing the uplink channel at the same time. Thus, by incorporating RSMA in the FL framework, it is expected that the aggregation latency can be reduced while maintaining model training quality. For example, multiple IoT devices in fog radio access networks can cooperate to perform a FL task by repeatedly uploading locally updated models to a cloud server. To overcome the performance limitations due to finite capacity front-haul links, a rate-splitting transmission scheme at IoT devices can be used [241]. With flexible hybrid edge and cloud decoding strategy achieved by RSMA, we can reduce the completion time of FL while maintaining a specific target global accuracy. Not only should the PHY layer benefit from the integration of RSMA and ML, but the focus should also be on cross-layer design such as aiding network orchestration, to truly assist wireless communication for intelligent 6G [26]. For instance, ML has been used for UAV deployment in combination with RSMA transmission [182]. RSMA was shown to require less power compared to other MA schemes. Question 14: How would RSMA work in Millimeterwave and Terahertz networks? Early research on millimeter-wave communication focused on the strong directionality benefits that could be attained using beamforming. The general idea was to communicate with each user using a narrow spatial beam. It was thought that the small beamwidths would in turn generate minimal interference between users, which would allow for extremely efficient SDMA designs. The results of millimeter-wave 5G deployments have begun to change this thinking. The millimeter-wave benefits to 5G NR has been relatively disappointing, with few devices being scheduled on the large, under-utilized frequency bands. It is thought that this problem is a result of the beam blockage issues, which appear to be a much bigger problem in a largescale deployment than was initially thought. To make millimeter-wave and higher frequencies practical for future broadband wireless access, it is likely that many of the tight spatial directivity design philosophies will need to change. Beamwidths may need to increase, and multi-beam communication may become a necessity. This will likely result in interference becoming the dominant limiting factor as it is in sub-6 GHz frequencies. Even in scenarios where blockage is unlikely, the everincreasing number of antennas will likely not lead to beamwidths predicted by theory. Ideally, the number of feedback bits should scale linearly with the number of users [95], [242]- [244]. From a standardization perspective, this is almost impossible to maintain in the long-term. This will result in a broadening of beams and a mismatch between the beamformers/precoders used and the CSI. This effect will cause millimeter-wave and terahertz systems to have interference challenges. There has already been some initial work on the application of RSMA at millimeter-wave frequency [91]. This work focused on the CSI feedback mismatch issue and showed a number of benefits that could simplify implementations in practice. Recently, RSMA has been shown to expand the coverage of terahertz systems [107]. VIII. CONCLUSIONS This paper has provided a tutorial on RSMA. To demonstrate how powerful RSMA is, the tutorial has departed from the oversimplisitc OMA vs NOMA discussion held in 5G and has re-centered the discussion and design of multiple access techniques around the key role of interference management. Building upon first principles, the tutorial has demonstrated how existing multiple access techniques are fundamentally limited by their inherent interference management strategy, namely orthogonalization in OMA, treat interference as noise in SDMA, and decode interference in NOMA. In contrast RSMA schemes build upon the RS principle which enables to partially decode interference and partially treat interference as noise. Consequently RSMA has been shown to provide unique benefits, including enhanced spectral, energy and computation efficiency; universality by unifying and generalizing OMA, SDMA, NOMA, physical-layer multicasting; flexibility by coping with any interference levels, network loads, services, traffic, user deployments; robustness to inaccurate channel state information (CSI) and resilience to mixed-critical quality of service; reliability under short channel codes and low latency. Future systems will see a growing demand for spectrum intensive applications and for integrated wireless systems. Moreover they will have to face growing concerns for congested and contested electromagnetic environments (in both civilian and defense networks). These will push network designers to adopt sophisticated interference management, multiuser, and multiple access techniques. Thanks to its deep root in information theory, numerous benefits and applications, and its superiority over previous generation multiple access techniques (OMA, SDMA, NOMA), RSMA will play a growing and underpinning role in next generation communication systems. It is hoped that the RSMA techniques presented in this article will help inspiring future research in this exciting new area and pave the way for designing and implementing RSMA in 6G and beyond. Fig. 1 . 1Two-user symmetric Gaussian interference channel. Fig. Fig. 2. RS for two-user SISO IC (HK scheme) [11]. Fig. 3 . 3Symmetric rate versus INR/SNR. INR=P |hc| 2 , SNR=P |h d | 2 =1000. Fig. 4 . 4The relationship between RS, treat interference as noise and decode interference strategies. Each set illustrates the optimization space of the corresponding interference management strategy. The optimization space of RS is larger such that other strategies are just subsets. Fig. 5 . 5Two-user MISO downlink (Gaussian MISO broadcast channel). Fig. 6 . 6Two-user downlink MISO RSMA without message combiner. Fig. 7 . 7Two-user downlink MISO RSMA with message combiner (1-layer RS). Fig. 8 . 8The relationship between OMA, NOMA, SDMA, physical-layer multicasting, and RSMA in the downlink two-user case. Each set illustrates the optimization space of the corresponding communication strategy. The optimization space of RSMA is larger such that SDMA, NOMA, and physicallayer multicasting are just subsets. Fig. 9 . 9A beam representation of RSMA and its sub-schemes SDMA, NOMA, OMA, and physical-layer multicasting. Fig. 10 . 10Cooperative downlink RSMA with user relaying[52]. s 0 sequentially using subband A observation by treating the private streams as noise. Secondly, after removing s 0 from subband B Fig. 11 . 11Downlink MIMO RSMA[35]. Fig. 12 . 12Two-user uplink SIMO RSMA. Fig. 13 . 13The relationship between OMA, NOMA, and RSMA in the uplink two-user case. Each set illustrates the optimization space of the corresponding communication strategy. The optimization space of RSMA is larger than OMA and NOMA. j ∈ {1, . . . , N } is split into two parts, W two N -dimensional vectors s 1,1 = [s Fig. 14 . 14Two-user uplink MIMO RSMA. T . The two stream vectors are linearly precoded and superposed to form the transmit signal x 1 . At user-2, the two messages in w 2 are directly encoded into two streams in s 2 = s ] k ] T for user-k. The common messages vectors of all users {w c,1 , . . . , w c,K } are combined into Q c (Q c ≤ min(M, N )) common messages w c ∈ C Qc×1 and encoded into one common stream vector s c = T , which is decoded by all users. The private message vector w p,k of user-k is independently encoded into the private stream vector s k = [s Fig. 15 . 15K-user downlink 1-layer MIMO RSMA. Fig. 16 . 164-user downlink MIMO HRS. Fig. 17 . 173-user downlink MIMO generalized RS. Fig. 18 . 18K-user downlink 1-layer MIMO DPCRS. Fig. 19 . 193-user downlink multi-layer MIMO DPCRS. Fig . 20 illustrates the proposed K-user uplink MIMO RSMA based on SC at the transmitters and SIC at the receiver.A Q k -dimensional message vector w k = [W (1) k , . . . , W (Q k ) k] T is transmitted from user-k to the receiver, where Q k ≤ min(M, N ). At user-k, each user message W Fig. 20 . 20K-user uplink MIMO RSMA. Fig. 21 . 21Multi-cell RSMA-enabled transmission[25]. Fig. 22 . 22Ergodic rate region comparison in MIMO BC with perfect CSIT, SNR = 20 dB, M = 4, K = 2, N = 2, and Qc = Q 1 = Q 2 = 2 [35].rics, namely, spectral efficiency, energy efficiency, generality, flexibility, robustness, reliability and latency.Unless stated otherwise, the elements of the channel matrix H k are generated as i.i.d. complex Gaussian random variables CN (0, σ 2 k ) and the noise variances are normalized. The imperfect CSIT model is H k = H k + H k , where both the channel estimate H k and the channel error H k have i.i.d. complex Gaussian random entries drawn from CN (0, σ 2 k − σ 2 e,k ) and CN (0, σ 2 e,k ), respectively. All results are averaged over 100 channel instances.1) Spectrally and Energy Efficient: Consider a downlink MU-MIMO with one transmitter equipped with M = 4 antennas and K = 2 users, each is equipped with N = 2 antennas. Each user requires Q 1 = Q 2 = 2 streams, which are split, combined, and encoded into a common stream vector of length Q c = 2 and two private stream vectors of length 2. Fig. 23 .[ 9 Fig. 24 . 23924Ergodic rate region comparison in MISO BC with imperfect CSIT, SNR = 20 dB, α = 0.6, M = 4, K = 2 Energy efficiency versus transmit power budget comparison in MISO BC with perfect CSIT. Fig. 26 . 26Operation regions for RS, SDMA, NOMA, OMA and Multicast with perfect CSIT[24]. Fig. 27 . 27Max-min rate vs. SNR comparison in MISO BC with imperfect CSIT, α = 0.5, K = 6, σ 2 k ∈ [0.1, 1] [7]. 3 Fig. 28 .Fig. 29 . 32829) M = 64, K = 8, Nr = 4, Qc = Qp = Throughput vs. user speed comparison in (massive) MIMO BC with outdated CSIT employing OFDM and a 3GPP channel model, and 10 ms CSI feedback delay[66]. MMF rate versus blocklength (solid lines) of downlink SDMA (red), NOMA (yellow), RSMA (blue) at SNR=20dB[34]. Dashed line is the upperbound achieved with infinite blocklength. Fig. 30 . 30A subset of promising scenarios and applications of RSMA. Fig. 31 . 31Relationship between existing MA strategies and the K-user RSMA framework. Fig. 32 . 32Sum-rate vs. SNR comparison in three-user SISO BC with perfect CSIT, K = 3, Fig. 33 . 33Max-min fairness rate vs. SNR comparison in MISO BC with imperfect CSIT, α = 0.5, M = 5, K = 6, σ 2 k = 1, ∀k ∈ K.[7]. Bruno Clerckx is a Professor, the Head of the Wireless Communications and Signal Processing Lab, and the Deputy Head of the Communications and Signal Processing Group, within the Electrical and Electronic Engineering Department, Imperial College London, London, U.K. He is also the Chief Technology Officer (CTO) of Silicon Austria Labs (SAL) where he is responsible for all research areas of Austria's top research center for electronic based systems. He received the MSc and Ph.D. degrees in Electrical Engineering from Université Catholique de Louvain, Belgium, and the Doctor of Science (DSc) degree from Imperial College London, UK. He has authored two books on "MIMO Wireless Communications" and "MIMO Wireless Networks", 250 peer-reviewed international research papers, and 150 standards contributions, and is the inventor of 80 issued or pending patents among which 15 have been adopted in the specifications of 4G standards and are used by billions of devices worldwide. His research spans the general area of wireless communications and signal processing for wireless networks. He served as an editor or guest editor for theIEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and the IEEE TRANSACTIONS ON SIGNAL PROCESSING, EURASIP Journal on Wireless Communications and Networking, IEEE ACCESS, the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, the IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, the PROCEEDINGS OF THE IEEE, and the IEEE Open Journal of the Communications Society. He was an Editor for the 3GPP LTE-Advanced Standard Technical Report on CoMP. He received the prestigious Blondel Medal 2021 from France for exceptional work contributing to the progress of Science and Electrical and Electronic Industries, the 2021 Adolphe Wetrems Prize in mathematical and physical sciences from Royal Academy of Belgium, multiple awards from Samsung, IEEE best student paper award, and the EURASIP (European Association for Signal Processing) best paper award 2022. He is a Fellow of the IEEE and the IET, and an IEEE Communications Society Distinguished Lecturer 2021-2023. Yijie Mao is an Assistant Professor at the School of Information Science and Technology, Shang-haiTech University (Shanghai, China). She received the B.Eng. degree from the Beijing University of Posts and Telecommunications (Beijing, China), the B.Eng. degree (Hons.) from the Queen Mary University of London (London, United Kingdom) in 2014, and the Ph.D. degree from the Electrical and Electronic Engineering Department, the University of Hong Kong (Hong Kong, China) in 2018. She was a Postdoctoral Research Fellow at the University of Hong Kong from 2018 to 2019 and a postdoctoral research associate with the Communications and Signal Processing Group, Department of the Electrical and Electronic Engineering at the Imperial College London (London, United Kingdom) from 2019 to 2021. Her research interests include the design of future wireless communications and artificial intelligence-empowered wireless networks. Dr. Mao receives the Best Paper Award of EURASIP Journal on Wireless Communications and Networking 2022 and the Exemplary Reviewer for IEEE Transactions on Communications 2021. She is currently serving as a guest editor for special issues of IEEE Journal on Selected Areas in Communications and IEEE Open Journal of the Communications Society. She has been a vice-chair of IEEE ComSoc WTC SIG on rate-splitting multiple access (RSMA) and a workshop co-chair for 2020-2022 IEEE ICC, 2021-2022 IEEE WCNC, 2020-2022 IEEE PIMRC, and 2022 IEEE SECON, and she has been a Technical Program Committee member of many symposia on wireless communication for several leading international IEEE conferences. Eduard A. Jorswieck is managing director of the Institute of Communications Technology and the head of the Chair for Information Theory and Communications Systems and Full Professor at Technische Universität Braunschweig, Brunswick, Germany. From 2008 until 2019, he was the head of the Chair of Communications Theory and Full Professor at TU Dresden, Germany. Eduard's main research interests are in the broad area of communications. He has co-authored some 160 journal papers, 15 book chapters, 4 monographs, and more than 300 conference papers. Since 2017, he serves as Editor-in-Chief of the Springer EURASIP Journal on Wireless Communications and Networking. He currently serves as editor for IEEE Transactions on Communications. He has served on the editorial boards for IEEE Transactions on Signal Processing, IEEE Transactions on Wireless Communications, IEEE Signal Processing Letters, and IEEE Transactions on Information Forensics and Security. In 2006, he received the IEEE Signal Processing Society Best Paper Award. Jinhong Yuan (M'02-SM'11-F'16) is a Professor and Head of Telecommunication Group with the School of Electrical Engineering and Telecommunications, The university of New South Wales, Sydney, Australia. He has published two books, five book chapters, over 300 papers in telecommunications journals and conference proceedings, and 50 industrial reports. He is a co-inventor of one patent on MIMO systems and four patents on low-densityparity-check codes. He has co-authored four Best Paper Awards and one Best Poster Award, including the Best Paper Award from the IEEE International Conference on Communications, Kansas City, USA, in 2018, the Best Paper Award from IEEE Wireless Communications and Networking Conference, Cancun, Mexico, in 2011, and the Best Paper Award from the IEEE International Symposium on Wireless Communications Systems, Trondheim, Norway, in 2007. He is an IEEE Fellow and currently serving as an Associate Editor for the IEEE Transactions on Wireless Communications and IEEE Transactions on Communications. He served as the IEEE NSW Chapter Chair of Joint Communications/Signal Processions/Ocean Engineering Chapter during 2011-2014 and served as an Associate Editor for the IEEE Transactions on Communications during 2012-2017. His current research interests include error control coding and information theory, communication theory, and wireless communications. are in the design and analysis of broadband wireless communication systems, beyond 5G wireless systems, multiple-input multiple-output (MIMO) communications, millimeter wave wireless, software defined radios and wireless networks, coding theory, and MIMO array processing. He holds 32 issued US patents. He served as a Senior Editor for IEEE Signal Processing Magazine, Editor for the IEEE Transactions on Communications, Associate Editor for the IEEE Transactions on Signal Processing, and Guest Editor for special issues of the IEEE Journal on Selected Areas in Communications and the EURASIP Journal on Wireless Communications and Networking. Since 2022, he is a Fellow of the American Association for the Advancement of Science (AAAS). Along with his co-authors, he won best paper awards from the IEEE Communications Society (2016 Stephen O. Rice Prize and 2020 Fred Ellersick Prize), the IEEE Signal Processing Society (2015 IEEE Signal Processing Society Best Paper Award), and the IEEE Vehicular Technology Society (2010 Jack Neubauer Memorial Award). Elza Erkip is an Institute Professor in the Electrical and Computer Engineering Department at New York University Tandon School of Engineering. She received the B.S. degree in Electrical and Electronics Engineering from Middle East Technical University, Ankara, Turkey, and the M.S. and Ph.D. degrees in Electrical Engineering from Stanford University, Stanford, CA, USA. Her research interests are in information theory, communication theory, and wireless communications. Dr. Erkip is a member of the Science Academy of Turkey and is a Fellow of the IEEE. She received the NSF CAREER award in 2001, the IEEE Communications Society WICE Outstanding Achievement Award in 2016, the IEEE Communications Society Communication Theory Technical Committee (CTTC) Technical Achievement Award in 2018, and the IEEE Communications Society Edwin Howard Armstrong Achievement Award in 2021. She was the Padovani Lecturer of the IEEE Information Theory Society in 2022. Her paper awards include the IEEE Communications Society Stephen O. Rice Paper Prize in 2004, the IEEE Communications Society Award for Advances in Communication in 2013 and the IEEE Communications Society Best Tutorial Paper Award in 2019. She was a member of the Board of Governors of the IEEE Information Theory Society 2012-2020, where she was the President in 2018. She was a Distinguished Lecturer of the IEEE Information Theory Society from 2013 to 2014. Dusit Niyato is the President's Chair Professor in the School of Computer Science and Engineering, at Nanyang Technological University, Singapore. He received B.Eng. from King Mongkuts Institute of Technology Ladkrabang (KMITL), Thailand in 1999 and Ph.D. in Electrical and Computer Engineering from the University of Manitoba, Canada in 2008.His research interests are in the areas of sustainability, edge intelligence, decentralized machine learning, and incentive mechanism design. TABLE I LIST IOF ABBREVIATIONS.BC Broadcast Channel MU-LP Multi-User Linear Precoding CDMA Code Division Multiple Access MU-MIMO Multi-User Multiple-Input Multiple-Output CoMP Coordinated Multi-Point NOMA Non-Orthogonal Multiple Access CSI Channel State Information NOUM Non-Orthogonal Unicast and Multicast CSIT/R Channel State Information at the Transmitter/Receiver OFDMA Orthogonal Frequency Division Multiple Access C-RAN Cloud-Radio Access Networks OMA Orthogonal Multiple Access DoF Degree-of-Freedom QoS Quality of Service DPC Dirty Paper Coding RF Radio Frequency DPCRS Dirty Paper Coded Rate-Splitting RIS Reconfigurable Intelligent Surfaces D2D Device-to-Device RS Rate-Splitting EE Energy Efficiency RSMA Rate-Splitting Multiple Access (F)eMBB (further-)enhanced Mobile Broadband Service SC Superposition Coding ER Ergodic Rate SDMA Space Division Multiple Access ESR Ergodic Sum Rate SE Spectral Efficiency FDD Frequency Division Duplex SIC Successive Interference Cancellation FDMA Frequency Division Multiple Access SISO Single-Input Single-Output F-RAN Fog-Radio Access Networks SNR Signal-to-Noise Ratio HK Han and Kobayashi SWIPT Simultaneous Wireless Information and Power Transfer IC Interference Channel SIMO Single-Input Multiple-Output IRS Intelligent Reconfigurable Surface TDD Time Division Duplex LLS Link-Level Simulation TDMA Time-Division Multiple Access MA Multiple Access THz TeraHertz MAC Multiple Access Channel UAV Unmanned Aerial Vehicles MIMO Multiple-Input Multiple-Output (e)URLLC (extremely) Ultra-Reliable Low-Latency Communication MISO Multiple-Input Single-Output VLC Visible Light Communication MMF Max-Min Fairness V2X Vehicle-to-Everything (u)mMTC (ultra) massive Machine-Type Communication WSR Weighted Sum Rate mmWave millimeter-Wave ZFBF Zero-Forcing Beamforming We discuss numerous myths/misunderstandings and frequently asked questions about RSMA that we have personally witnessed. Myths and questions cover a wide range of topics, ranging from RSMA relationship with NOMA, gain over MU-MIMO and massive MIMO, role and content of common streams, standardization status of RSMA, implementation complexity of RSMA, interplay with other techniques, role of machine learning in RSMA, etc.are shown particular instances of MIMO RSMA. • We demonstrate that, in contrast to previous generation MA schemes (as OMA, SDMA, NOMA), RSMA is spectrally and energy efficient, universal, flexible, robust and resilient, reliable and low latency. Those unique features of RSMA are discussed in over 40 applications of RSMA in different systems, scenarios, and services. We briefly describe how RSMA is beneficial for each of those applications to demonstrate how powerful RSMA is for next generation networks. • TABLE II MESSAGES II-TO-STREAMS MAPPING IN THE TWO-USER IC.s k s c,k Very weak W 1 - Weak W p,k W c,k Strong/Very strong - W 2 decoded by its intended Rx-k and decoded by treated as noise by Rx-j both Rx 10 -3 10 -2 10 -1 10 0 10 1 INR/SNR 0 1 2 3 4 5 6 7 8 9 Symmetric rate R sym [bits/s/Hz] RS Orthogonalization Treat interference as noise Decode interference very weak weak strong TABLE III COMPARISON IIIOF PREFERABLE INTERFERENCE LEVELS OF DIFFERENT INTERFERENCE MANAGEMENT STRATEGIES.Interference levels Very weak Weak Strong Very strong Orthogonalize × × × × Treat interf. as noise √ × × × Decode interference × × √ √ Rate-Splitting √ √ √ √ Notations: √ : Suited. ×: Not well suited. TABLE IV MESSAGES IV-TO-STREAMS MAPPING IN THE TWO-USER MISO BC[24]. TABLE VI COMPARISON VIOF DIFFERENT MA SCHEMES IN TERMS OF TRANSMIT-SIDE VS RECEIVE-SIDE INTERFERENCE CANCELLATION.Interf. cancel. transmit-side receive-side both sides SDMA/DPC √ × × NOMA × √ × RSMA × × √ Notations: √ : Relevant. ×: Not relevant. TABLE VII MESSAGES VII-TO-STREAMS MAPPING IN THE TWO-USER COOPERATIVE MISO DOWNLINK TABLE VIII TWO VIIIDIFFERENT CSIT PATTERNS.User-1 User-2 subband A √ √ subband B × × Non-alternating CSIT User-1 User-2 subband A × √ subband B √ × Alternating CSIT Notations: √ : Good CSIT quality. ×: Poor CSIT quality. TABLE IX IXMESSAGES-TO-STREAMS MAPPING IN THE TWO-USER MIMO BC. TABLE X XMESSAGES-TO-STREAMS MAPPING IN THE TWO-USER MULTIPLE ACCESS CHANNEL. TABLE XI MESSAGES XI-TO-STREAMS MAPPING IN THE TWO-USER MIMO MULTIPLE ACCESS CHANNEL. -24 show the SE and EE gains of RSMA in the0.5 1 1.5 2 Transmit power budget (dBm) 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 Weighted Sum Rate (Mbits/s) RSMA NOMA FDMA TDMA Fig. 25. Sum rate versus the transmit power budget of each user in SISO MAC with perfect CSIT [65]. , a specific channel realization h 1 = 1 √ 2 [1, 1] H and h 2 = γ √ 2 [1, e jθ ] H is considered. The x-axis is ρ = 1 −|h H 1 h2| 2 Question 8: What is the status of RSMA standardization? Why would RSMA succeed in 6G when NOMA was not well received in 5G? Note that if the two-user IC is used to represent a two-cell network, the inter-cell interference level is unlikely to be in the strong regime since a user is associated with its closest base station. We assume a sum power constraint but a per-antenna power constraint could also be considered.9 Only the receiver of user-1 is detailed inFig. 6. The receiver of user-2 follows the same principle as that of user-1 where two SIC layers are needed to decode and cancel the common streams before retrieving user-2's private message. 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Kim, "Optimal power allocation for rate splitting communications with deep reinforcement learning," IEEE Wireless Commun. Lett., vol. 10, no. 12, pp. 2820- 2823, 2021. Federated learning: Challenges, methods, and future directions. T Li, A K Sahu, A Talwalkar, V Smith, IEEE Trans. Signal Process. 373T. Li, A. K. Sahu, A. Talwalkar, and V. Smith, "Federated learning: Challenges, methods, and future directions," IEEE Trans. Signal Pro- cess., vol. 37, no. 3, pp. 50-60, Mar. 2020. Completion time minimization of fog-RANassisted federated learning with rate-splitting transmission. S.-H Park, H Lee, IEEE Trans. Veh. Technol. to appearS.-H. Park and H. Lee, "Completion time minimization of fog-RAN- assisted federated learning with rate-splitting transmission," IEEE Trans. Veh. Technol., to appear, 2022. On the performance of random vector quantization limited feedback beamforming in a MISO system. C K Au-Yeung, D J Love, IEEE Trans. Wireless Commun. 62C. K. Au-Yeung and D. J. Love, "On the performance of random vector quantization limited feedback beamforming in a MISO system," IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 458-462, Feb. 2007. Multiple antenna broadcast channels with shape feedback and limited feedback. P Ding, D J Love, M D Zoltowski, IEEE Trans. Signal Process. 557P. Ding, D. J. Love, and M. D. Zoltowski, "Multiple antenna broadcast channels with shape feedback and limited feedback," IEEE Trans. Signal Process., vol. 55, no. 7, pp. 3417-3428, July 2007. Trellis coded line packing: Large dimensional beamforming vector quantization and feedback transmission. C K Au Yeung, D J Love, S Sanayei, IEEE Trans. Wireless Commun. 106C. K. Au Yeung, D. J. Love, and S. Sanayei, "Trellis coded line packing: Large dimensional beamforming vector quantization and feedback transmission," IEEE Trans. Wireless Commun., vol. 10, no. 6, pp. 1844-1853, June 2011.

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{'abstract': 'Rate-Splitting Multiple Access (RSMA) has emerged as a powerful multiple access, interference management, and multi-user strategy for next generation communication systems. In this tutorial, we depart from the orthogonal multiple access (OMA) versus non-orthogonal multiple access (NOMA) discussion held in 5G, and the conventional multi-user linear precoding approach used in space-division multiple access (SDMA), multiuser and massive MIMO in 4G and 5G, and show how multi-user communications and multiple access design for 6G and beyond should be intimately related to the fundamental problem of interference management. We start from foundational principles of interference management and rate-splitting, and progressively delineate RSMA frameworks for downlink, uplink, and multicell networks. We show that, in contrast to past generations of multiple access techniques (OMA, NOMA, SDMA), RSMA offers numerous benefits: 1) enhanced spectral, energy and computation efficiency; 2) universality by unifying and generalizing OMA, SDMA, NOMA, physical-layer multicasting, multi-user MIMO under a single framework that holds for any number of antennas at each node (SISO, SIMO, MISO, and MIMO settings); 3) flexibility by coping with any interference levels (from very weak to very strong), network loads (underloaded, overloaded), services (unicast, multicast), traffic, user deployments (channel directions and strengths); 4) robustness to inaccurate channel state information (CSI) and resilience to mixed-critical quality of service; 5) reliability under short channel codes and low latency. We then discuss how those benefits translate into numerous opportunities for RSMA in over forty different applications and scenarios of 6G, e.g., multi-user MIMO with statistical/quantized CSI, FDD/TDD/cell-free massive MIMO, millimeter wave and terahertz, cooperative relaying, physical layer security, reconfigurable intelligent surfaces, cloud-radio access network, internetof-things, massive access, joint communication and jamming, non-orthogonal unicast and multicast, multigroup multicast, satellite, space-air-ground integrated networks, unmanned aerial vehicles, integrated sensing and communications, grant-free access, network slicing, cognitive radio, optical/visible light communications, mobile edge computing, machine/federated learning, etc. We finally address common myths and answer frequently asked questions, opening the discussions to interesting future research avenues. Supported by the numerous benefits and applications, the tutorial concludes on the underpinning role played by RSMA in next generation networks, which should inspire future research, development, and standardization of RSMA-aided communication for 6G.2 The Degrees-of-Freedom (DoF), or multiplexing gain, is a first-order approximation of the rate at high signal-to-noise ratio (SNR). It can be viewed as the pre-log factor of the rate at high SNR and be interpreted as the number or fraction of interference-free stream(s) that can be simultaneously communicated to a user (or multiple users). The DoF achieved depends on the communication strategy used. The larger the DoF, the faster the rate increases with the SNR. Hence, ideally a communication strategy should achieve the highest DoF possible. Readers are referred to[7]for more details on various definitions used to assess the DoF performance of multiple access schemes.3 Perfect CSIT is obtained by setting the channel estimation error in the imperfect CSIT model to zero.', 'arxivid': '2209.00491', 'author': ['Fellow, IEEEBruno Clerckx [email protected] ', 'Member, IEEEYijie Mao ', 'Fellow, IEEEEduard A Jorswieck ', 'Fellow, IEEEJinhong Yuan [email protected]. ', 'Fellow, IEEEDavid J Love ', 'Fellow, IEEEElza Erkip ', 'Fellow, IEEEDusit Niyato [email protected] ', '\nDepartment of Electrical and Electronic Engineering\nSchool of Information Science and Tech-nology\nand with Silicon Austria Labs (SAL)\nImperial College London\nSW7 2AZ, A-8010London, GrazUK, Austria\n', '\nSchool of Electrical Engineering and Telecommunica-tions\nShanghaiTech University\n201210ShanghaiChina\n', '\nSchool of Computer Engineering\nUniversity of New South Wales\n2052SydneyNSWAustralia\n', '\nNanyang Techno-logical University\nSingapore\n'], 'authoraffiliation': ['Department of Electrical and Electronic Engineering\nSchool of Information Science and Tech-nology\nand with Silicon Austria Labs (SAL)\nImperial College London\nSW7 2AZ, A-8010London, GrazUK, Austria', 'School of Electrical Engineering and Telecommunica-tions\nShanghaiTech University\n201210ShanghaiChina', 'School of Computer Engineering\nUniversity of New South Wales\n2052SydneyNSWAustralia', 'Nanyang Techno-logical University\nSingapore'], 'corpusid': 251979414, 'doi': '10.1109/jsac.2023.3242718', 'github_urls': [], 'n_tokens_mistral': 87567, 'n_tokens_neox': 76597, 'n_words': 47266, 'pdfsha': 'e4538f24922377ae6ec99663e182d6d76f3ca8fb', 'pdfurls': ['https://export.arxiv.org/pdf/2209.00491v2.pdf'], 'title': ['A Primer on Rate-Splitting Multiple Access: Tutorial, Myths, and Frequently Asked Questions', 'A Primer on Rate-Splitting Multiple Access: Tutorial, Myths, and Frequently Asked Questions'], 'venue': []}

arxiv

Strategic Behavior is Bliss: Iterative Voting Improves Social Welfare Joshua Kavner [email protected] Department of Computer Science Department of Computer Science Rensselaer Polytechnic Institute Troy Rensselaer Polytechnic Institute Troy 12180, 12180NY, NY Lirong Xia [email protected] Department of Computer Science Department of Computer Science Rensselaer Polytechnic Institute Troy Rensselaer Polytechnic Institute Troy 12180, 12180NY, NY Strategic Behavior is Bliss: Iterative Voting Improves Social Welfare Recent work in iterative voting has defined the additive dynamic price of anarchy (ADPoA) as the difference in social welfare between the truthful and worst-case equilibrium profiles resulting from repeated strategic manipulations. While iterative plurality has been shown to only return alternatives with at most one less initial votes than the truthful winner, it is less understood how agents' welfare changes in equilibrium. To this end, we differentiate agents' utility from their manipulation mechanism and determine iterative plurality's ADPoA in the worst-and average-cases. We first prove that the worst-case ADPoA is linear in the number of agents. To overcome this negative result, we study the average-case ADPoA and prove that equilibrium winners have a constant order welfare advantage over the truthful winner in expectation. Our positive results illustrate the prospect for social welfare to increase due to strategic manipulation. Introduction Voting is one of the most popular methods for a group of agents to make a collective decision based on their preferences. Whether a decision is for a high-stakes presidential election or a routine luncheon, agents submit their preferences and a voting rule is applied to select a winning alternative. One critical flaw of voting is its susceptibility to strategic manipulations. That is, agents may have an incentive to misreport their preferences (i.e. votes) to obtain a more favorable outcome. Unfortunately, manipulation is inevitable under any non-dictatorial single-round voting systems when there are three or more alternatives, as recognized by the celebrated Gibbard-Satterthwaite theorem [Gibbard, 1973, Satterthwaite, 1975. Consequently, decades of research sought to deter manipulation, especially by high computational barriers [Bartholdi et al., 1989, Faliszewski and Procaccia, 2010; see [Conitzer and Walsh, 2016] for a recent survey of the field. While there is a large body of literature on manipulation of single-round voting systems, sequential and iterative voting procedures are less understood. Indeed, these procedures occur in a variety of applications, such as Doodle or presidential election polls, where people finalize their votes after previewing others' responses [Meir et al., 2010, Desmedt and Elkind, 2010, Xia and Conitzer, 2010, Reijngoud and Endriss, 2012, Zou et al., 2015. Our key question is: What is the effect of strategic behavior in sequential and iterative voting? A series of work initiated by Meir et al. [2010] characterizes the dynamics and equilibria of iterative voting, where agents sequentially and myopically improve their reported preferences based on other agents' reports [Reyhani and Wilson, 2012, Lev and Rosenschein, 2012, Brânzei et al., 2013, Grandi et al., 2013, Obraztsova et al., 2013, Meir et al., 2014 35th Conference on Neural Information Processing Systems (NeurIPS 2021). 2015, Endriss et al., 2016, Meir, 2016, Tsang and Larson, 2016, Koolyk et al., 2017. While the convergence of iterative voting has been investigated for many commonly studied voting rules, the effect of strategic behavior, in terms of aggregate social welfare, remains largely unclear. A notable exception is Brânzei et al. [2013]'s work that introduced and characterized the additive dynamic price of anarchy (ADPoA) of iterative voting with respect to the plurality, veto, and Borda social choice functions. The (additive) DPoA measures the social welfare (difference) ratio between the truthful winner and an iterative policy's equilibrium winners when an adversary minimizes aggregate social welfare by controlling both the order in which agents make their strategic manipulations and agents' truthful preferences altogether. In particular, Brânzei et al. [2013] proved that under iterative plurality, the number of agents whose top preference is an equilibrium winner is at most one less than that of the truthful plurality winner. Therefore, strategic behavior does not have a significant negative impact on the social welfare measured by the sum plurality score of the winner. Nevertheless, it is unclear whether this observation holds for other notions of social welfare. Our Contributions We address the key question discussed above in the iterative voting framework, first proposed by Meir et al. [2010], by characterizing Brânzei et al. [2013]'s ADPoA under plurality dynamics and rank-based utility functions that differ from the iteration method. Given m ≥ 3 alternatives, a ranked-based utility function is characterized by a utility vector u such that each agent receives u i utility if their i-th ranked alternative wins, although this alternative may differ for each agent. We study iterative plurality due to its simplicity and popularity in practice. Moreover, our results absolve the need for the mechanism's center to know u exactly, thus conserving agents' privacy. Still, we assume this is constant for all agents. Our first main result (Theorem 1) states that, unfortunately, for any fixed m ≥ 3 and utility vector u, the ADPoA is Θ(n) for n agents. Therefore, the positive result achieved by Brânzei et al. [2013] is not upheld if u differs from plurality utility under the iterative plurality mechanism. To overcome this negative worst-case result, we introduce the notion of expected additive dynamic price of anarchy (EADPoA), which presumes agents' truthful preferences to be generated from a probability distribution. Our second main result (Theorem 2) is positive and surprises us: for any fixed m ≥ 3 and utility vector u, the EADPoA is −Ω(1) when agents' preferences are i.i.d. uniformly at random, known as Impartial Culture (IC) in social choice. In particular, our result suggests that strategic behavior is bliss because iterative voting helps agents choose an alternative with higher expected social welfare, regardless of the order of agents' strategic manipulations. Techniques. We compute the EADPoA by partitioning the (randomly generated) profiles according to their potential winners -the alternatives that can be made to win by incrementing their plurality scores by at most one. Conditioned on profiles with two potential winners, we show that iterative plurality returns the alternative that beats the other in a head-to-head competition (Lemma 1). This type of "self selection" improves the expected social welfare over truthful plurality winner by Ω(1) (Lemma 2). When there are three or more potential winners, we further show that the expected welfare loss is o(1) (Lemmas 3 and 4). Since the likelihood of k-way ties is exponentially small (in Xia, 2021]), the overall social welfare is improved in expectation. We provide an experimental justification of our second main result in Appendix B. fact, Θ n − k−1 2 [ Related Work and Discussions Sequential and iterative voting. Since iterative voting's inception in 2010, many researchers have studied its convergence properties under differing assumptions and iteration protocols. Meir et al. [2010] first established the convergence of iterative plurality with deterministic tie-breaking from any initial preference profile or with randomized tie-breaking from the truthful profile. However, this result appears quite sensitive to its assumptions, since the authors found counter-examples when allowing agents to manipulate simultaneously, using better-instead of best-replies, or weighing agents' votes unequally. Lev and Rosenschein [2012] and Reyhani and Wilson [2012] independently showed that no other scoring rule besides veto necessarily converges, while Koolyk et al. [2017] demonstrated the same for common non-scoring rules, such as Maximin, Copeland, and Bucklin. Similarly, Obraztsova et al. [2013 and each analyze the conditions for Nash equilibrium for iterative voting rules and their truth-biased or lazy voting counterparts . conclude that determining whether a given profile is a reachable Nash equilibrium is NP-complete. Most iterative voting rules require agents to have full information about each others' votes in order to compute their best-responses. To relax this strong assumption, Reijngoud and Endriss [2012] and Endriss et al. [2016], inspired by Chopra et al. [2004], introduce communication graphs and poll information functions that restricts the amount information each agent receives and better respects voter privacy. The researchers subsequently provide susceptibility, immunity, and convergence results according to different voting rules. Tsang and Larson [2016] use these concepts to simulate iterative plurality on a social network and allow agents to infer their best-responses based on their neighbors' reports. The authors demonstrate how correlating agents' preferences affects the PoA and DPoA of strategic outcomes. Sequential but non-iterative voting games have also been investigated in the literature. Desmedt and Elkind [2010] characterized the subgame perfect Nash equilibrium of a voting game where agents vote sequentially and are allowed to abstain from voting. Xia and Conitzer [2010] characterized the subgame perfect Nash equilibrium of a similar voting game where agents are not allowed to abstain, and proved that the equilibrium winner is highly unfavorable in the worst case, which can be viewed as an ordinal PoA. Our paper focuses on iterative voting setting proposed by Meir et al. [2010] and therefore differs from these works. Best-response mechanisms. A separate line of research from iterative voting has studied the convergence and acyclicity properties of best-response mechanisms. Monderer and Shapley [1996] first introduced the finite improvement property applied to games with agents that sequentially change their actions. Apt and Simon [2015] and Fabrikant et al. [2010] subsequently characterized betterresponse dynamics in weakly acyclic games, which encapsulate potential and dominance-solvable games, and demonstrate bounds on finding their Nash equilibrium. The relationship between iterative voting and best-response mechanisms was explored by Meir et al. [2014] and Meir [2016], who fully characterized the acyclicity and local dominance properties of iterative voting rules. Implications of our main results. Our results provide completeness and explanatory power to the empirical studies of Grandi et al. [2013] and Tsang and Larson [2016]. The former work shows an increase in additive social welfare using the Borda welfare vector due to plurality dynamics when agents have restricted manipulations and independent or correlated preferences [Berg, 1985]. The latter work shows a similar gain when agents have single-peaked preferences, are embedded on a social network, and make their manipulations based on estimates of their neighbors' reports. Put together, iterative voting provides a social welfare benefit that serves as an additional defense of strategic manipulation to those presented by Dowding and Hees [2008]. We believe our results will benefit a further study of non-strategyproof mechanisms in other social choice domains, such multi-issue voting [Bowman et al., 2014, Grandi et al., 2020. Preliminaries Basic setting. Let A = [m] {1, . . . , m} denote the set of m ≥ 3 alternatives and n ∈ N denote the number of agents. We denote by L(A) the set of all strict linear orders over A and use R j ∈ L(A), j ≤ n to represent agents' preference rankings. Preferences are aggregated into profiles P = (R 1 , . . . , R n ), and we use top(R j ) ∈ A to denote agent j's top preferred alternative. For any pair of alternatives a, b ∈ A, we use P [a b] to denote the number of agents that prefer a to b in P . Integer positional scoring rules. An (integer) positional scoring rule r s is characterized by an integer scoring vector s = (s 1 , . . . , s m ) ∈ Z m ≥0 with s 1 ≥ s 2 ≥ · · · ≥ s m ≥ 0 and s 1 > s m . For example, plurality uses the vector s plu = (1, 0, . . . , 0), veto uses (1, . . . , 1, 0), and Borda uses (m − 1, m − 2, . . . , 0). In this work we focus on the plurality rule r plu = r s plu and define the score of a ∈ A according to profile P as s P (a) = R∈P 1{top(R) = a}. We use the resolute function r plu (P ) = arg max a∈A s P (a) to select a single winning alternative, breaking ties lexicographically and favoring that with the smallest index. Assume r = r plu unless stated otherwise. Rank-based utility and additive social welfare. We assume that agents have additive utilities characterized by a rank-based utility vector u = (u 1 , . . . , u m ) ∈ R m ≥0 with u 1 ≥ . . . ≥ u m ≥ 0 and u 1 > u m . Like the scoring rule, each agent j gets u(R j , a) = u i utility for the alternative a ∈ A ranked i th in R j . Unlike prior work, however, we do not presume that u is the same as the scoring vector s. We define the additive social welfare of a according to P as SW u (P, a) = n j=1 u(R j , a). Iterative plurality voting. Given agents' truthful preferences P , we consider an iterative process of profiles (P t ) t≥0 that describe agents' reported preferences (i.e. votes) (R t 1 , . . . , R t n ) t≥0 over time. For each round t, one agent j is chosen by a scheduler φ to make a myopic improvement step, denoted by R t j j − → R j , to their report. This step is called a better-response if j prefers the new outcome r(R j , R t −j ) to the prior one r(R t j , R t −j ), whereas it is a best-response (BR) if, additionally, j could not have achieved a more preferred outcome than manipulating to R j from P t . 1 Following Brânzei et al. [2013], we limit our discussion to strategic manipulations beginning from the truthful profile P 0 = P . This guarantees that all improvement steps R j j − → R j from profile P to P are best-responses that change the iterative winner: r(P ) = top(R j ) ∧ r(P ) = top(R j ). 2 As a result, any sequence of BR steps converges in O(nm) rounds [Reyhani and Wilson, 2012]. The profiles {P * } with no further improvement steps are therefore Nash equilibrium (NE) with respect to P . We define EW(P ) as the set of equilibrium winning alternatives corresponding to all NE reachable from P via some BR sequence. That is, EW(P ) = {r(P * ) : ∃ a BR sequence from P leading to the NE profile P * } Lastly, we'll define the set of potential winning alternatives of any profile P as those who could become a winner if their plurality score were to increment by one, including the current winner. That is, some agent could make these alternatives win by taking a BR step that increases their plurality score, if the agent's ranking permits. Following , we have: PW(P ) = a ∈ A : s P (a) = s P (r(P )) − 1, a is ordered before r(P ) s P (a) = s P (r(P )), a is ordered after r(P ) ∪ {r(P )} where the ordering is lexicographical for tie-breaking. Reyhani and Wilson [2012] proved that the potential winning set is monotonic in t: ∀t ≥ 0, PW(P t+1 ) ⊆ PW(P t ), which implies EW(P ) ⊆ PW(P 0 ). As a result, iterative plurality voting acts like a sequential tie-breaking mechanism whose outcome follows from the scheduler φ. The following example demonstrates this section's concepts. Example 1. Let n = 9, m = 3, and consider the truthful profile P defined with R 1 = R 2 = R 3 = [1 3 2], R 4 = R 5 = [2 3 1], R 6 = [2 1 3], and R 7 = R 8 = R 9 = [3 2 1]. We observe from the plurality scores (s P (1), s P (2), s P (3)) = (3, 3, 3) that r(P ) = 1 and P W (P ) = {1, 2, 3}, representing a three-way tie. Next, Figure 1 describes the five BR sequences from P : Figure 1: Five BR sequences in Example 1. The tuples denote agents' reported top alternatives; the winner appears in curly brackets; arrows denote which agent makes each BR step and the updated report is emphasized. We therefore conclude EW(P ) = {2, 3}. Moreover, consider the utility vector u = (u 1 , u 2 , u 3 ). Then the social welfare for each alternative is (SW u (P, 1), SW u (P, 2), SW u (P, 3)) = (3u 1 + 1u 2 + 5u 3 , 3u 1 + 3u 2 + 3u 3 , 3u 1 + 5u 2 + 1u 3 ). 2 1 Note that φ must select a BR step if one exists [Apt and Simon, 2015]. 2 These are characterized as Type 1 or direct best replies in the literature [Meir, 2016]. Conversely, Type 3 best-responses (r(P ) = top(R j ) ∧ r(P ) = top(R j )) do not occur in improvement sequences from the truthful profile. Note that no BR step is of Type 2 (r(P ) = top(R j ) ∧ r(P ) = top(R j )). Additive Dynamic PoA under General Utility Vectors How bad are equilibrium outcomes, given that strategic manipulations inevitably occur by the Gibbard-Satterthwaite theorem [Gibbard, 1973, Satterthwaite, 1975? Brânzei et al. [2013] sought to answer this question by defining the additive dynamic price of anarchy (ADPoA) as the adversarial loss -the difference in welfare between the truthful winner r(P ) and its worst-case equilibrium winner in EW(P ) -according to the worst-case P . To motivate this concept, consider users of a website that can regularly log in and update their preferences for an election. Then the ADPoA bounds the welfare loss if a virtual assistant can recommend when users should make their changes. Brânzei et al. originally defined the ADPoA for a given positional scoring rule r s and an additive social welfare function respecting u = s. In this case, the ADPoA of plurality was found to be 1, while the (multiplicative) DPoA of veto is Ω(m) and Borda is Ω(n) for m ≥ 4 [Brânzei et al., 2013]. Although these results answer the authors' question and appear optimistic for plurality, they suggest more about the iteration mechanism than agents' collective welfare. For example, an ADPoA for plurality of 1 means that for any truthful profile, the difference in initial plurality scores of any equilibrium winner is at most one less that of the truthful winner. However, when we relax the utility vector u to differ from s, we find in Theorem 1 that the ADPoA is quite poor at Θ(n). First we recall Brânzei et al.'s definition of ADPoA using our notation and explicitly define the adversarial loss D + for a particular truthful profile P before proceeding to our first main result. Definition 1 (Additive Dynamic Price of Anarchy (ADPoA) [Brânzei et al., 2013]). Given a positional scoring rule r s , utility vector u = (u 1 , . . . , u m ) over m ≥ 3 alternatives, and truthful profile P , the adversarial loss starting from P is defined as D + r s , u (P ) = SW u (P, r s (P )) − min a∈EW(P ) SW u (P, a) The additive dynamic price of anarchy (ADPoA) of r s and u under n agents is defined as ADPoA(r s , u, n) = max P ∈L(A) n D + r s , u (P ) We will use ADPoA and D + to denote ADPoA(r plu , u, n) and D + r plu , u when the context is clear. 3 For example, we saw in Example 1 that r(P ) = 1 and EW (P ) = {2, 3}. Then D + (P ) = max{ SW u (P, 1) − SW u (P, 2), SW u (P, 1) − SW u (P, 3) } = max{ (3u 1 + 1u 2 + 5u 3 ) − (3u 1 + 3u 2 + 3u 3 ) , (3u 1 + 1u 2 + 5u 3 ) − (3u 1 + 5u 2 + 1u 3 ) } = −2(u 2 − u 3 ) ≤ 0 Therefore the social welfare of both equilibrium winners is at least that of the truthful winner. In Theorem 2 below we'll see this conclusion hold in expectation. For the worst case profile P , however, the following theorem proves that this is not the case -rather, the worst-case equilibrium winner of P has a social welfare linearly worse than the truthful winner. Theorem 1. Fix m ≥ 3 and utility vector u = (u 1 , . . . , u m ). Then ADPoA(r plu , u, n) is Θ(n). Specifically, ∀n > 2m, (u 2 − u m ) n m − 2 ≤ ADPoA(r plu , u, n) ≤ nu 1 Proof. The ADPoA is trivially upper bounded by the maximum social welfare attainable by any truthful profile P . For example, if P is defined with R j = (1, 2, . . . , m) ∀j ≤ n, then ∀P ∈ L(A) n , D + (P ) = SW u (P, r(P )) − min a∈EW(P ) SW u (P, a) ≤ SW u (P, r(P )) ≤ SW u (P, r(P )) = nu 1 To lower bound ADPoA, we will construct a profile P with a two-way tie between alternatives 1, 2 ∈ A such that D + (P ) = (u 2 − u m ) n m − 2 . This implies the desired lower bound of ADPoA = maxP ∈L(A) n D + (P ) ≥ D + (P ) = (u 2 − u m ) n m − 2 Fix m ≥ 3 and let n > 2m be even. We denote by k = arg mink ∈[2,m−1] (uk − uk +1 ) the position in u with the minimal difference in adjacent coordinates. Let α = 1 m (n + m − 2) and β = (α − 1)(m − 2), such that n = 2α + β. We will then construct P as follows, with α agents that prefer 1 first and 2 last, α agents that prefer 2 first and 1 second, ( β 2 − 1) agents that prefer 1 second and 2 last, and ( β 2 + 1) agents that prefer 2 in their ranking's k-th position and 1 in their ranking's (k + 1)-th position. We can see here that s P (1) = s P (2) = α, and ∀c > 2, s P (c) = α − 1, thus guaranteeing the two-way tie. Therefore r(P ) = 1 and P [2 1] = α + β 2 + 1 > α + β 2 − 1 = P [1 2]. This implies EW(P ) = {2} by the following lemma. Lemma 1. Let m ≥ 2 and a, b ∈ A such that a is ordered before b in tie-breaking. Suppose PW(P ) = {a, b} for some truthful profile P . Then EW (P ) = {a} if P [a b] ≥ P [b a]; otherwise EW(P ) = {b}. The lemma's proof can be found in Appendix A.1. As a result, D + (P ) = SW u (P, 1) − SW u (P, 2) = α(u 2 − u m ) + β 2 − 1 (u 2 − u m ) − β 2 + 1 (u k − u k+1 ) ≥ (u 2 − u m )(α − 2) = (u 2 − u m ) n − m − 2 m ≥ (u 2 − u m ) n m − 2 where the first inequality holds because (u k − u k+1 ) ≤ (u 2 − u m ). Expected Additive DPoA In this section we extend Brânzei et al.'s ADPoA notion to account for the average-case adversarial loss for a positional scoring rule r s , rather than only the studying worst-case. This expected additive dynamic price of anarchy (EADPoA) bounds the adversarial loss of strategic manipulation according to more typical distributions of agents' rankings. Here we distribute profiles i.i.d. uniformly over L(A) n , known as the Impartial Culture distribution π n = IC n . Definition 2 (Expected Additive DPoA (EADPoA)). Given a positional scoring rule r s , a utility vector u over m ≥ 3 alternatives, n agents, and a distribution π n over L(A) n for agents' preferences, the expected additive dynamic price of anarchy is defined as follows: EADPoA(r s , u, n, π n ) = E P ∼πn D + r s , u (P ) Like before, we will use EADPoA and D + to denote EADPoA(r plu , u, n, IC n ) and D + r plu , u respectively. We similarly fix a rank-based utility vector u that may differ from the scoring rule s, but we will not presume in the following theorem that this is known by the iterative plurality mechanism. In the subsequent proof, we will also drop the subscript "P ∼ IC n " to simplify notation when the context is clear. where α denotes the number of potential winners in P . It is straightforward to see that when α = 1, any profile P with |PW(P )| = 1 is already a NE, which implies D + (P ) = 0. The rest of the proof proceeds as follows. For any n ∈ N we will show in Lemma 2 that for ∀W ⊆ A with |W | = 2 PoA(W ) = −Ω(1). We will then demonstrate that PoA(W ) = o(1) ∀W ⊆ A with |W | = 3 (Lemma 3) and |W | ≥ 4 (if m ≥ 4; Lemma 4). Recalling that m is fixed, the total number of subsets of A is viewed as a constant. Finally, these results combine to conclude EADPoA = 0 α=1 − Ω(1) α=2 + o(1) α≥3 = −Ω(1) Profiles with two tied alternatives drive the EADPoA negative because of the self-selecting property of Lemma 1. For example, consider a truthful P with PW(P ) = {a, b} and r(P ) = a. When more agents prefer the non-truthful winner b in this setting, iterative plurality makes this correction by changing the winner to b and increases agents' social welfare on average. When more agents prefer the truthful winner a, rather, iterative plurality doesn't change this outcome and the adversarial loss remains zero. Without a sufficient counter-balance to the former α = 2 case by any of the α ≥ 3 cases, the adversarial loss overall remains negative in expectation. The remainder of this section is devoted to detailing the proof behind the α = 2 case (Lemma 2). We declare the α = 3 case without proof (Lemma 3) and briefly prove the α ≥ 4 case (Lemma 4). Lemma 2 (α = 2). Given m ≥ 3 and a utility vector u, for any W ⊆ A with |W | = 2 and any n ∈ N, we have PoA(W ) = −Ω(1). Proof. Without loss of generality let W = {1, 2} and suppose u 2 > u m . There are two possible cases of PW(P ) = {1, 2}: either s P (1) = s P (2) or s P (1) + 1 = s P (2), which we'll denote by E 1 and E 2 respectively. This suggests PoA (W ) = Pr(E 1 ) × E[D + (P ) | E 1 ] + Pr(E 2 ) × E[D + (P ) | E 2 ]. We'll focus on the first case where alternatives 1 and 2 are tied, since the latter's proof is similar. We believe this proof is challenging due to the dependence in agents' rankings once we condition on profiles that satisfy two-way ties (i.e. E 1 ). As a result, standard approximation techniques that assume independence, such as the Berry-Esseen inequality, no longer apply and may also be too coarse to support our claim. Instead, we will use a Bayesian network to further condition agents' rankings based on two properties: the top ranked-alternative and which of the two tied alternatives the agents prefer. Once we guarantee agents' rankings' conditional independence, we can identify the expected utility they gain for each alternative and then compute E[D + (P ) | E 1 ] efficiently. At a high level, there are two conditions for a profile P to satisfy E 1 and have non-zero adversarial loss. First, the profile must indeed be a two-way tie. This is represented in Step 1 below by identifying each agent j's top-ranked alternative t j ∈ A and conditioning D + (P ) on a specific vector of top-ranked alternatives t ∈ T 2 ⊆ A n , a set corresponding to all profiles satisfying E 1 . Second, by Lemma 1, the profile should satisfy P [2 1] ≥ P [1 2]. This is represented in Step 1 by identifying an indicator z j ∈ {1, 2} to suggest whether 1 j 2 or 2 j 1 respectively. We further condition D + (P ) on a specific vector z ∈ Z t,k , a set corresponding to all profiles in E 1 with k = P [2 1] ≥ P [1 2] = n − k. Once we condition D + (P ) to satisfy these two conditions, we identify the expected difference in welfare between the alternatives E tj ,zj for each agent j conditioned on t j , z j in Step 2, which follows from the Impartial Culture assumption. Finally, we compute D + (P ) by summing over all profiles satisfying the above two conditions and solve in Step 3, making use of Stirling's approximation. More precisely, for any j ≤ n, we represent agent j's ranking distribution (i.i.d. uniform over L(A)) by a Bayesian network of three random variables (see Figure 2). First, T j ∈ A represents j's top-ranked alternative and follows a uniform distribution. Second, Z j ∈ {1, 2} indicates whether (1 j 2) or (2 j 1) conditioned on T j , and has probability {0.5, 0.5} if T j / ∈ {1, 2}. Third, Q j follows the uniform distribution over linear orders that uphold both T j and Z j . It is not hard to verify that (unconditional) Q j follows the uniform distribution over L(A), which implies that Q = (Q 1 , . . . , Q n ) follows the same distribution as P . Step 1: Identify profiles that satisfy E 1 . Let T 2 ⊆ [m] n denote the set of top-ranked alternative vectors t = (t 1 , . . . , t n ) such that alternatives 1 and 2 have the maximum plurality score. Then E 1 holds for Q if and only if T takes a value in T 2 . Figure 2: Bayesian network representation of P as T , Z, and Q for the α = 2 case. T 2 = t ∈ [m] n : ∀3 ≤ i ≤ m, |{j : t j = 1}| = |{j : t j = 2}| > |{j : t j = i}| T 1 … Z 1 … Q 1 T 2 Z 2 Q 2 T n Z n Q n Conditioned on agents' top-ranked alternatives being t ∈ T 2 , we have by Lemma 1 that D + ( Q) is non-zero if and only if Q[2 1] > Q[1 2]. Let Id 1 ( t) = {j ≤ n : t j = 1}, Id 2 ( t) = {j ≤ n : t j = 2}, and Id 3 ( t) = {j ≤ n : t j / ∈ {1, 2}} be the respective set of first-, second-, and third-party agents for t. Since t ∈ T 2 implies |Id 1 ( t)| = |Id 2 ( t)|, there must be more third-party agents that prefer 2 1 than those that prefer 1 2. For every |Id3( t)|+1 2 ≤ k ≤ |Id 3 ( t)|, we thus define Z t,k ⊆ {1, 2} n as the set of all vectors z where the number of 2's in Id 3 ( t) is exactly k. Z t,k = { z ∈ {1, 2} n : ∀j ∈ Id 1 ( t) ∪ Id 2 ( t), z j = t j , and |{j ∈ Id 3 ( t) : z j = (2 1)}| = k} By the law of total expectation and noting the independence of Q's components, we have Pr(E 1 ) × E[D + (P ) | E 1 ] = t∈T |Id3( t)| k= |Id 3 ( t)|+1 2 z∈Z t,k Pr( T = t, Z = z) n j=1 E tj ,zj (2) where E tj ,zj = E Qj [ u(Q j , 1) − u(Q j , 2) | T j = t j , Z j = z j ] is the expected difference in welfare between alternatives 1 and 2 for an agent j with T j = t j and Z j = z j . Step 2: Compute expected welfare difference per agent. We note that E tj ,zj only depends on the values of t j and z j , but not j. The cases for (t j = z j = 1) and (t j = z j = 2) negate each other with E 1,1 + E 2,2 = 0. If t j / ∈ {1, 2} and z j = 1, then E tj ,1 = η > 0 because u 2 > u m . Similarly, it follows that if t j / ∈ {1, 2} and z j = 2, then E tj ,2 = −η. Therefore Equation (2) becomes t∈T |Id3( t)| k= |Id 3 ( t)|+1 2 z∈Z t,k Pr( T = t, Z = z) × (|Id 3 ( t)| − 2k)η(3) where we've inserted n j=1 E tj ,zj = |Id 1 ( t)|E 1,1 + |Id 2 ( t)|E 2,2 − kη + (|Id 3 ( t)| − k)η. Step 3: Simplify and solve. Note that Id 3 ( T ) is equivalent to the sum of n i.i.d. binary random variables, each of which is 1 with probability m−2 m ≥ 1 3 . By Hoeffding's inequality, with exponentially small probability we have Id 3 ( T ) < 1 6 n. Therefore, we can focus on the Id 3 ( T ) ≥ 1 6 n case in (3), which, by denoting β = |Id 3 ( t)| for ease of notation, becomes: Pr( T = t) × Θ( √ n) (5) = e −Ω(n) − η Pr T ∈ T 2 , Id 3 ( T ) ≥ 1 6 n × Θ( √ n) ≤ −Ω(1)(6) where Equation (4) follows from Claim 1 (see Appendix A.3) and Equation (5) follows from Stirling's approximation (see Appendix A.4). We get Equation (6) since Pr( T ∈ T 2 ) is equivalent to the probability of two-way ties under plurality w.r.t. IC, which is known to be Θ(n −1/2 ) [Gillett, 1977]. This concludes Lemma 2, and a more full proof can be found in Appendix A.2. Lemma 3 (α = 3). Given m ≥ 3 and a utility vector u, for any W ⊆ A with |W | = 3 and any n ∈ N, we have PoA(W ) = o(1). We defer the proof of Lemma 3 to Appendix A.5. Lemma 4 (α ≥ 4). Given m ≥ 4 and a utility vector u, for any W ⊆ A with |W | ≥ 4 and any n ∈ N, we have PoA(W ) = o(1). Proof. The lemma follows after noticing the following. Firstly, we note that Pr(PW(P ) = W ) = Θ(n −1.5 ) following a similar proof using the polyhedron representation as described in the proof of Lemma 3 (see Appendix A.5). Second, for any profile P , D + (P ) = O(n). Conclusions and Future Work This paper studies the effects of strategic behavior in iterative plurality voting in terms of its adversarial loss -the difference in social welfare between the truthful winner and the worst-case equilibrium winning alternative. Our results naturally extend those of Brânzei et al. [2013] by utilizing rank-based utility functions whose utility vector u differs from the iterative positional scoring rule r s . We prove that iterative plurality has an adversarial loss linear in the number of agents in the worst case (Theorem 1). By distributing agents' preferences according to the impartial culture, we overcome this negative result and prove a constant order improvement in social welfare regardless of the order of agents' repeated strategic manipulations (Theorem 2). Even through our main result only works for IC, we are not aware of previous theoretical work on the expected performance of iterative voting under any distribution. Generalizing this study to other dynamics, utility functions, and families of distributions are interesting and important directions for future work. For example, many iterative voting rules do not necessarily converge, but all games with bestresponse dynamics have cycles or steady-state equilibrium over agents' joint pure-strategy action space L(A) n [Young, 1993, Meir, 2016. Such games may instead be characterized by the worstcase ratio (or difference) between the social welfare of the game's truthful outcome and the average welfare of a stationary distribution over each cycle -known as the price of sinking [Goemans et al., 2005]. Bounding the welfare in each cycle could plausibly extend the DPoA, left for future work. A second branch of future work could compare the iterative voting equilibrium winners' social welfare to that of the optimal winner, rather than the truthful outcome -known as the price of stability [Anshelevich et al., 2004, Tsang andLarson, 2016]. This is related to work in distortion which modifies agents' utility functions to be normalized Rosenschein, 2006, Caragiannis andProcaccia, 2011] or embedded in a metric space [Anshelevich et al., 2018]. A third branch of future work could generalize the choice of agents' ranking distribution from IC, for example using smoothed analysis [Xia, 2020]. Determining the robustness of our theoretical results to other preference distributions, especially to those based on real-world data, would provide further insight into the effects of strategic manipulation on electoral outcomes. It would be interesting to see whether a greater proportion of i. denote the indices of agents who don't rank a or b highest but prefer (a b) or (b a) respectively. Since each BR sequence begins at P 0 = P , all BR steps are of Type 1 and must change the iterative winner each round, starting from r(P 0 ) = a. BR steps will therefore alternate whether they are taken by agents represented in Id (a) (P ) or Id (b) (P ). Agents from the former set will best-respond to rankings whose top preference is a, changing the winner to a, whereas agents from the latter set will best-respond to rankings whose top preference is b, changing the winner back to b. This alternation will continue until round t when either Id (a) (P t ) or Id (b) (P t ) are emptied of indices. If |Id (a) (P 0 )| ≥ |Id (b) (P 0 )|, the last BR step will make a the unique equilibrium winner, whereas if |Id (a) (P 0 )| < |Id (b) (P 0 )|, the last BR step will make b the unique equilibrium winner. Inverse reasoning holds if a and b differ by one initial plurality score and s P (a) = s P (b) − 1, implying r(P 0 ) = b. In this case, the last BR step will make a the unique equilibrium winner only if |Id (a) (P 0 )| > |Id (b) (P 0 )|, since the plurality score of a is initially disadvantaged by 1. If not, the unique equilibrium winner will be b. We therefore conclude that if P [a b] ≥ P [b a] across all n agents, then EW(P ) = {a}; otherwise EW(P ) = {b}. A.2 Proof of Lemma 2 Lemma 2 (α = 2). Given m ≥ 3 and a utility vector u, for any W ⊆ A with |W | = 2 and any n ∈ N, we have PoA(W ) = −Ω(1). Proof. Without loss of generality let W = {1, 2} and suppose u 2 > u m , since the case where u 2 = u m is covered in [Brânzei et al., 2013]. There are two possible cases of PW(P ) = {1, 2}: E 1 = 1{s P (1) = s P (2)}, where 1 is the truthful winner, and E 2 = 1{s P (1) = s P (2) − 1}, where 2 is the truthful winner. This suggests the following partition: PoA(W ) = Pr(E 1 ) × E[D + (P ) | E 1 ] + Pr(E 2 ) × E[D + (P ) | E 2 ] We'll focus on the former summand where 1 and 2 are tied and prove that Pr(E 1 )×E[D + (P ) | E 1 ] = −Ω(1). The proof for the latter summand can be done similarly. We believe this proof is challenging due to the dependence in agents' rankings once we condition on profiles that satisfy two-way ties (i.e. E 1 ). As a result, standard approximation techniques that assume independence, such as the Berry-Esseen inequality, no longer apply and may also be too coarse to support our claim. Instead, we will use a Bayesian network to further condition agents' rankings based on two properties: the top ranked-alternative and which of the two tied alternatives the agents prefer. Once we guarantee agents' rankings' conditional independence, we can identify the expected utility they gain for each alternative and then compute E[D + (P ) | E 1 ] efficiently. At a high level, there are two conditions for a profile P to satisfy E 1 and have non-zero adversarial loss. First, the profile must indeed be a two-way tie. This is represented in Step 1 below by identifying each agent j's top-ranked alternative t j ∈ A and conditioning D + (P ) on a specific vector of top-ranked alternatives t ∈ T 2 ⊆ A n , a set corresponding to all profiles satisfying E 1 . Second, . This is represented in Step 1 by identifying an indicator z j ∈ {1, 2} to suggest whether 1 j 2 or 2 j 1 respectively. We further condition D + (P ) on a specific vector z ∈ Z t,k , a set corresponding to all profiles in E 1 with k = P [2 1] ≥ P [1 2] = n − k. Once we condition D + (P ) to satisfy these two conditions, we identify the expected difference in welfare between the alternatives E tj ,zj for each agent j conditioned on t j , z j in Step 2, which follows from the Impartial Culture assumption. Finally, we compute D + (P ) by summing over all profiles satisfying the above two conditions and solve in Step 3, making use of Stirling's approximation. T 1 … Z 1 … Q 1 T 2 Z 2 Q 2 T n Z n Q n (a) α = 2 case T 1 … … Q 1 T 2 Q 2 T n Q n (b) α = 3 case More precisely, for any j ≤ n, we represent agent j's ranking distribution (i.i.d. uniform over L(A)) by a Bayesian network of three random variables: T j represents the top-ranked alternative, Z j represents whether (1 j 2) or (2 j 1), conditioned on T j , and Q j represents the linear order conditioned on T j and Z j . Formally, we have the following definition. Definition 3. For any j ≤ n, we define a Bayesian network with three random variables T j ∈ A, Z j ∈ {1, 2}, and Q j ∈ L(A), where T j has no parent, T j is the parent of Z j , and T j and Z j are Q j 's parents (see Figure 3a). Let T = (T 1 , , . . . , T n ), Z = (Z 1 , , . . . , Z n ), and Q = (Q 1 , , . . . , Q n ). The (conditional) distributions are: • T j follows a uniform distribution over A • Pr(Z j = 1 | T j = t) =    1, t = 1 0, t = 2 0.5, t ∈ [3, m] • Q j follows the uniform distribution over linear orders whose top alternative is T j and (1 j 2) if Z j = 1, or (2 j 1) if Z j = 2. It is not hard to verify that (unconditional) Q j follows the uniform distribution over L(A), which implies that Q follows the same distribution as P , namely IC n . Notice that if alternative 1 or 2 is ranked at the top, then Z j is deterministic and equals to T j . Furthermore, if T j ∈ {1, 2}, then Q j follows the uniform distribution over (m − 1)! linear orders; otherwise Q j follows the uniform distribution over (m − 1)!/2 linear orders. Example 2. Let m = 4 and W = {1, 2}. For every j ≤ n, T j is the uniform distribution over [4]. We have that Pr(Z j = 1 | T j = 1) = Pr(Z j = 2 | T j = 2) = 1 and Pr(Z j = 1 | T j = 3) = Pr(Z j = 1 | T j = 4) = 0.5. Given T j = Z j = 1, Q j is the uniform distribution over Step 1: Identify profiles that satisfy E 1 . Let T 2 ⊆ [m] n denote the set of vectors t = (t 1 , . . . , t n ) such that alternatives 1 and 2 have the maximum plurality score: T 2 = t ∈ [m] n : ∀3 ≤ i ≤ m, |{j : t j = 1}| = |{j : t j = 2}| > |{j : t j = i}| E 1 holds for Q if and only if T takes a value in T 2 , implying the following equality. Pr(E 1 ) × E[D + (P ) | E 1 ] = t∈T2 Pr T = t × E Q [D + ( Q) | T = t](7) Conditioned on agents' top-ranked alternatives being t ∈ T 2 , we have by Lemma 1 that D + ( Q) is non-zero if and only if Q[2 1] > Q[1 2] -thus EW ( Q) = {2} is unique. For any t ∈ T 2 , let • Id 1 ( t) ⊆ [n] denote the indices j such that t j = 1 • Id 2 ( t) ⊆ [n] denote the indices j such that t j = 2 • Id 3 ( t) ⊆ [n] denote the indices j such that t j / ∈ {1, 2} -we call these third-party agents E 1 implies |Id 1 ( t)| = |Id 2 ( t)|, so in order to uphold Q[2 1] > Q[1 2] there must be more thirdparty agents that prefer (2 1) than those that prefer (1 2). Specifically, for every |Id3( t)|+1 2 ≤ k ≤ |Id 3 ( t)|, we define Z t,k ⊆ {1, 2} n as the vectors z where the number of 2's among indices in Id 3 ( t) is exactly k: (1, 1, 2, 2, 1, 2, 2, 1, 2) (1, 1, 2, 2, 2, 2, 1, 1, 2) (1, 1, 2, 2, 2, 2, 2, 1, 1) Z t,k = { z ∈ {1, 2} n : ∀j ∈ Id 1 ( t) ∪ Id 2 ( t), z j = t j , and |{j ∈ Id 3 ( t) : z j = 2}| = k} where exactly two reports from agents 5, 7, or 9 are 2's: |{z j = 2 : j ∈ {5, 7, 9}}| = 2. 2 Continuing (7), we have Pr(E 1 ) × E[D + (P ) | E 1 ] = t∈T2 |Id3( t)| k= |Id 3 ( t)|+1 2 z∈Z t,k Pr( T = t, Z = z) × E Q [D + ( Q) | T = t, Z = z] = t∈T2 |Id3( t)| k= |Id 3 ( t)|+1 2 z∈Z t,k Pr( T = t, Z = z) n j=1 E Qj [ u(Q j , 1) − u(Q j , 2) | T = t, Z = z] = t∈T2 |Id3( t)| k= |Id 3 ( t)|+1 2 z∈Z t,k Pr( T = t, Z = z) n j=1 E tj ,zj(8) where E tj ,zj = E Qj [ u(Q j , 1) − u(Q j , 2) | T j = t j , Z j = z j ] The last equation holds because of the Bayesian network structure: for any j ≤ n, given T j and Z j , Q j is independent of other Q's. Pr( T = t) × Θ( √ n) = e −Ω(n) − η Pr T ∈ T 2 , Id 3 ( T ) ≥ 1 6 n × Θ( √ n) = e −Ω(n) − η Pr( T ∈ T 2 ) − Pr T ∈ T 2 , Id 3 ( T ) < 1 6 n × Θ( √ n) ≤ e −Ω(n) − η Pr( T ∈ T 2 ) − Pr Id 3 ( T ) < 1 6 n × Θ( √ n) ≤ e −Ω(n) − η Θ(n −1/2 ) − e −Ω(n) × Θ( √ n) = −Ω(1) where Pr( T ∈ T 2 ) is equivalent to the probability of two-way ties under plurality w.r.t. IC, which is known to be Θ(n −1/2 ) [Gillett, 1977]. This proves Lemma 2. A.4 Application of Stirling's Approximation for Lemma 2 Let u ∈ N and set v = u 2 . We can immediately see that u+1 2 = v + 1, and from Equation (11) in Lemma 2, we want to simplify the term (v + 1) u v+1 . Stirling's approximation states that for every n ∈ N, n! ∼ √ 2πn n e n If u is odd, then u = 2v + 1 and we have u v + 1 (v + 1) = u! v! 2 ∼ √ 2πuu u e −u √ 2πv (v+0.5) e −v 2 = √ u √ 2π (u u e −u ) v (2v+1) e −2v = √ u e √ 2π u v u = √ u e √ 2π 2 + 1 v u If u is even, then u = 2v and we have u v + 1 (v + 1) = u!v v! 2 ∼ √ 2πuu u e −u v √ 2πv (v+0.5) e −v 2 = √ u √ 2π (u u e −u ) v v (2v+1) e −2v = √ u2 u e √ 2π In both cases the objective scales as Θ( √ u2 u ). Step 1: Identify E. Let T 3 ⊆ [m] n denote the set of vectors t = (t 1 , . . . , t n ) such that alternatives 1, 2, and 3 have the maximum plurality score. Formally, T 3 = t ∈ [m] n : ∀4 ≤ i ≤ m, |{j : t j = 1}| = |{j : t j = 2}| = |{j : t j = 3}| > |{j : t j = i}| E holds for Q if and only if T takes a value in T 3 , implying the following equality. PoA({1, 2, 3}) = Pr PW( Q) = {1, 2, 3} × E[D + ( Q) | PW( Q) = {1, 2, 3}] = t∈T3 Pr( T = t) × E[D + ( Q) | T = t](13) Step 2: Upper-bound the conditional adversarial loss. We next employ the law of total expectation on Equation (13) by further conditioning on 1{D + ( Q) > n 0.6 }. This event represents whether the adversarial loss scales positively and at least sub-linearly in n. We will show this holds with high probability and establish the following conditional expectation to be o(n), term-by-term: E[D + ( Q) | T = t] = E[D + ( Q) | T = t, D + ( Q) > n 0.6 ] × Pr(D + ( Q) > n 0.6 | T = t) + E[D + ( Q) | T = t, D + ( Q) ≤ n 0.6 ] × Pr(D + ( Q) ≤ n 0.6 | T = t) Trivially, we note that E[D + ( Q) | T = t, D + ( Q) ≤ n 0.6 ] ≤ n 0.6(14) Second, for any t ∈ [m] and i 1 , i 2 ∈ {1, 2, 3} with i 1 = i 2 , we denote by D t i1,i2 the random variable representing the utility difference between alternatives i 1 and i 2 in Q j , conditioned on T j = t: D t i1,i2 = u(Q j , i 1 ) − u(Q j , i 2 ) Notice that D t i1,i2 does not depend on j. For any t ∈ [m] n and j ≤ n, D tj i1,i2 ∈ [u m − u 1 , u 1 − u m ], which implies D + ( Q) ≤ (u 1 − u m )n, and henceforth E[D + ( Q) | T = t, D + ( Q) > n 0.6 ] ≤ (u 1 − u m )n(15) Thirdly, we observe that E[D tj i1,i2 ] > 0 if t j = i 1 , E[D tj i1,i2 ] = −E[D i1 i1,i2 ] < 0 if t j = i 2 , and E[D tj i1,i2 ] = 0 otherwise. Let D t i1,i2 = n j=1 D tj i1,i2 . It follows that for any t ∈ T 3 we have E[D t i1,i2 ] = 0, since E implies |{j : t j = i 1 }| = |{j : t j = i 2 }|. Recalling that D tj i1,i2 is bounded, it follows from Hoeffding's inequality that Pr(|D t i1,i2 | > n 0.6 ) = exp(−Θ(n 0.2 )) Recall that as a result of only having Type 1 BR steps, the equilibrium winner must be among the initial potential winners of any truthful profile [Reyhani and Wilson, 2012]. Therefore, for any t ∈ T 3 , following the law of total probability, we have Pr D + ( Q) > n 0.6 | T = t ≤ 6 exp(−Θ(n 0.2 )) Combining Equations (14), (15), and (16) with Equation (13) yields our claim: PoA({1, 2, 3}) = t∈T3 Pr( T = t) × E[D + ( Q) | T = t] ≤t∈T3 Pr( T = t) 6n(u 1 − u m ) exp(−Θ(n 0.2 ))) + n 0.6 (1 − 6 exp(−Θ(n 0.2 ))) = Pr( T ∈ T 3 )o(n) Step 3. Determine the probability of three-way ties. Notice that Pr( T ∈ T 3 ) is equivalent to the probability of three-way ties under plurality w.r.t. IC, which is known to be Θ(n −1 ) [Gillett, 1977]. Alternatively, it can be proved by representing Pr( T ∈ T 3 ) as a polyhedra in R m! , which can be equivalently described by a system of linear inequalities, and then applying [Xia, 2021, Theorem 1], as in the proof of [Xia, 2021, Theorem 3]. This method can be easily extended to other cases where {1, 2, 3} are potential winners and not exactly tied, which is not covered by previous studies on the likelihood of ties [Gillett, 1977, Xia, 2021. For completeness, we recall from [Xia, 2021] the system of linear inequalities used to represent the winners being W under any integer positional scoring rule r s . Let x A = (x R : R ∈ L(A)) denote the vector of m! variables, each of which represents the multiplicity of a linear order in a profile. Therefore, Score s a,b · x A represents the score difference between a and b in the profile whose histogram is x A . For any W ⊆ A, we define the polyhedron H s,W as follows. Definition 6. For any integer scoring vector s and any W ⊆ A, we let E s,W denote the matrix whose row vectors are {Score s a,b : a ∈ W, b ∈ W, a = b}. Let S s,W denote the matrix whose row vectors are {Score s a,b : a ∈ W, b ∈ W }. Let A s,W = E s,W S s,T , b = ( 0, − 1), and let H s,W denote the corresponding polyhedron. For example, for W = {1, 2, 3}, H s plu ,W is represented by the following inequalities. 2, 1, 0), and varying the number of agents. For each n ∈ {100, 200, . . . , 1000}, we sampled 10 million profiles uniformly at random and determined, for each P ∼ IC n , its equilibrium winning set EW(P ). We then computed each profile's adversarial loss D + (P ) and averaged their values across all profiles with the same n. Experiments were run on an Intel Core i7-7700 CPU running Windows with 16.0 GB of RAM. Figure 4 demonstrates the sample average adversarial loss using these parameters. Figure 5 partitions the loss based on α-way ties, α ∈ {2, 3, 4}. We note the average adversarial loss decreases as n increases and takes the trend of the two-way tie case complexity. Since a significant proportion of profiles have no BR dynamics, the overall trend keeps close to zero. Therefore these results support our main theorem in this paper, that the welfare of the worst-case strategic equilibrium winner is greater than that of the truthful winner when agents' preferences are distributed according to IC. Theorem 2 . 2Fix m ≥ 3 and utility vector u = (u 1 , . . . , u m ). For any n ∈ N we have EADPoA(r plu , u, n, IC n ) = −Ω(1) Proof. The key idea is to partition L(A) n according to each profile's potential winner set. More precisely, for every W ⊆ A with W = ∅, we define: PoA(W ) = Pr(PW(P ) = W ) × E[D + (P ) | PW(P ) = W ] By the law of total expectation, then EADPoA = E[D + (P )] = m α=1 W ⊆A:|W |=α PoA(W ) . Let m ≥ 2 and a, b ∈ A such that a is ordered before b in tie-breaking. Suppose PW(P ) = {a, b} for some truthful profile P . Then EW(P ) = {a} if P [a b] ≥ P [b a]; otherwise EW(P ) = {b}.Proof. Suppose PW(P ) = {a, b} for some truthful profile P . First consider the case where a and b are tied with s P (a) = s P (b). Let • Id (a) (P ) = {j ∈ [n] : top(R j ) = a, b, and a j b} • Id (b) (P ) = {j ∈ [n] : top(R j ) = a, b, and b j a} Figure 3 : 3Bayesian network representation of P as T , Z, and Q by Lemma 1, the profile should satisfy P [2 1] ≥ P [1 2] Given T j = 4 and Z j = 2, Q j is the uniform distribution over{[4 2 1 3], [4 2 3 1], [4 3 2 1]} 2 Example 3 . 3Suppose m = 4, n = 9, and t = (1, 1, 2, 2, 3, 2, 4, 1, 3). Then, Id 1 ( t) = {1, 2, 8}, Id 2 ( t) = {3, 4, 6}, Id 3 ( t) = {5, 7, 9}. Moreover, for k = 2, we have Z t,2 = Definition 5 ( 5Score difference vector). For any scoring vector s and pair a, b ∈ A, let Score s a,b denote the m!-dimensional vector indexed by rankings in L(A): ∀R ∈ L(A), the R-element of Score s a,b is s(R, a) − s(R, b). ∀{i 1 , i 2 } ⊆ [3] s.t. i 1 = i 2 ,Other cases of PW(P ) = {1, 2, 3} can be characterized by modifying b accordingly. For example, s P (1) + 1 = s P (2) = s P (3) is represented by the following inequalities. generated by fixing m = 4 alternatives with the Borda utility vector u Borda = (3, Figure 4 : 4Average adversarial loss with m = 4, u Borda , and 10M samples. Error bars represent 95% confidence intervals, too small to see. Figure 5 : 5Average adversarial loss partitioned by α-way ties, α ∈ {2, 3, 4}. Error bars represent 95% confidence intervals. i.d. preference distributions yield EADPoA results similar or dissimilar to that of IC. Alan Tsang and Kate Larson. The echo chamber: Strategic voting and homophily in social networks. In Proceedings of the 2016 International Conference on Autonomous Agents and Multiagent Systems, AAMAS '16, page 368-375, 2016. Lirong Xia. The Smoothed Possibility of Social Choice. In Proceedings of NeurIPS, 2020.Lirong Xia. How Likely Are Large Elections Tied? In Proceedings of the 22nd ACM Conference on Economics and Computation, Budapest, Hungary, 2021. Lirong Xia and Vincent Conitzer. Stackelberg voting games: Computational aspects and paradoxes. In Proceedings of the National Conference on Artificial Intelligence (AAAI), pages 921-926, At- lanta, GA, USA, 2010. H Peyton Young. The Evolution of Conventions. Econometrica, 61(1):57-84, January 1993. James Zou, Reshef Meir, and David Parkes. Strategic voting behavior in doodle polls. In Proceed- ings of the 18th ACM Conference on Computer Supported Cooperative Work & Social Computing, CSCW '15, page 464-472, New York, NY, USA, 2015. The proof of Claim 1 can be found in Appendix A.3. We now apply Stirling's approximation to simplify Equation (11) as follows. See Appendix A.4, plugging in u ← β which we recall is Θ(n).e −Ω(n) − η t∈T2:β≥ 1 6 n The superscript '+' denotes an additive measure instead of multiplicative in the classical definition of PoA. AcknowledgementsWe thank anonymous reviewers for helpful comments. This work is supported by NSF #1453542, ONR #N00014-17-1-2621, and a giftfund from Google. J. Kavner acknowledges Abigail Jacobs for helpful discussions during the earliest stage of this work.Step 2: Computer expected welfare difference per agent. E tj ,zj only depends on the values of t j , z j but not j:• If t j = z j = 1, then E tj ,zj = u 1 − u2+...+um m−1 , the expected utility of alternative 2.• If t j = z j = 2, then E tj ,zj is the expected utility of alternative 1, which is u2+...+um m−1 , minus u 1 . Notice that E 2,2 + E 1,1 = 0.• If t j / ∈ {1, 2} and z j = 1, then η = E tj ,1 is the expected utility difference of alternatives 1 minus 2, conditioned on third-party agents and (1 2). Note that η > 0 since u 2 > u m .• If t j / ∈ {1, 2} and z j = 2, then E tj ,2 is the expected utility difference of alternative 1 minus 2, conditioned on third-party agents and (2 1). It follows that E tj ,2 = −η.As a result, equation(8)Step 3: Simplify and solve. 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R M David, Omer Thompson, Kevin Lev, Jeffrey Leyton-Brown, Rosenschein, Proceedings of the Thirteenth International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). the Thirteenth International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS)Paris, FranceDavid R.M. Thompson, Omer Lev, Kevin Leyton-Brown, and Jeffrey Rosenschein. Empirical analy- sis of plurality election equilibria. In Proceedings of the Thirteenth International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pages 391-398, Paris, France, 2013. . A.5 Proof of Lemma. 3A.5 Proof of Lemma 3 Given m ≥ 3 and a utility vector u, for any W ⊆ A with |W | = 3 and any n ∈ N, we have PoA(W ) = o. Lemma 3 (α = 3Lemma 3 (α = 3). Given m ≥ 3 and a utility vector u, for any W ⊆ A with |W | = 3 and any n ∈ N, we have PoA(W ) = o(1). The proof uses a similar and simpler technique than that of Lemma 2. Without loss of generality, suppose W = {1, 2, 3} and consider the case where the plurality scores for 1, 2, and 3 are equal, denoted E. The proofs for cases with alternatives 2 or 3 being truthful winners are similar. We first prove that conditioned on the vector t of all agents' top preferences that satisfy E, the maximum score difference between any pair of alternatives in {1, 2, 3} is o(n) with high probability that is close to 1. Secondly, we prove that PW(P ) = W with probability O(n −1Proof. The proof uses a similar and simpler technique than that of Lemma 2. Without loss of generality, suppose W = {1, 2, 3} and consider the case where the plurality scores for 1, 2, and 3 are equal, denoted E. The proofs for cases with alternatives 2 or 3 being truthful winners are similar. We first prove that conditioned on the vector t of all agents' top preferences that satisfy E, the maximum score difference between any pair of alternatives in {1, 2, 3} is o(n) with high probability that is close to 1. Secondly, we prove that PW(P ) = W with probability O(n −1 ). More precisely, for every j ≤ n, we represent agent j's ranking distribution (i.i.d. uniform over L(A)) by a Bayesian network of two random variables: T j represents agent j's top-ranked alternative, and Q j represents j's ranking conditioned on T j . Formally, we have the following definition. More precisely, for every j ≤ n, we represent agent j's ranking distribution (i.i.d. uniform over L(A)) by a Bayesian network of two random variables: T j represents agent j's top-ranked alterna- tive, and Q j represents j's ranking conditioned on T j . Formally, we have the following definition. we define a Bayesian network with two random variables T j ∈ A and Q j ∈ L(A), where T j has no parent and is the parent of Q j (see Figure 3b). . . , Q , Let T = (T 1. Definition 4. For any j ≤ n. The (conditional) distributions are: • T j follows a uniform distribution over ADefinition 4. For any j ≤ n, we define a Bayesian network with two random variables T j ∈ A and Q j ∈ L(A), where T j has no parent and is the parent of Q j (see Figure 3b). Let T = (T 1 , , . . . , T n ) and Q = (Q 1 , , . . . , Q n ). The (conditional) distributions are: • T j follows a uniform distribution over A follows the uniform distribution over linear orders whose top alternative is T j It is not hard to verify that (unconditional) Q j follows the uniform distribution over L(A). • Q J, Therefore, Q follows the same distribution as P. which is IC n• Q j follows the uniform distribution over linear orders whose top alternative is T j It is not hard to verify that (unconditional) Q j follows the uniform distribution over L(A). There- fore, Q follows the same distribution as P , which is IC n . Let m = 4 and W = {1, 2, 3}. For every j ≤ n, T j is the uniform distribution over. Example 4.Example 4. Let m = 4 and W = {1, 2, 3}. For every j ≤ n, T j is the uniform distribution over Given T j = 1, Q j is the uniform distribution over. Given T j = 1, Q j is the uniform distribution over

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{'abstract': "Recent work in iterative voting has defined the additive dynamic price of anarchy (ADPoA) as the difference in social welfare between the truthful and worst-case equilibrium profiles resulting from repeated strategic manipulations. While iterative plurality has been shown to only return alternatives with at most one less initial votes than the truthful winner, it is less understood how agents' welfare changes in equilibrium. To this end, we differentiate agents' utility from their manipulation mechanism and determine iterative plurality's ADPoA in the worst-and average-cases. We first prove that the worst-case ADPoA is linear in the number of agents. To overcome this negative result, we study the average-case ADPoA and prove that equilibrium winners have a constant order welfare advantage over the truthful winner in expectation. Our positive results illustrate the prospect for social welfare to increase due to strategic manipulation.", 'arxivid': '2106.08853', 'author': ['Joshua Kavner [email protected] \nDepartment of Computer Science\nDepartment of Computer Science\nRensselaer Polytechnic Institute Troy\nRensselaer Polytechnic Institute Troy\n12180, 12180NY, NY\n', 'Lirong Xia [email protected] \nDepartment of Computer Science\nDepartment of Computer Science\nRensselaer Polytechnic Institute Troy\nRensselaer Polytechnic Institute Troy\n12180, 12180NY, NY\n'], 'authoraffiliation': ['Department of Computer Science\nDepartment of Computer Science\nRensselaer Polytechnic Institute Troy\nRensselaer Polytechnic Institute Troy\n12180, 12180NY, NY', 'Department of Computer Science\nDepartment of Computer Science\nRensselaer Polytechnic Institute Troy\nRensselaer Polytechnic Institute Troy\n12180, 12180NY, NY'], 'corpusid': 235446917, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 22893, 'n_tokens_neox': 19966, 'n_words': 12221, 'pdfsha': '3efd811f401aa3021bca690eb3e18ef556f75f03', 'pdfurls': ['https://export.arxiv.org/pdf/2106.08853v3.pdf'], 'title': ['Strategic Behavior is Bliss: Iterative Voting Improves Social Welfare', 'Strategic Behavior is Bliss: Iterative Voting Improves Social Welfare'], 'venue': []}

arxiv

Classical Exchange Algebra of the Superstring on S 5 with 25 Sep 2007 February 2, 2008 Shogo Aoyama ×s5*e-mail:[email protected] Department of Physics Shizuoka University Ohya 836Shizuoka Japan Classical Exchange Algebra of the Superstring on S 5 with 25 Sep 2007 February 2, 20080220Uw0420Fy1110Lm1125Tq Keywords: Classical Yang-Baxter equationCanonical formalismSuperstringAdS 5 A classical exchange algebra of the superstring on S 5 with the AdS-time is shown on the light-like plane. To this end we use the geometrical method of which consistency is guaranteed by the classical Yang-Baxter equation. The Dirac method does not work, there being constraints which contain first-class and second-class and one can disentangle with each other keeping the isometry hardly. The AdS/CFT correspondence between the type IIB string theory on AdS 5 × S 5 and the N = 4 supersymmetric QCD is one of the subjects which have been discussed with great interest in the last decade. The study of such string/QCD duality has a long history going back to the late 70's [1]. The AdS/CFT correspondence [2] is the most clear-cut assertion of string/QCD duality given ever since. Correlation functions were extensively studied on both sides of the correspondence to check this. Later on people discovered a remarkable relationship between the N = 4 supersymmetric QCD and the Heisenberg spin-chain systems. Namely they claimed that chains of scalar fields of the former theory can be identified with spin-chains of the latter by calculating the anomalous dimension and the energy for the respective chains [3]. It is no doubt that integrablity of both chain systems is involved behind this phenomenon. From the AdS/CFT duality we are then lead to demand a corresponding integrability on the string side. It is wellknown that the XXX Heisenberg spin-chain equivalent to the O(3) non-linear σ-model at the thermodynamical limit [4]. Therefore it is quite natural to consider the the type IIB string theory on AdS 5 × S 5 as a non-linear σ-model and suspect its integrable structure. Indeed many works have been done about integrability of non-linear σ-models on S 5 and AdS 5 [5,6]. Exchange algebrae based on the Yang-Baxter equation are also characteristic for integrable systems. This aspect of the type IIB string theory on AdS 5 × S 5 [7] is currently attracting much attention in connection with higher-loop integrability of N = 4 supersymmetric QCD [8]. In this letter we will shed a light on the subject from a different angle. The following action is a part of the type IIB string theory on AdS 5 × S 5 S = d 2 ξ 6 a=1 ∂ µ M a ∂ µ M a − ∂ µ X 0 ∂ µ X 0 ,(1) where X 0 is the AdS-time and M a are fields constrained on S 5 by 6 a=1 M a M a = 1. It is supplimented by the Virasoro constraint ∂ ± M a ∂ ± M a = ∂ ± X 0 ∂ ± X 0 . With a gauge X 0 ∝ t the action (1) becomes the non-linear σ-model on S 5 [9]. The aim of this letter is to show, though at the classical level, in a rather simple way the exchange algebra for the non-linear σ-model on S 5 {M a (ξ + , ξ − ), M b (ξ + , ξ ′− )} = − 1 4 θ(ξ − − ξ ′− )r + + θ(ξ ′− − ξ − )r − ab cd M c (ξ + , ξ − )M d (ξ + , ξ ′− ).(2) Here { , } is the Poisson bracket on the light-cone plane ξ + = const. and r ± are the classical r-matrices given in the fundamental representation of SO (6), which satisfy the classical Yang-Baxter equation. As a deformation of (2) we might think of a quantum exchange algebra M a (ξ + , ξ − )M b (ξ + , ξ ′− ) = θ(ξ − − ξ ′− )R + + θ(ξ ′− − ξ − )R − ab cd M d (ξ + , ξ ′− )M c (ξ + , ξ − ), by extending the classical r-matrices to quantum R-matrices as R ± = 1 ⊗ 1 +hr ± + O(h 2 ). withh = − 1 4 . The similar classical exchange algebrae in the canonical formalism were discussed for the 2-dimensional effective gravity in the literature [10]. We will proceed the arguments exactly in the same way as for that case. We shall consider non-linear σ-models on the coset space G/H, which are given by the action S = d 2 ξ L = 1 2 d 2 ξ η µν g ij (X)∂ µ X i ∂ ν X j ,(3) in the 2-dimensional flat world-sheet. The energy-momentum tensor and the isometry current are respectively given by T µν = g ij (X)∂ µ X i ∂ ν X j − η µν L, J A µ = R A i (X)∂ µ X i . Here R Ai are the Killing vectors of the coset space G/H, which non-linearly realize the Lie-algebra of G as R Ai ,j R Bj − R Bi ,j R Aj = f AB C R Ci ,(4) and satisfy the Killing equations R A {i;j} ≡ R A i;j + R A j;i = 0.(5) The non-linear σ-models (3) have conformal invariance and isometry so that T +− = 0, ∂ + T −− = 0, ∂ − T ++ = 0, ∂ + J A − + ∂ − J A + = 0, due to the equation of motion ∇ µ ∂ µ X i ≡ ∂ µ ∂ µ X i + Γ i jk ∂ µ X j ∂ µ X k = 0.(6) We study the canonical structure of the models on the light-like plane setting up the Poisson brackets. It can be done by the geometrical method formulated in [10]. Namely we may set up the Poisson brackets {X i , X j } on the light-like plane ξ + = const. so as to be able to correctly reproduce the diffeomorphism and the isometry transformations as δ dif f X i (ξ + , ξ − ) ≡ ǫ(ξ − )∂ − X i (ξ + , ξ − ) = dζ − ǫ(ζ − ){X i (ξ + , ξ − ), T −− (X(ξ + , ζ − ))},(7)δ iso X i (ξ + , ξ − ) ≡ ǫ A R Ai (X(ξ + , ξ − )) = dζ − ǫ A {X i (ξ + , ξ − ), J A (X(ξ + , ζ − ))}.(8) Here ǫ(ξ − ) and ǫ A are respectively local and global parameters of the transformations. It turns out that they are given in the form {X i (ξ − ), X j (η − )} = −{X j (η − ) X i (ξ − )} = − 1 4 θ(ξ − − η − )t + AB R Ai (X(ξ − ))R Bj (X(η − )) + 1 4 θ(η − − ξ − )t + AB R Aj (X(η − ))R Bi (X(ξ − )).(9) The notation of this formula is as follows. θ(ξ − ) is the step function and R Ai (X(ξ + )) are the Killing vectors of the coset space G/H, given by (4) and (5). The world-sheet coordinate ξ + in X i (ξ + , ξ − ) was omitted to avoid a unnecessary complication. t + AB is the most crucial part for our argument in this letter. It is a set of the coefficients taken from the classical r-matrices r ± = α∈R sgn αE α ⊗ E −α + A,B t AB T A ⊗ T B ≡ t ± AB T A ⊗ T B ,(10) with T A the generators of the group G given in the Chevallay basis as {E ±α , H µ }, t + AB the corresponding Killing metric and sgn α = ± according as the roots are positive or negative. The r-matrix satisfies the classical Yang-Baxter equation [11] [r 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] = 0, which guarantees the Jacobi identities for the Poisson brackets (9), as will be shown later. First of all we will show that the diffeomorphism (7) can be reproduced by using the Poisson brackets (9). A little algebra yields {X i (ξ − ), T −− (X(ζ − ))} = δ(ξ − − ζ − )∂ − X j (ζ − )g jk (ζ − )t AB R Ai (ξ − )R Bk (ζ − ) − 1 4 θ(ξ − − ζ − )t + AB R Ai (ξ − )R B {j;k} (ζ − )∂ − X j (ζ − )∂ − X k (ζ − ) + 1 4 θ(ζ − − ξ − )t + AB R A {j;k} (ζ − )R Bi (ξ − )∂ − X j (ζ − )∂ − X k (ζ − ). This becomes {X i (ξ − ), T −− (X(ζ − ))} = δ(ξ − − ζ − )∂ − X i (ζ − ),(12) by the Killing equations (5) and the property of the Killing vectors [12] t AB R Ai R Bj = g ij . With (12) the diffeomorphism (7) is correctly reproduced. Similarly one can check that the isometry transformations (8) can be reproduced. We remark that the above demonstration would work even if the Poisson brackets (9) were simply given with the Killing metric t AB in place of t + AB . The specfic choice of t + AB given by (10) is required to show the Jacobi identities Q ijk ≡ {X i (ξ − ), {X j (ζ − ), X k (η − )}} + {X j (ζ − ), {X k (η − ), X i (ξ − )}} + {X k (η − ), {X i (ξ − ), X j (ζ − )}} = 0.(13) To show it, we calculate the quantities Q ijk by (9) assuming that ξ − > ζ − > η − . We then use the the Lie-Algebra for the Killing vectors (4) to find Q ijk = − 1 4 t + AB t + CD f AC E R Ei (ξ − ) R Dj (ζ − )R Bk (η − ) − 1 4 t + AB t CD R Ci (ξ − ) f AD E R Ej (ζ − ) R Bk (η − ) − 1 4 t + AB t + CD R Ci (ξ − )R Aj (ζ − ) f BD E R Ek (η − ) .(14) Note that eq. (4) can be put in the form R Ci ∝ f C AB R Ai ,j R Bj , by using f C AB f AB D ∝ δ C D . Replace all the Killing vectors in (14) by this formula. Then it becomes Q ijk ∝ 1 4 t + AB t + CD ([T A , T C ]) EF ⊗ (T B ) GH ⊗ (T D ) KL +(T A ) EF ⊗ ([T B , T C ]) GH ⊗ (T D ) KL +(T A ) EF ⊗ (T C ) GH ⊗ ([T B , T D ]) KL ×R Ei ,l (ξ − )R F l · (ξ − )R Gj ,m (ζ − )R Hm (ζ − ) · R Ki ,n (η − )R Ln (η − ). This is vanishing due to the classical Yang-Baxter equation (11) so that the Jacobi identities (13) are satisfied. Thus it has been shown that the Poisson brackets (9) are correct for the canonical formalism of the non-linear σ-models (3) on the light-like plane. We will use this to find the classical exchange algebra (2) in the non-linear σ-models. Our claim is that such an algebra exists if the models admit local composite fields M a (X), a = 1, 2 · · · , d which change as δM a (X) ≡ ǫ A R Ai (X)M a ,i (X) = ǫ A (T A ) a b M b ,(15) under the isometry transformation δX i = ǫ A R Ai (X). That is, the composite fields M a (X) belong to a d-dimensional representation of the isometry group G. It depends on the type of the coset space G/H whether such composite fields exist or not. We proceed with the argument assuming the existence of them. By using the Poisson brackets (9) we find that {M a (X(ξ − )), M b (X(ζ − ))} = M a ,i (X(ξ − )){X i (ξ − ), M b (X(ζ − ))} = M a ,i (X(ξ − )){X i (ξ − ), X j (ζ − )}M b ,j (X(ζ − )) = − 1 4 θ(ξ − − ζ − )t + AB (T A ) a c (T B ) b d M c (X(ξ − )))M d (X(ζ − )) + 1 4 θ(ζ − − ξ − )t + AB (T A ) a c (T B ) b d M c (X(ζ − )))M d (X(ξ − )). By using the property t + AB = −t + BA and the r-matrices (10) we get the classical exchange algebra (2). We apply the above arguments to the case of the non-linear σ-model on the coset space SO(6)/SO(5)(= S 5 ). For this model there exsits a set of composite fields M a transforming as (15) in the fundamental representation of SO (6). They are constrained by 6 a=1 M a M a = 1 and are parametraized as M 1 (X) = cos X 1 , M 2 (X) = sin X 1 cos X 2 , M 3 (X) = sin X 1 sin X 2 cos X 3 , M 4 (X) = sin X 1 sin X 2 sin X 3 cos X 4 , M 5 (X) = sin X 1 sin X 2 sin X 3 sin X 4 cos X 5 , M 6 (X) = sin X 1 sin X 2 sin X 3 sin X 4 sin X 5 . By this parametrization we rewrite the SO (6) generators to obtain the Killing vectors R [ab]i as iM a ∂ ∂M b − iM b ∂ ∂M a = R [ab]i ∂ ∂X i . Therefore the whole arguments leading us to the exchange algebra (2) go through for this special case. As the result we obtain the exchange algebra (2) with the r-matrices given in the fundamental representation of SO (6). The reader might think of studying the Poisson brackets (9) by using the Dirac method. However we can see that it does not work. For the model (3) we have π i ≡ δL δ∂ + X i = g ij ∂ − X j , which lead us to a set of constraints φ i (ξ) ≡ π i − g ij ∂ − X j = 0.(16) They are typical for the Lagrangian which is homogeneous of the first degree in the velocities ∂ + X i . According to the Dirac method [13] the Hamiltonian is merely given by H = dξ − λ i (ξ)φ i (ξ), with Lagrangian multipliers λ i (ξ). Setting the Poisson brackets as {X i (ξ − ), π j (η − )} P = δ i j δ(ξ − − η − ) on the light-like plane ξ + = const. we find that C ij (ξ − , η − ) ≡ {φ i (ξ − ), φ j (η − )} P = −g ij (ξ − )∂ ξ − δ(ξ − − η − ) − g ik,j (ξ − )∂ − X k (ξ − )δ(ξ − − η − ) +g ji (η − )∂ η − δ(ξ − − η − ) + g jk,i (η − )∂ − X k (η − )δ(ξ − − η − ).(17) Here again the coordinate ξ + was omitted from X i (ξ + , ξ − ), π i (ξ + , ξ − ), φ i (ξ + , ξ − ) and g ij (ξ + , ξ − ). They look like second-class constraints. But they are a mixture of firstand second-class constraints as follows. The consistency of the constraints requires that {φ i (ξ − ), H} = dη − λ j (η){φ i (ξ − ), φ j (η − )} = 0,(18) which becomes ∇ − λ i (ξ + , ξ − ) = 0. With λ i = ∂ + X i they are satisfied by means of the equation of motion (6). Therefore the quantity C ij (ξ − , η − ) is not invertible. It means that the constraints (16) contain first-class ones. A similar problem can be seen in the Green-Schwarz formulation of superparticle or superstring. The Green-Schwarz superparticle [14] is given by S ≡ dtL = 1 2 dte −1 (Ẋ M − iΘγ MΘ )(Ẋ M − iΘγ MΘ ). The problematic constraint takes the form φ ≡ π Θ − iΘ/ p = 0,(19) with π Θ = δL δΘ and p M = δL δẊ M . A simple algebra gives C αβ ≡ {φ α , φ β } +P = −2i(γ 0 / p) αβ . We have p 2 = 0 as the equation of motion for e. Therefore C αβ is not invertible. Thus constraints (19) are a mixture of first-class and second-class. We do not know how to disentangle the two classes of constraints with each other in a Lorentz-covariant manner. This is why the Green-Schwarz superparticle can not be quantized in a covariant way. We do not know either how to do the disentanglement of the constraints (16) in a isometrical way. Therefore the Dirac method hardly works for the non-linear σ-models on the light-like plane. The geometrical arguments in this letter gives an alternative way to the Dirac method. But note that dη − C ij (ξ − , η − ){X j (η − ), X k (ζ − )} = δ k i δ(ξ − − ζ − ) + 1 2 θ(ξ − − ζ − )t + AB R A i;j (ξ − )∂ − X j (ξ − )R Bk (ζ − ), with the Poisson brackets (9) and C ij (ξ − , η − ) given by (17). For the special case when the target space is flat, the second term in the r.h.s. vanishes owing to the equation of motion, which is solved by holomorphic functions X i (ξ + , ξ − ) = f i (ξ + ) + g i (ξ − ). Then C ij (ξ − , η − ) becomes invertible as C(ξ − , η − ) −1 ij ≡ {X i (ξ − ), X j (η − )} = − 1 4 δ ij θ(ξ − − η − ) − θ(η − − ξ − ) . Putting this in other words, eq. (18) has no other solution than the trivial ones λ i = ∂ + X i = 0 for this case, so that the constraints (16) do not contain first-class ones at all. Consequently the Dirac method works in the standard way. In this letter we have shown the exchange algebra of non-linear σ-models on the lightlike plane ξ + = const., setting the canonical structure by the geometrical arguments. 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{'abstract': 'A classical exchange algebra of the superstring on S 5 with the AdS-time is shown on the light-like plane. To this end we use the geometrical method of which consistency is guaranteed by the classical Yang-Baxter equation. The Dirac method does not work, there being constraints which contain first-class and second-class and one can disentangle with each other keeping the isometry hardly.', 'arxivid': '0709.3911', 'author': ['Shogo Aoyama ×s5*e-mail:[email protected] \nDepartment of Physics\nShizuoka University\nOhya 836Shizuoka Japan\n'], 'authoraffiliation': ['Department of Physics\nShizuoka University\nOhya 836Shizuoka Japan'], 'corpusid': 18656589, 'doi': '10.1088/1751-8113/47/7/075402', 'github_urls': [], 'n_tokens_mistral': 9208, 'n_tokens_neox': 7889, 'n_words': 4351, 'pdfsha': 'b8a94cb409007ee85dd5fd9490777ad70b2f7f6b', 'pdfurls': ['https://arxiv.org/pdf/0709.3911v2.pdf'], 'title': ['Classical Exchange Algebra of the Superstring on S 5 with', 'Classical Exchange Algebra of the Superstring on S 5 with'], 'venue': []}

arxiv

A sequence of quasipolynomials arising from random numerical semigroups 26 Sep 2018 Calvin Leng [email protected] Department of Mathematics Department of Mathematics and Statistics University of California Davis Davis 95616CAUSA Christopher O&apos;neill [email protected] San Diego State University 92182San DiegoCAUSA A sequence of quasipolynomials arising from random numerical semigroups 26 Sep 2018 A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generators can be expressed in terms of a doubly-indexed sequence of integers, denoted h n,i , that count generating sets with certain properties. We prove a recurrence that implies the sequence h n,i is eventually quasipolynomial when the second parameter is fixed. Introduction A numerical semigroup is a subset of Z ≥0 containing 0 that is closed under addition. The numerical semigroup generated by a set A = {a 1 , a 2 , . . . , a k } is the smallest numerical semigroup containing A, namely S = A = a 1 , . . . , a k = {a 1 x 1 + · · · + a k x k : x i ∈ Z ≥0 }. A generating set A is minimal if for all x ∈ A, we have A = A \ {x} . Every numerical semigroup S has a unique minimal generating set, and the embedding dimension of S is the size of its minimal generating set (see [3] for a thorough introduction). The authors of [2] introduce a model of randomly generating a numerical semigroup that is similar to the Erdős-Renyi model for random graphs. Their model takes two inputs M ∈ Z ≥1 and p ∈ [0, 1], and randomly selects a generating set A that includes each integer n = 1, 2, . . . , M with independent probability p. As an example, if M = 40 and p = 0.1, then one possible set is A = {6, 9, 18, 20, 32} (this is not unreasonable, as on average one would expect 4 generators to be selected). However, only 3 elements of A are minimal generators, since 18 = 9 + 9 and 32 = 20 + 6 + 6. As such, the resulting semigroup S = A = 6, 9, 20 has embedding dimension 3. One of the main results in [2] is that the expected number of minimal generators of a numerical semigroup S sampled with the above model can be expressed as E[e(S)] = M n=1 p(1 − p) ⌊n/2⌋ h n,0 + h n,1 p + h n,2 p 2 + · · · , where h n,i equals the number of sets A ⊂ [1, n/2) ∩ Z with |A| = i that minimally generate a numerical semigroup not containing n. Of interest is the asymptotic behavior of E[e(S)] for fixed p as M → ∞. Although this is currently out of reach, E[e(S)] can be approximated for fixed M using the above formula, so long as h n,i is known for n ≤ M. The doubly-indexed sequence h n,i is available on OEIS as A319608, computed for n ≤ 90. Figure 1 contains the values of h n,i for n = 68, . . . , 76, where each row is comprised of h n,0 , h n,1 , . . . , h n,dn from left to right. The following facts about the sequence h n,i are known: • h n,i is nonzero for n ≥ 1 and 0 ≤ i ≤ d n = ⌊n/2⌋ − ⌊n/3⌋; • h n,0 = 1; • h n,1 = ⌊(n + 1)/2⌋ − τ (n), where τ (n) denotes the number of divisors of n; and • The sum of the n th row equals the number of irreducible numerical semigroups with Frobenius number n [1,4], which appears in OEIS as A158206 [5]. Currently, computing the values of h n,i for large n is time-intensive; the fastest known algorithm computes the n th row by first computing the set of irreducible numerical semigroups with Frobenius number n and utilizing the last bullet point above [4]. This computation takes 3 days for n = 89 on the authors' machines. The more values of h n,i that are known, the more accurately E[e(S)] can be approximated. Due to the limited known values of h n,i , appoximations computed with the currently known values still differ drastically from those obtained from experimental data. In this paper, we examine the combinatorics of the sequence h n,i . Our main result is Corollary 2, which states that for fixed k the sequence h n,dn−k coincides with a polynomial in n ≫ 0 whose coefficients are 6-periodic functions of n, follows from the following recurrence. Theorem 1. Fix k ∈ Z ≥0 , b ∈ {0, 1, 2}, and m > 24k + 12 − 8b with m ≡ b mod 3. The recurrence h n,dn−k = k l=0 h m,dm−l d n − d m k − l holds for all n ≥ m satisfying n ≡ b mod 3. A quasipolynomial is a function q : Z → Z such that q(x) = c 0 (x) + c 1 (x)x + c 2 (x)x 2 + · · · + c d (x)x d where each c i (x) is a periodic function. The degree of q, denoted deg q, is the largest integer d for which c d is not identically 0, and the period of q is the smallest integer p such that c i (x + p) = c i (x) for every x and i. Corollary 2. For fixed k, the function n → h n,dn−k coincides with a quasipolynomial c k (n)n k + · · · + c 1 (n)n + c 0 (n) with degree k, period 6, and leading coefficient c k (n) = 2 k!6 k , if n ≡ 0, 1 mod 3; 1 k!6 k , if n ≡ 2 mod 3, for all n > 24k + 12 − 8b, where b ∈ {0, 1, 2} with n ≡ b mod 3. In the development of the proof of Theorem 1, we obtain an algorithm for computing the values h n,i appearing in Corollary 2 (Algorithm 19). Our algorithm has obtained h n,i values that were previously unobtained. With the improved algorithm and Theorem 1, explicit quasipolynomials have been provided for h n,dn−k for each k ≤ 7 (see Figure 2 for the quasipolynomials up to k = 4). Computing the quasipolynomial coefficients of h n,dn−7 requires computing the value of e.g., h 183,d 183 −7 = h 183,23 = 6423209, a task that would have been impossible with existing methods. h n,dn =                    2, if n ≡ 0 mod 6 and n ≥ 18; 2, if n ≡ 1 mod 6 and n ≥ 7; 1, if n ≡ 2 mod 6 and n ≥ 2; 2, if n ≡ 3 mod 6 and n ≥ 15; 2, if n ≡ 4 mod 6 and n ≥ 10; 1, if n ≡ 5 mod 6 and n ≥ 5. h n,dn−1 =                   h n,dn−2 =                    1 36 (n 2 + 108), if n ≡ 0 mod 6 and n ≥ 66; h n,dn−3 =                    1 648 (n 3 − 9n 2 + 342n − 3240), if n ≡ 0 mod 6 and n ≥ 90; h n,dn−4 =                    1 15552 (n 4 − 24n 3 + 828n 2 − 17280n + 419904), Setup Unless otherwise stated, throughout the rest of the paper assume n ∈ Z ≥1 and b n ∈ {0, 1, 2} with n ≡ b n mod 3. Let X n = n 3 , n 2 ∩ Z. Definition 3. Fix a set A ⊂ Z ≥1 . We say A works for n ∈ Z ≥1 if (i) n / ∈ A , (ii) x < n/2 for all x ∈ A, and (iii) A minimally generates a numerical semigroup. In particular, h n,i equals the number of sets A with |A| = i that work for n. To motivate the next several definitions, recall that for n ≥ 13, h n,dn = 2, if n ≡ 0, 1 mod 3; 1, if n ≡ 2 mod 3.(1) The set X n works for n and |X n | = d n . Let E 0,n and E 1,n denote the remaining working sets for n of size d n when b n = 0 and b n = 1, respectively. The key observation is that for all n, Since X n contains every integer in the interval (n/3, n/2), if we wanted to construct counted sets from X n , we could only adjoin elements from {1, 2, . . . , ⌊n/3⌋} to X n . Thus we thought of ⌊n/3⌋ as a sort of cutoff point. From this, it felt natural to express sets in terms of how offset the elements are from ⌊n/3⌋. This motivates the following. Definition 4. The offset form of a set A = {x 1 , x 2 , . . . , x k } ⊂ Z ≥1 is the set A (n) = A − ⌊n/3⌋ = {x 1 − ⌊n/3⌋ , x 2 − ⌊n/3⌋ , . . . , x k − ⌊n/3⌋}. After expressing the sets we computed in offset form, we noticed that we could go one step further. We noticed that if we instead expressed sets in terms of how different they are from X n and then take the offset form of the result, the expressions would be equal. This motivates the following. An RI-pair for n is a pair (R, I) of a removing set R and an inserting set I. There is a natural bijection between RI-pairs for n and the powerset of {1, 2, . . . , d n } given by the map φ(R, I) = (X n \ R) ∪ I. The inverse map is given by A → (X n \ A, A \ X n ). Since ϕ n gives a bijection between the two objects, we say the set corresponding to an RI-pair (R, I) is the set ϕ n (R, I), and vice-versa. Theorem 1 follows from the fact that for fixed k and large n, every RI-pair (R, I) corresponding to a working set for n of size d n − k satisfies I (n) ⊆ {p n (k), . . . , −1, 0}, where p n (k) = b n − 2k − 1 only depends on n modulo 3 (Theorem 15). As a consequence, the restrictions on removal sets corresponding to a given insertion set are independent of the size of n in this case. Theorem 11 classifies the possible RI-pairs that correspond to working sets for large n. Strongly bounded sets We begin by classifying the working sets for n that are strongly n-bounded (Definition 7). As it turns out, for k fixed and large n, every working set for n with size d n − k is strongly n-bounded (Theorem 16). Note that any strongly n-bounded set automatically satisfies parts (ii) and (iii) of Definition 3. Definition 7. We say a set A ⊂ Z ≥1 is strongly n-bounded if A ⊂ (n/4, n/2). Proposition 8. A strongly n-bounded set A works for n if and only if b n / ∈ 3A (n) . Proof. Any strongly n-bounded set automatically satisfies part (ii) and (iii) of Definition 3 since x + y > n 2 > z for any x, y, z ∈ A. As such, A works for n if and only if n / ∈ A . Moreover, since A is strongly n-bounded, we have x < n < y for any x ∈ 2A and y ∈ 4A, so n ∈ A if and only if n ∈ 3A. The claim now follows from the fact that n = 3 ⌊n/3⌋+b n . Definition 9. An RI-pair (R, I) is compatible for n (or, equivalently, R is compatible with I) if the corresponding set A satisfies b n / ∈ 3A (n) . The removal degree of an inserting set I, denoted r(I), is given by r(I) = min{|R| : (R, I) is compatible}. and the removal degree of an integer α ≤ 0 is given by r(α) = r({α}). Remark 10. Note that Proposition 8 does not imply that an RI-pair (R, I) compatible for n corresponds to a set A that works for n, as A need not be strongly n-bounded in general. Theorem 11 classifies the RI-pairs compatible for n in terms of I (n) and R (n) by examining the different ways for three integers to sum to b n ∈ {0, 1, 2}. Theorem 11. If A is a set and (R, I) is the corresponding RI-pair, then b n / ∈ 3A (n) if and only if for all α ∈ I (n) , the following hold: (i) b n − 2α ∈ R (n) ; (ii) (b n − α)/2 ∈ R (n) if α ≡ b n mod 2; (iii) b n − α − β ∈ R (n) for all β ∈ I (n) with β = α; and (iv) y ∈ R (n) or b n − α − y ∈ R (n) for all y satisfying 1 ≤ y < b n − α − y. Proof. If any of (i)-(iv) is violated for some α ∈ I (n) , then it is easy to check that b n ∈ 3A (n) . Conversely, suppose b n ∈ 3A (n) , meaning α + y + z = b n for some α, y, z ∈ 3A (n) with α ≤ y ≤ z. Since b n ∈ {0, 1, 2}, we must have α ≤ 0 and thus α ∈ I (n) . Let S = {α, y, z}. If every element of S is nonpositive, then α = y = z = b n = 0 so (i) fails to hold and we are done. As such, at most 2 elements of S are nonpositive, so z > 0. Similarly, if |S| = 1, then α = y = z = b n = 0 so (i) fails to hold and we are done. This leaves four distinct cases: • |S| = 2 and y ≤ 0, in which case y = α and (i) fails to hold; • |S| = 2 and y > 0, in which case y = z and (ii) fails to hold; • |S| = 3 and y ≤ 0, in which case α = y and (iii) fails to hold; or • |S| = 3 and y > 0, in which case y = z and (iv) fails to hold. This completes the proof. Remark 12. Given an inserting set I, Theorem 11 provides a systematic way to construct a removing set R such that the set A corresponding to (R, I) satisfies b n / ∈ 3A (n) . Most applications of Theorem 11 will involve starting with a set R = ∅ and systematically putting elements into R; see Example 20. Moreover, Theorem 11 yields a better-than-brute-force method of computing h n,dn−k for large n; see Algorithm 19. Lemma 13. We have r(α) = 1 + b n − α − 1 2 for any integer α ≤ 0. Proof. Fix α ≤ 0, let I = {α}, and suppose R is a removing set that is minimal among all removing sets compatible with I. We will apply Theorem 11, noting that for fixed α, parts (i)-(iv) each require distinct elements to lie in R (n) . Theorem 11(i) requires 1 element to lie in R, and Theorem 11(iv), which forces ⌊(b n − α − 1)/2⌋ additional elements to lie in R. Since |I| = 1, Theorem 11(iii) is vacuously satisfied. This leaves Theorem 11(ii), which only requires an additional element to lie in R if α ≡ b n mod 2. This completes the proof. Lemma 14. If A ⊂ Z ≥1 corresponds to an RI-pair (R, I) that is compatible for n, then |A| ≤ d n + 1 − r(m), where m = min I (n) . Proof. Let m = min I (n) , and apply Theorem 11 to α = m. Following the proof of Lemma 13, Theorem 11(i), (ii), and (iv) require r(m) elements to lie in R, and Theorem 11(iii) requires R to contain an additional |I| − 1 elements. We conclude |A| = d n + |I| − |R| ≤ d n + |I| − r(m) − |I| + 1 = d n + 1 − r(m), as desired. Theorem 15. Fix k ∈ Z ≥0 , and suppose A ⊂ Z ≥1 corresponds to an RI-pair (R, I) that is compatible for n. If |A| ≥ d n − k, then I (n) ⊂ {p n (k), p n (k) + 1, . . . , −1, 0}. Proof. Let m = min I (n) . By Lemma 14, we have d n − k ≤ |A| ≤ d n + 1 − r(m), meaning k ≥ r(m) − 1. Applying Lemma 13, we obtain k ≥ 1 + b n − m − 1 2 − 1 ≥ b n − m − 1 2 which can then be rearranged to yield m ≥ b n − 2k − 1 = p n (k). Theorem 16. If n > 24k + 12 − 8b n , then every set A with |A| = d n − k that works for n is strongly n-bounded. Proof. Fix a set A with |A| = d n − k that works for n. Theorem 15 implies min A − p n (k) ≥ n 3 = n − b n 3 = 1 4 n − b n 3 + n 4 > 6k + 3 − 3b n 3 + n 4 = 2k + 1 − b n + n 4 meaning A is strongly n-bounded. Proof of Theorem 1 We now have enough machinery to prove Theorem 1 and Corollary 2. Fix A ∈ A, and let (R, I) denote the corresponding RI-pair. Write R = R 1 ∪ R 2 with (R 1 ) (n) = {α ∈ R (n) : 1 ≤ α ≤ d m } and (R 2 ) (n) = {α ∈ R (n) : d m + 1 ≤ α ≤ d n }, and define f : A → B by f (A) = (ϕ m (R 1 , I), (R 2 ) (n) ). We first show f is well-defined. Let l = |R 1 | − |I|. It is clear that (R 2 ) (n) ∈ S k−l , and (R 1 , I) is an RI-pair for m, so it remains to show that (R 1 , I) is compatible for m. By Theorem 16, A is strongly n-bounded, so Theorem 15 implies min I (n) ≥ p n (k) = p m (k). The key observation is that the criteria in Theorem 11(i)-(iv) only involve I (n) and R (n) , so tracing through each part, the fact that (R, I) is compatible for n implies (R 1 , I) is compatible for m. Hence, f is well-defined. To prove f is a bijection, we observe that basic set-theoretic arguments verify the map ((R 1 , I), R 2 ) → (R 1 ∪ R 2 , I), is the inverse function of f , thereby completing the proof. The authors used a C++ implementation, now posted on Github at the following URL, to compute the quasipolynomial functions in Corollary 2 up to k = 7, the last of which took 6 hours to complete. https://github.com/calvinleng97/rnsg-qp-coeffs Example 20. Suppose n = 60, and consider the insertion set I = {17, 18}. Theorem 11 provides a systematic method of constructing all removal sets R that are compatible with I. Since min I > n/4, the resulting set will be strongly n-bounded. This ensures the resulting sets A corresponding to (R, I) will work for n. We check every item of Theorem 11 with every element α ∈ I (n) to construct R (n) . We first compute the offset form I (n) = {−3, −2} and initialize R (n) = ∅. Note that b n = 0 since 60 ≡ 0 mod 3. We begin by applying Theorem 11(i)-(iii) to each α ∈ I (n) , since Theorem 11(iv) requires additional decisions. For α = −2, we see that 1, 4, 5 ∈ R (n) , and for α = −3, we must have R (n) = {1, 4, 5, 6}. Lastly, we deal with Theorem 11(iv), which is vacuously satisfied for α = −2, and for α = −3 implies either 1 ∈ R (n) or 2 ∈ R (n) , the first of which is already required from above. As such, R (n) = {1, 4, 5, 6} yields a removal R = {1, 4, 5, 6} that is compatible with I. Moreover, any removal set R ′ ⊃ R is also compatible with I. Problem 21. Characterize the sets counted by h n,dn−k for all n in terms of those counted by h n,dn−k for n sufficiently large. Algorithm 19 has the potential to be parallelized (with different threads handling different insertion sets), but the current implementation does not take advantage of this fact. Doing so would likely extend the current limits of computation, which would be especially useful if Problem 21 has a positive answer. Problem 22. Write a parallelized implementation of Algorithm 19. Figure 1 : 1Values of h n,i for n = 68 through n = 76. ), if n ≡ 5 mod 6 and n ≥ 23. 1 36 1(n 2 + 16n + 19), if n ≡ 1 mod 6 and n ≥ 55;1 72 (n 2 + 26n + 160), if n ≡ 2 mod 6 and n ≥ 50; 1 36 (n 2 + 6n + 117), if n ≡ 3 mod 6 and n ≥ 63; 1 36 (n 2 + 10n − 20), if n ≡ 4 mod 6 and n ≥ 58; 1 72 (n 2 + 32n + 247), if n ≡ 5 mod 6 and n ≥ 47. 1 648 1(n 3 + 15n 2 − 69n + 5885), if n ≡ 1 mod 6 and n ≥ 79;1 1296 (n 3 + 30n 2 + 264n − 1952), if n ≡ 2 mod 6 and n ≥ 74; 1 648 (n 3 + 315n − 2268), if n ≡ 3 mod 6 and n ≥ 87; 1 648 (n 3 + 6n 2 − 132n + 6200), if n ≡ 4 mod 6 and n ≥ 82; 1 1296 (n 3 + 39n 2 + 471n − 863), if n ≡ 5 mod 6 and n ≥ 71. if n ≡ 0 mod 6 and n ≥ 114; 1 15552 (n 4 + 8n 3 − 282n 2 + 24728n + 413225), if n ≡ 1 mod 6 and n ≥ 103; 1 31104 (n 4 + 28n 3 + 204n 2 − 10256n + 454912), if n ≡ 2 mod 6 and n ≥ 98; 1 15552 (n 4 − 12n 3 + 666n 2 − 12852n + 374949), if n ≡ 3 mod 6 and n ≥ 111; 1 15552 (n 4 − 4n 3 − 300n 2 + 26528n − 490112), if n ≡ 4 mod 6 and n ≥ 106; 1 31104 (n 4 + 40n 3 + 510n 2 − 8168n + 426817), if n ≡ 5 mod 6 and n ≥ 95. Figure 2 : 2Quasipolynomial expressions for h n,dn−k with k = 0, 1, . . . , 4. X n − n 3 = 3{1, 2, . . . , d n }, E 0,n − n 3 = {−1, 1, 3, 4, . . . , d n }, and E 1,n − n 3 = {0, 2, 3, . . . , d n }. Definition 5 . 5A set I ⊆ Z is an inserting set for n ∈ Z ≥1 if I (n) ⊆ {− ⌊n/3⌋ , . . . , −1, 0}, and a set R ⊆ Z is a removing set for n if R (n) ⊆ {1, 2, . . . , d n }. Example 6 . 6If n = 11 and k = 1, then h n,dn−k = h 11,1 = 4 and X n = {4, 5}. The sets A with |A| = 1 that work for 11 are A = {2} = (X n \ {4, 5}) ∪ {2}, A = {4} = (X n \ {5}) ∪ {}, A = {3} = (X n \ {4, 5}) ∪ {3}, and A = {5} = (X n \ {4}) ∪ {}. Proof of Theorem 1 . 1Fix n, m ∈ Z satisfying n ≡ m mod 3 and n ≥ m > 24k + 12 − 8b n . Let S k denote the set of k-subsets of {d m + 1, d m + 2, . . . , d n }. We will prove the claim combinatorially by constructing a bijection between A := {A : A works for n and |A| = d n − k} and B := k l=0 {A : A works for m and |A| = d m − l} × S k−l . Remark 18. It is interesting to note that the eventual quasipolynomial form of h n,dn−3 would not be impossible to compute using the "standard" method of finding polynomial coefficients. Indeed, the values of h n,i have only been successfully computed for n ≤ 90, and since the quasipolynomial behavior of h n,dn−3 only holds for n ≥ 87, the standard methods of finding the coefficients of a cubic require knowning h 87,d 87 −3 , h 90,d 90 −3 , . . ., most of which have yet to be computed. The above method, on the other hand, only relies on h 87,d 87 −i for 0 ≤ i ≤ 3.Algorithm 19. Theory developed in Section 3 yields an algorithm to compute h m,dm−k for m > 24k+12−8b m . In particular, for each possible inserting set I ⊂ {p m (k), . . . , −1, 0} for m, Theorem 11 determines precisely which removal sets R are compatible with I. Example 20 demonstrates the main idea of the algorithm. Future WorkAlthough the quasipolynomials in Corollary 2 only hold for n sufficiently large, the machinery developed in Section 3 describes the sets counted by h n,dn−k and the relations between them as n varies. For n just below the start of quasipolynomial behavior, computations indicate the sets counted by h n,dn−k are simply those predicted by Theorem 11 that still mimimally generate a numerical semigroup. A better understanding of this phenomenon could allow Algorithm 19 to be extended to all n ≥ 1, rather than just sufficiently large n. Proof of Corollary 2. Fix k ≥ 0 and b ∈ {0, 1, 2}, and let m = min{x > 24k + 12 − 8b : x ≡ b mod 3}.For any n ≥ m satisfying n ≡ b mod 3, we obtain the expressionfrom Theorem 1, wherein each binomial coefficient is a polynomial in d n of degree at most k.Since d n is a quasilinear function of n with period 6, we conclude h n,dn−k is a quasipolynomial in n of degree k and period 6. It remains to verify the leading coefficient of h n,dn−k has the desired form. The highest degree term in(2)isCombined with the fact that d n has constant leading coefficient 1/6, we obtain the leading coefficient h m,dm /k!6 k , and the claim now follows from examination ofFigure 2. Repeating this process for b = 1, 2 yields the function given inFigure 2. On the number of semigroups of natural numbers. J Backelin, Math. Scand. 662J. Backelin, On the number of semigroups of natural numbers, Math. Scand. 66 (1990), no. 2, 197-215. Random numerical semigroups and a simplicial complex of irreducible semigroups, preprint. J De Loera, C O&apos;neill, D Wilbourne, J. De Loera, C. O'Neill, and D. Wilbourne, Random numerical semigroups and a simplicial complex of irreducible semigroups, preprint, 2017. Available at https://arxiv.org/abs/1710.00979 J Rosales, P García-Sánchez, Numerical semigroups, Developments in Mathematics. New YorkSpringer-Verlag20J. Rosales and P. García-Sánchez, Numerical semigroups, Developments in Mathemat- ics, Vol. 20, Springer-Verlag, New York, 2009. Fundamental gaps in numerical semigroups. J Rosales, P García-Sánchez, J García-García, J Jiménez Madrid, J. Pure Appl. Algebra. 1891-3J. Rosales, P. García-Sánchez, J. García-García, J. Jiménez Madrid, Fundamental gaps in numerical semigroups, J. Pure Appl. Algebra 189 (2004), no. 1-3, 301-313. N J A Sloane, The On-Line Encyclopedia of Integer Sequences. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, https://oeis.org. Mathematics Subject Classification: Primary 20M14; Secondary 05E40. Keywords: numerical semigroup, quasipolynomial. (Concerned with sequences A158206, A319608). Mathematics Subject Classification: Primary 20M14; Secondary 05E40. Keywords: numerical semigroup, quasipolynomial. (Concerned with sequences A158206, A319608)

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{'abstract': 'A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generators can be expressed in terms of a doubly-indexed sequence of integers, denoted h n,i , that count generating sets with certain properties. We prove a recurrence that implies the sequence h n,i is eventually quasipolynomial when the second parameter is fixed.', 'arxivid': '1809.09915', 'author': ['Calvin Leng [email protected] \nDepartment of Mathematics\nDepartment of Mathematics and Statistics\nUniversity of California Davis Davis\n95616CAUSA\n', 'Christopher O&apos;neill [email protected] \nSan Diego State University\n92182San DiegoCAUSA\n'], 'authoraffiliation': ['Department of Mathematics\nDepartment of Mathematics and Statistics\nUniversity of California Davis Davis\n95616CAUSA', 'San Diego State University\n92182San DiegoCAUSA'], 'corpusid': 49255312, 'doi': None, 'github_urls': ['https://github.com/calvinleng97/rnsg-qp-coeffs'], 'n_tokens_mistral': 8850, 'n_tokens_neox': 7461, 'n_words': 4570, 'pdfsha': '93d0bf07d1ae8fcadabe81fc1b047a3456373084', 'pdfurls': ['https://arxiv.org/pdf/1809.09915v1.pdf'], 'title': ['A sequence of quasipolynomials arising from random numerical semigroups', 'A sequence of quasipolynomials arising from random numerical semigroups'], 'venue': []}

arxiv

Programming Is Hard -Or at Least It Used to Be: Educational Opportunities And Challenges of AI Code Generation Brett A Becker [email protected] University College Dublin Dublin Ireland Paul Denny The University of Auckland AucklandNew Zealand James Finnie-Ansley [email protected] The University of Auckland AucklandNew Zealand Andrew Luxton-Reilly [email protected] The University of Auckland AucklandNew Zealand James Prather [email protected] Abilene Christian University Abilene TexasUSA Eddie Antonio Santos [email protected] University College Dublin Dublin Ireland Programming Is Hard -Or at Least It Used to Be: Educational Opportunities And Challenges of AI Code Generation CCS CONCEPTS • Social and professional topics → Computing educationCom- puter science educationCS1• Computing methodologies → Artificial intelligence KEYWORDS AIAlphaCodeAmazonartificial intelligencecode generationCodeWhispererCodexCopilotCS1CS2GitHubGoogleGPT-3introductory programmingmachine learningMidjourneynovice programmersOpenAIprogrammingTabnine Figure 1: An image generated by Midjourney with the prompt "robot writing computer code while student watches, computer screens, computer programming, computer code, realistic, highly detailed, cinematic -aspect 16:9" ABSTRACT The introductory programming sequence has been the focus of much research in computing education. The recent advent of several viable and freely-available AI-driven code generation tools present several immediate opportunities and challenges in this domain. In this position paper we argue that the community needs to act quickly in deciding what possible opportunities can and should be leveraged and how, while also working on how to overcome or otherwise mitigate the possible challenges. Assuming that the effectiveness and proliferation of these tools will continue to progress rapidly, without quick, deliberate, and concerted efforts, educators will lose advantage in helping shape what opportunities come to be, and what challenges will endure. With this paper we aim to seed this discussion within the computing education community. INTRODUCTION Recent months have seen the release of several AI models that represent step-changes in their respective domains. Text-to-image models such as OpenAI's DALL-E 2 [35] and Midjourney 1 (see Figure 1) are revolutionizing how images are created, with the latter being called "the greatest artistic tool ever built, or a harbinger of doom for entire creative industries" [23]. In July 2022, it was announced that DeepMind's AlphaFold predicted the structure of nearly all 200 million proteins known to science and is making them freely available [13]. Also in the last year, OpenAI and Deep-Mind -among others -have released groundbreaking models that generate computer code. The model for use so far is that most of these tools will cost money to use professionally but often be free for educational use and to students [44]. It is safe to assume that some computing students are already using AI code completion to generate large chunks of code that could be used in various ways during the completion of assignments. The introductory programming sequence has been the focus of much research over several decades [5,26] and the challenges of programming at the level required of a first-year computing student have been debated extensively [3,4,25]. One particular sticking point is that students should gain extensive practice writing code through dozens of small exercises checked against automated assessment tools [34,42]. However students face numerous barriers. With the recent advent of several viable AI-driven code generation tools, 'writing' code that will suffice to pass traditional first-year programming assignments and even exams seems to have become much easier [12,36]. What does an introductory computing course look like when we can assume that students will be able to easily auto-generate code solutions to their lab and assignment tasks by merely pasting problem descriptions into an AI-powered tool? Further, how will this affect the delivery of computing curricula in general? Our view is that these tools stand to change how programming is taught and learned -potentially significantly -in the near-term, and that they present multiple opportunities and challenges that warrant immediate discussion as we adapt to the use of these tools proliferating. BACKGROUND AND CONTEXT Although long-sought, viable AI-driven code generation has only become a reality for the general public since 2021. Three large-scale AI-powered code generation tools -OpenAI Codex, DeepMind AlphaCode, and Amazon CodeWhisperer -have recently come into the spotlight. The advantages such systems claim to provide include making "programming more productive and accessible" [22]. They also bring several challenges. OpenAI Codex In September 2021 the New York Times published an article titled "A.I. Can Now Write Its Own Computer Code. That's Good News for Humans" describing OpenAI's 2 Codex model [29]. OpenAI is a nonprofit "AI research and deployment company" 3 set up in 2015 with a $1 billion pledge from several tech leaders and investors [37]. Codex is a descendant of GPT-3 (third-generation Generative Pre-trained Transformer) which is one of the most advanced natural language models available today. To demonstrate the kind of output that GPT-3 can generate from a simple prompt, we asked GPT-3 4 the following (after the outline of this paper was determined): Write a paragraph on the educational opportunities and challenges presented by automated code generation tools. Its response was: There are both educational opportunities and challenges presented by automated code generation tools. On the one hand, these tools can help students learn to code more quickly and efficiently. On the other hand, they can also lead to a false sense of understanding and proficiency, as students may become reliant on the tools to do the heavy lifting for them. Additionally, automated code generation tools can sometimes produce code that is difficult to read and understand, making it more challenging for students to debug and troubleshoot their programs. It is worth noting that this single paragraph from GPT-3 includes some of the opportunities and challenges (in relation to AI-generated code) that we address in this paper. Codex is trained on more than 50 million GitHub repositories including the vast majority of GitHub Python code, totaling 159 GB. 2 openai.com 3 openai.com/about/ 4 text-davinci-002 model Files deemed as likely to be auto-generated, those with an average line count greater than 100, those with maximum line length greater than 1000, and those containing a small percentage of alphanumeric characters were filtered [8]. Codex can take Englishlanguage prompts and generate code in several languages including JavaScript, Go, Perl, PHP, Ruby, Swift, TypeScript, and shell, but is "most capable" in Python 5 . It can also translate code between programming languages, explain (in English) the functionality of code provided as input, and return the time complexity of code it generates. It also has the ability to generate code that uses APIs, allowing it to, for example, send emails and access information in various databases. Codex is available via the OpenAI API 6 and also powers GitHub Copilot 7 which is billed as "Your AI pair programmer" -an intentional reference to pair programming, a well-known software engineering [2] and programming education approach [28]. Copilot is now available for free to verified students and teachers. 8 The Codex model has been shown to perform well when solving programming tasks presented in plain English. The paper announcing Codex solved 29% of the problems in a new evaluation set developed by the Codex authors to measure functional correctness for synthesizing programs from Python docstrings. This performance increased to 70% when repeated sampling is employed [8]. The first evaluation of Codex on introductory programming problems was reported by Finnie-Ansley et al. [12], who compared its performance on summative exam questions to that of students in an introductory course, and found that it outperformed almost 80% of the students in the course. In addition, it comfortably solved various definitions of the classic Rainfall Problem [39], including one novel variation that had never been published. DeepMind AlphaCode In February 2022, DeepMind 9 announced AlphaCode 10 which, like Codex, utilizes a transformer-based model that "writes computer programs at a competitive level" 11 . It is trained on over 715 GB of GitHub code including programs written in C++, C#, Go, Java, JavaScript, Lua, PHP, Python, Ruby, Rust, Scala, and TypeScript [22]. All files larger than 1 MB or with lines longer than 1000 characters, and duplicates of the same file (ignoring whitespace) were filtered from training data. Unlike Codex, AlphaCode was fine-tuned on a curated set of publicly released competitive programming problems called CodeContests. 12 In introducing AlphaCode, Li et al. claim that a stripped-down version of AlphaCode, without the modifications described in their paper, performs similarly to Codex, "however, problems used in the Codex paper and similar work consist of mostly simple task descriptions with short solutions -far from the full complexity of real-world programming" [22]. AlphaCode ranked in the top 54% of over 5,000 programming competition participants from the Codeforces platform, solving new problems requiring a combination of critical thinking, logic, algorithms, coding, and natural language understanding [22]. Based 5 openai.com/blog/openai-codex 6 beta.openai.com 7 copilot.github.com 8 github.com/pricing 9 deepmind.com 10 alphacode.deepmind.com 11 deepmind.com/blog/competitive-programming-with-alphacode 12 github.com/deepmind/code_contests on these results, the authors estimate that AlphaCode has a Codeforces 13 rating of 1238 which is within the top 28% of users that participated in a contest in the last 6 months [22]. Li et al. also showed that AlphaCode does not duplicate sections of code from the training dataset when producing solutions, instead relying heavily on natural language problem descriptions to create original solutions. AlphaCode is not currently available as an API or otherwise. Amazon CodeWhisperer Amazon CodeWhisperer was announced in June 2022 [1]. Unsurprisingly a Google Scholar search (July 27, 2022) returned only four results for amazon codewhisperer none of which pertain to the tool itself. CodeWhisperer is billed as "the ML-Powered Coding Companion" 14 which "helps improve developer productivity by providing code recommendations based on developers' natural comments and prior code" [1]. Based on (for instance) a developer comment describing a task, CodeWhisperer attempts to determine which cloud services and public libraries are best for the task, generates code, and presents this as a recommendation to the developer within the IDE. Like Codex and AlphaCode, it is trained on public data. It is also claimed that accuracy is directly proportional to the size of the training data [1] -a finding similar to that of Codex [40]. CodeWhisperer is currently available for free, subject to a waitlist. 15 Other AI code generation products Although Codex, AlphaCode, and CodeWhisperer are the most publicized AI-driven code generation platforms, several others exist including Tabnine 16 , Code4Me 17 and FauxPilot 18 (based on Sales-Force CodeGen [32]). Most of these tools are commercial offerings aimed at professional software developers, as one of the oft-touted (although currently unproven) advantages of AI-driven code generation is increased development productivity. POSITION Our position is the following: AI-generated code presents both opportunities and challenges for students and educators in introductory programming and related courses. The sudden viability and ease of access to these tools suggest educators may be caught unaware or unprepared for the significant impact on education practice resulting from AI-generated code. We therefore urgently need to review our educational practices in the light of these new technologies. We take it as given that these tools will continue to be readily available to students (as they are currently), that adoption will increase, and that the capabilities of the tools will improve. In the following sections we describe some of the opportunities and challenges presented by AI code generating tools in the context of university-level novices learning to program in the present time. We largely focus on opportunities and challenges that are already well-documented in the computing education literature, and discuss how AI-generated code is likely to affect the landscape of areas that are already well-studied. Where available, we include evidence and results from the literature although the literature on the effects of cutting-edge AI code generation tools is in its infancy. We intend this presentation of opportunities and challenges to form the basis for the inevitable discussions about the role of code generation tools in our education practices. In this work we do not discuss wider societal (e.g., economic [8], political [17]) considerations presented by AI-generated code, nor those specific to advanced/professional programmers (e.g., [9,31]). While important issues, our focus is on how AI-generated code is likely to impact students and educators in introductory programming (and related) classrooms in the near term. While any new technology brings with it both positive and negative impacts, the hope is always that long-term net effects are positive. The developers of Codex specifically note that they do not "expect the impact of this class of technologies to be net-negative; rather, risks merit particular attention ... because they may be subtle or require deliberate effort to address, whereas [they] expect the benefits to be more obvious and "automatic" from the perspective of most users and affected stakeholders" [8]. The challenges and opportunities presented here are not exhaustive, but rather starting points for ongoing discussions that we hope lead to best educational practices involving code generation tools. OPPORTUNITIES Any new tool, when effective and widely-available, poses opportunities for learning. Handheld calculators have been ubiquitous in mathematics education since the 1980s. A study in (US) grades K-12 statistically analyzing 524 effects from 79 separate studies recommended the use of calculators in all mathematics classes from kindergarten (approx. age 5) on, including in testing situations for grades 5 (approx. age 10) and up [14]. It remains to be seen if AI-powered code generation will follow a similar path. Opportunities noted by the developers of Codex include "the potential to be useful in a range of ways" including: "help onboard users to new codebases; reduce context switching for experienced coders; enable non-programmers to write specifications; have [such tools] draft implementations; and aid in education and exploration" [8]. The developers of AlphaCode also see obvious opportunities in such tools, suggesting "the potential for a positive, transformative impact on society, with a wide range of applications including computer science education, developer tooling, and making programming more accessible" [22]. In this section we offer a number of avenues where AI-generated code tools present clear opportunities for computer science education. Although some opportunities bring related challenges, in this section we focus on their benefits. Code Solutions for Learning 4.1.1 Exemplar solutions. Students learning to program are typically encouraged to practice writing code by completing short problems. However, students are sometimes unable to complete the exercises, and even when successful often seek exemplar solutions. Unfortunately, instructors do not always have the time to prepare and publish model solutions for all the programming exercises that students engage in (which may include test and exam questions). AIgenerated solutions provide a low-cost way for students to generate exemplar solutions to check their work when practicing [12]. Variety of solutions. Code generation tools can also be used to help expose students to the variety of ways that a problem can be solved. There are usually many different approaches for solving a programming problem, although novices do not always appreciate this. Thompson et al. argue that providing appropriate variation in programming instruction is important, because it helps learners to appreciate the efficiencies and differences in approaches to writing code [41]. Eckerdal and Thuné make a similar argument, drawing on variation theory to state that teachers should make available resources that highlight dimensions of variation in concepts being studied [11]. For most non-trivial problems, code generation tools produce a variety of correct solutions that are offered to the programmer for selection. This was illustrated by Finnie-Ansley et al. who observed a great deal of variation in the solutions that Codex generated when solving the classic Rainfall problem [12]. Code review of solutions. Current assessment approaches in introductory programming courses often focus on code correctness, rather than code quality or style. With the ability to generate syntactically-correct solutions automatically, assessment can focus on the differences between multiple correct solutions, and making judgments on the style and quality of solutions. Extensive literature on peer review, including code reviews [15,24], outline the many benefits from looking at a variety of solutions to a given problem. These benefits are reportedly present even when the code is flawed -there are benefits from looking at good solutions as well as poor ones. Code generation models could be used to generate solutions of varying, or unknown quality, and these could be used for assessment tasks focusing on the evaluation of code quality to engage students at the highest level of Bloom's taxonomy. This may prove useful for generating discussions around alternative approaches and the quality of solutions, and provide the basis for refactoring exercises [12]. Current models are effective at generating correct code, but to our knowledge, no studies have looked at the style of AI-generated code. We believe that future models will have more sophisticated methods of selecting high-quality code that adheres to style conventions. The developers of AlphaCode note that automatic code generation could make programming more accessible and help educate new programmers [22]. These models could suggest alternative, and more efficient or idiomatic ways of implementing programs, which could help learners to improve their coding style. Producing Learning Resources Generating high-quality learning resources is very time consuming and typically requires a high level of expertise. The potential for generating novel learning resources, like programming exercises, explanations of code, and worked examples at essentially an unlimited scale is an exciting avenue for future work. Exercise generation. Very recent work by Sarsa et al. has shown that the Codex model is capable of producing novel learning resources from a single priming example [36]. They explored the generation of two types of resources -programming exercises and code explanations -finding that most of the generated exercises were sensible and novel and included an appropriate sample solution [36]. They also reported that the Codex model was very effective at producing contextualized problem statements, targeting certain thematic topics that were simply specified as input to the model as part of the priming example. Code explanations. High quality explanations of code are a useful type of resource for helping learners develop a robust understanding of programming concepts. One of the widely publicized features of Codex is that it can generate explanations of complicated pieces code. The example of this functionality provided as part of the OpenAI playground uses the prompt: "Here's what the above class is doing: 1. " (with the number at the end prompting the model to produce an enumerated list when describing a code fragment). In their study of learning resource generation, Sarsa et al. found that most of the explanations generated by Codex were thorough and correct [36]. Recent work by MacNeil et al. also explored different kinds of prompts and the diverse code explanations they lead to when using the GPT-3 language model [27]. Illustrative examples. Texts and other learning resources typ- ically provide examples that are used to learn the relationship between a described programming problem and a solution. These can be used to illustrate a given programming construct, algorithmic pattern, data structure, or mapping from problem to solution. Students use these examples as models that help them learn, and frequently express a desire for more examples than are available. Code generation tools provide a means of satisfying this desire and providing as many examples as needed. As AI code generation tools improve, this could lead to worked examples that include reasoning for coding decisions, similar to those generated by Minerva for mathematics problems [21]. Such examples are believed to lower cognitive load and result in more effective learning [18]. New Pedagogical Approaches Teaching in CS1 typically focuses initially on syntax and basic programming principles, and it usually takes time for students to master these fundamentals. If code generation models can be used to solve the low level implementation tasks, this may allow students to focus on higher level algorithms. In a way, this is similar to the use of block-based environments that remove the complexities of syntax and allow students to focus on algorithmic issues. Teaching could initially focus more on algorithms and problems solving, relying on automatic code generation for implementation, and then delay discussions of syntax until later. Explaining algorithmic concepts clearly. The way that prompts to code generation models are constructed affects their performance. Simplifying the problem description was found to significantly increase the success rate of AlphaCode. On a sample of difficult problems, simplifying the description to make the required algorithm more explicit increased the percentage of correct samples from 12% to 55% [22]. It was also found that sensitivity to consistent variable naming decreases with model size -random changes to variable names in problem descriptions mattered less. That is, models are increasingly able to capture relevant relationships between variables described in the problem formulation. Students can focus more on how to communicate algorithmic problems clearly, thereby providing a better description to code generation models that can then generate working solutions. Alleviating programmer's writer's block. Anecdotally, students sometimes struggle with programmer's writer's block -that is, they don't know how to get started. Vaithalingam et al. [43] found that Copilot helped students to get started with programming assignments by producing some starter code, thereby offering the opportunity to extend code rather than struggling with a blank page. Such an approach may require us to shift focus towards rewriting, refactoring, and debugging code; however, this provides the opportunity to help students maintain forward momentum in an authentic environment where the need for evaluating, rewriting, and extending code is perhaps more important than writing every line of code from scratch [30]. Overcoming traditional barriers. Novices face many barriers in learning to program [5]. For instance, programming (compiler) error messages are a known barrier to student progress [4,16]. Recent work has demonstrated that Codex is capable of explaining error messages in natural language -often effectively -and that that it can also provide correct fixes based on input code and error messages [20]. It is likely that the efficacy of these approaches will improve in time, and that other barriers to novice learning may be similarly mitigated by such models. CHALLENGES The availability AI-based code generation raise concerns that it could be used in ways that limit learning, or in ways that make the work of educators more difficult. The developers of Codex note that their tool "raises significant safety challenges, does not always produce code that is aligned with user intent, and has the potential to be misused" [8]. Similarly, the developers of AlphaCode note that "like most technologies, these models might enable applications with societal harms which we need to guard against, and desire to have a positive impact is not itself a mitigation against harm" [22]. In this section we present a number of challenges presented by AI-generated code tools. Although some of these may also present opportunities, in this section we focus on their challenges. Ethical Issues Academic integrity in computing is a complex issue, particularly when software development encourages reuse of code and collaborative practices [38]. The use of auto-generated code raises significant issues with respect to academic integrity and code reuse. Academic misconduct. Prior work has shown that AI-generated code tools can achieve better than average marks on actual student exams, can perform well on both standard programming questions such as Rainfall [12], and reliably generate correct code for common algorithms such as insertion sort and tree traversal [9]. We can assume that AI-generated code tools will be capable of completing assignments that we give to students learning programming. Simon et al. [38] note that contract cheating is growing in prevalence and increasingly difficult to detect. This suggests that there is student desire to outsource graded work to others. Traditional outsourced solutions have risks that communication between student and provider may be breached, or that the solution may be shared (or reused) by the provider resulting in duplicate submissions that can be detected. AI-generated solutions vary [12], and do not require communicating with another person, producing similar results to contracted outsourcing for students with fewer inherent risks. This provides a low-risk/high-reward avenue for students focused on short-term grades rather than developing a deep understanding of content. This may exacerbate existing issues with detection of academic misconduct. Attribution. Simon et al. [38] surveyed academics about the use of attribution for code obtained from outside sources, finding a diverse range of views on the acceptability of code reuse. This academic integrity quagmire becomes more complex with relatively opaque differences between standard code completion tools present in IDEs and plugins such as Copilot that will provide code suggestions that are indistinguishable from IDE code completion. In other contexts, we use spell-checkers, grammar-checking tools that suggest rewording, predictive text and email auto-reply suggestions -all machine-generated. In a programming context, most development environments support code completion that suggests machine-generated code. Distinguishing between different forms of machine suggestions may be challenging for academics, and it is unclear if we can reasonably expect introductory programming students who are unfamiliar with tool support to distinguish between different forms of machine-generated code suggestions. If students are unable to distinguish these, then it would be unjust to treat use of machine-generated code as academic misconduct. This raises a key philosophical issue: how much content can be machine-generated while still attributing the intellectual ownership to a human? This calls into question the very concept of plagiarism [10] and how we should interpret plagiarism and intellectual contribution with machine-supported generation of content. Code reuse and licensing. There are also potential licensing issues that arise when new content is produced using code generation models, even when the model data is publicly-available [22]. Many different licenses apply to much of the publicly-available code and typically these require authors to credit the code they used, even when the code is open-source. When the use of that code comes about via an AI model, developers may end up using code that requires license compliance and not be aware that it does. 19 This is clearly an issue that extends beyond educational use of software, but as educators it is our role to inform students of their professional responsibilities when reusing code. Sustainability. The sustainability of our education practices, and in particular the impact on our environment, is an ethical issue that we must acknowledge. Training AI models can consume significant energy. Brown et al. [7] report that GPT-3/Codex required more than several thousand petaflop/s-days of computation during the pre-training process. Further, due size these models are hosted centrally and accessed remotely. At present these tools are likely not as efficient in terms of compute power and network traffic than more established web services and their environmental costs are a sustainability concern that should be known to those using them. Bias and Bad Habits The issue of bias in AI is well known [6]. In addition to general bias (subtle or overt) that applies to almost all AI-generated outputs such as only representing certain groups of people, genders, etc., there are likely biases specific to AI code generation. Appropriateness for beginners. Given that most of the code that these models are trained on is public, it is reasonable to question if the public code used for training is appropriate for students who are starting to learn programming. For example, professionals (and the over-confident) are likely more amenable to posting their code publicly. This can be used to support an argument that, despite myriad examples that public code is not very good, it is nonetheless on average of higher quality than non-public code. At least most public code is complete and is subject to public scrutiny -something that can not be said for private code. In addition to this quality bias, the code styles of public code is likely different -and possibly more advanced than that of a typical "blank slate" novice. However, these styles and approaches may not match those of the instructor. Harmful biases. The developers of Codex note found that code generation models raise bias and representation issues beyond problematic natural language -notably that Codex can generate code with structure that reflects stereotypes about gender, race, emotion, class, the structure of names, and other characteristics [8]. Additionally, Codex "can be prompted in ways that generate racist, denigratory, and otherwise harmful outputs as code comments" [8]. Security. Although largely ignored for much of the short history of computing education, the requirement for novices programmers to begin learning secure coding practices has been welldocumented in recent years [19]. Given this, the security of AIgenerated code is extremely important, even in educational settings. It has been shown that code generated by these models can be insecure [33], and human oversight is required for the safe use of AI code generation systems [8]. CodeWhisperer claims to tackle security head-on by providing the ability to run scans on code to detect security vulnerabilities [1], although this is currently untested. Chen et al. noted that although future code generation models may be able to be trained to produce more secure code than the average developer, this is far from certain [8]. Over-reliance The Codex developers noted that a key risk of using code generation models in practice is users' over-reliance on the generated outputs [8]. Novices using such models, especially with tools such as Copilot that embed support in an IDE, may quickly become accustomed to auto-suggested solutions. This may lead to students not reading problem statements carefully, or at all, and therefore not thinking about the computational steps needed to solve a problem. Reinforcing behaviors that reduce learning. An analysis of solutions generated by AlphaCode revealed that 11% of Python solutions were syntactically-incorrect (produced a SyntaxError) and 35% of C++ solutions did not compile [22]. It is not known what the average compilation rate of submitted solutions for average introductory programming students studying these languages are, however it is clear that students using AlphaCode and other AI-generated code tools would be dealing with code that has a high probability of being incorrect in some way. The Codex developers noted that it can recommend syntactically-incorrect code including variables, functions, and attributes that are undefined or outside the scope of the codebase. Chen et al. [8] observe "Codex may suggest solutions that superficially appear correct but do not actually perform the task the user intended. This could particularly affect novice programmers, and could have significant safety implications depending on the context. " If suggested code is incorrect, students may lose trust in the feedback provided by IDEs, including error messages, warnings and other auto-generated forms of feedback. CONCLUSIONS AI-generated code is now firmly part of the education landscape, but we do not yet know how to adapt our practices to overcome the challenges and leverage the benefits. What we confidently predict is that software development of the future will include an increasing amount of auto-generated code and this includes those training for such roles and jobs, such as our students. We believe this minimally suggests a shift in emphasis towards code reading and evaluating rather than code generation -a pedagogical approach consistent with the theory of instruction advanced by Xie et al. [46]. Beyond pedagogy, it also demands we examine the ethical implications of the use of these tools and that we guide our students through such ethical reflection. In a 2022 ITiCSE keynote, Titus Winters, a principal software engineer at Google, suggested it's at least as important to be an ethically-aware person as it is be a good programmer [45]. We believe AI-generated code coupled with demands from industry will force us to face ethical issues in computing education from the very beginning of the curriculum. Without quick, concerted efforts, educators will lose advantage in helping shape what opportunities come to be, and what challenges will endure. midjourney.com/home arXiv:2212.01020v1 [cs.HC] 2 Dec 2022 codeforces.com 14 aws.amazon.com/codewhisperer 15 pages.awscloud.com/codewhisperer-sign-up-form.html 16 tabnine.com 17 code4me.me 18 github.com/moyix/fauxpilot githubcopilotinvestigation.com/ Introducing Amazon CodeWhisperer, the MLpowered Coding Companion. Desai Ankur, Deo Atul, Desai Ankur and Deo Atul. 2022. 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{'abstract': 'Figure 1: An image generated by Midjourney with the prompt "robot writing computer code while student watches, computer screens, computer programming, computer code, realistic, highly detailed, cinematic -aspect 16:9" ABSTRACT The introductory programming sequence has been the focus of much research in computing education. The recent advent of several viable and freely-available AI-driven code generation tools present several immediate opportunities and challenges in this domain. In this position paper we argue that the community needs to act quickly in deciding what possible opportunities can and should be leveraged and how, while also working on how to overcome or otherwise mitigate the possible challenges. Assuming that the effectiveness and proliferation of these tools will continue to progress rapidly, without quick, deliberate, and concerted efforts, educators will lose advantage in helping shape what opportunities come to be, and what challenges will endure. With this paper we aim to seed this discussion within the computing education community.', 'arxivid': '2212.01020', 'author': ['Brett A Becker [email protected] \nUniversity College Dublin Dublin\nIreland\n', 'Paul Denny \nThe University of Auckland\nAucklandNew Zealand\n', 'James Finnie-Ansley [email protected] \nThe University of Auckland\nAucklandNew Zealand\n', 'Andrew Luxton-Reilly [email protected] \nThe University of Auckland\nAucklandNew Zealand\n', 'James Prather [email protected] \nAbilene Christian University Abilene\nTexasUSA\n', 'Eddie Antonio Santos [email protected] \nUniversity College Dublin Dublin\nIreland\n'], 'authoraffiliation': ['University College Dublin Dublin\nIreland', 'The University of Auckland\nAucklandNew Zealand', 'The University of Auckland\nAucklandNew Zealand', 'The University of Auckland\nAucklandNew Zealand', 'Abilene Christian University Abilene\nTexasUSA', 'University College Dublin Dublin\nIreland'], 'corpusid': 253202773, 'doi': '10.1145/3545945.3569759', 'github_urls': [], 'n_tokens_mistral': 14995, 'n_tokens_neox': 12712, 'n_words': 7760, 'pdfsha': 'ee49e271141ed7c25c959f7557664e855ef97250', 'pdfurls': ['https://export.arxiv.org/pdf/2212.01020v1.pdf'], 'title': ['Programming Is Hard -Or at Least It Used to Be: Educational Opportunities And Challenges of AI Code Generation', 'Programming Is Hard -Or at Least It Used to Be: Educational Opportunities And Challenges of AI Code Generation'], 'venue': []}

arxiv

Angular-dependence of magnetization switching for a multi-domain dot: experiment and simulation 27 May 2004 O Fruchart J.-C Toussaint P.-O Jubert W Wernsdorfer R Hertel J Kirschner D Mailly Max Plank Institut für Mikrostrukturphysik Laboratoire Louis Néel CNRS BP166, F-38042, D-06120Grenoble Cedex 9, HalleFrance, Germany Laboratoire de Photonique et de Nanostructures CNRS Route de NozayF-91460MarcoussisFrance Angular-dependence of magnetization switching for a multi-domain dot: experiment and simulation 27 May 2004(Dated: March 22, 2022) We have measured the in-plane angular variation of nucleation and annihilation fields of a multidomain magnetic single dot with a micro-SQUID. The dots are Fe/Mo(110) self-assembled in UHV, with sub-micron size and a hexagonal shape. The angular variations were quantitatively reproduced by micromagnetic simulations. Discontinuities in the variations are observed, and shown to result from bifurcations related to the interplay of the non-uniform magnetization state with the shape of the dot. Coherent rotation of magnetization is the simplest model of magnetization reversal, proposed by Stoner and Wohlfarth in 1948 [1]. Coherent rotation predicts the value of the switching field H swi of a single-domain system as a function of the direction of the external field H ext . For a two-dimensional system with uniaxial anisotropy the polar plot H swi (ϕ) falls on the wellknown astroid [2]. The full experimental proof for coherent rotation was given only recently, when nanoparticles of high quality and of size small enough to roughly satisfy the hypothesis of uniform magnetization could be investigated individually [3,4]. Starting from this proof, it is now a challenge to understand magnetization reversal in increasingly large (and thus complex) systems. The simplest ingredient to add to coherent rotation is to allow minor deviations from strictly homogeneous magnetization. The consequences on magnetization processes were addressed by numerical micromagnetics [5], investigated analytically [6] and checked experimentally [7]. The next step is now to tackle quantitatively more strongly non-uniform systems, those that may display magnetic domains and domain walls [8]. In such systems a switching field H swi is not the signature of the full reversal of magnetization, but instead reflects events like nucleation, propagation and annihilation [9]. Few and only partial experimental [9] and numerical [10] reports are found on this issue. A more detailed study would open the door to understanding microscopic details of magnetization reversal processes in macroscopic materials. In this Letter we present such a study in a model system: submicrometer Fe faceted dots self-assembled in UHV, that have a high structural quality and display simple multidomain states [11,12]. The angular dependence of the H swi 's of a single dot was studied with the micro-SQUID technique [13]. This can be seen as the first experimental generalization of astroids for an individual multi-domain system. A striking feature is the occurrence of disconti-nuities (hereafter named jumps) in H swi (ϕ) plots. These jumps were reproduced and understood with the help of numerical micromagnetism. They result from bifurcations, related to the interplay of the non-uniform magnetization with the shape of the dot. This also shows that a complex H swi behavior does not necessarily result from defects, whose role may have been overestimated in the literature of magnetic dots made by lithography. The Fe(110) epitaxial dots were fabricated with pulsed laser deposition in ultra-high vacuum by self-assembly on Mo(110)[8 nm]/Al 2 0 3 (1120). The dots display the shape of ingots with atomically-flat facets, bulk lattice parameter and bulk cubic magneto-crystalline anisotropy K 1 favoring < 100 > axes, however of magnitude much smaller than 1 2 µ 0 M 2 s [11]. The inter-dot dipolar fields are negligible with respect to H swi . The remanent state consists of flux-closure domains, resulting from demagnetizing fields within each dot [12]. Such domains can occur due to size of the dots being well above the exchange length Λ ex. [14,15]. The in-plane H swi 's of a single Fe dot were measured below 4 K using the micro-SQUID technique [13]. For these measurements, the dots were covered in UHV by Mo[2 nm], followed by Al[2 nm] (then 12 hours airoxidized), and a Si[2 nm]\Nb[15 nm]\Si[2 nm] tri-layer. Arrays of square micro-SQUIDs with edge 1 × 1 µm were patterned by e-beam lithography and SF 6 reactive ion etching of the tri-layer. The oxidized Al layer prevents ferromagnetic-superconductor proximity effects between the dots and the micro-SQUIDs. Although the dots are randomly distributed on the surface, their large number yields a significant probability to find one suitably coupled to a micro-SQUID. The location and shape of the single dots under investigation were checked a posteriori by AFM. The size of the dot selected here (FIG. 1a) is 420 × 200 × 30 nm (FIG. 1b). Micromagnetic simulations were performed for 0 K (no thermal activation) using custom-developed codes, either based on integrat-ing the LLG equation in a finite differences code (rectangular prisms) [16] or on energy minimization in a finite elements code (tetrahedra) [17]. The applied field was increased step-wise in hysteresis loops. In finite differences the sample was divided into cells with uniform lateral and vertical size ∆ x = ∆ y = 4.70 nm and ∆ z = 3.75 nm, respectively. For finite elements 83310 tetrahedra of irregular but similar shape were used, with a maximum (resp. minimum) volume of 42.29 nm 3 (resp. 12.50 nm 3 ). We set K 1 = 4.8 × 10 4 J.m −3 , A = 2 × 10 −11 J.m −1 and M s = 1.73 × 10 −6 A.m −1 in the calculation. In the following we call ϕ the angle between the in-plane H ext and the in-plane long axis of the dot [001] (FIG. 1b). Due to a shape effect, in-plane [110] (ϕ = 90 • ) is a magnetically-harder direction than [001]. The insets of FIG. 1c show micro-SQUID hysteresis loops for two angles (ϕ = +6; +90 • ). Such loops with negligible remanence although with significant hysteresis, are characteristic of multidomain systems with a limited number of domains. Starting from positive saturation the first H swi , named hereafter H nuc , is expected to reveal a nucleation event, e.g. the entry of a magnetic vortex [18] in the dot. The second H swi , occurring at negative fields and named H ann , is expected to reveal an annihilation event, i.e. the expulsion from the dot of a previously-nucleated vortex or wall. FIG. 1c shows the experimental angular variation H nuc (ϕ) and H ann (ϕ). The two-fold symmetry results from the elongated shape of the dot. Two striking features are observed, that shall be explained in the course of the discussion. First, jumps of both H nuc and H ann occur at some angles. Second, depending on the range of angles, one or two H nuc and/or H ann are observed. The jumps of H swi can be understood qualitatively by simple arguments. Let us sketch in a quasistatic picture the evolution of magnetization M(r) close to an edge during the first stages of a hysteresis loop (FIG. 2a-b). Starting from saturation, upon decrease of H ext the relative importance of the dipolar energy E d increases. As a result M progressively rotates towards the edge to reduce surface charges, and thus reduce E d . The direction of rotation, clockwise or anticlockwise, depends on the initial direction of M with respect to the normal to the edge (imposed by the direction of H ext ), due to the torque exerted by the dipolar field H d on M. With this picture at least two different slightly inhomogeneous magnetization states, so-called 'leaf states' [6], are expected to appear upon decrease of H ext , e.g. when starting from saturation along ϕ = 0 • or ϕ = 90 • (FIG. 2c-d). Bifurcation must occur for at least one intermediate angle between these two paths. Then, it is obvious that for H ext applied on either side of this angle, the magnetization pattern will evolve towards very different states, each characterized by a different entry point for vortices-and thus of edge orientation, explaining a jump in H nuc (ϕ). These ideas were confirmed by micromagnetic simulation. In a first attempt the simulations were performed on a dot with vertical facets and symmetric ends. FIG. 3ab shows the static magnetization states just before and after H nuc . For ϕ = 40 • , before H nuc the state belongs to the class sketched in FIG. 2c, as expected. Then two regions of strongly non-uniform magnetization develop simultaneously, ending up in the entry of two vortices at H nuc . As H ext is further decreased the two vortices with opposite circulation move towards the inner part of the dot, ending up in a diamond state (FIG. 3a). For ϕ = 50 • the state before nucleation looks like that in FIG. 2d. The loci of the entering vortices are thus modified with respect to the above situation, explaining the jump of H nuc , but ending as well in a diamond state, i.e. with two vortices (FIG. 3b). The above arguments explain the jumps of H nuc , but fail to explain (1) the existence of either one or two H nuc and H ann for some angles (FIG. 1c) (2) the experimental observation of both diamond and Landau states [12], i.e. with two or one wall or vortex. Indeed in the simulations for any ϕ two vortices appear simultaneously on opposite loci, as the dot was assumed to be perfectly symmetric. Due to the large dot size these vortices interact weakly with each other, thus both enter the dot, ending up in a diamond state. In order to refine our interpretations, we now report simulations performed on dots with a slightly asymmetric shape, similar to that of the AFM observation of the measured dot (FIG. 1a). Opposite loci are no more equivalent due to the point-reversal symmetry breaking. Two situations occur. If opposite loci are similar two vortices still enter the dot, one slightly before the other in terms of H ext , however still ending in the diamond state (e.g. for ϕ = +50 • ,FIG. 3d). If opposite loci are significantly different, then one of the vortices may enter the dot at a much higher field than the other. It then moves towards its centre, delaying and possibly preventing the entry of a second main vortex. This ends up in a Landau state (e.g. for ϕ = +40 • , FIG. 3d). Notice that the vortex may continuously change its shape into a Bloch wall, provided that the dot is long and thick enough [12,19]. Thus, an asymmetric feature (like shape) is necessary to explain the experimental observation of one-wall/vortex state [12]. Notice also that more vortices may appear depending on the detailed shape of the dot (FIG. 3e,g). FIG. 1c (upper part) shows that these simulations reproduce experimental H nuc (ϕ) convincingly. As a further step some simulations were performed in the finite-element scheme to avoid numerical roughness on the surfaces. This yielded results quantitatively similar to the case of vertical facets. Simulations of H ann (ϕ) (lower part of FIG. 1c) also reproduce experimental data. The jumps of H ann (ϕ) again result from the bifurcation before nucleation, implying different states before annihilation. The occurrence of one versus two H ann may be associated with the occurrence of different flux-closure states at low field. A more general experiment consists in proceeding to nucleation with the field decreased along a given angle ϕ 1 , followed by annihilation with the field increased along a different direction ϕ 2 . The plot H ann (ϕ 2 ), measured while keeping ϕ 1 fixed i.e. trying to prepare the system always in the same starting state, can be viewed as a signature of this state. FIG. 4a displays such plots for several values of ϕ 1 , chosen in different branches of the experimental H nuc (FIG. 4b). The plots are not identical, which confirms that the remanent state depends on the angle of H ext . The fact that different plots roughly consist of different parts of a common set of two branches is also easily understood. For a given ϕ 2 a wall or a vortex of given circulation will always be pushed towards the same locus of the dot, be it alone at remanence (vortex or Landau state) or having a companion (diamond state). In the latter case one point is found on each branch, whereas in the former case only one branch is revealed. Finally, the values of H swi depend only weakly on the algorithm used (finite differences or finite elements). However, sometimes fine differences appear upon nucleation, such as the magnetization direction of vortices' cores, or the occurrence of more than two vortices (see FIG. 3, c versus e; d versus f; g), although both codes were benchmarked successfully one against each other on time-resolved magnetization reversal issues [20]. This underlines that describing the fine details of nucleation with simulations remains a challenge and that results should still be taken with care. To conclude, we reported the measurements, quantitative reproduction and understanding of angular nucleation and annihilation fields H nuc (ϕ) and H ann (ϕ) in a multi-domain magnetic particle. This is an example of a generalization to multi-domain states of the well-known Stoner-Wohlfarth astroid. The main feature is the occurrence of jumps in both plots, which result from the interplay of a non-uniform magnetization state with the shape of the dot. Thus, such jumps should not be automatically ascribed to defects when observed in experiments. We thank Ph. David and V. Santonacci for technical support, and C. Meyer and W. Wulfhekel for critically reading the manuscript. * Electronic address: [email protected] † Present address: IBM Research -Zurich Research Laboratory, 8803. Rüschlikon, Switzerland* Electronic address: [email protected] † Present address: IBM Research -Zurich Research Labo- ratory, 8803 Rüschlikon, Switzerland . E C Stoner, E P Wohlfarth, Phil. Trans. Royal Soc. London. 240599E. C. Stoner and E. P. Wohlfarth, Phil. Trans. Royal Soc. London A240, 599 (1948). . J C Slonczewski, Research Memo RM 003.111.224IBM Research Center. unpublishedJ. C. Slonczewski, Research Memo RM 003.111.224, IBM Research Center, Poughkeepsie, NY (unpublished). . W Wernsdorfer, E B Orozco, K Hasselbach, A Benoit, B Barbara, N Demoncy, A Loiseau, H Pascard, D Mailly, Phys. Rev. Lett. 781791W. Wernsdorfer, E. B. Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, and D. Mailly, Phys. Rev. Lett. 78, 1791 (1997). . E Bonet-Orozco, W Wernsdorfer, B Barbara, A Benoit, D Mailly, A Thiaville, Phys. Rev. Lett. 834188E. Bonet-Orozco, W. Wernsdorfer, B. Barbara, A. Benoit, D. Mailly, and A. Thiaville, Phys. Rev. Lett. 83, 4188 (1999). . M A Schabes, H N Bertram, J. Appl. Phys. 641347M. A. Schabes and H. N. Bertram, J. Appl. Phys. 64, 1347 (1988). . R P Cowburn, M E Welland, Phys. Rev. B. 589217R. P. Cowburn and M. E. Welland, Phys. Rev. B 58, 9217 (1998). . R P Cowburn, A O Adeyeye, M E Welland, Phys. Rev. Lett. 815414R. P. Cowburn, A. O. Adeyeye, and M. E. Welland, Phys. Rev. Lett. 81, 5414 (1998). Magnetic domains. The analysis of magnetic microstructures. A Hubert, R Schäfer, SpringerBerlinA. Hubert and R. Schäfer, Magnetic domains. The analy- sis of magnetic microstructures (Springer, Berlin, 1999). 3: (a-d) Mid-height views of micromagnetic finite differences simulations of dots with vertical facets. for ϕ =FIG. 3: (a-d) Mid-height views of micromagnetic finite dif- ferences simulations of dots with vertical facets, for ϕ = Hext is decreased from positive saturation towards the flux-closure state. Nucleation loci are indicated with full dots (e-f) Flux-closure state obtained on the same element with finite elements simulations (surface views) (g) Flux-closure states for ϕ = −40 • , for finite differences (left) and finite elements (right). The color codes the perpendicular component of magnetization. • (left) and ϕ = 50 • (right). for (a-b) symmetric and (cd) asymmetric in-plane shape. see scale• (left) and ϕ = 50 • (right), for (a-b) symmetric and (c- d) asymmetric in-plane shape. Hext is decreased from positive saturation towards the flux-closure state. Nucleation loci are indicated with full dots (e-f) Flux-closure state obtained on the same element with finite elements simulations (surface views) (g) Flux-closure states for ϕ = −40 • , for finite differ- ences (left) and finite elements (right). The color codes the perpendicular component of magnetization (see scale). . W Wernsdorfer, K Hasselbach, A Sulpice, A Benoit, J.-E Wegrowe, L Thomas, B Barbara, D Mailly, Phys. Rev. B. 533341W. Wernsdorfer, K. Hasselbach, A. Sulpice, A. Benoit, J.-E. Wegrowe, L. Thomas, B. Barbara, and D. Mailly, Phys. Rev. B 53, 3341 (1996). . Z H Wei, C R Chang, N A Usov, M F Lai, J C Wu, J. Magn. Magn. Mater. 2391Z. H. Wei, C. R. Chang, N. A. Usov, M. F. Lai, and J. C. Wu, J. Magn. Magn. Mater. 239, 1 (2002). . P.-O Jubert, O Fruchart, C Meyer, Phys. Rev. B. 64115419P.-O. Jubert, O. Fruchart, and C. Meyer, Phys. Rev. B 64, 115419 (2001). . P O Jubert, J C Toussaint, O Fruchart, C Meyer, Y Samson, Europhys. Lett. 63135P. O. Jubert, J. C. Toussaint, O. Fruchart, C. Meyer, and Y. Samson, Europhys. Lett. 63, 135 (2003). W Wernsdorfer, Advances in Chemical Physics. I. Prigogine and S. A. RiceWiley118W. Wernsdorfer, in Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice (Wiley, 2001), Vol. 118. . E H Frei, S Shtrikman, D Treves, Phys. Rev. 106446E. H. Frei, S. Shtrikman, and D. Treves, Phys. Rev. 106, 446 (1957). . H Kronmüller, Z. Physik. 168478H. Kronmüller, Z. Physik 168, 478 (1962). . J C Toussaint, A Marty, N Vukadinovic, J Ben Youssef, M Labrune, Comput. Mater. Sci. 24175J. C. Toussaint, A. Marty, N. Vukadinovic, J. Ben Youssef, and M. Labrune, Comput. Mater. Sci. 24, 175 (2002). . R Hertel, J. Appl. Phys. 905752R. Hertel, J. Appl. Phys. 90, 5752 (2001). . K Y Guslienko, K L Metlov, Phys. Rev. B. 63100403K. Y. Guslienko and K. L. Metlov, Phys. Rev. B 63, 100403(R) (2001). . R Hertel, H Kronmüller, Phys. Rev. B. 607366R. Hertel and H. Kronmüller, Phys. Rev. B 60, 7366 (1999). Polar plots Hann(ϕ2) starting from different zerofield states, each state being prepared by initial saturation along a given direction ϕ1, shown in (b) with respect to the Hnuc(ϕ) plot (c) Polar plot of Hann(ϕ) for radial field sweeping. same data as on FIG. 2)FIG. 4: (a) Polar plots Hann(ϕ2) starting from different zero- field states, each state being prepared by initial saturation along a given direction ϕ1, shown in (b) with respect to the Hnuc(ϕ) plot (c) Polar plot of Hann(ϕ) for radial field sweep- ing (same data as on FIG. 2).

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{'abstract': 'We have measured the in-plane angular variation of nucleation and annihilation fields of a multidomain magnetic single dot with a micro-SQUID. The dots are Fe/Mo(110) self-assembled in UHV, with sub-micron size and a hexagonal shape. The angular variations were quantitatively reproduced by micromagnetic simulations. Discontinuities in the variations are observed, and shown to result from bifurcations related to the interplay of the non-uniform magnetization state with the shape of the dot.', 'arxivid': 'cond-mat/0405651', 'author': ['O Fruchart ', 'J.-C Toussaint ', 'P.-O Jubert ', 'W Wernsdorfer ', 'R Hertel ', 'J Kirschner ', 'D Mailly ', '\nMax Plank Institut für Mikrostrukturphysik\nLaboratoire Louis Néel\nCNRS\nBP166, F-38042, D-06120Grenoble Cedex 9, HalleFrance, Germany\n', '\nLaboratoire de Photonique et de Nanostructures\nCNRS\nRoute de NozayF-91460MarcoussisFrance\n'], 'authoraffiliation': ['Max Plank Institut für Mikrostrukturphysik\nLaboratoire Louis Néel\nCNRS\nBP166, F-38042, D-06120Grenoble Cedex 9, HalleFrance, Germany', 'Laboratoire de Photonique et de Nanostructures\nCNRS\nRoute de NozayF-91460MarcoussisFrance'], 'corpusid': 119434683, 'doi': '10.1103/physrevb.70.172409', 'github_urls': [], 'n_tokens_mistral': 5784, 'n_tokens_neox': 4921, 'n_words': 3053, 'pdfsha': '64cc574ae3b8c66436df974dbd7387fa387c6525', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0405651v1.pdf'], 'title': ['Angular-dependence of magnetization switching for a multi-domain dot: experiment and simulation', 'Angular-dependence of magnetization switching for a multi-domain dot: experiment and simulation'], 'venue': []}

arxiv

An Optimal Stochastic Algorithm for Decentralized Nonconvex Finite-sum Optimization 19 Nov 2022 Luo Luo Haishan Ye An Optimal Stochastic Algorithm for Decentralized Nonconvex Finite-sum Optimization 19 Nov 2022 This paper studies the decentralized nonconvex optimization problem min x∈R d f (x)is the local function on the i-th agent of the network. We propose a novel stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST), which integrates the techniques of variance reduction, gradient tracking and multi-consensus. We construct a Lyapunov function that simultaneously characterizes the function value, the gradient estimation error and the consensus error for the convergence analysis. Based on this measure, we provide a concise proof to show DEAREST requires at most O(mn + √ mnLε −2 ) incremental first-order oracle (IFO) calls and O(Lε −2 / 1 − λ2(W ) ) communication rounds to find an ε-stationary point in expectation, where L is the smoothness parameter and λ2(W ) is the second-largest eigenvalue of the gossip matrix W . We can verify both of the IFO complexity and communication complexity match the lower bounds. To the best of our knowledge, DEAREST is the first optimal algorithm for decentralized nonconvex finite-sum optimization. Introduction We study decentralized optimization problem min x∈R d f (x) 1 m m i=1 f i (x),(1) where m agents form a connected network and each f i (x) is the local function on the i-th agent that can be written as the average of n individual functions as follows f i (x) 1 n n j=1 f i,j (x).(2) We suppose each individual function f i,j (x) is smooth but possibly nonconvex. This formulation includes a lot of applications in statistics, signal processing and machine learning [3,5,10,24,33,34,36,39]. In decentralized scenario, all of agents target to collaboratively minimize the global function f (x) and each of them can only communicate with its neighbours. This paper focus on designing the efficient algorithm to find an expected ε-stationary point x out ∈ R d such that E ∇f (x out ) ≤ ε. Incremental first-order oracle (IFO) algorithms [1,2,9,11,13,15,19,25,26,29,30,44,45] are the most popular methods to solve the optimization problem with finite-sum objective. They evaluate the first-order information by accessing the individual functions to reduce the computational cost. For the optimization on single machine, the algorithms with stochastic recursive gradient estimator [11,19,25,26] achieve the optimal IFO complexity of O N + √ N Lε −2 , where N is the total number of individual functions and L is the average-smoothness parameter. In decentralized setting, the topology of the network leads to the local machine cannot access the global information at each round and the communication cost could be expensive. The gradient tracking [23,27,28] is a useful technique to track the average of local gradients and make the first-order information accurate. Ye et al. [40,41,42] showed that combining the gradient tracking with multi-consensus [4,17] encourages the analysis of decentralized algorithm be close to the centralize counterpart and it is useful to decentralized convex optimization. Table 1: We compare the order of upper bound complexities for our algorithm with related work. Here we use λ 2 to abbreviate the second-largest eigenvalue of gossip matrix λ 2 (W ). The column entitled with "#IFO" presents the IFO complexity on per-agent. Algorithms #IFO #Communication D-GET [32] n + √ nLε −2 (1 − λ 2 ) 2 same as #IFO GT-SARAH [38] n + 1 (1 − λ 2 ) 2 + n m + (n/m + 1) 1/3 1 − λ 2 Lε −2 same as #IFO DESTRESS [18] n + n m Lε −2 √ mn + Lε −2 √ 1 − λ 2 DEAREST (this work) n + n m Lε −2 Lε −2 √ 1 − λ 2 Directly extending (stochastic) gradient descent methods to decentralized setting [7,8,20,22,31,35,43] does not take the advantage of finite-sum structure in our local function (2). Sun et al. [32] first applied variance reduction and gradient tracking to decentralized nonconvex finite-sum optimization and proposed the algorithm called Decentralized Gradient Estimation and Tracking (D-GET). Later, Xin et al. [38] proposed GT-SARAH, which improved the complexity of D-GET in terms of the dependency on m and n. Li et al. [18] further improved the result of GT-SARAH by proposing DEcentralized STochastic REcurSive gradient methodS (DESTRESS), which requires O n + n/mLε −2 per-agent IFO calls and O ( √ mn + Lε −2 )/ 1 − λ 2 (W ) rounds of communication to obtain an ε-stationary point in expectation, where λ 2 (W ) is the second-largest eigenvalues of the gossip matrix W . Since the formulations (1) and (2) mean there are mn individual functions, the total IFO upper bound of DESTRESS for all agents is O mn + √ mn Lε −2 , matching the best result for single machine optimization and it is optimal. However, the lower bound of communication complexity for finding an ε-stationary point of smoothed nonconvex function by first-order algorithm [22,31] is Ω Lε −2 / 1 − λ 2 (W ) . The additional term related to √ mn in the communication complexity of DESTRESS means the algorithm could be communication inefficient when mn is extremely large. In this paper, we proposed a novel algorithm named as DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST), which simultaneously matches the IFO lower bound [11,19] and communication lower bound [22,31] for decentralized nonconvex finite-sum optimization. We compare the theoretical guarantees of this paper and related work in Table 1. The design of DEAREST is based on the framework of loopless variance reduced iteration [12,14,16,19,41], gradient tracking [23,27,28] and multi-consensus [21]. We construct a Lyapunov function to characterize the function value, the global gradient estimation error, the local gradient estimation error and the consensus error. By verifying this measure is non-increasing, we provide a concise analysis to show the tightness for the IFO upper bound and the communication upper bound of DEAREST. To the best of our knowledge, DEAREST is the first optimal algorithm for decentralized nonconvex finite-sum optimization. Notations and Assumptions We use · to denote the Frobenius norm of a matrix or the ℓ 2 -norm of a vector. We let 1 = [1, · · · , 1] ⊤ ∈ R m and denote I as the identity matrix. We define aggregate variables for all agents as x =    x(1) ⊤ . . . x(m) ⊤    ∈ R m×d , where each x(i) ∈ R d are the local variable on the i-th agent. We use the lower case with the bar to represent the mean vector, such thatx = 1 m m i=1 x(i) ∈ R d . We also introduce the matrix of aggregate gradients as ∇f (x) =    ∇f 1 (x(1)) ⊤ . . . ∇f m (x(m)) ⊤    ∈ R m×d . We use the gossip (mixing) matrix W ∈ R m×m to characterize the behavior of agents updating local variables by weighted sum of information form the neighbors. We suppose the matrix W satisfies the following assumption. Assumption 1. We suppose the gossip (mixing) matrix W = [w ij ] ∈ R m×m is symmetric and satisfies w ij = 0 if the i-th agent and the j-th agent are not connected. Furthermore, we suppose it holds that W 1 = 1. We assume the objective function in (1) and the local function (2) satisfy the following assumptions. Assumption 2. We suppose f (x) is lower bounded. That is, we have f * = inf x∈R d f (x) > −∞. Assumption 3. We suppose the individual functions {f i,j } n j=1 on each agent are L-average smooth for some L > 0. That is, we have 1 n n j=1 ∇f i,j (x) − ∇f i,j (x ′ ) 2 ≤ L 2 x − x ′ 2 for any x and x ′ ∈ R d . The Algorithm and Main Results We propose DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST) in Algorithm 1. It uses the subroutine FastMix (Algorithm 2) and gradient tracking (line 11) to reduce the consensus error. The update rules of local variables can be written as x t+1 = FastMix(x t − ηs t , K t ), s t+1 = FastMix(s t + g t+1 − g t , K t ),(3) which has the following properties. 1 m 1 ⊤ u (K) =ū (0) and u (K) − 1ū (0) ≤ 1 − 1 − λ 2 (W ) K u (0) − 1ū (0) ,whereū (0) = 1 m 1 ⊤ u (0) . Using Lemma 1, we can analyze the update (3) by focusing on the relationship of the corresponding mean vectors, that isx t+1 = 1 m 1 ⊤ FastMix(x t − ηs t , K t ) = 1 m 1 ⊤ (x t − ηs t ) =x t − ηs t =x t − ηḡ t .(4) Algorithm 1 DEAREST 1: Input: initial parameterx 0 ∈ R d , stepsize η > 0, probability p ∈ (0, 1], mini-batch size b, numbers of communication rounds K in , K andK. 2: x 0 = 1x 0 , g 0 = ∇f (x 0 ) 3: s 0 = FastMix(g 0 , K in ) 4: for t = 0, . . . , T − 1 do 5: y t ∼ Bernoulli(p) 6: K t = K, if y t = 1, K, otherwise, 7: x t+1 = FastMix(x t − ηs t , K t ) 8 : parallel for i = 1, . . . , m do 9: g t+1 (i) =        ∇f i (x t+1 (i)), if y t = 1, g t (i) + 1 b b j=1 ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) , otherwise, where each ξ i,ξj is uniformly and independently sampled from {1, . . . , n} 10: end parallel for 11: s t+1 = FastMix(s t + g t+1 − g t , K t ) 12: end for 13: Output: x out by uniformly sampling from {x t (i) : t = 0, . . . , T − 1; i = 1, . . . , m} Algorithm 2 FastMix(u (0) , K) 1: Initialize: u (−1) = u (0) , η u = 1− √ 1−λ 2 2 (W ) 1+ √ 1−λ 2 2 (W ) . 2: for k = 0, 1, . . . , K do 3: u (k+1) = (1 + η u )W u (k) − η u u (k−1) 4: end for 5: Output: u (K) . The equation (4) can be viewed as the step of inexact gradient descent on the mean variablex t with gradient estimators t and stepsize η. Additionally, Lemma 2 leads to each x t (i) and g t (i) be close tox t andḡ t respectively. The construction of gradient estimator g t (i) follows the probabilistic recursive way [19]. Intuitively, it estimates the gradient recursively by using the mini-batch gradient with high probability 1 − p and computing the exact gradient with low probability p, where p = Ω(1/ √ mn ). By taking the mini-batch size b = O n/m , the stepsize η = Θ(1/L) and the iteration number T = O Lε −2 , the IFO complexity on per-agent is at most O n + n/m Lε −2 in expectation. Note that each agent uses the same random seed to generate the Bernoulli distributed variable y t , which enforces all agents always share the identical y t . The communication rounds K t at each iteration is also decided by the random variable y t . Concretely, we let K t = K = Θ log(n/m)/ 1 − λ 2 (W ) with probability p and K t =K = O 1/ 1 − λ 2 (W ) with probability 1 − p. Combining with the initial communication rounds K in = O log(1/(mε))/ 1 − λ 2 (W ) , the total communication complexity is O Lε −2 / 1 − λ 2 (W ) in expectation. Based on above settings, we guarantee the output of DEAREST is an ε-stationary point in expectation. We formally describe our main result in Theorem 1. Theorem 1. Under Assumption 1, 2 and 3, we run Algorithm 1 with η = 1 2L , b = 6 n m , p = b b + n , T = 16L(f (x 0 ) − f * ) ε 2 , K = max 12, log(( √ mn + 6)/(24m)) 2 1 − λ 2 (W ) ,K = 6 1 − λ 2 (W ) and K in = log g 0 − 1ḡ 0 2 /(mε 2 ) 1 − λ 2 (W ) . Then the output x out satisfies E ∇f (x out ) ≤ ε. Furthermore, the algorithm requires at most O n + n/m Lε −2 IFO calls on per-agent and O Lε −2 / 1 − λ 2 (W ) rounds of communication in expectation. Now we give a brief sketch for the proof of Theorem 1 and the details are shown in Section 4. We first introduce the following quantities: • the global gradient estimation error: U t = 1 m m i=1 (g t (i) − ∇f i (x t (i))) 2 ; • the local gradient estimation error: V t = 1 m E g t − ∇f (x t ) 2 ; • the consensus error: C t = x t − 1x t 2 + η 2 s t − 1s t 2 . Then we define the Lyapunov function: Φ t f (x t ) + η p U t + η mp V t + 1 mη C t . By establishing the recursion for each terms of Φ t , we can achieve η 2 E ∇f (x t ) 2 ≤ E Φ t − Φ t+1 − 1 mη C t .(5) Taking the average on (5) over t = 0, . . . , T − 1, we have η 2T T t=0 E ∇f (x t ) 2 ≤ E Φ 0 − Φ T T − 1 mηT T t=0 C t .(6) On the other hand, the multi-consensus steps (FastMix) in DEAREST guarantee the difference between E ∇f (x out ) 2 and 1 T T −1 t=0 E ∇f (x t ) 2 could be controlled by the term related to T t=0 C t in (6). As a result, we can prove the output x out is an ε-stationary point in expectation by taking η = Ω(1/L) and T = O(Lε −2 ). Complexity Analysis We first provide some useful lemmas. Lemma 3 ([40, Lemma 3]). For any z ∈ R m×d , we have z − 1z ≤ z wherez = 1 m 1 ⊤ z. Lemma 4. Under Assumption 3, we have ∇f (x) − ∇f (x ′ ) ≤ L x − x ′ for any x, x ′ ∈ R m×d . Proof. Assumption 3 means each f i (·) is L-Lipschitz continuous. Then we have ∇f (x) − ∇f (x ′ ) = m i=1 ∇f i (x(i)) − ∇f i (x ′ (i)) 2 ≤L 2 m i=1 x(i) − x ′ (i) 2 =L 2 x − x ′ 2 . We describe the decease of function value in following lemma. Lemma 5. Algorithm 1 holds that f (x t+1 ) ≤ f (x t ) − η 2 ∇f (x t ) 2 + ηU t + L 2 η m C t − 1 2η − L 2 x t+1 −x t 2 .(7) Proof. Using Lemma 2 of Li et al. [19] and the relation (4), we have f (x t+1 ) ≤ f (x t ) − η 2 ∇f (x t ) 2 − 1 2η − L 2 x t+1 −x t 2 + η 2 s t − ∇f (x t ) 2 .(8) We bound the last term of (8) as follows s t − ∇f (x t ) 2 = 1 m m i=1 g t (i) − ∇f i (x t ) 2 ≤2 1 m m i=1 g t (i) − ∇f i (x t (i)) 2 + 2 1 m m i=1 ∇f i (x t (i)) − ∇f i (x t ) 2 ≤2 1 m m i=1 g t (i) − ∇f i (x t (i)) 2 + 2 m m i=1 ∇f i (x t (i)) − ∇f i (x t ) 2 ≤2 1 m m i=1 g t (i) − ∇f i (x t (i)) 2 + 2L 2 m m i=1 x t (i) −x t 2 =2 1 m m i=1 g t (i) − ∇f i (x t (i)) 2 + 2L 2 m x t − 1x t 2 ≤2U t + 2L 2 m C t ,(9) where the equality uses Lemma 1; the first inequality uses Young's inequality; the second inequality uses the fact 1 m m i=1 a i 2 ≤ 1 m m i=1 a i 2 for a 1 , . . . , a m ∈ R d ; the third inequality is based on the Assumption 3; the last step is based on the definitions of U t and C t . We finish the proof by combining the results of (8) and (9). We define the variables ρ t = 1− 1 − λ 2 (W ) Kt , ρ = 1− 1 − λ 2 (W ) K andρ = 1− 1 − λ 2 (W ) K to characterize the effect of FastMix (Algorithm 2). Note that the setting of K t in Theorem 1 means ρ ≤ ρ t ≤ρ < 1 √ 296 .(10) Then we provide the recursion of consensus error as follows. Lemma 6. Under the setting of Theorem 1, we have E [C t+1 ] ≤ E 1 4 C t + 4ρ 2 pη 2 mV t + 6 ρ 2 (1 − p) b + 2ρ 2 p m x t+1 −x t 2 . Proof. The relation of (4) means x t+1 − 1x t+1 = FastMix(x t − ηs t , K t ) − 1 m 11 ⊤ FastMix(x t − ηs t , K t ) ≤ρ t (x t − ηs t ) − 1 m 11 ⊤ (x t − ηs t ) =ρ t x t − ηs t − 1(x t − ηs t ) ≤ρ t x t − 1x t + η s t − 1s t , where the first inequality is based on Lemma 2 and the last step is due to triangle inequality. Consequently, we use Young's inequality to obtain x t+1 − 1x t+1 2 ≤ 2ρ 2 t x t − 1x t 2 + 2ρ 2 t η 2 s t − 1s t 2 = 2ρ 2 t C t ≤ 2ρ 2 C t .(11) The update of g t+1 (i) means E ρ 2 t g t+1 (i) − g t (i) 2 ≤ρ 2 pE ∇f i (x t+1 (i)) − g t (i) 2 +ρ 2 (1 − p) b E ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) 2 ≤2ρ 2 pE ∇f i (x t+1 (i)) − ∇f i (x t (i)) 2 + 2ρ 2 pE ∇f i (x t (i)) − g t (i) 2 +ρ 2 (1 − p)L 2 b E x t+1 (i) − x t (i) 2 ≤2ρ 2 pL 2 E x t+1 (i) − x t (i) 2 + 2ρ 2 pE ∇f i (x t (i)) − g t (i) 2 +ρ 2 (1 − p)L 2 b E x t+1 (i) − x t (i) 2 =2ρ 2 pE ∇f i (x t (i)) − g t (i) 2 + ρ 2 (1 − p) b + 2ρ 2 p L 2 E x t+1 (i) − x t (i) 2 ,(12) where the first inequality is based on the update rule; the second inequality is based on Young's inequality and the last inequality is due to Assumption 3 and (10). Summing over (12) above result over i = 1, . . . , m, we obtain E ρ 2 t g t+1 − g t 2 ≤2ρ 2 pE ∇f (x t ) − g t 2 + ρ 2 (1 − p) b + 2ρ 2 p L 2 E x t+1 − x t 2 ≤2ρ 2 pE ∇f (x t ) − g t 2 + 3 ρ 2 (1 − p) b + 2ρ 2 p L 2 E x t+1 − 1x t+1 2 + 1x t+1 − 1x t 2 + x t − 1x t 2 ≤2ρ 2 pE ∇f (x t ) − g t 2 + 3 ρ 2 (1 − p) b + 2ρ 2 p L 2 2ρ 2 x t − 1x t 2 + η 2 s t − 1s t 2 + 3 ρ 2 (1 − p) b + 2ρ 2 p mE x t+1 −x t 2 + E x t − 1x t 2 =2ρ 2 pE ∇f (x t ) − g t 2 + 9 ρ 2 (1 − p) b + 2ρ 2 p L 2 x t − 1x t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p L 2 η 2 s t − 1s t 2 + 3 ρ 2 (1 − p) b + 2ρ 2 p L 2 mE x t+1 −x t 2 ,(13) where the second inequality is based on Young's inequality and the third inequality uses the result of (11). Using Lemma 2, we have s t+1 − 1s t+1 2 = FastMix(s t + g t+1 − g t , K) − 1 m 11 ⊤ FastMix(s t + g t+1 − g t , K) 2 ≤ρ 2 t s t + g t+1 − g t − 1 m 11 ⊤ (s t + g t+1 − g t ) 2 ≤2ρ 2 t s t − 1s t 2 + 2ρ 2 t g t+1 − g t − 1 m 11 ⊤ (g t+1 − g t ) 2 ≤2ρ 2 t s t − 1s t 2 + 2ρ 2 t g t+1 − g t 2 ,(14) where the second inequality is based on triangle inequality and the last step uses Lemma 3. Combining the results of (13) and (14), we have E s t+1 − 1s t+1 2 ≤E 2ρ 2 t s t − 1s t 2 + 4ρ 2 p ∇f (x t ) − g t 2 + 18 ρ 2 (1 − p) b + 2ρ 2 p L 2 x t − 1x t 2 + 12 ρ 2 (1 − p) b + 2ρ 2 p L 2 η 2 s t − 1s t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p L 2 m x t+1 −x t 2 ≤E 18 ρ 2 (1 − p) b + 2ρ 2 p L 2 x t − 1x t 2 + 2 6ρ 2 (1 − p) b + 12ρ 2 p + ρ 2 t s t − 1s t 2 + 4ρ 2 p ∇f (x t ) − g t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p L 2 m x t+1 −x t 2 ≤E 18 ρ 2 (1 − p) b + 2ρ 2 p L 2 x t − 1x t 2 + 38ρ 2 s t − 1s t 2 + 4ρ 2 pE ∇f (x t ) − g t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p L 2 m x t+1 −x t 2 .(15) We introduce the vector z t = [ x t − 1x t 2 ; η 2 s t − 1s t 2 ]. Connecting inequalities (11), (15) and the definitions of ρ andρ, we achieve E[z t+1 ] ≤2ρ 2   1 1 9 1 − p b + 2p 25   E[z t ] + E   0 4ρ 2 pη 2 E ∇f (x t ) − g t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p m x t+1 −x t 2   ≤ 1 4 E[z t ] + E   0 4ρ 2 pη 2 E ∇f (x t ) − g t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p m x t+1 −x t 2   , where we use 9 1 − p b + 2p ≤ 27, 1 1 27 25 2 ≤ 37 and (10). Hence, we have C t+1 ≤E 1 4 x t − 1x t 2 + η 2 s t − 1s t 2 + 4ρ 2 pη 2 E ∇f (x t ) − g t 2 + 6 ρ 2 (1 − p) b + 2ρ 2 p mE x t+1 −x t 2 =E 1 4 C t + 4ρ 2 pη 2 mV t + 6 ρ 2 (1 − p) b + 2ρ 2 p m x t+1 −x t 2 . Now we provide the recursion for local and global error of gradient estimation. Lemma 7. Under the setting of Theorem 1, we have V t+1 ≤ E (1 − p)V t + 9(1 − p)L 2 mb C t + 3(1 − p)L 2 b x t+1 −x t 2 .(16) Proof. The update of g t+1 (i) means E g t+1 (i) − ∇f i (x t+1 (i)) 2 =pE ∇f i (x t+1 (i)) − ∇f i (x t+1 (i)) 2 + (1 − p)E g t (i) + 1 b b j=1 ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) − ∇f i (x t+1 (i)) 2 =(1 − p)E g t (i) − ∇f i (x t (i)) 2 + (1 − p)E 1 b b j=1 ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) − ∇f i (x t+1 (i)) + ∇f i (x t (i)) 2 ≤(1 − p)E g t (i) − ∇f i (x t (i)) 2 + 1 − p b E ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) 2 ≤(1 − p)E g t (i) − ∇f i (x t (i)) 2 + (1 − p)L 2 b E x t+1 (i) − x t (i) 2 , where the first equality is based on the update rule; the second equality uses the property of Martingale [11,Proposition 1]; the first inequality use the property of variance and independence of ξ 1 , . . . , ξ b ; the last step is based on Assumption 3. Taking the average over on above result over i = 1, . . . , m, we obtain E[V t+1 ] = 1 m E g t+1 − ∇f (x t+1 ) 2 ≤ 1 − p m E g t − ∇f (x t ) 2 + (1 − p)L 2 mb x t+1 − x t 2 ≤ 1 − p m E g t − ∇f (x t ) 2 + 3(1 − p)L 2 mb E x t+1 − 1x t+1 2 + E 1x t+1 − 1x t 2 + E x t − 1x t 2 ≤(1 − p)E V t + 9(1 − p)L 2 mb C t + 3(1 − p)L 2 b x t+1 −x t 2 , where the last step use inequality (11). Lemma 8. Under the setting of Theorem 1, we have E[U t+1 ] ≤ E (1 − p)U t + 9(1 − p)L 2 m 2 b C t + 3(1 − p)L 2 mb x t+1 −x t 2 . Proof. The update of g t+1 (i) means E[U t+1 ] =pE 1 m m i=1 (∇f i (x t+1 (i)) − ∇f i (x t+1 (i))) 2 + (1 − p)E 1 m m i=1   g t (i) + 1 b b j=1 ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) − ∇f i (x t+1 (i))   2 =(1 − p)E 1 m m i=1 (g t (i) − ∇f i (x t (i))) 2 + (1 − p)E 1 mb m i=1 b j=1 ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) − ∇f i (x t+1 (i)) + ∇f i (x t (i)) 2 ≤(1 − p)E[U t ] + 1 − p m 2 b 2 m i=1 b j=1 E ∇f i,ξj (x t+1 (i)) − ∇f i,ξj (x t (i)) 2 ≤(1 − p)E[U t ] + (1 − p)L 2 m 2 b 2 m i=1 b j=1 E x t+1 (i) − x t (i) 2 =(1 − p)E[U t ] + (1 − p)L 2 m 2 b E x t+1 − x t 2 ≤(1 − p)E[U t ] + 3(1 − p)L 2 m 2 b E x t+1 − 1x t+1 2 + 1x t+1 − 1x t 2 + x t − 1x t 2 ≤(1 − p)E U t + 6ρ 2 (1 − p)L 2 m 2 b C t + 3(1 − p)L 2 mb x t+1 −x t 2 + 3(1 − p)L 2 m 2 b x t − 1x t 2 ≤(1 − p)E U t + 9(1 − p)L 2 m 2 b C t + 3(1 − p)L 2 mb x t+1 −x t 2 , where the second equality use the property of Martingale [11,Proposition 1]; the first inequality is based on the property of variance; the second inequality is based on Assumption 3; the third inequality use Young's inequality; the last two steps use the upper bound (11) and the definition of C t . Finally, we give the proof of Theorem 1. Proof. Combining results of Lemma 5, 6, 7 and 8, we have E[Φ t+1 ] =E f (x t+1 ) + η p U t+1 + η mp V t+1 + 1 mη C t+1 ≤E f (x t ) − η 2 ∇f (x t ) 2 + ηU t + L 2 η m C t − 1 2η − L 2 x t+1 −x t 2 + η(1 − p) p U t + 9(1 − p)ηL 2 m 2 bp C t + 3(1 − p)L 2 η mbp x t+1 −x t 2 + (1 − p)η mp V t + 9(1 − p)L 2 η m 2 bp C t + 3(1 − p)L 2 η mbp x t+1 −x t 2 + 1 4mη C t + 4ρ 2 pη 2 η V t + 6 η ρ 2 (1 − p) b + 3ρ 2 p x t+1 −x t 2 =E f (x t ) − η 2 ∇f (x t ) 2 + η + η(1 − p) p U t + (1 − p)η mp + 4ρ 2 pη V t + L 2 η m + 9(1 − p)ηL 2 m 2 bp + 9(1 − p)L 2 η m 2 bp + 1 4mη C t − 1 2η − L 2 − 3(1 − p)L 2 η mbp − 3(1 − p)L 2 η mbp − 6 η ρ 2 (1 − p) b + 3ρ 2 p x t+1 −x t 2 ≤E Φ t − η 2 ∇f (x t ) 2 − 3 8mη C t , where the last step is based on the settings of η, p, b, K andK in the theorem. Taking the average on above inequality over t = 0, . . . , T − 1, we have E 1 T T −1 t=0 ∇f (x t ) 2 ≤ 2 ηT T −1 t=0 E Φ t − Φ t+1 − 3 8mη C t ≤ 2 ηT T −1 t=0 E Φ t − Φ t+1 − 3 8mη x t − 1x t 2 = 2E[Φ 0 − Φ T ] ηT − 3 4mη 2 T T −1 t=0 x t − 1x t 2 .(17) Based on Assumption 2, we have Φ 0 − Φ T =f (x 0 ) + η p U 0 + η mp V 0 + 1 mη C 0 − f (x T ) + η p U T + η mp V T + 1 mη C T =f (x 0 ) − f (x T ) + η p 1 m m i=1 (g 0 (i) − ∇f i (x 0 (i))) 2 + η mp g 0 − ∇f (x 0 ) 2 + η m s 0 − 1s 0 2 ≤f (x 0 ) − f * + η m s 0 − 1s 0 2 .(18) Connecting the inequalities (17) and (18), we have E 1 T T −1 t=0 ∇f (x t ) 2 ≤ 2(f (x 0 ) − f * ) ηT + 2 mT s 0 − 1s 0 2 − 2 mη 2 T T −1 t=0 E x t − 1x t 2 .(19) Hence, the output x out satisfies that E ∇f (x out ) 2 = 1 mT m i=1 T −1 t=0 E ∇f (x t (i)) 2 ≤ 2 mT m i=1 T −1 t=0 E ∇f (x t ) 2 + ∇f (x t (i)) − ∇f (x t ) 2 ≤ 2 mT m i=1 T −1 t=0 E ∇f (x t ) 2 + L 2 x t (i) −x t 2 = 2 T T −1 t=0 E ∇f (x t ) 2 + 2L 2 mT T −1 t=0 E x t − 1x t 2 ≤ 4(f (x 0 ) − f * ) ηT + 4 mT s 0 − 1s 0 2 − 6L 2 mT − 2L 2 mT x t − 1x t 2 ≤ 4(f (x 0 ) − f * ) ηT + 4 m s 0 − 1s 0 2 ≤ ε 2 ,(20) where the first step uses Young's inequality; the second inequality is based on Assumption 3; the third inequality uses inequality (19); the other steps are based on the parameter settings. Using Jensen's inequality, we obtain E ∇f (x out ) ≤ E ∇f (x out ) 2 ≤ ε. Furthermore, the total expected IFO calls on per-agent is O n + T (pn + (1 − p)b) = O n + n m Lε −2 and the total expected number of communication rounds is O K in + T (pK + (1 − p)K) = O Lε −2 1 − λ 2 (W ) . Remark 1. The terms of x t+1 −x t 2 are crucial to making the analysis be concise. In the proof of Theorem 1, we take the advantage of the negative coefficient before x t+1 −x t 2 in (7) to cancel the term x t+1 −x t 2 appears in the recursion of C t , U t and V t , which directly leads to the decrease of Φ t . The Optimality We first consider the IFO complexity. Theorem 1 means the total IFO complexity of DEAREST for all of m agents is O mn + √ mn Lε −2 , which matches the lower bound of finding ε-stationary point in nonconvex finite-sum optimization with mn individual functions by IFO algorithm on single machine [11,19]. Since there are mn individual functions in our settings (1)-(2), the IFO complexity of DEAREST is optimal. Then we consider the communication complexity. Lu et al. [22] study the lower bound for solving decentralized nonconvex problem by using unbiased stochastic first-order oracle (SFO) with bounded variance. Concretely, they considered the decentralized optimization problem E[g i (x)] = ∇f i (x) and E ∇f i (x) −g i (x) 2 ≤ σ 2(21) for all x ∈ R d and gossip protocol associate withŴ requires at least Ω   Lσ 2 ε −4 m + Lε −2 1 − λ 2 (Ŵ )  (22) Related Work and Discussion We summarize the results of our algorithm and related work in Table 1. The proposed algorithm DEAREST is the only one with both optimal IFO complexity and optimal communication complexity. D-GET [32] and GT-SARAH [37] are also based on variance reduction and gradient tracking but they only take single consensus step when communication takes place. DESTRESS includes the multi-consensus steps like DEAREST while its procedure has double-loop structure, leading to the unnecessary complicated implementation and more communication cost. In contrast, our DEAREST enjoys the single-loop framework, which results to the easier implementation, the simpler convergence analysis and the optimal IFO/communication complexity. Numerical Experiments In this section, we provide the empirical study on regularized logistic regression for binary classification [3] The individual function for the model is defined as f ij (x) log 1 + exp(−b ij a ⊤ ij x) + λ d k=1 x 2 k 1 + x 2 k , where a j ∈ R d and b j ∈ {−1, 1} are the feature and label of the j-th sample on the i-th agent; λ > 0 is the regularization parameter; x = [x 1 , . . . , x d ] ⊤ ∈ R d is the vector of the classifier. We compare the proposed DEAREST with baseline algorithms GT-SARAH [37] and DESTRESS [18] on three real-world datasets "a9a" (mn = 32, 560, d = 123), "w8a" (mn = 49, 740, d = 300) and "rcv1" (mn = 20, 420, d = 47, 236). All of the datasets can be downloaded from LIBSVM repository [6]. We set the number of agents m = 20 and split the dataset uniformly at random to all of the agents. We conduct the experiments on two communication graphs: a) the random graph where each pair of agents has a connection with the probability of 0.15; b) the circle graph. We set W = I − L/λ 1 (L), where L is the Laplacian matrix associated with the adjacency matrix and λ 1 (L) is the largest eigenvalue of L. We observe that 1 − λ 2 (W ) = 0.0482 for the random graph and and 1 − λ 2 (W ) = 0.0245 for the circle graph. We set λ = 10 −4 in experiments and the parameters of all algorithms are well-tuned. We report the results on random graph and circle graph in Figure 1-2 and Figure 3-4 respectively. We observe that DEAREST outperforms the other two algorithms is all of the settings. Especially, the communication cost of DEAREST is apparently less than the ones of baselines. This phenomenon matches our theoretical result that the upper bound of communication complexity for DEAREST is tighter than the ones of GT-SARAH and DESTRESS. The results also show that both DEAREST and DESTRESS achieve better performance than GT-SARAH. The reason is GT-SARAH lacks the multi-consensus step, which leads to more computation cost and communication cost when 1 − λ 2 (W ) is close to zero. Additionally, DEAREST and DESTRESS has the similar computational cost in most of the cases, which also validates the theoretical results that both of these algorithms has the optimal IFO complexity. Conclusion In this paper, we have proposed a novel stochastic algorithm DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST) for decentralized nonconvex finite-sum optimization. The algorithm has single-loop structure and easy to be implemented. We have proved DEAREST achieves both optimal incremental firstorder oracle (IFO) complexity and optimal communication complexity. Our convergence analysis depends on constructing the Lyapunov function consists of function value, global/local gradient estimation error and consensus error, which is intuitive and easy to follow. The results of numerical experiments also support our theoretical analysis. We believe the idea of DEAREST is not limited to the IFO algorithm for nonconvex optimization and it also can be extended to solve other types of decentralized optimization problems. Lemma 1 ([ 40 , 40Lemma 2]). For Algorithm 1, we haves t =ḡ t .Lemma 2 ([21, Lemma 2]). Under Assumption 1, Algorithm 2 holds that i (x) is the L-smooth local function on the i-th agent and there are m agents form a network with gossip matrixŴ . It is shown that there exist some {f i (x)} m i=1 andŴ such that finding an ε-stationary point off (x) by SFOs {g i (x)} satisfying rounds of communication [ 22 , 22Corollary 1]. For the setting of decentralized finite-sum optimization in this paper, consider the formulations (1)-(2) with f ij (x) f i (x) and network topology with gossip matrixŴ , wheref i (x) andŴ correspond to the ones we mentioned in last paragraph. We let σ = 0 for (21), then the SFOs {g i (x)} satisfying (21) are equivalent to our IFOs {∇f ij (x)}. Hence, the lower bound(22) directly means there exists some {f i (x)} m i=1 andŴ such that using IFO algorithm to find an ε-stationary point for finite-sum problem (1) with f i of communication, which matches the upper bound of DEAREST shown in Theorem 1. Figure 1 : 1The comparison for the number of IFO calls vs. ∇f (x) on random graph. Figure 2 : 2The comparison for the number of communication rounds vs. ∇f (x) on random graph. 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{'abstract': 'This paper studies the decentralized nonconvex optimization problem min x∈R d f (x)is the local function on the i-th agent of the network. We propose a novel stochastic algorithm called DEcentralized probAbilistic Recursive gradiEnt deScenT (DEAREST), which integrates the techniques of variance reduction, gradient tracking and multi-consensus. We construct a Lyapunov function that simultaneously characterizes the function value, the gradient estimation error and the consensus error for the convergence analysis. Based on this measure, we provide a concise proof to show DEAREST requires at most O(mn + √ mnLε −2 ) incremental first-order oracle (IFO) calls and O(Lε −2 / 1 − λ2(W ) ) communication rounds to find an ε-stationary point in expectation, where L is the smoothness parameter and λ2(W ) is the second-largest eigenvalue of the gossip matrix W . We can verify both of the IFO complexity and communication complexity match the lower bounds. To the best of our knowledge, DEAREST is the first optimal algorithm for decentralized nonconvex finite-sum optimization.', 'arxivid': '2210.13931', 'author': ['Luo Luo ', 'Haishan Ye '], 'authoraffiliation': [], 'corpusid': 253107801, 'doi': '10.48550/arxiv.2210.13931', 'github_urls': [], 'n_tokens_mistral': 16764, 'n_tokens_neox': 14069, 'n_words': 8285, 'pdfsha': 'd645cc8a750375578b135d328bec004f480682a2', 'pdfurls': ['https://export.arxiv.org/pdf/2210.13931v3.pdf'], 'title': ['An Optimal Stochastic Algorithm for Decentralized Nonconvex Finite-sum Optimization', 'An Optimal Stochastic Algorithm for Decentralized Nonconvex Finite-sum Optimization'], 'venue': []}

arxiv

The CoRoT ⋆ target HD 49933: 1 -Effect of the metal abundance on the mode excitation rates 18 Nov 2009 November 18, 2009 November 18, 2009 R Samadi UMR 8109 Observatoire de Paris LESIA CNRS Université Pierre et Marie Curie Université Denis Diderot 5 pl. J. JanssenF-92195MeudonFrance H.-G Ludwig UMR 8111 Observatoire de Paris GEPI CNRS 5 pl. J. JanssenF-92195MeudonFrance K Belkacem UMR 8109 Observatoire de Paris LESIA CNRS Université Pierre et Marie Curie Université Denis Diderot 5 pl. J. JanssenF-92195MeudonFrance Institut d'Astrophysique et de Géophysique de l' Université de Liège Allée du 6 Août 17 -B 4000LiègeBelgium M J Goupil UMR 8109 Observatoire de Paris LESIA CNRS Université Pierre et Marie Curie Université Denis Diderot 5 pl. J. JanssenF-92195MeudonFrance M.-A Dupret UMR 8109 Observatoire de Paris LESIA CNRS Université Pierre et Marie Curie Université Denis Diderot 5 pl. J. JanssenF-92195MeudonFrance Institut d'Astrophysique et de Géophysique de l' Université de Liège Allée du 6 Août 17 -B 4000LiègeBelgium The CoRoT ⋆ target HD 49933: 1 -Effect of the metal abundance on the mode excitation rates 18 Nov 2009 November 18, 2009 November 18, 2009Astronomy & Astrophysics manuscript no. 11867 c ESO 2009convection -turbulence -atmosphere -Stars: oscillations -Stars: individual: HD 49933 -Sun: oscillations Context. Solar-like oscillations are stochastically excited by turbulent convection at the surface layers of the stars. Aims. We study the role of the surface metal abundance on the efficiency of the stochastic driving in the case of the CoRoT target HD 49933. Methods. We compute two 3D hydrodynamical simulations representative -in effective temperature and gravity -of the surface layers of the CoRoT target HD 49933, a star that is rather metal poor and significantly hotter than the Sun. One 3D simulation has a solar metal abundance, and the other has a surface iron-to-hydrogen, [Fe/H], abundance ten times smaller. For each 3D simulation we match an associated global 1D model, and we compute the associated acoustic modes using a theoretical model of stochastic excitation validated in the case of the Sun and α Cen A. Results. The rate at which energy is supplied per unit time into the acoustic modes associated with the 3D simulation with [Fe/H]=-1 is found to be about three times smaller than those associated with the 3D simulation with [Fe/H]=0. As shown here, these differences are related to the fact that low metallicity implies surface layers with a higher mean density. In turn, a higher mean density favors smaller convective velocities and hence less efficient driving of the acoustic modes.Conclusions. Our result shows the importance of taking the surface metal abundance into account in the modeling of the mode driving by turbulent convection. A comparison with observational data is presented in a companion paper using seismic data obtained for the CoRoT target HD 49933.⋆ The CoRoT space mission, launched on December 27 2006, has been developped and is operated by CNES, with the contri- Introduction Using the measured linewidths and the amplitudes of the solar acoustic modes, it has been possible to infer the rate at which energy is supplied per unit time into the solar acoustic modes. Using these constraints, different models of mode excitation by turbulent convection have been extensively tested in the case of the Sun (see e.g. recent reviews by Samadi et al. (2008b) and Houdek (2006)). Among the different approaches, we can distinguish pure theoretical approaches (e.g. Samadi & Goupil 2001;Chaplin et al. 2005), semi-analytical approaches (e.g. Samadi et al. 2003b,a) and pure numerical approaches (e.g. Nordlund & Stein 2001;Stein et al. 2004;Jacoutot et al. 2008). The advantage of a theoretical approach is that it easily allows massive computation of the mode excitation rates for a wide variety of stars with different fundamental parameters (e.g. effective temperature, gravity) and different surface metal abundance. However, pure theoretical approaches are based on crude or simplified descriptions of turbulent convection. On the other hand, a semi-analytical approach is gener-ally more realistic since the quantities related to turbulent convection are obtained from 3D hydrodynamical simulation. 3D hydrodynamical simulations are at this point in time too time consuming, so that a fine grid of 3D models with a sufficient resolution in effective temperature (T eff ), gravity (log g) and surface metal abundance (Z) is not yet available. In the present paper, we study and provide a procedure to interpolate for any value of Z the mode excitation rates P between two 3D simulations with different Z but the same T eff and log g. With such interpolation procedure it is no longer required to have at our disposal a fine grid in Z of 3D simulations. The semi-analytical mode that we consider here is based on Samadi & Goupil (2001)'s theoretical model with the improvements proposed by Belkacem et al. (2006a). This semi-analytical model satisfactorily reproduces the solar seismic data (Samadi et al. 2003a;Belkacem et al. 2006b). Recently, the seismic constraints obtained for α Cen A (HD 128620) have provided an additional validation of the basic physical assumptions of this theoretical model (Samadi et al. 2008a). The star α Cen A has a surface gravity (log g = 4.305) lower than that of the Sun (log g ⊙ = 4.438), but its effective temperature (T eff = 5810 K) does not significantly differ from that of the Sun (T eff,⊙ = 5780 K). The higher T eff , the more vigorous the convective velocity at the surface and the stronger the driving by turbulent convection (see e.g. Houdek et al. 1999). For main sequence stars with a mass M 1.6 M ⊙ , an increase of the convective velocity is expected to be associated with a larger turbulent Mach number, M t (Houdek et al. 1999). However, the theoretical models of stochastic excitation are strictly valid in a medium where M t is -as in the Sun and α Cen A -rather small. Hence, the higher M t , the more questionable the different approximations and the assumptions involved in the theory (see e.g. Samadi & Goupil 2001). It is therefore important to test the theory with another star characterized by a T eff significantly higher than in the Sun. Furthermore, the star α Cen A has an iron-to-hydrogen abundance slightly larger than the Sun, namely [Fe/H]= 0.2 (see Neuforge-Verheecke & Magain 1997). However, the modeling performed by Samadi et al. (2008a) for α Cen A assumes a solar iron abundance ([Fe/H]=0). According to Houdek et al. (1999), the mode amplitudes are expected to change with the metal abundance. However, Houdek et al. (1999)'s result was obtained on the basis of a mixing-length approach involving several free parameters and by using a theoretical model of stochastic excitation in which a free multiplicative factor is introduced in order to reproduce the maximum of the solar mode excitation rates. Therefore, it is important to extend Houdek et al. (1999)'s study by using a more realistic modeling based on 3D hydrodynamical simulation of the surface layers of stars and a theoretical model of mode driving that reproduces -without the introduction of free parameters -the available seismic constraints. To this end, the star HD 49933 is an interesting case for three reasons: First, this star has T eff = 6780 ± 130 K (Bruntt et al. 2008), log g ≃ 4.25 ± 0.13 (Bruntt et al. 2008) and [Fe/H] ≃ −0.37 dex (Solano et al. 2005;Gillon & Magain 2006). The properties of its surface layers are thus significantly different from those of the Sun and α Cen A. Second, HD 49933 was observed in Doppler velocity with the HARPS spectrograph. A seismic analysis of these data performed by Mosser et al. (2005) has provided the maximum of the mode surface velocity (V max ). Third, the star was more recently observed continuously in intensity by CoRoT during 62 days. Apart from observations for the Sun, this is the longest seismic observation ever peformed both from the ground and from space. This long term and continuous observation provides a very high frequency resolution (∼ 0.19 µHz). The seismic analysis of these observations undertaken by Appourchaux et al. (2008) or more recently by Benomar et al. (2009) have provided the direct measurements of the mode amplitudes and the mode linewidths with an accuracy not previously achieved for a star other than the Sun. We consider two 3D hydrodynamical simulations representative -in effective temperature and gravity -of the surface layers of HD 49933. One 3D simulation has [Fe/H]=0, while the second has [Fe/H]= -1. For each 3D simulation, we match an associated global 1D model and compute the associated acoustic modes and mode excitation rates, P. This permits us to quantify the variation of P induced by a change of the surface metal abundance Z. From these two sets of calculation, we then deduce P for HD 49933 by taking into account the observed iron abundance of the star (i.e. [Fe/H]=-0.37). In a companion paper (Samadi et al. 2009, hereafter Paper II), we will use these theoretical calculations of P and the mode linewidths ob-tained from the seismic analysis of HD 49933 performed with the CoRoT data to derive the expected mode amplitudes in HD 49933. These computed mode amplitudes will then be compared with the observed ones. This comparison will then constitute a test of the stochastic excitation model with a star significantly different from the Sun and α Cen A. It will also constitute a test of the procedure proposed here for deriving P for any value of Z between two 3D simulations with different Z. The present paper is organised as follows: we first describe in Sect. 2 the method to compute the theoretical mode excitation rates associated with the two 3D hydrodynamical simulations. Next, the effects on P of a different surface metal abundance are presented in Sect. 3. Then, by taking into account the actual iron abundance of HD 49933, we derive theoretical values of P expected for HD 49933. Finally, Sect. 5 is dedicated to our conclusions. Calculation of mode excitation rates Model of stochastic excitation The energy injected into a mode per unit time P is given by the relation (see Samadi & Goupil 2001;Belkacem et al. 2006b): P = 1 8 I C 2 R + C 2 S ,(1) where C 2 R and C 2 S are the turbulent Reynolds stress and entropy contributions, respectively, and I = M 0 dm |ξ r | 2(2) is the mode inertia, ξ r is the adiabatic radial mode displacement and M is the mass of the star. The expressions for C 2 R and C 2 S are given for a radial mode with frequency ω osc by C 2 R = 64π 3 15 dmρũ 4 k 3 0 ω 0 K w 3 f r S R (r, ω osc ) ,(3)C 2 S = 16π 3 3 ω 2 osc dm (α ssũ ) 2 ρ k 3 0 ω 0 g r S s (r, ω osc )(4) where we have defined the "source functions": S R (r, ω osc ) = k 3 0 ω 0 u 4 dk k 2 E 2 (k) × dω χ k (ω + ω osc ) χ k (ω) (5) S s (r, ω osc ) = k 3 0 ω 0 u 2s2 dk k 2 E(k) E s (k) × dω χ k (ω + ω osc ) χ k (ω)(6) where P is the gas pressure, ρ the density, s the entropy, ρ the equilibrium density profile, α s ≡ (∂P/∂s) ρ , f r ≡ (dξ r /dr) 2 and g r are two functions that involve the first and second derivatives of ξ r respectively, k is the wavenumber, E(k) is the turbulent kinetic energy spectrum, E s (k) is the spectrum associated with the entropy fluctuations (s),s is the rms of s, χ k is the time-correlation function associated with the velocity,ũ is a characteristic velocity defined in a way that 3ũ 2 = u 2 , . refers to horizontal and time average, u is the turbulent velocity field, and finally K w ≡ u 4 z / u 2 z 2 is the Kurtosis (see Belkacem et al. 2006a,b, for details). Furthermore, we have introduced for convenience the characteristic frequency ω 0 and the characteristic wavenumber k 0 : ω 0 ≡ k 0ũ (7) k 0 ≡ 2π Λ (8) where Λ is a characteristic size derived from E(k) as explained in Samadi et al. (2003b). Note that the introduction of the term k 3 0 ω 0ũ −4 in the RHS of Eq. (5) and the term k 3 0 ω 0ũ −2s−2 in the RHS of Eq. (6) ensure dimensionless source functions. The kinetic spectrum E(k) is derived from the 3D simulation as detailled in Samadi et al. (2003b). As shown by Samadi et al. (2003b), the k-dependence of E s (k) is similar to that of the E(k). Accordingly, we assume E s ∝ E. In Samadi et al. (2008a), two different analytical functions for χ k (ω) have been considered, namely a Lorentzian function and a Gaussian one. In the present study we will in addition derive χ k (ω) directly from the 3D simulations as detailled in Samadi et al. (2003a). Once χ k (ω) is derived from the 3D simulation, it is implemented in Eq. (5) and Eq. (6). We compute the mode excitation as detailled in Samadi et al. (2008a): all required quantities -except ξ r , I and ω osc -are obtained directly from two 3D hydrodynamical simulations representative of the outer layers of HD 49933, whose characteristics are described in Sect. 2.2 below. The quantities related to the modes (ω osc , I and ξ r ) are calculated using the adiabatic pulsation code ADIPLS (Christensen-Dalsgaard & Berthomieu 1991) from 1D global models. The outer layers of these 1D models are derived from the 3D simulation as described in Sect. 2.3. The 3D simulations We computed two 3D radiation-hydrodynamical model atmospheres with the code CO 5 BOLD (Freytag et al. 2002;Wedemeyer et al. 2004). One 3D simulation had a solar iron-to-hydrogen [Fe/H]=0.0 while the other had [Fe/H]=-1.0. The 3D model with [Fe/H]=0 (resp. [Fe/H]=-1) will be hereafter referred to as model S0 (resp. S1). The assumed chemical composition is similar (in particular for the CNO elements) to that of the solar chemical composition proposed by Asplund et al. (2005). The abundances of the α-elements in model S1 were assumed to be enhanced by 0.4 dex. For S0 we obtain Z/X = 0.01830 and Y=0.249, and for S1 Z/X = 0.0036765 and Y =0.252. Both 3D simulations have exactly the same gravity (log g =4.25) and are very close in effective temperature (T eff ). Both models employ a spatial mesh with 140 × 140 × 150 grid points, and a physical extent of the computational box of 16.4 × 16.4 × 24.2 Mm 3 . The equation of state takes into account the ionisation of hydrogen and helium as well as the formation of H 2 molecules according to the Saha-Boltzmann statistics. The wavelength dependence of the radiative transfer is treated by the opacity binning method (Nordlund 1982;Ludwig 1992;Vögler et al. 2004) using five wavelength bins for model S0 and six for model S1. Detailed wavelengthdependent opacities were obtained from the MARCS model atmosphere package (Gustafsson et al. 2008). Table 1 summarizes the characteristics of the 3D models. The effective temperature and surface gravity correspond to the parameters of HD 49933 within the observational uncertainties, while the two metallicities bracket the observed value. For each 3D simulation, two time series were built. One has a long duration (38h and 20h for S0 and S1, respectively) and a low sampling frequency (10 mn). This time series is used to compute time averaged quantities (ρ, E(k), etc.). The second time series is shorter (8.8h and 6.8h for S0 and S1, respectively), but has a high sampling frequency (1 mn). Such high sampling frequency is required for the calculation of χ k (ω). Indeed, the modes we are looking at lie between ν ≈ 1.25 mHz and ν ≈ 2.4 mHz. Label [Fe/H] Y Z Z/X T eff [K] S0 0 0.249 13.5 10 −3 0.018305 6725 ± 17 S1 -1 0.252 2.74 10 −3 0.003676 6730 ± 12 Table 1. Characteristics of the 3D simulations. The two 3D simulations extend up to T = 100 000 K. However, for T 30 000 K, the 3D simulations are not completely realistic. First of all, the MARCS-based opacities are provided only up to a temperature of 30 000 K; for higher temperatures the value at 30 000 K is assumed. Note that we refer to the opacity per unit mass here. For the radiative transfer the opacity per unit volume is the relevant quantity, i.e. the product of opacity per mass unit and density. Since in the simulation the opacity is still multiplied at each position with the correct local density, the actual error we make when extrapolating the opacity is acceptable. Another limitation of the simulations is the restricted size of the computational box which does not allow for a full development of the largest flow structures, again in the layers above T ≃ 30 000 K. Two hints make us believe that the size of the computational domain is not fully sufficient: i) in the deepest layers of the simulations there is a tendency that structures align with the computational grid; ii) the spatial spectral power P of scalar fields in a horizontal layer does not tend towards the expected asymptotic behaviour P × k for low spatial wavenumber k. We noticed this shortcoming only after the completion of the simulation runs. To mitigate its effect in our analysis, we will later by default integrate the mode excitation rates up to T = 30 000 K. However, for comparison purposes, some computations have been extended down to the bottom of the 3D simulations. For S0, the layers located below T ≃ 30 000 K contribute only by 10 % to the excitation of the modes lying in the frequency range where modes have the most chance to be detected (ν ≃ 1.2−2.5 mHz). For S1, the contribution of the deep layers is even smaller (∼ 5 %). Finally, one may wonder how the treatment of the smallscales or the limited spatial resolution of the simulation can influence our calculations. Dissipative processes are handled in CO 5 BOLD on the one hand side implicitely by the numerical scheme (Roe-type approximate Riemann solver), and on the other hand explicitely by a sub-grid model according to the classical Smagorinsky (1963) formulation. Jacoutot et al. (2008) found that computed mode excitation rates significantly depend on the adopted sub-grid model. Samadi et al. (2007) have found that solar mode ex-citation rates computed in the manner of Nordlund & Stein (2001), i.e., using data directly from the 3D simulation, decrease as the spatial resolution of the solar 3D simulation decreases. As a conclusion the spatial resolution or the subgrid model can influence computed mode excitation rates (see a discussion in Samadi et al. 2008a). However, concerning the spatial resolution and according to Samadi et al. (2007)'s results, the present spatial resolution (1/140 of the horizontal size of the box and about 1/150 of the vertical extent of the simulation box) is high enough to obtain accurate computed energy rates. The increased spatial resolution of our models in comparison to the work of Jacoutot et al. (2008) reduces the impact of the unresolved scales. The 1D global models For each 3D model we compute an associated 1D global model. The models are built in the manner of Trampedach (1997) as detailled in Samadi et al. (2008a) in such way that their outer layers are replaced by the averaged 3D simulations described in Sect. 2.2. The interior of the models are obtained with the CESAM code assuming standard physics: Convection is described according to Böhm-Vitense (1958)'s local mixing-length theory of convection (MLT), and turbulent pressure is ignored. Microscopic diffusion is not included. The OPAL equation of state is assumed. The chemical mixture of the heavy elements is similar to that of Asplund et al. (2005)'s mixture. As in Samadi et al. (2008a), we will refer to these models as "patched" models hereafter. The two models have the effective temperature and the gravity of the 3D simulations. One model is matched with S0 and has [Fe/H]=0, while the second is matched with S1 and has [Fe/H]=-1. The 1D models have the same chemical mixture as their associated 3D simulations. The parameters of the 1D patched models are given in Table 2. The stratification in density and temperature of the patched 1D models are shown in Fig. (1). At any given temperature the density is larger in S1 as a consequence of its lower metal abundance. Indeed, the lower the metal abundance, the lower the opacity ; then, at a given optical depth (τ ), the density is larger in S1 compared to S0. The photosphere corresponds to the optical depth τ = 2/3. Since the two 3D simulations have approximatively the same effective temperature, the density in S1 is larger at optical depth τ = 2/3. Since the density in S1 increases with depth even more rapidly than in S0, the density in S1 remains larger for τ > 2/3 than in S0. Effects of the metal abundance on excitation rates The mode excitation rates (P) are computed for the two 3D simulations according to Eqs. (1)-(6). The integration is performed from the top of the simulated domains down to T = 30 000 K (see Sect. 2.2). In the following, P 1 (resp. P 0 ) corresponds to the mode excitation rates associated with the 3D model with [Fe/H]=-1 (resp. [Fe/H]=0) Figure 2 shows the effect of the assumed metal abundance of the stellar model on the mode excitation rates. P 1 is found to be three times smaller than P 0 , i.e. p modes associated with the metal poor 3D model (S1) receive approximatively three times less energy per unit time than those associated with the 3D model with the solar metal abundance (S0). For both 3D models, the dominant part of the driving is ensured by the Reynolds stresses. The entropy fluctuations contribute by only ∼ 30 % of the total power for both S0 and S1. By comparison, in the case of the Sun and α Cen A it contributes by only ∼ 15 %. Furthermore, we find that the contribution of the entropy source term is -as for the Reynolds stress term -about three times smaller in S1 than in S0. We conclude that the effect of the metal abundance on the excitation rates is almost the same for the Reynolds stress contribution and the entropy source term. Results Interpretation From Eqs. (1), (2), (3), (7) and (8) we show that at a given layer the power supplied to the modes -per unit mass -by the Reynolds stress is proportional to F kin Λ 4 S R /M, where F kin is the flux of the kinetic energy, which is proportional toρũ 3 , Λ is a characteristic length (see Sect. 2.1) and M is the mode mass defined as: M = I ξ 2 r (9) where ξ r is the mode displacement evaluated at the layer in the atmosphere where the mode is measured. The power supplied to the modes -per unit massby the entropy source term is proportional toρũ 3 Λ 4 R 2 S s where ω osc is the mode frequency, R ∝ F conv /F kin , where F conv ∝ w α ss is the convective flux, and finallys is the rms of the entropy fluctuations (see Samadi et al. 2006). We recall that the higher R, the higher the relative contribution of the entropy source to the excitation. We study below the role of M, F kin , Λ, S R , S s and R: Table 2. Characteristics of the 1D "patched" models. α is the mixing-length parameter. Mode mass (M): The frequency domain, where modes are strongly excited, ranges between ν ≈ 1.2 mHz and ν ≈ 2.5 mHz. In this frequency domain, the mode masses M associated with S0 are quite similar to those associated with S1 (not shown). Consequently the differences between P 1 and P 0 do not arise from the (small) differences in M. Kinetic energy flux (F kin ): The larger F kin , the larger the driving by the Reynolds stress. However, we find that the two 3D models have very similar F kin . This is not surprising since the two 3D models have very similar effective temperatures. This means that the differences between P 1 and P 0 do not arise from the (small) differences in F kin . Characteristic length (Λ): In the manner of Samadi et al. (2003b) we derive from the kinetic energy spectra E(k) of the two 3D simulations the characteristic length Λ (Λ = 2π/k 0 , see Eq. (8)) for each layer of the simulated domain. We find that the differences in Λ between the two 3D simulations is small and does not play a significant role in the differences in P. This can be understood by the fact that S0 and S1 have the same gravity. Indeed, as shown by Samadi et al. (2008a) -at a fixed effective temperature -Λ scales as the inverse of g. We conclude that the differences between P 1 and P 0 do not originate from the (small) differences in Λ. Source functions (S R and S s ): The dimensionless source functions S R and S s are defined in Eqs. (5) and (6) respectively. Both source functions involve the eddy timecorrelation function χ k (ω). We define ω k as the frequency width of χ k (ω). As shown by Samadi et al. (2003a) and as verified in the present case, ω k can be evaluated as the product k u k where u k is given by the relation (Stein 1967): u 2 k = 2k k dk E(k)(10) where E(k) is normalised as: +∞ 0 dk E(k) = 1 2 u 2 ≡ 1 2ũ 2 .(11) According to Eqs. (10) and (11), u k is directly proportional toũ. At a fixed k/k 0 , we then have ω k ∝ũ k 0 = ω 0 . As seen above, ω 0 controls ω k , the frequency width of χ k . Then, at fixed ω osc , we can easly see from Eqs. (5) and (6) that the smaller ω 0 , the smaller S R (ω osc ) and S s (ω osc ). Since ω 0 =ũ k 0 = 2πũ/Λ and since both 3D simulations have approximately the same Λ, smallerũ results directly in smaller ω 0 and hence in smaller source functions. We have plotted in Fig. 3 the characteristic velocityũ. This quantity is found to be up to 15 % smaller for S1 compared with S0. In other words, the metal poor 3D model is characterized by lower convective velocities. Consequently, the source functions are smaller for S1 compared to S0. Although the convective velocities differ between S0 and S1 by only 15 %, the excitation rates differ by a factor ∼ 3. The reason for this is that he source functions, which are non-linear functions ofũ, decrease very rapidly withũ. This is the consequence of the behavior of the eddy-time correlation χ k . Indeed, this function varies with the ratio ω osc /ω k approximately as a Lorentzian function. This is why χ k varies rapidly withũ (we recall that ω k ∝ũ k 0 ). In conclusion, the differences between P 1 and P 0 are mainly due to differences in the characteristic velocityũ. In turn, the low convective velocity in S1 is a consequence of the larger density compared to S0. Indeed, as shown in Fig. 1, the density is systematically higher in S1. At the layer where the modes are the most excited (i.e. at T ∼ 10000K), the density is ∼ 50 % higher. Since the two 3D models have a similar kinetic energy flux (see above), it follows that a larger density for S1 then implies lower convective velocities. Relative contribution of the entropy source term (R): The convective flux F conv in S1 is almost identical to that of S0. This is due to the fact that the two 3D simulations have almost the same effective temperature. Furthermore, as pointed out above, the differences in F kin between S1 and S0 are small. As a consequence, the ratio R ∝ F conv /F kin does not differ between the two 3D simulations. Accordingly, as for the Reynolds contribution, the variation of the excitation rates with the metal abundance is only due to the source term S S . The latter varies with ω 0 in the same manner as S R , which is turn the reason for As a consequence, the contribution of the entropy fluctuations to show the same trend with the metal abundance as the Reynolds stress term. Fig. 2. The dot-dashed line corresponds to the solid line multiplied by γ 1 , where γ 1 (T ) ≡ (ρ 0 /ρ 1 ) 1/3 andρ 0 (resp.ρ 1 ) is the mean density stratification of S0 (resp. S1)(see Appendix A). Theoretical calculation of P for HD 49933 We derive the mode excitation rates P for HD 49933. According to Gillon & Magain (2006) As seen in Sect. 3.2, differences in P between S0 and S1 are a direct consequence of the differences in the source functions S R and S S . It follows that in order to derive P for HD 49933, we only have to derive the expected values for S R and S S . As seen in Sect. 3.2, differences in S R (or in S S ) between S0 and S1 are related to the surface metal abundance through the surface densities that impact the convective velocities (ũ). The determination of the HD 49933 convective velocities allows us to determine its source function. To this end, we use the fact that the kinetic flux is almost unchanged between S1 and S0 (see Sect. 3.2) to derive the profile ofũ(T ), expected at the surface layers of HD 49933. This is performed by interpolating in Z between S0 and S1, the surface density stratification representative of the surface layers of HD 49933. The whole procedure is described in Appendix A. In order to compute P for HD 49933, we then need to know Z for this star. Since we do not know its surface helium abundance, we will assume by default the solar value for Y : Y = 0.249 ± 0.003 (Basu 1997). Gillon & Magain (2006)'s analysis shows that the chemical mixture of HD 49933 does not significantly differ from that of the Sun. According to Asplund et al. (2005), the new solar metal to hydrogen ratio is (Z/X) ⊙ = 0.0165 Accordingly, since [Fe/H]=-0.37 ± 0.03 dex, we derive Z = 5.3 10 −3 ± 0.4 10 −3 for HD 49933. Note that assuming Grevesse & Noels (1993)'s chemical mixture yields Z = 7.8 10 −3 ± 0.5 10 −3 . The result of the calculation is shown in Fig. 2. The maximum P is 1.08 ± 0.05 10 17 J/s when Asplund et al. (2005)'s chemical composition is assumed (see Appendix A). This is about 30 times larger than in the Sun and about 14 times larger than in α Cen A. When Grevesse & Noels (1993)'s chemical mixture is assumed, the maximum in P is in that case equal to 1.27 ± 0.05 10 17 J/s, that is about 30 % larger than with Asplund et al. (2005)'s solar chemical mixture. We note that the uncertainties in the knowledge of [Fe/H] set uncertainties on P which are the on order of 10 % in the frequency domain of interest. Conclusion We have built two 3D hydrodynamical simulations representative in effective temperature (T eff ) and gravity (g) of the surface layers of an F type star on the main sequence. One model has a solar iron-to-hydrogen abundance ([Fe/H]=0) and the other has [Fe/H]=-1. Both models have the same T eff and g. For each 3D simulation, we have computed an associated "patched" 1D full model. Finally, we have computed the mode excitation rates P associated with the two "patched" 1D models. Mode excitation rates associated with the metal poor 3D simulation are found to be about three times smaller than those associated with the 3D simulation which has a solar surface metal abundance. This is explained by the following connections: the lower the metallicity, the lower the opacity. At fixed effective temperature and surface gravity, the lower the opacity, the denser the medium at a given optical depth. The higher the density, the smaller are the convective velocities to transport the same amount of energy by convection. Finally, smaller convective velocities result in a less efficient driving. On the other hand, a surface metal abundance higher than the solar metal abundance will result in a lower surface density, which in turn will result in a higher convective velocity and then in a more efficient driving. Our result can then be qualitatively generalised for any surface metal abundance. By taking into account the observed surface metal abundance of the star HD 49933 (i.e. [Fe/H]=-0.37), we have derived, using two 3D simulations and the interpolation procedure developed here, the rates at which acoustic modes are expected to be excited by turbulent convection in the case of HD 49933. These excitation rates P are found to be about two times smaller than for a model built assuming a solar metal abundance. These theoretical mode excitation rates will be used in Paper II to derive the expected mode amplitudes from measured mode linewidths. We will then be able to compare these amplitudes with those derived for HD 49933 from different seismic data. This will constitute an indirect test of our procedure which permits us to interpolate for any value of Z the mode excitation rates P between two 3D simulations with different Z but the same T eff and log g. We must stress that a more direct validation of this interpolation procedure will be to compute a third 3D model with the surface metal abundance of the star HD 49933 and to compare finally the mode excitation rates obtained here with the interpolation procedure with that obtained with this third 3D model. This represents a long term work since several months (about three to four months) are required for the calculation of this additional 3D model, which is in progress. fixed T and between S0 and S1 -a linear interpolation of ρ(T ) with respect to Z. A.3. Derivation of M As shown in Sect. 3.2 above in the frequency domain where modes are detected in HD 49933, M does not change sig-nificantly between S0 and S1. This suggests that the mode masses associated with a patched 1D model with the metal abundance expected for HD 49933 would be very similar to those associated with S0 or S1. Consequently we will assume for the case of HD 49933 the same mode masses as those associated with S1, since this 3D model has a Z abundance closer to that of HD 49933. A.4. Derivation of P Before deriving P for HD 49933, we check that, from S0 and the knowledge of P 0 , we can approximately reproduce P 1 , the mode excitation rates, associated with S1 following the procedure described above. Let γ 1 ≡ (ρ 0 /ρ 1 ) 1/3 . As seen in Fig. 3, when we multiplyũ 0 by γ 1 (T ) we matcheũ 1 . Then, using γ 1 (T ) and following the procedure described above, we derive P 01 , the mode excitation rates associated with S1 but derived from S0. The result is shown in Fig. A.1. P 01 matches P 1 rather well. However, there are differences remaining in particular in the frequency domain ν=1.2-1.5 mHz. Nevertheless, the differences between P 01 and P are in any case not significant compared to the accuracy at which the mode amplitudes are measured with the CoRoT data (see Paper II). This validates the procedure, at least at the level of the current seismic precisions. Mode excitation rates P as a function of the mode frequency ν. The thin dot-dashed line corresponds to P 01 , the mode excitation rates derived for S1 from S0 (see Appendix A4). The other lines have the same meaning as in Fig. 2. Since the metal abundance Z of HD 49933 is closer to that of S1 than that of S0, we derive the mode excitation rates P associated with HD 49933 from S1 following the procedure detailled above. The result is shown in Fig. 2. As expected, the mode excitation rates P associated with HD 49933 lie between those of S0 and S1, while remaining closer to S1 than to S0. Note that the differences between P 1 and the excitation rates derived for HD 49933 (P) are of the same order as the differences seen locally between P 1 and P 01 . These differences remain small compared to the current seismic precisions. On the other hand the differences Fig. 1 . 1Mean densityρ as a function of temperature, T . The solid line corresponds to the 3D model with the metal abundance (S0) and the dashed line to metal poor 3D model (S1). The filled dots show the location where the 1D models have been matched to the associated 3D simulation. Fig. 2 . 2Mode excitation rates P as a function of the mode frequency, ν. The solid line corresponds to the 3D model with the canonical metal abundance (S0) and the dashed line to the metal poor 3D model (S1). The dot-dashed line corresponds to the mode excitation rates derived for the specific case of HD 49933 as explained in Appendix A. Fig. 3 . 3Characteristic velocityũ defined in Eq. (7) as a function of temperature, T . The solid and dashed lines have the same meaning as in Fig. A. 1 . 1Fig. A.1. Mode excitation rates P as a function of the mode frequency ν. The thin dot-dashed line corresponds to P 01 , the mode excitation rates derived for S1 from S0 (see Appendix A4). The other lines have the same meaning as in Fig. 2. , HD 49933 has [Fe/H]=-0.37 ± 0.03 dex, while we only have two 3D simulations with values of [Fe/H], respectively [Fe/H]=0 and [Fe/H]=-1. R.Samadi et al.: The CoRoT target HD 49933: 1 -Role of the metal abundance, Online Material p 3 between P and P 0 are significant and have an important impact on the mode amplitudes (see Paper II). Acknowledgements. We thank C. Catala for useful discussions concerning the spectrometric properties of HD 49933. We are indebted to R. Samadi et al.: The CoRoT target HD 49933: 1 -Role of the metal abundance 7 J. Leibacher for his careful reading of the manuscript. K.B. acknowledged financial support from Liège University through the Subside Fédéral pour la Recherche 2009.Appendix A: Theoretical calculation of the mode excitation rates for HD 49933The mode excitation rate P is inversely proportional to the mode mass M (see Eqs.(9),(2)and(2)). This is why we can derive M and M P separately in order to derive P for HD 49933.A.1. Derivation of M P As pointed out in Sect. 3.2, the kinetic flux F kin =ρũ 3 is almost unchanged between S1 and S0 because both 3D models have the same T eff . This has also to be the case for HD 49933 (same T eff and same log g than S0 and S1). Therefore, the calculation of M P for HD 49933 relies only on the evaluation of the values reached -at a fixed mode frequency -by the source functions S R and S S . As seen in Sect. 3.2, ω 0 = k 0ũ controls the width of χ k in a way that the source functions S R (ω osc ) and S S (ω osc ) can be seen as functions of the dimensionless ratio ω osc /ω 0 . The variation of E with k as well as the variation of χ k with ω/ω 0 are shown to be similar in the two 3D simulations. Furthermore, S0 and S1 have approximately the same characteristic length Λ and hence approximately the same k 0 ≡ 2π/Λ. Therefore, the source function S R (resp. S S ) associated with S0 only differs from that of S1 by the characteristic velocityũ. This must then also be the case for HD 49933. Further, in order to evaluate the source functions in the case of HD 49933, we only need to know the factor γ by whichũ is modified in HD 49933 with respect to S1 or S0. According to Eq. (5) (resp. Eq. (6)), multipling u by γ is equivalent to replace S R (ω osc ) (reps. S S (ω osc )) by γ S R (ω osc /γ) (resp. γ S s (ω osc /γ)).Since the kinetic flux F kin in HD 49933 must be the same for S0 or S1, the characteristic velocityũ can be derived for HD 49933 according toũ * (T ) =ũ 1 γ * with γ * (T ) ≡ (ρ 1 /ρ * ) 1/3 whereρ 1 (T ) is the mean density stratification of S1,ũ 1 (T ) the characteritic velocity of S1 andρ * the mean density of HD 49933. Once γ * and thenũ * are derived for HD 49933, we then compute the source functions associated with HD 49933. Finally, we compute M P by keeping F kin constant. We now turn to the derivation of the factor γ * .A.2. Derivation of γ *To derive γ * at a given T , we need to know how the mean densityρ varies with the metal abundance Z. In order to this we consider five "standard" 1D models with five different values of Z. These 1D models are built using the same physics as described in Sect. 2.3. Two of these models have the same abundance as S0 and S1. 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R Samadi, Å Nordlund, R F Stein, M J Goupil, I Roxburgh, A&A. 403303Samadi, R., Nordlund,Å., Stein, R. F., Goupil, M. J., & Roxburgh, I. 2003b, A&A, 403, 303 . J Smagorinsky, Monthly Weather Review. 9199Smagorinsky, J. 1963, Monthly Weather Review, 91, 99 . E Solano, C Catala, R Garrido, AJ. 129547Solano, E., Catala, C., Garrido, R., et al. 2005, AJ, 129, 547 . R Stein, D Georgobiani, R Trampedach, H.-G Ludwig, Å Nordlund, Sol. Phys. 220229Stein, R., Georgobiani, D., Trampedach, R., Ludwig, H.-G., & Nordlund,Å. 2004, Sol. Phys., 220, 229 R F Stein, Solar Physics. 2385Stein, R. F. 1967, Solar Physics, 2, 385 Master's thesis, Master's thesis. R Trampedach, Aarhus UniversityTrampedach, R. 1997, Master's thesis, Master's thesis, Aarhus University (1997) . A Vögler, J H M J Bruls, M Schüssler, A&A. 421741Vögler, A., Bruls, J. H. M. J., & Schüssler, M. 2004, A&A, 421, 741 . 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{'abstract': 'Context. Solar-like oscillations are stochastically excited by turbulent convection at the surface layers of the stars. Aims. We study the role of the surface metal abundance on the efficiency of the stochastic driving in the case of the CoRoT target HD 49933. Methods. We compute two 3D hydrodynamical simulations representative -in effective temperature and gravity -of the surface layers of the CoRoT target HD 49933, a star that is rather metal poor and significantly hotter than the Sun. One 3D simulation has a solar metal abundance, and the other has a surface iron-to-hydrogen, [Fe/H], abundance ten times smaller. For each 3D simulation we match an associated global 1D model, and we compute the associated acoustic modes using a theoretical model of stochastic excitation validated in the case of the Sun and α Cen A. Results. The rate at which energy is supplied per unit time into the acoustic modes associated with the 3D simulation with [Fe/H]=-1 is found to be about three times smaller than those associated with the 3D simulation with [Fe/H]=0. As shown here, these differences are related to the fact that low metallicity implies surface layers with a higher mean density. In turn, a higher mean density favors smaller convective velocities and hence less efficient driving of the acoustic modes.Conclusions. Our result shows the importance of taking the surface metal abundance into account in the modeling of the mode driving by turbulent convection. A comparison with observational data is presented in a companion paper using seismic data obtained for the CoRoT target HD 49933.⋆ The CoRoT space mission, launched on December 27 2006, has been developped and is operated by CNES, with the contri-', 'arxivid': '0910.4027', 'author': ['R Samadi \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n', 'H.-G Ludwig \nUMR 8111\nObservatoire de Paris\nGEPI\nCNRS\n5 pl. J. JanssenF-92195MeudonFrance\n', "K Belkacem \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n\nInstitut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium\n", 'M J Goupil \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n', "M.-A Dupret \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n\nInstitut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium\n", 'R Samadi \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n', 'H.-G Ludwig \nUMR 8111\nObservatoire de Paris\nGEPI\nCNRS\n5 pl. J. JanssenF-92195MeudonFrance\n', "K Belkacem \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n\nInstitut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium\n", 'M J Goupil \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n', "M.-A Dupret \nUMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance\n\nInstitut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium\n"], 'authoraffiliation': ['UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', 'UMR 8111\nObservatoire de Paris\nGEPI\nCNRS\n5 pl. J. JanssenF-92195MeudonFrance', 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', "Institut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium", 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', "Institut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium", 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', 'UMR 8111\nObservatoire de Paris\nGEPI\nCNRS\n5 pl. J. JanssenF-92195MeudonFrance', 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', "Institut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium", 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', 'UMR 8109\nObservatoire de Paris\nLESIA\nCNRS\nUniversité Pierre et Marie Curie\nUniversité Denis Diderot\n5 pl. J. JanssenF-92195MeudonFrance', "Institut d'Astrophysique et de Géophysique de l'\nUniversité de Liège\nAllée du 6 Août 17 -B 4000LiègeBelgium"], 'corpusid': 118548934, 'doi': '10.1051/0004-6361/200911867', 'github_urls': [], 'n_tokens_mistral': 14333, 'n_tokens_neox': 12164, 'n_words': 7813, 'pdfsha': '91144042ebc34417f9344449d56d0ba9b75c28cc', 'pdfurls': ['https://arxiv.org/pdf/0910.4027v3.pdf'], 'title': ['The CoRoT ⋆ target HD 49933: 1 -Effect of the metal abundance on the mode excitation rates', 'The CoRoT ⋆ target HD 49933: 1 -Effect of the metal abundance on the mode excitation rates', 'The CoRoT ⋆ target HD 49933: 1 -Effect of the metal abundance on the mode excitation rates', 'The CoRoT ⋆ target HD 49933: 1 -Effect of the metal abundance on the mode excitation rates'], 'venue': []}

arxiv

Contact2Grasp: 3D Grasp Synthesis via Hand-Object Contact Constraint Haoming Li Key Lab of CS&AUS Zhejiang University HangzhouChina Xinzhuo Lin Key Lab of CS&AUS Zhejiang University HangzhouChina Yang Zhou OPPO US Research Center Palo AltoUSA Xiang Li [email protected] OPPO US Research Center Palo AltoUSA Yuchi Huo [email protected] State Key Lab of CAD&CG and Zhejiang Lab Zhejiang University HangzhouChina Jiming Chen Key Lab of CS&AUS Zhejiang University HangzhouChina Qi Ye [email protected] Key Lab of CS&AUS Zhejiang University HangzhouChina Contact2Grasp: 3D Grasp Synthesis via Hand-Object Contact Constraint 3D grasp synthesis generates grasping poses given an input object. Existing works tackle the problem by learning a direct mapping from objects to the distributions of grasping poses. However, because the physical contact is sensitive to small changes in pose, the high-nonlinear mapping between 3D object representation to valid poses is considerably non-smooth, leading to poor generation efficiency and restricted generality. To tackle the challenge, we introduce an intermediate variable for grasp contact areas to constrain the grasp generation; in other words, we factorize the mapping into two sequential stages by assuming that grasping poses are fully constrained given contact maps: 1) we first learn contact map distributions to generate the potential contact maps for grasps; 2) then learn a mapping from the contact maps to the grasping poses. Further, we propose a penetrationaware optimization with the generated contacts as a consistency constraint for grasp refinement. Extensive validations on two public datasets show that our method outperforms state-of-the-art methods regarding grasp generation on various metrics. Introduction 3D grasp synthesis studies the problem of generating grasping poses given an input object. It has wide applications ranging from animation, human-computer interaction to robotic grasping. Though it has been researched for many years, only a limited number of works about 3D grasp generation using deep learning have been proposed due to the lack of large grasping data [Corona et al., 2020;Taheri et al., 2020;Jiang et al., 2021;Karunratanakul et al., 2020;Zhang et al., 2021;Taheri et al., 2021]. Recently, a dataset for human grasping objects with annotations of full body meshes and objects meshes have been collected by a multi-view capture rig, and a coarse-to-fine hand pose generation network based on a conditional autoencoder (CVAE) is proposed [Taheri et al., 2020]. In [Karunratanakul et al., 2020], a new implicit representation is proposed for hand and object interactions, * Corresponding author. and a similar CVAE method is used for static grasps generation. Taheri et al. [Taheri et al., 2021] take a step further to learn dynamic grasping sequences including the whole body motion given an object, instead of static grasping poses. Existing methods treat the generation as a black box mapping from an object to its grasp pose distribution. However, this formulation has its defects. On one hand, the mapping from the 3D object space to the pose space represented by rotations is highly non-linear. On the other hand, physical contact is sensitive to small changes in pose, e.g., less than a millimeter change in the pose of a fingertip normal to the surface of an object can make the difference between the object being held or dropped on the floor [Grady et al., 2021]. Therefore, the mapping between 3D object representation to valid poses is non-smooth, as a small change in the pose could make a valid pose invalid. These defects raise a challenge for the network to learn the sparse mapping and generalize to unseen valid poses in the highly non-linear space. In robotics, contact areas between agents and objects are found to be important [Deng et al., 2021;Roy and Todorovic, 2016;Zhu et al., 2015] because localizing the position of possible grasps can greatly help the planning of actions for robotic hands Mandikal and Grauman, 2021;Mandikal and Grauman, 2022]. For ex- Figure 2: The framework of our method. It consists of three stages: ContactCVAE, GraspNet and Penetration-aware Partial Optimization. ContactCVAE takes an object point cloud O as input and generates a contact map C . GraspNet estimates a grasp parameterized by θ from the contact map C . Finally, penetration-aware partial (PAP) optimization refines θ to get the final grasp. ample, and first estimate the contact points for parallel-jaw grippers and plan paths to grasp the target objects. The common assumption in the literature is that the contact area is a point and the contact point generation is treated as a per-point (or pixel voxel) detection problem, i.e. classifying each 3D object point to be a contact or not, which cannot be applied to dexterous hand grasps demonstrating much more complex contact. For dexterous robotic hand grasping, recent work [Mandikal and Grauman, 2021] finds that leveraging contact areas from human grasp can improve the grasping success rate in a reinforcement learning framework. However, it assumes an object only affords one grasp, which contradicts the real case and limits its application. To tackle the limitations, we propose to leverage contact maps to constrain the grasp synthesis. Specifically, we factorize the learning task into two sequential stages, rather than taking a black-box hand pose generative network that directly maps an object to the possible grasping poses in previous work. In the first stage, we generate multiple hypotheses of the grasping contact areas, represented by binary 3D segmentation maps. In the second stage, we learn a mapping from the contact to the grasping pose by assuming the grasping pose is fully constrained given a contact map. The intermediate segmentation contact maps align with the smooth manifold of the object surface: for example, a small change in a valid contact map would likely produce another valid solution (as illustrated in Figure 1), then the corresponding pose can be deterministically established by the following GraspNet and PAP optimization. This manner reduces the challenging pose generation to an easier map generation problem in a low-dimension and smooth manifold, benefiting generation efficiency and generality. The other benefit of the intermediate contact representation is enabling the optimization from the contacts. Different from the optimization for the full grasps from scratch [Brahmbhatt et al., 2019b;Xing et al., 2022], we propose a penetrationaware partial (PAP) optimization with the intermediate contacts. It detects partial poses causing penetration and leverages the generated contact maps as a consistency constraint for the refinement of the partial poses. The PAP optimization constrains gradients from wrong partial poses to affect these poses requiring adjustment only, which results in better grasp quality than a global optimization method. In summary, our key contributions are: 1) we tackle the high non-linearity problem of the 3D generation problem by introducing the contact map constraint and factorizing the generation in two stages: contact map generation and mapping from contact maps to grasps; 2) we propose a PAP optimization with the intermediate contacts for the grasp re-finement; 3) benefiting from the two decomposed learning stages and partial optimization, our method outperforms existing methods both quantitatively and qualitatively. Related Works Human grasp generation is a challenging task due to the higher degrees of freedom of human hands and the requirement of the generated hands to interact with objects in a physically reasonable manner. Most methods use models such as MANO [Romero et al., 2017] to parameterize hand poses, aiming to directly learn a latent conditional distribution of the hand parameters given objects via large datasets. The distribution is usually learned by generative network models such as Conditional Variational Auto-Encoder [Sohn et al., 2015], or Adversarial Generative Networks [Arjovsky et al., 2017]. To get finer poses, many existing works adopt a coarse-to-fine strategy by learning the residuals of the grasping poses in the refinement stage. [Corona et al., 2020] uses a generative adversarial network to obtain an initial grasp, and then an extra network to refine it. [Taheri et al., 2020] follows a similar strategy but uses a CVAE model to output an initial grasp. In recent works, contact maps are exploited to improve robotic grasping, hand object reconstruction, and 3D grasp synthesis. [Brahmbhatt et al., 2019b] introduces a loss for robotic grasping optimization using contact maps captured from thermal cameras [Brahmbhatt et al., 2019a; to filter and rank random grasps sampled by GraspIt! [Miller and Allen, 2004]. It concludes that synthesized grasping poses optimized directly from the contact demonstrate superior quality to other approaches which kinematically re-target observed human grasps to the target hand model. In the reconstruction of the hand-object interaction, [Grady et al., 2021] propose a differentiable contact optimization to refine the hand pose reconstructed from an image. In the 3D grasp synthesis, [Jiang et al., 2021] also exploits contact maps but they only use them to refine generated grasps during inference. Our work differs from these works using contact maps in three aspects: 1) these works use contact maps as a loss for the grasp optimization or post-processing for further grasp refinement while our work exploits the contact maps as an intermediate constraint for the learning of the grasp distribution; 2) in contrast to the learning-based works with contact maps which treat objects-to-grasps as a black box, our work factorizes the grasp synthesis into objects-tocontact maps and contact maps-to-grasps; 3) moreover, these works refine the whole grasps with global optimization methods using contact maps while our penetration-aware partial optimization detects the partial poses causing the penetration and leverages the contact map constraint to optimize the par- = (V, F ) (V ∈ R 778×3 , F ∈ R 1538 denotes the mesh vertices and faces) by the shape parameters β ∈ R 10 and pose parameters θ ∈ R 51 , i.e. M = M(θ, β). In the work, we use the mean shape and use M = M(θ) for brevity. In the first stage, ContactCVAE aims to learn a contact map distribution represented by a latent vector z given an input object using a conditional variational autoencoder. The network takes an object point cloud O ∈ R N ×3 and the contact map C ∈ R N ×1 as the input and learns to make the output contact map C ∈ R N ×1 as close to the input contact map as possible. N is the number of points in O. Each point in the point cloud is represented by its normalized 3D positions. Each point in the contact map takes a value in [0, 1] representing the contact score. During inference, given an object, a contact map C can be generated by sampling from z. In the second stage, GraspNet learns a mapping from the contact map C to the hand mesh M constrained by the map. The pose θ of the predicted mesh M from GraspNet is refined with PAP optimization in the third stage. Figure 3 demonstrates the architecture of the ContactCVAE network, which is a generative model based on CVAE [Sohn et al., 2015]. It consists of two blocks: a Condition-Encoder and a Generator. Condition-Encoder The Condition-Encoder E θc is built on PointNet [Qi et al., 2017]. It takes a point cloud as input to extract local features f l ∈ R N ×64 and global featuresf g ∈ R 1×1024 . f g are then duplicated N times to make a feature map f g ∈ R N ×1024 for matching the shape of f l . These two features are then concatenated as f lg for the conditional inputs for the generator below. Generator The generator G φg follows an encoder-decoder architecture. As shown on the top of Figure 3, the encoder, E θe : et al., 2017] architecture which takes both an object point cloud O and a contact map C as inputs and outputs the latent code z ∈ R 64 . The encoder is only employed in training and is discarded in inference. The latent code z represents a sample of the learned distribution Q(z|µ, σ 2 ) and is used to generate the contact map, where µ, σ 2 denotes the mean and variance of the distribution. We then duplicate the latent code z N times to make the latent feature f z for all the points. The decoder D θ d :(f i z , f i lg ) C i is a classifier for a point i which merges three different features (global f i g , local f i l and latent f i z ) to classify whether the point belongs to a contact map or not. The decoder D θ d uses the MLP architecture and the weights are shared for all points. Testing Stage During inference, as shown in the bottom of Figure 3, we only employ the Conditional-Encoder and decoder D θ d . A latent code z is randomly sampled from a Gaussian distribution and forms the latent feature f z . At the same time, the Condition-Encoder takes an object point cloud to output the global and local features. With these features (f z , f g , f l ), D θ d outputs the grasp contact C for the object. Contact Loss The goal of training the model is to optimize θ e , θ d in order to reconstruct the contact map well. We simplify the goal as a binary classification task. Thus, we adopt the binary cross-entropy loss for the model over all the points, named as L c1 . However, some samples have small contact regions and it is hard for the model to learn those samples well by simply adopting the BCE loss. To address this problem, we additionally introduce the dice loss [Milletari et al., 2016] to train the model. It can assist the model in paying attention to small target regions. In our work, we adopt the dice loss for the same purpose and name as L c2 . the formulation of the two loss is defined as: whereŷ i and y i represent the predicted contact and ground truth of a point i, respectively. Following the training of CVAE [Sohn et al., 2015], we use the KL-Divergence loss regularizing the latent distribution to be close to a standard Gaussian distribution. The loss term is named as L kl . The overall loss function of the ContactCVAE network, L contact , is represented as: ContactCVAE (C, O) z, is based on PointNet [QiL c1 = − N i=0 [y i log(ŷ i ) + (1 − y i )log(1 −ŷ i )],(1)L c2 = 1 − 2 N i=0 yiŷi N i=0 yi+ N i=0ŷ i ,(2)L contact = γ 0 L c1 + γ 1 L c2 + γ 2 L kl ,(3) where the γ 0 = 0.5, γ 1 = 0.5 and γ 2 = 1e − 3 are constants for balancing the loss terms. GraspNet With the assumption of hands full constrained by a contact map, we adopt a mapping function to get the grasping pose from the generated contact from the first stage. As shown in Figure 4, the model takes an object point cloud O and its generated (or reconstructed) contact C as the input to predict the hand mesh for the grasping pose, which is represented by the MANO model [Romero et al., 2017]. Specifically, we employ a PointNet [Qi et al., 2017] to extract the feature and then use an MLP with four hidden layers to regress the MANO parameters. Given the parameters, the MANO model forms a differentiable layer that outputs the hand mesh M . During the training period, we use both ground truth and reconstructed contact maps to train the GraspNet. During inference, we only use the generated contact map to predict the grasp mesh. Both reconstructed and generated contact maps are from the ContactCVAE model in the first stage. Reconstruction Loss We simply adopt the reconstruction loss (L 2 distance) for the predicted vertices, named as L v . The loss on MANO parameters is divided into two parts. We use the L1 loss for the translation parameter and the geodesic loss [Mahendran et al., 2017] for the pose parameter, named as L t and L p respectively. The final reconstruction error can be represented as L R = λ v L v + λ t L t + λ p L p , where λ v =35, λ t =0. 1 and λ p =0.1 are constants balancing the losses. We also use the penetration loss L ptr = 1 |O h in | o∈O h in min i o − V i 2 which punishes pene- trations between the hand and object. O h in denotes the object point subset that is inside the hand. Consistency Loss Similar to the previous work [Jiang et al., 2021], we introduce the contact consistency loss L cst = C − C 2 . Based on the distance between the object and the grasp mesh M , the contact map C can be inferred by normalizing the distance between O and their nearest hand prior vertice. If the grasp mesh M is predicted correctly from the GraspNet, the input contact map C should be consistent with the contact map C . The overall loss of GraspNet, L grasp , is the weighted sum of all the above loss terms: L grasp = L R + λ ptr L ptr + λ cst L cst ,(4) where λ ptr =5 and λ cst =0.05 denote the corresponding loss weights. Figure 5: Left: illustration of our penetration-aware partial (PAP) optimization and the refined pose θ * p . Penetration is detected in the index finger and therefore partial poses θ p on the index finger are to be optimized. Right: the refined pose of global optimization. Penetration-aware Partial Optimization Though GraspNet gives plausible grasps for most cases, physically feasible grasps are sensitive to small errors in poses. For example, small penetration of a fingertip to the surface of an object can make the object drop to the floor. Hence, we propose the penetration-aware partial (PAP) optimization with generated contact maps to provide further constraints for small-scale partial pose refinement. PAP aims to detect the penetration and refine the partial poses causing it while keeping other partial poses of good quality unchanged. To this end, the full hand mesh is divided into six parts: five fingers and the palm. If penetration is detected in the palm area, all the poses are adjusted. If penetration is detected in a finger part and no penetration happens in the palm area, only the partial poses of the finger are adjusted. The loss for the PAP optimization is formulated as: L opt (θ p ) =ω 0 L cst (C , C ) + ω 1 L ptr (M(θ p ), O)+ +ω 2 L h (θ p , θ p ).(5) L cst , similar to the contact consistency loss defined above, is the difference between the generated contact map C and the contact map C . , 2020], defined above. L h = θ p − θ p regularizes the pose hypothesis θ p to stay close to the generated pose θ p . We set ω 0 =0.1, ω 1 =2 and ω 2 =0.2. Figure 5 (Left) shows an example of our partial optimization for the poses θ p of the finger. During the optimization, as the loss mainly results from local wrong partial poses, the global optimization shown on the right side of Figure 5 has two issues 1) the gradient from local wrong partial poses affects other good poses, 2) the gradient cannot take full effect on the refinement for the wrong partial poses, which together results in many failures of small scale refinement. In contrast, our PAP only optimizes the poses causing the errors to get rid of these issues. Experiment Implementation Details We sample N = 2048 points on an object mesh as the input object point cloud. Our method is trained using a batch size of 32 examples, and an Adam optimizer with a constant learning rate of 1e-4. The training dataset is randomly augmented with [−1, 1]cm translation and rotation at three (XYZ) dimensions. All the experiments were implemented in PyTorch, in which our models ran 130 epochs in a single RTX 3090 GPU with 24GB memory. In the Obman dataset [Hasson et al., 2019], all the ground truth contact map is derived by normalizing the distance between the ground truth of the hand and the object. For the inference refinement (both PAP and global optimization), the Adam optimizer with a learning rate of 2.0 × 10 −4 is used. In the refinement process, each input is optimized for 200 steps. Datasets Obman We first validate our framework on the Obman dataset [Hasson et al., 2019], which is a large-scale synthetic dataset, including 3D hand interacting with objects. The hands are generated by a physical-based optimization engine GraspIt! [Miller and Allen, 2004], and are parameterized by the MANO model [Romero et al., 2017]. The dataset contains 8 categories of everyday objects selected from ShapeNet [Chang et al., 2015] with a total of 2772 meshes. The model trained on this dataset will benefit from the diversified object models. The object contact map is derived as [Taheri et al., 2020] by thresholding the normalized distance between the object points and their nearest hand vertices. Points with the distance smaller than a threshold are marked as contact points. ContactPose The ContactPose dataset is a real dataset for studying hand-object interaction, which captures both ground-truth thermal contact maps and hand-object poses. Though the dataset contains only 25 household objects and 2306 grasp contacts, it captures more real interactions. For example, the contact in ContactPose spreads across large sections of the hand, as opposed to that at the fingertips for most cases in Obman. We manually split the dataset into a training and test group according to the object type. Specifically, we use 4 objects (cup, toothpaste, stapler, and flashlight) with 336 grasp contacts as a test set, and the rest for training the model. ContactPose uses the thermal camera-based method to capture the contact region. Evaluation Metrics A good generated pose should be physically stable and should be in contact with the object without penetration. In this work, we adopt three metrics to evaluate the quality of generated grasp poses: (1) Penetration (Ptr) The penetration is measured by the depth (Dep, cm) and the volume (Vol, cm 3 ) between the objects and generated hand meshes. The depth is the maximum or mean of the distances from the hand mesh vertices to the surface of the object if penetration occurs. Following [Jiang et al., 2021;Karunratanakul et al., 2020], the volume is measured by voxelizing the hand-object mesh with voxel size 0.5cm. (2) Simulation Displacement (SD) The simulation displacement is adopted to measure the stability of the generated grasp. We report the average (Mean, cm) and variance (Var, cm) of the Figure 6: Scatter plots about metrics of Ptr-Vol vs. SD for the GT and sampled grasps of the ContactPose test set. Dots in the green box denote the positive samples grasping objects successfully during the simulation. simulation displacement as measured by a physics-based simulator following the same settings as [Jiang et al., 2021;Karunratanakul et al., 2020]. The displacement is the Euclidean distance between the object centers before and after applying a grasp on the object. Though used in the existing work, the results of previous work [Karunratanakul et al., 2020] indicate that a high penetration might correspond to a low simulation value and therefore we suggest readers use it for a rough reference only. (3) Contact Rate (CR, %) A physically plausible hand-object interaction requires contact between the hand and the object. We define a sample as positive if the hand-object contact exists, which means that there exists at least a point on the hand surface is on or inside the surface of the object. The contact rate is the percentage of those positive samples over all the test samples. In addition to the metrics used in the hand generation work, we introduce two more metrics to evaluate the quality of the grasping pose distributions. (4) Grasp Success Rate (GSR, %) The grasp success rate aims to evaluate the rate of grasp success. Specifically, we define the positive sample as the one with Ptr-Vol< 5cm 3 and SD-Mean < 2cm. The success rate is the percentage of those positive samples over all the test samples. (5) Diversity (Div, cm) It is also significant to evaluate the diversity for the generation task. In this work, we use MAE to measure the diversity of generated results. Specifically, we measure the divergence between each generated sample and all other samples and then average them. The formulation of the metrics mentioned above can be found in the supplementary material (Section B). Comparison with state-of-arts To illustrate the advantages of the proposed method, we compared our method with three state-of-art methods: Grab-Net [Taheri et al., 2020], GraspField [Karunratanakul et al., 2020] and GraspTTA [Jiang et al., 2021]. We train the stateof-art methods on ContactPose using their public code, as they do not provide the results or metrics on this dataset. As for Obman, only the results of GraspField are quoted from [Jiang et al., 2021]. The results on Obman and ContactPose dataset in Table 1 show that our proposed method achieves the best performance on all the metrics. Specifically, our method yields the best penetration depth (0.44cm and 0.36cm) and volume (3.94cm 3 and 4.15cm 3 ). More importantly, our method achieves the best performance on diversity (10.14cm and 7.91cm) and GSR (61.37% and 58.97%), indicating the robustness and variety of the results generated from our method. Note that due to the limitation of simulation, grasps with large penetration in many cases can still hold objects while a reasonable grasp pose should embody both low penetration and simulation displacement. Thus the GSR is a more comprehensive metric considering both of them. To verify this, Figure 6 plots penetration volume and simulation displacement of the GT grasps, sampled grasps from GraspTTA [Jiang et al., 2021] and our method for the objects in the testing set of ContactPose. We can see that grasps in the red box of Figure 6 exhibiting larger penetration volume still demonstrate good grasp stability (small simulation displacement). Considering both metrics, the generated grasps from our method are closer to the origin, indicating the results have better stability with smaller penetration (see the comparisons of the green box). Methods Ptr ( Ablation Study To verify the effectiveness of our proposed factorization and PAP optimization, we construct three variants of our method and a baseline, comparing their performances on ContactPose test set. The results are shown in Effectiveness of Factorization By comparing between Param2Mesh and Ours w/o PAP, we can see that our method achieves significant improvement on all metrics, indicating the effectiveness of the two-stage factorization. Especially, the penetration volume, depth and GSR are improved by 57%, 38% and 68% respectively. In addition, the improvement on diversity (Div) indicates that our method generates more diverse samples. To further demonstrate the diversity and generalization performance of our method, we select two test objects (toothpaste and stapler) of ContactPose and visualize the distribution of generated grasp poses for each of them. As shown in Figure 7, we generate 200 grasp poses with small Ptr-Vol (< 5cm 3 ) and SD (< 2cm) for each object, and adopt the t-SNE technique to visualize the distribution of the pose parameters θ. The sample distribution of our method is closer to the distribution of ground truth grasps, indicating better generalization performance. More results are shown in the supplementary material. Effectiveness of PAP Optimization By comparing between Ours and Ours (global opt), we can see that our PAP optimization strategy achieves better performance than global refinement over all the metrics, which presents the effectiveness of our PAP optimization. Quality of Generated Contact Maps When compared with Our (GT) and Our w/o PAP (GT), the metrics for penetration of our methods (Ours and Our w/o PAP) are worse but the margin of corresponding improvement is relatively small. For example, the penetration depth of our methods are 0.36mm and 0.60mm while those of Our (GT) and Our w/o PAP (GT) are 0.40mm and 0.56mm. The comparison indicates the generated maps in the first stage are of high quality. The examples in Figure 8 also show that the generated contacts convey meaningful information for grasping. We can observe that the generated contact map is reasonable, corresponding to the grasp pose. Although there are some failure examples (including unstable grasps and serious penetration), the hand pose is substantially natural as human behavior. Semantic Analysis of Latent Contact Map Space Using the generated object contacts to formulate the hand grasp is one of the contributions and here we show whether our ContactC-VAE model can learn the latent distribution for contact well. In point generation work [Achlioptas et al., 2018], it demonstrates the quality of the generation model by showing that the learned representation is amenable to intuitive and semantically rich operations. Inspired by the work, we conduct the semantic analysis of the latent space learned from our Con-tactCVAE model. The detail of the procedure can be found in Figure 8: The visualization of generated contact maps and grasps for objects from the Obman test set and ContactPose test set. For each example, we present both the predicted grasp pose (left) and the corresponding contact map (right), which is presented in the form of the heat map. the supplementary material (Section D). Figure 1 exemplifies the contact maps and grasping poses generated by interpolating the latent z of the contact maps for TypeA and TypeB. Notice that the grasps change gradually in alignment with the contact maps between Type A and Type B. For example, in Figure 1 (bottom), the yellow arrow and circle on the mouse, denote small differences between the contact maps and the grasp poses. As the contact region gradually appears, the middle finger moves to the corresponding position smoothly. Similar interesting observation can be found for manipulating the phone in Figure 1 (top) where the hand poses change from holding to pressing gradually. Conclusion In this paper, we propose a novel framework for human grasps generation, which holds the potential for different deep architectures. The highlight of this work is exploiting the object affordance represented by the contact map, to formulate a functionality-oriented grasp pose and using penetrationaware partial optimization to refine partial-penetrated poses without hurting good-quality ones. The proposed method is extensively validated on two public datasets. In terms of diversity and stability, both quantitative and qualitative evaluations support that, our method has clear advantages over other strong competitors in generating high-quality human grasps. Although our method achieves great performance over all the metrics, limitations still exist. In some cases, the generated contact maps are ambiguous, resulting in more than one plausible grasping pose. Therefore, our assumption of the grasp is fully constrained by the contact map does not hold. Contact maps with more detailed hand-part segmentation can provide stronger constraints and help to reduce ambiguity. Figure 1 : 1Interpolated contact maps and grasps between different generated contacts (TypeA and TypeB) by our method. Note that the grasping poses (e.g. finger positions denoted in the yellow circle and arrow) change with transitions between two types of contacts and a small change in a valid contact map produces another valid grasp. The intermediate contact maps reduce the non-smooth highnonlinear pose generation problem to a map generation problem in a low-dimension and smooth manifold, benefiting generation efficiency and generality. Figure 3 :Figure 2 32The architecture of ContactCVAE. (a) In the training stage, it takes both an object point cloud and a contact map as input to reconstruct the contact map; (b) In the testing stage, by sampling from the latent distribution, it generates grasp contacts with an object point cloud as the conditional input only. ⊗ means concatenation. tial poses only rather than the whole poses. shows our method pipeline. It generates maps for contact areas by a network naming ContactCVAE, maps the contact maps to grasping poses by the other network naming GraspNet, and refines the generated grasp by a penetration-aware optimization module. In the work, we adoptMANO [Romero et al., 2017] to represent grasps. The MANO model M parameterizes the hand mesh M Figure 4 : 4The architecture of GraspNet. It takes the concatenation of the generated (reconstructed) object contact C and the point cloud O as input to predict the grasp mesh parameterized by MANO. C is obtained by normalizing the distance between the O and their nearest point in M(θ p ). L ptr penalizes the penetration between the hand M(θ p ) and object O as similar in [Jiang et al., 2021] and [Karunratanakul et al. Figure 7 : 7Visualization of the distribution of the generated grasp poses in the test objects (toothpaste and stapler) using t-SNE. Table 2 : 2Self comparison on ContactPose test set. Table 2 . 2Param2Mesh: A baseline for grasp generations learning an end-to-end grasp generation model. It learns the latent distribution of MANO parameter directly. Specifically, we use the same architecture of our ContactCVAE to make a fair comparison but replace the encoder E θe with fully connected layers to take the MANO parameters as inputs. Given a 3D object point cloud and a random sample for the distribution, the decoder D θ d generates MANO parameters directly, which is similar to the previous work[Taheri et al., 2020]. Ours w/o PAP: A variant of our method removing the PAP optimization at the third stage. Ours global opt: A variant of our method simply adopting global refinement at the third stage. Ours (GT) skips ContactCVAE stage, training and testing GraspNet with GT contact maps directly, and Ours w/o PAP (GT) without the PAP optimization. Acknowledgments Contactdb: Analyzing and predicting grasp contact via thermal imaging. Achlioptas, arXiv:1512.03012Guillem Alenya, Francesc Moreno-Noguer, and Grégory Rogez. Ganhand: Predicting human grasp affordances in multi-object scenes. Chengcheng Tang, Christopher D Twigg, Charles C Kemp, and James Hays; Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran SongAlbert PumarolaarXiv preprintProceedings of the IEEE/CVF Conference on Computer Vision and Pattern RecognitionReferences [Achlioptas et al., 2018] Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning rep- resentations and generative models for 3d point clouds. In International Conference on Machine Learning, pages 40- 49, 2018. [Arjovsky et al., 2017] Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial net- works. In International conference on machine learning, pages 214-223, 2017. [Brahmbhatt et al., 2019a] Samarth Brahmbhatt, Cusuh Ham, Charles C Kemp, and James Hays. Contactdb: An- alyzing and predicting grasp contact via thermal imaging. 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{'abstract': '3D grasp synthesis generates grasping poses given an input object. Existing works tackle the problem by learning a direct mapping from objects to the distributions of grasping poses. However, because the physical contact is sensitive to small changes in pose, the high-nonlinear mapping between 3D object representation to valid poses is considerably non-smooth, leading to poor generation efficiency and restricted generality. To tackle the challenge, we introduce an intermediate variable for grasp contact areas to constrain the grasp generation; in other words, we factorize the mapping into two sequential stages by assuming that grasping poses are fully constrained given contact maps: 1) we first learn contact map distributions to generate the potential contact maps for grasps; 2) then learn a mapping from the contact maps to the grasping poses. Further, we propose a penetrationaware optimization with the generated contacts as a consistency constraint for grasp refinement. Extensive validations on two public datasets show that our method outperforms state-of-the-art methods regarding grasp generation on various metrics.', 'arxivid': '2210.09245', 'author': ['Haoming Li \nKey Lab of CS&AUS\nZhejiang University\nHangzhouChina\n', 'Xinzhuo Lin \nKey Lab of CS&AUS\nZhejiang University\nHangzhouChina\n', 'Yang Zhou \nOPPO US Research Center\nPalo AltoUSA\n', 'Xiang Li [email protected] \nOPPO US Research Center\nPalo AltoUSA\n', 'Yuchi Huo [email protected] \nState Key Lab of CAD&CG and Zhejiang Lab\nZhejiang University\nHangzhouChina\n', 'Jiming Chen \nKey Lab of CS&AUS\nZhejiang University\nHangzhouChina\n', 'Qi Ye [email protected] \nKey Lab of CS&AUS\nZhejiang University\nHangzhouChina\n'], 'authoraffiliation': ['Key Lab of CS&AUS\nZhejiang University\nHangzhouChina', 'Key Lab of CS&AUS\nZhejiang University\nHangzhouChina', 'OPPO US Research Center\nPalo AltoUSA', 'OPPO US Research Center\nPalo AltoUSA', 'State Key Lab of CAD&CG and Zhejiang Lab\nZhejiang University\nHangzhouChina', 'Key Lab of CS&AUS\nZhejiang University\nHangzhouChina', 'Key Lab of CS&AUS\nZhejiang University\nHangzhouChina'], 'corpusid': 256459915, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11469, 'n_tokens_neox': 9926, 'n_words': 6741, 'pdfsha': '019392870bb1c5e4d79de51fcb162406e6f1225f', 'pdfurls': ['https://export.arxiv.org/pdf/2210.09245v3.pdf'], 'title': ['Contact2Grasp: 3D Grasp Synthesis via Hand-Object Contact Constraint', 'Contact2Grasp: 3D Grasp Synthesis via Hand-Object Contact Constraint'], 'venue': []}

arxiv

3D Feature Prediction for Masked-AutoEncoder-Based Point Cloud Pretraining Siming Yan The University of Texas at Austin Yuqi Yang Microsoft Research Asia Yuxiao Guo Microsoft Research Asia Hao Pan Microsoft Research Asia Peng-Shuai Wang Peking University Xin Tong Microsoft Research Asia Yang Liu Microsoft Research Asia Qixing Huang The University of Texas at Austin 3D Feature Prediction for Masked-AutoEncoder-Based Point Cloud Pretraining Masked autoencoders (MAE) have recently been introduced to 3D self-supervised pretraining for point clouds due to their great success in NLP and computer vision. Unlike MAEs used in the image domain, where the pretext task is to restore features at the masked pixels, such as colors, the existing 3D MAE works reconstruct the missing geometry only, i.e, the location of the masked points. In contrast to previous studies, we advocate that point location recovery is inessential and restoring intrinsic point features is much superior. To this end, we propose to ignore point position reconstruction and recover high-order features at masked points including surface normals and surface variations, through a novel attention-based decoder which is independent of the encoder design. We validate the effectiveness of our pretext task and decoder design using different encoder structures for 3D training and demonstrate the advantages of our pretrained networks on various point cloud analysis tasks. Introduction Self-supervised pretraining has recently gained much attention. It starts from a pretext task trained on large unlabeled data, where the learned representation is fine-tuned on downstream tasks. This approach has shown great success in 2D images [8,15,18,4,17,61] and natural language processing (NLP) [13,5] . Recently, people started looking into self-supervised pretraining on point cloud data due to its importance in 3D analysis and robotics applications. An important self-supervised pretraining paradigmmasked signal modeling (MSM), including BERT [6], BEiT [4], and masked autoencoders (MAE) [17], has recently been adopted to 3D domains. MSM has a simple setup: a randomly-masked input is fed to the encoder, and a decoder strives to recover the signal at the masked region. MSM is highly scalable and exhibits superior performance in many downstream vision and NLP tasks, outperforming their * Work done when interning at Microsoft Research Asia. fully supervised equivalents. Additionally, it does not require extensive augmentation, which is essential and critical to another self-supervised pretraining paradigm -contrastive learning. In images, a mask refers to a randomly selected portion of the pixels, and the pixel colors or other pixel features in the masked region are to be reconstructed by the decoder. For 3D point clouds, the PointBERT approach [56] masks point patches and recovers patch tokens that are pretrained by a point cloud Tokenizer. As reconstruction features are associated with patches of points, the learned features at the point level are less competitive. MAE-based pretraining schemes [31,19,58,57,28] tackle this problem by pointwise pretext tasks. However, their decoders are designed to recover the positions of the masked points in Cartesian coordinates or occupancy formats ( Fig. 1-left). These designs make an intrinsic difference from 2D MSMs, where there is no need to recover masked pixel locations. This key difference makes MSM pay more attention to capturing the irregular and possibly noisy point distribution and ignore the intrinsic surface features associated with points, which are essential for 3D point cloud analysis. In the presented work, we propose to recover intrinsic point features, i.e., point normals, and surface variations [33] at masked points, where point normals are first-order surface features and surface variations are related to local curvature properties. We clearly demonstrate that the recovery of highorder surface point features, not point locations, is the key to improving 3D MSM performance. Learning to reconstruct high-order geometric features forces the encoder to extract distinctive and representative features robustly that may not be captured by learning to reconstruct point positions alone. Our study justifies the importance of designing signal recovery for 3D MSMs. It aligns 3D MSM learning with MSM development in vision, where feature modeling plays a critical role [46]. To recover point signals, we design a practical attentionbased decoder. This new decoder takes masked points as queries, and stacks several transformer blocks. In each block, self-attention is used to propagate context features over the masked points and cross-attention is applied to fabricate the point features with the encoder's output (As shown in Fig. 1-right and Fig. 2). This design is separable from the encoder design. Therefore, common 3D encoders, such as sparse CNNs, point-based networks, and transformer-based networks, can all be adopted to strengthen the pretraining capability. Another benefit of this decoder design is that the masked point positions are only accessible by the decoder, thus avoiding leakage of positional information in the early stage of the network, as suggested by [31]. We conducted extensive ablation studies to verify the efficacy of our masked feature design and decoder. Substantial improvements over previous approaches and the generalization ability of our pretraining approach are demonstrated on various downstream tasks, including 3D shape classification and 3D shape part segmentation. Our implementation will be released to the public and we hope that our study can stimulate future research on designing strong MAE-based 3D backbones. We summarize the contributions of our paper as follows: -We propose a novel masked autoencoding method for 3D self-supervised pretraining that predicts intrinsic point features at masked points instead of their positions. -We introduce a unique attention-based decoder that can generate point features without relying on any particular encoder architecture. -Our experiments demonstrate that restoring intrinsic point features is superior to point location recovery in terms of Point cloud MAE, and we achieve state-of-the-art performance on various downstream tasks. Related Work Self-supervised pretraining in 3D Self-supervised pretraining is an active research topic in machine learning [29]. The early adoption of self-supervised pretraining for 3D is to use autoencoders [53] and generative adversarial networks [48] to learn shape-level features, mainly for shape classification and retrieval tasks. Other self-supervised pretext tasks, such as clustering and registration, are also developed for 3D pretraining. Later, due to the great ability to learn features at both the instance and pixel levels in a self-supervised manner, contrastive learning [50,15,18,5,9] was introduced into the 3D domains to extract distinctive instance and point-wise features for various downstream tasks [44,51,21,59]. However, contrastive learning requires data augmentation heavily to form positive or negative pairs for effective feature learning. Masked signal modeling in 3D Masked signal modeling using transformer-based architectures for self-supervised learning (SSL) has shown great simplicity and superior performance. PointBERT [56] and PointMAE [31] are two such works that inherit from this idea. PointBERT partitions a point cloud into patches and trains a transformer-based autoencoder to recover masked patches' tokens. In contrast, PointMAE directly reconstructs point patches without costly tokenizer training, using Chamfer distance as the reconstruction loss. Other works like [58,57,28] and [19] explore different strategies for point cloud reconstruction or classification with masking. As discussed in Sec. 1, the pretext tasks of most previous works focus only on masked point locations. Signal recovery in masked autoencoders Masked autoencoders for vision pretraining typically use raw color information in masked pixels as the target signal [17]. However, Wei et al. [46] have found that using alternative image features, such as HOG descriptors, tokenizer features, and features from other unsupervised and supervised pretrained networks, can improve network performance and efficiency. In contrast, existing 3D MAE methods have limited use of point features and struggle with predicting the location of masked points. Our approach focuses on feature recovery rather than position prediction, selecting representative 3D local features such as point normals and surface variation [33] as target features to demonstrate their efficacy. Our study allows for leveraging more advanced 3D features in 3D masked autoencoders, while further exploration of other types of 3D features [24] is left for future work. Figure 2: The pretraining pipeline of our masked 3D feature prediction approach. Given a complete input point cloud, we first separate it into masked points and unmasked points (We use cube mask here for better visualization). We take unmasked points as the encoder input and output the block feature pairs. Then the decoder takes the block feature pairs and query points(i.e., masked points) as the input, and predicts the per-query-point features. Masked 3D Feature Prediction In this section, we present our masked 3D feature prediction approach for self-supervised point cloud pretraining. Our network design follows the masked autoencoder paradigm: a 3D encoder takes a point cloud whose points are randomly masked as input, and a decoder is responsible for reconstructing the predefined features at the masked points. The network architecture is depicted in Fig. 2. In the following sections, we first introduce the masking strategy and 3D masked feature modeling in Sec. 3.1 and 3.2, and then present our encoder and decoder design in Sec. 3.3 and 3.4. Here, the key ingredients of our approach are the design of prediction targets and the decoder, which govern the quality of the learned features. 3D Masking We follow the masking strategy proposed by PointBERT [56] to mask out some portions of an input point cloud and feed it to the encoder. Denote the input point cloud as P ∈ R N ×3 , where N is the number of points. We sample K points using farthest point sampling (FPS). For each sample point, its N/K -nearest neighbor points form a point patch. For a given mask ratio m r , 0 < m r < 1, we randomly select M patches and remove them from the input, where M = min( m r · K , K − 1). In the following, the masked points and the remaining points are denoted by P M and P U , respectively. Target Feature Design As argued in Sec. 1, we advocate against using point locations as the reconstructed target. We choose to reconstruct normal and surface variation at each point, which reflect differential surface properties. On the other hand, our decoder design (to be introduced in Se. 3.4) takes query points as input and output predicted pointwise features. Therefore, the decoder implicitly carries positional information for learning meaningful features through the point cloud point normal surface variation encoder. Given a point cloud, both point normal and surface variations are defined using local principal component analysis (PCA). We first define a covariance matrix C r over a local surface region around p: C r := x∈S Sr(p) (p − x)(p − x) T dx x∈S Sr(p) 1 · dx ,(1) where S S r (p) is the local surface region at p, restricted by a sphere centered at p with radius r. We set r = 0.1 in our case. The ablation details are shown in the supplement. The normal n(p) at p is estimated as the smallest eigenvector of C r . The sign of each normal is computed by using the approach of [20]. Surface variation [33] at p is denoted by σ r (p), in the following form: σ r (p) = λ 1 λ 1 + λ 2 + λ 3 ,(2) where λ 1 ≤ λ 2 ≤ λ 3 are the eigenvalues of C r . Surface variation is a geometric feature that measures the local derivation at point p in a neighborhood of size r on a given surface S . Its original and modified versions have been used as a robust feature descriptor for a variety of shape analysis and processing tasks, such as saliency extraction [32], curved feature extraction [34], shape segmentation [22], and shape simplification [33]. In the limit, i.e., when r → 0, σ r (p) is related to the mean curvature [11]. By varying the radii of S r , multiscale surface variation descriptors can be constructed. In our work, we chose only single-scale surface variation for simplicity. Although both surface normal and surface variation are derived from local PCA, they are complementary to each other in the sense that surface normal carries firstorder differential property while surface variation carries second-order differential property due to its relation to mean curvature. We visualize both features in Fig. 3 and show more examples in supplement. In Sec. 4.3, we show that reconstructing surface normal and surface variation leads to better learned features than reconstructing one of them. Loss function Point normals and surface variations represent first-and second-order surface properties. Their value intervals are also bounded: surface normal has unit length; surface variation is non-negative and not greater than 1 3 . Their value-bounded properties are suitable for easy minimizing the deviation from the prediction to their ground truths, compared to using unbounded features such as curvatures. We denote the point normals and surface variations of P M by N M ∈ R M ×3 and V M ∈ R M , respectively. The loss function for pretraining the masked autoencoders is composed of two terms: L n = N M − N M 2 2 ; (3) L v = V M − V M 1 ;(4) where N M and V M are the predicted versions of N M and V M , respectively. The total loss function L = λ 1 L n + λ 2 L v , where λ 1 = 1, λ 2 = 1 in our case. Encoder Design Unlike most MAE-based approaches that are limited to ViT-based encoders, our approach is not restricted to any specific type of encoder. Common 3D encoders for point clouds are all supported, as long as the encoder outputs a set of learned features bind to spatial blocks, where spatial blocks could be point patches used for ViT-like transformer encoders [56,31,28,58], set abstractions used by PointNet++-like encoders [36,37], or coarse voxels used by sparse CNN-based encoders [43,14,10]. In the following, we briefly review these typical encoders and their adaption for our pretraining. ViT-based encoders These encoders first embed point patches via PointNet [35], then send these patch tokens to a standard transformer that includes several multihead selfattention layers and feedforward layers. The transformer outputs the fabricated token features, corresponding to every input point patch. The token feature f i and the patch center c i form a block feature pair B i = {f i , c i }, which is needed by our decoder. Here we can call f i block feature and c i block centroid. PointNet++-like encoders In these encoders, the network features are aggregated through a number of set abstraction levels. We take the learned features and the centroids at the coarsest set abstractions as block feature pairs. Sparse CNN-based encoders These encoders apply 3D convolution on sparse voxels from the finest level to the coarsest level. Multiple convolution layers and resblocks are commonly used. We interpolate the coarse voxel features at the centroids of the unmasked patches and use these interpolated features and the patch centroids to form our block feature pairs. As suggested by [31], the early leaking of masked point information to the network could jeopardize feature learning. We adopt this suggestion: feed the unmasked points to the encoder only, and leave the masked points to the decoder. Decoder Design Decoder structure We design an attention-based decoder to restore the target features at masked regions. The decoder takes the block feature pairs B : = {B i } b i=1 from the encoder and a query point set Q, i.e. ,the masked point set P M . It is composed of a stack of l transformer blocks, where l = 4 in our case (See Fig. 2). Each block contains a self-attention layer and a cross-attention layer. The self-attention layer takes the query points and their positional embeddings as input and outputs the per-query point features, denoted by S in . Then S in and the encoder block features B are passed into the cross-attention layer, where S in serves as attention query, the block features serve as attention key and value, and the block centroids are the positional embedding of the block features. The output per-point features from the last block go through an MLP head to predict the target features at the query points. Efficacy of self-attention layers At first glance, it is sufficient to use cross-attention layers only for predicting perpoint features. The recent masked discrimination work [28] obeys this intuition for its decoder design, no information exchanged between different query points. Instead, we introduce the self-attention layer to propagate information over query points and use multiple attention blocks to strengthen the mutual relationship progressively. We found that our design significantly improves feature learning, as verified by our ablation study (see Sec. 4.3). Supporting of various encoders In the above design, the decoder needs block feature pairs only from the encoder, thus having great potential to leverage various encoder MaskDiscr [28], MaskSurfel [58]. Here, note that the absolute colors from different methods are not comparable as their feature spaces are not the same, instead the point color difference from the same method is a good visual measurement for assessing feature discriminativity. First row: By comparing the colors of the two front legs, we can see our pretrained network produces more discriminative features than PointBERT and MaskSurfel. Second row: It is clear the features on the different chair legs from our methods, as well as the features of two chair arms, are more discriminative. structures, not limited to ViT-based transformer structures. This advantage is verified by our experiments (see Sec. 4). Feature reconstruction versus position reconstruction Note that our decoder and loss design do not explicitly model point positions, which are zero-order surface properties complementary to surface normals and surface variations. Instead, the decoder predicts feature values at query points. Therefore, the zero-order positional information is already encoded implicitly. This explains why our approach is superior to baseline approaches that reconstruct point positions for feature learning (See Sec. 4.2). Query point selection Due to the quadratic complexity of self-attention, the computational cost for a full query set could be much higher. In practice, we can randomly choose a point subset from P M as the query set during training. By default, we use all masked points as queries. Experiment Analysis We conducted a series of experiments and ablation studies to validate the efficacy and superiority of our masked 3D feature prediction approach, in short MaskFeat3D, for point cloud pretraining. Experiment Setup Pretraining dataset We choose ShapeNet [7] dataset for our pretraining, following the practice of PointBERT [56] and previous 3D MAE-based approaches [31,58,28]. ShapeNet [7] contains 57 748 synthetic 3D shapes from 55 categories. We sample 50 000 points uniformly on each shape and select 128 nearest points from them for each point in the point cloud for constructing the local region to approximate surface variation. During pretraining, 2048 points are randomly sampled to create the point cloud. Network training We integrated different encoders with our masked 3D feature prediction approach, including the ViT-based transformer used by [31], sparse-CNN-based encoder [10], and PointNeXt encoder [37] which is an advanced version of PointNet++. We implemented all pretraining models in PyTorch and used AdamW optimizer with 10 −4 weight decay. We use PointBERT's masking strategy for ShapeNet pretraining, and the best masking ratio is 60% empirically. The number of transformer blocks in the decoder is 4. The learning rates of the encoder and the decoder are set to 10 −3 and 10 −4 , respectively. Standard data augmentation such as rotation, scaling, and translation are employed. All models were trained with 300 epochs on eight 16 GB Nvidia V100 GPUs. The total batch size is 64. Downstream Tasks We choose shape classification and shape part segmentation tasks to validate the efficacy and generalizability of our pretrained networks. - each model has 2∼6 parts. Following the standard evaluation protocol [36], 2048 points are sampled on each shape. For evaluation, we report per-class mean IoU (cls. mIoU) and mean IoU averaged over all test instances (ins. mIoU). The training-and-test split of the above tasks follows existing works. For these downstream tasks, we employ the task-specific decoders proposed by PointMAE [31] and reload the pretrained weights for the encoder. Training details are provided in the supplemental material. Efficacy of 3D Feature Prediction We evaluate the efficiency of 3D feature prediction for MAE-based pretraining from two aspects: feature discriminativity and network performance. Feature visualization We assess the discriminativity of learned features by visualizing point features as follows. For a given point on a point cloud, we linearly interpolate block features at this point, using the inverse of its distance to the block centroids as the interpolation weights. We then project all the interpolated point features into the 3D space, via T-SNE embedding [41]. The 3D space is treated as the RGB color space for assigning colors to points. Here, the block features are from the pretrained encoder, not finetuned on downstream tasks. The color difference between different points characterizes their feature differences, and distinguishable features are preferred by many downstream tasks. The first column of Fig. 4 shows the point features of two chair shapes learned by our approach using the ViTbased encoder. We also visualize the point features learned by PointBERT [56] and other MAE-based pretraining approaches: PointMAE [31], MaskDiscr [28], MaskSurfel [58], where all these methods use a similar ViT-based encoder. Fig. 4 shows that our learned features are more discriminative than other methods. For instance, the legs of the chair in the first row from our method -MaskFeat3D, are more distinguishable than the results of PointBert and MaskSurfel, because of their clear color difference; the two arms and different legs in the second row from MaskFeat3D are also more discriminative than those from other methods. More examples are provided in the Method Acc. (%) 3D-GAN [47] 83.3 Latent-GAN [1] 85.7 SO-Net [26] 87.3 MAP-VAE [16] 88.4 Jigsaw [39] 84.1 FoldingNet [54] 88.4 DGCNN + OcCo [42] 89.7 DGCNN + STRL [23] 90.9 DGCNN + IAE [52] 92.1 Point-M2AE [57] 92.9 Transformer + OcCo * [42] 89. supplementary material. Network performance The advantages in learning discriminative features by our masked feature prediction approach are further verified by its superior performance in downstream tasks. We also compare the performance of our approach with PointBERT [56] ,PointMAE [31], and MaskDiscr [28] on ShapeNetPart segmentation with less labeled data. In this experiment, we randomly select 1% labeled data from each category, and finetune the network with all selected data. The performance is reported in Tab. 1, which shows that using our pretrained network leads to much better performance than the baseline methods. Comparison with supervised approaches Compared with state-of-the-art supervised methods, our approach again achieves superior performance than most existing works as seen from Tab. 2, including PointNet++ [36] ,PointCNN [27], DGCNN [45], MinkowskiNet [10], PointTransformer [60] and PointMLP [30]. It is only inferior to the approaches that use advanced encoder structures such as stratified transformer [25] and PointNeXt [37]. Encoder replacement To make a more fair comparison, we replaced the PointViT encoder with the PointNeXt's encoder, and retrained our pretraining network, denoted as MaskFeat3D (PointNeXt). From Tab. 2, we can see that our pretraining approach with this enhanced encoder can yield SOTA performance on all the downstream tasks, surpassing PointNeXt trained from scratch. Its performance is also much better than the recent 3D MAE work-Point-M2AE [57] which employs multiscale autoencoder and the multiscale masking strategy to improve pretraining. We also used MinkowskiNet [10] as our pretraining encoder, the performance gain over MinkowskiNet trained from scratch is +0.7% overall accuracy improvement on ScanObjectNN classification, and +0.3% on ShapeNetPart segmentation. Please refer to the supplementary material for details. Linear SVM We also evaluate the performance of different pretrained networks by fixing the pretrained weights and using linear SVMs on ModelNet40 classification (see Tab. 3). Compared with other MAE-based approaches that use the same ViT-based encoder structure, our approach achieves the best classification accuracy. Few-shot Classification To perform few-shot classification on ModelNet40, we adopt the "K-way N-shot" settings as described in prior work [42,56,31]. Specifically, we randomly choose K out of the 40 available classes and sample N+20 3D shapes per class, with N shapes used for training and 20 for testing. We evaluate the performance of MaskFeat3D under four few-shot settings: 5-way 10-shot, 5-way 20-shot, 10-way 10-shot, and 10-way 20-shot. To mitigate the effects of random sampling, we conduct 10 independent runs for each few-shot setting and report the mean accuracy and standard deviation. Our approach shows significant improvement on all settings. Overall, the improvements of our approach are consistent across different backbone encoders and datasets. Ablation Study We proceed to present an ablation study to justify various design choices. For simplicity, we choose the shape classification task on ScanObjectNN, where the gaps under different configurations are salient and provide meaningful insights on the pros and cons of various design choices. Decoder design The primary question that arises is whether it is essential to disregard point position recovery. PointMAE's decoder follows a standard ViT-like architecture, utilizing a fully connected (FC) layer to directly predict the masked point coordinates. We implemented this decoder to predict our target features. However, since their decoder design does not encode masked point position, it cannot solely predict target features without predicting point position. To address this, we follow the approach proposed in [58] and employ position-index matching for feature loss computation. As shown in Tab. 6, even though incorporating point features as the predicting target can enhance performance, the overall performance still significantly lags behind our design. This experiment highlights the significance of both point feature prediction and disregarding point position recovery. Target feature choice In Tab. 6, the experiment shows that: (1) All combinations of point normal and surface variation can yield significant improvements over existing MAE approaches that recover point positions (cf. Tab. 1); (2) using both point normals and surface variations yields the best performance. As discussed in Sec. 3.2, this is due to the fact that they correspond to first-and second-order differential properties. They are relevant but complementary to each other. Therefore, reconstructing them together forces the encoder to learn more informative features than merely reconstructing one of them. Decoder depth Tab. 5-a varies the number of transformer blocks (decoder depth). A sufficient deep decoder is necessary for feature learning. Increasing the number of blocks from 2 to 4 provides +1.5% improvement on ScanObjectNN classification task. The performance drops when increasing the depth further, due to the overfitting issue. Interestingly, we note that a 1-block decoder can strongly achieve 85.8% accuracy, which is still higher than the runnerup method (PointMAE). Data augmentation Tab. 5-b studies three traditional data augmentation methods: rotation, scaling, and translation. Since the standard scaling could change the surface normal and variation, we scale the shape by using the same factor on 3 different axis. The experiments show that rotation and scaling play a more important role. Masking ratio. Tab. 5-c varies the masking ratio of input point cloud, which is another important factor on our approach. When the masking ratio is too large, e.g. , 90%, the remaining part contains too limited information, which makes the task too hard to complete. When masking ratio is too small, e.g. , 40%, the task becomes too simple and impedes the feature learning. In our experiments, masking ratio=60% shows the best performance. Decoder block design We tested whether the self-attention layer in our decoder is essential. By simply removing selfattention layers and using cross-attention layers only, we find that the performance has a large drop (-2.0), see Tab. 5-d. Number of query points Finally, we varied the number of query points used by our decoder to see how it affects the network performance. Tab. 5-e shows that more query points lead to better performance. Here, "query/mask" is the ratio of selected query points with respect to the total number of masked points. Scene-level Pretraining Extension In principle, masked point cloud autoencoders could be scaled to noisy, large-scale point clouds. Additionally, we conducted an extension experiment on real-world scene-level data to evaluate our approach. Specifically, we pretrained our model on the ScanNet [12] dataset and evaluated its performance on 3D object detection and indoor semantic segmentation tasks using the S3DIS [3] Table 7: Area 5 Semantic segmentation and detection results on S3DIS. † represents the from scratch results and MaskFeat3D in the same section represents the fine-tuning results using pretrained weights under same backbone. experiment, we observed that surface normal has a minor influence on the pre-training, while surface variation remains a robust feature. Moreover, we discovered that color signal could be an effective target feature. Hence, we pre-trained our model with surface variation and color as the target features, and then fine-tuned the pre-trained encoder on the downstream tasks. As shown in Table 7, we choose two most recent works, PointNeXt [37] and FCAF3D [38] on objection detection and semantic segmentation tasks as the network backbones respectively, and our model exhibits consistent improvements in both tasks, which further proves the generalizability of our approach on noisy, large-scale point clouds. Although the concrete scene-level experiments are not the main focus of this paper, the results indicate that this is a promising direction. Conclusion Our study reveals that restoration of masked point location is not essential for 3D MAE training. By predicting geometric features such as surface normals and surface variations at the masked points via our cross-attention-based decoder, the performance of 3D MAEs can be improved significantly, as evaluated through extensive experiments and downstream tasks. Moreover, the performance gains remain consistent when using different encoder backbones. We hope that our study can inspire future research in the development of robust MAE-based 3D backbones. Figure 1 : 1Comparison of standard Point-MAE and our proposed method. Unlike standard Point-MAE that uses masked points as the prediction target, our method use a novel attentionbased decoder to leverage masked points as an additional input and infer the corresponding features. Figure 3 : 3Visualization of point features.. The point normal is color-coded by the normal vector. The surface variation is colorcoded where white indicates low value and red indicates high value. Figure 4 : 4Feature visualization of different pretrained approaches including our MaskFeat3D, PointBERT[56], PointMAE[31], Comparison with MAE-based approaches We compare our approach with other MAE-based approaches that use the same encoder structure. Tab.1 reports that: (1) the performance of all MAE-based methods surpasses their supervised baseline -PointViT; (2) our strategy of reconstructing point features instead of point positions yields significant improvements in ScannObjectNN classification, improving overall accuracy on the most challenging split, PB-T50-RS, from 85.7% (MaskSurfel) to 87.7%, and showing consistent improvements on other splits and ShapeNetPart segmentation. Shape classification: The experiments were carriedTable 1: Performance comparison of MAE-based approaches on downstream tasks. All the methods use the same transformer backbone architecture. † represents the from scratch results and all other methods represent the fine-tuning results using pretrained weights.out on two different datasets: ModelNet40 [49] and ScanObjectNN [40]. ModelNet40 is a widely used synthetic dataset that comprises 40 classes and contains 9832 training objects and 2468 test objects. In contrast, ScanObjectNN is a real-world scanned dataset that includes approximately 15 000 actual scanned objects from 15 classes. It is divided into three evaluation splits: OBJ-BG, OBJ-ONLY, and PB-T50-RS, with PB-T50- RS being the most challenging for recognition. As the domain gap between ShapeNet and ScanObjectNN is larger than that between ShapeNet and ModelNet40, the evaluation on ScanObjectNN is a good measure of the generalizability of pretrained networks. -Shape part segmentation ShapeNetPart Dataset [55] contains 16 880 models from 16 shape categories, and Method ScanObjectNN ShapeNetPart ShapeNetPart(1% labels) OBJ-BG OBJ-ONLY PB-T50-RS ins. mIoU cls. mIoU ins. mIoU cls. mIoU Transformer † [56] 79.9 80.6 77.2 85.1 83.4 77.6 72.2 PointBERT [56] 87.4 88.1 83.1 85.6 84.1 79.2 73.9 MaskDiscr [28] 89.7 89.3 84.3 86.0 84.4 78.8 72.3 MaskSurfel [58] 91.2 89.2 85.7 86.1 84.4 - - PointMAE [31] 90.0 88.3 85.2 86.1 - 79.1 74.4 MaskFeat3D 91.7 90.0 87.7 86.3 84.9 80.0 75.1 Method ScanObjectNN ShapeNetPart OBJ-BG OBJ-ONLY PB-T50-RS ins. mIoU cls. mIoU PointNet [35] 73.3 79.2 68.0 - - PointNet++ [36] 82.3 84.3 77.9 85.1 81.9 PointCNN [27] 86.1 85.5 78.5 86.1 84.6 DGCNN [45] 82.8 86.2 78.1 85.2 82.3 MinkowskiNet [10] 84.1 86.1 80.1 85.3 83.2 PointTransformer [60] - - - 86.6 83.7 PointMLP [30] 88.7 88.2 85.4 86.1 84.6 StratifiedTransformer [25] - - - 86.6 85.1 PointNeXt [37] 91.9 91.0 88.1 87.1 84.7 Point-M2AE [57] 91.2 88.8 86.4 86.5 84.9 MaskFeat3D 91.7 90.0 87.7 86.3 84.9 MaskFeat3D (MinkowskiNet) 85.1 87.0 80.8 85.6 83.5 MaskFeat3D (PointNeXt) 92.7 92.0 88.6 87.4 85.5 Table 2 : 2Comparison with supervised methods. Methods in the first section are supervised approaches. Table 3 : 3Linear evaluation for shape classification onModelNet40. This task is sensitive to the encoder backbone. Different * methods use the same Transformer encoder backbone. Table 4 : 4Few-shot classification on ModelNet40. We report theaverage accuracy (%) and standard deviation (%) of 10 independent experiments. ( a ) aDecoder depth# blocks ScanNN 1 85.8 2 86.2 4 87.7 8 87.5 12 87.1 (b) Data augmentation rot scale trans ScanNN √ - - 87.0 - √ - 85.9 √ √ - 87.7 - √ √ 85.1 √ √ √ 86.7 (c) Mask ratio ratio ScanNN 40% 86.8 60% 87.7 90% 86.5 (d) Decoder attention attention type ScanNN cross only 85.7 cross+self 87.7 (e) Query point ratio query/mask ScanNN 25% 85.7 50% 86.2 75% 86.6 100% 87.7 Table 5 : 5Ablation studies of our design choices. Please refer to Sec. 4.3 for a detailed analysis.Method Target Feature ScanNN PointMAE position only 85.2 position + normal * 85.7 position + surface variation * 85.9 position + normal + variation * 86.0 MaskFeat3D normal 86.5 surface variation 87.0 normal + variation 87.7 Table 6 : 6Ablation study on different features. * uses positionindex matching[58] for feature loss computation. dataset. The training details can be found in the supplementary material. In thisMethod Sem Seg Detection mIoU OA mAP 0.25 mAP 0.5 PointNeXt † [37] 70.8 90.7 - - MaskFeat3D 71.7 91.7 - - FCAF3D † [38] - - 66.7 45.9 MaskFeat3D - - 71.6 49.2 Learning representations and generative models for 3d point clouds. Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, Leonidas Guibas, International conference on machine learning. PMLRPanos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representations and generative models for 3d point clouds. 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{'abstract': 'Masked autoencoders (MAE) have recently been introduced to 3D self-supervised pretraining for point clouds due to their great success in NLP and computer vision. Unlike MAEs used in the image domain, where the pretext task is to restore features at the masked pixels, such as colors, the existing 3D MAE works reconstruct the missing geometry only, i.e, the location of the masked points. In contrast to previous studies, we advocate that point location recovery is inessential and restoring intrinsic point features is much superior. To this end, we propose to ignore point position reconstruction and recover high-order features at masked points including surface normals and surface variations, through a novel attention-based decoder which is independent of the encoder design. We validate the effectiveness of our pretext task and decoder design using different encoder structures for 3D training and demonstrate the advantages of our pretrained networks on various point cloud analysis tasks.', 'arxivid': '2304.06911', 'author': ['Siming Yan \nThe University of Texas at Austin\n\n', 'Yuqi Yang \nMicrosoft Research Asia\n\n', 'Yuxiao Guo \nMicrosoft Research Asia\n\n', 'Hao Pan \nMicrosoft Research Asia\n\n', 'Peng-Shuai Wang \nPeking University\n\n', 'Xin Tong \nMicrosoft Research Asia\n\n', 'Yang Liu \nMicrosoft Research Asia\n\n', 'Qixing Huang \nThe University of Texas at Austin\n\n'], 'authoraffiliation': ['The University of Texas at Austin\n', 'Microsoft Research Asia\n', 'Microsoft Research Asia\n', 'Microsoft Research Asia\n', 'Peking University\n', 'Microsoft Research Asia\n', 'Microsoft Research Asia\n', 'The University of Texas at Austin\n'], 'corpusid': 258170309, 'doi': '10.48550/arxiv.2304.06911', 'github_urls': [], 'n_tokens_mistral': 17538, 'n_tokens_neox': 15232, 'n_words': 8559, 'pdfsha': 'cc50e939de93bd2716f2e4c770660558eecfae83', 'pdfurls': ['https://export.arxiv.org/pdf/2304.06911v1.pdf'], 'title': ['3D Feature Prediction for Masked-AutoEncoder-Based Point Cloud Pretraining', '3D Feature Prediction for Masked-AutoEncoder-Based Point Cloud Pretraining'], 'venue': []}

arxiv

Graphene-based spin switch device via modulated Rashba Field and Strain G S Diniz E Vernek F M Souza Instituto de Física Curso de Física Universidade Federal de Uberlândia 38400-902UberlândiaMGBrazil Instituto de Física Universidade Federal de Goiás 75801-615JataíGOBrazil Universidade Federal de Uberlândia 38400-902UberlândiaMGBrazil Graphene-based spin switch device via modulated Rashba Field and Strain graphene nanoribbonspin polarized FETspin-orbituniaxial strain We investigate the spin-resolved transport in a two-terminal zigzag graphene nanoribbon device with two independent gate induced Rashba spin-orbit coupling regions and in the presence of strain. By employing a recursive Green's function technique to the tight-binding model for the graphene nanoribbon, we calculate the spin-resolved conductance of the system. We show that by switching the sign of one of the gates it is possible to select which spin component will be transmitted. Moreover, our results show that an uniaxial strain applied to the nanoribbon plays a significant role in the transport, providing and additional manner to control the spin-polarized conductance. This makes the present system a potential candidate for future implementations of spin-based mechanical strain sensors. electron-electron interaction. In Ref. [27] a finite chain of quantum circular rings was used to investigate the electronic transport in the presence of modulated Rashba SOC. They demonstrated that periodic modulations of Rashba SOC were able to widen up transport gaps and produce an exotic nearly square-wave conductance [27]. As compared to the III-V nanostructures, graphene-based devices have the advantage of displaying several intriguing phenomena [28] such as a versatile band structure with localized edge states, [29] zero-conductance Fano resonances, [30] quantum spin Hall effect, [10,31,32] exceptional mechanical properties [33] and spin-valley filtering. [34] All these properties are handy for the control of their transport properties. Along graphene-based structures, several proposals for using graphene as spin and valley filters exploring SOC effects in a single barrier have emerged in the past few years [34,35,36]. In this work we consider a simple device composed of a ZGNR deposited on top of two spatially separated gates. These gates are used to induce spatially modulated Rashba SOC. [22] By using a Green's function method, we calculate the differential conductance and show that by appropriately tuning the Rashba SOC with the gate voltages, it is possible to control the spin-polarized conductance of the system. Moreover, we have studied the effect of an uniaxial strain along different directions of the ribbon. We observe a strong dependence of the spin-polarized conductance with strain, [37] by an appropriate combination of gate voltages. The strain can modify the transport of a selected spin component and can be used to design spintronic devices, e.g. spin-based mechanical sensors. Theoretical Model For concreteness, we consider a device composed by a ZGNR deposited on the top of a substrate with underneath gates that induce a Rashba SOC with tunable parameters V g1 and V g2 (see illustration in Fig. 1). The device is attached to pristine semi-infinite ZGNR leads of identical chirality at both ends. To inject (collect) spin-polarized electrons into (from) the device, an induced ferromagnetic ZGNR lead can be used for this purpose. [38,39,40,41,42,43] Figure 1: Schematic representation of the nanoribbon device. A single layer ZGNR is deposited on an appropriated substrate; local gates V g1 and V g2 underneath the region l R (cyan) of the ZGNR control the Rashba SOC. The region lp (gray) between the gates is pristine ZGNR width. Proximity effect to exchange coupled ferromagnetic insulators Fm1 and Fm2 [38,39,40] or electric fields [42,43] can be used to polarize the electrons in the ZGNR lead. The modulated Rashba SOC system generated by the gates underneath the ZGNR (see Fig. 1) is modeled using a π-orbital orthogonal tight-binding Hamiltonian, H = i,j σσ t ij δ σσ + iλ Rẑ · ( s × d ij ) c † iσ c jσ + H.c.,(1) where c † iσ (c iσ ) is the π-orbital creation (annihilation) operator for an electron in the i-th site with spin σ, d ij is a lattice vector pointing from j-th to the i-th site of the ZGNR, s is a vector whose components are the Pauli matrices andẑ is a unit vector perpendicular to the ZGNR plane. t ij =t 0 is the nearest neighbors hopping amplitude on the honeycomb lattice. Finally, λ R represents the Rashba SOC strength, [22] that is induced by the strong electric fields generated only within the region right above the gates with width l R , otherwise we set λ R to zero. Strain effects -In addition to the Rashba field we are also interested in strain effects. The application of an external uniaxial stress to the ZGNR, or the deposition of ZGNR on top of a substrate may induce an uniaxial strain. [44,45,46] To achieve a controllable uniaxial stress the graphene nanoribbons can be deposited on flexible substrates, which can be perfectly stretched along specific directions. [47,48,49,50] To simulate this uniaxial strain in our system, we will consider a strain-dependent hopping parameter in our tight-binding model. [51] This simple model is capable of capturing the main consequences of uniaxial strain on the band structure of graphene and ZGNR. [52,53,54] Here the strain modified distances between carbon atoms are described by d s i = (I + ) d i , with d i (i= 1, 2, 3) the unstrained vectors for nearest-neighbors, I is the identity matrix and is the strain tensor defined as [51] = ε   cos 2 θ − ν sin 2 θ (1 + ν) cos θ sin θ (1 + ν) cos θ sin θ sin 2 θ − ν cos 2 θ   .(2) Here, ν (= 0.165) is the Poisson's ratio with the value known for graphite, [51] θ is the direction of strain and ε is the strain modulus. The hopping matrix element is affected by the strain as t ij = t 0 e −3.37(d s i /a0−1) , in which t 0 = 2.7eV is the unstrained hopping parameter [51] and a 0 (set as the unity) is the C-C distance. The θ= 0 direction is parallel to the zigzag chain, and θ= π/2 is along armchair direction. It is known that strain can induce band gap in armchair GNR, [55,37] although no band gap is observed in ZGNR. [37] To illustrate the uniaxial strain effects on the conductance profiles of ZGNR with modulated Rashba field, we assume that the entire system, composed by the leads and the central conductor are under the influence of stress, so that we avoid any lattice mismatch at the interface. [56] Conductance-To calculate the spin-resolved conductance, we use a surface Green's function approach in real space. [57] For this purpose, we divide the two G r/a C (E) = (ω ± − H C − Σ L − Σ R ) −1 ,(3) where a/r represents the advanced/retarded Green's function (with energy ω ± = E ± iη, respectively, η → 0), and E is the energy of the injected electron (the Fermi energy). H C denotes the Hamiltonian in the central conductor and Σ µ=L,R are the self-energies for the connected left/right leads, Σ µ = H † µC g µ H µC , where g µ is the local Green's function at the end of the semi-infinite left and right leads. [57] The matrix element H µC gives the coupling between leads and central conductor. The spin-dependent conductance through the central conductor is then calculated by, G σσ = G 0 T r Γ L σ G r C,σσ Γ R σ G a C,σσ ,(4) where the trace runs all the lattice sites in the central conductor. Here G 0 = e 2 /h is the quantum of conductance and Γ µ σ are the coupling matrices for the leads, associated to the spin-diagonal self-energies Γ µ = i Σ r µ − Σ a µ . [58,57] When Rashba SOC is turned on in the device, the conductance profile will have two different spin-dependent components: the spin-conserving component G σσ ), with σ =↑ or ↓, Numerical Results and Discussions Throughout this work we will assume a device composed of a 26-ZGNR (with width (3/2N Z − 1), N Z = 26) with a specific width of 5.4nm and length of 21.9nm. For wider ribbons, there are more conducting channels available at moderated energy of injected electron, henceforth, the device will increase its complexity due to possible inter-channel scattering. Although, close to the Fermi level there will be the same amount of conducting channels. We have also checked the length dependence for a fixed region of Rashba field λ R , and increasing l P , and it is indeed relevant to the device prototype, but only for higher energy doping (beyond 0.15t 0 ). Therefore, for electrons injected with energies close to the Fermi level, the separation between the underneath gates is irrelevant, as the injected electron can not feel such fields in a longer length scale beyond λ R region, which is a characteristic that might be relevant in the experimental setup. Notice that although we choose a specific width, the results presented show a general behavior of the ZGNR devices prototypes at low energy regime, which is interesting for electronic transport. In what follows we set the Rashba parameter λ R = 0.1t 0 and V g1 = V 0 (vertical dashed line in Fig. 2), that produces a z-dependent potential V (z), leading to a Rashba SOC λ R = −(e/2m 2 v f )(dV /dz), where dV /dz = E is the electric field perpendicular to the ZGNR plane. we see that G σσ = G 0 for the entire range of energy shown while G σσ = 0. G σσ G σσ l R =W l R =2W l R =W l R =2W (a) (c) (b) (d) (e) V g2 >0 V g2 <0 dashed: λ R =0, solid: λ R =0.1t 0 G σσ G σσ G σσ G σσ When λ = 0.1t 0 , however both conductances oscillate with opposite phase (note that the maximum of G σσ corresponds to the minimum of G σσ = 0 and vice versa). These oscillations result from the spin rotation produced by the SOC. The period of oscillation is due to competing effects. As the energy increases the SOC becomes more pronounced, however the electrons travel faster across the device, having less time to precess. This turns into slower oscillations in the conductance for increasing energy. [9] Besides, for larger energies more conducting channels contributes to the conductance. The system is more tunable in the case of two independently gated regions. In Fig. 3(b) and 3(c) we show the spin dependent conductances G σσ and G σσ vs energy for l R = W and l R = 2W , respectively and for V g2 = V 0 . When comparing these results with those of Fig. 3(a) we see that for l R = W the conductances oscillate much slower with the energy because the influence of the SOC is smaller (remember that while traveling across the central region l p the electrons preserve their spin because there is no SOC in this region). Note that when l R increase for l R = 2W [as in Fig. 3(c)] the profile of the curves approaches that of the Fig. 3(a). orientation when going through the region over V g2 . This switch between G σσ and G σσ as V g2 changes sign can be useful to control the current intensity when both leads are ferromagnetic. For instance, when the magnetization of the leads are parallel aligned, the transport will be dominated by G σσ so the current will be higher for V g2 > 0 than for V g2 < 0. For antiparallel alignment of the leads magnetization, we expect the opposite, i.e., higher current for V g2 < 0 than for V g2 > 0, since in this configuration the transport will be dominated by G σσ . Another appealing behavior of our device is revealed when it is under strain. 1.0 (a) ε =0.1 θ =0 V g2 >0 V g2 <0 (c) ε =0.1 θ =0 (b) ε =0.1 θ =π/2 (d) ε =0.1 θ =π/2G/G 0 (a) V g2 >0 (b) V g2 <0 (c) ε =0.1 solid: G σσ dashed: G σσ black: θ =0 red: θ =π/2 solid: G σσ black: V g2 >0 dashed: G σσ red: V g2 <0 To appreciate this, in Fig. 4 we show the effect of the uniaxial strain effects on the spin-polarized conductance. The strain parameter is taken as ε = 0.1 in two different directions: (i) θ = 0 (along the ribbon) and (ii) θ = π/2 (transversal to the ribbon). Here only l R = W is considered. For θ = 0 [ Fig. 4(a)] the conductances G σσ and G σσ preserve their main features seen in the unstrained case, for both positive and negative V g2 . For a strain along θ = π/2 [ Fig. 4(c)] the variation of G σσ and G σσ with energy is faster as compared with the unstrained case. Because for θ = π/2 the conductance is more sensitive to the energy we note also that additional structures in the conductance are seen for higher energies that were note seen in the case of θ = 0. Physically, this is because a positive ε along θ = π/2 increases the width of the ribbon (for a fixed number of carbon atoms), reducing the transversal confinement. In Fig. 5(a) and 5(b) we also fixed the energy at E = 0.05t 0 and plot the conductances G σσ (solid) and G σσ (dashed) vs strain ε, respectively, for θ = 0 (black) and θ = π/2 (red). For V g2 = V 0 and θ = 0, as ε increases the spin-conserving conductance G σσ increases while the spin-flip component G σσ decreases. The opposite is seen for θ = π/2. For V g2 = −V 0 [ Fig. 5(b)], although the conductances are less sensitive to ε, the picture is fully reversed. In this situation G σσ (G σσ ) decreases (increases) as the strain enhances along θ = 0, while for θ = π/2, G σσ (Gσ) increases (decreases) with strain. Finally, figure 5(c) shows how G σσ and G σσ evolves with θ and fixed strain parameter ε = 0.1. For V g2 > 0 (black lines) G σσ remains close to one, while G σσ is close to zero. As strain is rotated both G σσ and G σσ approaches the same value around 0.5G 0 . For V g2 = −V 0 the conductance G σσ (G σσ ) is only slightly suppressed (enhanced). These features could be exploited as a possible manner to control the spin-polarized transport via strain, with potential application as a spintronic mechanical strain sensors. Conclusion Summarizing, we have studied the spin transport through ZGNR device composed of two local gates with Rashba SOC field generators. We demonstrate the feasibility of spin-charge current flow manipulation by means of the spatially modulate Rashba field and strain. By changing the sign of the underneath gates, it is possible to control the spin selective current intensity by considering spinpolarized leads. Furthermore, we also show that strain plays an important role in the spin polarized current control being an additional tuning parameter. In the light of our numerical simulations, we expect our results to be of great applicability in the GNR spin-based devices as well as for other 2D-related materials in which similar setups can be performed. terminal ZGNR device into three well defined regions: the left lead, the central conductor and the right lead. The central conductor corresponds to the region with gates. The Rashba SOC takes place only in the two regions on top of the gates. The Green's function of the central conductor G C (the spin index is omitted) is then that is associated with the injection and detection of electrons with the same spin and the spin-flip component (G σσ ), resulting from spin rotation by the Rashba field. It is important to mention that if there is no polarized electrons being injected or drained by a ferromagnetic leads, time-reversal symmetry is preserved in the device, therefore G ↑↑ = G ↓↓ and G ↑↓ = G ↓↑ , resulting in no polarized net current. The unpolarized electron flux can also be resulting of a multichannel lead, as reported in Ref. [59] in two-terminal device. For ferromagnetic leads, two possible configurations can be used: (i) parallel alignment of the leads magnetization and (ii) anti-parallel alignment. In the former case the transport is carried on by electrons with the same spin in the source and drain, while in the latter the transport is dominated by electrons with opposite spins in the source and the drain. Figure 2 : 2For metallic armchair graphene nanoribbons, we were able to obtain similar results in the absence of uniaxial strain, as depending on the strain direction there is an induced transport gap close to the Fermi level,[37] which is not our proposed effect: spin-selective filtering at low energy. It is important to mention that our formalism is not able to capture possible valley filtering, as all the analysis is over the energy of injected electrons that has contributions of both valleys (with no separation).The contour plots of the spin-resolved conductance G σσ (Rashba parameter vs energy of injected electron) is displayed inFig. 2: (a) and (b) spin conserving and spin-flip conductance with two gated regions with width l R = W =10 unit cells long, respectively; (c) and (d) are similar but with twice the width l R = 2W =20 unit cells long. Hereafter, we set the width of the SO region W = 2.46nm (corresponding to 10 unit cells of the ZGNR). From the figures one can notice that while widening the l R width it causes the reduction of the period in the oscillation pattern of the conductance profiles for fixed E, a behavior attributed to the additional unit cells along the device with Rashba SOC field, which is responsible for the electron spin precession along the device. It is also clear the opposite behavior for the spin-conserving and spin-flip conductance; Contour plots (Rashba parameter vs energy of injected electron) of the spindependent conductance for different set of configurations. (a) Spin-conserving conductance Gσσ with width l R =W , (b) spin-flip conductance Gσσ with width l R =W . Panels (c) and(d) are for Gσσ and Gσσ, but with different width l R =2W . In all panels V g1 = V g2 > 0 is assumed and we set the width of the SO region W = 2.46nm. enhancement in one of the components (brighter region) reflects in the reduction of the other (dark regions). Another remarkable phenomenon is the oscillatory dependence of the spin components of G σσ on the value of λ R for fixed E, which can be observed in all panels ofFig. 2. This oscillatory behavior is reminiscent of the spin field effect transistor (FET) and has a similar origin,[9] as the spin precesses as it propagates in the presence of the Rashba field, acquiring a net phase that is proportional to λ R and the total length of the central conductor. Figure 3 : 3(Color online) Spin dependent conductances Gσσ (black) and Gσσ (red) vs energy. (a) λ R = 0 (dashed) λ R = 0.1t 0 (solid) for V g1 = V g2 applied to the entire region underneath the device. (b) and (c) for width l R = W and l R = 2W , respectively; (d) and (e) similar (b)and (c) but for V g2 < 0, respectively. For all panels (b)-(e), λ R =0.10t 0 . Fig. 3 3(a) the conductance vs energy for the situation in which the gate voltage in applied to the entire region underneath the device. For λ R = 0 (dashed lines) Now, in Fig. 3(d) and 3(e) we switch the sign of V g2 such that V g2 = −V g1 = −V 0 . In this case, by comparing these results with those of Figs. 3(b) and 3(c) we note a completely different profile. Note, for example, that the behavior of G σσ and G σσ are inverted. This is best seen for l R = 2W [Figs. 3(c) and 3(e)].Observe that for E ≈ 0.1t 0 , while G σσ ≈ G 0 and G σσ ≈ 0 for V g2 = V 0 , the opposite occurs for V g2 = −V 0 . Naively, one could think that the spins of the electrons rotated by V g1 in the first region would be rotated back to their original Figure 4 : 4(Color online) Gσσ (black) and Gσσ (red) vs E for strained ZGNR with ε=0.1 and θ=0 for V g2 > 0 and V g2 < 0, respectively. (c) and (d) are the results for the same values of parameter as in the panels (a) and (b) except that θ = π/2. For all cases we fixed l R =W and λ R =0.1t 0 . 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{'abstract': "We investigate the spin-resolved transport in a two-terminal zigzag graphene nanoribbon device with two independent gate induced Rashba spin-orbit coupling regions and in the presence of strain. By employing a recursive Green's function technique to the tight-binding model for the graphene nanoribbon, we calculate the spin-resolved conductance of the system. We show that by switching the sign of one of the gates it is possible to select which spin component will be transmitted. Moreover, our results show that an uniaxial strain applied to the nanoribbon plays a significant role in the transport, providing and additional manner to control the spin-polarized conductance. This makes the present system a potential candidate for future implementations of spin-based mechanical strain sensors.", 'arxivid': '1512.05436', 'author': ['G S Diniz ', 'E Vernek ', 'F M Souza ', '\nInstituto de Física\nCurso de Física\nUniversidade Federal de Uberlândia\n38400-902UberlândiaMGBrazil\n', '\nInstituto de Física\nUniversidade Federal de Goiás\n75801-615JataíGOBrazil\n', '\nUniversidade Federal de Uberlândia\n38400-902UberlândiaMGBrazil\n'], 'authoraffiliation': ['Instituto de Física\nCurso de Física\nUniversidade Federal de Uberlândia\n38400-902UberlândiaMGBrazil', 'Instituto de Física\nUniversidade Federal de Goiás\n75801-615JataíGOBrazil', 'Universidade Federal de Uberlândia\n38400-902UberlândiaMGBrazil'], 'corpusid': 119119553, 'doi': '10.1016/j.physe.2016.06.008', 'github_urls': [], 'n_tokens_mistral': 17377, 'n_tokens_neox': 13553, 'n_words': 6354, 'pdfsha': '3169ebf4788f4358aa191d8f37696a03baec4545', 'pdfurls': ['https://arxiv.org/pdf/1512.05436v2.pdf'], 'title': ['Graphene-based spin switch device via modulated Rashba Field and Strain', 'Graphene-based spin switch device via modulated Rashba Field and Strain'], 'venue': []}

arxiv

What is the scale of new physics behind the muon g − 2 ? Lukas Allwicher Physik-Institut Universität Zürich CH-8057ZürichSwitzerland Luca Di Luzio Dipartimento di Fisica e Astronomia 'G. Galilei' Università di Padova Italy Istituto Nazionale Fisica Nucleare Sezione di Padova Italy Marco Fedele Institut für Theoretische Teilchenphysik Karlsruhe Institute of Technology D-76131KarlsruheGermany Federico Mescia Departament de Física Quàntica i Astrofísica Institut de Ciències del Cosmos (ICCUB) Universitat de Barcelona Martí i Franquès 1E-08028BarcelonaSpain Marco Nardecchia Physics Department and INFN Sezione di Roma La Sapienza, Piazzale Aldo Moro 500185RomaItaly What is the scale of new physics behind the muon g − 2 ? We study the constraints imposed by perturbative unitarity on the new physics interpretation of the muon g − 2 anomaly. Within a Standard Model Effective Field Theory (SMEFT) approach, we find that scattering amplitudes sourced by effective operators saturate perturbative unitarity at about 1 PeV. This corresponds to the highest energy scale that needs to be probed in order to resolve the new physics origin of the muon g −2 anomaly. On the other hand, simplified models (e.g. scalar-fermion Yukawa theories) in which renormalizable couplings are pushed to the boundary of perturbativity still imply new on-shell states below 200 TeV. We finally suggest that the highest new physics scale responsible for the anomalous effect can be reached in non-renormalizable models at the PeV scale.A Unitarity bounds in the SMEFTIn this Appendix we expand on some aspects of the calculation of unitarity bounds in the SMEFT. The case of the operator O eW = ( L σ µν e R )τ I HW I µν is analyzed in detail, since it offers the possibility of discussing several non-trivial aspects, like the multiplicity of the scattering amplitude in SU(2) L space, the contribution of higher-partial waves and that of 2 → 3 scatterings. The calculations of the unitarity bounds for O eB and O T follow in close analogy and are not reported here.A.1 2 → 2 scatteringConsider the 2 → 2 scattering W I a L → H †,b e R sourced by O eW , where we have explicitly written the SU(2) L indices (I = 1, 2, 3 in the adjoint and a, b = 1, 2 in the fundamental). Introduction The recent measurement of the muon anomalous magnetic moment, a µ ≡ (g µ − 2)/2, by the E989 experiment at Fermilab [1], in agreement with the previous BNL E821 result [2], implies a 4.2σ discrepancy from the Standard Model (SM) ∆a µ ≡ a µ (Exp) − a µ (SM) = (251 ± 59) × 10 −11 , (1.1) following the Muon g − 2 Theory Initiative recommended value for the SM theory prediction [3]. Although a recent lattice determination of the SM hadron vacuum polarization contribution to a µ claims no sizeable deviation from the SM [4], we will work here under the hypothesis that ∆a µ is due to new physics. In particular, we will focus on the case in which new physics states are so heavy that their effects can be parameterized via the so-called SM Effective Field Theory (SMEFT) and ask the following question: What is the scale of new physics behind ∆a µ ? This question is of practical relevance, given the futuristic possibility of resolving the new physics origin of ∆a µ via direct searches at high-energy particle colliders. As explored recently in [5][6][7], a muon collider seems to be the best option for this goal. However, while the very existence of the SMEFT operators contributing to ∆a µ could be tested via processes like µ + µ − → Z(γ)h or µ + µ − → tt at a multi-TeV-scale muon collider [6], it is less clear whether the origin of the muon g − 2 SMEFT operators can be resolved via the direct production of new on-shell states responsible for ∆a µ . This is the question that we want to address in the present work, using the tools of perturbative unitarity. Unitarity bounds on the new physics interpretation of ∆a µ were previously considered in [5,7] focusing however on a specific class of renormalizable models. Here, we will consider instead the most conservative case in which unitarity limits are obtained within the SMEFT and reach a more pessimistic conclusion about the possibility of establishing a no-lose theorem for testing the origin of ∆a µ at a future high-energy particle collider. Generally speaking, given a low-energy determination of an EFT coefficient, unitarity methods can be used either within an EFT approach, in order to infer an upper bound on the scale of new physics unitarizing EFT scattering amplitudes, or within explicit new physics (renormalizable) models. In the latter case, one obtains a perturbativity bound on certain renormalizable couplings that can be translated into an upper bound on the mass of new on-shell degrees of freedom. In the present work we will be interested in both these approaches. First, we will consider a SMEFT analysis in which ∆a µ is explained in terms of a set of Wilson coefficients normalized to some cut-off scale 2 , C i /Λ 2 , and later deal with renormalizable models featuring new heavy mediators that can be matched onto the SMEFT. Schematically, ∆a µ ∼ C i Λ 2 = (loops) × (couplings) M 2 on−shell ,(1.2) where M on−shell denote the mass of new on-shell states and we included possible loop factors in the matching between the new physics model and the SMEFT operators. Hence, by fixing the value of the SMEFT coefficients C i /Λ 2 in terms of ∆a µ , we will consider high-energy scatterings sourced by the associated effective operators, determine the √ s that saturates perturbative unitarity (according to a standard criterium to be specified in Sect. 2) and interpret the latter as an upper bound on the scale of new physics responsible for the muon g−2 anomaly. Analogously, in the case of new physics models, we will use the unitarity tool in order to set perturbativity bounds on the new physics couplings and in turn (given Eq. (1.2)) an upper limit on M on−shell . While the first approach is model-independent (barring possible degeneracies in the choice of the effective operators) and yields the most conservative bound on the scale of new physics, the second approach relies on further assumptions, but it directly connects to new on-shell degrees of freedom which are the prime targets of direct searches at high-energy particle colliders. The paper is structured as follows. We start in Sect. 2 with a brief review of partial wave unitarity, in order to set notations and clarify the physical interpretation of unitarity bounds. Next, we consider unitarity bounds within a SMEFT approach (Sect. 3) and within renormalizable models matching onto the SMEFT operators (Sect. 4). Finally, we comment in Sect. 5 on nonrenormalizable realizations which can saturate the unitarity bounds obtained in the SMEFT. Our main findings and implications for the direct resolution of the muon g − 2 anomaly at high-energy particle colliders are summarized in the conclusions (Sect. 6). Technical aspects of partial wave unitarity calculations, both in the SMEFT and in renormalizable setups, are deferred to Apps. A-B. Partial wave unitarity We start with an instant review of partial wave unitarity, which will serve to set notations and discuss the physical significance of unitarity bounds. The key point of our analysis is the study of scattering amplitudes with fixed total angular momentum J, the so-called partial waves. Here we focus only on the case of 2 → 2 partial waves (while the 2 → 3 scattering is discussed in App. A.2) defined as a J f i = 1 32π 1 −1 d cos θ d J µ i µ f (θ) T f i ( √ s, cos θ) , (2.1) with θ the scattering angle in the centre-of-mass frame, (2π ) 4 δ (4) (P i − P f )iT f i ( √ s, cos θ) = f | S−1 |i and S the S-matrix. Here, d J µ i µ f is Wigner's d-function that arises in the construction of the two-particle incoming (outcoming) state of helicities µ i (µ f ) onto angular momentum J [8]. The S-matrix unitarity condition S † S = 1 then yields the relation 1 2i (a J f i − a J * if ) = h a J * hf a J hi =⇒ Im (a J ii ) = h |a J hi | 2 ≥ |a J ii | 2 , (2.2) where we have restricted ourselves to the elastic channel h = i = f . The equation on the right hand side of (2.2) defines a circle in the complex plane inside which the amplitude must lie at all orders, Re a J ii 2 + Im a J ii − 1 2 2 ≤ 1 4 , (2.3) suggesting the following bound, under the assumption of real tree-level amplitudes: |Re a J ii | ≤ 1 2 . (2.4) Hence, in order to extract the bound, one needs to fully diagonalize the matrix a J . Once this is achieved, every eigenvalue will give an independent constraint. In the presence of multiple scattering channels, it follows from Eq. (2.4) that the strongest bound arises from the largest eigenvalue of a J . When the latter bound is saturated, it basically means that one needs a correction of at least 40% from higher orders to get back inside the unitarity circle, thus signaling the breakdown of perturbation theory (see e.g. [9,10]). Here, a J stands for the leading order expansion of the partial wave, both in the coupling constants and in external momenta over cut-off scale for the case of an EFT. Although the criterium is somewhat arbitrary, and hence Eq. (2.4) should not be understood as a strict bound, we stick to that for historical reasons [11]. Strictly speaking, a violation of the perturbative unitarity criterium in Eq. (2.4) should be conservatively interpreted as the onset of a regime of incalculability due to the breakdown of the perturbative expansion either in couplings or external momenta. More specifically, in the case of an EFT (where scattering amplitudes grow with energy) the scale of unitarity violation, hereafter denoted as √ s = Λ U =⇒ |Re a J ii | = 1 2 ,(2.5) can be associated with the onset of "new physics", where on-shell new degrees of freedom should manifest themselves and be kinematically accessible. Although one can conceive exotic UV completions where this is not the case [12], well-known physical systems behave in this way. 1 Unitarity methods can be employed in renormalizable setups as well. In this case, the unitarity limit corresponds to the failure of the coupling expansion and hence the bound on the renormalizable coupling can be understood as a perturbativity constraint. SMEFT In this section we present the unitarity bounds for the new physics interpretation of the muon g −2 anomaly within a SMEFT approach. The strategy consists in fixing the Wilson coefficients (C i /Λ 2 ) in terms of the observable ∆a µ and determine next the energy scale √ s that saturates the unitarity bounds derived from the tree-level scattering amplitudes sourced by the effective operator. The shorthand 1/Λ 2 i ≡ C i /Λ 2 is understood in the following. SMEFT approach to ∆a µ Assuming a short-distance new physics origin of ∆a µ , the leading SMEFT operators contributing up to one-loop order are (see Refs. [6,13] for a more systematic discussion) L SMEFT g−2 = C eB Λ 2 ( L σ µν e R )HB µν + C eW Λ 2 ( L σ µν e R )τ I HW I µν + C q T Λ 2 ( a L σ µν e R )ε ab (Q b L σ µν u R ) + h.c. , (3.1) which results in [6] ∆a 4m v e √ 2Λ 2 Re C eγ − 3α 2π c 2 W − s 2 W s W c W Re C eZ log Λ m Z − q=t,c,u 4m m q π 2 Re C q T Λ 2 log Λ m q , (3.2) where C eγ = c W C eB − s W C eW and C eZ = −s W C eB − c W C eW , in terms of the weak mixing angle. For the Wilson coefficients of the dipole operators that contribute at tree level to ∆a , one can consistently include one-loop running, obtaining [14,15] C eγ (m ) C eγ (Λ) 1 − 3y 2 t 16π 2 log Λ m t − 4α π log Λ m . (3.3) A convenient numerical parameterization reads ∆a µ 2.5 × 10 −9 277 TeV Λ 2 Re C µ eγ (Λ) − 0.28 Re C µt T (Λ) − 0.047 Re C µ eZ (Λ) ,(3.4) where we have kept only the leading top-quark contribution for C T (since we are interested on scenarios which maximize the scale of new physics) and the logs have been evaluated for Λ = 277 TeV. Note, however, that the full log dependence will be retained in the numerical analysis below. In the following, we will drop the scale dependence of the Wilson coefficients, which are understood to be evaluated at the scale Λ. Operator Λ U i → f Channels J 1 Λ 2 eB ( L σ µν e R )HB µν 2 √ π|Λ eB | Be R → H † L 1/2 1 Λ 2 eW ( L σ µν e R )τ I HW I µν 2 √ π 2 3 1/4 |Λ eW | W L → H † e R 1/2 1 Λ 2 T, ( a L σ µν e R )ε ab (Q b L σ µν u R ) 2 π 3 √ 2 |Λ T, | e R u R → Q L L 0 Unitarity bounds Given Eq. (3.1), we can compute the scale of unitarity violation Λ U (defined via Eq. (2.5)) associated with each of the dimension-6 operators involved. To do so, we consider here only 2 → 2 scattering processes, since the 2 → 3 processes (mediated by O eW ) turn out to be suppressed by the weak gauge coupling and the 3-particle phase space, as shown in Appendix A.2. The results obtained by switching one operator per time are collected in Table 1, where the bound in correspondence of different initial and final states (i = f ) comes from the diagonalization of the scattering matrix (cf. discussion below Eq. (2.4)). In App. A we present the full calculation of the unitarity bounds stemming from the SU(2) L dipole operator, which presents several nontrivial aspects, like the presence of higher than J = 0 partial waves, the multiplicity in SU(2) L space and the possibility of 2 → 3 scatterings. We next make contact with the physical observable ∆a µ , whose dependence from the Wilson coefficients can be read off Eq. (3.4). Turning on one operator per time, we find the following numerical values for the unitarity violation scales: • O µ eB ≡ ( L σ µν e R )HB µν Λ U 277 TeV 2 √ π c W + 0.047s W 930 TeV . (3.5) • O µ eW ≡ ( L σ µν e R )τ I HW I µν Λ U 277 TeV 2 √ π 2 3 1/4 s W − 0.047c W 590 TeV . (3.6) • O µt T ≡ ( a L σ µν e R )ε ab (Q b L σ µν u R ) Λ U 277 TeV 2 π 3 √ 2 √ 0.28 240 TeV . (3.7) Hence, the scale of new physics is maximized if the origin of ∆a µ stems from a dipole operator oriented in the U(1) Y direction. If more than one operator is switched on, correlations can arise between the Wilson coefficients whenever they couple same sectors of the theory. For instance, in the case in which both the dipole operators O µ eW and O µ eB are present one can derive a combined bound (see Eq. (A.13)) which leads to the region displayed in Fig. 1 . Note that for Λ eB → ∞ (Λ eW → ∞) we reproduce the bound with O µ eW (O µ eB ) only. However, if both operators contribute sizeably to ∆a µ , the unitarity bound can be slightly relaxed above the PeV scale. Renormalizable models We next consider simplified models featuring new heavy states, which after being integrated out match onto the dipole and tensor SMEFT operators contributing to ∆a µ (cf. Eq. (3.4)). We will One-loop matching onto the dipole operator In order to match onto the dipole operator at one loop we consider a simplified model with a new complex scalar S = (1, 1, Y + 1) and two vector-like fermions F = (1, 2, Y + 1 2 ) and F e = (1, 1, Y ) allowing for a mixing via the SM Higgs (see e.g. [5,7,16,17]) L g−2 FFS = λ L F L S + λ R F e e R S + F (y L P L + y R P R )F e H + h.c. − M F F − M e F e F e − m 2 S |S| 2 − κ |H| 2 |S| 2 − λ S |S| 4 . (4.1) The FFS model allows for a chirality flip of the external leptons via the product of couplings λ * L y L,R λ R (cf. Fig. 2), which can be used to maximize the scale of new physics. For vy L,R M , M e , m S , we can integrate out the new physics states and find at one loop C µ eγ Λ 2 = − eλ * L λ R 32π 2 m 2 S √ x x e (x − x e ) Q S y R (g S (x ) − g S (x e )) + y L x x e g S (x ) − x e x g S (x e ) + Q F y R (g F (x ) − g F (x e )) + y L x x e g F (x ) − x e x g F (x e ) , (4.2) where Q S = Y + 1, Q F = Y , x ,e = M 2 ,e /m 2 S and the loop functions are given by g F (x) = x 2 − 4x + 3 + 2 log x 2(x − 1) 3 , g S (x) = x 2 − 2x log x − 1 2(x − 1) 3 . (4.3) This result agrees with Ref. [18] in which the special case y L = y R was considered. Note that in Eq. (4.2) we already matched onto the photon dipole operator at the scale Λ, while the connection with the low-energy observable ∆a µ is given in Eq. (3.4). Our goal is to maximize the mass of the lightest new physics state for a fixed value of the Wilson coefficient. This is achieved in the degenerate limit e R ℓ L S F ℓ F ℓ F e F e H γ ℓ L λ * L λ * L λ R y L,R C µ eγ ℓ L e R H γ M ℓ , M e , m S ≃ Λ vy L,R ≪ Λm S = M = M e , yielding C µ eγ Λ 2 = − eλ * L λ R 384π 2 m 2 S [(1 + 2Y ) y L − (1 + 4Y ) y R ] eY λ * L λ R 192π 2 m 2 S (2y R − y L ) ,(4.4) where in the last expression we took Y 1. The unitarity bounds for the FFS model are summarized in Table 2, where in the case of multiple scattering channels the bound corresponds to the highest eigenvalue of a J . We refer to App. B for further details on their derivation. Applying these bounds, the maximum value of the combination |Re (λ * L λ R (2y R − y L ))| entering Eq. (4.4) is ≈ 121, while |eY | 3.5. Hence, we obtain both chiralities (see e.g. [19]), thus maximizing the effect on ∆a µ via a top-mass insertion. ∆a µ 2.5 × 10 −9 131 TeV m S 2 eY 3.5 Re λ * L λ R (2y R − y L ) 121 ,(4. Unitarity bound i → f Channels J Re (λ * L λ R ) < 8π e R F R → e R F R 0 Re (y * L y R ) < 8π/ √ 2 F e R F e L → F e R F e L 0 Re (y * L y R ) ± 4|λ L | 2 |λ R | 2 + (y * L ) 2 y 2 R < 16π i, f = F R F e L , e R L 0 2|λ L | 2 + |λ R | 2 < 8π i, f = F R L , F e L e R 0 |y R | < √ 8π HF L → HF L 1/2 |λ R | 2 + 2|y L | 2 < 16π i, f = Se R , H † F ,R 1/2 Re (y L λ * L ) < 8π/ √ 2 i, f = F e R S † , e R H 1/2 |λ R | 2 + 32|y L | 2 |λ R | 2 + |λ R | 4 < 32π i, f = SF e L , H † L 1/2 |Re (λ L y L )| < 16π/ √ 2 L F e L → SH † 1 Re (y * L λ R ) < 16π/ √ 2 F R e R → HS 1 |κ| < 8π/ √ 2 HH † → SS † 0 |g Y (Y + 1)| < √ 6π SB µ → SB µ 1/2e R R 2 t ℓ L λ * L λ R t C µt T ℓ L e R t t m R2 ≃ Λ ≫ m t Massive vectors can also lead to renormalizable extensions, but they result at least into a m b /m t suppression compared to scalar extensions (see e.g. [20]). Let us focus for definiteness on the R 2 case (similar conclusions apply to S 1 ). The relevant interaction Lagrangian reads 2 L g−2 R 2 ⊃ λ L t R a L ε ab R b 2 + λ R q a L µ R R 2a + h.c. (4.6) where a and b are SU(2) L indices and ε = iσ 2 . Upon integrating out the leptoquark with mass m R 2 v (cf. Fig. 3), one obtains [13,21] C µt T Λ 2 = − λ * L λ R 8m 2 R 2 . (4.7) The unitarity bounds for the R 2 model (see App. B for details) are collected in Table 3 and they imply |Re (λ * L λ R )| 12. Hence, we can recast the contribution to ∆a µ via Eq. (3.4) as ∆a µ 2.5 × 10 −9 180 TeV m R 2 2 Re (λ * L λ R ) 12 . (4.8) Hence, we conclude that in the leptoquark model one expects m R 2 180 TeV (the same numerical result is obtained for S 1 ), thus providing the largest new physics scale among the renormalizable extensions responsible for ∆a µ . Moreover, since the matching with the tensor operator is at tree level, the leptoquark model fairly reproduces the unitarity bound from the SMEFT operator (see Eq. (3.7)). Unitarity bound i → f Channels J |λ L | 2 + |λ R | 2 < 8π i, f = t R L , q L µ R 0 Re (λ R λ * L ) < 8π/ √ 3 µ R L → q L t R 0 |λ R | 2 < 8π/3 q L R * 2 → q L R * 2 1/2 |λ L | 2 < 16π/3 t R R * 2 → t R R * 2 1/2 Raising the scale of new physics via multiplicity? Naively, one could be tempted to increase the upper limit on the scale of new physics by adding N copies of new physics states contributing to ∆a µ . However, while both C eγ and C T increase by a factor of N , the unitarity bounds on the couplings gets also stronger due to the correlation of the scattering channels, so that larger new physics scales cannot be reached. In order to see this, consider e.g. the FFS model with N copies of F , F e and S. The scaling of the unitarity bounds is most easily seen in processes where the SM states are exchanged in the s-channel, for example S i F i R → S j F j R . Since L is coupled to all copies in the same way, the T -matrix can be written as T J=1/2 = 1 32π |λ L | 2 J N ,(4.9) where J N is a N × N matrix filled with 1. Given that the largest eigenvalue of J N is N , the unitarity bound on λ L reads |λ L | < 16π N . (4.10) Similar processes can be considered for all the couplings in Eq. (4.1), leading to a 1/ √ N scaling for each Yukawa coupling. Hence, the overall N contribution to ∆a µ ∝ N Re (λ * L λ R y L,R )/m 2 S is compensated by the 1/ √ N scaling of the unitarity bounds on the couplings and, for fixed ∆a µ , the mass of extra states gets even lowered at large N . In this respect, we reach a different conclusion from the analysis in Ref. [7]. The same considerations apply if we consider just one new scalar and N new fermions. The situation is different with N scalars and just one family of fermions, since S does not couple directly to the Higgs (barring the portal coupling κ in Eq. (4.1), which however does not contribute to ∆a µ ). This implies that only λ L and λ R will scale as 1/ √ N , which in turn means that ∆a µ does not change. Similar arguments apply when considering larger SU(2) L representations, thus implying that the minimal choice we made for the FFS model ensures that m S is maximized. The case of the leptoquark R 2 is analogous to what we have just described for N new scalars, with the new fermions of the FFS model replaced by SM fields. Given that λ L and λ R would scale as 1/ √ N , there is no gain in taking N copies of leptoquarks. Non-renormalizable models Till now we focused on renormalizable extensions of the SM addressing ∆a µ and showed that they predict on-shell new physics states well below the unitarity bound obtained from the SMEFT dipole operators, suggesting instead that new physics can hide up to the PeV scale. Nonetheless, the SMEFT bound should be understood as the most conservative one and applies if the origin of ∆a µ can be for instance traced back to a strongly-coupled dynamics. While such a scenario could have calculability issues, we want to provide here an intermediate step in which the SMEFT dipole operators are generated via a tree-level exchange of a new vector resonance from a strongly-coupled sector taking inspiration from the case of the ρ meson in QCD, but whose UV origin we leave unspecified. Spin-1 vector resonances are conveniently described via the two-index anti-symmetric tensor field V µν , following the formalism of Ref. [22]. In particular, we consider a composite spin-1 state featuring the same gauge quantum numbers of the SM Higgs doublet and described via the effective Lagrangian Lagrangian, with cut-off scale Λ V above m V . The free Lagrangian of Eq. (5.1) propagates three degrees of freedom describing a free spin-1 particle of mass m V , with propagator [22][23][24] i∆ µν;ρσ (q) = 2i L V = −D µ V † µν D ρ V ρν + 1 2 m 2 V V † µν V µν + c HB V † µν HB µν + c HW V †m 2 V − q 2 I µν;ρσ (q) − q 2 m 2 V P µν;ρσ (q) , (5.2) where I µν;ρσ = g µρ g νσ − g µσ g νρ /2 and P µν;ρσ = P µρ T P νσ T − P µσ T P νρ T /2 with P µν T = g µν − q µ q ν /q 2 . Assuming that there is a calculable regime where one can parametrically keep m V Λ V (in analogy to the chiral approach to the ρ meson in QCD, for which m ρ Λ χ ∼ 1 GeV) we can integrate V µν out and get the following tree-level matching contribution with the photon dipole operator (cf. also Fig. 4) where we normalized m V at the PeV scale, that is in the ballpark of the unitarity bound obtained from the SMEFT dipole operators. It should be noted that although the operators in the second line of Eq. (5.1) have canonical dimension equal to 4, scattering amplitudes involving the c HB,HW, e couplings, as e.g. HB → e R L , grow like s/m 2 V due to the high-energy behaviour of the propagator in Eq. (5.2). Hence, the effective description of the vector resonance breaks down not far above m V , being the theory non-renormalizable. 3 C µ eγ Λ 2 = − 2 (c W c HB − s W c HW ) c e Conclusions Unitarity bounds are a useful tool in order to infer the regime of validity of a given physical description. In EFT approaches, the energy scale at which unitarity is violated in tree-level scattering amplitudes can be often associated to the onset of the new physics completing the effective description. Instead, within renormalizable setups unitarity bounds are a synonym of perturbativity bounds on the size of the adimensional couplings. In this work we have investigated unitarity constraints on the new physics interpretation of the muon g − 2 anomaly. Assuming a short-distance SMEFT origin of the latter, we have first computed unitarity bounds considering a set of leading (dipole and tensor) operators contributing to ∆a µ . It turns out that the scale of tree-level unitarity violation is maximized in the case of dipole operators and reaches 3 Another way to generate the dipole operators relevant for ∆a µ at tree level is to consider non-renormalizable models, involving for example a new vector-like fermion F = (1, 2, − 1 2 ) [25]. the PeV scale when both U(1) Y and SU(2) L dipoles are switched on (cf. Fig. 1). Hence, most conservatively, in order to resolve the new physics origin of the SMEFT operators behind ∆a µ one would need to probe high-energy scales up to the PeV. This most pessimistic scenario, clearly outside from the direct reach of next-generation high-energy particle colliders, can be understood as a no-lose theorem for the muon g − 2 puzzle. Of course, the new physics origin of ∆a µ might reside well below the PeV scale, as it is indeed suggested by simplified models based on renormalizable scalar-Yukawa theories. In the latter case we have considered a couple of well-known scenarios matching either on the tensor (at tree level) or the dipole (at one loop) operators of the SMEFT analysis. In both cases, we have computed unitarity bounds on renormalizable couplings, thus allowing the mass of the new on-shell states to be maximized. The latter are found to be M on−shell 130 TeV and 180 TeV, respectively for the dipole and the tensor operators. Moreover, we have shown that multiplicity does not help to relax those bound because unitarity limits scale as well with the number of species. Since the bound obtained within renormalizable models is well below the SMEFT bound, it is fair to ask which UV completions could lead to a new physics resolution of the muon g − 2 puzzle hidden at the PeV scale. In fact, one could imagine a strongly-coupled dynamics at the PeV scale that is equivalent to writing the SMEFT Lagrangian. Here, we have provided instead an intermediate step in which the SMEFT dipole operators are generated via the tree-level exchange of a new spin-1 vector resonance described by a two-index anti-symmetric tensor field V µν with the same quantum numbers of the SM Higgs and whose origin should be traced back to the dynamics of a strongly-coupled sector. This effective scenario provides a non-trivial example in which the dipole effective operators are generated via tree-level matching, thus suggesting that the SMEFT unitarity bound can be saturated with new on-shell states hidden at the PeV scale. It would be interesting to investigate whether a UV dynamics leading to such effective scenario can be explicitly realized. Taking a W with positive helicity, the lowest partial wave is J = 1/2. The only possible source for a multiplicity of states in this sector is given by SU(2) L , giving a total of 3 × 2 + 2 = 8 states, so the J = 1/2 sector is a 8 × 8 matrix, with entries given by (τ I ) ab . Ordering the states as {W 1 1 L , W 1 2 L , W 2 1 L , W 2 2 L , W 3 1 L , W 3 2 L , H †,1 e R , H †,2 e R }, we have a J=1/2 f i = a 1/2      0 0 0 τ 1 0 0 0 τ 2 0 0 0 τ 3 τ 1 τ 2 τ 3 0      , (A.1) where a 1/2 = √ 2 16π s Λ 2 eW encodes the result of Eq. (2.1) (and whose calculation is reported below). The largest eigenvalue of this matrix is a J=1/2 ii = √ 3a 1/2 , leading to the bound √ s < Λ U = 2 √ π 2 3 1/4 |Λ eW | . (A.2) We now report the computation of the amplitude a 1/2 of Eq. (A.1). The process is W (p, +) + L (k) → H(p ) + e R (k ) , (A.3) with p chosen along theẑ direction and the scattering angle θ the one formed by p and p , and we have suppressed SU(2) L indices. The T -matrix element is T f i = 1 Λ 2 eW (p µ ε (+) ν ( p) − p ν ε (+) µ ( p))(v (R) ( k)σ µν v (L) ( k )) = 2 √ 2 s Λ 2 eW cos θ 2 . (A.4) Since the lowest partial wave is J = 1/2, and µ i = µ f = 1/2, we need the d-function d a 1/2 = 1 32π 2 √ 2 s Λ 2 eW 1 −1 d cos θ cos 2 θ 2 = √ 2 16π s Λ 2 eW . (A.5) A.2 2 → 3 scattering Here we show how the unitarity bound for the 2 → 3 scattering is weaker than the one obtained for 2 → 2 processes, in the special case of the operator O eW . This is due to the presence of the weak gauge coupling g 2 0.6, in addition to the phase-space suppression of the 3-particle final state. Extracting the 2 → 3 partial wave is slightly more involved, since one needs to construct the three-particle states at fixed total J, which in the centre-of-mass frame have five degrees of freedom we have to integrate over, instead of the only two polar angles of the two-particle case. In particular, a convenient set of variables is the one obtained by combining 2 particles together (as it is done e.g. for semi-leptonic hadron decays, in which one usually considers the lepton pair). Fixing their mass m 2 R , and boosting to the frame in which these are back-to back, one can construct a state with fixed J R (and helicity λ R ) out of the two-particles, and then combine this with the third to form the eigenstates of the total angular momentum J. The explicit expression is given by | √ s, m 2 R ; JM ; λ = N(3)J ( λ) J R ,λ R dΩ 1 dΩ R D J * M,λ R −λ 3 (φ R , θ R , −φ R ) × D J * R λ R ,λ 1 −λ 2 (φ 1 , θ 1 , −φ 1 ) | √ s, m 2 R ; θ R φ R ; θ 1 φ 1 ; λ , (A.6) where D M M (α, β, γ) are Wigner's D-matrices, with α, β, γ Euler angles in the z-y-z convention and N (3) J ( λ) = √ 2J + 1 4π   J R ,λ R 1 2J R + 1   −1/2 (A.7) is a normalisation factor, the angles θ 1 and φ 1 are the polar angles of particle 1 in the centreof-mass of particles 1 and 2, 4 θ R and φ R the polar angles of p 1 + p 2 in the centre-of-mass of the three particles (i.e. p 1 + p 2 + p 3 = 0), and λ = (λ 1 , λ 2 , λ 3 ) are the helicities. The dependence on λ in the normalization factor is implicit, since the helicities determine over which values the sum over J R , λ R runs. This will have to be considered case by case, depending on the type of particles involved and the partial wave one wants to obtain. With these states, we can extract the 2 → 3 partial wave at fixed m 2 R , in the case of massless particles, as follows: a J f i = s − m 2 R 256π 2 √ s   J R 1 2J R + 1   −1/2 J R d cos θ 1 d cos θ R × d J µ i ,µ i (θ R )d J R µ i +λ 3 ,λ 1 −λ 2 (θ 1 )M f i ( √ s, m 2 R ; θ 1 , θ R ; r, s, λ) , (A.8) where r, s are the helicities of the incoming particles, and µ i is their sum. The largest eigenvalue is then given by ξ = s 0 dm 2 R a J f i (2 → 3; m 2 R ) 2 . (A.9) Finally, the full diagonalisation of the T -matrix is then achieved by considering the multiplicities in helicity and gauge space, which can lead to further enhancements. =ã 1/2     0 0 1 Λ 2 eW A 0 0 1 Λ 2 eB 1 2×2 1 Λ 2 eW A † 1 Λ 2 eB 1 2×2 0     A =   τ 1 τ 2 τ 3   , (A.12) 4 Theẑ axis is chosen along the direction of p 1 + p 2 in the 3-particle centre-of-mass. Hence, as shown in Fig. 1, we can constrain simultaneously C eB and C eW . It is worth noticing that, following the same procedure with the scattering Be R → H † L (which minimizes the bound for O eB ), i.e. considering also W e R → H † L , we would still find two independent bounds for the two operators. 5 This is due to the fact that the state W e R transforms as (1, 3, −1), which cannot mix into the SU(2) L singlet configuration formed by Be R . B Unitarity bounds in renormalizable models In this section we provide some details about the computation of the unitarity bounds for the simplified models of Sect. 4. Staring from the case of the R 2 leptoquark, whose interactions relevant for the anomalous magnetic moment are described by the lagrangian (4.6), L g−2 R 2 ⊃ λ L t R a L ε ab R b 2 + λ R q a L µ R R 2a + h.c. , (B.1) one can see that several 2 → 2 scattering processes can be considered, both scalar and fermion mediated. The goal is therefore to analyse all of them, in order to identify which channel gives the strongest bound. In general, since there is more than one coupling (two in this case), the different channels will yield independent (combined in general) bounds, as in Table 3. The overall bound on the couplings λ L and λ R can then be visualised as the region defined by the intersection of all the individual constraints. In particular, if the interest lies in one specific combination of said couplings, as for example in Eq. (4.7), one can maximise the function over this region. The best way to proceed in order to compute the unitarity bounds is to classify the possible scattering sectors according to their quantum numbers under the SM gauge symmetry, exploiting the fact that different sectors cannot mix due to gauge invariance. As an example, we show here how the bound is obtained when the leptoquark R 2 is exchanged in the s-channel, i.e. the gauge quantum numbers are (3, 2, 7 6 ). The lowest partial wave, giving the strongest bound, is J = 0 in this case, and the T -matrix takes the form T J=0 (3,2,7/6) = 1 16π |λ L | 2 λ * L λ * R λ L λ R |λ R | 2 , (B.2) where we have ordered the incoming and outgoing states as {t R L , q L µ R } and we have taken the high-energy limit. Diagonalising, the unitarity bound for the highest eigenvalue reads |λ L | 2 + |λ R | 2 < 8π . (B.3) All other bounds in Table 3 are obtained in a similar way. The case of the simplified models with one extra scalar and two extra fermions (FFS) is very similar to the case just described, with some complication due to the presence of more fields and couplings, which increases the number of channels one needs to consider. The philosophy, however, is the same: consider all possible processes and identify the strongest independent bounds (the results are in Table 2). Once this is done, one can extract a bound on the specific combination of the couplings entering the formula for ∆a µ . Finally, the bounds on the parameter 5 Using this process to give a bound on O eW alone, we would find, after considering all SU(2) L multiplicities, Λ U = 2 √ π|Λ eW |, which is slightly weaker than the one given in Table 1. Y entering the hypercharges of the fields F e , F and S have been obtained by considering scattering channels that are completely separated from the ones where the new Yukawa couplings are involved, i.e. considering initial and final states containing the B boson. This has the twofold advantage of giving an independent bound on Y while also avoiding issues of unphysical singularities arising in the exchange of a massless vector boson. Figure 1 : 1In blue, the region in the (Λ eB , Λ eW ) plane that is needed to reproduce the experimental value of ∆a µ at the 2σ level (with the central line corresponding to the central value of ∆a µ ). The dashed iso-lines represent the unitarity bound Λ U , defined according to Eq. (A.13).then use unitarity methods to set perturbativity limits on renormalizable couplings and in turn set an upper bound on the mass of the new on-shell physics states. To maximize the scale of new physics, we will focus on two renormalizable setups based scalar-fermion Yukawa theories, allowing for a left-right chirality flip that is either entirely due to new physics (Sect. 4.1) or with a top Yukawa insertion (Sect. 4.2). Figure 2 : 2Sample diagram of the FFS model matching onto C µ eγ at the scale Λ. that the ∆a µ explanation in the FFS model requires an upper bound on the mass of the new on-shell states of about 130 TeV. On the other hand, due to the extra loop suppression, it is not possible to saturate the unitarity bound that was obtained within the SMEFT (see Eq. (3.5)). Figure 3 : 3Sample diagram of the leptoquark model matching onto C µt T at the scale Λ. Figure 4 : 4µν τ I HW I,µν + c e V µν ( L σ µν e R ) + . . . , (5.1) where we neglected V µν self-interactions as well as other higher-dimensional operators. In fact, Eq. (5.1) should be understood as the leading term of an effective non-renormalizable Tree-level matching onto the photon dipole operator via the exchange of a spin-1 vector resonance. 2 Re ((−c W c HB + s W c HW ) c e ) Table 1 : 1Unitarity violation scale for the SMEFT operators contributing to ∆a µ . Table 2 : 2Unitarity bounds for the FFS model.4.2 Tree-level matching onto the tensor operator We now consider a simplified model that matches onto the tensor operator O µq T . The scalar leptoquarks R 2 = (3, 2, 7 6 ) and S 1 = (3, 1, − 1 3 ) allow for a coupling to the top-quark with Table 3 : 3Unitarity bounds for the couplings of the leptoquark model defined in Eq. (4.6). In the case of the operator O eW , the largest channel is the J = 1/2 scattering He R → L W W , yielding the bound Comparing this with the 2 → 2 bound in Eq. (A.2), Λ 2→2 A.3 A combined bound with O eW and O eB Let us examine now the case in which both the operators O eW and O eB are switched on. Consider again the scattering W L → H † e R mediated by O eW . From the point of view of SM gauge symmetry, the final state forces the process to occur in the (1,2,1/2) representation. The same applies to the process B L → H † e R . We can therefore construct the T -matrix in a similar manner as above. Now ordering the states as {W L , B L , H † e R }, we find√ s < Λ 2→3 U = 32π √ g 2 1 8 + π 2 1 √ 3 Λ eW . (A.10) U = 2 √ πΛ eW , one finds Λ 2→2 U Λ 2→3 U 0.3 √ g 2 . (A.11) a J=1/2 f i 16π . The largest eigenvalue is awithã 1/2 = s √ 2 J=1/2 ii =ã 1/2 3 Λ 4 eW + 1 Λ 4 eB , thus we find √ s < Λ U = min 2 √ 2π 3 Λ 4 eW + 1 Λ 4 eB −1/4 , 2 √ π|Λ eB | . (A.13) Most notably, ππ scattering in chiral perturbation theory yields Λ U = √ 8πf π 460 MeV which is not far from the mass of the ρ meson resonance. Note that the leptoquark models in Eq. (4.6) can be understood as a variant of the FFS model in Eq. (4.1),where F and F e are replaced by the SM states q L and t R , whereas S is the scalar leptoquark (that is the only new physics state). Substituting instead S with the SM Higgs and integrating out the heavy F and F e fermions gives contributions to ∆a µ through dimension-9 SMEFT operators. AcknowledgmentsWe thank Paride Paradisi and Bartolomeu Fiol for useful discussions. Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm. B Abi, Muon g-2 Collaboration10.1103/PhysRevLett.126.141801arXiv:2104.03281Phys. Rev. Lett. 12614141801hep-exMuon g-2 Collaboration, B. Abi et al., "Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm," Phys. Rev. Lett. 126 no. 14, (2021) 141801, arXiv:2104.03281 [hep-ex]. 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{'abstract': 'We study the constraints imposed by perturbative unitarity on the new physics interpretation of the muon g − 2 anomaly. Within a Standard Model Effective Field Theory (SMEFT) approach, we find that scattering amplitudes sourced by effective operators saturate perturbative unitarity at about 1 PeV. This corresponds to the highest energy scale that needs to be probed in order to resolve the new physics origin of the muon g −2 anomaly. On the other hand, simplified models (e.g. scalar-fermion Yukawa theories) in which renormalizable couplings are pushed to the boundary of perturbativity still imply new on-shell states below 200 TeV. We finally suggest that the highest new physics scale responsible for the anomalous effect can be reached in non-renormalizable models at the PeV scale.A Unitarity bounds in the SMEFTIn this Appendix we expand on some aspects of the calculation of unitarity bounds in the SMEFT. The case of the operator O eW = ( L σ µν e R )τ I HW I µν is analyzed in detail, since it offers the possibility of discussing several non-trivial aspects, like the multiplicity of the scattering amplitude in SU(2) L space, the contribution of higher-partial waves and that of 2 → 3 scatterings. The calculations of the unitarity bounds for O eB and O T follow in close analogy and are not reported here.A.1 2 → 2 scatteringConsider the 2 → 2 scattering W I a L → H †,b e R sourced by O eW , where we have explicitly written the SU(2) L indices (I = 1, 2, 3 in the adjoint and a, b = 1, 2 in the fundamental).', 'arxivid': '2105.13981', 'author': ['Lukas Allwicher \nPhysik-Institut\nUniversität Zürich\nCH-8057ZürichSwitzerland\n', "Luca Di Luzio \nDipartimento di Fisica e Astronomia 'G. Galilei'\nUniversità di Padova\nItaly\n\nIstituto Nazionale Fisica Nucleare\nSezione di Padova\nItaly\n", 'Marco Fedele \nInstitut für Theoretische Teilchenphysik\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany\n', 'Federico Mescia \nDepartament de Física Quàntica i Astrofísica\nInstitut de Ciències del Cosmos (ICCUB)\nUniversitat de Barcelona\nMartí i Franquès 1E-08028BarcelonaSpain\n', 'Marco Nardecchia \nPhysics Department and INFN\nSezione di Roma La Sapienza, Piazzale Aldo Moro 500185RomaItaly\n'], 'authoraffiliation': ['Physik-Institut\nUniversität Zürich\nCH-8057ZürichSwitzerland', "Dipartimento di Fisica e Astronomia 'G. Galilei'\nUniversità di Padova\nItaly", 'Istituto Nazionale Fisica Nucleare\nSezione di Padova\nItaly', 'Institut für Theoretische Teilchenphysik\nKarlsruhe Institute of Technology\nD-76131KarlsruheGermany', 'Departament de Física Quàntica i Astrofísica\nInstitut de Ciències del Cosmos (ICCUB)\nUniversitat de Barcelona\nMartí i Franquès 1E-08028BarcelonaSpain', 'Physics Department and INFN\nSezione di Roma La Sapienza, Piazzale Aldo Moro 500185RomaItaly'], 'corpusid': 235247852, 'doi': '10.1103/physrevd.104.055035', 'github_urls': [], 'n_tokens_mistral': 16570, 'n_tokens_neox': 14140, 'n_words': 8801, 'pdfsha': 'e5513d27893395c8f7f25d2172203a809ca9f4c2', 'pdfurls': ['https://arxiv.org/pdf/2105.13981v1.pdf'], 'title': ['What is the scale of new physics behind the muon g − 2 ?', 'What is the scale of new physics behind the muon g − 2 ?'], 'venue': []}

arxiv

Plaquette-centered rotation symmetry and octet-nodal superconductivity in KFe 2 As 2 22 Sep 2017 Guo-Yi Zhu Department of Physics State Key Laboratory of Low-Dimensional Quantum Physics Tsinghua University 100084BeijingChina Guang-Ming Zhang Department of Physics State Key Laboratory of Low-Dimensional Quantum Physics Tsinghua University 100084BeijingChina Collaborative Innovation Center of Quantum Matter 100084BeijingChina Plaquette-centered rotation symmetry and octet-nodal superconductivity in KFe 2 As 2 22 Sep 2017(Dated: September 25, 2017) A plaquette-centered rotation symmetry C p 4 is identified to play a significant role in determining and stabilizing the Fermi-surface structure of Fe-based superconductors. Together with the S4 symmetry previously found, we are able to sort out the tangling orbitals and solve the puzzle of pairing symmetry of superconductivity in KFe2As2 in a simple but comprehensive way. By modeling the material with a strong coupling t − J1 − J2 model, we find phase transitions of pairing symmetry driven by the competition between the local spin antiferromagnetic couplings from nodal d x 2 −y 2 × s x 2 +y 2 -wave to nodeless s x 2 y 2 -wave through the intermediate s + id × s mixed pairing phase, which is consistent with the observation of pressure experiments. The emergent d-wave form factor inevitably arises from the projection of inter-orbital Cooper pairing onto the Fermi surface and is inherited from the electronic structure in the representation of C p 4 symmetry. Moreover, the S4 symmetry dictates 2 copies of d-wave pairing condensates, counting 8 nodes in total. We further show that weakly breaking C p 4 naturally leads to the octet nodal gap as precisely observed in laser angle resolved photoemission spectroscopy. The octet nodes reflect the collaboration of the C p 4 and S4 symmetries, which sheds new light on the enigma of the pairing symmetry in KFe2As2. arXiv:1707.04009v2 [cond-mat.supr-con] As the first family of high temperature superconductivity (SC), the cuprates have been under elaborate studies for nearly thirty years [1][2][3][4], and a consensus has been reached by the majority of condensed matter physicists that the intriguing superconducting phase of cuprates are rooted in the electronic strong antiferromagnetic (AF) correlation. The SC in optimally doped compounds is confirmed by a vast majority experiments to be d-wave pairing symmetry [5][6][7]. For the theorists, the complex reality arising from both the 3d orbital on Cu ions and 2p orbital on O ions is compromised by formation of the Zhang-Rice singlets, which justifies a rather successful description of a single-band model for the cuprates -the celebrated t-J model [3,4,[8][9][10]. The phase diagrams of Fe-based superconductors show significant similarity with those of the cuprates, suggesting deep connection of the essential physics between them. However, distinct from the cuprates, the family of Fe-based superconductors has a seemingly less unified picture as to their Fermi surface (FS) structure, pairing symmetries and pairing glues [11][12][13][14][15][16]. From the perspective of band structure, they owe their complexity to the multi-orbital character. Some Fermi pockets could vanish under doping or pressure and are less robust than the others [17,18]. Regarding the pairing symmetry, the s ±wave pairing symmetry has been established for many iron-pnictides from different approaches [19][20][21], but it fails to account for all the materials observed experimentally. When it comes to the interactions responsible for pairing, there are weak coupling theories that count on the antiferromagnetic spin fluctuation [19], but they are less satisfying when FS nesting is absent, nor when there exists strong correlation between local magnetic moments [11][12][13][14][15][16]. After all these considerations, a simple but unifying theoretical recipe for the Fe-based superconductors is still elusive, and we still need deep reflection upon the electronic structure and interactions of Fe-based superconductors. To model the Fe-based SCs, we have to firstly understand the electronic structure. There have been many theoretical treatments by taking all five or even ten 3dorbitals of Fe into consideration, but the necessity for such a huge parameter space is hardly conceivable, because the pairing symmetries are so robust. From this point of view, we especially appreciate the S 4 symmetry found by Hu and Hao [22], which provides a rational way to sort out the tangling multi-orbitals of Fe-based SCs and reduce the parameters of the electronic structure. In the presence of S 4 , the band structure of multi-orbitals is then reduced to a minimal two-band model. Based on this S 4 symmetry, the model with a few parameters is capable of accommodating most FS structures of both ironpnictides and iron-chalcogenides [22], potentially unifying the electronic structure of Fe-based SCs. However, the heavily hole doped limit of iron-pnictides KFe 2 As 2 is left behind. The representative of heavily hole doped Ba 1−x K x Fe 2 As 2 family -KFe 2 As 2 is a particular intriguing case among the Fe-based superconductors, regarding its pairing symmetry and interaction. Angle resolved photoemission spectroscopy (ARPES) experiments [17,18] report the absence of electron pockets and leave only three hole Fermi pockets around the Brillouin zone (BZ) center (Γ point). Usually the nodeless s-wave pairing symmetry is favorable. But it turns out that nodal gap structure was observed by thermal conductivity measurement [23], penetration depth measurement [24], and NMR probe [25][26][27]. The d-wave pairing symmetry was then proposed after functional renormalization group calculation [28], which seems consistent with the measurement of heat conduction [29]. Later on Okazaki et. al. utilized the ultra-high energy resolution of laser ARPES to map the gap structure much more explicitly [30]. Quite dramatically, their report showed octet nodal gap on the middle hole pocket while nodeless gap on the inner hole pocket, in striking contrast to the theoretical expectation. Then there were attempts to resort to accidental nodes to account for the octet nodes. But as far as we are concerned, the nodes robustly observed in various experiments deserve far more convincing explanation. If the octet nodes was given by an effective g-wave pairing condensate phenomenologically, it is a natural question how the Cooper pair obtains angular momentum as high as l = 4 microscopically. Although the so-called concealed d-wave scenario was proposed to explain the formation of the effective g-wave Cooper pair, such a theory starts from an orbital selective pairing form, and their orbital triplet pairing vector field is required to be locked with the orbital Rashba vector field and to rotate oppositely [31]. In this paper, we will provide a mechanism based on a microscopic minimal model, which is much simpler but more transparent. Our idea is inspired by another experiment. Taillefer's group reported their pressure study on KFe 2 As 2 , in which they witnessed a non-monotonic V-shaped tendency of the superconducting transition temperature with increasing pressure [32]. Their results clearly indicated a phase transition near the point of the sudden change of T c . The possibility of FS transition was ruled out by their Hall coefficient measurement, pointing towards a phase transition of pairing symmetry. It should be emphasized that the KFe 2 As 2 is regarded as a strongly correlated system with strong AF fluctuation evidenced by the remarkable mass enhancement [17,33] and high incoherent spectral weight [34]. Since the main ingredients determining the pairing symmetry are the FS structure and the electronic interactions responsible for pairing, it can be concluded that the phase transitions are essentially driven by the competition between the local spin AF exchanges, because the FS remains almost the same. First of all, we identify a plaquette-centered rotation symmetry C p 4 which is complementary to the S 4 symmetry, providing a simple and potentially universal organizing principle for the tangling orbitals of Fe-based superconductors. We explicitly show that, while S 4 plays the role of organizing two groups of orbitals, C p 4 is responsible for determining and stabilizing the FS structure of ironpnictides, including the heavily hole doped limit where the electron Fermi pockets are absent. The joint cooperation of the S 4 and C p 4 symmetries manifest its power particularly in settling the controversy of pairing symmetry and nodeness in KFe 2 As 2 . To study the pairing symmetry of SC in KFe 2 As 2 , we take the strong coupling approach by modeling this ma-terial with the t − J 1 − J 2 model subjected to the particle occupancy constraint, in contrast to the models with the on-site Coulomb repulsion and Hund's coupling [35]. Since we are mainly concerned with the pairing symmetry of superconductivity, we can neglect the quantum fluctuation for the moment and adopt the slave-boson mean-field approximation. We find that, by decreasing the ratio of J 1 /J 2 , the KFe 2 As 2 samples could experience phase transitions of pairing symmetry from the d x 2 −y 2 ×s x 2 +y 2 -wave SC to s x 2 y 2 -wave SC through a narrow intermediate s+id×s mixed pairing phase. Although the d-wave pairing symmetry is indeed energetically unfavorable compared with the s-wave pairing, the electronic structure imposes an indispensable d-wave form factor on the Cooper pairs glued by the local spin AF coupling J 1 . This d-wave form factor is inherited from the orbital hybridization in the representation of C p 4 and becomes the characteristic nature of multi-orbital iron-pnictide superconductors. Moreover, together with the S 4 symmetry, we are in fact bestowed with 2 copies of d-wave gap structure, counting 8 nodes in total. By weakly breaking C p 4 we are naturally led to the so-called "octet-noded monster" observed by laser ARPES. We thus propose a symbolic equation "8 = 4 + 4" that captures the essential physics ruled by S 4 and C p 4 symmetries, exhibiting the origin of nodes in a rather simple and comprehensive way. Moreover, we find that a different representation of this plaquette-centered rotation symmetryC p 4 can stabilize the FS structure of iron-chalcogenides, including the monolayer FeSe on SrTiO 3 substrate where a tiny hole Fermi pocket around Γ vanishes [36,37]. So both families of Fe-based superconductors share the plaquette-centered rotation symmetry, and the importance of this symmetry lies in its capability to understand the band topology and robustness of FSs, and to reconcile some seemingly contradictory experimental observations in a rather simple, inevitable, and comprehensive manner. RESULTS C p 4 symmetry and band topology Due to the weak out-of-plane coupling (along c-axis), the electronic properties of Fe-based SCs are mainly contained in the FeAs plane, where the Fe atoms form a square lattice and the As atoms alternate above or below the Fe plaquette center (Fig.1a). Because of the checkerboard pattern of As lattice, the FeAs plane does not respect the site-centered C s 4 symmetry, but is invariant under the site-centered C s 4 rotation followed by a mirror reflection with respect to the plane [22]: S 4 ≡ C s 4 × R z . We further discover that the plaquette-centered rotation symmetry C p 4 is preserved. Among the five d-orbitals, the low-energy physics near the FS is mainly contributed by the d xz , d yz , d xy orbitals. Instead of a direct tunneling via the wave function over-lap, the hopping between d xz -and d yz -orbitals is primarily contributed by their hybridization with the p-orbitals on the As atoms, whereas the d xy orbitals do not have this privilege and are not included in our minimal model. The As atoms on plaquette centers polarize the d xz -and d yz -orbitals into d x z -and d y z -orbitals to maximize energy gain from hopping ( Fig.1a and Fig.1b). Therefore the square lattice with d x z and d y z orbitals on each site is effectively factorized into the top and bottom layers where each site has one orbital but the unit cell has to be doubled. As shown in Fig.1b, the top layer has d x z living on the odd lattice sites (denoted as A) and d y z on the even lattice sites (denoted as B). Likewise, the bottom layer has d y z living on the odd lattice sites (denoted as C) and d x z on the even lattice sites (denoted as D). For convenience we will denote the top layer as "AB layer" while the bottom layer as "CD layer". Albeit weakly coupled by inter-layer tunneling, the two layers are related by the S 4 symmetry, while the sublattice degrees of freedom inside each layer are rotated by C p 4 . To demonstrate the symmetries, we introduce the notion τ µν ≡ I µ ⊗ I ν , µ, ν = 0, 1, 2, 3, where the first Pauli matrix I µ acts on the S 4 spinor space spanned by those states living on either layer, and the latter Pauli matrix I ν acts on the C p 4 spinor space composed of states living on each sublattice. By choosing the simple gauge as shown in Fig.1b, the representation of S 4 and C p 4 symmetries can be explicitly expressed as S 4 :     A k,σ B k,σ C k,σ D k,σ     →     C k ,σ −D k ,σ −A k ,σ B k ,σ     = iτ 23     A k ,σ B k ,σ C k ,σ D k ,σ     , C p 4 :     A k,σ B k,σ C k,σ D k,σ     →     B k ,σ −A k ,σ −D k ,σ C k ,σ     = iτ 32     A k ,σ B k ,σ C k ,σ D k ,σ     , where k = C 4 k. Since the spin-orbit coupling is not concerned, the spin degeneracy is always present. The rotation factor contributed by the spin rotation (1 + iσ z )/ √ 2 does not affect the physics and can be absorbed by a basis transformation. Moreover, we would like to point out that these two symmetries commute with each other. Formally, the C p 4 symmetry operation is indeed diagonal in the S 4 spinor space and mainly rotates the intralayer sublattices, while the S 4 operation is diagonal in each sublattice but primarily rotates the layers. It is worth noticing that the S 4 spinor and C p 4 spinor look like dual to each other. But the inter-sublattice hopping is much stronger than the inter-layer tunneling, which makes the S 4 doublet weakly coupled but C p 4 doublet strongly hybridized. The power of the S 4 symmetry lies in that, once the dynamics of one layer is obtained, it is straightforward to derive the other. Therefore, in the presence of S 4 symmetry, we are bestowed with a minimal x' C p 4 spinor model living on either layer. In the following we focus on the properties of the top layer before paying a revisit to the complete S 4 spinor. A B A + B B A C C + D C D D (c) w = 2 w = +2+ + - - + + - - As x'(y') z A(B) A(B) y'(x') z + + - - + + - - As C(D) C(D) top bottom C p 4 C s 4 S 4 = C s 4 ⇥ R z t1x t1y t2 t 0 2 J 1 J 2 J 0 2 d xz d x 0 z d yz d y 0 z d xz d x 0 z d yz d y 0 z As shown in Fig.1b, the kinetic part of the model Hamiltonian mainly involves anisotropic nearest neighbor (NN) hopping t 1 and the next nearest neighbor (NNN) hopping t 2 or t 2 , which can be recombined and decomposed into s-wave and d-wave representation: t 1s = (t 1x + t 1y ) /2, t 1d = (t 1x − t 1y ) /2, t 2s = (t 2 + t 2 ) /2, and t 2d = (t 2 − t 2 ) /2. In terms of the C p 4 spinor Ψ k,σ ≡ (A k,σ , B k,σ ) T , it is then expressed as H AB t = 1 2 k,σ Ψ † k,σ [ 0 (k) + x (k)I 1 + z (k)I 3 ] Ψ k,σ ,(1) where k ranges in the unfolded BZ and 0 (k) = 4t 2s cos k x cos k y − µ, z (k) = −4t 2d sin k x sin k y , x (k) = 2t 1d (cos k x − cos k y ) + 2t 1s (cos k x + cos k y ) ≡ xd (k) + xs (k). Note that z (k) denotes the sublattice energy difference and x (k) is the energy gain from the NN hopping process. A vector can be defined by h(k) ≡ ( x , 0, z ), which acts as a "magnetic field" in the momentum space and pinning the C p 4 spinor (Fig.1c). It should be noted that the sublattice degree of freedom in AB layer is locked with the atomic internal angular momentum i.e. the odd sites carry d x z orbitals while the even sites carry d y z orbitals, so that the C p 4 spinor is actually a composite of the sublattice degree of freedom and the internal atomic angular momentum. For instance, when t 1s = 0, t 1d < 0, t 2s > 0, and t 2d > 0, the spinor along k x with the lowest energy is composed of the equal superposition of d x z -orbitals on odd sites and d y z -orbitals on even sites (Fig.1c), equivalent to d xz -orbitals. But along the BZ diagonal, the spinor with the lowest energy consists of the d x z -orbitals on odd sites or d y z -orbitals on even sites. Before diagonalization, there are several remarks about the symmetries in this kinetic part. The d-wave and s-wave NN hopping correspond to two distinctive symmetry representations of the plaquette-centered rotation symmetry, respectively. Based on the sketch of a snapshot of the wave function distribution in Fig.1b, we naturally expect |t 1d | |t 1s | ≈ 0. Indeed, adding a nonzero t 1s would break the symmetry C p 4 . So the ideal case of t 1s = 0 and t 1d = 0 respects the symmetry C p 4 , which can faithfully characterize the FS structure of iron-pnictides. In the helicity basis, H AB t can be directly diagonalized H AB t = 1 2 k,σ ξ e (k)α † k,σ α k,σ + ξ h (k)β † k,σ β k,σ ,(2) where ξ e(h) (k) = 0 ± 2 x + 2 z represents the electron (hole) band with ± helicity, respectively: α † k,σ = cos θ k 2 A † k,σ + sgn ( x ) sin θ k 2 B † k,σ , β † k,σ = cos θ k 2 B † k,σ − sgn ( x ) sin θ k 2 A † k,σ ,(3) and the hybridization angle is given by θ k = tan −1 | x| z ∈ [0, π]. The band dispersion and the corresponding FS structures can be easily obtained. Fig.2a exhibits the coexistence of a hole pocket around Γ and an electron pocket around X point in the absence of the s-wave NN hopping, a characteristic FS structure of most ironpnictides. As the d-wave NN hopping is growing, the electron band is gradually pushed upwards, shrinking the electron pocket. It finally leads to the FS structure of KFe 2 As 2 (Fig.2b), where the electron pocket completely vanishes. This evolution is also related to the doping process of Ba 1−x K x Fe 2 As 2 iron-pnictides [39]. Note that the shape of the hole pocket has the C 4 rotational symmetry thanks to the symmetry C p 4 , and the t 1d =-0.3 , t 1s =0 , t 2d =1 , t 2s =0.5 two bands touch each other quadratically at zone center Γ point, which is not accidental but rather due to the topological nature. Namely, the helical spinor is pinned by the vector field h(k) to wind around the Γ point twice, yielding a topological number w = 2sgn (t 1d t 2d ) (Ref. [40]). It is this topological number that forces the electron and hole bands to touch quadratically at the vortex core of h(k), which becomes the source of the Berry flux experienced by the helical spinor (see Fig.1c). Consequently, the hole pocket surrounding the vortex core is protected by topology, whereas the electron pocket is less robust. As long as the time reversal (TR) symmetry is present, the occurrence of I 2 is forbidden and this vortex core is robust. However, the TR symmetry cannot prevent the double Dirac point at Γ from splitting apart into two separated Dirac nodes, which could be driven by a weak C p 4 -breaking perturbation. This is indeed the case when the s-wave hoping comes in mixing the d-wave NN hopping (Fig.2c), where the hole pocket experiences a "nematic force" and is simultaneously elongated. When the C p 4 symmetry breaking term is large enough, the hole pocket around Γ is going to be torn apart, resulting in two pockets surrounding the Dirac nodes, which is not seen in the real materials. So the iron-pnictides with hole pockets around Γ robustly observed in experiments must preserve the C p 4 symmetry, at least approximately. In other words, the FS structure of iron-pnictides is stabilized by the C p 4 symmetry which helps the TR symmetry protect and confine the double Dirac point to the hole pocket center. In this sense, the C p 4 symmetry is one of the key relevant features of iron-pnictides, which can tolerate only tiny amount of s-wave hopping. In the next section, we will focus on the ideal case of d-wave NN hopping limit by sending t 1s → 0 for KFe 2 As 2 . Specifically, we choose the realistic hopping parameters as t 1d = −1.2, t 2s = 0.5, and t 2d = 1, whose corresponding band structure is given by Fig.2b. When things are clear in the ideal case, we will introduce a weak s-wave NN hopping to model the more realistic material later on. C p 4 protected d-wave renormalized pairing SC Provided the electronic structure of KFe 2 As 2 , we need to include the electronic interactions. The strong correlation evidenced by experiments [17,33,34] requires a strong coupling approach. Particularly, the optical measurement [34] had showed the incoherent spectral weight of KFe 2 As 2 as high as 10% hole-doped La 2−x Sr x CuO 4 , revealing the strong local AF correlation similar to the cuprates. Therefore, we would like to model this system with AF super-exchanges containing the NN and NNN interactions as shown in Fig.1b. Distinct from the t − J 1 − J 2 model in the previous form [20], our model Hamiltonian is subjected to a constraint that projects out double-occupancy. Thus, the model Hamiltonian for AB layer is described by H AB = PH AB t P + H AB J with interaction terms: H AB J = J 1 r,δ S A r · S B r+δ +J 2 r S A r · S A r+(x+ŷ) + S B r · S B r+(x−ŷ) +J 2 r S A r · S A r+(x−ŷ) + S B r · S B r+(x+ŷ) ,(4) where δ = ±x,ŷ is the NN vector. As the low-energy descendant of the on-site Hubbard interaction, the AF super-exchange J-terms rely on the corresponding hopping integrals. Like cuprates, the parameters can be chosen as J 2 = 0.5 and J 2 = 0.2 as roughly one third of the corresponding hopping integrals and the dopant concentration is fixed at 0.05. Note that although J 2 differs from J 2 , only the combination J 2s ≡ 2J 2 J 2 /(J 2 + J 2 ) contributes to pairing. So that the physics is essentially captured by the competition between J 1 and J 2s . The parameter J 1 is chosen as the tuning parameter that mimics the inverse pressure when compared with the pressure experiments [32]. The projector P declares the particle occupancy constraints: σ A † r,σ A r,σ ≤ 1 and σ B † r,σ B r,σ ≤ 1, which lead to emergent spin-charge separation physics. To tackle the constraint, we adopt the well-established slaveboson decomposition to factorize electron into fermionic spinon and bosonic holon: A r,σ = h † r a r,σ , B r,σ = h † r b r,σ ,(5) in which way the constraint becomes equalities: σ a † r,σ a r,σ + h † r h r = 1 and σ b † r,σ b r,σ + h † r h r = 1, and can be enforced via a Lagrangian multiplier. After the bosonic holons are condensed h † r = h r = √ x, we introduce the uniform valence bond and singlet pairing order parameters κ ij = − J 4 σ a † i,σ a j,σ , ∆ ij = J 4 a i,↑ a j,↓ − a i,↓ a j,↑ , to decouple the local AF interactions. Moreover, the NN and NNN valence bond and singlet pairing order parameters in real space can be rearranged into the s-wave and d-wave representations: κ 1 is automatically of d-wave symmetry required by C p 4 symmetry and other valence bond orders are κ 2s = (κ 2 + κ 2 )/2, κ 2d = (κ 2 − κ 2 )/2, ∆ 1s = (∆ 1x + ∆ 1y )/2, ∆ 1d = (∆ 1x − ∆ 1y )/2, ∆ 2s = (∆ 2 + ∆ 2 )/2, ∆ 2d = (∆ 2 − ∆ 2 )/2.(6) Finally, a mean-field Hamiltonian can be obtained H ab MF = H ab t + H ab ∆ ,(7) which describes the superconducting quasiparticles on the AB layer. The kinetic part H ab t has the same band structure as we discussed in the last section. H AB t → H ab t requires replacing the electrons with the corresponding quasiparticles Ψ k,σ = (A k,σ , B k,σ ) T → ψ k,σ ≡ (a k,σ , b k,σ ) T , while renormalizing the hopping integrals and the chemical potential as t 1d →t 1d = (t 1d x + κ 1 ) , t 1s →t 1s = t 1s x, t 2s →t 2s = (t 2s x + κ 2s ) , t 2d →t 2d = (t 2d x + κ 2d ) , µ 0 → µ = µ 0 − λ − (J 2 + J 3 ) /4.(8) The hybridization relation Eq. (3) also holds by replacing A(B) with a(b), and the mean-field pairing terms can be compactly expressed in terms of the C p 4 spinor: H ab ∆ = 1 2 k ψ † k,↑ (∆ 0 + i∆ x I x + ∆ z I z ) (ψ † k,↓ ) T + h.c.(9) with ∆ 0 (k) = 4∆ 2s cos k x cos k y , ∆ z (k) = 4∆ 2d sin k x sin k y , ∆ x (k) = 2∆ 1s (cos k x + cos k y ) + i2∆ 1d (cos k x − cos k y ) . Provided with all the form factors available from the interactions, we can eliminate some of them which are apparently unfavorable energetically. In this minimal model for KFe 2 As 2 , within the AB layer there is only a small hole pocket around Γ point. The Cooper pair formed on the FS is more likely to be scattered onto the same FS with small momentum transfer. And the pairing interactions J 1 (q) ∝ −2J 1 (cos q x + cos q y ) and J 2s (q) ∝ −4J 2s cos q x cos q y are attractive when q ≈ 0, while J 2d (q) ∝ 4J 2d sin q x sin q y tends to vanish. Therefore, the pairing components ∆ 1d and ∆ 2d are energetically unfavorable compared to ∆ 1s and ∆ 2s , which endow the FS with largest possible energy gain. In fact, we did some self-consistent calculation numerically to verify the results ∆ 1d = ∆ 2d = 0. Thus we are left with two swave pairing components ∆ 1s and ∆ 2s to be determined self-consistently by minimizing the ground state energy. As there is only single hole pocket FS in the low energy excitations, we can project the effective Hamiltonian onto the hole band and especially focus on the vicinity of FS. Turned to the band basis Γ k,σ ≡ (α k,σ , β k,σ ) T , the meanfield Hamiltonian can be straightforwardly obtained H ab eff = 1 2 k,σ ξ h (k)β † k,σ β k,σ(10)+ 1 2 k ∆ 0 (k) − i xd (k)∆ x (k) 2 xd + 2 z β † k,↑ β † −k,↓ + h.c.. Here we can see that the s-wave pairing component ∆ x arising from the NN spin exchange interaction is renormalized by a d-wave form factor xd ∝ (cos k x − cos k y ), which is inherited from the NN hopping integral in the C p 4 symmetry representation. Actually it should be no surprise, since we can physically understand this renormalization factor in the following way. The NN pairing interaction glues the inter-sublattice particles, which coexist in the hole band with probability proportional to the hybridization energy gain. As a result, the effective intra-hole-band pairing condensate is supposed to be renormalized by xd . As shown in Fig.1c, along the zero lines of hybridization energy xd , there is no coexistence of the sublattice degrees of freedom so that there are no inter-orbital Cooper pairs, regardless of their internal pairing symmetry. Such a mechanism is parallel to the effective intra-band p-wave pairing induced on the helical electrons in proximity to the s-wave superconductors [38]. Then the Bogoliubov quasi-particle spectrum can be derived E(k) = ± ξ 2 h (k) + ∆ 2 0 (k) +∆ 2 x (k),(11)where∆ x (k) ≡ xd (k) √ 2 xd + 2 z ∆ x (k) and E(k) exhibits the nodal excitations when ∆ 0 (k) = 0. Note that the interband pairing between hole band and high energy electron band only contributes second order perturbation corrections to the gap structure, but is unable to alter its symmetry. The d-wave gap nodes carrying one unit of vorticity in the Nambu space are topologically protected [42,43]. Within the AB layer, there are only two ways to destroy these nodes. The first one is to generate a mass upon the massless Bogoliubov quasiparticles [44] which is forbidden by the TR symmetry in pairing sector. The mass term is an additional pairing component with phase difference that cannot be gauged away. The other way is to move the nodes with opposite vorticity to annihilate each other, which is prevented by the C p 4 symmetry that confines the nodes to the unfolded BZ diagonal. Now we have two competing pairing components: the NNN Cooper pair condensate ∆ 0 (k) with the s x 2 y 2 -wave form factor and the NN Cooper pair condensate∆ x (k) carrying the d x 2 −y 2 × s x 2 +y 2 -wave form factor. They can be varied by the NN interaction J 1 for a fixed NNN interaction J 2s 0.3. The detailed self-consistency calculation explicitly shows phase transitions of pairing symmetry displayed in Fig.3. When J 1 J 2s , the NNN interaction overwhelms the NN interaction, leading to the s x 2 y 2 -wave pairing, whereas J 1 J 2s results in the d x 2 −y 2 × s x 2 +y 2 -wave pairing. In between, the SC with a mixed pairing s + id × s is energetically more favorable, which spontaneously breaks the TR symmetry. To compare with the pressure experiments [32], we notice that, upon increasing pressure, both t 2 and J 2 are expected to grow faster than t 1 and J 1 . Since the dopant concentration is fixed and the small hole pocket FS does not change qualitatively, we expect that decreasing the ratio of J 1 /J 2s is adequate to capture the essential physics during the period of increasing pressure. When the superfluid density does not vary drastically, the superconducting critical temperature would be roughly proportional to the maximum gap on the FS, and it turns out the tendency of maximum gap on the FS along J 1 shown by the black solid line in Fig.3 concurs qualitatively well with the T c trend under decreasing pressure in experiments [32]. Moreover, the phase transition from the d × s-wave nodal SC to the s + id × s nodeless SC belongs to the TR breaking mass generation scenario of destroying the pairing gap nodes. For the CD layer, we can simply apply the S 4 symmetry to obtain its low-energy effective Hamiltonian: where γ † k,σ = cos θ k 2 c † k,σ − sgn ( xd ) sin θ k 2 d † k,σ . η embodies the global phase difference between the pairing condensates of the two layers, and is restricted to be either ±1 or ±i as the one-dimensional representations of the S 4 point group [22]. The relative phase difference can be pinned down by inter-layer couplings for the benefit of energetics. H cd eff = 1 2 k,σ ξ h (k)γ † k,σ γ k,σ(12)+ η 2 k ∆ 0 (k) + xd (k) 2 xd + 2 z i∆ x (k) γ † k,↑ γ † −k,↓ + h.c. Octet nodal SC from distorting d-wave pairing We are now in a good position to focus on the nodal phase at ambient pressure, which appears like a sphinx in a variety of experiments. Based on what we obtained, we have two S 4 -related C p 4 symmetric hole pockets which are absolutely degenerate. They can form d-wave pairing condensates with degenerate quartet nodes residing on the unfolded BZ diagonal (Fig.4a and 4b). Such degeneracy is highly unstable against any arbitrary perturbation, so we are obliged to return to the more realistic materials by involving the weak C p 4 -breaking hopping t 1s and inter-layer tunneling t c . As for the inter-layer tunneling, the s-wave NN tunneling would be the dominant term, from the estimate of orbitals overlap and symmetry analysis (see Fig.1b). Note that the on-site tunneling is suppressed because of the orbital orthogonality. Surprisingly, the S 4 symmetry survives this tunneling process. When t c t 1s t 1d , the effective Hamiltonians for the AB and CD layers are modified by t 1s in their respective ways: H ab eff → 1 2 k,σ ξ + h (k)β † k,σ β k,σ(13)+ 1 2 k xd (k) + xs (k) ( xd + xs ) 2 + 2 z ∆ x (k)β † k,↑ β † −k,↓ + h.c. H cd eff → 1 2 k,σ ξ − h (k)γ † k,σ γ k,σ(14)+ η 2 k − xd (k) + xs (k) ( xd − xs ) 2 + 2 z ∆ x (k)γ † k,↑ γ † −k,↓ + h.c. where the global phase of ∆ 1s has been gauged to absorb the phase factor −i. Meanwhile, the normal state dispersion and the hybridization angle for AB/CD layer are given by ξ ± h (k) = 0 (k) − ( xd ± xs ) 2 + 2 z ,(15) and θ ± = tan −1 | xd ± xs | z , so that the quasiparticles, β † k,σ and γ † k,σ are adapted (see Supplementary Material for detail). As a result, the d-wave nodal lines that used to lie across the unfolded BZ diagonal are now twisted and dragged away from the Γ point by a "nematic force", resulting in a simultaneous movement of nodes along the elongating direction of the hole Fermi pockets (Fig.4a and 4b). Since the nodes with opposite vorticity need to travel a finite path along the FS before annihilation, they can survive a finite weak C p 4 -breaking term depending on the size of the FS. The hole pockets from the AB and CD layers are required by S 4 symmetry to intersect at the unfolded BZ diagonal (Fig.4c). Degeneracy at this point is unstable against any arbitrarily weak perturbation of inter-layer tunneling: H c = 1 2 k,σ c (k) a † k,σ d k,σ + b † k,σ c k,σ + h.c. → 1 2 k,σ ˜ c (k)β † k,σ γ k,σ + h.c. .(16) So the degenerate Fermi-points at the unfolded BZ diagonal are split by˜ c (k) into two Fermi-points, whose corresponding quasi-particles are given by the bonding and anti-bonding states of the two layers. The split Fermi points smoothly join the Fermi sheets far away from the unfolded BZ diagonal, where˜ c (k) amounts only up to second order perturbation corrections (Fig.4c). Next let's consider how the pairing matrix changes with the interlayer tunneling. It is expected that η = 1 to avoid the destructive interference along the BZ diagonal where two layers strongly hybridize. Therefore, the pairing condensates on the unfolded BZ diagonal were identical for hole pockets from both layers, guaranteed by the S 4 symmetry + = t 1s CD layer In the AB layer, the mixing weak s-wave NN hopping acts like a "nematic force" that elongates the FS and distorts the d-wave nodal lines, leading to shifted quartet nodes. (b) As the CD layer is related to AB layer by the S4 symmetry, the distortion occurs in the perpendicular direction. (c) Near unfolded BZ diagonal the quasi-particles from the AB and CD layers are strongly hybridized with each other by any infinitesimal inter-layer tunneling. The intersecting elliptic hole pockets are therefore reconstructed into inner and outer pockets. The inner one has nodeless pairing gap whereas the outer one shows octet nodal gap structure. The origin of nodes can be attributed to a rather simple symbolic equation "8=4+4". ( Fig.4c). The bonding of the two layers does not affect the pairing matrix along unfolded BZ diagonal, which maintains diagonal in the form of identity (see Supplementary Material for detail). As a result, the two reconstructed bands yield the outer and inner hole pockets and are decoupled in pairing sector to the leading order. The outer pocket inherits the total octet nodes (Fig.4c), which still carry vorticity in the Nambu space and are protected by topology. On the inner pocket the pairing gap is nodeless. It should be made clear that our outer pocket corresponds to the "middle pocket" termed in experimental report [30], because the d xy orbital has been neglected in our minimal model. In this picture, we have seen how the octet nodes come out naturally as observed in the laser ARPES [30], and we can understand why every two of the octet nodes are located so close to the unfolded BZ diagonal in experiment, because they are essentially born of the d-wave representation of C p 4 albeit distorted by the weak C p 4 -breaking term. Therefore, the so-called "octet-noded monster" is neither accidental nor some crazy Cooper pair of angu-lar momentum as high as g-wave, but the combination of two distorted d-wave related by the S 4 symmetry. In short, the origin of nodes can be attributed to a symbolic equation "8 = 4+4". The nodes share the same fate with the FS structure that protests against strong breaking of C p 4 symmetry. Therefore, we settle the disagreement between d-wave gap structure and the observation of octet nodes on one of the hole pockets. All in all, the key feature lies in the multi-orbital character. DISCUSSION Inspired by this organizing principle governed by C p 4 and S 4 symmetries, we can gain some insight into the electronic structure and pairing symmetry of ironselenide (FeSe). While the d-wave representation C p 4 of plaquette-centered rotation symmetry stabilizes the FS structure of iron-pnictides, the s-wave representatioñ C p 4 captures the key feature of FSs in FeSe as shown in Fig.5a, where the NN s-wave hopping dominates. In Fig.5b, it can be shown that, when the d-wave NN hopping is completely replaced by the s-wave hoping followed by a particle-hole transform, we can obtain the FS with a robust electron pocket around X point without hole pocket around Γ point, which is characteristic of FeSe [41]. In this case, the electron pocket and its central double Dirac point are protected by the TR symmetry together with the s-wave representation of the plaquette-centered rotation symmetryC p 4 ≡ τ 31 . Thus, the band structure of FeSe is related to that of iron-pnictides by a gauge transform combined with a particle-hole transform. FIG. 1 : 1Lattice configuration of the iron-pnictides. (a) The transverse view of the Fe-As plane shows the hopping of 3d x (y )z orbital via hybridization with p x (y ) orbital on arsenic atoms, either above or below the Fe square lattice plane. (b) The 3d x (y )z orbitals on the Fe lattice can be approximately decoupled into two groups which hop around via the arsenic atom on top or bottom of the plane respectively. Thus the lattice can be viewed as "factorized" into the top and bottom layers, which are related by S4 symmetry. (c) For d-wave representation of plaquette-centered rotation symmetry C p 4 , the Γ point is the source of Berry flux carrying topological winding number w = ∓2 for AB (CD) layer. The red arrow denotes the polarization of C p 4 spinor on the Fermi pocket, where spin up represents A(C) orbitals and spin down represents B(D) orbitals. FIG. 2 : 2Band structures and Fermi surfaces of the minimal model without considering any electron interactions for various systems of Fe-based superconductors. Left panel shows the band structure for varying parameters while right panel shows the corresponding FS in the unfolded BZ. (a) The parameters are t 1d = −0.3, t1s = 0, t2s = 0.5, in unit of t 2d = 1. Hole pocket is located around Γ and electron pocket around X, describing the FS structure for iron-pnictides. The Γ point is a double Dirac point with quadratic band touching dispersion. (b) The parameters are t 1d = −1.2, t1s = 0, t2s = 0.5, in unit of t 2d = 1. Electron pocket vanishes, while the hole pocket around Γ is robust, standing for the heavily hole doped iron-pnictides. (c) The parameters are t 1d = −1.2, t1s = −0.3, t2s = 0.5, in unit of t 2d = 1. Mixing d-wave NN hopping with s-wave component can split the double Dirac point into two Dirac points while elongating the hole pocket. Hole pocket could be further torn apart. FIG. 3 : 3Phase diagram of the superconducting phases for KFe2As2 under pressure (decreasing J1 mimics the trend of increasing pressure). The parameters are x = 0.05, t 1d = −1.2, t2s = 0.5, J2 = 0.5, J 2 = 0.2 in unit of t 2d = 1. Red line is for the NN Cooper pair |∆1s| that obtains s x 2 y 2 form factor in momentum space, while blue line is for the NNN Cooper pair condensate |∆2s| who is endowed with d x 2 −y 2 × s x 2 +y 2wave form factor. The d x 2 −y 2 × s x 2 +y 2 -wave SC comes to replace the s x 2 y 2 -wave SC when J1 overwhelms J2s. The grey color region marks the intermediate phase region with s x 2 y 2 + id x 2 −y 2 × s x 2 +y 2 mixed pairing SC. Black solid line shows the maximal gap along the FS, which is roughly proportional to the critical temperature when superfluid density does not vary drastically. Insets show the pairing gap structure on the FS, characteristic of the three pairing phases, respectively. FIG. 4 : 4Distorted d-wave nodes on the Fermi surfaces. The white diamond emphasizes the folded BZ; the orange dashed lines denote the nodal lines of the effective d x 2 −y 2 -wave pairing condensate on the FS distorted by s x 2 +y 2 -wave factor (the effect of weak s-wave is moderately exaggerated for illustration); The FS is separated by pairing nodal lines into the segments with positive (negative) pairing condensate marked by red (blue) color, respectively. (a) choice characteristic of FeSe, followed by a particle-hole transform. (b) Band dispersion typical of ironchalcogenides, where the electron pocket around (π, 0) is robust while hole pocket around (0,0) is dispensable. The parameters for the demonstration are chosen to be t 1d = 0, t1s = 0.45, t2s = 0.5 in unit of t 2d = 1. AcknowledgmentGMZ acknowledges the support of National Key Research and Development Program of China (2016YFA0300300).Author contributions GMZ initiated and supervised this project, GYZ conducted the derivation and calculation, and GYZ and GMZ wrote the paper. Parallel to the above discussion for KFe 2 As 2 , there are two possible pairing symmetries for the monolayer FeSe: one is the s x 2 y 2 -wave Cooper pair glued by the dominant NNN AF coupling J 2 and the other one is the effective s x 2 +y 2 × d x 2 −y 2 -wave Cooper pair on the electron pockets around the X point for a relatively large NN AF coupling J 1 . Although the inter-orbital Cooper pairs glued by J 1 has d x 2 −y 2 -wave symmetry, whose nodal line avoids the FS to maximize the energy gain, the projection of the interorbital Cooper pairs onto the FS yields an additional s x 2 +y 2 form factor inherited from the electronic structure. However, when J 2 overwhelms J 1 , the pairing symmetry is the s x 2 y 2 -wave. Without more interactions, the two layers tolerating weak inter-layer tunneling tend to lock their respective phases of Cooper pairs to be identical, otherwise there would be destructive interference. This supports the plain s-wave pairing symmetry as observed by the STM measurement. 45the pairing symmetry is considered, we can just focus on the monolayer FeSe with only electron Fermi pockets around X point[41. But when J 1 dominates over J 2 , the effective s x 2 +y 2 × d x 2 −y 2 -wave pairing condensate would show a gap minimum near the folded BZ boundary and a gap maximum along the unfolded BZ boundary, which seems to concur with the superconducting gap anisotropy measured by ARPES[41the pairing symmetry is considered, we can just focus on the monolayer FeSe with only electron Fermi pockets around X point[41]. Parallel to the above dis- cussion for KFe 2 As 2 , there are two possible pairing sym- metries for the monolayer FeSe: one is the s x 2 y 2 -wave Cooper pair glued by the dominant NNN AF coupling J 2 and the other one is the effective s x 2 +y 2 × d x 2 −y 2 - wave Cooper pair on the electron pockets around the X point for a relatively large NN AF coupling J 1 . Al- though the inter-orbital Cooper pairs glued by J 1 has d x 2 −y 2 -wave symmetry, whose nodal line avoids the FS to maximize the energy gain, the projection of the inter- orbital Cooper pairs onto the FS yields an additional s x 2 +y 2 form factor inherited from the electronic struc- ture. However, when J 2 overwhelms J 1 , the pairing sym- metry is the s x 2 y 2 -wave. Without more interactions, the two layers tolerating weak inter-layer tunneling tend to lock their respective phases of Cooper pairs to be iden- tical, otherwise there would be destructive interference. This supports the plain s-wave pairing symmetry as ob- served by the STM measurement[45]. But when J 1 dom- inates over J 2 , the effective s x 2 +y 2 × d x 2 −y 2 -wave pairing condensate would show a gap minimum near the folded BZ boundary and a gap maximum along the unfolded BZ boundary, which seems to concur with the superconduct- ing gap anisotropy measured by ARPES[41]. 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{'abstract': 'A plaquette-centered rotation symmetry C p 4 is identified to play a significant role in determining and stabilizing the Fermi-surface structure of Fe-based superconductors. Together with the S4 symmetry previously found, we are able to sort out the tangling orbitals and solve the puzzle of pairing symmetry of superconductivity in KFe2As2 in a simple but comprehensive way. By modeling the material with a strong coupling t − J1 − J2 model, we find phase transitions of pairing symmetry driven by the competition between the local spin antiferromagnetic couplings from nodal d x 2 −y 2 × s x 2 +y 2 -wave to nodeless s x 2 y 2 -wave through the intermediate s + id × s mixed pairing phase, which is consistent with the observation of pressure experiments. The emergent d-wave form factor inevitably arises from the projection of inter-orbital Cooper pairing onto the Fermi surface and is inherited from the electronic structure in the representation of C p 4 symmetry. Moreover, the S4 symmetry dictates 2 copies of d-wave pairing condensates, counting 8 nodes in total. We further show that weakly breaking C p 4 naturally leads to the octet nodal gap as precisely observed in laser angle resolved photoemission spectroscopy. The octet nodes reflect the collaboration of the C p 4 and S4 symmetries, which sheds new light on the enigma of the pairing symmetry in KFe2As2. arXiv:1707.04009v2 [cond-mat.supr-con]', 'arxivid': '1707.04009', 'author': ['Guo-Yi Zhu \nDepartment of Physics\nState Key Laboratory of Low-Dimensional Quantum Physics\nTsinghua University\n100084BeijingChina\n', 'Guang-Ming Zhang \nDepartment of Physics\nState Key Laboratory of Low-Dimensional Quantum Physics\nTsinghua University\n100084BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100084BeijingChina\n'], 'authoraffiliation': ['Department of Physics\nState Key Laboratory of Low-Dimensional Quantum Physics\nTsinghua University\n100084BeijingChina', 'Department of Physics\nState Key Laboratory of Low-Dimensional Quantum Physics\nTsinghua University\n100084BeijingChina', 'Collaborative Innovation Center of Quantum Matter\n100084BeijingChina'], 'corpusid': 103666677, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 20135, 'n_tokens_neox': 17344, 'n_words': 10825, 'pdfsha': '451ea9d4149fed3f8ef0521f84265b9e755ce426', 'pdfurls': ['https://arxiv.org/pdf/1707.04009v2.pdf'], 'title': ['Plaquette-centered rotation symmetry and octet-nodal superconductivity in KFe 2 As 2', 'Plaquette-centered rotation symmetry and octet-nodal superconductivity in KFe 2 As 2'], 'venue': []}

arxiv

Joule-Thomson Expansion of Charged AdS Black Holes Ozgürökcü Department of Physics Faculty of Science İstanbul University 34134İstanbulTurkey Ekrem Aydıner Department of Physics Faculty of Science İstanbul University 34134İstanbulTurkey Joule-Thomson Expansion of Charged AdS Black Holes (Dated: January 24, 2017) In this paper, we study Joule-Thomson effects for charged AdS black holes. We obtain inversion temperatures and curves. We investigate similarities and differences between van der Waals fluids and charged AdS black holes for the expansion. We obtain isenthalpic curves for both systems in T − P plane and determine the cooling-heating regions. I. INTRODUCTION It is well known that black holes as thermodynamic systems have many interesting consequences. It sets deep and fundamental connections between the laws of classical general relativity, thermodynamics, and quantum mechanics. Since it has a key feature to understand quantum gravity, much attention has been paid to the topic. The properties of black hole thermodynamics have been investigated since the first studies of Bekenstein and Hawking [1][2][3][4][5][6]. When Hawking discovered that black holes radiate, black holes are considered as thermodynamic systems. Black hole thermodynamics shares similarities with general thermodynamics systems. Specifically, black holes in AdS space have common properties with general systems. The study of AdS black hole thermodynamics began with pioneering paper of Hawking and Page [7]. They found a phase transition between Schwarzschild AdS black hole and thermal AdS space. Up to now, thermodynamic properties of AdS black holes have been widely studied in the literature . In [8,9], authors studied the thermodynamics of charged AdS black holes and they found analogy between phase diagrams of black hole and van der Waals fluids. When cosmological constant and its conjugate quantity are, respectively, considered as a thermodynamic pressure P = − Λ 8π ,(1) and thermodynamic volume V = ( ∂M ∂P ) S,Q,J , this analogy gains more physical meaning. Particularly, in the extended phase space (including P and V terms in the first law of black hole thermodynamics), charged AdS black holes phase transition is remarkable coincidence with van der Waals liquid-gas phase transition [16]. This type of transition is not limited with charged AdS black holes, various kind of black holes in AdS space show the same phase transitions [17][18][19][20][21][22][23][24][25][26][27]. It is also possible to consider heat cycle for AdS black holes [33][34][35][36][37][38][39][40]. In [33,40], author suggested two kind of heat cycles and obtained exact efficiency formula for black holes. Variable cosmological constant notion has some nice features such as phase transition, heat cycles and compressibility of black holes [41]. Applicabilities of these thermodynamic phenomena to black holes encourage us to consider Joule-Thomson expansion of charged AdS black holes. In this letter, we study the Joule-Thompson expansion for chraged AdS black holes. We find similarities and differences with van der Waals fluids. In Joule-Thomson expansion, gas at a high pressure passes through a porous plug to a section with a low pressure and during the expansion enthalpy is constant. With the Joule-Thomson expansion, one can consider heatingcooling effect and inversion temperatures. The paper arranged as follows. In Section II, we briefly review the charged AdS black hole. In Section III, we firstly review Joule-Thompson expansion for van der Waals gases and then we investigate Joule Thomson expansion for charged AdS black holes. Finally, we discuss our result in Section IV. (Here we use the units G N = = k B = c = 1.) II. CHARGED ADS BLACK HOLES In this section, we briefly review charged AdS black hole and we present its thermodynamic properties. Charged black hole in four dimensional space is defined with the metric ds 2 = −f (r)dt 2 + f −1 (r)dr 2 + r 2 dΩ 2 ,(2) where dΩ 2 = dθ 2 + sin 2 (θ)dφ 2 and f (r) is given by f (r) = 1 − 2M r + Q 2 r 2 + r 2 l 2 .(3) In these equations, l, M and Q are the AdS radius, mass, and charge of the black hole, respectively. One can obtain black hole event horizon as largest root of f (r + ) = 0. The mass of black hole in Eq. (3) is given by M = r + 2 1 + Q 2 r 2 + + r 2 + l 2(4) which satisfies the first law of black hole thermodynamics dM = T dS + ΦdQ + V dP (5) and corresponding Smarr relation is given by M = 2(T S − P V ) + ΦQ .(6) arXiv:1611.06327v2 [gr-qc] 21 Jan 2017 One can derive Smarr relation by scaling argument [15]. The first law of black hole thermodynamic includes P and V , when the cosmological constant is considered as a thermodynamic variable. The cosmological constant corresponds to the pressure, P = − 1 8π Λ = 3 8π 1 l 2(7) and cosmological constant's conjugate quantity corresponds to thermodynamic volume. The expression for entropy is given by S = A 4 = πr 2 + , A = 4πr 2 +(8) and the corresponding Hawking temperature T = ∂M ∂S P,Q = l 2 (r 2 + − Q 2 ) + 3r 4 + 4πl 2 r 3 + .(9) On the other hand, the electric potential is given by Φ = Q r+ and equation of state P = P (V, T ) for charged AdS black hole is obtained from Eqs. (7) and (9) as P = T 2r + − 1 8πr 2 + + Q 2 8πr 4 + , r + = ( 3V 4π ) 1 3 .(10) The critical points [16] obtained from ∂P ∂r + = 0, ∂ 2 P ∂r 2 + = 0,(11) which leads to T c = √ 6 18πQ , r c = √ 6Q, P c = 1 96πQ 2 .(12) Other thermodynamic properties can be obtained by using above relations. For example, heat capacities at constant pressure and constant volume are, respectively, given by C P = T ∂S ∂T P,Q = 2πr 2 3r 4 + − l 2 Q 2 + l 2 r 2 + 3r 4 + + 3l 2 Q 2 − l 2 r 2 + ,(13) and C V = T ∂S ∂T V,Q = 0 .(14) In this section, we give thermodynamic definitions for charged AdS black hole. In the next section, we will review Joule-Thomson expansion for van der Waals fluids and investigate Joule-Thomson expansion for charged AdS black holes. III. JOULE-THOMSON EXPANSION In this section, we review the well-known Joule-Thomson expansion [42,43]. In Joule-Thomson expansion, gas at a high pressure passes through a porous plug or small valve to a section with a low pressure in a thermally insulated tube and enthalpy remains constant during the expansion process. One can describe temperature change with respect to pressure and this change is given by µ = ∂T ∂P H .(15) Here µ is called the Joule-Thomson coefficient. It is possible to determine whether cooling or heating will occur by evaluating the sign of Eq. (15). In Joule-Thomson expansion, pressure decreases so change of pressure is negative but change of temperature may be positive or negative. If the change of temperature is positive (negative) µ is negative (positive) and so gas warms (cools). It is also possible to express Eq. (15) in terms of volume and heat capacity at constant pressure. From the first law of thermodynamics, one can write the fundamental relation for constant particle number N dU = T dS − P dV . Using the relation H = U + P V , Eq. (16) is given by dH = T dS + V dP .(17) Since dH = 0, Eq. (17) is given by 0 = T ∂S ∂P H + V .(18) Since entropy is a state function, the differential dS is given by dS = ∂S ∂P T dP + ∂S ∂T P dT(19) which can be rearranged to give ∂S ∂P H = ∂S ∂P T + ∂S ∂T P ∂T ∂P H .(20) If one can substitute this expression into Eq. (18), one can obtain the following expression: 0 = T ∂S ∂P T + ∂S ∂T P ∂T ∂P H + V .(21) Substituting Maxwell relation ∂S ∂P T = − ∂V ∂T P and C P = T ∂S ∂T P into Eq. (21) gives 0 = −T ∂V ∂T P + C P ∂T ∂P H + V(22) and it can be rearranged to give the Joule-Thomson coefficient [42] as follows: µ = ∂T ∂P H = 1 C P T ∂V ∂T P − V .(23) At the inversion temperature, µ equals zero and inversion temperature is given by T i = V ∂T ∂V P(24) which is useful to determine the heating and cooling regions in the T − P plane. A. van der Waals Fluids The van der Waals equation is a generalized form of ideal gas equation, which usually describes the liquid-gas phase transition behaviours for real fluids [43,44]. It takes into account the size of molecules and attraction between them. It is given by P = k B T v − b − a v 2 .(25) Here v = V N , P , T and k B denote the specific volume, pressure, temperature, and Boltzmann constant. a > 0 constant is a measure of attraction between particles and b > 0 is a measure of molecule volume. a and b constants are determined from experimental data. Before more proceeding to the Joule-Thomson expansion, it is useful to give some thermodynamic properties of van der Waals equation. Following [16,45], free energy is given by F (T, v) = −k B T 1 + ln (v − b)T 3 2 Φ − a v .(26) Here φ is a constant characterizing the gas. Now, entropy can be obtained from Eq. (26) S(T, v) = − ∂F ∂T v = k B 5 2 + ln (v − b)T 3 2 Φ . (27) Using Eqs. (26) and (27), we can calculate the internal energy U (T, v) = F + T S = 3k B T 2 − a v(28) and from Eqs. (25) and (28), enthalpy is H(T, v) = U + P V = 3k B T 2 + k B T v v − b − 2a v .(29) Now, let us calculate the inversion temperature for van der Waals equation. Using Eq. (24), inversion temperature is given by T i = 1 k B P i v − a v 2 (v − 2b)(30) where P i denotes the inversion pressure. From Eq. (25), one can get T i = 1 k B P i + a v 2 (v − b) .(31) Subtracting Eq. (30) from Eq. (31) yields bP i v 2 − 2av + 3ab = 0 (32) and, solving this equation for v(P i ), one can obtain two roots v = a ± √ a 2 − 3ab 2 P i bP i .(33) Substituting these roots into Eq. (31), one can obtain (35) which give lower and upper inversion curves, respectively. In Fig. (1), lower and upper inversion curves are presented. At the point P i = 0, we can obtain the minimum and maximum inversion temperatures T lower i = 2 5a − 3b 2 P i − 4 √ a 2 − 3ab 2 P i 9bk ,(34)T upper i = 2 5a − 3b 2 P i + 4 √ a 2 − 3ab 2 P i 9bkT min i = 2a 9bk , T max i = 2a bk .(36) The critical temperature for van der Waals fluids is given by T c = 8a 27bk and hence T min Using Eqs. (25) and (29), we can obtain the isenthalpic curves in T − P plane. In Fig. (2), isenthalpic and inversion curves are presented. When the isenthalpic curves cross inversion curves, their slopes change sign. Isenthalpic curves have positive slopes inside the inversion curves, otherwise their slopes are negative. As a result the Joule-Thomson coefficient is positive inside the inversion curves and cooling occurs inside this region. i T c = 3 4 , T max i T c = 27 4 .(37) B. Charged AdS Black Holes In this section, we will consider Joule-Thomson expansion for charged AdS black holes. In [15], authors suggested that black hole mass is considered as the enthalphy in AdS space. It means that our isenthalpic curves are actually constant mass curves in AdS space. We can consider black hole mass not to change during the Joule-Thomson expansion. For a fixed charge, similar steps in the previous section can be used to obtain the Joule-Thomson coefficient. Hence µ = ∂T ∂P M = 1 C P T ∂V ∂T P − V .(38) The charged AdS black hole equation of state can be given in terms of thermodynamic volume, T = 1 3 3 4π 2 3 V 1 3 8π 3 4π 2 3 P + 1 V 2 3 − 4π 3 2 3 Q 2 V 4 3(39) and evaluating this in the right hand side of Eq. (38), the inversion temperature is given by T i = 1 3 6 π 1 3 V 1 3 π 6 1 3 Q 2 V 4 3 − 6 π 1 3 1 12V 2 3 + P i = Q 2 4πr 3 + − 1 12πr+ + 2Pir+ 3 .(40) From Eq. (39), one can get and solving this equation for r + gives us four roots but only one root is physically meaningful, other roots are complex or negative. A positive and real root is T i = 1 3 3 4π 2 3 V 1 3 8π 3 4π 2 3 P i + 1 V 2 3 − 4π 3 2 3 Q 2 V 4 3 = − Q 2 4πr 3 + + 1 4πr+ + 2P i r + .(41)r + = 1 2 √ 2 1 + 24P i πQ 2 P i π − 1 P i π .(43) If we substitute this root into Eq. (41), the inversion temperature is given by T i = √ P i √ 2π 1 + 16πP i Q 2 − 1 + 24πP i Q 2 −1 + 1 + 24πP i Q 2 3 2 .(44) When P i is zero, we have T min i T min i = 1 6 √ 6πQ (45) and ratio between minimum inversion and critical temperatures is T min i T c = 1 2 .(46) In Fig (3), inversion curves are presented for various values of charge Q. There is only a lower inversion curve. In contrast to van der Waals fluids, the expression inside the square root in Eq. (44) is always positive, so this curve does not terminate any point. Now, we can plot isenthalpic, i.e. constant mass, curves in T −P plane. From Eq. (4), one can obtain event horizon and substituting event horizon into Eq. (10) gives isenthalpic curves in T − P plane. In Fig. (4), inversion curves and isenthalpic curves are presented. Isenthalpic curves have positive slope above the inversion curves so cooling occurs above the inversion curves. The sign of slope changes under the inversion curves and heating occurs in this region. It is also interesting to talk about naked singularities for charged AdS black holes. In Fig. (5), we plot event horizon versus mass and pressure. We introduce four graphics, which correspond to Q = 1, 2, 10, 20. The regions can be seen that denote the naked singularities in Fig. (5). One cannot consider Joule-Thomson expansion due to the lack of event horizon for naked singularity. For example, we cannot define event horizon for Q = 20 and M ≤ 20. For these values, event horizon is imaginary and it corresponds to naked singularity so isenthalpic curves in T − P plane are imaginary. IV. CONCLUSION In this paper, we studied the well known Joule-Thomson expansion for charged AdS black hole. The black hole mass in AdS space is identified with enthalpy due to variable cosmological constant notion, so one can consider that mass does not change during the expansion. First, we reviewed Joule-Thomson expansion for van der Waals fluids and then we investigated Joule-Thomson expansion for charged AdS black holes. We only found one inversion curve that corresponds to the lower curve. It means that black holes always cool above the inversion curve during the Joule-Thomson expansion. Cooling and heating regions were shown for various values of charge Q and mass M . We also denoted naked singularity which is not sensible for Joule-Thomson expansion due to the lack of event horizon. Both systems are not well behaved for low temperatures. Unfortunately, isenthalpic curves have positive slopes under the lower inversion curves for both systems. It is also known that van der Waals equation does not too well agree with experiments. Thus Joule-Thomson expansion have been investigated for various equations of state. In charged AdS case, it needs further investigation. ACKNOWLEDGMENTS We thank Can Onur Keser with improving the figures in this work. This work was sported by Scientific Research Projects Coordination Unit of Istanbul University. Project number is FYL-2016-20615. FIG. 1 . 1Lower (dashed blue line) and upper (solid orange line) inversion curves. We fix a = b = kB = 1. FIG. 2 . 2Dashed blue line and orange line are inversion curves. Red lines are isenthalpic curves. The enthalpies of isenthalpic curves increase from bottom to top and correspond to H 10. We fix a = b = kB = 1. FIG. 4 . 4Inversion and isenthalpic curves for charged AdS black hole. From bottom to top, the isenthalpic curves correspond to increasing values of M . Red and black lines are isenthalpic and inversion curves, respectively. (a) Q = 1 and M = 1.5, 2, 2.5, 3 (b) Q = 2 and M = 2.5, 3, 3.5, 4 (c) Q = 10 and M = 10.5, 11, 11.5, 12 (d) Q = 20 and M = 20.5, 21, 21.5, 22 FIG. 5 . 5Event horizon of charged AdS black hole. (a) Q = 1 (b) Q = 2 (c) Q = 10 (d) Q = 20 . J. D. Bekenstein, Lett. Nuovo Cimento. 4737J. D. Bekenstein, Lett. 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{'abstract': 'In this paper, we study Joule-Thomson effects for charged AdS black holes. We obtain inversion temperatures and curves. We investigate similarities and differences between van der Waals fluids and charged AdS black holes for the expansion. We obtain isenthalpic curves for both systems in T − P plane and determine the cooling-heating regions.', 'arxivid': '1611.06327', 'author': ['Ozgürökcü \nDepartment of Physics\nFaculty of Science\nİstanbul University\n34134İstanbulTurkey\n', 'Ekrem Aydıner \nDepartment of Physics\nFaculty of Science\nİstanbul University\n34134İstanbulTurkey\n'], 'authoraffiliation': ['Department of Physics\nFaculty of Science\nİstanbul University\n34134İstanbulTurkey', 'Department of Physics\nFaculty of Science\nİstanbul University\n34134İstanbulTurkey'], 'corpusid': 119387109, 'doi': '10.1140/epjc/s10052-017-4598-y', 'github_urls': [], 'n_tokens_mistral': 8063, 'n_tokens_neox': 6683, 'n_words': 3809, 'pdfsha': '84abfc02cbabc054c13f86b13bf7fa90e305030d', 'pdfurls': ['https://arxiv.org/pdf/1611.06327v2.pdf'], 'title': ['Joule-Thomson Expansion of Charged AdS Black Holes', 'Joule-Thomson Expansion of Charged AdS Black Holes'], 'venue': []}

arxiv

Surface acoustic wave coupled to magnetic resonance on a multiferroic CuB 2 O 4 R Sasaki Department of Basic Science University of Tokyo 153-8902TokyoJapan Y Nii Institute for Materials Research Tohoku University 980-8577SendaiJapan Y Onose Institute for Materials Research Tohoku University 980-8577SendaiJapan Surface acoustic wave coupled to magnetic resonance on a multiferroic CuB 2 O 4 We observed surface acoustic wave (SAW) propagation on a multiferroic material CuB 2 O 4 with use of two interdigital transducers (IDTs). The period of IDT fingers is as short as 1.6 µm so that the frequency of SAW is 3 GHz, which is comparable with that of magnetic resonance. In antiferromagnetic phase, the SAW excitation intensity varied with the magnitude and direction of the magnetic field, owing to the dynamical coupling between SAWs and antiferromagnetic resonance of CuB 2 O 4 . The microscopic mechanism is discussed based on the symmetrically allowed magentoelastic coupling. arXiv:1811.01527v1 [cond-mat.mes-hall] Multiferroics are materials where magnetism and ferroelectricity coexist. Novel electromagnetic phenomena are frequently observed, thanks to the interplay between the magnetism and ferroelectricity. For example, they show giant magnetoelectric effects, which is polarization change induced by a magnetic field, and reciprocally magnetization change induced by an electric field 1,2 . The interplay is valid even for the dynamical state. The magnetoelectric correlation in optical frequency range gives rise to nonreciprocal directional dichroism [3][4][5][6] . The electric-field active magnon mode has also been observed in multiferroics 7 . Here, we study the dynamical coupling between an antiferromagnetic magnon and surface acoustic wave (SAW) in a multiferroic CuB 2 O 4 . The SAW is an elastic wave localized on a surface of media 8 . The amplitude decays exponentially with the depth from the surface. The SAW can be excited and detected on a piezoelectric substrate with use of interdigital transducers (IDTs). The SAW can carry electromagnetic signals between two separated IDTs when the wavelength coincides with the IDT finger period. The SAW devices composed of two IDTs on a piezoelectric substrate are industrially used as bandpass filters or delay lines. The combination with magnetism seems useful for making these devices more functional. In fact, introducing ferromagnetic thin film between two IDTs gives rise to emergent functionality such as acoustically driven ferromagnetic resonance [9][10][11] , acoustic spin pumping 12,13 , and nonreciprocal SAW propagation 14 . A more direct way of introducing magnetism is replacing piezoelectric substrate with multiferroic one. Quite recently, we succeeded in fabricating SAW device based on a multiferroic material BiFeO 3 in collaboration with other researchers 15 . We observed that the SAW intensity and velocity were modulated due to the static magnetostructural change in the magnetic fields. One might think the dynamical coupling between SAW and magnon should show more rich phenomena but magnon mode in BiFeO 3 is too high to be coupled to SAW 16,17 . In order to elucidate this issue, we investigate SAW coupled to magnon mode on another multiferroic material CuB 2 O 4 . CuB 2 O 4 has the non-centrosymmetric but non-polar crystal structure with space group of I42d. According to symmetry analysis, the piezoelectric tensor is expressed as  We fabricated a SAW device on CuB 2 O 4 substrate ( Fig. 1(a)). The CuB 2 O 4 single crystal was synthesized by flux method 23 . Two Al IDTs with the thickness of 50 nm were fabricated on the substrate using electron beam lithography and electron beam evaporation.     0 0 0 d 1 0 0 0 0 0 0 d 1 0 0 0 0 0 0 d 2      ,(1) One finger width of the IDT and space between the fingers were designed to be 400 nm ( Fig. 1(b)) so that the wavelength of SAWs is 1.6 µm, which is determined by the periodicity of and z components, and Rayleigh-type SAW has xx , zz , zx components 8 . In this coordinate system, the piezoelectric tonsor can be discribed as 1 2      0 0 0 0 2d 1 0 0 0 0 −2d 1 0 0 d 2 −d 2 0 0 0 0      .(2) Because xx , zx components can be induced by the IDT electric field, the SAW can be excited in this configuration. All the measurements were performed at T = 10 K. As shown in Fig. 1 (a), the magnetic field is applied parallel to the device surface, and the azimuth angle of the magnetic field from the SAW propagation direction is defined as φ. this is the SAW signal, we perform a time domain analysis. Figure 1 spectra |S SAW 21 |(f ) by performing Fourier transformation only on the SAW transmission time region (the colored region in Fig. 1 (e)). In this spectrum, the ripples are removed, and the background level is decreased. Hereafter, we used the |S SAW 21 |(f ) spectra for the analysis of the SAW transmission. ; for the precise definition, see Appendix A)). The SAW transmission gradually decreases with increasing magnetic field from 0 mT. It shows rapid increase when the magnetic resonance frequency coincides with the SAW frequency. Then it decreases toward the high field. This characteristic magnetic field dependence seems caused by the interaction between the SAW and magnetic resonance. In Fig. 2(b), we also plotted SAW excitation intensity at IDT 1 normalized by the 200 mT value (−S 11 /S 0 11 (H); for the precise definition, see Appendix A)), which is estimated by the decrease of reflection due to SAW generation. Because both the transmission and excitation show similar dependence on the magnetic field, it seems that the magnetic field dependence is caused by the excitation process. In order to discuss the microscopic origin of magnetic field dependence, we show the SAW transmission as a function of the magnetic field along various directions φ in Fig. 3(a). The magnetic field dependence became broad and the magnitude decreased with increasing or decreasing φ from 0 • . Nevertheless, the characteristic magnetic field dependence around 40 mT was still discerned even at φ = ± 90 • . Irrespective of the field angle, the magnetic field dependence is almost unchanged when the field direction or propagation direction is reversed (S SAW 12 /S 0 12 (H) is the normalized transmission for the opposite SAW propagation direction). Figure 3(c) shows the magnetic field angle dependence of SAW transmission at 40 mT. The SAW transmission varied with φ as C 1 − C 2 cos 2 φ (C 1 and C 2 are constants). Let us theoretically discuss the microscopic origin of the magnetic field dependence of the SAW signal. When the frequency of magnetic resonance is close to that of SAW, the SAW excited state |SAW and magnetic excited state |Mag are expected to be hybridized with each other through the magnetoelastic coupling H me . The SAW state hybridized with the magnetic excited state is expressed as SAW ≈ |SAW + Mag|H me |SAW ω Mag − ω SAW |Mag ,(3) where ω Mag , ω SAW , and are the frequencies of magnetic excitation and SAW, and Planck constant divided by 2π, respectively. When ω Mag is increased from 0, the magnitude of the second term in the right hand side increases. It shows steep sign change at ω Mag = ω SAW . Then the magnitude decreases toward 0 with ω Mag . The magnetic field dependence of SAW excitation may be related to this. The angle dependence is more closely related to the microscopic mechanism. To discuss this issue we should consider the explicit form of magnetoelastic coupling. The magnetoelastic coupling energy in antiferromagnetic ordered state is F me = p,q=1,2 i,j,k,l=1,2,3 b pqijkl m pi m qj kl ,(4)1 γ ∂ ∂t (δm 1X − δm 2X ) + b a 2 (δm 1Y + δm 2Y ) = m 0 2 B 1 + b a 2 B 2 31 cos φ,(5)1 γ ∂ ∂t − iaH 0 ((δm 1Y + δm 2Y ) + ia(δm 1Z + δm 2Z )) = m 0 2 (B 2 31 cos φ + iaB 3 11 sin 2φ) ,(6)1 γ ∂ ∂t + iaH 0 ((δm 1Y + δm 2Y ) − ia(δm 1Z + δm 2Z )) = m 0 2 (B 2 31 cos φ − iaB 3 11 sin 2φ) .(7) Here, γ and H 0 are gyromagnetic ratio and magnitude of external magnetic field, respectively. δm p and m 0 are defined by δm p = m p − m 0 p and m 0 = |m 0 p |, where m 0 p is static part of magnetic moments. XY Z coordinate system is defined as in Fig. 3(b), so that X-axis is parallel to the magnetic field. Dimensionless constants a and b are defined by a = − K m 0 cos ψ + H 0 /H 0 and b = − K m 0 + 2m 0 Λ sin ψ/H 0 , where K, Λ and ψ are the uniaxial magnetic anisotropy constant, molecular field constant, and angle between the sublattice magnetic moments and magnetic field, respectively. B 1 ,B 2 and B 3 are constants defined by B 1 = 8 sin ψ ((b 111313 + b 121313 ) cos ψ − b 121323 sin ψ) ,(8)B 2 = −8 cos ψ ((b 111313 + b 121313 ) cos ψ − b 121323 sin ψ) ,(9)B 3 = −2 ((b 111111 − b 111122 ) cos 2ψ + b 121111 − b 121122 ) .(10) In the absence of SAW excitation, the right hand sides of Eq.(5)- (7) vanish. In this case, these equations stand for the pure antiferromagnetic excitation. Because ω Mag > 0, only right-hand ellipsoidal polarized components (δm 1Y + δm 2Y ) + ia(δm 1Z + δm 2Z ) exhibits resonance behavior at ω Mag = γaH 0 . Eq. (7) does not show any resonance behavior, and Eq. (5) stand for constraint condition describing the relationship between magnetic moment components parallel (δm 1X , δm 2X ) and perpendicular (δm 1Y , δm 2Y ) to the magnetic field during the precession motion of magnetic moments. The right hand sides of these equations are the magnetic torque due to the SAW excitation. In particular, the right hand side of Eq. (6) is the direct coupling between the SAW and magnetic resonance mode. The magnitude of hybridization is proportional to |B 2 31 cos φ + iaB 3 11 sin 2φ| 2 .(11) The experimentally obtained magnetic field dependence of C 1 − C 2 cos 2 φ indicates that B 3 is negligible, which is composed of longitudinal type magnetoelastic coupling constants. It seems that the strain 31 most effectively excite the acoustic antiferromagnetic resonance. Note that nonreciprocity should be caused by the mixture of phase different effective magnetic fields induced by the shear-type and longitudinal acoustic strains, similarly to the case of Ni/LiNbO 3 14 . In this case, however, the longitudinal type magnetoelastic coupling is absent, and the magnetic field is linearly polarized. That is why the nonreciprocity is negligible in this system. The constant term C 1 cannot be deduced by the present theoretical analysis, in which the magnitude of magnetic moments is fixed. Perhaps, the constant term is caused by the magnetic field change of the magnitude. In conclusion, we could successfully excite and detect the surface acoustic wave on mul- propagation direction, i.e. φ = 0 • . There are three components of decrease of reflection (absorption); a sharp dip, broad dip, and background signal from the device. While the sharp peak and background do not show large magentic field variation, the frequency of broad peak increases with the magnetic field. Therefore, we ascribed the origins of sharp and broad peaks to SAW and magnetic resonance, respectively. In order to accurately estimate the SAW contribution, we fitted the contribution of magnetic resonance and background to the Lorentzian and quadratic function, respectively, and subtract them from |S 11 |(f ). Figure A.1(b) shows the minus subtracted reflection −S SAW 11 (f ) at µ 0 H = 40 and 200 mT. Because the frequency of magnetic resonance is far above 3 GHz at 200 mT, the SAW excitation is not affected by the magnetic resonance. When the magnetic field is decreased to 40 mT, the SAW intensity decreases (Fig. A.1.(b)). Similarly, the transmission at 40 mT is a little smaller than that at 200 mT as shown in Fig. A.1 (c). The relative change of SAW reflection and transmission is defined as S SAW 11 /S 0 11 (H) = max |S SAW 11 |(f ) (H)/max |S SAW 11 |(f ) (H = 200 mT/µ 0 ),(A1)S SAW 21 /S 0 21 (H) = max |S SAW 21 |(f ) (H)/max |S SAW 21 |(f ) (H = 200 mT/µ 0 ) (A2) where max{} stands for the maximum value. From the definition of SAW reflection, the value −S SAW 11 /S 0 11 (H) can be interpreted as SAW excitation. These quantities are plotted in Fig. 2(b) in the main text. Appendix B: magnetoelastic conpling energy As discussed above, the magnetoelastic coupling energy in staggered antiferromagnetic state can be expressed as Here we estimate the effective magnetic field due to the small strain. Therefore, we neglect the dynamical component of magnetization (δm p ). Since the magnetic order in CuB 2 O 4 at T = 10 K is easy-plane Néel type magnetic order, we assume m 0 p3 = 0. From the fact that the device was designed so that its sagittal plane is parallel to the mirror plane of the crystal, we can assume that Rayleigh type SAW mode is excited in our measurement 25 . We neglected the strain components other than 11 , 33 , and 31 , of which Rayleigh type SAW is composed. By diagonalizing Eq. (C3), we get the six equations for antiferromagnetic resonance as follows; 1 γ ∂ ∂t (δm 1X − δm 2X ) + b a 2 (δm 1Y + δm 2Y ) = m 0 2 B 1 + b a 2 B 2 31 cos φ,(C5)1 γ ∂ ∂t − iaH 0 ((δm 1Y + δm 2Y ) + ia(δm 1Z + δm 2Z )) = m 0 2 (B 2 31 cos φ + iaB 3 11 sin 2φ) ,(C6)1 γ ∂ ∂t + iaH 0 ((δm 1Y + δm 2Y ) − ia(δm 1Z + δm 2Z )) = m 0 2 (B 2 31 cos φ − iaB 3 11 sin 2φ) ,(C7)1 γ ∂ ∂t ((δm 1X + δm 2X ) + c(δm 1Y − δm 2Y )) = m 0 2 (B 4 + cB 5 ) 13 cos φ,(C8)1 γ ∂ ∂t − ie(2mΛ cos ψ − H 0 ) (d(δm 1X + δm 2X ) + (δm 1Y − δm 2Y ) − ie(δm 1Z − δm 2Z )) = m 0 2 (dB 4 + B 5 ) 13 cos φ − ie(B 6 33 + 11 (B 7 cos 2φ + B 8 )), between the [001] axis and the magnetic field. As the angle θ decrease, magnetic resonance spectra shifted to low frequency and its intensity decreases. This is quit consistent with Figure 1 ( 1d) shows the absolute value of complex forward transmission from left IDT (IDT 1) to right IDT (IDT 2) (|S 21 |(f )) measured by a vector network analyzer (Agilent E5071C FIG. 1 . 1(a) Top view of CuB 2 O 4 SAW device used in this research. Magnetic field was applied along the surface of the device. (b) Enlarged view of Al IDT. (c) Schematic diagrams of the device. the surface of device is (001) plane of the CuB 2 O 4 crystal, and SAW propagation direction is parallel to the [110] axis. (d) Transmission between two IDTs. Solid line is the raw data measured and dashed line shows the transmission due to SAW deduced by the Fourier transformation analysis (see text). (e) Transmission as a function of time obtained by the inverse Fourier transformation. Data in the colored region is used for the Fourier transformation to the SAW transmission S SAW 21 . FIG. 2 . 2(e) shows absolute value of the time domain transmission complex amplitudeS 21 (t) obtained by an inverse Fourier transformation 24 . The large impulse observed around 0 ns is due to the direct electromagnetic transmission between two IDTs. In addition, we have observed the delayed impulse around 120 ns. From the distance between IDTs, the velocity of the delayed transmission signal is estimated as 4.8 × 10 3 m/s, which almost coincides with the phase velocity estimated from the frequency and IDT finger period. Because the velocity is comparable with SAW velocity of other piezoelectrics (e.g. 3488 m/s for YZ-cut LiNbO 3 8 ), the delayed signal can be attributed to the SAW signal. We obtained the absolute value of SAW transmission (a) Magnetic field dependence of microwave absorption due to magnetic resonance of CuB 2 O 4 . The external magnetic field is parallel to [110] axis. Thick line shows the SAW frequency. (b) Magnetic field dependence of normalized SAW transmission S SAW 21 Figure 2 ( 2a) shows the magnetic field dependence of microwave absorption spectra at T = 10 K, which is measured in the measurement system similar to that of ref.22 22 . The static and alternating magnetic fields were parallel to [110] axis and (110) plane, respectively. We have found a magnetic resonance mode in this frequency region. The frequency increases almost linearly with the magnetic field. As discussed in appendix, the origin is ascribed to the acoustic mode of antiferromagnetic resonance. The magnetic resonance frequency coincided with the frequency of the SAW around 40 mT. directions. Solid lines show the transmission from port 1 to port 2 (S SAW 21 /S 0 21 ) and dashed lines are transmission for opposite direction (S SAW 12 /S 0 12 ) . (b) Illustration of xyz-and XYZcoordinate systems. (c) Magnetic field angle dependence of normalized SAW transmission S SAW 21 at µ 0 H = 40 mT. Solid line shows C 1 − C 2 cos 2 φ. Figure 2 ( 2b) shows magnetic field dependence of SAW transmission normalized by that at 200 mT, where the magnetic resonance frequency is far above the SAW frequency (S SAW 21 /S 0 21 (H) where m pi represents the i-component of magnetization on p sublattice of the antiferromagnet, and kl is the strain tensor on the surface of the device. The subscripts 1,2,3 for i, j, k, lindicate x, y, z components, respectively. The coefficients b pqijkl represent magnetoelastic coupling constants. The nonvanishing component of b pqijkl is determined by the symmetry analysis (see appendix B). The effective magnetic field h me p due to the strains acting on the magnetic moments at p-sublattice m p is given by h me p = −∇ mp F me . To understand the acoustic antiferromagnetic resonance under SAW excitation, we consider Landau-Lifshitz (LL) equation without damping term, in which the magnetic moments are driven by h me p as well as H and anisotropy field. By partially diagonalizing 6 components of the equation, we can deduce 3 effective equations regarding the acoustic antiferromagnetic resonance as follows (see appendix) ; FigureFIG tiferroic material CuB 2 O 4 by fabricating the interdigital transducers on a CuB 2 O 4 single crystal substrate. In the antiferromagnetic phase, SAW excitation and transmission exhibited characteristic change due to the coupling to the antiferromagnetic resonance. This magnetic field dependence was explained by the analysis using the effective magnetic field caused by magnetoelastic coupling. This research may pave a new path to increase the controllability of the SAW device. ACKNOWLEDGMENTS This work was in part supported by the Grant-in-Aid for Scientific Research (Grants No. 17H05176, No. 16H04008) and for Young Scientists (18K13494) from the Japan Society for the Promotion of Science and the Noguchi Institute. R.S. is supported by the Grant-in-Aid for JSPS Research Fellow (No. 18J12130). Appendix A: Definitions of relative SAW excitation and transmission −S A.1(a) shows reflection of the device |S 11 |(f ) under the external magnetic fields µ 0 H = 40, 45, 50, and 200 mT. The external magnetic field is applied parallel to the SAW . A.1. (a) Microwave reflection from the SAW device |S 11 |(f ) at several magnetic fields. The magnetic fields are applied parallel to the SAW propagation direction. The solid line shows the observed data, and the dotted and dashed lines are the result of fitting of magnetic resonance and background, respectively (see text). (b) SAW excitation −S SAW 11 at 40 mT and 200 mT and the difference between 40 mT and 200 mT values (∆S 11 ). (c) SAW transmission S SAW 21 at 40 mT and 200 mT and their difference (∆S 21 ). ∆S 11 and ∆S 21 in (b) and (c) are multiplied by 5. b pqijkl m pi m qj kl . (B1) Because the magnetoelastic coupling energy is unchanged by the symmetry operation belong to the space group I42d, some of constants b pqijkl should vanish and the number of independent magnetoelastic coupling constants decrease. The reduced magnetoelastic coupling energy can be represented by the following equation; F me = 11 (b 111111 m 2 11 + 2b 121111 m 21 m 11 + b 121122 m 12 m 22 + 2b 121211 (m 11 m 22 − m 12 m 21 ) + 2b 123311 m 13 m 23 ) + 12 (4b 111212 (m 11 m 12 + m 21 m 22 ) + 4b 121212 (m 12 m 21 + m 11 m 22 )) + 13 (4(m 21 (b 111313 m 23 + b 121313 m 13 ) + m 11 (b 111313 m 13 + b 121313 m 23 ) + b 121323 (m 13 m 22 − m 12 m 23 ))) + 22 (b 111111 m 2 12 + 2b 121111 m 22 m 12 + b 111111 m 2 22 + b 111122 m 2 11 + m 2 21 + b 113311 m 2 13 + m 2 23 + 2b 121122 m 11 m 21 + 2b 121211 (m 11 m 22 − m 12 m 21 ) + 2b 123311 m 13 m 23 ) + 23 (4(m 22 (b 111313 m 23 + b 121313 m 13 ) + m 12 (b 111313 m 13 + b 121313 m 23 ) + b 121323 (m 11 m 23 − m 13 m 21 ))) + 33 (b 111133 m 2 11 + 2m 11 (b 121133 m 21 + b 121233 m 22 ) + b 111133 m 2 12 + b 111133 m 2 21 + m 2 22 + b 113333 m 2 13 + m 2 23 + 2m 12 (b 121133 m 22 − b 121233 m 21 ) + 2b 123333 m 13 m 23 ). (B2) where p, q = 1 or 2 and p = q. Based on the reduced magnetoelastic coupling energy (B2) , the effective magnetic field components form h me 11 = − 2m 0 11 (b 111111 11 + b 111122 22 + b 111133 33 ) − 4b 111212 m 0 12 12 − 2m 0 21 (b 121111 11 + b 121122 22 + b 121133 33 ) − 2m 0 22 (b 121211 ( 11 + 22 ) + 2b 121212 12 + b 121233 33 ) , h me 12 = − 2b 111111 m 0 12 22 − 2b 111122 m 0 12 11 − 2b 111133 m 0 12 33 − 4b 111212 m 0 11 12 − 2b 121111 m 0 22 22 − 2b 121122 m 0 22 11 − 2b 121133 m 0 22 33 + 2m 0 21 (b 121211 ( 11 + 22 ) − 2b 121212 12 + b 121233 33 ) , h me 13 = − 4b 111313 m 0 11 31 − 4b 111313 m 0 12 23 − 4b 121313 m 0 22 23 − 4b 121323 m 2b 121323 23 − 2b 121313 31 ) , h me 21 = − 2m 0 21 (b 111111 11 + b 111122 22 + b 111133 33 ) − 2m 0 11 (b 121111 11 + b 121122 22 + b 121133 33 ) + 2 m 0 12 (b 121211 ( 11 + 22 ) − 2b 121212 12 + b 121233 33 ) − 2b 111212 m 0 22 12 , h me 22 = − 2b 111111 m 0 22 22 − 2b 111122 m 0 22 11 − 2b 111133 m 0 22 33 − 4b 111212 m 0 21 12 − 2b 121111 m 0 12 22 − 2b 121122 m 0 12 11 − 2b 121133 m 0 12 33 − 2m 0 11 (b 121211 ( 11 + 22 ) + 2b 121212 12 + b 121233 33 ) , h me 23 = − 4b 111313 m 0 21 31 − 4b 111313 m 0 22 23 − 4b 121313 m 0 12 23 + 4b 121323 m 0 12 31 − 4m 0 11 (b 121313 31 + b 121323 23 ) . ie(2mΛ cos ψ − H 0 ) (d(δm 1X + δm 2X ) + (δm 1Y − δm 2Y ) + ie(δm 1Z − δm 2Z )) = m 0 2 (dB 4 + B 5 ) 13 cos φ + ie(B 6 33 + 11 (B 7 cos 2φ + B 8 )),(C10)where δm 1X = δm 11 cos φ+δm 12 sin φ, δm 2X = δm 21 cos φ+δm 22 sin φ, δm 1Y = −δm 11 sin φ+ δm 12 cos φ, δm 2Y = −δm 21 sin φ + δm 22 cos φ, δm 1Z = δm 13 , δm 2Z = δm 23 . Dimensionless constants c, d, and e are defined by c =K sin ψ (2Λ(m 0 ) 2 +K) cos ψ−m 0 H 0 , d = 2m 0 Λ sin ψ 2m 0 Λ cos ψ−H 0 , e = √ (H 0 ) 2 −(4Λ(m 0 )+K/m 0 ) cos ψH 0 +2Λ(Λ(m 0 ) 2 +(Λ(m 0 ) 2 +K) cos 2ψ) 2m 0 Λ cos ψ−H 0 . B 4 ,B 5 and B 6 are constants de-. A.2. Microwave absorption spectra in magnetic fields with various directions within the (110) plane and the fixed magnitude of µ 0 H = 40 mT. θ is defined as the angle between the [001] axis and the magnetic field.fined byB 4 = 8 sin ψ ((b 121313 − b 111313 ) sin ψ + b 121323 cos ψ) ,(C11)B 5 = −8 cos ψ ((b 121313 − b 111313 ) sin ψ + b 121323 cos ψ) ,(C12)B 6 = −4 (b 121133 sin 2ψ + b 121233 cos 2ψ) , (C13) B 7 = −2 sin 2ψ (b 111111 − b 111122 ) , (C14) B 8 = −2 ((b 121111 + b 121122 ) sin 2ψ + 2b 121211 cos 2ψ) . (C15) Eqs.(C5)-(C7) and Eqs.(C8)-(C10) are relevant to the acoustic mode and optical mode of antiferromagnetic resonance, respectively. Appendix D: Antiferromagnetic resonance measurements We measured the microwave absorption of CuB 2 O 4 in order to study the origin of magnetic resonance. While the magnetic field is along the (001) plane in the previous work 22 , it is within the (110) plane in the present study. The other experimental configurations are similar to those in the previous study. Figure A.2 shows microwave absorption in magnetic fields with various directions and the fixed magnitude of 40 mT. θ is defined as the angle where d 1 and d 2 are non-vanishing constants. As discussed later, the SAW can be generated in some choice of the device plane and propagation direction. At low temperature, CuB 2 O 4 exhibits two successive magnetic phase transition at T N = 21 K and T * = 9 K. At T N , it shows easy-plane Néel type antiferromagnetic order18 . Ferroelectric polarization depending on the direction of antiferromagnetic moments is emergent in this magnetic phase. Several unusual optical phenomena have been reported such as magnetic field induced second harmonic generation 19 and magneto-optical dichroism 5 . The second transition at 9 K corresponds to the incommensurate helical ordering. In these magnetic states, a magnetic resonance mode was observed in low frequency range of several GHz 20-22 , owing to the small magnetic anisotropy of Cu 2+ S = 1/2 moment. SAW excitation in this frequency range can be achieved by means of conventional electron beam lithography technique. Therefore, this material is suitable for the investigation of the SAW coupled to the magnetic resonance. IDTs. The distance between the center of the two IDTs is 580 µm. We designed the CuB 2 O 4SAW device so that the surface of the substrate is perpendicular to crystal [001] axis, and SAW propagation direction is parallel to [110] axis (Fig. 1(c)). When x, y, z coordinate is defined as x||[110], y||[110], and z||[001], alternating electric field induced on the IDT has x The effective magnetic field on the magnetization m p (p = 1,2) is defined byThe subscript p represents the index of two sublattice of ordered spins.Assuming that h me p acts on the magnetic moments similarly to the real magnetic field , LL equation without damping term is represented bywhereẑ is the unit vector along z-axis. By putting m p = m 0 p + δm p and linearizing the 6 equations with respect to δm and h me p , we get 1 γ ∂δm p ∂t + m 0 p × (−Λδm q + K (m 0 ) 2 (δm p ·ẑ)ẑ) + δm p × (−Λm 0 q + H) = −m 0 p × h me p , (C3) In the previous work, we tentatively assigned this magnetic resonance to the paramagnetic resonance of Cu(B) site because this absorption peak is smoothly connected to the magnetic resonance peak above T N . Nevertheless. 26. this new experimental result strongly supports the antiferromagnetic resonance origin26 . In the previous work, we tentatively assigned this magnetic resonance to the paramagnetic resonance of Cu(B) site because this absorption peak is smoothly con- nected to the magnetic resonance peak above T N . Nevertheless, this new experimental result strongly supports the antiferromagnetic resonance origin. . S W Cheong, M Mostovoy, 10.1038/nmat1804Nature Materials. 613S. W. Cheong and M. Mostovoy, Nature Materials 6, 13 (2007). Reports on progress in physics. Y Tokura, S Seki, N Nagaosa, 10.1088/0034-4885/77/7/076501Physical Society (Great Britain). 7776501Y. Tokura, S. Seki, and N. Nagaosa, Reports on progress in physics. Physical Society (Great Britain) 77, 076501 (2014). . M Kubota, T Arima, Y Kaneko, J P He, X Z Yu, Y Tokura, 10.1103/PhysRevLett.92.137401Physical Review Letters. 92137401M. Kubota, T. Arima, Y. Kaneko, J. P. He, X. Z. Yu, and Y. Tokura, Physical Review Letters 92, 137401 (2004). . 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M Weiler, H Huebl, F S Goerg, F D Czeschka, R Gross, S T B Goennenwein, 10.1103/PhysRevLett.108.1766011110.1187Physical Review Letters. 108M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross, and S. T. B. Goennenwein, Physical Review Letters 108, 1 (2012), 1110.1187. . M Xu, J Puebla, F Auvray, B Rana, K Kondou, Y Otani, 10.1103/PhysRevB.97.180301Physical Review B. 97180301M. Xu, J. Puebla, F. Auvray, B. Rana, K. Kondou, and Y. Otani, Physical Review B 97, 180301 (2018). . R Sasaki, Y Nii, Y Iguchi, Y Onose, 10.1103/PhysRevB.95.0204071605.04058Physical Review B. 9520407R. Sasaki, Y. Nii, Y. Iguchi, and Y. Onose, Physical Review B 95, 020407 (2017), 1605.04058. . Y Ishii, R Sasaki, Y Nii, T Ito, Y Onose, 10.1103/PhysRevApplied.9.034034Physical Review Applied. 934034Y. Ishii, R. Sasaki, Y. Nii, T. Ito, and Y. Onose, Physical Review Applied 9, 034034 (2018). . 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Melkov, Magnetization Oscillations and Waves (Taylor & Francis, 1996).

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{'abstract': 'We observed surface acoustic wave (SAW) propagation on a multiferroic material CuB 2 O 4 with use of two interdigital transducers (IDTs). The period of IDT fingers is as short as 1.6 µm so that the frequency of SAW is 3 GHz, which is comparable with that of magnetic resonance. In antiferromagnetic phase, the SAW excitation intensity varied with the magnitude and direction of the magnetic field, owing to the dynamical coupling between SAWs and antiferromagnetic resonance of CuB 2 O 4 . The microscopic mechanism is discussed based on the symmetrically allowed magentoelastic coupling. arXiv:1811.01527v1 [cond-mat.mes-hall]', 'arxivid': '1811.01527', 'author': ['R Sasaki \nDepartment of Basic Science\nUniversity of Tokyo\n153-8902TokyoJapan\n', 'Y Nii \nInstitute for Materials Research\nTohoku University\n980-8577SendaiJapan\n', 'Y Onose \nInstitute for Materials Research\nTohoku University\n980-8577SendaiJapan\n'], 'authoraffiliation': ['Department of Basic Science\nUniversity of Tokyo\n153-8902TokyoJapan', 'Institute for Materials Research\nTohoku University\n980-8577SendaiJapan', 'Institute for Materials Research\nTohoku University\n980-8577SendaiJapan'], 'corpusid': 119529390, 'doi': '10.1103/physrevb.99.014418', 'github_urls': [], 'n_tokens_mistral': 12501, 'n_tokens_neox': 9629, 'n_words': 5697, 'pdfsha': '0ddf6965e1911cc3938a938a2b844ccf9ceb90f9', 'pdfurls': ['https://arxiv.org/pdf/1811.01527v1.pdf'], 'title': ['Surface acoustic wave coupled to magnetic resonance on a multiferroic CuB 2 O 4', 'Surface acoustic wave coupled to magnetic resonance on a multiferroic CuB 2 O 4'], 'venue': []}

arxiv

Femtoscopy via Lévy sources with PHENIX at RHIC October 30, 2021 Máté Csanád PHENIX Collaboration Eötvös Loránd University Pázmány P. s. 1/AH-1117BudapestHungary Femtoscopy via Lévy sources with PHENIX at RHIC October 30, 2021 Charged pion two-particle correlation functions were measured in 0−30% centrality √ s N N = 200 GeV Au+Au collisions with the PHENIX experiment at RHIC. The measured correlation functions can be statistically well described based on the assumption of Lévy-shaped source distributions. In this proceedings paper we present the Lévy parameters of the measured correlation functions: correlation strength parameter λ, Lévy index α and Lévy scale parameter R as a function of pair transverse mass mT , in 31 bins from 228 to 871 MeV, separately for positive and negative pion pairs. We discuss the physical interpretation of the mT dependence of the parameters. Introduction In ultra-relativistic collisions of heavy ions, strongly coupled Quark Gluon Plasma (sQGP) is formed [1][2][3][4] for a very short amount of time, and after a quark-hadron freeze-out, hadrons are created. The measurement of Bose-Einstein correlations (i.e. femtoscopy) can be used to gain knowledge about the space-time geometry of the particle emitting source, as originally observed by [5,6], and in radio and optical astronomy by R. Hanbury Brown and R. Q. Twiss (HBT) [7]. In an interaction-free case, the two-particle Bose-Einstein correlation functions are related to the Fourier transform of the source function (S(x, k), describing the probability density of particle creation at the space-time point x and with four-momentum x): C (0) 2 (Q, K) 1 + S(Q, K) S(0, K) 2 ,(1) where S(q, k) = S(x, k)e iqx d 4 x is the Fourier-transformed of S, and Q = p 1 − p 2 is the momentum difference, K = (p 1 + p 2 )/2 is the average momentum, and we assumed, that q K holds for the investigated kinematic range. Usually, correlation functions are measured versus Q, for a well-defined K-range, and then properties of the correlation functions are analyzed as a function of the average K of each range. In an expanding Gaussian source, then 1+exp −(QR) 2 correlations are thus measured, where the observed Gaussian radius R is a homogeneity length, depending on the average momentum K or the related transverse mass m T . The approximate dependence of R −2 ∝ a + bm T is observed, rather universally (for various collision systems, collision energies and particle types) [8,9], which can be interpreted as a consequence of hydrodynamical expansion [10,11]. See Ref. [12] (and references therein) for details. It is important to note, that a significant fraction of pions are secondary, coming from decays. Hence the source will have two components: a core of primordial pions, stemming from the hydrodynamically expanding sQGP, and a halo, consisting of the decay products of long lived resonances (such as η, η , K 0 S , ω): S = S core + S halo . These two components have characteristically different sizes ( 10 fm for the core, > 50 fm for the halo, based on the half-lives of the above mentioned resonances). In particular, the halo component is so narrow in momentum-space, that it cannot be resolved experimentally. This leads to the following apparent correlation function: lim q→0 C (0) 2 (Q, K) = 1 + λ(K),(2) where λ = N core /(N core + N halo ) was introduced, being related related to the fraction of primordial pions among all (primordial plus decay) pions. One of the motivations for measuring λ is that it is related [23] to the η meson yield, expected [24] to increase in case of chiral U A (1) symmetry restoration in heavy-ion collisions (due to the expected in-medium mass decrease of the η ). Note that a study [25] reported the compatibility of existing λ(m T ) data and predictions based on a decreased in-medium η mass. Experimental results show [13,14], that the source function does not always exhibit a Gaussian shape. In an expanding hadron resonance gas, increasing mean free paths lead to a Lévy-flight, anomalous diffusion, and hence to spatial Lévy distributions [15][16][17] This leads to a correlation function of Q) ε (1+ × N × ;Q) α ,R, λ ( 2 C Q) ε (1+ × N × ;Q) α ,R, λ ( 2 (0) C Q) ε (1+ × N =C (0) 2 (Q, K) = 1 + λ(K) · e −(QR(K)) α(K) ,(3) where α is the (K-dependent) Lévy-exponent, which is conjectured [18] to be identical to the critical exponent η, conjectured to take a value of 0.5 or even lower, identivally to the universality class of the 3D Ising model (possibly with random external fields) [18][19][20][21][22]. Since the exploration the search for the QCD critical endpoint is one of the major goals of experimental heavy ion physics nowadays, we gain additional motivation for the measurement and analysis of of Bose-Einstein correlation functions. Hence, in the following we utilize a generalization of the usual Gaussian shape of the Bose-Einstein correlations, namely we analyze our data using Lévy stable source distributions. In this proceedings paper we omit the discussion of final-state interactions, in particupar the effect of the Coulomb interaction. The handling of this is described in detail in Ref. [12]. Results We analyzed √ s N N = 200 GeV Au+Au collisions from the 2010 running period of the PHENIX experiment, selecting about 2.2 billion 0 − 30% centrality events from the recorded 7.3 billion Minimum Bias events. Note that in the original conference presentation, the Minimum Bias data were presented (shown also e.g. in Ref. [27]). In this paper we present the final data from Ref. [12], which yield the same conclusion. Two-particle correlation functions of π − π − and π + π + pairs (versus the momentum difference length in the longitudinally comoving system, Q) were measured 31 m T bins ranging from 228 to 871 MeV/c 2 (where m T denotes the transverse momentum variable related to the average momentum K). We fitted these correlation functions with the Coulomb-effect incorporated, based on Lévy-shaped sources, as described in the previous section and in Ref. [12]. Additionally, we introduced a linear background, as indicated in Fig. 1, where an example fit is shown. The fits in all m T bins and for both charges yield statistically acceptable descriptions of the measured correlation functions, indicating that the fit parameters of R, α and λ indeed represent the measurements. The m T dependence of the fit parameters is shown in Fig. 2. We may observe that α is approximately constant (within systematic uncertainties), and takes an average value of 1.207, being far from the Gaussian assumption of α = 2, but also far from the conjectured α = 0.5 value at the critical point. The results are furthermore incompatible with the exponential assumption of α = 1. We also see, that despite being far from the hydrodynamic limit of Gaussian distributions, the hydro prediction of 1/R 2 a + bm T still holds. The correlation function strength λ is shown after a normalization by λ max = λ m T =0.5−0.7GeV/c 2 . This clearly indicates a decrease at small m T , which may be explained by resonance effects, and is in particular not incompatible with predictions based on a reduced η mass. We also show, that a new, empirically found scaling parameter R = R/(λ(1 + α)) may be defined with decreased statistical uncertainties, exhibiting a clear linear scaling with m T . 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{'abstract': 'Charged pion two-particle correlation functions were measured in 0−30% centrality √ s N N = 200 GeV Au+Au collisions with the PHENIX experiment at RHIC. The measured correlation functions can be statistically well described based on the assumption of Lévy-shaped source distributions. In this proceedings paper we present the Lévy parameters of the measured correlation functions: correlation strength parameter λ, Lévy index α and Lévy scale parameter R as a function of pair transverse mass mT , in 31 bins from 228 to 871 MeV, separately for positive and negative pion pairs. We discuss the physical interpretation of the mT dependence of the parameters.', 'arxivid': '1711.05605', 'author': ['Máté Csanád \nPHENIX Collaboration Eötvös Loránd University\nPázmány P. s. 1/AH-1117BudapestHungary\n'], 'authoraffiliation': ['PHENIX Collaboration Eötvös Loránd University\nPázmány P. s. 1/AH-1117BudapestHungary'], 'corpusid': 119091919, 'doi': '10.1393/ncc/i2017-17195-1', 'github_urls': [], 'n_tokens_mistral': 3723, 'n_tokens_neox': 3013, 'n_words': 1696, 'pdfsha': 'e4b2d6cfb90f3074cc30e1edacaf6ca913e82c58', 'pdfurls': ['https://arxiv.org/pdf/1711.05605v1.pdf'], 'title': ['Femtoscopy via Lévy sources with PHENIX at RHIC', 'Femtoscopy via Lévy sources with PHENIX at RHIC'], 'venue': []}

arxiv

Near-Infrared 3D Imaging with Upconversion Detection Compiled November 1, 2022 H E Zhang Department of Physics Stevens Institute of Technology 07030HobokenNJUSA Center for Quantum Science and Engineering Stevens Institute of Technology 07030HobokenNJUSA Santosh Kumar Department of Physics Stevens Institute of Technology 07030HobokenNJUSA Center for Quantum Science and Engineering Stevens Institute of Technology 07030HobokenNJUSA Yong Meng Sua Department of Physics Stevens Institute of Technology 07030HobokenNJUSA Center for Quantum Science and Engineering Stevens Institute of Technology 07030HobokenNJUSA Shenyu Zhu Department of Physics Stevens Institute of Technology 07030HobokenNJUSA Center for Quantum Science and Engineering Stevens Institute of Technology 07030HobokenNJUSA Yu-Ping Huang Department of Physics Stevens Institute of Technology 07030HobokenNJUSA Center for Quantum Science and Engineering Stevens Institute of Technology 07030HobokenNJUSA Near-Infrared 3D Imaging with Upconversion Detection Compiled November 1, 202210.1364/PRJ.XX.XXXXXXResearch Article Photonics Research 1 We demonstrate a photon-sensitive, three-dimensional camera by active near-infrared illumination and fast time-of-flight gating. It uses pico-second pump pulses to selectively up-convert the backscattered photons according to their spatiotemporal modes via sum-frequency generation in a χ 2 nonlinear crystal, which are then detected by electron-multiplying CCD with photon sensitive detection. As such, it achieves sub-millimeter depth resolution, exceptional noise suppression, and high detection sensitivity. Our results show that it can accurately reconstruct the surface profiles of occluded targets placed behind highly scattering and lossy obscurants of 14 optical depth (round trip), using only milliwatt illumination power. This technique may find applications in biomedical imaging, environmental monitoring, and wide-field light detection and ranging. INTRODUCTION Three-dimensional (3D) imaging has been a long and actively pursued technology due to its important applications in medical diagnosis [1][2][3], remote sensing [4,5], facial recognition [6][7][8], environmental monitoring [9,10], and so on. A handful of techniques and realizations have thus far been demonstrated, including those based on structured-light imaging [8,11], light detection and ranging (LiDAR) with raster scanning [6,7,12,13], and stereophotogrammetry [14]. Recently, the time-correlated singlephoton counting technology has been deployed to boost the detection sensitivity [15][16][17]. In general, those active-illumination systems can generate 3D profiles of target object with higher accuracy than those based on passive sensing. Meanwhile, infrared (IR) imaging and detection has been studied extensively in the past decade, which allows sensitive detection of many biomolecular and chemical signals [18,19] compared to that of visible light [20,21]. However, existing IR detection techniques are mostly based on thermal sensors, which have low sensitivity and high noise, even with cryogenic cooling [22]. On the other hand, visible detectors have much lower noise and high sensitivity without the need for cryogenic cooling. Instead of the direct IR imaging, the parametric frequency upconversion imaging [23][24][25] plays a critical role for hyper-spectral IR imaging, where the IR signal is frequency upconverted into visible wavelength [20,[26][27][28][29] and detected by a silicon-based detector or camera with high sensitivity and low noise. Many unique nonlinear optical systems have also been developed for 2D imaging, such as noise-less optical parametric amplification imaging [30,31], non-degenerate two-photon absorption [32][33][34], spontaneous parametric down conversion imaging [35,36]. Some are deployed to NIR or MIR imaging regimes. With parametric frequency up-conversion, it has been shown that near-unity conversion efficiency can be achieved, and also can preserve the quantum characteristics of infrared photons [37]. This will facilitate NIR or MIR imaging at a few photon levels with low dark noise [20]. For example, by utilizing a sensitive detector such as silicon electron-multiplying charge-coupled device (EMCCD) to directly record upconversion photons in the visible or near-infrared region [24,25]. The natural extension of the existing 2D parametric upconversion imaging system to 3D imaging can have great potential, and may offer promising applications that require infrared multidimensional imaging. One attempt on 3D IR imaging with parametric upconversion process is shown in [38]. Here they used a chirped ultrashort pulse as the laser source and utilized the principle of ultrafast conversion between space, time, and frequency to obtain a single shot classical 3D image. However, the weakness of their study was due to the trade-off between the measurement range and depth resolution, as well as the low spatial resolution due to tight focusing at the crystal plane. This makes it difficult to extract critical features of the objects. Some common challenges in most of the back-reflected imaging systems are strong background noise and multiple scattering [33,[39][40][41]. In order to strip the background noise from the contaminated signal, traditional methods such as time-frequency filters are commonly used [42,43], but inherently limited by the trade-off between signal detection efficiency and noise rejection. Realizing this, quantum parametric mode sorting has been proposed and demonstrated [43][44][45]. Some other optical techniques are also present in the literature to address imaging through multi-scattered media. Most of these techniques can be broadly classified as relying on ballistic photons [39,46], diffuse optical tomography [47,48], etc. Those techniques offer potential applications for medical imaging, communications and security [2,49]. However, most studies in the field of overcoming strong background noise and multiple scattering have focused on 2D imaging. In the past, QPMS has been utilized for single-pixel 3D single photon imaging system in photon-starved, noise-polluted environments as well as imaging through strongly scattering medium [7,50]. With raster scanning methods, it demonstrated imaging of a 3D object through a highly reflective obscuring scene with 36 dB advantage in noise rejection. However, the requirement of raster scanning has severely impeded the image acquisition speed while the traverse spatial resolution of the 3D image is limited by the field of view of single pixel detector [51]. In this paper, we extend these promising studies and explore an active NIR 3D imaging system using EMCCD with upconversion detection through a highly scattered medium amid strong spatio-temporal background noise. Our system is based on the nonlinear frequency upconversion process via sum-frequency (SF) generation of time-correlated pump and signal optical pulses. Combined with single-photon sensitive EMCCD, we capture spatial and temporal information of the scene of interest, which can be used to reconstruct a 3D surface profiles of the target object. To surmount the imaging through strong scattering medium, we utilize time-resolving photon detection by detecting the frequency up-conversion image confined temporally within the time window of the pump pulse, which shows excellent behavior in noise rejection. Compare with the raster scanning methods [7,12,13], in our work, we could significantly improve the spatial resolution up to 48 µm and effectively reduce the 3D image acquisition time. Besides, we could reconstruct the 3D profile of the target object through strong spatio-temporal background noise (SNR about -20 dB). This system can be deployed in applications that require ultra-sensitive imaging, such as medical diagnostics and quantum optics at single or few-photons level [52,53], and find values in biomedical imaging, non-destructive label free diagnosis and quantum communications. In the future, selective 3D edge enhancements can be implemented by imprinting the spatial phase patterns on the pump beam using a spatial light modulator (SLM) [54,55]. . EXPERIMENT SETUP The nonlinear optical setup for the 3D image reconstructor is shown in Fig. 1(a). The signal and pump pulses are derived from a 50 MHz femtosecond mode-locked laser (MLL) using two inline narrowband wavelength division multiplexers (WDM) of 0.8 nm linewidth to pick two separate wavelengths, one at 1545 nm as the signal and another at 1558 nm as the pump. The pump pulses are sent to a programmable optical delay line (ODL) and then guided into free space with collimated beam size of 0.65 mm FWHM. The collimated signal beam is magnified by a beam expander to 10.8 mm FWHM. The intensity of the horizontally polarized signal beam is tuned by the combination of a half and a quarter waveplates (HWP & QWP) with a polarizing beam splitter (PBS). The signal light, after PBS, then passes through another QWP and scattering media before incident on the object. The back reflected or scattered light from the object then changes the polarization and pass through a telescope with lenses FL1= 300 mm and FL2 = 25 mm which reduce the beam size to ∼ 0.53 mm FWHM. After that, a beam splitter (BS) com- RESULTS First, we evaluate the transverse spatial resolution of the 3D imager with two different experimental settings, as shown in Figure 2. A 1951 USAF resolution test chart (USAF-RTC) is used at the object position to test the spatial resolution of this system. There are 54 target elements provided in the USAF-RTC, and each element consists of three bars which are separated by the bar width. We define the feature size as the width of the bar in the USAF-RTC, which is half the distance between the centers of the two bars. The spatial resolution result shown in Fig. 2(a) is the up-conversion image of group 1 in USAF-RTC, which was obtained from the experimental setup depicted in Fig. 1(a). The intensity vs position line plot shown in Fig. 2(b) reflects the 1D spatially resolved blue dashed line in Fig. 2(a). It shows that our setup is able to easily resolve features with size ∼144 µm, which is also consistent with its actual value 140.31 µm (group 1, element 6). The decrease in the intensity at the edges is due to the Gaussian intensity distribution of the probing signal beam and the pump beam. We have validated that the size of the spot hitting on the object and the subsequent imaging system affect the final resolvable resolution of our 3D imager system, since they affect the point spread function of back reflected signal from the object. So, another version of the experiment setup has been constructed, whose spatial resolution result is shown in Fig. 2(c), which is the up-conversion image of group 3 in USAF-RTC. In this case, the signal beam size is reduced to 3.6 mm FWHM. Besides, the FLs of the first 4f system are replaced as FL1 = 100 mm and FL2 = 25 mm. The intensity profile along the red dashed line in Fig. 2(c) is shown in Fig. 2(d), where we can easily resolve objects with spatial resolution 48 µm. This feature size is matched with its actual value 49.37 µm (group 3, element 5). In this case, the resolvable feature size is improved, but the field of view from the observed object is reduced. In order to image bigger objects, we present the following results in this work using the setup shown in Fig. 1(a), with the spatial resolution of 144 µm. Our 3D object image reconstruction technique utilizes the spatial and temporal photon information enabled by the nonlinear frequency conversion process. Figure 3 shows an example of the 3D reconstruction process. In Fig. 3 (a1), the blue curve indicates the normalized photon intensity as a function of the arrival time for the reflected/back-scattered signal photons from the object to reach the crystal, and the red solid curve represents the pump pulse arrival time. By sweeping the optical delay line, the pump pulses are swept along the temporal domain, which can overlap with the returning signal pulses at a certain arrival time. These two overlapped pulses interact inside the PPLN crystal, and the SF light is generated as shown in the Fig. 3 (a2). The signal can be up-converted efficiently only if it is spatially and temporally overlap with the pump. At each delay step, the SF image was captured to reconstruct the depth z (= c × t/2) of the 3D object. Therefore, the optical delay time indicates the arrival time of the back-scattered signal photons. In Fig. 3 (a3)-(a5), three SF images collected at optical delay time t = 75 ps, t = 105 ps, and t = 140 ps respectively. The collected data is a 3D data set shown in Fig. 3 (a6). Each pixel has one histogram of photon's arrival time versus the counts. After converting photon's arrival time into distance, we show two histograms for two different pixels in Fig. 3 (a6). The peak value (z1 and z2) of the curve provides the information of relative depth for each pixel on the object. By post-processing the data, the reconstructed image with depth information is shown in Fig. 3 (a7) followed by applying a median filter (4 × 4 pixels) to smoothen the reconstructed image. In the experiment, a washer and a bolt are used as the target objects to perform the measurement. The dimensional measurement of the washer and the bolt are shown in Fig.3 (b1) and (c1), respectively. The results of the reconstructed 3D images are shown in Fig.3 Fig. 3. (a1)-(a7) illustrate the present 3D imaging method. (a1) presents the intensity measurement for the input signal and pump at different arrival times, with the corresponding SF intensity shown in (a2). (a3)-(a5) shows the spatial information at different arrival time I, I I, and I I I, respectively. (a6) shows the reconstruction procedure of the 3D data set collection. At each pump delay, 2D image data are acquired by the camera in each frame, and the z-dimension represents the photon's fight distance. (a7) shows post-processing data of (a6), in which the z-axis gives the depth information of the target object. The object photos with profile data for washer and bolt are shown in (b1) and (c1), respectively. After performing 3D imaging measurement, the results for washer and bolt are shown in (b2) and (c2), respectively. (b2) and (c2). The X-axis and Y-axis give the cross-section of the target object in mm, and Z-axis shows the depth information of the target object in mm. In the Fig.3 (b2), the measured outer diameter of the reconstructed washer is 7.92 mm, inner diameter is 3.65 mm, and depth is 1.6 mm. In the Fig.3 (c2), the measured stub height and diameter of reconstructed bolt is 5.9 mm and 2.3 mm, the height and the diameter of the bottom parts is 3.2 mm and 6.45 mm. We set the EMCCD exposure time of each image to 1 s. The total acquisition time for a full 3D image reconstruction is ∼30 s. The reconstructed 3D image in Fig.3 (b2) and (c2) agree well with the ground truth shown in (b1) and (c1). Next, we test the performance of our technique through scattering media. The scattering media are made from epoxy resin and Titanium Oxide (TiO2) pigment (∼220 nm particle size). We examine two different pieces of scattering media (SM1 and SM2) in our setup. The thickness, mean free path (l s ) and optical depth of the scattered media are shown in Table 1. SM1 with thickness 4.3 mm and optical depth 7.29l s (14.58l s round trip) is more scattered compare to SM2 with thickness 3.2 mm and optical depth 4.04l s (8.08l s round trip). In our case, the scattering media is placed in front of the object, so the signal propagates twice through the scattering media before upconversion detection. The back-reflected photons coming from the different surface of the target object are upconverted in different time intervals using the pump pulse time gating. It could allow us to isolate the other back-scattered noise photons arriving in different time intervals. Figure 4 shows the 3D image reconstruction of the target objects, bolt ((a) and (c)) and washer ((b) and (d)) through scattering media (SM1 and SM2), respectively. To effectively reconstruct the 3D image, we used a time windowing procedure to post-select several continuous EMCCD images that are captured at different temporal delay of the pump pulse. The time window could discard most of the background noise coming from the scattering, and process only those within the time window defined by the pump pulses, following the procedure shown in Fig.3(a6) to (a7). This procedure is programmed via Matlab to improve the SNR of the reconstructed 3D image. It reduces the speckle noise induce by the scattering medium, thus better resolve the shape of the target object. The reconstructed image of the bolt through SM1 without and with time windowing are shown in Fig.4 (a1) and (a2), respectively. Here, the edge and depth variation of the cap bolt can be distinguished clearly. The washer image without and with temporal windowing are also reconstructed through SM1 in Fig. 4 (b1) and (b2), respectively. It can also reconstruct the image of the washer. Similar results are shown in the third row of Fig. 4 for relatively weaker scattering media (SM2). In both cases, we can effectively reconstruct the 3D image of the target object by carving the temporal window. After that, we inject the amplified spontaneous emission (ASE) noise that is temporally and spectrally overlapping with the signal, as shown in 1 (a). This noise is generated from an erbium doped fiber amplifier (EDFA). Both the signal and ASE noise pass twice through SM2 before the upconversion detec-tion. To ensure that the ASE noise is in the same time-frequency and spatial profile, we choose the same WDM filter bandwidth and spatial beam size, as the signal. In this measurement, we use the washer in Fig. 3 (b1) as the target object. Figure 5 (a) shows the image of the signal mixed with the temporal noise before up-conversion, as taken by the IR camera. At a low SNR (about -20 dB), it is impossible to reconstruct the image of the target object with direct detection. Yet, our system captures an image with up to 8 × 10 7 converted photons per second using the EMCCD as shown in Fig.5 (b). By time windowing, the noise is effectively suppressed, and the reconstructed washer image gives ∼16 dB improvement in the SNR. Figure 5(c) shows the 3D reconstruction image of the target object by temporally scanning the pump pulses. The exposure time of the EMCCD is the same as the other previous measurements (= 1 s). This result clearly demonstrates the noise rejection advantage of our system. We now briefly discuss the effects of the EMCCD exposure time on the reconstructed 3D images. Figure 6 shows the results for three different exposure times: 0.1 s, 1 s and 2 s. As the exposure time increases, the 3D image of the washer can be easily recognizable with better contrast. For the exposure time of 2 s on EMCCD, the total acquisition time in reconstructing a 3D image is ∼ 60 s, which is considerably long. Yet, as the above improvement comes from the increased number of photons collected by the EMCCD, one can instead increase the nonlinear conversion efficiency or use detector arrays with higher quantum efficiency, to enhance the image contrast while reducing the acquisition time. Thus far, we have used highly-reflective metal objects as our testing targets. To further assess the capability of our photon sensitive 3D imager for general objects, we now switch to a target with diffusive surface, shown in Fig. 7(a). This diffusive surface, a digit '3', is homemade using PLA plastic Filament by 3D printer. For 3D reconstruction of this object, we set the EMCCD exposure time to 2s. Figure 7(b) shows the reconstructed 3D image without any scattered media or ASE noise. It shows that the 3D reconstructed image is clearly recovered. When we add the same scattering media and ASE noise as used in Fig. 5, our 3D imager performance degrades. Nonetheless, we are still able to recover the object by properly choosing the temporal window. Figure 7(c) presents the results with the full time scanning, while Fig. 7(d) gives the manicured data by carving the temporal window. In these cases, we need to increase the EMCCD exposure time to 10 s to collect more photons. As seen, the unrecognizable 3D image in (c) can be well recovered in (d) by properly tuning the temporal window. CONCLUSION We have experimentally demonstrated a high-performance 3D imager for photon-sensitive detection using optical frequency upconversion pumped by picosecond pulses. It obtains millimeter depth resolution and 140 µm spatial resolution, while effectively rejecting background noise from ambient environment and obscurants. As such, the present technique could potentially find applications in biomedical imaging, remote sensing over low visibility. On the other hand, in the current method, the acquisition time of photon-sensitive imaging is longer than what is needed for typical real-time target object identification. This shortcoming is mainly due to the low conversion efficiency of the current nonlinear process, which can be increased by using a high-power laser or a longer nonlinear crystal. Also, to improve the imaging sensitivity through different kind of scattered mate- (Table 1), the results of SM1 with optical depth 14.58l s for double passes shown in second row, and the results of SM2 with optical depth 8.08l s shown in third row. The time window can partially discard the redundant noise in the temporal scans and improve the reconstructed image contrast. rials, one could use spatially modulated pump beams to further improving the 3D image contrast, similar to demonstrated in 2D imaging cases [54,55]. Fig. 1 . 1(a) Experiment setup. Mode-locked laser pulses are separated into two arms by using WDM filters, with signal and pump wavelengths at 1545 and 1558 nm, respectively. The signal beam is incident on an obscured object. The back-scattered signal photons are combined with the pump, then up-converted in a nonlinear crystal to generate SF output centered at wavelength 775.5 nm. The time-resolved measurements can faithfully reconstruct the 3D object image captured by the EMCCD camera. (b) The picture of the object (a washer) attached on an aluminum block. (c) The picture of the obscured object, i.e. the washer obscured by the scattering media (SM). WDM: wavelength division multiplexers, EDFA: Erbium-Doped Fiber Amplifier, QWP: Quarter Waveplate, HWP: Half Waveplate, BS: Beamsplitter, FL: Fourier lens, PPLN crystal: Magnesium-doped Periodic Poled Lithium Niobate crystal, EMCCD: electron multiplying silicon charged coupled device, ASE: amplified spontaneous emission. Fig. 2 . 2Field of view and spatial resolution of upconverted images using two different experimental situations. (a) The upconversion image of group 1 in USAF resolution test chart from the experiment setup depicted inFig. 1(a). (b) The intensity profile along the blue dashed line in (a). (c) The up-conversion image of group 3 in USAF resolution test chart with another experimental settings. In this case, the signal beam size is reduced to 3.6 mm FWHM and the first Fourier lens(FL) of the object 4f imaging system is changed to F1 = 100 mm. (d) The intensity profile along the red dashed line in (c).bines the collimated signal and pump beams which incident into a temperature-stabilized PPLN crystal with the poling period of 19.36 µm (5 mol.% MgO doped PPLN, 2 mm length, 3 mm width, and 1 mm height from HC Photonics) for frequency conversion process. The normalized conversion efficiency in our case is 9 × 10 −4 %/W, which is restricted by three factors: (a) low pump power, (b) short crystal length (2 mm) and (c) not in the optimum focusing condition for signal and pump (both are collimated beam) inside the crystal. The 4f system after the crystal consists of two Fourier lenses with focal length FL3 = 25 mm and FL4 = 100 mm, imaging the SF output onto a EMCCD (iXon Ultra 897, Andor) with 512 × 512 pixels and 16 µm pixel size. The quantum efficiency of this EMCCD is measured to be 7.5% at upconversion wavelength, calibrated against a silicon avalanche photo diode at mean photon number ≈ 0.01 per pulse. On the other output of the BS, an IR camera (FIND-R-SCOPE Model No. 85700) is used to capture the image via direct signal detection. Fig. 4 . 43D reconstructed image through scattering media. Two SM samples are used Fig. 5 . 53D image reconstruction through addition noises in the time-frequency and spatial domain. (a) The signal image before upconversion captured using the IR camera, which reflect from the target object and mixes with time-frequency and spatial noises. (b) The up-converted SF image at a certain arrival time, which can captured using the EMCCD. (c) The reconstructed 3D image. Fig. 6 . 6Reconstructed 3D image of washer for a series of exposure times (a) 0.1 s, (b) 1 s, and (c) 2 s. Fig. 7 . 73D imaging measurement for the target object with diffusive surface. (a) A photo for target object. (b) A reconstructed image without scattering media. (c) A reconstructed image of the object with scattering media placed in front of it. (d) Post processing image of (c). Table 1 . 1The parameters of scattering media.Sample Mean free path Optical depth (l s ) (round trip) SM1 4.3 mm 0.58 mm 14.58l s SM2 3.2 mm 0.66 mm 8.08l s Acknowledgement. This material is based upon work sup-Disclosures. The authors declare no conflicts of interest.Data Availability. 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{'abstract': 'We demonstrate a photon-sensitive, three-dimensional camera by active near-infrared illumination and fast time-of-flight gating. It uses pico-second pump pulses to selectively up-convert the backscattered photons according to their spatiotemporal modes via sum-frequency generation in a χ 2 nonlinear crystal, which are then detected by electron-multiplying CCD with photon sensitive detection. As such, it achieves sub-millimeter depth resolution, exceptional noise suppression, and high detection sensitivity. Our results show that it can accurately reconstruct the surface profiles of occluded targets placed behind highly scattering and lossy obscurants of 14 optical depth (round trip), using only milliwatt illumination power. This technique may find applications in biomedical imaging, environmental monitoring, and wide-field light detection and ranging.', 'arxivid': '2210.17286', 'author': ['H E Zhang \nDepartment of Physics\nStevens Institute of Technology\n07030HobokenNJUSA\n\nCenter for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA\n', 'Santosh Kumar \nDepartment of Physics\nStevens Institute of Technology\n07030HobokenNJUSA\n\nCenter for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA\n', 'Yong Meng Sua \nDepartment of Physics\nStevens Institute of Technology\n07030HobokenNJUSA\n\nCenter for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA\n', 'Shenyu Zhu \nDepartment of Physics\nStevens Institute of Technology\n07030HobokenNJUSA\n\nCenter for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA\n', 'Yu-Ping Huang \nDepartment of Physics\nStevens Institute of Technology\n07030HobokenNJUSA\n\nCenter for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA\n'], 'authoraffiliation': ['Department of Physics\nStevens Institute of Technology\n07030HobokenNJUSA', 'Center for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA', 'Department of Physics\nStevens Institute of Technology\n07030HobokenNJUSA', 'Center for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA', 'Department of Physics\nStevens Institute of Technology\n07030HobokenNJUSA', 'Center for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA', 'Department of Physics\nStevens Institute of Technology\n07030HobokenNJUSA', 'Center for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA', 'Department of Physics\nStevens Institute of Technology\n07030HobokenNJUSA', 'Center for Quantum Science and Engineering\nStevens Institute of Technology\n07030HobokenNJUSA'], 'corpusid': 253237866, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14017, 'n_tokens_neox': 11914, 'n_words': 6956, 'pdfsha': 'c3834243cc4c13bcb123cb80b536575eed06a103', 'pdfurls': ['https://export.arxiv.org/pdf/2210.17286v1.pdf'], 'title': ['Near-Infrared 3D Imaging with Upconversion Detection', 'Near-Infrared 3D Imaging with Upconversion Detection'], 'venue': []}

arxiv

Restless pions: orbifold boundary conditions and noise suppression in lattice QCD 1 Aug 2007 (Dated: February 1, 2008 -17:37) Paulo F Bedaque *[email protected]†[email protected] Maryland Center for Fundamental Physics Department of Physics University of Maryland 20742College ParkMD André Walker-Loud Maryland Center for Fundamental Physics Department of Physics University of Maryland 20742College ParkMD Restless pions: orbifold boundary conditions and noise suppression in lattice QCD 1 Aug 2007 (Dated: February 1, 2008 -17:37)arXiv:0708.0207v1 [hep-lat] The study of one or more baryons in lattice QCD is severely hindered by the exponential decay in time of the signal-to-noise ratio. The rate at which the signal-to-noise decreases is a function of the the pion mass. More precisely, it depends on the minimum allowed pion energy in the box, which, for periodic boundary conditions, is equal to its mass. We propose a set of boundary conditions, given by a "parity orbifold" construction, which eliminates the zero momentum pion modes, raising the minimum pion energy without altering the QCD ground state, and thereby improving the signal-to-noise ratio of (multi)-baryon correlation functions at long Euclidean times.We discuss variations of these "restless pions" boundary conditions and focus on their impact on the study of nuclear forces. I. INTRODUCTION Lattice QCD studies of heavy systems are plagued by large statistical noise. The signalto-noise ratio of a correlation function created by an operator with n quark-anti-quark pairs decreases with (Euclidean) time as e −(E− n 2 mπ)t , where E is the mass of the state under consideration. This is a particularly nasty problem for the recent studies of nucleon-nucleon [1,2] and hyperon-nucleon [3] forces with lattice QCD. The large statistical error renders the numerical information at large times useless. Compounded with this problem, at early times the correlators are contaminated by excited states and so there is only a very narrow range of time slices left containing useful information. In unquenched calculations [1,3] the statistical noise allows for semi-quantitative results only, even after the computation of thousands of fermion propagators. These errors are also much larger than finite-volume [4] and finite lattice spacing [5] effects in these observables. We propose here a scheme to alleviate this signal-to-noise problem. We start with the simple observation that the statistical noise is dominated by the energy of the lightest pion states. Periodic boundary conditions allow a pion zero mode and thus the lowest pion energy is equal to its mass. If one were to impose anti-periodic boundary conditions for all three pions, then the pion zero-modes are forbidden and the minimum energy is given by (assuming anti-periodic boundary conditions in all three spatial directions) E π = 3 π L 2 + m 2 π , with L the size of each spatial direction. Thus it is clear that the signal can be improved by using these "restless pions" boundary conditions. There are other applications for which these restless pions are useful. In addition to the obvious benefit to spectroscopy studies [6], the advantage of anti-periodic boundary conditions for the pion in the extraction of the K → ππ amplitude using the Lellouch-Lüscher method [7] were pointed out in references [8,9]. In lattice calculations, one does not have direct control over the hadronic boundary conditions. What can be controlled at will are the boundary conditions of the quark and gluon fields. However, it is not obvious which modifications of the quarks and gluons at the boundary implies an anti-periodic boundary condition for all three of the pions; antiperiodic boundary conditions for the neutral pion have remained elusive. Obvious choices get tantalizingly close to the desired restless pions but upon close inspection, have undesired consequences. For instance, twisted boundary conditions [10] allow for continuous momentum transfer by providing hadrons a momentum kick at the boundary. But as we will explain in sec. II, these twisted boundary conditions do not effect the signal-to-noise issue we are interested in. The G-parity boundary condition suggested in references [8,9] breaks both the spatial subset of the hypercubic rotation invariance as well as chiral symmetry [11]. In Ref. [9], an isospin boundary condition was used, q(L) = τ 3 q(0), but this leaves the neutral pion unaffected. This allows for an extraction of the ∆ I = 3/2 K → ππ amplitude (in which the pions are in an I = 2 final state) but it does not serve our purpose of reducing the statistical noise for baryon calculations. Various "hybrid boundary conditions" have been employed, first in the numerical study of penta-qaurk states [12]. In this first implementation, the u and d quarks were given anti-periodic boundary conditions while the s quark was given periodic boundary conditions, allowing for a definitive identification of bound vs. scattering states by forcing the nK-system to have non-zero relative momentum (the scattering state) while leaving a possible Θ + (uudds) resonant state unaffected (the bound state). In a second variation of the hybrid boundary condition, used in the numerical study of charmonium [13] and a possible tetra-quark state [14], an anti-periodic boundary condition is imposed upon the quarks while the anti-quarks are given periodic boundary conditions. This allows for the same identification of bound and scattering states for these systems as the first variant, however this second hybrid boundary condition violates charge conjugation invariance. Furthermore, neither variant of these hybrid boundary conditions will help with the signal to noise issue we want to address. 1 An axial twisted boundary condition, q(L) = γ 5 q(0) (and similar choices) provide for anti-periodic pions but additionally make σ ∼ qq anti-periodic and alter the QCD pattern of symmetry breaking. We propose a novel approach to this problem making use of an orbifold boundary condition. Similar constructions have been employed in the context of the "chirality problem" in extra-dimensional extensions of the Standard Model [15], domain-wall fermions [16,17] and the Schrödinger functional formalism [18]. Instead of relating the field values at the two ends of the box (z = 0 and z = L), we impose periodic boundary conditions on an extended box −L < z < L. However, the fields at negative values of z are not independent, but are MeV. The signal-to-noise estimate of eq. (5) was normalized to the lattice calculation at t = 11. determined from those with positive z. By appropriately choosing a relation between the quark and gluon fields in these two halves of the lattice (the orbifold condition), we can enforce a π(−z) = −π(z) condition, eliminating the zero momentum mode for all pions, making them restless. II. SIGNAL-TO-NOISE RATIO ESTIMATES Here we review the argument estimating the statistical noise for lattice QCD correlation functions [19]. Consider first a nucleon correlator C(t) = q(t)q(t)q(t)q(0)q(0)q(0) , where, for clarity, we have suppressed the Dirac, flavor and color indices. At large times, C(t) is dominated by the intermediate state of lowest energy with the quantum numbers of the nucleon: C(t) t→∞ −→ Ae −M t ,(1) where M is the nucleon mass. In a Monte Carlo calculations, C(t) is estimated by an average over N gauge configurations C(t) ∼ =C(t) = 1 N A S A (t)S A (t)S A (t) ≡ S 3 A (t) ,(2) where S A (t) is the quark propagator in each one of the gauge configurations, A. The variance in this estimate is given by σ 2 C (t) = 1 N A |S A (t)S A (t)S A (t) −C(t)| 2 = S 3 A (t)S † 3 A (t) − |C(t)| 2 .(3) For large times, C 2 (t) ∼ e −2M t , while the large time behavior of S 3 A (t)S † 3 A (t) can be found by noticing that S 3 A (t)S † 3 A (t) = q 3 (t)Q 3 (t)q 3 (0)Q 3 (0) t→∞ −→ Be −3mπ t ,(4) where Q is a fictitious quark with identical quantum numbers and properties of the q quarks. 2 The long time behavior of the correlator in eq. (4) is then dominated by the intermediate state with the lowest energy with the quantum numbers of three qQ mesons. Since they have the same mass as the qq mesons, this lowest energy state is given by three times the pion mass. Thus, for sufficiently light pions, S 3 A (t)S † 3 A (t) decays at a rate smaller than C 2 (t). The signal-to-noise ratio of the nucleon correlator is then given by C(t) 1 N σ 2 C (t) t→∞ −→ A √ N e −M t e − 3 2 mπ t ∼ √ N e −(M − 3 2 mπ )t .(5) We show in fig. (1) the signal-to-noise ratio in an actual lattice QCD calculation (details of the simulation can be found in Refs. [1,20]) as well as the estimate in eq. (5). The estimate in eq. (5) is easily generalized for correlation functions of multi-baryons and baryons with strange quarks. In the case of two-nucleon correlators, for example, the 2 This explains why the twisted and hybrid boundary conditions do not help the signal-to-noise problem. The fictitious Q quarks have the same boundary conditions as the q quarks, and thus the qQ and Qq mesons have periodic boundary conditions and are allowed a zero-momentum mode. signal-to-noise ratio is proportional to √ Ne −(2M −3mπ )t . Recent lattice studies of nuclear forces (and hyperon-nucleon interactions) were severely hindered by the fast decrease of the signal-to-noise ratio with time [1,3]. The correlators at short times cannot be used for fitting purposes since it is contaminated by excited states 3 while at later times the statistical noise overwhelms the signal, leaving only a very narrow plateau from which the physics is extracted. This is in stark contrast to lattice calculations of ππ interactions [21] and other two-meson systems [22]. III. PARITY ORBIFOLDS Let us now describe the basic idea of the orbifold construction in the case when only one dimension is orbifolded. Consider a lattice whose z coordinate belongs to the interval Then identify the points z and −z by relating φ(z) to φ(−z), effectively transforming the circle into a line segment (including the boundary) as shown in fig. (2). In the simplest case, φ(z) = ±φ(−z). If the plus sign is chosen, φ(z) will be a linear combination of spatially symmetric wavefunctions, φ + (z) = ∞ n=0 A (n) + cos nπz L .(6) If, however, the minus sign is chosen then the φ(z) will be a linear combination of antisymmetric wavefunctions, φ − (z) = ∞ n=1 A (n) − sin nπz L ,(7) and consequently there is no zero mode for this field. The lowest momentum allowed is k min = π L with an energy of (π/L) 2 + m 2 . This upward shift in the minimum allowed energy value is the desired result. In order to eliminate the pions at rest we will require that π(z) = −π(−z). The ways to achieve this by imposing orbifold conditions on the quark and gluon fields and the generalization to higher dimensions will be discussed next. 3 In the single baryon sector, a significant improvement in the isolation of the ground state and excited states at early times has been achieved with the use of multiple operators combined with quark and gluon smearing [6]. The equivalent study for operators coupling to multi-nucleon states has not been performed and is anticipated to be significantly more challenging and costly given the larger number of operators and quark contractions. the "parity orbifolding" condition (the issues we discuss here belong to the infrared regime and we use a continuum notation) A µ (t, x, y, z) = A µ (t, x, y, −z), for µ = 3 A 3 (t, x, y, z) = −A 3 (t, x, y, −z), q(t, x, y, z) = P z q(t, x, y, −z), q(t, x, y, z) =q(t, x, y, −z)P z ,(8) where P z = iγ 5 γ 3 is the z-parity operator corresponding to a reversal of the z direction and we work in Euclidean space. 4 The z-parity operator P z is obtained from the usual parity operator γ 0 , corresponding to a simultaneous reversal of all three spatial axes, combined with a rotation by π around the z-axis. The conditions in eq. (8) relate the QCD fields in one side of the box to their parity conjugates in the opposite side. Notice that, since parity is a symmetry of the theory, the contribution to the action from the z < 0 region is exactly the same as the z > 0 region and the computational cost of using the extended box, [−L, L] is the same as that of the smaller box, [0, L]. The only effect of the orbifold condition is on the link connecting the z < 0 and z > 0 regions. In other words, it acts as a boundary 4 We use the conventions γ 2 5 = γ 2 µ = 1, γ † µ = γ µ . condition at z = 0. In fact, consider the orbifolded action in the case of Wilson quarks S = κ [q −1 (γ 3 − r)q 1 −q 1 (γ 3 + r)q −1 ] + a 4 (q 1 q 1 +q −1 q −1 ) + · · · = −2κq 1 (γ 3 + r)P z q 1 + 2a 4q 1 q 1 + · · · ,(9) where κ is the hopping parameter, the index on the quark fields denotes the position in z Notice that we could have equally used the opposite z-parity operator, −P z , implementing a reversal of all three spatial axis followed by a rotation by −π about the z-axis. The difference between rotating by π in the positive or in the negative direction amounts to a 2π rotation which, for spin-1/2 fermions, leads to a minus sign difference between P z and −P z . Physical observables, being quark bilinears, generally do not depend on this sign. As can be seen in eq. (9), however, the boundary terms are linear in P z and are able to distinguish between the choice in sign of P z . This shows that the orbifold condition breaks the z → −z symmetry. The parity orbifold condition on the quark and gluon fields implies orbifold conditions for the hadronic fields. If we identify the pion field with the π ∼qγ 5 τ q interpolating field, we see that it satisfies the desired π(t, x, y, z) = −π(t, x, y, −z) orbifold condition. In fact, the same condition will follow if any other pion interpolating field is used like, for instance, π ∼qτ qF µνFµν , since it depends only on the fact that the pion has negative intrinsic parity. In fact, all parity odd operators will satisfy a condition similar to eq. (10) while the parity even operators will satisfy the analogue equation without the minus sign. In particular, the σ field σ ∼ qq has a zero mode and the QCD pattern of 5 Notice that, contrary to the continuum case, the boundary conditions in lattice field theory are already contained in the action. Different lattice action terms localized at the boundary imply different boundary conditions in the continuum and the relation between them is, in general, a complicated dynamical question. symmetry breaking is not affected by the orbifolding procedure. The nucleon fields satisfy N(t, x, y, z) = −P z N(t, x, y, −z), N (t, x, y, z) = −N (t, x, y, −z)P z ,(11) as can be seen using the interpolating field N ∼ qq T τ 2 Cγ 5 q. In the non-relativistic domain, P z = iγ 5 γ 3 reduces to σ 3 and the allowed modes for the nucleon are N(x, y, z) = e i nxπx 2L x+i ny πy 2L y                      cos( nzπz L )   1 0   , n x , n y , n z = 0, 1, · · · sin( nzπz L )   0 1   , n x , n y = 0, 1, · · · , n z = 1, 2, · · · .(12) Notice that only spin up nucleons can be at rest. Consequently we can construct a spin triplet two-nucleon state, like the deuteron, with zero momentum but a spin singlet twonucleon state will necessarily have a minimum momentum equal to π/L. This asymmetry between spin up and down is a consequence of the breaking of the z → −z symmetry discussed above. Unfortunately, the boundary term shown in eq. (9) is not γ 5 -Hermitian and the fermion determinant is not positive definite. This makes simulations with dynamical quarks satisfying the parity orbifold condition impractical. However, this method is perfectly suited to implementation in the valence sector only, i.e. only on the propagators generated in the background of dynamical configurations. In refs. [23,24], it was argued that up to exponentially suppressed corrections, for many channels of interest including baryon-baryon channels, different boundary conditions can be used in the valence and sea sectors of the theory, known as "partially twisted boundary conditions". Therefore, gauge configurations generated with sea quarks satisfying periodic boundary conditions can be used with valence quarks satisfying "parity orbifold" boundary conditions. Intuitively, the possibility of using different boundary conditions for sea and valence quarks follows from the observation that sea quarks can "notice" their different boundary conditions only if they propagate around the lattice. But, for observables without annihilation diagrams, the propagation of sea quarks around the lattice is suppressed by e −mL , where m is the mass of the lightest hadron made of sea quarks or a mixture of valence and sea quarks. In our case, this is the pion mass. This argument is better appreciated by looking at the graphs in fig. (3), which display examples of processes contributing to baryon-baryon scattering. Only diagrams containing a baryon-baryon intermediate state give rise to power law volume dependence (below the inelastic threshold). These two intermediate baryons are made of valence quarks and therefore satisfy the orbifold boundary condition. We stress that the rate at which the signal-to-noise decreases is set by the valence nucleon and pion masses. The increase on the pion minimum energy has an additional benefit. With the exception of the relation between two-particle energy levels and the S-matrix, described by the Lüscher formula, finite volume effects are suppressed by factors of e −EπL . An increase on the value of E π is then clearly beneficial. This is specially important for the exponentially supressed correction to the Lüscher formula where the suppression factor, formally of order e −EπL , can be sizable for realistic lattices and periodic pions with E π = m π [4]. These finite volume corrections can be estimated using an extension of chiral perturbation theory adapted to the case where valence and sea quarks obey different boundary conditions in the molds of [23,24,25,26,27]. B. Three-dimensional parity T 3 /Z 2 orbifold The method of the previous section can be generalized in order to remove the zeromomentum modes of the pions in all three directions, further improving the signal-to-noise ratio. The simplest generalization of eq. (8) is A 0 (t, r) = A 0 (t, −r), A i (t, r) = −A i (t, r), for i = 1, 2, 3 q(t, ) = Pq(t, −r), q(t, r) =q(t, −r)P,(13) where P = γ 0 is the usual parity operator corresponding to the reversal of all three space condition π(t, r) = −π(t, −r) but now their minimum energy is 3( π L ) 2 + m 2 π . Nucleons obey the same conditions as the quarks, N(t, r) = γ 0 N(t, −r). Since in the non-relativistic limit γ 0 reduces to 1, non-relativistic nucleons satisfy periodic boundary conditions and contain zero modes. This property is very convenient when extracting low-energy phase shifts on the lattice, as with the 3-D parity-orbifolding, there is no restriction on the spinisospin channels one can study in the ground state and the standard Lüscher formula relating energy levels to phase shifts is unchanged. As it will be exemplified below, the increase in the signal-to-noise ratio is dramatic. IV. IMPACT ON LATTICE CALCULATIONS A. Nuclear force studies In order to provide an explicit example, we use the values of the parameters used in [1] to estimate the impact of the method advocated here in the expected rate with which the signal-to-noise ratio decreases with increasing time. We disregard the interaction energy between the hadrons and approximate the energy of the two-nucleon state by ≈ 2M. The energy of the three-pions is approximated by ≈ 3m π when periodic boundary conditions are used, 3 ( π L ) 2 + m 2 π if the S 1 /Z 2 orbifold is used and 3 3( π L ) 2 + m 2 π if the T 3 /Z 2 orbifold is used. The result is plotted in fig. (4). The inclusion of the interaction energy between the two nucleons would change the figure by very little. In fact, for pion masses above 350 MeV the energy shifts found [1] are of order of 10 − 20 MeV. It is expected, however, that in a narrow band close to the physical value of m π the energy shift should be larger [28], corresponding to the diverging scattering lengths, but still much smaller than the rest mass of the nucleons. Even the modest increase in the pion minimum energy found in the onedimensional orbifolding has a potential significant impact by noise limited measurements. In the case of the three-dimensional orbifolding that potential improvement is enormous (notice the log scale in the corresponding graph). B. Impact on K → ππ As pointed out in [8,9], the extraction of the K → ππ amplitude with the Lellouch-Lüscher method [7] can benefit from eliminating pion zero modes. The method to eliminate pions at rest discussed here can only be applied to the I = 2 channel. In the I = 0 channel, the use of different boundary conditions in the valence and sea sectors alters the amplitude by factors that are not exponentially suppressed. Of course, a modified chiral perturbation theory taking into account the differences of the valence and sea sectors can still be used to relate the results of such a lattice calculation with the real world QCD amplitude. V. DISCUSSION We have introduced "restless pions" boundary conditions designed to reduce the rapid degradation of the signal-to-noise ratio which plagues studies of heavy systems with lattice QCD. We have shown how these boundary conditions can be implemented with a parityorbifold construction in either one or three spatial dimensions. Unfortunately, the action at the boundary is not γ 5 -Hermitian and so this particular construction is not suitable for the sea sector. However, this method is perfectly suited for implementation of the valence fermions. For non-scalar channels, the difference in sea and valence boundary conditions is felt only by exponentially small terms. The numerical cost of implementing these parityorbifolded valence propagators is the same for propagators with (anti)-periodic boundary conditions, as the fields in each half of the bulk are not independent, and therefore the implementation is achieved with a special boundary condition on the non-doubled lattice. FIG. 1 : 1Log of signal-to-noise ratio of the two-nucleon correlator in the spin singlet channel as a function of Euclidean time (from the calculation described in[20]). The pion mass is about 350 [ 0 , 0L]. Extend it to [−L, L] and identify the points z = −L and z = L, effectively turning the interval [−L, L] into a circle. Let all fields, φ(z), satisfy the periodic condition φ(L) = φ(−L). FIG. 2 : 2Identification of z and −z points reduces the circle to a line segment. A. One-dimensional S 1 /Z 2 parity orbifold In the simplest version of our proposal the orbifold trick is used in only one of the spatial directions. Consider QCD fields in the periodic box [0, L] × [0, L] × [−L, L] × [0, β] satisfying ( the remaining coordinates are implicit) and the dots denote the contributions from the two sides of the bulk, z > 0 and z < 0 (which are equal to each other). We see then that the orbifolded [−L, L] lattice is equivalent to a [0, L] lattice with some extra terms residing at the boundary, as is the case with any boundary condition.5 directions. While the boundary conditions in eq. (8) can be seen as a mirror placed at z = 0, the conditions in eq. (13) can be visualized as a pin hole located x = y = z = 0 with a lattice [−L/2, L/2] × [−L/2, L/2] × [−L, L] × [0, β]. Again, all three pions obey the odd orbifold FIG. 3 : 3Examples of two-nucleon graphs containing sea quarks. The left column shows the graphs at QCD level (dotted lines represent sea quarks) and the right column represents the same graphs at the low energy effective theory level. The graphs on the first row are proportional to e −Λ QCD L , the second and third are proportional to e −mπ L . The last row shows a graph with a power law dependence on the volume. LL Estimate of the signal-to-noise ratio with the S 3 /Z 2 orbifold condition ) 2 +m 2 )t (solid line) and with periodic boundary conditions e −(2M −3m)t (dashed line) as a function of t. Right: Log plot of the signal-to-noise ratio with the T 3 /Z 2 orbifold condition ) 2 +m 2 )t (solid line) and with periodic boundary conditions e −(2M −3m)t (dashed line) as a function of t. In both figures the pion mass is 350 MeV and the box size is L = 2.5 fm. See section II for details. AcknowledgmentsWe would like to thank T. Cohen and K. Orginos for conversations on this subject and the NPLQCD collaboration for the use of their data infig. (1). This research was supported of Energy under grant no. DE-FG02-93Er-40762. the U.S. Deptpart by the U.S. Dept. of Energy under grant no. DE-FG02-93Er-40762. . 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{'abstract': 'The study of one or more baryons in lattice QCD is severely hindered by the exponential decay in time of the signal-to-noise ratio. The rate at which the signal-to-noise decreases is a function of the the pion mass. More precisely, it depends on the minimum allowed pion energy in the box, which, for periodic boundary conditions, is equal to its mass. We propose a set of boundary conditions, given by a "parity orbifold" construction, which eliminates the zero momentum pion modes, raising the minimum pion energy without altering the QCD ground state, and thereby improving the signal-to-noise ratio of (multi)-baryon correlation functions at long Euclidean times.We discuss variations of these "restless pions" boundary conditions and focus on their impact on the study of nuclear forces.', 'arxivid': '0708.0207', 'author': ['Paulo F Bedaque *[email protected]†[email protected] \nMaryland Center for Fundamental Physics Department of Physics\nUniversity of Maryland\n20742College ParkMD\n', 'André Walker-Loud \nMaryland Center for Fundamental Physics Department of Physics\nUniversity of Maryland\n20742College ParkMD\n'], 'authoraffiliation': ['Maryland Center for Fundamental Physics Department of Physics\nUniversity of Maryland\n20742College ParkMD', 'Maryland Center for Fundamental Physics Department of Physics\nUniversity of Maryland\n20742College ParkMD'], 'corpusid': 119563949, 'doi': '10.1016/j.physletb.2008.01.011', 'github_urls': [], 'n_tokens_mistral': 11535, 'n_tokens_neox': 9436, 'n_words': 5342, 'pdfsha': '07bd522d51440b1ae42890386b17c05f46fd0b0b', 'pdfurls': ['https://arxiv.org/pdf/0708.0207v1.pdf'], 'title': ['Restless pions: orbifold boundary conditions and noise suppression in lattice QCD', 'Restless pions: orbifold boundary conditions and noise suppression in lattice QCD'], 'venue': []}

arxiv

A modern description of Rayleigh's criterion 11 Jul 2018 Sisi Zhou Liang Jiang Departments of Applied Physics and Physics Yale University 06511New HavenConnecticutUSA Yale Quantum Institute Yale University 06520New HavenConnecticutUSA A modern description of Rayleigh's criterion 11 Jul 2018(Dated: July 12, 2018)2 Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a oneor two-dimensional image can be precisely resolved up to its second moment in the subdiffraction limit, i.e. the Fisher information (FI) is non-zero. But the FI with respect to higher order moments always tends to zero polynomially as the size of the image decreases, for any type of non-adaptive measurement. We call this phenomenon a modern description of Rayleigh's criterion. For PSFs under certain constraints, the optimal measurement basis estimating all moments in the subdiffraction limit for 1D weak-source imaging is constructed. Such basis also generates the optimal-scaling FI with respect to the size of the image for 2D or strong-source imaging, which achieves an overall quadratic improvement compared to direct imaging. arXiv:1801.02917v2 [quant-ph] 11 Jul 2018 where p k (n) = J j=1 ∂ ∂xj k p(n) xj =X is equal to the k-th order derivative of 0|ψX E(n)ψ †X |0 wrtX and M k are normalized moments defined by M k = I. INTRODUCTION Rayleigh's criterion, as a long-standing textbook theorem, puts a fundamental limit on the power of optical resolution [1,2]. It states that when two points are separated from each other by a distance smaller than the width of pointspread function (PSF) of the optical system, namely in the subdiffraction limit, it becomes essentially difficult to distinguish them. Recently however, researchers made a breakthrough towards surpassing Rayleigh's criterion using a new type of measurement technique, by looking at the imaging problem from the perspective of quantum metrology [3][4][5][6][7][8][9][10]. In metrology, Fisher information (FI) characterizes the ultimate precision of parameter estimation through Cramér-Rao bound [11][12][13]. When estimating the separation between two equal strength incoherent sources, it can be shown that FI tends to zero as they become closer when we use direct imaging approach (i.e. counting photons at different positions on the imaging plane). However, the quantum Fisher information (QFI, equal to the maximum FI over all possible quantum measurements) remains a constant, implying the possibility of superresolution [3]. In fact, many types of measurement have been proposed to achieve this kind of superresolution [3][4][5][6][14][15][16][17][18] and some of these approaches have already been demonstrated experimentally [19][20][21][22]. For example, when the PSF is Gaussian, it is possible to achieve the highest estimation precision by projecting the optical field onto Hermite-Gaussian modes [3,16,17]. While this new approach appears to be a promising candidate to substantially improve imaging resolution, many questions are yet to be answered: (1) What is the ultimate precision one can achieve, in a general imaging scenario, given experimentalists access to all types of measurement? (2) Which type of measurement achieves such precision? In this paper, we tackle these two questions by conducting a comprehensive Fisher information analysis in the general scenario where the incoherent source distribution on the object plane is arbitrary. A direct way to parametrize an image is to use positions and intensities of each point as parameters to be estimated. However, it may not be the perfect choice because the position of one specific point does not tell much about the structure of the whole image. Instead, we can use moments to characterize an image which has wide applications in image analysis [23]. Since the difficulty involved in calculating QFI increases significantly as the number of sources increase, we only consider the limiting values of QFIs as the size of the image tends to zero (much smaller than the width of PSF) which we call "the subdiffraction limit". In this paper, we choose normalized moments (normalized so that it has dimension of length) as parameters to be estimated, where detailed calculations for Gaussian PSFs and the spatial-mode demultiplexing (SPADE) measurement scheme are contained in Refs. [7,17]. We obtain the fundamental precision limit of estimating moments in the subdiffraction limit which formulated a modern description of Rayleigh's criterion, as opposed to the traditional Rayleigh's criterion restricted by direct imaging. We find that the FI with respect to (wrt) second moment remains a positive value in the subdiffraction limit, in accordance with previous work on estimating the separation between two coherent source. However, the FI wrt higher order moments always vanishes in the subdiffraction limit for nonadaptive measurements, answering question (1). This result shows the capability of going beyond direct imaging will not provide unlimited power and only push image resolution one step forward -from the first moment (the centroid of the image) to the second moment. To be specific, if we use s to represent the size of an image, the FI wrt to the K-th order moment vanishes as O(s K−2 ) (O(s K−1 )) when K is even (odd), compared with O(s 2K−2 ) using direct imaging. Based on the FI analysis, we also obtain optimal quantum measurements (in the subdiffraction limit) corresponding to the optimal FI. It is shown in this paper that when PSF is under certain constraints, the optimal measurement basis is strongly related to its derivatives. Roughly speaking, the probability from projecting the optical field onto the K-th order derivative of the PSF provides information of the 2K-th order moment of the image. And choosing derivatives as the measurement basis successfully classifies information of different moments into different measurement outcomes, which will provide optimal FIs wrt these moments in the subdiffraction limit. In this paper, we partially answer question (2) by first providing optimal quantum measurement scheme for second moment. For higher order moments, we prove the optimality of this scheme for 1D weak-source imaging. For 2D imaging or for strong-source imaging, such scheme only provides the optimal scaling of FI wrt s, but the coefficient may be further improved. II. SUMMARY OF RESULTS Here we briefly summarize our results on Fisher information analysis for incoherent optical imaging. • In Sec. III, we provide the formalism of the far-field imaging of incoherent optical sources, where we use P representation of optical states to express the Fisher information matrix (FIM). • In Sec. IV, we consider imaging for weak incoherent sources in one-dimensional imaging. We show that the Fisher information (FI) with respect to normalized moments decreases polynomially as the size of the image decrease, by order-of-magnitude analysis. To be specific, the FI wrt second moments remains a constant as the size of the image tends to zero, and the FI wrt to higher order moments drops to zero. • In Sec. V, we generalize the statement in Sec. IV to sources with arbitrary strength, again by order-of-magnitude analysis. • In Sec. VI, we detail the FI analysis wrt to second moments by providing the exact value of FI and corresponding optimal measurements, as FI wrt second moment is not influenced by Rayleigh's criterion. • In Sec. VII, we generalize all discussions about one-dimensional imaging to two-dimensional imaging, including calculating FI wrt to second moments in 2D. • In Sec. VIII, we detail the FI analysis wrt to all moments and show how the optimal scaling of FI can be achieved wrt all moments, which is improved quadratically when compared to direct imaging. Sec. VIII also serves as a justification of the order-of-magnitude analysis in Sec. IV and Sec. V. We also summarize the contents of each appendix here: • Appendix (A) discusses the condition under which the series expansion of probabilities and FIs. For a wellbehaved point spread function, the series expansion of probability converges uniformly and therefore the FIs can also be expanded wrt different orders of the size of the image. We also point out that our analysis can only be applied to non-adaptive measurements in order for the series expansion to be valid. • Appendix (B) provides the first three terms in the series expansion of measurement probability for arbitrary incoherent sources, which is not explicitly given in Sec. V. • Appendix (C) provides an alternative way to parametrize second moments in 2D imaging, as opposed to the one in Sec. VII. • Appendix (D), Appendix (E) and Appendix (F) complement discussions in Sec. VIII in terms of optimizing FI wrt odd moments for weak incoherent sources in 1D imaging, 2D imaging, generalization to arbitrary strengths. • Appendix (G) discusses the pre-estimation of the centroid. We provide a measurement scheme which is optimal for weak sources and at least 96.4% efficient for strong sources. The main results in this paper are also summaried in Table I and Table II for further reference. Weak source ( 1) Strong source (arbitrary ) Moments M k = j γ j (x j −X) k 1/k Eq. (15) Probability for outcome n P (n; {x j , Γ j }) = E[ ψα|E(n)|ψα ] Eq. (12) P (n; {x j , Γ j }) = (1 − ) 0|E(n)|0 + p(n) + O( 2 ) P (n; {x j , Γ j }) = ∞ k=0 Q k (n; {M , ≤ k}) Eq. (13) Eq. (21) p(n) = ∞ k=0 p k (n) k! (M k ) k Appendix (B) Eq. (14) FI F k = n 1 P (n;{x j ,Γ j }) ∂P (n;{x j ,Γ j }) ∂M k 2 = O(s k−2 ) Eq. (17) Maximum FI max {E(n)} F kk = O(s k−2 ) k is even, O(s k−1 ) is odd. max {E(n)} lim s→0 F 2 2 = q 2 (2 ) 2 (M 2 ) 2 −2 max {E(n)} lim s→0 F 22 = 4 ∆k 2 Eq. (55) max {E(n)} lim s→0 F 2 +1 2 +1 = 4 q 2 +1 (2 + 1) 2 (M 2 +1 ) 4 (M 2 ) 2 Eq. (31) Eq. (60) Optimal Measurement B w 0 , B w 1 and B w 2 For M 2 , E(N 0 ) = ∞ k=0 (ψ † X ) k ψ (1) † X |0 0|ψ (1) X (ψX ) k k! 0|ψ (1) X ψ (1) † X |0 Sec. VIII Eq. (32) & Appendix (F)Moments M k = j γ j (x j −X) k (y j −Ȳ ) 1/(k+ ) Eq. (33) Probability for outcome n P (n; {x j , y j , Γ j }) = E[ ψα|E(n)|ψα ] Eq. (12) P (n; {x j , y j , Γ j }) = (1 − ) 0|E(n)|0 + p(n) + O( 2 ) P (n; {x j , y j , Γ j }) = ∞ K=0 Q K (n; {M k , + k ≤ K}) Eq. (13) p(n) = ∞ k =0 p k (n) k! ! (M k ) k+ p k (n) = ∂ k X ∂ Ȳ 0|ψXȲ E(n)ψ †XȲ |0 FI F k k = n 1 P (n;{x j ,y j ,Γ j }) ∂P (n;{x j ,y j ,Γ j }) ∂M k 2 = O(s k+ −2 ) ,Eq. (37) Maximum FI max {E(n)} F L (K−L) L (K−L) = O(s K−2 ) k is even, O(s K−1 ) is odd. Sec. VIII & Appendix (E) Optimal Measurement B w 0,1,2,3,4,5,6 For M 20 , M 11 and M 02 , see Sec. VII. III. FORMALISM Consider a one-dimensional object composed of J points on the object plane. The original field on the object plane can be expressed using P representation [24], ρ 0 = DαP Γ0 (α) |α α| ,(1) where α = (α 1 , . . . , α J ) T is the column vector of complex field amplitudes for J optical spatial modes and |α = J j=1 e − |α j | 2 2 e αj a † j |0 ,(2) where |0 is the vacuum state, a † j and a j are the canonical creation and annihilation operators at position x j . Suppose the fields are uncorrelated at different points on the object plane, then P Γ0 (α) is the independent Gaussian distribution of the J modes: P Γ0 (α) = J j=1 1 π(Γ 0 ) j exp − J j=1 |α j | 2 (Γ 0 ) j ,(3) where (Γ 0 ) j ≥ 0 is the average photon number emitted at the jth point and Γ 0 = ((Γ 0 ) 1 , . . . , (Γ 0 ) J ) T . The imaging system maps the source operators a j , a † j into the image operators ψ j , ψ † j with an attenuation factor η: a † j → √ ηψ † j + 1 − ηv † j .(4) Here η is the transmission probability. ψ † j = dxψ PSF (x − x j )a † x is described by the point-spread function ψ PSF (x) (normalized) where a † x is the canonical creation operator at position x and v † j is the creation operator of the auxiliary environmental modes [6]. Moreover, we assume the PSF satisfies the following assumption ∞ −∞ d dx ψ * PSF (x) d +1 dx +1 ψ PSF (x) dx = 0, ∀ ≥ 0.(5) which will later be used in determining the optimal measurement basis. This assumption is easily satisfied, for example, when PSFs are real (real PSFs can be implemented by a two-lens system [25]), e.g. ψ PSF (x) ∝ e −x 2 /4σ 2 ; or when they are even, e.g. ψ PSF (x) ∝ e ikx 2 /2z sinc(x/σ). The field on the image plane is expressed as ρ = tr env DαP Γ0 (α) J j=1 e − |α j | 2 2 e αj ψ † j e √ 1−ηαj v † j |0 0| J j=1 e − |α j | 2 2 e αj ψj e √ 1−ηαj vj = DαP Γ (α) |ψ α ψ α | ,(6) where |ψ α = J j=1 e − |α j | 2 2 e αj ψ † j |0 0| J j=1 e −|αj | 2 e α * j ψj e αj ψ † j |0 1/2 ,(7) and Γ := ηΓ 0 is the average photon number received from each mode. We also define the average photon number on the image plane := J j=1 Γ j (which is usually a small number) and the relative source strength γ j := Γ j / for later use. We can see that after integrating all phases in α, only those photon number diagonal terms will survive and we may write ρ = ∞ m=0 π m ρ m(8) where π m is the probability of having m photons in the state and ρ m is an m-photon multimode Fock state. Our goal is to extract information of the image from ρ. We use a set of positive operators {E(n)} satisfying n E(n) = I to represent the positive-operator valued measure (POVM) performed on ρ [12,26]. The resultant probability distributions are P (n; {x j , Γ j }) = tr(ρE(n)) ≡ E[ ψ α |E(n)|ψ α ],(9) where E[·] represents expectation values under Gaussian distribution P Γ (α). The Cramér-Rao bound [11] Σ F −1 (10) provides the ultimate precision limit in terms of parameter estimation, where " " means the LHS minus the RHS is positive semi-definite, Σ k is the error covariance matrix wrt parameters {M k } k≥1 and F k = n 1 P (n; {x j , Γ j }) ∂P (n; {x j , Γ j }) ∂M k ∂P (n; {x j , Γ j }) ∂M(11) is the corresponding Fisher information matrix (FIM). M k are some functions of {x j , Γ j }, later chosen to be the normalized moments. IV. THE ULTIMATE RESOLUTION LIMIT FOR WEAK INCOHERENT SOURCES The probability of measurement outcome n is P (n; {x j , Γ j }) = E[ ψ α |E(n)|ψ α ] = E 0|e α † ψ E(n)e ψ † α |0 0|e α † ψ e ψ † α |0 ,(12) where ψ = (ψ 1 , . . . , ψ J ) T is the column vector of annihilation operators ψ j . In the limit where the average photon number on the image plane is small (the value of is considered known because it is easy to measure), we can expand it as a series in : P (n; {x j , Γ j }) = (1 − ) 0|E(n)|0 + p(n) + O( 2 ),(13) where p(n) := J j=1 γ j 0|ψ j E(n)ψ † j |0 . Since the first term contains no information of the object, the FIM will be dominated by the second term, which corresponds to the situation where only one photon is detected. To study the behavior of FIM in the subdiffraction limit, we expand ψ j around its centroidX. One should be careful with the convergence radius of the series expansion though, which has a lower bound independent of the measurement E(n) (see Appendix (A)). The second term in Eq. (13) becomes p(n) = ∞ k=0 p k (n) k! (M k ) k ,(14) example, if the object contains only two points, there are only three degrees of freedom -the positions of two points and the ratio of their strengths, then we choose the first three moments as the parameters to be measured. We use s = max i,j |x j − x i | to characterize the size of the image and conduct FI analysis in the subdiffraction limit when s → 0. Here we assume the centroid of the imageX = J j=1 γ j x j is known accurately either based on existing telescopic data or pre-estimation. In this case, we have M 1 = 0. In Appendix (G), we provide a measurement scheme for pre-estimation ofX. In 1D imaging, the scheme is optimal for weak sources and at least 96.4% efficient for strong sources. The methodology behind this scheme is not clear until Sec. VIII. Therefore we are not going to explain it in detail here. Since any converging power series is dominated by its first non-zero term as s → 0, we have ∂P (n; {x j , Γ j }) ∂M k = O(s k−1 ) and 1 P (n; {x j , Γ j }) ∂P (n; {x j , Γ j }) ∂M k = O(s −1 ).(16) Note that when the terms of lower order than k in P (n; {x j , Γ j } does not vanish, 1 P (n;{xj ,Γj }) ∂P (n;{xj ,Γj }) ∂M k should be bounded by a power of s with higher order than O(s −1 ). From Eq. (16), the FI for k ≥ 2 would be F kk = n 1 P (n; {x j , Γ j }) ∂P (n; {x j , Γ j }) ∂M k 2 = O(s k−2 ),(17) which indicates the following theorem: Theorem 1 (Modern Rayleigh's criterion for one-dimensional imaging): For imaging of incoherent point sources in the subdiffraction limit, the estimation variance of moment M k>2 increases inverse-polynomially as s decreases; meanwhile, the estimation variance of the second moment M 2 is bounded by a constant independent of s. Note that we only need to bound the diagonal element of the FIM because the variance in estimation M k satisfies Σ kk ≥ (F −1 ) kk ≥ F −1 kk .(18) where the equality holds true when F is diagonal. A simple schematic illustration of above theorem is shown in Fig. 1. Further justifications are contained in Sec. V, Sec. VI and Sec. VIII. We discuss the validity of this order-of-magnitude analysis in Appendix (A). We emphasize here that the measurement is assumed to be non-adaptive in this paper and our analysis does not include the case where measurement can be adaptively modified (Appendix (A)) assuming prior knowledge on the moments to be estimated. And the adaptive measurement is excluded because it requires demanding experimental techniques. A more general analysis through direct calculation of quantum Fisher information, which can be applied to all type of measurement, can be found in Ref. [27]. Consider two point sources with equal source strengths. The distance between them equal to 2M2 can be measured precisely, therefore it shall be easy to distinguish (a1) and (a2). (b) Images (b1) and (b2) have the same M2 but different M4. Consider four point sources with equal source strengths. It is difficult to estimate the third and higher moments to distinguish the two images from each other. V. THE ULTIMATE RESOLUTION LIMIT FOR INCOHERENT SOURCES WITH ARBITRARY STRENGTHS In this section, we generalize the above discussion in weak source limit to sources with arbitrary strengths. In Eq. (12), we replace ψ † α with its expansion J j=1 α j dxψ PSF (x − x j )a † x ≡ ∞ k=0 A (k) k! ψ (k) † X , where A (k) = J j=1 α j (x j −X) k and ψ (k) † X = d k dX k dxψ PSF (x −X)a † x .(19) According to Wick's theorem (Isserlis' theorem) [28], any moment of Gaussian distributions can be calculated using the values of second order moments E[A ( 1 ) A ( 2 ) * ] = J j=1 Γ j (x j −X) 1 + 2 ; E[A ( 1) A ( 2) ] = 0.(20) Here E[A ( 1) A ( 2) ] vanishes when integrating wrt phases of α. We observe that P (n; {x j , Γ j }) can be decomposed into a power series of O(s), like in Eq. (14), P (n; {x j , Γ j }) = ∞ k=0 Q k (n; {M , ≤ k}),(21) where Q k (n; {M , ≤ k}) is a function of the moments M with ≤ k so that Q k (n; {M }) = O(s k ). Explicit expressions of Q 0,1,2 (n) are provided in Appendix (B). For example, Q 0 (n) = ∞ k=0 k (1 + ) k+1 0|(ψX ) k E(n)(ψ †X ) k |0 ,(22) which is the probability of outcome n when all J points are located at the centroidX with thermal average 'excitation' number . Hence, we have shown that order-of-magnitude analysis is still valid. Specially, for 1, the expansion of Q k (n) depends solely on p k (n) and M k : Q 0 (n) = 0| E(n) |0 + O( ), Q k (n) = p k (n) k! (M k ) k + O( 2 ), ∀k ≥ 1,(23) and Eq. (21) simplifies to Eq. (13) for weak incoherent sources. We also notice that Q 2 (n)/Q 0 (n) = O( s 2 ) (see Appendix (B)), which means the subdiffraction limit (requiring Q 2 (n) Q 0 (n)) needs smaller s as increases. VI. FI WRT SECOND MOMENT AND CORRESPONDING OPTIMAL MEASUREMENT In Sec. IV, we have shown that there is a possibility to obtain a non-zero FI wrt M 2 . We are now going to find the exact value of the optimal FI wrt second moment and corresponding measurement basis. First, let's consider the weak-source scenario, F 22 = n 1 P (n; {x j , Γ j }) ∂P (n; {x j , Γ j }) ∂M 2 2 .(24) As s → 0, P (n; {x j , Γ j }) and ∂P (n;{xj ,Γj }) ∂M2 will be dominated by its first non-zero term, therefore according to Eq. (14), lim s→0 F 22 = n∈N w 0 1 p2(n) 2 (M 2 ) 2 p 2 (n)(M 2 ) 2 = 4 0|ψ (1) X E(N w 0 )ψ (1) † X |0 + Re[ 0|ψ (2) X E(N w 0 )ψ †X |0 ] ,(25) where we define a set of 0-null measurement outcomes N w 0 = {n| 0|E(n)|0 = 0|ψX E(n)ψ †X |0 = 0} and E(N w 0 ) = n∈N w 0 E(n). We also note that p 0 (n) = 0 implies p 1 (n) = 0. Since E(N w 0 ) is Hermitian and non-negative, its eigenstates corresponding to non-vanishing eigenvalues must be orthogonal to ψ †X |0 and Re[ 0|ψ (2) X E(N w 0 )ψ †X |0 ] must be zero. Therefore, max {E(n)} lim s→0 F 22 = 4 0|ψ (1) X ψ (1) † X |0 = 4 |∂ x ψ PSF (x)| 2 dx ≡ 4 ∆k 2 ,(26) where the first equality is achieved when ψ (1) † X |0 is an eigenstate of E(N w 0 ) with an eigenvalue equal to one. For example, E(N w 0 ) = ψ (1) † X |0 0| ψ (1) X 0|ψ (1) X ψ (1) † X |0(27) is optimal, in accordance with the optimality of the SPADE measurement scheme for Gaussian PSFs [3]. Furthermore, if ψ PSF (x) is an even function, its derivative will be odd and we can also choose E(N w 0 ) to be I−P 2 where P is the parity operator satisfying P · f (x) = f (−x), which is the so-called SLIVER measurement scheme [14]. This type of measurement does not depend on the specific expressions of the point-spread functions. We emphasize that above discussions are only applicable in the subdiffraction limit and the optimal measurement should be modified for finite s. When we consider the special case where there are only two equal strength point sources, however, Eq. (27) remains optimal even when s is large [3]. When we use direct imaging approach, i.e. {E(n)} = {a † x a x dx}, the 0-null measurement outcomes have zero measure and lim s→0 F 22 = 0, because the probability density of the photon position x is 0|ψX a † x a x ψ †X |0 = |ψ PSF (x −X)| 2 = 0 almost everywhere,(28) which explains the traditional Rayleigh's criterion. For an arbitrary source strength lim s→0 F 22 = n∈N0 1 Q 2 (n) ∂Q 2 (n) ∂M 2 2 ,(29) where the 0-null measurement outcome N 0 = {n|Q 0 (n) = ∞ k=0 k k!(1+ ) k+1 0|(ψX ) k E(n)(ψ †X ) k |0 = 0} = {n| 0|(ψX ) k E(n)(ψ †X ) k |0 0, ∀k}. We also note that Q 0 (n) = 0 implies Q 1 (n) = 0 (see Appendix (B)). A detailed calculation of Eq. (12) shows that when n ∈ N 0 , Q 2 (n) = ∞ k=0 k+1 k!(1 + ) k+1 0|(ψX ) k ψ (1) X E(n)ψ (1) † X (ψ †X ) k |0 M 2 2 ,(30) and hence max {E(n)} lim s→0 F 22 = 4 |∂ x ψ PSF (x)| 2 dx = 4 ∆k 2 .(31) It has the exact same expression as Eq. (26), meaning FI wrt the second moment grows linearly as the source strength grows, following the standard quantum limit [29]. Our results agree with previous work on estimating the separation between two incoherent sources for arbitrary source strengths [5,6]. The measurement is optimal when (ψ †X ) k ψ (1) † X |0 are all eigenstates of E(N 0 ) with eigenvalues equal to one. For example, E(N 0 ) = ∞ k=0 (ψ †X ) k ψ (1) † X |0 0| ψ (1) X (ψX ) k k! 0|ψ (1) X ψ (1) † X |0 (or when ψ PSF (x) is even, E(N 0 ) = 1 − P 2 )(32) is optimal, in accordance with the optimality of fin-SPADE and pix-SLIVER [5]. VII. GENERALIZATION TO TWO-DIMENSIONAL IMAGING Results in previous sections can be directly generalized to two-dimensional imaging. Suppose there are J point sources at positions (x j , y j ). The normalized moments are redefined as following: M k = J j=1 γ j (x j −X) k (y j −Ȳ ) 1 k+ (33) which fully characterizes the object configuration. Also, the size of the image s : = max ij (x i − x j ) 2 + (y i − y j ) 2 and the centroid (X,Ȳ ) := ( J j=1 γ j x j , J j=1 γ j y j ). We can expand the creation and annihilation operators around the centroid (∂ k X denotes the k-th order derivative wrtX) ψ † j = dxdyψ PSF (x − x j , y − y j )a † xy = ∞ k, =0 dxdy∂ k X ∂ Ȳ ψ PSF (x −X, y −Ȳ )a † xy k! ! (x j −X) k (y j −Ȳ ) ≡ ∞ k, =0 ψ (k ) † XȲ k! ! (x j −X) k (y j −Ȳ ) ,(34) and calculate the probability distribution P (n; {x j , y j , Γ j }) which is a series of O(s k ) P (n; {x j , y j , Γ j }) = ∞ K=0 Q K (n; {M k , k + ≤ K}).(35) Similar to Eq. (16), we have the following order-of-magnitude analysis: ∂P (n; {x j , y j , Γ j }) ∂M k = O(s k+ −1 ) and 1 P (n; {x j , y j , Γ j }) ∂P (n; {x j , y j , Γ j }) ∂M k = O(s −1 );(36) and similar to Eq. (17), the diagonal elements of the FI matrix is F k k = n 1 P (n; {x j , y j , Γ j }) ∂P (n; {x j , y j , Γ j }) ∂M k 2 = O(s k+ −2 ),(37) thus extending the modern description of Rayleigh's criterion to 2D imaging: Theorem 2 (Modern Rayleigh's criterion for two-dimensional imaging): For imaging of incoherent point sources in the subdiffraction limit, the estimation variance of any moment M k with k + > 2 increases inversepolynomially as s decreases; however, the estimation variance of the second moment M 20 , M 11 and M 02 are bounded by a constant independent of s. A simple schematic illustration above theorem is shown in Fig. 2. We are now going to find the exact values of FI wrt M 20 , M 11 and M 02 and corresponding optimal measurements. For simplicity we consider the weak source scenario. For arbitrary source strength, the FIs are still the same and the optimal measurements E(n) should be replaced with ∞ k=0 1 k! (ψ †X ) k E(n)(ψX ) k because Q 2 (n) = ∞ k=0 k+1 k!(1 + ) k+1 0| (ψX ) k ψ (10) XȲ E(n)ψ (10) † XȲ (ψ †X ) k |0 M 2 20 + 2Re[ 0| (ψX ) k ψ (10) XȲ E(n)ψ (01) † XȲ (ψ †X ) k |0 ]M 2 11 + 0| (ψX ) k ψ (01) XȲ E(n)ψ (01) † XȲ (ψ †X ) k |0 M 2 02 , (38) which is a generalization of Eq. (30) from 1D to 2D. Suppose [ψ (10) † XȲ , ψXȲ ] = [ψ (01) † XȲ , ψXȲ ] = 0 and 0|ψ (10) XȲ ψ (01) † XȲ |0 ∈ R. This assumption is satisfied, for example, when the PSF is real. The second order term of P (n; {x j , y j , Γ j }) is Q 2 (n) = 0| ψ (10) XȲ E(n)ψ (10) † XȲ |0 M 2 20 + 2Re[ 0| ψ(10)XȲ E(n)ψ (01) † XȲ |0 ]M 2 11 + 0| ψ (01) XȲ E(n)ψ (01) † XȲ |0 M 2 02 + O( 2 ).(39) We only consider 0-null measurement outcome n ∈ N w 0 = {n| 0|E(n)|0 = 0|ψXȲ E(n)ψ †XȲ |0 = 0, ∀k} because for n / ∈ N w 0 , the zeroth order term of P (n; {x j , y j , Γ j }) would be positive and does not contribute to the FI as s → 0. Furthermore, we assume E(n) = ΠE(n)Π where Π is the projection onto the space span{ψ (10) † XȲ |0 , ψ (01) † XȲ |0 } because any component of E(n) perpendicular to it does not contribute to Q 2 (n) in the first order expansion of and consequently only affects the value of the FI in higher order terms of . Then we can write every operator as a two-dimensional matrix in basis |e 1 = 1 2(1 + r) ψ (10) † XȲ ∆k x + ψ (01) † XȲ ∆k y |0 , |e 2 = 1 2(1 − r) ψ (10) † XȲ ∆k x − ψ (01) † XȲ ∆k y |0 ,(40) where ∆k 2 x := 0|ψ (10) XȲ ψ (10) † XȲ |0 = dxdy ∂ x ψ PSF (x, y) 2 , ∆k 2 y := 0|ψ (01) XȲ ψ (01) † XȲ |0 = dxdy ∂ y ψ PSF (x, y) 2 and r := 0|ψ (10) XȲ ψ (01) † XȲ |0 /(∆k x ∆k y ) = 1 ∆kx∆ky dxdy ∂ x ψ * PSF (x, y)∂ y ψ PSF (x, y) ∈ (−1, 1). Therefore, Q 2 (n) ≈ tr(E(n)ρ 2 ),(41) where ρ 2 = 1 2 (∆k 2 x M 2 20 + ∆k 2 y M 2 02 )(I + rσ z ) + 2∆k x ∆k y M 2 11 (rI + σ z ) + 1 − r 2 (∆k 2 x M 2 20 − ∆k 2 y M 2 02 )σ x .(42) Note that ρ 2 depends not only on the PSF via (∆k x , ∆k y , r) but also on the second moments. The FIM can be then be calculated using Q 2 (n) for any specific POVM {E(n)}. One way to parametrize the second moments is to define M 20 = X 2 , M 02 = Y 2 and M 11 = βXY , where X, Y is the standard deviation along x-and y-axis and β is the correlation between the distributions along x-and y-axis. (a) (b) (c) (d) If we approximate the image by a Gaussian distribution P (x, y) = √ 1−β 2 exp(− 1 2(1−β 2 ) (x y)C −1 (x y) T ), where the covariance matrix C =   M 2 20 M 2 11 M 2 11 M 2 02   =   X 2 βXY βXY Y 2   (43) the contour lines of P (x, y) will be ellipses described by x 2 X 2 + y 2 Y 2 − 2βxy XY = constant. Different distributions can be distinguished from each other if we can precisely estimate the values of (X, Y, β). Another way to parametrize the second moments is to use C =   M 2 20 M 2 11 M 2 11 M 2 02   =   cos θ − sin θ sin θ cos θ     Λ 2 1 0 0 Λ 2 2     cos θ sin θ − sin θ cos θ   ,(44) The major and minor length of the ellipses correspond to the square root Λ 1,2 of the eigenvalues of C and the orientation θ is associated with the direction of its eigenvectors. Estimation wrt (Λ 1 , Λ 2 , θ) is discussed in Appendix (C). First let's consider the singular case where β = 1, |M 2 11 | = M 2 20 M 2 02 and ρ 2 is pure. It happens when all points sources are aligned on the same line, e.g. when there are only two point sources [4]. The optimal measurement can be determined by calculating quantum Fisher information matrix (QFIM) wrt X and Y : J µν = tr( L µ L ν + L ν L µ 2 ρ 2 ), µ, ν = X, Y(45) where the Hermitian operator L µ is the symmetric logarithmic derivative of ρ 2 wrt µ defined via ∂ µ ρ 2 = 1 2 (L µ ρ 2 +ρ 2 L µ ) [13]. The QFIM derived from Eq. (42) is J [X, Y ] = 4   ∆k 2 x r∆k x ∆k y r∆k x ∆k y ∆k 2 y   .(46) The optimal measurement can be chosen to be any rank-one projection onto an orthonormal basis of the real space For 2D PSF satisifying the following more strict assumption (generalized from Eq. (5)): } where the parity operators P 1(2) satisfies P 1 (2) f (x, y) = f (−x, y) (f (x, −y)) (the same as 2D-SLIVER [4]). This type of measurement does not depend on the specific expressions of PSFs. In fact, any measurement E(n) = µν=+,− m µ,ν |e µ e ν | can be transformed into a PSF-independent version by replacing |e + e + | with When M 20 , M 11 and M 02 are indepedent parameters, β < 1, |M 2 11 | < M 2 20 M 2 02 and ρ 2 is a mixed state. The QFIM wrt (X, Y, β) is ∞ −∞ d 1 dx 1 d 2 dy 2 ψ * PSF (x, y) d 3 dx 3 d 4 dy 4 ψ PSF (x, y) dxdy = 0, when | 1 − 3 | = 1 or | 2 − 4 | = 1,(47)J [X, Y, β] = 4      ∆k 2 x 0 0 0 ∆k 2 y 0 0 0 ∆k 2 x ∆k 2 y X 2 Y 2 (∆k 2 x X 2 +∆k 2 y Y 2 )(1−β 2 )      .(48) However, the QFIM is not simultaneously achievable for (X, Y, β), meaning the quantum Cramér-Rao bound Σ J −1 it not attainable. The optimal measurement for (X, Y ) is {ΠE(n 1 )Π = |e + e + | , ΠE(n 2 )Π = |e − e − |} and the optimal measurement for β is {ΠE(n 1 )Π = |e 1 e 1 | , ΠE(n 2 )Π = |e 2 e 2 |}, where |e 1 = cos θ |e 1 + sin θ |e 2 ,(49)|e 2 = − sin θ |e 1 + cos θ |e 2 ,(50) and θ = 1 2 tan −1 β(X 2 −Y 2 ) 2XY . We note that when β = 0, the optimal measurement basis for (X, Y ) and β are mutually unbiased. In fact, any three parameters characterizing ρ 2 can never be measured simultaneously using projection-valued measurement (PVM) because ρ 2 is only rank two. In practice, we can switch between different types of measurements during the measurement process. The resultant FIM will be the average of FIMs wrt each measurement. VIII. ESTIMATION OF ALL MOMENTS IN THE SUBDIFFRACTION LIMIT Even though the information of normalized moments with an order higher than two is jeopardized in the subdiffraction limit, it is worth figuring out the maximum FI achievable and the optimal measurement corresponding to it as one may still need to measure the high-order normalized moments even when the FI is low and the estimation cost is expensive. In this section, we will assume all moments are inpedendent variables and we only consider weak source scenario here. Generalization to sources with arbitrary strengths is contained in the Appendix (F). Ref. [17] contains a detailed discussion on the special case where the source is weak and the PSF is Gaussian, but the optimality was not proved there. Eq. (5) For simplicity, let's first look at the one-dimensional case with weak sources ( 1), (The analysis for arbitrary source strengths is detailed in Appendix (F).) According to Eq. (16) and Eq. (17), the lowest power of s F k can attain is max{k, } − 2 if and only if there is an E(n) such that P (n; {x j , Γ j }) is zero until the min{k, }-th order of s. However, this condition is not necessarily satisfiable for each moments. In order for p 0 (n) = 0| ψX E(n)ψ †X |0 = 0, E(n) has to be orthogonal to ψ †X |0 (ψ †X |0 is not in the support of E(n)). Similarly, according to Eq. (30), in order for Q 2 (n) (up to the first order of ) to be zero, E(n) has to be orthogonal to ψ (1) † X |0 . We define -null measurement outcomes N w = {n| 0| E(n) |0 = 0| ψ (k) X E(n)ψ (k) † X |0 = 0, ∀k ≤ },(51) and we have N w ⊆ N w −1 for all , that is, -null measurement is ( − 1)-null. Then for all ≥ 0, Q 2 = O( 2 ) requires n ∈ N w . Suppose n ∈ N w −1 , then Q k (n; {M k , k ≤ k}) = O( 2 ), ∀k ≤ 2 − 1(52) and Q 2 (n; {M k , k ≤ 2 }) = ! 2 0|ψ ( ) X E(n)ψ ( ) † X |0 (M 2 ) 2 + O( 2 ).(53) We assume derivatives of the PSF {∂ k X ψ PSF (x −X), k ≥ 0} form a linear independent subset in L 2 (C). An orthonormal set {b (k) (x), k ≥ 0} can be generated via Gram-Schmidt process such that b ( ) (x) is orthogonal to every ∂ k X ψ PSF (x −X) with k ≤ − 1 and q := 1 ! b ( ) * (x)∂ X ψ PSF (x −X)dx ∈ R.(54) For example, when the PSF is Gaussian, {b (k) (x), k ≥ 0} are the Hermite-Gaussian modes; when the PSF is a sinc function, {b (k) (x), k ≥ 0} are the spherical Bessel functions of the first kind. We also notice that, according to Eq. (5), span{b (k) (x), k is even} = span{∂ k X ψ PSF (x −X), k is even}, span{b (k) (x), k is odd} = span{∂ k X ψ PSF (x −X) , k is odd} and they are orthogonal subspaces. Then F 2 2 is maximized when b ( ) † X |0 is an eigenstate of E(n) with an eigenvalue equal to one. The resultant FI is max {E(n)} F 2 2 ≈ q 2 (2 ) 2 (M 2 ) 2 −2 = O(s 2 −2 ).(55) For example when = 1, b (1) (x) = 1 ∆k ∂X ψ PSF (x −X) and Eq. (55) gives Eq. (26). We can show that it is possible for the FI to attain the lowest power of s (the highest precision) for even moments. To be specific, if k = 2 is even, by projecting quantum states on the image plane onto basis {b ( ) † X |0 , ≥ 0} (b ( ) † X = dxb ( ) (x)a † x ) , F kk is maximized and proportional to the (2 − 2)-th power of s, as indicated in Eq. (17). Moreover, according to the Cramér-Rao bound (Eq. (10)), Σ 2 2 ≥ (F −1 ) 2 2 ≥ (F 2 2 ) −1 .(56) The estimation precision of M 2 is lower bounded by the value of (F 2 2 ) −1 . Meanwhile, the choice of measurement basis {b ( ) † X |0 , ≥ 0} not only minimizes the value of (F 2 2 ) −1 but also makes F diagonal, which means that the second equality in Eq. (56) holds true. Therefore, we conclude that {b ( ) † X |0 , ≥ 0} is an optimal basis for estimation of even moments for weak incoherent sources. Note that {b ( ) † X |0 , ≥ 0} may not be a complete basis, but any POVM is optimal as long as it contains projections onto them and other terms E(n) contained in {E(n)} is irrelevant because they do not affect the FIM in the lowest order approximation. We do not write out the irrelevant part of POVM in our discussion. For odd moments, however, the above arguments do not apply. If we require n ∈ N w to satisfy Q 2 (n; {M k , k ≤ 2 }) = O( 2 ),(57) then E(n) is not supported by ψ (k) † X |0 for all k ≤ . Consequently, we have Q 2 +1 (n; {M k , k ≤ 2 + 1}) = 2 !( + 1)! Re[ 0|ψ ( ) X E(n)ψ ( +1) † X |0 ](M 2 +1 ) 2 +1 + O( 2 ) = O( 2 ),(58) which implies negligible contribution from weak sources. Therefore, in order to take odd moments into account, we need to relax Eq. (57) by choosing n ∈ N w −1 \N w +1 to keep the O( ) term in Q 2 +1 (n). The coefficient of (M 2 +1 ) 2 +1 can be non-zero when E(n) is supported by both ψ ( +1) † X |0 and ψ ( ) † X |0 . Meanwhile, Q 2 (n; {M k , k ≤ 2 }) = ! 2 0|ψ ( ) X E(n)ψ ( ) † X |0 (M 2 ) 2 + O( 2 )(59) would be non-zero at O( ) too. In the subdiffraction limit (s → 0), the denominator in Eq. (17) is dominated by Q 2 when n ∈ N w −1 \N w +1 . As shown in Appendix (D), we can maximize F 2 +1 2 +1 and in the meantime make the estimation of odd moments independent from the estimation of even moments (by letting F 2 +1 2 = F 2 2 +1 = O(s 2 )). Then analogous to Eq. (56), F 2 +1 2 +1 fully characterizes the estimation precision of M 2 +1 . It is maximized when E(n) are projections onto { b ( ) † X ±b ( +1) † X √ 2 |0 }. Up to the lowest order of s and , max {E(n)} F 2 +1 2 +1 ≈ 4 q 2 +1 (2 + 1) 2 (M 2 +1 ) 4 (M 2 ) 2 = O(s 2 ).(60) In the meantime, we can also calculate F 2 2 which is exactly its optimal value as in Eq. (55). Therefore, { b ( ) † X ±b ( +1) † X √ 2 |0 } achieves the optimal precision for both M 2 and M 2 +1 simultaneously. To conclude, we can use the following two subsets of measurement basis: B w 1 = { b ( ) † X ±b ( +1) † X √ 2 |0 , is even} and B w 2 = { b ( ) † X ±b ( +1) † X √ 2 |0 , is odd} (divided into two subsets so that they don't overlap) to estimate {M k |k = 4k or 4k + 1, k ≥ 1} and {M k |k = 4k + 2 or 4k + 3, k ≥ 0}, respectively. Each moment can be measured with the optimal precision and independently from other moments (the FIM is diagonal). However, each one of B w 1,2 can only extract half of the whole moment information: B w 1 estimates moments with orders equal to multiples of 4 plus 0 or 1; B w 2 estimates moments with orders equal to multiples of 4 plus 2 or 3. If one only needs to estimate even moments, B w 0 = {b ( ) † X |0 , ≥ 0} is optimal. Now let's consider the two-dimensional case. Similar to the one-dimensional case, we define N w K = { 0| E(n) |0 = 0| ψ (k ) X E(n)ψ (k ) † X |0 = 0, ∀k, , s.t. 0 ≤ k + ≤ K}. (61) Suppose n ∈ N w K−1 , the O(s 2K ) term in P (n; {x j , Γ j }) would be Q 2K (n; {M k , k + ≤ 2K}) = K , =0 ! !(K − )!(K − )! 0|ψ ( K− ) XȲ E(n)ψ ( K− ) † XȲ |0 (M ( + )(2K− − ) ) 2K + O( 2 ). (62) Q 2K is derived from Taylor expansion of Eq. (12). We notice that Q 2K can be written as E[ Ψ K | E(n) |Ψ K ] for some unnormalized state |Ψ K . Hence Q 2K is always non-negative and is equal to zero (up to the first order of ) if and only if n ∈ N w K . Based on the method of induction, we conclude that Q 2K = O( 2 ) if and only if n ∈ N w K . Therefore, by choosing proper measurement basis for n ∈ N w K−1 \N w K , one can estimate M L 2K−L with an FI up to O(s 2K−2 ) for all 0 ≤ L ≤ 2K. In general, the optimal measurement basis depends on the value of each moments. For M L 2K+1−L , consider the O(s 2K+1 ) term in P (n; {x j , Γ j }): Q 2K+1 (n; {M k , k + ≤ 2K + 1}) = K , =0 2 ! !(K − )!(K + 1 − )! Re[ 0|ψ ( K− ) XȲ E(n)ψ ( K+1− ) † XȲ |0 ](M ( + )(2K+1− − ) ) 2K+1 + O( 2 ). (63) Clearly, if n ∈ N w K , Q 2K+1 (n; {M k , k + ≤ 2K + 1}) = 0. Therefore we should focus on measurement E(n) such that n ∈ N w K−1 \N w K+1 . Similar to 1D imaging, the optimal scaling we can obtained for M L 2K+1−L is O(s 2K ). Again we assume derivatives of the PSF {∂ k X ∂ Ȳ ψ PSF (x −X, y −Ȳ ), k, ≥ 0} form a linear independent subset in L 2 (C). An orthonormal set {b (k ) (x), k ≥ 0} can be generated such that b (k ) (x) is orthogonal to every ∂ k X ∂ Ȳ ψ PSF (x− X, y −Ȳ ) with k + ≤ k + , (k, ) = (k , ) and q k := 1 k! ! b (k ) * (x)∂ k X ∂ Ȳ ψ PSF (x −X, y −Ȳ )dxdy ∈ R.(64) Suppose ψ PSF (x, y) is separable and ψ PSF (x, y) = ψ 1,PSF (x)ψ 2,PSF (y). One can generate two orthonormal sets {b b (k ) (x) = b (k) 1 (x)b ( ) 2 (x). Similar to 1D imaging, one can project ρ onto these basis to extract information of moments (see Table III) and achieve the optimal scaling of s (but not necessarily the optimal coefficients). As before, one type of measurement can only estimate part of all the moments (1/4 to be specific) and by combining different types of measurements one Types of measurement Measurement basis L Moments estimated can grasp information of all moments. In practice, combining {B w i } 6 i=1 will be enough to extract all the information of moments from ρ. For further justifications and calculations of FIs see Appendix (E). B w 0 b (L K−L) † XȲ |0 N M2L,2K−2L B w 1 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L−1) † XȲ ) |0 even M2L+1,2K−2L−1, B w 2 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L−1) † XȲ ) |0 odd (q 2 L,K−L (M2L,2K−2L) 2K + q 2 L+1,K−L−1 (M2L+2,2K−2L−2) 2K ) 1 2K B w 3 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L) † XȲ ) |0 even M2L+1,2K−2L, B w 4 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L) † XȲ ) |0 odd M2L,2K−2L B w 5 1 √ 2 (b (K−L L) † XȲ ± b (K−L L+1) † XȲ ) |0 even M2K−2L,2L+1, B w 6 1 √ 2 (b (K−L L) † XȲ ± b (K−L L+1) † XȲ ) |0 odd M2K−2L,2L In the case of sources with arbitrary strenghs, we show in Appendix (F) that the same scaling wrt s is still achievable by replacing every E(n) with k=0 1 k! (ψ †X ) k E(n)(ψX ) k (or k=0 1 k! (ψ †XȲ ) k E(n)(ψXȲ ) k for 2D imaging) which also give the same FIs as in the weak source scenario. However, the coefficient may be further improved using other basis, due to the fact that information of high order moments can be obtained by detecting several low order derivative operators simultaneously, which is neglectable when the source is weak. In contrast to estimation of the second moment, when estimating higher order moments, the optimal precision increases superlinearly (instead of linearly) as the source strength grows in the subdiffraction limit. IX. CONCLUSION We have performed a comprehensive Fisher information analysis on general imaging scenarios in the subdiffraction limit, where the improvement of image resolution is considered difficult due to the positive width of point spread functions. We conclude that, for any incoherence sources, a 1D or 2D image can be precisely estimated up to its second moment and the higher order moments are difficult to estimate in the sense that the error increase inversepolynomially as the size of image decreases. The imaging situation considered in the paper is quite general where both the number of point sources and source strengths can be arbitrary. The problem of pre-estimation of centroid is also worked out. For real point spread functions, we put forward a measurement scheme which provides the optimal Fisher information in the subdiffraction limit. The measurement basis is constructed based on the derivates of the point spread function, which are closely related to moments of an image. The optimal measurement scheme for second moment is discussed in detail. For higher order moments, compared with direct imaging approach, our measurement scheme guarantees at least a quadratic improvement of Fisher information in terms of the scaling wrt the size of the image. The coefficient of Fisher information is also optimal for weak sources, but can be further improved for strong sources. It is not clear, though, which measurement basis is optimal in terms of the exact value of Fisher information for strong sources. The generality of our results has a cost though -the Fisher information is only calculated in the limiting case where the size of the image tends to zero. Direct calculations for a positive size can be difficult and it remains unsolved how to identify the optimal measurement scheme when the size is not too small (in the subdiffraction limit) and also not too large (the point spread function can be viewed as a delta function). Our results, however, is an important theoretical result towards the ultimate resolution limit for incoherent optical imaging. Note added.-Recently, Ref. [27] appeared, which directly calculates the quantum Fisher information wrt moments for subdiffraction incoherent optical imaging. This approach can be applied to all types of measurements, without the non-adaptivity restriction in our analysis. Our results on arbitrary source strength, generalization to two-dimensional imaging and optimal scaling achieving measurement, however, are not covered in Ref. [27]. In this section, we justify the series expansion of the probability P (n; {x j , Γ j }) around its centroid. For simplicity, we only consider weak sources in 1D imaging. For single-photon measurement, P (n; {x j , Γ j }) = J j=1 γ j 0|ψ j E(n)ψ † j |0 . (A1) We want to know when the following series will converge uniformly to P (n; {x j , Γ j }): ∞ k=0 P k (n)(M k ) k ? = P (n; {x j , Γ j }),(A2) where P k (n) = k! ∂ k ∂x k j 0|ψ j E(n)ψ † j |0 xj =X .(A3) Let the radius of convergence R = (lim sup k→∞ |P k (n)| 1/k ) −1 , then Eq. (A2) converges uniformly as long as s < R [30]. Next we show that the radius of convergence R ≥ R 0 where R 0 independent of E(n). R 0 = sup ψ ( ) PSF ! 1/ −1 ,(A4) where ψ ( ) PSF represents the -th order derivative of f and ψ ( ) PSF = ∞ −∞ |ψ ( ) PSF (x)| 2 dx. Then R −1 = lim sup k→∞ |P k (n)| 1/k ≤ lim sup k→∞ k =0 1 !(k − )! ψ (k− ) PSF ψ ( ) PSF 1/k ≤ R −1 0 .(A5) Therefore when s < R 0 ≤ R, Eq. (A2) uniformly converges. For example, for a Gaussian PSF ψ PSF (x) = 1 (2πσ 2 ) 1/4 exp − x 2 4σ 2 . (A6) From +∞ −∞ e x 2 d dx e −x 2 2 dx = √ π !2 ,(A7) we see that R 0 ≥ σ from Eq. (A4). Therefore in the subdiffraction limit (s σ), the series expansion is always valid. However, things may break down when s > R 0 which may happen if ψ PSF (x) has complex sub-wavelength structure. When s < R 0 , the diagonal element of the Fisher information matrix is F kk = n 1 P (n; {x j , Γ j }) ∂P (n; {x j , Γ j }) ∂M k 2 = n P k (n)kM k−1 k 2 P k (n)M k k b 2 k a k ,(A8) where we assume P (n; {x j , Γ j }) P k (n)M k k = a k , and ∂P (n; {x j , Γ j })/∂M k P k (n)kM k−1 k = b k . (A9) Suppose b 2 k a k ≤ c k , we have F kk < n (P k (n)k |M k | k−1 ) 2 |P k (n)| |M k | k c k = c k k 2 |M k | k−2 n |P k (n)| = c k k 2 |M k | k−2 k! ∂ k ∂x k j 0|ψ j E(N + ) − E(N − ) ψ † j |0 xj =X ≤ 2c k k 2 k =0 !(k − )! ψ (k− ) PSF ψ ( ) PSF |M k | k−2 = O(s k−2 ),(A10) where N + = {n : P k (n) ≥ 0}, N − = {n : P k (n) < 0} and E(N ± ) = n∈N± E(n). The order-of-magnitude analysis above is valid only when c k = P k (n)(M k ) k ∞ k =0 P k (n)(M k ) k ∞ k =k P k (n) ∂(M k ) k ∂M k 2 P k (n)k(M k ) k−1 2 (A11) is reasonably small when s is small. We argue that this is usually true for non-adaptive measurements: • Consider first the case when P k (n)(M k ) k k >k P k (n)(M k ) k , then clearly c k ≈ P k (n)(M k ) k ∞ k =0 P k (n)(M k ) k · 1 1. (A12) • When P k (n)(M k ) k k >k P k (n)(M k ) k = O(s k+1 ), c k ≈ O(s k+1 ) |P k (n)k(M k ) k−1 | (A13) may be large. However, the contribution to F kk P k (n)kM k−1 k 2 P k (n)M k k c k = O(s k )(A14) is negligible. • When P k (n)(M k ) k ≈ k >k P k (n)(M k ) k and (when P k (n) = 0 for all k ≤ k) the first and second terms in P (n; {x j , Γ j }) = P k (n)(M k ) k + k >k P k (n)(M k ) k (A15) cancel each other out, up to the lowest order of s. Above analysis could become invalid. However, it requires a special design of measurement based on prior knowledge of the moments. We exclude this type of adaptive measurement in our discussion. Appendix B: First three terms in the series expansion of measurement probability for arbitrary incoherent sources We aim to expand P (n, {x j , Γ j }) in series of O(s k ) where s is the size of the image. To do this we replace ψ † α with ∞ k=0 A (k) k! ψ (k) † X in Eq. (12), where A (k) = J j=1 α j (x j −X) k and ψ (k) † X = d k dX k dxψ PSF (x −X)a † x . First of all, we calculate the value of denominator which gives 0|e α † ψ e ψ † α |0 = e dx| j αj ψPSF(x−xj )| 2 .(B1) Therefore, P (n, {x j , Γ j }) = E[e − dx| j αj ψPSF(x−xj )| 2 ∞ k=0 1 k! 2 0|(α † ψ) k E(n)(ψ † α) k |0 ]. (B2) The zeroth order term is Q 0 (n) = ∞ k=0 1 k! 2 E[e −|A (0) | 2 |A (0) | 2k ] 0|ψ k X E(n)(ψ †X ) k |0 = ∞ k=0 k k!(1 + ) k+1 0|ψ k X E(n)(ψ †X ) k |0 ,(B3) where we use E[e −|A (0) | 2 |A (0) | 2k ] = k! k (1+ ) k+1 . The first order term is Q 1 (n) = ∞ k=1 1 k! 2 (2k)E[(e −|A (0) | 2 )(A (0) * ) k−1 A (1) * (A (0) ) k ]Re[ 0|(ψX ) k−1 ψ (1) X E(n)(ψ †X ) k |0 ] = ∞ k=0 2 k+1 k!(1 + ) k+2 Re[ 0|(ψX ) k ψ (1) X E(n)(ψ †X ) k+1 |0 ]M 1 , (B4) where we use E[(e −|A (0) | 2 )(A (0) * ) k−1 A (1) * (A (0) ) k ] = k! k M1 (1+ ) k+1 . The second order term is Q 2 (n) = ∞ k=0 1 k! 2 E[(−e −|A (0) | 2 )(Re[A (0) * A (2) ] + |A (1) | 2 )|A (0) | 2k ] 0|ψ (1) X ψ (1) † X |0 0|(ψX ) k E(n)(ψ †X ) k |0 + ∞ k=1 1 k! 2 kE[(e −|A (0) | 2 )(A (0) * ) k−1 A (2) * (A (0) ) k ]Re[ 0|(ψX ) k−1 ψ (2) X E(n)(ψ †X ) k |0 ] + k(k − 1)E[(e −|A (0) | 2 )(A (0) * ) k−2 (A (1) * ) 2 (A (0) ) k ]Re[ 0|(ψX ) k E(n)(ψ (1) † X ) 2 (ψ †X ) k−2 |0 ] + k 2 E[(e −|A (0) | 2 )|A (0) | 2k−2 |A (1) | 2 ] 0|(ψX ) k−1 ψ (1) X E(n)ψ (1) † X (ψ †X ) k−1 |0 = ∞ k=0 k+1 k!(1 + ) k+1 ( 0|(ψX ) k ψ (1) X E(n)ψ (1) † X (ψ †X ) k |0 + Re[ 0|(ψX ) k ψ (2) X E(n)(ψ †X ) k+1 |0 ] − (k + 2) 0|ψ (1) X ψ (1) † X |0 0|(ψX ) k E(n)(ψ †X ) k |0 )M 2 2 + ∞ k=0 k+2 k!(1 + ) k+3 0|(ψX ) k+2 E(n)(ψ (1) † X ) 2 (ψ †X ) k |0 M 2 1 + ∞ k=0 k+1 (k − ) k!(1 + ) k+2 0|(ψX ) k ψ (1) X E(n)ψ (1) † X (ψ †X ) k |0 M 2 1 ,(B5)where we use E[(e −|A (0) | 2 )|A (0) | 2(k−1) |A (1) | 2 ] = (k−1)! k (M 2 2 −M 2 1 ) (1+ ) k + k! k M 2 1 (1+ ) k+1 , E[(e −|A (0) | 2 )|A (0) | 2(k−1) A (2) * A (0) ] = k! k M 2 2 (1+ ) k+1 and E[(e −|A (0) | 2 )(A (0) * ) k−2 (A (1) * ) 2 (A (0) ) k ] = k! k M 2 1 (1+ ) k+1 . Suppose the centroid is accurately known, we have M 1 = 0 and Q 1 (n) = 0. If we define N 0 = {n|Q 0 (n) = ∞ k=0 k k!(1+ ) k+1 0|(ψX ) k E(n)(ψ †X ) k |0 = 0} = {n| 0|(ψX ) k E(n)(ψ †X ) k |0 = 0, ∀k}. For n ∈ N 0 , Q 0 (n) = Q 1 (n) = 0 and only the first term in Q 2 (n) survives, which gives Eq. (30). The second term Re[ 0|(ψX ) k ψ (2) X E(n)(ψ †X ) k+1 |0 ] in Eq. (B5) vanishes for n ∈ N 0 because E(n) is Hermitian and non-negative and its eigenstates corresponding to non-vanishing eigenvalues must be orthogonal to (ψ †X ) k |0 for all k. intuition. The QFIM wrt (Λ 1 , Λ 2 , θ) calculated from Eq. (42) is J [Λ 1 , Λ 2 , θ] =      4 ∆k 2 0 0 0 4 ∆k 2 0 0 0 4 ∆k 2 (Λ 2 1 −Λ 2 2 ) 2 Λ 2 1 +Λ 2 2      . (C1) It is clear from Eq. (C1) that when Λ 1 = Λ 2 , the QFI is zero, which means when the image is circular-uniformly distributed (up to its second moment), we are not able to estimate θ in the subdiffraction limit. The corresponding optimal measurements found from the QFIM calculation are E(n 1 ) = (cos(θ + π/4) |e 1 − sin(θ + π/4) |e 2 )(cos(θ + π/4) e 1 | − sin(θ + π/4) e 2 |), E(n 2 ) = (sin(θ + π/4) |e 1 + cos(θ + π/4) |e 2 )(sin(θ + π/4) e 1 | + cos(θ + π/4) e 2 |). for estimation of (Λ 1 , Λ 2 ) and E(n 1 ) = (cos θ |e 1 − sin θ |e 2 )(cos θ e 1 | − sin θ e 2 |), E(n 2 ) = (sin θ |e 1 + cos θ |e 2 )(sin θ e 1 | + cos θ e 2 |). for estimation of θ. We note here that Eq. (C2) and Eq. (C3) are mutually unbiased. The measurement basis |Φ ± also leads to F 2 +1 2 = F 2 2 +1 = O(s 2 ) which means F is effectively diagonal and the second equality in the above inequality holds, because up to the lowest order of s we have Q 2 (n + ; {M k , k ≤ 2 }) = Q 2 (n − ; {M k , k ≤ 2 }),(D6)Q 2 +1 (n + ; {M k , k ≤ 2 + 1}) = −Q 2 +1 (n − ; {M k , k ≤ 2 + 1}),(D7)F 2 +1 2 ≈ E(n)=|Φ+ Φ+| E(n)=|Φ− Φ−| 1 Q 2 (n; {M k , k ≤ 2 }) ∂Q 2 +1 (n; {M k , k ≤ 2 + 1}) ∂M 2 +1 ∂Q 2 (n; {M k , k ≤ 2 }) ∂M 2 = 0. (D8) Appendix E: Measurement basis and corresponding FIs for weak incoherent sources in 2D imaging According to Eq. (62), by choosing measurement basis B w 0 = {b (L K−L) † XȲ |0 , ∀K ≥ 0, 0 ≤ L ≤ K} (E1) where b (k ) † XȲ = dxdyb (k) 1 (x −X)b ( ) 2 (y −Ȳ )a † xy , one can achieve the optimal scaling of s (but not necessarily the optimal coefficients) for FIs wrt M 2L 2K−2L for all K and L ≤ K: F 2L 2K−2L,2L 2K−2L | B w 0 ≈ q 2 L,K−L (2K) 2 (M 2L 2K−2L ) 2K−2 = O(s 2K−2 ).(E2) By choosing measurement basis B w 1 = { 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L−1) † XȲ ) |0 , ∀K ≥ 0, 0 ≤ L ≤ K − 1 is even} (E3) (or B w 2 = { 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L−1) † XȲ ) |0 , ∀K ≥ 0, 0 ≤ L ≤ K − 1 is odd}) , one can achieve the optimal scaling of s for FIs wrt M 2L+1 2K−(2L+1) for all K is even (or odd) and L < K: F 2L+1 2K−(2L+1),2L+1 2K−(2L+1) | B w 1,2 ≈ 4 (q 2 L,K−L (M 2L,2K−2L ) 2K + q 2 L+1,K−L−1 (M 2L+2,2K−2L−2 ) 2K )q 2 L,K−L q 2 L+1,K−L−1 (2K) 2 (M 2L+1 2K−2L−1 ) 4K−2 (q 2 L,K−L (M 2L,2K−2L ) 2K + q 2 L+1,K−L−1 (M 2L+2,2K−2L−2 ) 2K ) 2 − 4q 2 L,K−L q 2 L+1,K−L−1 (M 2L+1,2K−2L−1 ) 4K = O(s 2K−2 ). (E4) Meanwhile, (q 2 L,K−L (M 2L,2K−2L ) 2K + q 2 L+1,K−L+1 (M 2L+2,2K−2L−2 ) 2K ) 1 2K as a parameter can be estimated simultaneously with precision O(s −2K ) and independently of M 2L+1 2K−(2L+1) . Here we have used the property that b (k ) (x) is orthogonal to ∂ k X ∂ Ȳ ψ PSF (x −X, y −Ȳ ) as long as k and k (or and ) do not have the same parity (i.e. are not both even or odd). To conclude, B w 1,2 cover the estimation of moments whose orders on x-and y-axis are both even or both odd. The optimal FI scaling is O(s 2K−2 ) in this case, where 2K is the sum of orders on x-and y-axis. For moments who have different parities on x-and y-axis, we can use basis B w 3 = { 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L) † XȲ ) |0 , ∀K ≥ 0, 0 ≤ L ≤ K is even}; (E5) B w 4 = { 1 √ 2 (b (L K−L) † XȲ ± b (L+1 K−L) † XȲ ) |0 , ∀K ≥ 0, 0 ≤ L ≤ K is odd}; (E6) B w 5 = { 1 √ 2 (b (K−L L) † XȲ ± b (K−L L+1) † XȲ ) |0 , ∀K ≥ 0, 0 ≤ L ≤ K is even}; (E7) B w 6 = { 1 √ 2 (b (K−L L) † XȲ ± b (K−L L+1) † XȲ ) |0 , ∀K ≥ 0, 0 ≤ L ≤ K is odd}. (E8) Based on Eq. (63), we can calculate the following FIs (up to the lowest order of s): F 2L+1 2K−2L,2L+1 2K−2L | B w 3,4 ≈ 4 q 2 L+1,K−L (2K + 1) 2 (M 2L+1 2K−2L ) 4K (M 2L 2K−2L ) 2K = O(s 2K ); (E9) F 2K−2L 2L+1,2K−2L 2L+1 | B w 5,6 ≈ 4 q 2 K−L,L+1 (2K + 1) 2 (M 2K−2L 2L+1 ) 4K (M 2K−2L 2L ) 2K = O(s 2K ).(E10) and M 2L 2K−2L (M 2K−2L 2L ) can be estimated simultaneously and independently with M 2L+1 2K−2L (M 2K−2L 2L+1 ): F 2L 2K−2L,2L 2K−2L | B w 3,4 ≈ q 2 L,K−L (2K) 2 (M 2L 2K−2L ) 2K−2 = O(s 2K−2 ); (E11) F 2K−2L 2L,2K−2L 2L | B w 5,6 ≈ q 2 K−L,L (2K) 2 (M 2K−2L 2L ) 2K−2 = O(s 2K−2 ). (E12) which are exactly their optimal values as in Eq. (E2). Appendix F: Estimation of higher order moments with arbitrary source strengths Here we only consider 1D imaging, the discussion can be easily generalized to 2D imaging. As already shown in Sec. VI. Only 0-null measurement n ∈ N 0 = {n|Q 0 (n) = 0|(ψX ) k E(n)(ψ †X ) k |0 = 0, ∀k} contributes to the FI wrt M 2 . Using the method of induction, we define -null measurement N = {n | Φ| E(n) |Φ = 0, ∀ |Φ ∈ B (k) , k ≤ },(F1) where B ( ) = {( k ψ ( k ) † X ) |0 , ∀{ k ≥ 0, k ∈ N}, s.t. k k = }. When n ∈ N −1 , M 2 first apprears in Q 2 (n, {M k , k ≤ 2 }). Up to the lowest order of s, F 2 2 = n 1 P (n; {x j , Γ j }) ∂P (n; {x j , Γ j }) ∂M 2 2 ≈ n∈N −1 1 Q 2 (n, {M k , k ≤ 2 }) ∂Q 2 (n, {M k , k ≤ 2 }) ∂M 2 2 , (F2) where Q 2 (n, {M k , k ≤ 2 }) is the O(s 2 ) order term of P (n; {x j , Γ j }) = E 0|e α † ψ E(n)e ψ † α |0 0|e α † ψ e ψ † α |0 = E[e − dx| j αj ψPSF(x−xj )| 2 ∞ k=0 1 k! 2 0|(α † ψ) k E(n)(ψ † α) k |0 ].(F3) When n ∈ N −1 , Q 2 (n; {M k , k ≤ 2 }) has the following form: Q 2 (n; {M k , k ≤ 2 }) = 1 ! 2 ∞ k=0 k+1 k!(1 + ) k+1 0|(ψX ) k ψ ( ) X E(n)ψ ( ) † X (ψ †X ) k |0 (M 2 ) 2 + Q R 2 (n; {M k , k ≤ 2 − 1}),(F4) where the remainder term Q R 2 (n; {M k , k ≤ 2 − 1}) contains only moments with orders lower than 2 . We note that Q 2 (n, {M k , k ≤ 2 }) contains only terms like (summing over k ≥ max{K + 0 , K − 0 }) 1 ( + 1 ! · · · + m+ !)( − 1 ! · · · − m− !) 1 K + 1 ! · · · K + m+ !(k − K + 0 )! K − 1 ! · · · K − m− !(k − K − 0 )! E[e −|A (0) | 2 A (0) * k−K + 0 A ( + 1 ) * K + 1 · · · A ( + m + ) * K + m + A (0)k−K − 0 A ( − 1 )K − 1 · · · A ( − m − )K − m − ] 0|(ψX ) k−K + 0 (ψ ( + 1 ) X ) K + 1 · · · (ψ ( + m + ) X ) K + m + E(n)(ψ ( − 1 ) † X ) K − 1 · · · (ψ ( − m− ) † X ) K − m − (ψ †X ) k−K − 0 |0 , (F5) where K ± m , ± m ∈ N + , m = 1, . . . , m ± , K ± 0 = m± m =1 K ± m and = m± m =1 ± m . From Wick's theorem and Eq. (20), it is clear that the only term dependent on M 2 corresponds to K + 0 = K − 0 = 1, m ± = 1 and m± = . When m ± = 1 and m± = , we have E[e −|A (0) | 2 |A (0) | 2(k−1) |A ( ) | 2 ] = (k − 1)! k (M 2 ) 2 (1 + ) k + (k − 1)!(k − 1 − ) k M 2 (1 + ) k+1 ,(F6) proving Eq. (F4). Therefore, by choosing the modified measurement B 0 = ∞ k=0 1 k! (ψ †X ) k b ( ) † X |0 0| b ( ) X (ψX ) k , ∀ ≥ 0 ,(F7) Q R 2 (n; {M k , k ≤ 2 − 1}) = 0 and the same FI (Eq. (55)) wrt M 2 is recovered using the modified measurement B 0 . We can also modify other basis analogously by allowing multi-photon detection of ψ †X and it will provide the same HereM 1 andM 2 is redefined using X R as the centroid. According to Appendix (D), the optimal measurement in terms of estimatingM 1 =X − X R can be an arbitrary projection onto two orthonormal basis in the real span of {ψ † X R |0 , ψ (1) † X R |0 } as long as Q 0 (n) Q 1 (n) Q 2 (n) is satisfied. For example, E(n ± ) = ψ † X R ± 1 ∆k ψ (1) † X R √ 2 |0 0| ψ X R ± 1 ∆k ψ (1) X R √ 2 (G2) is optimal. The corresponding FI is F 11 = 4 ∆k 2 . (G3) which is the same as Eq. (26). Therefore, if we want to estimate both the second moment M 2 and the centroidX, a straightforward method is to first use half of the whole resource to locateX such that δX s and then use the rest half to estimate M 2 as described in Sec. VI. The effective FIM would be half of the optimal ones Eq. (G3) and Eq. (26), F(M 1 , M 2 ) =   2 ∆k 2 0 0 2 ∆k 2   ,(G4) When we want to estimate even higher order moments, the resource required to locateX is neglectable. Now we show Eq. (G4) is the optimal precision we can get (in the subdiffraction limit) and the QFIM Eq. (G5) is not achievable. For any POVM {E(n)}, the only case when P (n, {x j , Γ j }) does not lead to a zero-FI wrt M 2 is when there is an E(n) such that P (n, {x j , Γ j }) ≈ A 0 (n) + A 1 (n)M 1 + A 2 (n)(M 2 2 +M 2 1 ) = O(s 2 ) (G6) where A 0 (n) = 0| ψ X R E(n)ψ † X R |0 = O(s 2 ),(G7)A 1 (n) = 2 Re[ 0| ψ (1) X R E(n)ψ † X R |0 ] = O(s),(G8)A 2 (n) = ( 0| ψ (1) X R E(n)ψ (1) † X R |0 + 2Re[ 0| ψ (2) X R E(n)ψ † X R |0 ]) = O(1),(G9) and we use the relationM 2 2 = M 2 2 +M 2 1 . Note that A 2 (n) ≈ 0| ψ (1) X R E(n)ψ (1) † X R |0 , because Re[ 0| ψ (2) X R E(n)ψ † X R |0 ] = O(s) can be negelected. Since 1 P (n, {x j , Γ j }) ∂P (n, {x j , Γ j }) ∂M 1 2 = (A 1 (n) + 2A 2 (n)M 1 ) 2 A 0 (n) + A 1 (n)M 1 + A 2 (n)(M 2 2 +M 2 1 ) ,(G10)1 P (n, {x j , Γ j }) ∂P (n, {x j , Γ j }) ∂M 2 2 = 4A 2 (n) 2 M 2 2 A 0 (n) + A 1 (n)M 1 + A 2 (n)(M 2 2 +M 2 1 )(G11) and A 1 (n) 2 ≤ 4A 2 (n)A 0 (n), we have 1 P (n, {x j , Γ j }) ∂P (n, {x j , Γ j }) ∂M 2 2 + 1 P (n, {x j , Γ j }) ∂P (n, {x j , Γ j }) ∂M 1 2 4A 2 (n) ≈ 4 0| ψ (1) X R E(n)ψ (1) † X R |0 . (G12) When P (n, {x j , Γ j }) is dominated by Q 0 (n), we also have 1 P (n, {x j , Γ j }) ∂P (n, {x j , Γ j }) ∂M 2 2 + 1 P (n, {x j , Γ j }) ∂P (n, {x j , Γ j }) ∂M 1 2 ≈ A 1 (n) 2 A 0 (n) ≤ 4 0| ψ (1) X R E(n)ψ (1) † X R |0 . (G13) Therefore, any achievable FIM must satisfies F 11 + F 22 ≤ 4 n 0| ψ (1) X R E(n)ψ (1) † X R |0 = 4 ∆k 2 ,(G14) and (δM 1 ) 2 + (δM 2 ) 2 = tr(Σ) ≥ tr(F −1 ) ≥ 2 i=1 F −1 ii ≥ 4 tr(F) ≥ 1 ∆k 2 .(G15) Clearly the last three equalities are simultaneously satisfied when FIM is Eq. (G4), implying the optimality of our measurement scheme. The situation becomes a bit more complicated for arbitrary source strengths. First, we expand P (n; {x j , Γ j }) around X R up to O(s) P (n; {x j , Γ j }) ≈ Q 0 (n) + Q 1 (n) = ∞ k=0 k k!(1 + ) k+1 0|ψ k X R E(n)(ψ † X R ) k |0 + ∞ k=0 2 k+1 k!(1 + ) k+2 Re[ 0|ψ k X R ψ (1) X R E(n)(ψ † X R ) k+1 |0 ]M 1 . (G16) Since the quantum state is photon number diagonal, the optimal measurement estimatingM 1 must also be photon number diagonal [13], that is, {E(n)} should contains {E(n k ), k ≥ 1} where E(n k ) = Π k E(n k )Π k and Π k is projection onto k-photon subspace. In this case, we shall write F 11 = ∞ k=0 {E(n k )} 2 k+1 k!(1+ ) k+2 Re[ 0|ψ k X R ψ (1) X R E(n k+1 )(ψ † X R ) k+1 |0 ] 2 k+1 (k+1)!(1+ ) k+2 0|ψ k+1 X R E(n k+1 )(ψ † X R ) k+1 |0 ≤ 4 ∆k 2 ,(G17) where the equality holds when {E(n k )} is an arbitrary projection onto two orthonormal basis in the real span of {(ψ † X R ) k−1 ψ (1) † X R |0 , (ψ † X R ) k |0 } as long as Q 0 (n) Q 1 (n) Q 2 (n) is satisfied. For example, E(n k,± ) = 1 2 1 √ k! (ψ † X R ) k ± 1 ∆k (k − 1)! ψ (1) † X R (ψ † X R ) k−1 |0 0| 1 √ k! ψ k X R ± 1 ∆k (k − 1)! ψ (1) X R ψ k−1 X R (G18) is optimal. Therefore, if we want to estimate both the second moment M 2 and the centroidX =M 1 + X R , a straightforward method is to first use half of the whole resource to locateX such that δX ≤ s and then use the rest half to estimate M 2 as described in Sec. VI. Note that to achieve the optimal precision wrt M 1 , one has to count the number of detected photons by projecting the quantum state ontõ The resource required to locateX when we want to estimate even higher order moments is still neglectable as in the weak source scenario. Now we consider the possiblity of further improving Eq. (G20), here we show that above scheme is at least 96.4% efficient. According to Appendix (B), we have, up to O(s 2 ), P (n, {x j , Γ j }) = A 0 (n) + A 1 (n)M 1 + A 2 (n)M 2 2 + A 3 (n)M 2 1 . B = 1 2 1 √ k! (ψ † X R ) k ± 1 ∆k (k − 1)! ψ (1) † X R (ψ † X R ) k−1 |0 0| 1 √ k! ψ k X R ± 1 ∆k (k − 1)! ψ (1) X R ψ k−1 X R , k ≥ 1( For different measurement outcome n, there are only two situations: • P (n, {x j , Γ j }) = O(s 2 ), then A 0 (n) = ∞ k=0 k+1 (k + 1)!(1 + ) k+2 0|ψ k+1 X R E(n)(ψ † X R ) k+1 |0 ,(G23)A 1 (n) = ∞ k=0 2 k+1 k!(1 + ) k+2 Re[ 0|ψ k X R ψ (1) X R E(n)(ψ † X R ) k+1 |0 ],(G24)A 2 (n) = ∞ k=0 k+1 k!(1 + ) k+1 0|ψ k X R ψ (1) X R E(n)ψ (1) † X R (ψ † X R ) k |0 ,(G25)A 3 (n) = ∞ k=0 (k + 1) k+1 k!(1 + ) k+2 0|ψ k X R ψ (1) X R E(n)ψ (1) † X R (ψ † X R ) k |0 .(G26) Other terms can be ignored in the subdiffraction limit. • P (n, {x j , Γ j }) = O(1), then A 0 (n) = ∞ k=0 k+1 (k + 1)!(1 + ) k+2 0|ψ k+1 X R E(n)(ψ † X R ) k+1 |0 ,(G27)A 1 (n) = ∞ k=0 2 k+1 k!(1 + ) k+2 Re[ 0|ψ k X R ψ (1) X R E(n)(ψ † X R ) k+1 |0 ],(G28) and A 2 (n) and A 3 (n) can be ignored in the subdiffraction limit. For simplicity we can assume Eq. (G25) and Eq. (G26) are also true. One important property derived from this relation is that n A 2 (n) = n A 3 (n) = ∆k 2 . The entries of the FIM are where M 1 =M 1 + A 1 (n)/(2A 3 (n)) and A 0 (n) = A 0 (n) − A 1 (n) 2 /(4A 3 (n)) ≥ 0. We define another 2-by-2 matrix F by replacing all A 0 (n) above with 0. Clearly, tr(F −1 ) ≥ tr(F −1 ) because F F. Using Eq. (G29), we have (G35) F 11 + M 2 M 1 F 12 = F 22 + M 1 M 2 F 12 = 4 ∆k 2 ,(G33) We conclude that our measurement scheme is at least 1+e 4 ≈ 96.4% efficient for arbitrary in the sense that if one achieve certain estimation precision (δM 1 ) 2 + (δM 2 ) 2 by repeating our measurement N times, the optimal measurement scheme requires at least 96.4% · N times to achieve such precision. We can easily generalize above measurement scheme to 2D imaging when the PSF is separable. ψ 1 k! (ψ † X R Y R ) k ψ † X R Y R ± 1 ∆kx ψ (10) † X R Y R √ 2 |0 0| ψ X R Y R ± 1 ∆kx ψ (10) X R Y R √ 2 (ψ X R Y R ) k (G37) and 1 k! (ψ † X R Y R ) k ψ † X R Y R ± 1 ∆ky ψ (01) † X R Y R √ 2 |0 0| ψ X R Y R ± 1 ∆ky ψ (01) X R Y R √ 2 (ψ X R Y R ) k (G38) with optimal FIs equal to F 10 10 = 4 ∆k 2 x , F 01 01 = 4 ∆k 2 y . We won't discuss simultaneous estimation of the centroidM 10 ,M 01 and the second moments M 20 , M 11 , M 02 here. Figure 1 . 1(a) Images (a1) and (a2) have different M2. Figure 2 .. 2Three point sources with equal source strengths. Given the values of (M20, M11, M02), (a) Images (a1) and (a2) are distinguishable due to different standard deviations along x-axis X = M20 (b) Images (b1) and (b2) are distinguishable due to different standard deviations along y-axis Y = M02. (c) Images (c1) and (c2) are distinguishable due to different x-y correlations β = M 2 11 /(M20M02). (d) Images (d1) and (d2) have the same (M20, M11, M02). It is difficult to distinguish them from each other. spanned by {|e 1 , |e 2 }, such as {E(n 1 ) = |e 1 e 1 | , E(n 2 ) = |e 2 e 2 |} (the same as 2D-SPADE for Gaussian PSFs [4]) or {E(n 1 ) = |e + e + | , E(n 2 ) = |e − e − |} where |e + = 1 ∆kx ψ (10) † XȲ |0 and |e − = 1 ∆ky ψ (01) †XȲ |0 , because they will always satisfy the QFIM-achievable condition E(n i i = 1, 2, µ = X, Y and some real constant c i,µ . the FIM and corresponding measurement can be obtained in a simpler form (otherwise, the FIM can have offdiagonal terms). Eq. (47) is still quite general. When ψ PSF (x, y) is separable, i.e. ψ PSF (x, y) = ψ 1,PSF (x)ψ 2,PSF (y), Eq. (47) is automatically satisifed when ψ 1,PSF (x) and ψ 2,PSF (y) satisify Eq. (5), e.g. ψ PSF (x, y) ∝ e −(x 2 +y 2 )/4σ 2 or ψ PSF (x, y) ∝ e ik(x 2 +y 2 )/2z sinc(x/σ 1 )sinc(y/σ 2 ). When ψ PSF (x, y) is a circularly-symmetric function, i.e. ψ PSF (x, y) = ψ PSF ( x 2 + y 2 ), Eq. (47) is also true, e.g. ψ PSF (x, y) ∝ e ik(x 2 +y 2 )/2z J1 √ x 2 +y 2 /σ √ x 2 +y 2 (J 1 (·) is the first order Bessel function of the first kind). We assume from now on that Eq. (47) is satisfied for any 2D PSF. In this case, r = 0.Note that if the projection {ΠE(n)Π} of measurements {E(n)} onto the complex space spanned by {|e 1 , |e 2 } is optimal, {E(n)} is also optimal. In particular, when ψ PSF (x, y) is circularly symmetric, any measurement satisfying {ΠE(n 1 )Π = |e + e + | , ΠE(n 2 )Π = |e − e − |} is optimal, including {E(n 1 ) = (I−P1) , |e + e − | with (I−P1)(I+P2) 4 S 12 , |e − e + | with S 12 (I−P1)(I+P2) 4 and |e − e − | with (I+P1)(I−P2) 4where S 12 f (x, y) = f (y, x). and Eq. (47) are satisfied for PSFs in this section and the main result in this section can be summarized in this theorem: Theorem 3 (Optimal precision scaling wrt all moments): For any moment M K (M L (K−L) ) in 1D (2D) imaging with arbitrary source strengths, the estimation variance is at least O(s 2−K ) when K is even and O(s 1−K ) when K is odd. For directly imaging, the denominator in Eq. (17) and Eq. (37) are always O(1) and the Fisher information wrt M K or M L (K−L) will be O(s 2K−2 ) which is O(s K ) (O(s K−1 )) times smaller than the maximum FI O(s K−2 ) (O(s K−1 )) for even (odd) moments we obtain here. 2) (x), ∀k ≥ 0} via Gram-Schmidt process from the derivatives of ψ 1(2),PSF (x) as in 1D imaging. Then we have Eq. (32) to estimate M 2 where we don't need to count the number of photons. Similar to the weak soure scenario, this measurement scheme provides an effective FIM which is half of the optimal ones Eq. (G17) and Eq. (31), F(M 1 , M 2 ) = X R Y R and ψ (01) † X R Y R are orthogonal. As in 1D imaging,M 10 =X − X R ,M 01 =Ȳ − Y R (G36)are estimated by Table I . IA summary of the main results (1D)Weak source ( 1) Strong source (arbitrary ) Table III IIIAppendix (F) Table II . IIA summary of the main results (2D) Table III . IIIMeasurement basis and corresponding moments in 2D imaging, details in Appendix (E). andF 11 + F 22 ≤ 4 Therefore (δM 1 ) 2 + (δM 2 ) 2 ≥ tr(F −1 ) ≥ tr(F −1 ) = F 11 + F 22 F 11 F 22 − F 2n max{A 2 (n), A 3 (n)} ≤ 4∆k 2 k=0 k+1 (1 + ) k+1 + ∞ k= +1 (k + 1) k+1 (1 + ) k+2 = 4∆k 2 + + 1 1 + +2 ≤ 4(1 + 1 e ) ∆k 2 . (G34) 12 = 1 4 ∆k 2 1 − 4 ∆k 2 F 11 +F 22 ≥ 1 + e 4 1 ∆k 2 . ACKNOWLEDGMENTSWe acknowledge support from the ARL-CDQI (W911NF-15-2-0067), ARO (W911NF-14-1-0011, W911NF-14-1-0563), ARO MURI (W911NF-16-1-0349), AFOSR MURI (FA9550-14-1-0052, FA9550-15-1-0015), NSF (EFMA-1640959), the Alfred P. Sloan Foundation (BR2013-049), and the Packard Foundation (2013-39273).Appendix C: An alternative way to parametrize second moments in 2D imagingHere we calculation the optimal FIM wrt (Λ 1 , Λ 2 , θ) as defined in Sec. VII. We only consider the situation where ∆k x = ∆k y = ∆k and r = 0 as the form of FIM becomes quite complicated otherwise and provides no physical Appendix D: Optimization of FI wrt odd moments for weak incoherent sources in 1D imaging Up to the lowest order of s and ,First we note that, in order to maximize F 2 +1 2 +1 , we can assume E(n) is a rank-one projector for each n, because for anyaccording to Cauchy-Schwarz inequality. Therefore deviding any POVM into corresponding projective measurements will only increase FI. Furthermore, if E(n) = |Φ n Φ n |,We can, for example, choose the measurement basis to be(D4)Here we use the property that b ( ) (x) is orthogonal to ∂ +1 X ψ PSF (x −X) (based on Eq.(5)). Moreover, according to Eq. (10),expression of FIs as in the weak source scenario. Note that here each component of B 0 is not a POVM but a PVM because we don't need to distinguish the number of ψ †X photon we detect. However if we choose to distinguish them, that is, usingthe FI will be no smaller (easily proven using Cauchy-Schwarz inequality) and the FIM is still effectively diagonal. However, even B 0 is not optimal when estimating M 2 . Physically, the reason is that the information of high order moments can be obtained by detecting several low order derivative operators simultaneously, which is neglectable when the source is weak. For = 1, the only lower order moment is M 1 = 0, therefore strong source strength does not make a difference when calculating the FI, as shown in Sec. VI.We provide a simple example showing B 0 is not an optimal measurement basis by replacing it with a better basis. Consider = 4 (and we want to estimate the value of M 2 = M 8 ). Suppose s > 0. For simplicity, we only consider the replacement in 2-photon subspace, i.e. we don't change any k + 1-photon basis in B 0 with k = 1 and their contributions to F 88 will remain the same. For 2-photon subspace, we consider the possiblity of choosing another basis in B 4,whereWe can easily find a non-trivial image such that A 44 A 2 2 2 2 − A 2 42 2 > 0, then we maximize F 88 in this 2-dimensional subspace by doing QFI calculation, which givesNow we've proven b 4 b 4 † does not generate the maximum FI wrt M 8 and B 0 is not optimal. Meanwhile, we also note that the FIM is effectively diagonal in the subdiffraction limit, thus F 88 fully characterizes the measurement precision of M 8 . In general, any non-zero off-diagonal term (A 42 2 in this case) in the same photon number subspace would lead to the same result. It means the precision of high order moments estimation could be enhanced by utilizing the detection of several low order derivative operators simultaneously.For odd moments M 2 +1 ( ≥ 1), suppose n ∈ N −1 \N +1 , we haveThe modified measurementalso leads to the same FI Eq. (60), as for the even moments.Appendix G: Pre-estimation of the centroidThe procedure to estimate the centroid can be divided into two step: (1) Find a reference point X R such that X − X R s; (2) Precisely locateX within the subdiffraction limit. The resource required for step (1) is neglectable (it's a coarse estimation) and we only consider the resource required for step(2). Normally, to fully resolve an image, we need to achieve a degree of precision where δM k s (k ≥ 2) and here we analyze the resource required to achieve δX s so that it won't induce a significant error in the estimation of higher other moments. We first consider 1D weak source scenario. 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{'abstract': "Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a oneor two-dimensional image can be precisely resolved up to its second moment in the subdiffraction limit, i.e. the Fisher information (FI) is non-zero. But the FI with respect to higher order moments always tends to zero polynomially as the size of the image decreases, for any type of non-adaptive measurement. We call this phenomenon a modern description of Rayleigh's criterion. For PSFs under certain constraints, the optimal measurement basis estimating all moments in the subdiffraction limit for 1D weak-source imaging is constructed. Such basis also generates the optimal-scaling FI with respect to the size of the image for 2D or strong-source imaging, which achieves an overall quadratic improvement compared to direct imaging. arXiv:1801.02917v2 [quant-ph] 11 Jul 2018 where p k (n) = J j=1 ∂ ∂xj k p(n) xj =X is equal to the k-th order derivative of 0|ψX E(n)ψ †X |0 wrtX and M k are normalized moments defined by M k =", 'arxivid': '1801.02917', 'author': ['Sisi Zhou ', 'Liang Jiang ', '\nDepartments of Applied Physics and Physics\nYale University\n06511New HavenConnecticutUSA\n', '\nYale Quantum Institute\nYale University\n06520New HavenConnecticutUSA\n'], 'authoraffiliation': ['Departments of Applied Physics and Physics\nYale University\n06511New HavenConnecticutUSA', 'Yale Quantum Institute\nYale University\n06520New HavenConnecticutUSA'], 'corpusid': 119232415, 'doi': '10.1103/physreva.99.013808', 'github_urls': [], 'n_tokens_mistral': 30905, 'n_tokens_neox': 27607, 'n_words': 15551, 'pdfsha': '008b2a4a408983e31e62b28dacba56e3174baee5', 'pdfurls': ['https://arxiv.org/pdf/1801.02917v2.pdf'], 'title': ["A modern description of Rayleigh's criterion", "A modern description of Rayleigh's criterion"], 'venue': []}

arxiv

The hot Neptune WASP-166 b with ESPRESSO III: A blue-shifted tentative water signal constrains the presence of clouds 2022 M Lafarga Department of Physics University of Warwick Gibbet Hill RoadCV4 7ALCoventryUnited Kingdom Centre for Exoplanets and Habitability University of Warwick CV4 7ALCoventryUK M Brogi Department of Physics University of Warwick Gibbet Hill RoadCV4 7ALCoventryUnited Kingdom INAF -Osservatorio Astrofisico di Torino Via Osservatorio 2010025 Pino Torinese Italy Dipartimento di Fisica Università degli Studi di Torino via Pietro Giuria 1I-10125TorinoItaly S Gandhi Department of Physics University of Warwick Gibbet Hill RoadCV4 7ALCoventryUnited Kingdom Centre for Exoplanets and Habitability University of Warwick CV4 7ALCoventryUK Leiden Observatory Leiden University 9513, 2300 RAPostbus, LeidenThe Netherlands H M Cegla Department of Physics University of Warwick Gibbet Hill RoadCV4 7ALCoventryUnited Kingdom Centre for Exoplanets and Habitability University of Warwick CV4 7ALCoventryUK † J V Seidel European Southern Observatory Alonso de Córdova 3107VitacuraRegión MetropolitanaChile L Doyle Department of Physics University of Warwick Gibbet Hill RoadCV4 7ALCoventryUnited Kingdom Centre for Exoplanets and Habitability University of Warwick CV4 7ALCoventryUK R Allart Department of Physics Trottier Institute for Research on Exoplanets Université de Montréal H3T 1J4MontréalCanada N Buchschacher Observatoire Astronomique de l'Université de Genève Chemin Pegasi 51bCH-1290VersoixSwitzerland M Lendl Observatoire Astronomique de l'Université de Genève Chemin Pegasi 51bCH-1290VersoixSwitzerland C Lovis Observatoire Astronomique de l'Université de Genève Chemin Pegasi 51bCH-1290VersoixSwitzerland D Sosnowska Observatoire Astronomique de l'Université de Genève Chemin Pegasi 51bCH-1290VersoixSwitzerland The hot Neptune WASP-166 b with ESPRESSO III: A blue-shifted tentative water signal constrains the presence of clouds MNRAS 0002022Accepted XXX. Received YYY; in original form ZZZPreprint 10 February 2023 Compiled using MNRAS L A T E X style file v3.0instrumentation: spectrographs -methods: observational -techniques: spectroscopic -exoplanets -Planets and satellites: atmospheres -planets and satellites: individual: WASP-166 b With high-resolution spectroscopy we can study exoplanet atmospheres and learn about their chemical composition, temperature profiles, and presence of clouds and winds, mainly in hot, giant planets. State-of-the-art instrumentation is pushing these studies towards smaller exoplanets. Of special interest are the few planets in the 'Neptune desert', a lack of Neptune-size planets in close orbits around their hosts. Here, we assess the presence of water in one such planet, the bloated super-Neptune WASP-166 b, which orbits an F9-type star in a short orbit of 5.4 days. Despite its close-in orbit, WASP-166 b preserved its atmosphere, making it a benchmark target for exoplanet atmosphere studies in the desert. We analyse two transits observed in the visible with ESPRESSO. We clean the spectra from the Earth's telluric absorption via principal component analysis, which is crucial to the search for water in exoplanets. We use a cross-correlation-to-likelihood mapping to simultaneously estimate limits on the abundance of water and the altitude of a cloud layer, which points towards a low water abundance and/or high clouds. We tentatively detect a water signal blue-shifted ∼5 km s −1 from the planetary rest frame. Injection and retrieval of model spectra show that a solar-composition, cloud-free atmosphere would be detected at high significance. This is only possible in the visible due to the capabilities of ESPRESSO and the collecting power of the VLT. This work provides further insight on the Neptune desert planet WASP-166 b, which will be observed with JWST. INTRODUCTION High-resolution spectroscopy is used to detect and characterise the atmospheres of transiting planets, giving us information about their chemical composition, temperature profiles, and the presence of clouds and winds, mainly in hot, giant planets (see e.g. Birkby 2018, for a review). State-of-the-art instrumentation is pushing the precision of our measurements towards the detection and characterisation of the atmospheres of cooler and smaller exoplanets (Neptune and Earth-sized planets). One of the best studied chemical species with high-resolution instruments is water. Water analyses have mainly been focused in the infrared wavelength range, because its spectrum presents several strong absorption bands, while in the optical range, there are only few weaker absorption bands in the red. Water ★ E-mail: [email protected] † UKRI Future Leaders Fellow vapour has been targeted by several ground-based, high-resolution infrared spectrographs such as CRIRES (Kaeufl et al. 2004), NIRSPEC (McLean et al. 1998), GIANO (Origlia et al. 2014), CARMENES NIR (Quirrenbach et al. 2016), and SPIRou (Donati et al. 2020). Observations with these instruments have led to the detection of water vapour in the atmospheres of several transiting and non-transiting exoplanets (e.g. Birkby et al. 2013;Brogi et al. 2014Brogi et al. , 2016?;Birkby et al. 2017;Brogi et al. 2018;Alonso-Floriano et al. 2019;Sánchez-López et al. 2019;Webb et al. 2020;Boucher et al. 2021;Webb et al. 2022). However, detections of water in the visible range remain challenging. Esteves et al. (2017) and Jindal et al. (2020) studied the presence of water in the super-Earth 55 Cancri e with several transits obtained with the optical, high-resolution spectrographs HDS (wavelength range 5240-7890 Å, Noguchi et al. 2002) Gemini North telescope. They did not detect water and ruled out the presence of water-rich atmospheres if cloud-free. Deibert et al. (2019) studied HDS and GRACES observations of HAT-P-12 b and WASP-69 b, two warm sub-Saturns with inflated radii. They also did not detect water, but injection tests suggest a cloudy atmosphere with a small amount of absorption, in agreement with other studies. Allart et al. (2017) used HARPS (3780-6910 Å, Mayor et al. 2003) on ESO's La Silla 3.6 m telescope to look for water in the gas giant HD 189733 b, focusing on the 6500 Å absorption band. The data used is too noisy to constrain the presence of water and the authors estimated that over 10 HARPS transits would be needed to have a constrain at a significant level. However, they also estimated that a significant detection would be feasible with only a single ESPRESSO transit, due to its increased collecting power and the fact that its wavelength range includes a stronger water band at ∼7400 Å. With ESPRESSO (3782-7887 Å, Pepe et al. 2021) on ESO's 8.2 m VLT, Allart et al. (2020) studied WASP-127 b, a super-Neptune-mass planet with a radius larger than that of Jupiter, which makes it an extremely bloated planet. No water was found but, together with lowresolution data, the authors were able to constrain the pressure of a cloud-deck. Also with ESPRESSO, Sedaghati et al. (2021) observed the hot Jupiter WASP-19 b, which orbits a G8 V star in less than 1 day. Water was again not detected, but in this case, injection tests showed that it would only be detectable at high abundances and not feasible with ESPRESSO on an 8-m class telescope. The authors argued that this is the case due to the relatively faint host star ( = 12.3 mag) and the short transit duration (1.6 hours, Cortés-Zuleta et al. 2020), which result in few in-transit observations with low signal-to-noise ratio (S/N). Finally, Sánchez-López et al. (2020) reported a water detection in one out of three transits of HD 209458 b using the 7000 to 9600 Å absorption bands present in the red part of the visible arm of CARMENES VIS (5200-9600 Å, Quirrenbach et al. 2016). Injection tests indicated that the lack of detection in the other two nights could be due to a lower S/N and a higher degree of telluric variability, which results in a worse telluric removal that hinders the detection of water. As seen from the previous results, an important feature observed in the atmospheres of both hot and cool planets is the presence of clouds and/or hazes. Clouds and hazes reduce the strength of the features observed in an exoplanet spectrum, affecting the detectability of species such as water. Low-resolution observations of planets over a range of temperatures have shown muted water spectral features compared to what is expected for cloud-free atmospheres with solar metallicity (e.g. Sing et al. 2016;Stevenson et al. 2016; Barstow et al. 2017;Wakeford et al. 2017aWakeford et al. ,b, 2019Pinhas et al. 2019;Benneke et al. 2019a,b;Kreidberg et al. 2020), and some low-mass, low-temperature planets even show completely featureless spectra (Kreidberg et al. 2014;Knutson et al. 2014a,b). These muted features can be attributed to either the presence of thick, high-altitude clouds, or to inherently low water abundances. As opposed to low-resolution observations, which are sensitive to broad-band spectral features, high-resolution spectroscopy is able to resolve individual lines. The cores of absorption lines are formed higher up in the atmosphere than their wings. Therefore, highresolution data is sensitive to high-altitude regions of the atmosphere and can probe above clouds. The abundance of the species present in exoplanet atmospheres has been typically derived from low-resolution spectroscopy, which is sensitive to broad-band features over the continuum. Opposite to that, high-resolution observations do not preserve that continuum flux needed to measure abundances. However, it has recently been shown that by using a Bayesian framework, it is possible to recover abundances from the line-to-line and line-to-continuum contrast ratio alone (Brogi & Line 2019;Gibson et al. 2020;Line et al. 2021;Pelletier et al. 2021). Therefore, high-resolution spectroscopy is both sensitive to clouds and water abundance, and can break degeneracy between the two (Gandhi et al. 2020b;Hood et al. 2020). In this work, we use optical, high-resolution spectroscopy to study the presence of water and clouds on the transiting planet WASP-166 b, a bloated super-Neptune. WASP-166 b orbits a relatively bright ( = 9.36 mag), F9-type star in a close orbit of 5.4 days, at 0.06 AU (Hellier et al. 2019;Bryant et al. 2020, see Table 1 for the system parameters adopted here). The planet has a mass of 0.101 ± 0.005 M J (1.9 M Nep ) and a radius of 0.63 ± 0.03 R J (1.8 R Nep ) (Hellier et al. 2019), and its orbit has been found to be aligned with the stellar spin (Hellier et al. 2019;Doyle et al. 2022;Kunovac Hodžić 2022). It is located in the so-called 'Neptune desert', a dearth of Neptune-size planets in close orbits around their host stars. The study of such planets can provide insight to their formation and evolution, and the existence of the desert. Despite its close-in orbit, the planet has preserved its atmosphere, making it a benchmark target for exoplanet atmosphere studies in the Neptune desert. Seidel et al. (2020Seidel et al. ( , 2022 recently confirmed the presence of sodium in the atmopshere of WASP-166 b with high-resolution ground-based transit observations obtained with the spectrographs HARPS (Mayor et al. 2003) and ESPRESSO (Pepe et al. 2021), respectively. In the optical, other than sodium, we also expect the presence of potassium (although its signature usually overlaps with strong absorption from telluric oxygen, challenging its detection) and water, which we study here (e.g. Fortney et al. 2008;Madhusudhan 2012;Moses et al. 2013;Woitke et al. 2018;Drummond et al. 2019). Other species with signatures in the optical such as CH 4 or NH 3 do not have reliable opacities below 0.5 -1.0 micron, and hence have not been considered here. The planet is not hot enough to have other species with optical signatures such as Fe, TiO, or VO. WASP-166 is scheduled to be observed from space in the near-infrared with JWST, which should constrain the presence of molecules such as H 2 O, CO, CH 4 , CO 2 , C 2 H 2 , HCN, and NH 3 in the planetary atmosphere (Mayo et al. 2021). In section 2 we describe the ESPRESSO observations used. Section 3 details the analysis performed, and in Section 4 we show and discuss the results obtained. We summarise our findings and conclude in Section 5. OBSERVATIONS We observed two full transits of WASP-166 b, on 31st December 2020 and 18th February 2021, with the high-resolution, optical (wavelength range 3782 -7887 Å) spectrograph ESPRESSO (Pepe et al. 2021) installed on the VLT at the ESO Paranal Observatory, in Chile (ESO programme ID: 106.21EM, PI: H. M. Cegla). The observations were carried out in the 1-UT configuration (using UT1 on the first night and UT4 on the second) and high-resolution mode with 2 × 1 readout binning (HR21 mode, median resolving power of = 138 000). The target was observed with fibre A while fibre B was used to monitor the sky (i.e. simultaneous sky mode). The observations were reduced with the ESPRESSO Data Reduction Software 1 (DRS) version 2.3.1, which performs standard reduction steps for echelle spectra, including bias and dark subtraction, optimal order (2D spectra) extraction, bad pixel correction, flat-fielding and de-blazing, wavelength calibration, as well as extraction of sky spectrum from fibre B (see Pepe et al. 2021, for details). In our analysis, we used the blaze-corrected and sky-subtracted (i.e. corrected for telluric sky emission) 2D spectra. The observations of each night cover the full planetary transit (transit duration 3.608 h, Doyle et al. 2022) and 2 to 3 hours of outof-transit baseline (in total, before and after the transit). The exposure time was set to 100 s to ensure a S/N sufficiently high to have photonnoise dominated spectra (S/N∼ 50 at 550 nm) and to obtain a good temporal cadence to sample the transit. These observations were initially obtained to perform a study of the Rossiter-McLaughlin effect, which requires a fine temporal cadence during the transit (see Doyle et al. 2022). In the first night, we obtained 80 in-transit observations and 26/40 observations before/after the transit, and in the second night, 81 in-transit observations and 30/27 out-of-transit observations before/after the transit. For the two nights, most of the observations were taken at low airmass (< 1.5, see Figure 1 for an overview of the observing conditions). We discarded the first 8 observations of the first night (all outof-transit observations) because they were taken at an airmass larger than 2.2, which is the maximum value for which the ESPRESSO ADC (Atmospheric Dispersion Corrector) is calibrated for. Additionally, in the second night, we discarded 3 observations taken during the post-transit baseline due to telescope vignetting. We note that in the stellar RVs there is an offset of about 10 m s −1 in the systemic velocities of the two nights. These stellar RVs are obtained with the ESPRESSO DRS by computing the cross-correlation function with a suitable stellar mask. The reason for this offset is unknown but we attribute it to instrumental effects or differences in the observing conditions between the two nights. Regardless of the origin, the offset is too small to have any effect on our analysis (our precision is of about 1 km s −1 , 100 times larger than the offset). In the following, we consider as the systemic velocity of the system the average of the systemic velocities of each night. The same observations have been used in Doyle et al. (2022) to study the Rossiter-McLaughlin effect, characterise centre-to-limb convection-induced variations, and refine the star-planet obliquity, and in Seidel et al. (2022) to detect the presence of sodium in the planetary atmosphere. METHODS Telluric correction: PCA Spectroscopic observations taken from the ground are affected by spectral features produced by the Earth's atmosphere, known as telluric contamination. The ESPRESSO wavelength range is affected mainly by water (H 2 O) and oxygen (O 2 ), which produce absorption lines at specific wavelength ranges with varying strength, from shallow lines called microtellurics to deep and strong lines with completely saturated cores. The strength of the lines can vary depending on the observing conditions, such as the airmass or the atmospheric water vapour content. The effect of tellurics is especially relevant when trying to study water in exoplanet atmospheres. This is because the planetary water absorption lines can overlap in wavelength space with the telluric water (see e.g. Figure 2). Hence, we need to correct our observed spectrum from telluric lines. To correct for telluric effects, we used a principal component analysis ( ?, for other examples of works implementing PCA to study exoplanet atmospheres). We design our own automated algorithm to select the number of PCA components (described in Section 3.1.1) and to only feed into the PCA the spectral channels most affected by tellurics (Section 3.1.2). The use of PCA to remove tellurics is based on the fact that, during the transit observations, the Earth and the target star remain stationary or quasi-stationary, while the target planet moves tens of km s −1 as it orbits around the star. Therefore, telluric and stellar spectral lines are always approximately located in the same pixels in the detector CCD, as they only experience a small shift in RV, while the planetary signal will shift noticeably in pixel space (see Figure 1). The PCA method consists in finding an orthogonal basis for the covariance matrix of the data in which the eigenvectors (also called principal components, PC) represent the direction of decreasing variance in the data. That is, the first vector or PC of the new basis has the direction of the maximum variance in the data, the second one has the direction of the second largest variance, and so on. Since the first PCs are the ones that describe most of the variance in the data, we can remove them to clean the data of the strongest telluric, stellar, and instrumental time-dependent variations. In our case, the data matrix is composed of the different observations or frames as rows ( ) and the pixels or spectral channels as columns ( ). We work slice-by-slice, therefore, the steps described below are repeated for each slice, and for each night, separately. We note here that, conversely to other echelle spectrographs, ESPRESSO uses an APSU (anamorphic pupil slicer unit) that divides each order into two slices (i.e., the two slices corresponding to a specific order cover the same wavelength range). We treat each of the slices separately. We detail our PCA implementation in Appendix A. To briefly summarise it here, we first cleaned the spectra from flux anomalies, standarised the data matrix , and then performed the PCA. Instead of directly decomposing the covariance matrix of the data as in Giacobbe et al. (2021), we applied the PCA via a singular value decomposition (SVD). Left, top to bottom panels: Airmass (mean between start and end of each observation), seeing (mean between start and end of each observation), integrated water vapour (mean between start and end of each observation), and ambient humidity. Right, top to bottom panels: S/N at ∼550 nm, barycentric Earth radial velocity (BERV, note the offset between nights), star RV from the DRS CCF, and planet RV. The sys of the system has been subtracted from the star and planet RVs (see Table 1). All parameters obtained from the observations FITS headers, except for the planet RV, which is computed from the orbital parameters of the system (see text). Grey areas indicate out-of-transit phases. In this work, we are only studying the presence of water in the planetary atmosphere of WASP-166 b. Therefore, we are mostly concerned in the removal of telluric lines from the observed data. The host star, WASP-166, is too warm to display any water in the stellar spectrum (spectral type F9 V and eff = 6050, Hellier et al. 2019). The star is not especially active and we do not expect the presence of cool spots on the photosphere to be significant. Even if spots were present, their temperature contrast with the quiet photosphere is expected to be small, and hence, not sufficiently cool to display water either. Nevertheless, we want to note here that, in transmission spectroscopy, when studying planetary species that are also present in the stellar photosphere, one needs to account for the Rossiter-McLaughlin effect and centre-to-limb variations (CLV) across the stellar disc. This is because, during the transit, the planet occults different areas of the rotating stellar disc, which results in the in-transit stellar spectra being distorted (mainly depending on the projected stellar rotational velocity, the stellar obliquity, and the impact parameter). These distortions need to be accounted for to derive accurate and precise estimates of the planetary transmission spectra (see e.g. Brogi et al. 2016;Yan et al. 2017;Hoeijmakers et al. 2020;Casasayas-Barris et al. 2021;Seidel et al. 2022;Maguire et al. 2022, for more details on such effects and strategies to account and correct for them). Optimisation of the number of PCA components per slice Since different orders are differently affected by tellurics, we performed a per-slice optimisation of the number of components to be removed when applying the PCA, which we describe in this section. To perform this optimisation, we made use of the crosscorrelation function (CC) of the observed spectra with a water model. We refer the reader to the following Section 3.2 for all the details on the CC computation. For each slice affected by tellurics, we started by removing the first 2 components in the PCA. We then computed the CC of the resulting spectra with a water model and coadded the CCs of the in-transit observations in the barycentric rest frame. Coadding in the barycentric frame maximises the presence of telluric residuals in the CC, which is what we are focusing on at this stage. We then assessed the significance of the telluric signal by taking the value of minimum or maximum CC flux in the region ±10 km s −1 (to cover the full telluric feature) around the mean BERV of the observations, and comparing it with the scatter (standard deviation) of the CC flux outside of this region. The atmospheric water vapour changes during the observations, increasing and decreasing from the overall trend dictated by the change in airmass. This causes negative and positive residuals in the processed spectra, which result in correlation and anti-correlation with the CC water template used. Therefore, when looking at the telluric signal in the CC, we considered both minima and maxima features (i.e. anti-correlation and correlation with the template). We considered a signal at the telluric position of the CC to be significant if the minimum (or maximum) flux is below (or above) 3.5 times the standard deviation of the flux of the rest of the CC. If the telluric signal is significant, we repeat the process but removing an additional PCA component. This goes on until the signal is not significant, or until the algorithm reaches the maximum number of components allowed. We set the maximum number of components to be removed to 15 (after removing over ∼ 15 components, injected planetary signals start to decrease in significance). Although the aforementioned CC functions of our individual observations can show a minimum or a maximum at the expected telluric position, if we coadd the CC functions of the individual observations, the dominant feature at the telluric position in our case is a minimum, i.e., anti-correlation. This means that the observations with telluric residuals that anti-correlate with the models are more prominent than those observations with a positive correlation. The coadding of anti-correlated and correlated CCFs can result in a smearing of the overall signal. To check for that, we also computed the significance of the telluric peak in each individual observation. We observe that for all the cases where we have a significant signal in the 'observation-coadded' CC function, more than half of the individual observations also show a significant signal. Additionally, if more than half of the observations contain a significant telluric signal, so does the coadded CC function. We note again that here, instead of coadding all the available observations, we coadded only the in-transit ones. This is because these observations are the only ones we use in the planet analysis, and therefore we are mostly concerned about the telluric effects in them. Aside from this, we noticed that the observations at high airmass (airmass higher than 2 at the beginning of the first night, and airmass close to 1.7 at the end of the second night) are the ones that show the strongest telluric signals in the CC function, being very distinct than those immediately after or before. If we included these high airmass observations in the coadded CC function, they heavily biased the significance of the telluric signal, so that the algorithm keeps removing components even though the in-transit telluric signal is not significant. Selectively feeding telluric lines into the PCA To try to further improve the telluric removal, instead of using the whole spectral range of each slice, we tested feeding into the PCA only the pixels affected by tellurics, i.e. pixels containing telluric lines. By doing this, the PCA should better trace the variability due to telluric changes. To determine the telluric-affected pixels, we used the ESO Sky Model Calculator 2 based on the Cerro Paranal Advanced Sky Model (Noll et al. 2012) to generate a telluric absorption model in the ESPRESSO wavelength range (see Figure 2, middle panel). We interpolated the model to the observed wavelength grid of each slice and continuum-normalised it by fitting a cubic spline (we do this slice by slice). To fit the spline, we selected the pixel with maximum flux in windows of 25 pixels and avoided strong telluric bands that would bias the determination of the continuum. This results in a flat telluric spectrum normalised to one. After this normalisation, we flagged as telluric-affected all the pixels that overlap with a telluric line. We set the threshold to pixels where the telluric flux is below 0.998, which allows us to select most of the lines present in the ESPRESSO spectral range. The slices affected are 80-83, 96-103, 108-123, 128-141, 146-169 (slice numbering starts at 0 for the first slice of the bluest order). When applying the PCA, only these pixels are used in the SVD. In orders with no telluric lines, we still used all the pixels to remove any systematics. High-resolution cross-correlation spectroscopy After correcting for tellurics with the PCA, we used the highresolution cross-correlation spectroscopy (HRCCS) method to search for the presence of water in the atmosphere of WASP-166 b. Planetary water produces thousands of molecular absorption lines in the planetary transmission spectrum. This water signal, however, is below the noise level of the data. The HRCCS method coadds all the lines present in the transmission spectrum by cross-correlating the processed observations with an adequate spectral template of the planetary atmosphere. To compute the CC, the template is Dopplershifted by a range of RV values, and, for each shift, we take the dot product with the observed data. This operation results in a crosscorrelation function with much higher S/N than a single spectral absorption line, which enhances the planetary signal. This is because the S/N of the CC function scales with the square root of the number of lines coadded when computing the CC. In Section 3.2.1, we describe the different sets of CC models used, and in Sections 3.2.2 and 3.2.3, we explain the formalism used to compute the CC and assess the significance of the results within the cross-correlation-to-log likelihood (CC-to-log ) framework. Planetary atmosphere water models We generated primary eclipse spectra of WASP-166 b using GENE-SIS adapted for transmission spectroscopy (Gandhi & Madhusudhan 2017;Pinhas et al. 2018). GENESIS is a line-by-line numerical radiative transfer code that computes the transmission spectrum of the atmosphere given the atmospheric temperature and chemical abundance profile. The opacity of each species is computed on a grid of pressure-temperature ( -) values for each wavelength to determine the overall optical depth of rays passing through the atmosphere and therefore the transit depth at each wavelength. We use a grid of fixed pressure values, between 100 to 10 −7 bar and evenly spaced in log . We assumed an isothermal temperature profile consistent with the equilibrium temperature of WASP-166 b, ∼ 1270 K. The chemical abundances are set as volume mixing ratios (VMR) assumed to be vertically constant throughout the atmosphere. We also included a wavelength-independent cloud deck at different pressures by setting all wavelengths to a very high opacity. The models spanned a grid in H 2 O abundance and cloud pressure, encompassing log 10 (H 2 O) = −1 (highest abundance, in VMR) to −5 (lowest abundance), and cloud deck pressures of log 10 ( cloud /bar) = 0 (lowest altitude) to −5 (highest altitude), both in steps of 0.5 dex (see Figure 3 for examples). In total, we computed two grids of model spectra, one using an ExoMol POKAZATEL (Polyansky et al. 2018) line list and the other with a HITEMP (Rothman et al. 2010) line list (see Gandhi et al. 2020a, for further details on opacities). In addition, all models across both grids include collisionally-induced absorption from H 2 -H 2 and H 2 -He interactions (Richard et al. 2012) and Rayleigh scattering due to H 2 . Each model was generated at a spectral resolution of R=500 000 between 0.38-0.8 m. The models already include intrinsic pressure and temperature broadening. To better match the line shape of the expected observed planetary signal, we further broadened these model spectra by the instrument profile of the observations; for this, we used a Gaussian kernel with FWHM corresponding to the R=140,000 resolution of ESPRESSO (of ∼ 2.14 km s −1 ). We also computed the broadening due to planetary rotation (assuming it is tidally locked), which is of only 0.58 km s −1 . This is negligible compared to the instrument profile broadening, and hence, we do not consider it here (i.e. including it would only change the broadening from ∼ 2.14 km s −1 to ∼ 2.22 km s −1 ). We used two different line lists because published water lines in the optical have not been extensively empirically verified. In the optical, water absorption bands are weaker than in the near-infrared. Due to this reduced strength, the accuracy and completeness of the model lines in the optical is expected to be worse than in the nearinfrared, because their experimental verification is more challenging. Therefore, there could be differences between different line lists. To check for systematics due to these potential differences we decided to repeat our analysis using the two sets of line lists. CC-to-log framework To assess the significance of any planetary signals, we followed the cross-correlation to log-likelihood framework (CC-to-log , Zucker 2003;Brogi & Line 2019;Gibson et al. 2020). This is a Bayesian framework based on mapping the cross-correlation function to a log likelihood function. This allows us to accurately assess the significance of any detections by deriving confidence intervals, as well as to compare the performance of different models. We used the CC-to-log mapping proposed by Brogi & Line (2019) log( ) = − 2 log[ 2 − 2 + 2 ],(1) where 2 is the variance of the observed spectrum, 2 is the variance of the model used, is the cross-covariance between the observed spectrum and model, and is the number of points in the spectrum. The cross-correlation is contained in the above equation, since the correlation coefficient is proportional to the cross-covariance as C = √︃ 2 2 .(2) In our case, all these refer to each individual spectral slice, because we are working slice-by-slice. The broadened models are sliced so that they are within the wavelength range of each order. We also spline-interpolated the models to the wavelength grid of each order, so that the number of data points of the observed spectrum and model are the same. This interpolation is performed for every RV shift for which we compute the CC and log functions. We followed two different approaches to compute the CC and log functions. Both methods lead to the same final result but have different advantages and drawbacks, as we describe in the following paragraphs. For more details on the implementation of each approach, we refer the reader to Appendix B. In our initial or 'fast' approach, we compute the CC and log functions of each slice for a fixed RV grid. Then, for each observation, the log function of all the slices considered are coadded. Finally, the log functions of the in-transit observations are coadded in time along the planet RV, as a function of p , from which we can then build the usual p − sys (or p − rest if sys has been subtract) maps. In the second or 'slow' approach, instead of computing the full CC and log functions for a fixed grid of RV values, we only shift the model once to the expected planet RV (which is given by a pair of p and sys values), and compute a single CC and log value. We repeat this for a range of p and sys pairs, which also results in the usual p − sys maps. With the slow approach, we are building the p − sys map pixel-by-pixel, while in the fast one, we directly get a full row of the map for each p considered. The main advantage of the slow approach is that it allows us to process the model used in the cross-correlation through the same PCA as the data, which is not possible in the fast approach (see Appendix B for details about the model processing). This is important because the PCA might alter the planetary signal contained in the data by changing the line strength and shape. The models used in the fast approach do not contain any change due to the PCA, therefore, the match with a possible planetary signal will not be as good as if the model has also been altered in the same way as the data. Due to this mismatch, a potential planetary detection might be weaker and biased in p and sys . Moreover, when performing the model comparison (see below Section 3.3), we could also misinterpret the water abundance and cloud deck pressure because the line depths do not match between model and data. Therefore, we expect more accurate results with the slow approach than with the fast one, because 1) we are computing the log value at the exact p − sys , and not shifting and interpolating the whole function computed for a different p and sys pair, and 2) we are processing the model in the same way as the PCA modifies the data, which should result in a better match between model and data. However, the implementation of the fast approach is significantly faster than the slow one. Moreover, the fast approach allows us to study the behaviour of the telluric signal directly in the CC and log functions, which is not possible with the slow approach. In the following, we refer to the two approaches as 'fast/unprocessed-model' and 'slow/processed-model'. Confidence intervals The CC-to-log framework allows us to estimate confidence intervals for the p − sys maps (Brogi & Line 2019;Pino et al. 2020), to know which p − sys pair is more likely compared to all the pairs tested. According to Wilks' theorem (Wilks 1938), minus twice the difference between the log values of two models (Δ log = −2(log 1 − log 2 )) follows a 2 distribution with number of degrees of freedom equal to the number of explored parameters. In our case, we are comparing the log value of each p − sys pair (2 parameters) with the maximum log of the map. I.e., we subtracted each log value from the maximum log of the map, which, if detected, should correspond to the planetary signal. We can then compute the p-value of this 2 distribution, from which we can finally derive the confidence interval value in units of standard deviation ( ) for a Normal distribution. Then, the model with the highest log will have a of 0, with less likely models having increasing values. We computed the confidence intervals for the data of each night separately and also on both nights combined. To combine the nights, we summed the log values of each p − sys pair of both nights, and then computed the confidence intervals on this coadded log . Model comparison We also performed a likelihood ratio test for each of the 2 (POKAZA-TEL and HITEMP line lists) grids of 99 models (9 water abundances × 11 cloud top pressures) computed (see Section 3.2.1). This allows us to derive confidence intervals in both cloud top pressure and water abundance. We computed the p − rest map (we have subtracted the expected sys ) for each of the 99 models. To compute the log functions we followed the CC approach 2 as explained in Appendix B2, in which we modify the template in the same way as the PCA processes the data. We used a grid ranging from 90 to 150 km s −1 in p , in steps of 3 km s −1 , and from −20 to 5 km s −1 in rest , in steps of 1 km s −1 . This grid results in a reasonable computational time, is sufficiently fine to resolve any signals, and covers the expected planetary position as well as any tentative detections seen in our initial tests. To identify the model with the highest significance, we compared their log values, following the same idea as when computing confidence intervals for the different p − sys pairs explained above. Now, we have again two parameters: the water abundance log 10 (H 2 O) and the cloud deck pressure log 10 ( cloud /bar). In the p − rest map obtained for each model, we computed the maximum log of an area around where the planet is expected. We used an area spanning ±10 km s −1 from the expected p , and −10 and +5 km s −1 from the expected rest (see Table 1). We used +5 km s −1 in rest instead of +10 km s −1 (i.e. which would be symmetric around the expected rest ) because we are limited by the rest range covered. We tested smaller and larger areas (from ±5 km s −1 up to ±20 km s −1 in both p and rest ) and the results obtained do not change significantly. This gives us a log max for each model. We can then apply Wilks' theorem to obtain confidence intervals for the grid of models. As before, we computed twice the difference of the log max of each model from the absolute maximum of all models, derived the p-value from this distribution of Δlog , and finally computed the confidence intervals in . This likelihood ratio test informs us about how likely the 99 models tested are compared to each other. The best model will then have a of 0, and the rest of the models will have larger values of . We performed this analysis on each night individually, as well as on both nights coadded (for which we used the p − rest log map obtained by summing the log values of each night). Injection tests We also tested the detectability of the water signal in our data by performing several injection tests using the H 2 O models described in Section 3.2.1. We note that we do not use these injection tests to optimise our data analysis, but rather to assess the sensitivity of our data to a water signal. We tested different strengths of the model by scaling it to different values. To scale the model, we subtracted the mean of the model flux, which removes the average transit depth and leaves only the effect of the planet atmosphere. We then multiplied the residual spectrum (which is now only due to the planet atmosphere) by a scaling factor. We then brought back the original flux level by adding the original mean. We note that this scaling factor does not correspond to an increase or decrease in the H 2 O abundance of the model. Increasing the abundance could lead strong lines to saturate, while this will not happen with a scaling factor. By using the scaling factor we only want to study its detectability. Right before applying the PCA, we injected the scaled model to the in-transit observations. We performed this process slice-by-slice. To do this, we first Doppler-shifted the wavelength of the model by the corresponding RV of the planet at the time of each observation, so that the model shift reflect that of the actual planet, and interpolated the shifted model to the wavelength grid of each observation. We then multiplied the flux of each observation by the flux of the corresponding model. This way, each in-transit observation includes now a model of the planetary spectrum. We performed this step after the observed flux had been cleaned of bad pixels. After the injection, the standarisation and the PCA are performed as explained above (see Section 3.1). We then computed the CC as explained above with the same model as injected. RESULTS AND DISCUSSION In this section, we first present the results from the tests performed to optimise the PCA algorithm. We then apply the optimum PCA algorithm to constrain the presence of water and clouds in the data via model comparison with a likelihood ratio test. PCA optimisation As explained in the methodology section (3.1), we performed several tests with the goal of optimising the performance of the PCA to minimise the presence of tellurics in the CC and log functions. Here, we detail the results obtained. Unless explicitly stated, all figures in this section display CC functions and p − rest maps obtained using the fast/unprocessed-model CC approach (Appendix B1). This is because we wanted to directly study the shape of the CC functions, and in particular, the presence of telluric residuals, which is not possible with the slow/processed-model CC approach (Appendix B2). Moreover, to cover the same p − rest parameter space, the slow/processed-model method takes significantly longer computational time than the fast/unprocessed-model one, which in practice limits the p − rest values that we can sample, as well as the number of tests we can do. Therefore, to perform our initial tests, we decided to follow the first approach. This allowed us to test the optimal parameters for the PCA and identify any tentative planetary signals. Fixed number of PCA components Applying the PCA algorithm removing a fixed number of components for all the slices results in a strong telluric feature in the CC functions. We show this in the top panels of Figure 4 and Figure 5 (black and orange lines), where we indicate the position of the telluric residuals in red. Figure 4 shows the CC functions obtained for all the observations as a function of the orbital phase for the two nights (columns), for different tests performed (rows). Figure 5 shows the in-transit CC functions coadded in planet rest frame for the two nights (columns) and different tests (different lines in both rows). We know that the observed feature is due to telluric contamination because it appears at the expected BERV and spans the entire sequence of observations (i.e. it is not phase dependent and is present in both in-transit and out-of-transit observations). In the CC functions, we see that the signal evolves in time from correlated (maxima) to anticorrelated (minima), as a result of the positive and negative residuals in the processed spectra. These residuals, in turn, come from the change in airmass and the changes in the atmospheric integrated water vapour column that changes during the observations, which increase and decrease the amount of water vapour above or below the overall trend. The telluric residuals do not perfectly correlate with changes in airmass and water vapour because we have applied the PCA and removed the first components prior to computing the CC functions. That is, the first PCA components removed contain part or most of the airmass and water vapour variations, and hence, the correlation is broken. We do not show them here, but if we calculate the same maps using the log function (Equation 1) rather than CC (Equation 2), we see the same residuals. The telluric feature is also clearly seen in the form of maxima and minima in the p − rest maps produced after coadding the in-transit log functions in planet-rest-frame, see e.g. top panels in Figure 6, where we plot the confidence intervals obtained for each night and for both nights coadded (columns). As expected, the telluric signal decreases as we increase the number of components removed during the PCA, see top panels in Figure 5 for examples removing 2 and 6 components (black and orange lines), but it is never completely removed. We notice that the removal seems to work better in the first night than in the second one, i.e., when the integrated water vapour is higher. We tested removing between 2 to 15 components on all the slices, but found no significant improvement, i.e., the telluric signal did not decrease further, after removing more than ∼ 6 components. We qualitatively explain the inability of PCA to de-trend telluric lines as follows. In optical observations where telluric lines are not prominent, their contribution to the total variance within one slice is also negligible. Since the SVD algorithm 'ranks' components based on their contribution to the variance, telluric residuals might potentially be absent in the first 15 components, which would instead be dominated by throughput and continuum variations. The residual level of correlation is similar to that expected from standard telluric removal algorithms e.g. direct modelling of the telluric spectrum, unless these residuals are heavily masked prior to correlation. To improve the correction, we revised the SVD algorithm as explained in Section 4.1.3. Injection tests We also tested the behaviour of the PCA algorithm when injecting a planetary model (water abundance log 10 (H 2 O) = −3, in VMR, and cloud deck pressure log 10 ( cloud /bar) = 0 based on the POKAZATEL line list) in the data. The p − sys maps (Figure 7, top) show that an injected planetary signal with a scaling of ×1 (i.e., original strength) is recovered with high significance, despite the presence of tellurics in the data. The injected signal is clear in each night individually, with a higher confidence in the first night that increases when combining both nights. From the CC maps (Figure 4), we see that in both nights, the expected planetary RV and the BERV do not overlap, which might help in obtaining a significant detection. Even when only removing 2 PCA components, the injected planetary signal is clearly detected in each night in the form of a peak in the CC and log functions, see top panels in Figure 5, where we compare removing 2 and 6 components (blue and green dashed lines, respectively). From these same tests we also see that the injected planetary signal is not affected by increasing the number of components removed. This indicates that the PCA components are not selecting the injected planetary signal, which is the behaviour we expect. We also note from Figure 5 that the telluric residuals in the CC are slightly different if the model has been injected in the data (black, orange lines) or not (blue, green dashed lines), which indicates that part of the injected planetary signal could impact the PCA, despite the fact that its significance does not decrease. Figure 4). CCs computed using the fast/unprocessed-model approach. Red dashed lines indicate the expected p and rest . Left to right are the maps for the first, second, and coadded nights. Middle row: Same as top, but in this case, only the CCs of slices 84-95, 104-107, 124-127, 142-145 (slices with no or small telluric contamination) have been coadded (i.e. corresponds to coadding the in-transit CC shown in the above Figure 4, second row). Bottom row: Same as top, but in this case only the pixels affected by tellurics have been used in the PCA (i.e. corresponds to coadding the in-transit CC shown in the above Figure 4, bottom row). Neglecting orders affected by tellurics The orders where the telluric effect is stronger are those for which we see strong telluric absorption lines. These orders are also those where the planetary water shows the strongest absorption lines (see e.g. Figure 2). Coadding the CC (or log ) functions discarding these telluric-affected order slices (i.e., using only slices 84-95, 104-107, 124-127, 142-145) results in a decrease in the telluric signal, see the CC functions in the second row of Figure 4 and the top panels in Figure 5, red line, which show no significant feature at the telluric position in RV space. The telluric residuals also disappear in the p − rest maps, see middle panels in Figure 6. We note here that in these maps, all data points are within ∼ 4 (or less) of one another. This means that none of the data points, i.e., none of the p − rest pairs, is more significant than any other. In other words, the points with the highest likelihood in maps without the telluric orders maps (i.e. confidence interval close to 0) are not significant. If we now look at the cases where we have injected a water model, we note that the planetary signal that is clearly detected using all the orders also disappears. We see this in the top panels of Figure 5, purple dashed line, where the clear signal at the injected RVs is no longer there, and the CC looks as flat as in the case where we have not injected a planet, as well as in the p − rest in the second row of Figure 7. As happens in the case without any signal injected, now all data points in the p − rest maps are also within ∼ 4 of one another, meaning that no data point is significant with respect to the others. The fact that the injected planetary signal is not seen here is not surprising. Despite the fact that the exoplanet temperature is significantly higher than the Earth's temperature, the main H 2 O features are similar, and thus removing orders containing telluric H 2 O also removes exoplanetary H 2 O. Optimisation of the number of PCA components per slice We also optimised the number of components to be removed per slice using the method described in Section 3.1.1. The slices that are optimised, i.e., those that result in an increased number of components removed with respect to the initial, are in general those that contain strong telluric absorption lines: slices 80 -83, 98, 99, 108 -111, 116 -121, 134 -139, 147 -155, 158 -161, 168, 169, see Figure 2 above. There is a relatively strong telluric band covering slices 128 -131, and the strongest band of saturated O 2 lines in slices 162 -165, for which the number of components are not optimised. In the case of the saturated band in slices 162 -165, it is possible that the telluric lines are simply too strong for the PCA to be able to remove its effect, however this argument does not explain why the weaker band in slices 128-131 is not being properly removed. Further analysis is needed to understand these results and the behaviour of the PCA algorithm. In most of the order slices that contain tellurics, both slices have an increased number of tellurics removed, but the final number of components is not always the same for both slices of the same order. This is not necesarrily expected, and suggests that the PCA is selecting additional correlated noise different in both slices, rather than purely telluric signals, which should be the same in both slices. Again, further analysis of the PCA behaviour is needed to understand this difference. For the two nights, most of the slices mentioned above are optimised. However, the final number of components also differs between nights for the same slices, which is expected since the tellurics behave differently in the two nights. Figure 4, third row, shows the CC functions obtained when applying this optimisation. We notice that, despite removing a significantly higher number of components for the telluric-affected orders, this results is a CC map very similar to the one we obtain by removing a lower, fixed number of components. The p − rest map is also similar to this case, hence we have not included it here. This indicates that, despite still having strong telluric residuals in the CC, removing a higher number of components does not result in a significant telluric removal. Selectively feeding telluric lines into the PCA As explained in Section 3.1.2, we modified the PCA algorithm to focus on the pixels affected by telluric lines, rather than using the full spectral order. We show the CC function and p − rest maps that we obtain in the bottom panels of Figures 4 and 6. In the CC function maps, we see some telluric residuals at the beginning of the first night, including part of the transit, as well as some faint residuals during the transit of the second night. This translates into a very faint signal in the p − rest map at the position where we expect the telluric residuals to be for the first night, and in a stronger residual for the second one. Compared to the results obtained using the full spectral order (top panels in Figures 4 and 6), in this case, the telluric contamination is significantly removed in both the CC functions and p − rest maps. This means that the PCA components removed are more effective in tracing the telluric variability if we only use the regions of the spectrum affected by tellurics rather than the whole wavelength range. This is again expected, because in the (sub-)matrix containing only telluric lines the latter will have a more noticeable contribution to the variance, and therefore will be ranked higher by the SVD algorithm. In addition to using the telluric-affected pixels in PCA, we also performed the optimisation of the number of components to be removed per slice, as done above. In this case, since the initial PCA components are already removing most of the telluric signal, the optimisation algorithm did not detect a signal strong enough to continue removing components. Hence, for almost all the slices, the algorithm stops after the initial number of components considered has been removed. This means that the CC and p − rest maps look very similar if we apply the optimisation and if we do not, and are not shown here. Injection tests With the new SVD algorithm, injected planetary signals are still recovered at high significance, see bottom panels of Figure 7. As happened when using the full spectral range (Section 4.1.1), the telluric residuals look different if the planetary signal has been injected in the data or not; see bottom panels of Figure 5, orange and green lines, and bottom panels in Figures 6 (non-injected) and 7 (injected case). The telluric residuals are stronger if the planetary signal has been injected. Similarly, if we now compare the case where only the telluric-affected regions are used in the PCA with the initial case where the full spectral range of the order is used (both with an injected planetary signal), the telluric residuals are different, see again Figure 5, bottom, and top and bottom panels in Figure 7. In general, for both nights, the tellurics are more significant if only the telluric-affected pixels are used in the PCA (bottom panels of Figure 7) compared to the whole spectral order (top panels of Figure 7), which is the opposite as what happens when no planetary signal is injected. As mentioned before, this indicates that the injection of a planetary model affects the telluric identification in the PCA algorithm, but this does not seem to affect the planetary signal itself, as it is clearly detectable in both cases. A tentative H 2 O signal from WASP-166 b? The p − rest map of the first night obtained with the analysis in Section 4.1.3 above (i.e. with the modified PCA algorithm) shows a correlated signal close to the expected planetary position, about 5 km s −1 blue-shifted from the expected rest and extending about −30 km s −1 from the expected p (bottom left panel in Figure 6). The signal is slightly significant with respect to its neighbouring points. While this is outside of the uncertainties of rest (or sys ) and p (see Table 1), unaccounted atmospheric circulation at the km s −1 level has been shown to potentially alter rest and p measurements. The second night shows a similar structure but less prominent and not significant. This could be affected by the fact that the tellurics are less removed in the second night than in the first one, and hence, a possible planetary signal might be hidden in the telluric residuals. In addition, in RV space, the tellurics are closer to the expected sys in the second night compared to the first one. Despite this difference, the signal is still present when coadding both nights. It is also more significant with respect to its neighbouring points than in the first night alone. We will further discuss this candidate signal in Section 4.2. Model comparison In the previous section, we see that the modified PCA algorithm in which we use only the spectral regions affected by tellurics in the SVD results in less significant telluric residuals than any of the other tests performed. Therefore, to perform the model comparison between different theoretical models (as explained in Section 3.3), we used the data processed using the modified SVD algorithm, since it minimises the telluric residuals. Moreover, to be able to compare the different models, it is important to correctly reproduce the line strength of the data. We can only guarantee this if the model used in the CC has been processed by the same PCA as the data, which we do here by using the slow/processed-model CC approach. Figure 8 shows the confidence intervals obtained for the grid of 99 models tested. These results correspond to the models created with the POKAZATEL line list, but we obtain equivalent results for the HITEMP line lists. In each night separately, and also when coadding both nights, models with a high content of water and a cloud deck at high pressure (log 10 (H 2 O) −3 and log 10 ( cloud /bar) −2, bottom-right quadrant of Figure 8) are rejected with high confidence compared to the other models tested ( 6 , up to 15 for some models). Models with the lowest cloud deck pressures and lowest water abundances (upper-left quadrant of the plot), are also excluded but only with ∼ 4 confidence. Grid search Overall, the preferred model is that with log 10 (H 2 O) = −4 and log 10 ( cloud /bar) = 0 (i.e. no cloud deck). There is a preference for intermediate models with low water content and high cloud deck pressure, or higher water content and lower cloud deck pressure (models coloured in light-green in Figure 8). These models are within a confidence interval of ∼ 2 of the preferred model. This happens in all cases, i.e., for the two nights individually and both nights coadded. The fact that the intermediate models are preferred over those with the lowest cloud deck pressure and lowest water content (upper-left quadrant of the plot) points towards a tentative detection of a water signal. If there was no planetary signal present, the preferred models will be those that have the shallowest absorption lines, i.e., those that are compatible with an almost flat spectrum (see models in Figure 3). These are the models with the lowest water abundance and low-pressure clouds, i.e, models in the upper-left quadrant of Figure 8), which are not preferred here. In other words, a non-detection would only exclude the bottom-right quadrant of the grid, but not the upper-left, as happens here. This is in qualitative agreement with the predictions of Gandhi et al. (2020b). We note that at low cloud pressures, models with water abundance log 10 (H 2 O) −2 are more strongly rejected than those with higher water abundances (log 10 (H 2 O) = −1). This is due to the higher mean molecular weight of the atmosphere with log 10 (H 2 O) = −1 compared to the one with log 10 (H 2 O) = −2. As we increase the abundance of water, for log 10 (H 2 O) −2, the mean molecular weight of water-rich atmospheres becomes higher than at lower water abundances. This higher mean molecular weight reduces the scale height, which results in a decrease in the strength of the water absorption features (see Figure C1 in Appendix C). Hence, due to the fact that for water abundances log 10 (H 2 O) −2 the absorption features are stronger than for log 10 (H 2 O) = −1, models with log 10 (H 2 O) −2 are more strongly rejected (also see e.g. Gandhi et al. 2020b). Our confidence interval analysis is relative to the 'best' model (the one with a highest likelihood). That is, the best model has by default a of 0 and the other models have then values relative to the best one. In general, it is not clear how to assess the absolute significance of the best model. Even commonly-used signal-to-noise approaches do not assess how good a model fits to the data in an absolute sense. To try to obtain a 'baseline' likelihood and assess the absolute significance of our models, we have performed an extra test with a 'flat' model, i.e., a model with flux equal to 1 at all wavelengths. We have used this flat model to compute the CC and log functions with the data processed with the best PCA algorithm using the slow/processed-model method (including model processing), as done with the model grid above. We have then compared the log obtained with this model with that of the 'best' model according to our grid analysis (log 10 (H 2 O) = −4 and log 10 ( cloud /bar) = 0) in the same way as we compared the different models in the grid above. That is, we performed a likelihood ratio and computed how many s away the models are from each other. For the first night, the flat model is rejected with 3.9 compared with the best one, for the second night, 4.2 , and for both nights coadded, 4.9 . This tells us that in the data, there is a signal (regardless of its origin, planetary or telluric) that is ∼ 5 above a flat model. This test is similar to comparing the best model with models close to a flat line (i.e. those in the top left corner of our grid). Indeed, the difference in sigma obtained between the best model and those in the top left corner is similar to that obtained with respect to the flat model. p − sys maps of the preferred model In Figure 9 we show a comparison of the p − rest maps obtained with the model favoured by our grid search in the previous section (model with log 10 (H 2 O) = −4 and log 10 ( cloud /bar) = 0), obtained with each of the two line lists considered, POKAZATEL (top) and HITEMP (bottom). As mentioned above, here we used the slow/processed-model CC approach, including model processing, as opposed to the results we show in the bottom panels of Figure 6, in which we use the fast/unprocessed-model CC approach. Note also the smaller p and rest ranges explored here with the slow/processedmodel approach, which are around a blue-shifted signal close to the expected planet position. This blue-shifted signal close to the expected planet position, but with lower p value than expected, was already seen in the initial tests with the fast/unprocessed-model CC approach (Figure 6, bottom panels) and is still present in the first night. For the second night, this signal was less significant than in the first night in the initial tests. Now, with the slow/processed-model CC approach including the model processing, a signal also blue-shifted ∼ 5 km s −1 appears, but it is shifted towards higher p values. This difference between the fast/unprocessed-model and slow/processed-model approach is something expected. With the fast/unprocessed-model method, we process the data through the PCA and then directly cross-correlate it with a model. However, in the slow/processed-model approach, we additionally process the model through the same PCA as the data before computing the cross-correlation. By doing this extra processing of the model, we modify the model in the same way as the data has been modified by the PCA. The fact that this signal is clearer in the second night using the slow/processed-model approach highlights the importance of model processing. By altering the model in the same way the PCA has altered the data, the CC should result in a better match between data and model, which is what we see here. Coadding both nights results in a ∼ 5 km s −1 blue-shifted signal at the expected p , which may be the result of the original signals in both nights being displaced in p in opposite directions. This blueshifted signal is favoured with respect to the neighbouring points with ∼ 7 . For both line lists, the results obtained are equivalent, i.e., both nights show a blue-shifted signal displaced from the expected p , with very similar p − rest maps, and the signal is still present at high significance when both nights are coadded. Despite being at different p , the individual night signals are within 1 and 2 (for the second and first night, respectively), of the coadded nights signal. Therefore, the signal observed when coadding both nights is not rejected by the results obtained for each night individually. We note that p is not strongly constrained in the transit observations that we are considering here, because they only span a small part of the total Keplerian orbit. Hence, it is hard to obtain good constraints in p , which could be the cause of the discrepant p values observed in these maps. The −5 km s −1 shift in rest-frame velocity is outside of the uncertainties on the measured rest (or sys ), which has been obtained from the same observations as we use here (see Table 1). The observed blue-shift could potentially be due to the presence of winds in the planetary atmosphere. The planetary Na lines detected by Seidel et al. (2022) in the same observations show a significant broadening with respect to the instrument profile, of 9.37±0.95 km s −1 , which suggest that the Na is moving at high velocities, similarly to what we might be seeing here with H 2 O. The telluric residuals for the first and second nights are at ∼ −50 km s −1 and ∼ −25 km s −1 of this signal, respectively. Therefore, we do not expect the pixels neighbouring the blue-shifted signal to be significantly affected by tellurics. SUMMARY AND CONCLUDING REMARKS In this work, we have analysed two transits of the inflated super-Neptune WASP-166 b observed with the optical, highresolution spectrograph ESPRESSO. Using the high-resolution cross-correlation technique, we study the presence of water vapour and clouds in the atmosphere of the planet. To correct for the presence of telluric signals which may interfere with a potential planetary signal, we start by applying a PCA algorithm on the observed spectra. We noticed that a standard PCA algorithm results in very strong telluric residuals in the CC and log functions, as well as in the p − rest maps. Consequently, we performed several tests changing different parameters controlling the PCA to optimise the algorithm and obtain the best possible telluric removal. In particular, we explore the number of components removed, the spectral slices coadded, and the specific wavelengths (or pixels) used to compute the PCA components. We note here that our PCA optimisation, differently from other studies in the infrared, is model-independent, i.e. it is not performed by optimising an injected signal but rather by minimising the residual telluric noise. While this is arguably not the best choice to maximise S/N, it is a conservative choice to avoid any optimisation biases as highlighted by e.g. Cabot et al. (2019) and Spring et al. (2022). A full comparison with alternative telluric removal methods such as telluric fitting with Molecfit (Kausch et al. 2015;Smette et al. 2015) or polynomial detrending (e.g. Snellen et al. 2010) is out of the scope of this work and will be the subject of future studies. Increasing the number of components removed, whether if fixed or variable for each slice, slightly reduces the significance of the telluric residuals, but no improvement is found after removing more than ∼ 6 components. In all cases, relatively strong telluric residuals remain in the processed data. As expected, removing the spectral orders that are strongly affected by tellurics from the final coadded log results in a reduction of these telluric residuals. However, injection tests show that the injected planetary signal, which is clearly detected using all the orders, even when strong telluric residuals are present, also disappears. This occurs because both telluric and planetary water show the strongest absorption lines (and hence, the strongest signal) in similar spectral ranges. Finally, we find that modifying the PCA algorithm so that it uses only the specific parts of the spectrum affected by telluric absorption (i.e. pixels that capture telluric lines), rather than using the whole spectral range, results in a significant decrease of the telluric residuals. This happens because, by feeding the algorithm only with telluric-affected regions, telluric-related variations are more noticeable, and hence, are ranked higher than other effects in the PCA components. Therefore, in our case, avoiding the ranges where tellurics are the strongest in order to mitigate telluric residuals and enhance a potential planetary detection is not a good solution because the planetary signal is also suppressed. Instead, a PCA algorithm fed with pre-defined wavelength ranges where tellurics are known to be present results in a significantly stronger telluric mitigation, whilst preserving any potential H 2 O signals. We then cross-correlated the spectra resulting from the optimised PCA algorithm with a grid of models covering a range of water abundances and cloud deck levels. We use the CC-to-log Bayesian framework which allows us to robustly assess the significance of our results. We see that models with high water abundances and high cloud deck pressures, and models with low water abundances and low cloud deck pressures are significantly rejected. The preferred models are those with intermediate abundances and cloud deck pressures. These results are compatible with a potential detection of water in the atmosphere of WASP-166 b. If no water was detected, the preferred models would be those compatible with an almost flat spectrum, i.e., models with low water abundances and low cloud deck pressures, and only models with high water abundance and high cloud deck pressure would be excluded. We further tried to assess the significance of our best model by computing the CC function with a flat model. Compared to the best model, the flat one is rejected with 4 to 5 , meaning that, regardless of the origin, the data contain a signal ∼ 5 above a flat line. In the p − rest maps, we observe a correlated signal blue-shifted by about 5 km s −1 from the expected planetary RV. The signal observed in the two individual nights is shifted from the expected p by a few tens of km s −1 in opposite directions for each night. However, when both nights are coadded, the signal sits at the expected p and its significance is increased. The signals in the individual nights are within 1 and 2 from the coadded nights signal, meaning that the coadded nights signal is not strongly rejected by the individual night ones. Moreover, the transit observations analysed do not strongly constrain p because they only cover a small part of the total Keplerian orbit. The shift observed in the planetary rest could be due to winds in the planetary atmosphere. Global blue-shifts of the transmission spectrum of hot giant exoplanets have been predicted by several works (e.g. Miller-Ricci Kempton & Rauscher 2012;Showman et al. 2013;Rauscher & Kempton 2014). Such shifts have been observed in the optical through the Na doublet (e.g. Wyttenbach et al. 2015;Louden & Wheatley 2015) and tentatively reported in the infrared through CO (Snellen et al. 2010) andCO andH 2 O (Brogi et al. 2016;Flowers et al. 2019). The study of the Na doublet at ∼589 nm with the same observations as those analysed here (Seidel et al. 2022) shows that the Na lines are significantly broadened. This suggests the presence of winds, which seems compatible with what we might be observing here with H 2 O. An important step in the likelihood-ratio analysis is that the models are processed through the same PCA algorithm as the data. This is necessary to avoid biases introduced by the PCA modifying any potential planetary signal during the telluric correction performed initially, since a PCA can alter both the strength and the shape of the planetary lines, resulting in spurious shifts in p and sys . By processing the models through the same PCA as the data, both data and models should have been modified in the same way, which should result in a better match when performing the cross-correlation. In our case, we see that if we use a model without processing it through the same PCA as the data, then the tentative blue-shifted signal is very weak in the second night compared to what we obtain if the model has been adequately processed. The slow/processed-model method is the only method attempting to reproduce the effects of the telluric removal on the model, and therefore it should be taken as the most reliable reference when quoting a detection. The fast/unprocessedmodel method, despite being still common in the literature, is subject to biases with a large variety of telluric-removal algorithms, especially important when retrieving abundances, but also potentially affecting the measured value of p . Therefore, it is not surprising that the two methods give potentially different results, and such discrepancy does not imply that the tentative signal obtained with the slow method cannot be trusted. The biases of the fast/unprocessed-model method have been known for a few years now (e.g. the simulated tests in Brogi & Line 2019), and the slow/processed-model approach is standard among many research groups applying Bayesian analysis on high-resolution spectroscopy infrared data (e.g. recently Giacobbe et al. 2021;Line et al. 2021;Gibson et al. 2022;van Sluijs et al. 2022). To create the grid of models covering several H 2 O abundances and cloud deck pressures we used two different line lists, POKAZATEL and HITEMP, resulting in two sets of 99 models each. Since we are working with ESPRESSO observations, our water models cover optical wavelengths, a range for which published line lists have not been extensively empirically verified, as mentioned above. Hence, we can expect worse accuracy in general and differences between the two different line lists. Water lines are known but the line positions are not necessarily accurate. This is key in high-resolution studies such as the one performed here. A lack of accuracy in the line positions could result in Doppler shifts of any expected signal. Incomplete line lists could make any existing planetary water signal weaker, but we do not expect any possible incompleteness to create shifts in the planetary signal. In the data analysed here, we see that both the individual p − rest maps obtained for each model and the final grid of models are very similar for both POKAZATEL and HITEMP models, which points towards a good agreement between both sets of lines. Despite that, lines could still be inaccurate or incomplete in similar ways, and this agreement does not add evidence to support a planetary origin for the tentative signal observed. We note that when creating the models, we fixed their temperature and scaling factor, and only explored a range of water abundances and cloud deck pressures. We did not consider other sources of opacity. In other words, we did not perform a full atmospheric retrieval and have assumed that the parameters used to create our models are true, because our main goal was only to perform an initial assessment of the presence of water and clouds in the planetary atmosphere. Based on the tentative detection that we obtain, a full atmospheric retrieval is warranted to confirm the results reported here. Further observations of upcoming transits of WASP-166 b could also shed light on the differences obtained between the two nights studied here. To summarise, we have analysed the presence of water and clouds in WASP-166 b using two transits observed with ESPRESSO. We use the cross-correlation technique with a grid of models covering a range of water abundances and cloud deck pressures. We find a tentative planetary signal blue-shifted 5 km s −1 from the expected planet velocity in the p − rest maps, which could be caused by winds in the atmosphere. A comparison of the different models used favours those with intermediate water abundances and cloud deck pressures. Models with a high water abundance and low cloud deck pressure are strongly rejected, and models with low water abundance and high cloud deck pressure are also not preferred. If no planetary signal was present, we would expect models compatible with a flat spectrum (i.e. low water abundance and high cloud deck pressure) to be favoured, which is not what we observe, hence reinforcing the tentative signal observed in the p − rest maps. corrected the identified spikes and the adjacent pixels on each side by linear interpolation of the neighbouring points. After this, we fitted the continuum of the slice with a linear polynomial and divided by the best fit to remove instrumental slopes. We also flag channels with low S/N (channels with median flux lower than 2% of the overall median flux of all channels), which will not be used when computing the PCA eigenvectors. We then normalised (divided) each observation (each row) by its median flux value. This is done to account for variations in the light throughput in the different observations, so that all of them now have the same baseline flux. Each pixel or channel (each column) has its mean subtracted, so that the data matrix is centred. Then each channel is divided by its standard deviation, so that the matrix is now standardised. Hence, each channel has mean equal 0 and a standard deviation of 1. We note that the previous step of fitting and dividing by a linear polynomial to remove instrumental offsets is not strictly necessary, because any instrumental offset is removed afterwards when standarising each channel. However, removing these instrumental slopes is needed to correctly flag channels with low S/N (if not normalised, the flagged channels would be biased due to the instrumental slope). We then performed a PCA, but instead of directly decomposing the covariance matrix of the data as in Giacobbe et al. (2021), we performed it via a singular value decomposition (SVD) of the standarised data matrix (e.g. de Kok et al. 2013). If is the matrix with our standarised data, with dimensions × (i.e. rows × columns), the covariance matrix of the data is then given by T /( − 1). This can be diagonalised into T /( − 1) = T /( − 1), where contains the eigenvectors or principal components, and is a diagonal matrix containing the eigenvalues of a new basis. We can decompose the data matrix via an SVD into = Σ T , where Σ is an × diagonal matrix containing the singular values of , is an × matrix whose columns contain the left singular vectors of , and is an × matrix whose columns contain the right singular vectors of . The singular vectors are a set of orthogonal unit vectors, hence making a new orthonormal basis. If we now consider in terms of its SVD, it can be shown that its covariance matrix is T /( − 1) = ( Σ T )( Σ T ) T /( − 1) = ( Σ T ) ( Σ T )/( − 1) = Σ 2 T /( − 1). That is, the singular vectors of are equivalent to the principal components of the covariance matrix. We then created a new matrix containing the first columns of (i.e., the first eigenvectors of the SVD of or, what is the same, the first eigenvectors or principal components of the covariance matrix), where stands for number of components. We used this new matrix to perform a multi-linear regression with the initial matrix of data in order to determine the best-fit coefficients (i.e. the eigenvalues) for the linear combination of the chosen components. This resulted in a fit to the data that should contain most of the telluric, stellar, and instrumental variations, as captured by the first components of the PCA. We then divided the initial matrix of data by this fit and subtract 1. By doing this, we obtained the residuals of the observed data where the telluric, stellar, and instrumental variations captured by the components considered have been removed. Additionally, we applied a high-pass filter in the spectral direction to remove residual instrumental effects from the final processed data. Specifically, we first filter out pixels whose scatter (standard deviation) is larger than 2 times the median scatter of that specific pixel in all observations. We then fit 2-degree polynomials to each filtered observation to capture any residual instrumental effects, and subtract them out of each observation. APPENDIX B: CROSS-CORRELATION IMPLEMENTATION DETAILS B1 Fast/unprocessed-model CC approach In this approach, we compute the full CC and log functions over a grid of −100 to 100 km s −1 , in steps of 0.5 km s −1 (which corresponds to the ESPRESSO pixel width). That is, we shifted the model to each RV step, interpolated it to the wavelength grid of the observed spectra, and computed the CC and log functions following Equations 1 and 2. This is performed slice-by-slice for all the observations of each night, using the different models described in Section 3.2.1. For a specific model, this results in a CC and log function per slice, per observation, and per night. For each observation, we then combined the log functions of each slice by simply coadding them. There is no need to weight the different slices because the log already contains information about the different S/N of each slice. This results in a single log function per observation, per night. To enhance the planet signal, the log functions of the in-transit observations (where we expect the planetary signal to be) need to be coadded in the planet rest frame. To do this, we shift them by the corresponding planetary orbital velocity p , which we computed with the following equation (for which we assume that the planet has no eccentricity, Hellier et al. 2019) p ( ) = sys + p sin [2 ( )] ,(B1) where sys is the systemic velocity of the system, p is the planet orbital radial velocity semi-amplitude, and ( ) is the planet orbital phase. The phase is defined as ( ) = − 0 (B2) where 0 is the mid-transit time, and the orbital period of the planet, so that = 0 corresponds to mid-transit. After shifting all the log functions by the corresponding p , we only need to coadd them. When performing the shift to planet rest frame, we splineinterpolated the log functions of each observation to a common RV grid. This way, each point of the log functions of each observation can be directly summed. This results in a single log function per night. We perform this coadding for different p , computed using the expected sys and a range of p from 0 km s −1 to twice the expected value in steps of 1 km s −1 (following Equation B1). We obtain the expected p using the following equation with the most up-to-date literature values (see Table 1) p = 2 sin( p ),(B3) where is the planet semi-major axis, and p , the orbital inclination. By doing this, we can then produce the usual p − sys maps (or p − rest if we subtract sys ), since the RV grid of the log function is equivalent to sampling different sys (see Equation B1). In our case, since we included sys in the computation of p , the maps are in the planet rest frame, rather than in the systemic frame. B2 Slow/processed-model CC approach: precise implementation and model processing In the second approach, we compute a single CC and log value for each pair of p and sys considered. So rather than performing the cross-correlation with the same model shifted by a range of RV steps, we only shift the model once for each pair of p and sys values, and use this shifted model to compute a single point of the CC and log functions. As mentioned before 3.2.2, this approach allows us to process the model through the same PCA as the data prior to performing the cross-correlation. This is key to avoid biases and should result in a better match between model and data. In the following, we explain this model processing and the computation of the log with this slow/processed-model approach. To process the planetary water model through the same PCA as the data, we first created a data matrix with the same dimensions as the original spectra ( × ) containing the model that will be used to compute the CC (instead of the observed data). For the rows corresponding to in-transit observations, the model matrix contains the model shifted to the expected planet RV, interpolated to the same wavelength grid as each observation. For the rows corresponding to out-of-transit observations, the data matrix contains only ones. As explained in Section 3.1, the linear regression of the data with the selected PCA components results in a fit matrix that should only contain (if the PCA works as expected) the fitted tellurics and stellar lines, and changes in flux due to varying airmass and throughput. It also contains the overall drop in flux due to the planet transit, i.e., the broadband transmission planet spectrum. We want to inject the planetary water model to this fit matrix so that the model contains the same variability as the data. However, our model matrix already contains the drop in flux due to the planet transit, because the models are expressed in units of 1 − ( p / ★ ) 2 . Therefore, before injecting the models, we need to normalise them to remove this effect. We do this by dividing the in-transit observations in the model matrix by their mean (we do not need to apply any change to the out-of-transit observations, which are simple a flat spectrum at flux one). After this, we injected the normalised model to the fit from the data by multiplying the two matrices. We then apply the full PCA processing to this last matrix (including the out-of-transit observations), as done originally with the data. That is, we use the same number of components and bad-pixel masks, and perform the centering and standarisation, singular value decomposition, and linear regression. This results in a matrix with the processed model per observation, which should have been altered by the PCA in the same way as the real data. We then computed the CC and log of the in-transit observations using the same method as in the first approach (i.e. Equations 1 and 2). In this case, however, we have already shifted the template to the expected planet RV (for a specific pair of p and sys values). Therefore, we only compute the CC and log once for each observation. To get a single log value per observation, we then directly sum the log values that we obtain for each slice. Since the log has been computed with the model already shifted to the expected planet RV, we can directly sum the log of all the observations, as we are already in planet rest frame, and there is no need for interpolation as in the first approach. This directly gives us a data point on the p − sys maps. The processing of the model depends on the chosen p and sys values, therefore, we repeated this whole process (model processing and computation of a single CC and log value) for each pair of p and sys values considered, resulting in the full p − sys map (or again, p − rest if we subtract sys ). APPENDIX C: POKAZATEL MODELS FOR HIGH WATER ABUNDANCES This paper has been typeset from a T E X/L A T E X file prepared by the author. on the 8.2 m Subaru telescope, ESPaDOnS (5060-7950 Å, Donati 2003) on the 3.6 m CFHT, and GRACES (3990-10480 Å, Chene et al. 2014) on the 8.1 m PCA) on the observed spectral time series inspired by Giacobbe et al. (2021) (see also de Kok et al. 2013; Piskorz et al. 2016, 2017; Figure 1 . 1Observing conditions, S/N, and Earth, star, and planet RVs for the 2 transits observed, as a function of the planetary phase, where phase 0 corresponds to the mid-transit. Figure 3 . 3Left: POKAZATEL H 2 O templates for WASP-166 b covering the ESPRESSO wavelength range for a range of water abundances (log 10 (H 2 O) = −1 top, log 10 (H 2 O) = −4 middle, and log 10 (H 2 O) = −5 bottom), and a range of cloud deck pressures (depicted by various colours in all panels). Right: Zoom in on a region with strong absorption lines. This figure shows that the strength of the water absorption lines decreases as we decrease the abundance and/or decrease the cloud deck pressure. Figure 4 . 4RV (K p =134 km/s) + V sys =24 km/s Star RV + V sys =24 km/s BERV Comparison of telluric removal algorithms (Section 3.1 and sub-sections). Top row: CC of each observation as a function of the observation phase. Left and right panels correspond to the two transits observed. The white, yellow, and red dashed lines correspond to the planetary, stellar, and barycentric Earth RVs, respectively. The short-dash white lines indicate the transit ingress and egress. The CCs shown correspond to the coadding of the CCs of all the slices considered. The CCs have been computed with the log 10 (H 2 O) = −3 and log 10 ( cloud /bar) = 0 model using the fast/unprocessed-model approach, and in the PCA processing we removed 6 components for all slices. Second row: Same as top, but in this case only the CCs of slices 84-95, 104-107, 124-127, 142-145 (slices with no or small telluric contamination) have been coadded. Third row: Same as top, but in this case the number of components removed per slice has been optimised. Bottom row: Same as top, but in this case only the pixels affected by tellurics have been used in the PCA. Figure 5 . 5CC functions from a selected sub-set of tests performed to optimise the telluric removal via PCA. The CC shown are the result of coadding the CC of each in-transit observation in the planet rest frame, at the expected p and sys . Shaded grey, yellow, and red areas correspond to the the expected planetary, stellar, and barycentric Earth RVs, respectively. Left and right correspond to the first and second night of observations, and top and bottom correspond to different sets of tests (see the legend). Figure 6 . 6Top row: p − rest confidence interval maps obtained when removing 6 components from the PCA for all slices, using the log 10 (H 2 O) = −3 and log 10 ( cloud /bar) = 0 model based on the POKAZATEL line list to compute the CC, and coadding all the slices (this corresponds to coadding the in-transit CC shown in the top panel of Figure 7 . 7Same asFigure 4, but in this case, the planetary model used in the CC has been initially injected in the observations with no scaling. Figure 8 . 8slow/processed-model approach, POKAZATEL (NC 6, all orders, PCA telluric pixels) Top: Confidence intervals (colour) for the POKAZATEL model grids of different water abundances (log 10 (H 2 O) = −1 to −5, in VMR, x-axis) and cloud deck pressures (log 10 ( cloud /bar) = 0 to −5, y-axis) obtained with the slow/processed-model approach to compute the CCs. Left and middle panels are the results for each of the transits, and right, for both transits coadded. Bottom: Same as top, but here contour levels have been added to help visualise the confidence intervals. Figure 9 . 9slow/processed-model approach, HITEMP (NC 6, all orders, PCA telluric pixels) p − sys confidence interval maps obtained with the PCA algorithm that uses only telluric-affected regions, removing 6 PCA components, CCs computed with the slow/processed-model approach, and using the log 10 (H 2 O) = −4 and log 10 ( cloud /bar) = 0 model based on the POKAZATEL (top) and HITEMP (bottom) line list to compute the CC, and coadding all the order slices. Red dashed lines indicate the expected p and sys . Left to right are the maps for the first, second, and coadded nights. Figure C1 . C1Left: POKAZATEL H 2 O templates for WASP-166 b covering the ESPRESSO wavelength range for a range of cloud deck pressures (log 10 ( cloud /bar) = 0 top, log 10 ( cloud /bar) = −1 middle, log 10 ( cloud /bar) = −2 bottom), and a range of water abundances (depicted by the various colours in all panels). Right: Zoom in on a region with strong absorption lines. This figure shows the decrease in absorption strength for water rich atmospheres (log 10 (H 2 O) = −1, purple) compared to lower abundances (log 10 (H 2 O) = −2, blue) due to the increase in mean molecular weight for water-rich atmospheres (see Section 4.2.1). For lower water abundances (log 10 (H 2 O) = −3, green), the strength of the absorption features decreases as expected due to the decrease in water content. Table 1 . 1WASP-166 system properties used in this work.Parameter Value Reference p / ★ 0.05177 +0.00063 −0.00035 Doyle et al. (2022) p [ J ] 0.6155 +0.0306 −0.0307 Doyle et al. (2022) / ★ 11.83 +0.29 −0.68 Doyle et al. (2022) [AU] 0.0668 +0.0040 −0.0044 Doyle et al. (2022) p [ • ] 88.85 +0.74 −0.94 Doyle et al. (2022) 0 [BJD] 2458524.40869201 +0.00030021 −0.00029559 Doyle et al. (2022) dur [hours] 3.608 +0.020 −0.015 Doyle et al. (2022) [days] 5.44354215 +0.00000307 −0.00000297 Doyle et al. (2022) 0.0 Hellier et al. (2019) sys [km s −1 ] 23.532 ± 0.012 Doyle et al. (2022) p [km s −1 ] 134.1 ± 8.1 This work eff [K] 6050±50 Hellier et al. (2019) eq [K] 1270±30 Hellier et al. (2019) Notes: Values from Doyle et al. (2022) have been derived using the same ESPRESSO observations as here, as well as TESS and NGTS photometry. In particular, sys has been measured from the out-of-transit cross-correlation functions of the ESPRESSO data and here we use the mean sys of the two nights (Doyle et al. 2022, see their Table 1). p has been computed here based on the parameters from Doyle et al. (2022) (see Section 3.2 and Appendix B). Figure 2. Example of an observed spectrum of, telluric template used to select telluric-affected regions (middle), and one of the H 2 O models used to compute the CC functions (bottom). Grey and blue shaded regions and numbers indicate the different ESPRESSO orders (also colour-coded in the WASP-166 spectrum in the top panel). The numbers correspond to the ESPRESSO slices (each order has two slices), starting at 0 for the first slice of the bluest order). We only show the wavelength range covering the spectral region used (where planetary model shows stronger absorption.1000 2000 Obs. spectrum [Norm. flux] r.ESPRE.2021-02-19T08:23:15.856_S2D_SKYSUB_A 80 82 84 86 88 90 92 94 96 98 100102104106108110112114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 0.00 0.25 0.50 0.75 1.00 Telluric model [Transm. fract.] 80 82 84 86 88 90 92 94 96 98 100102104106108110112114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 5000 5500 6000 6500 7000 7500 Wavelength [Å] 0.9966 0.9968 0.9970 0.9972 H 2 O model [1 (R p /R ) 2 ] log10(H2O)=-3, log10(Pcloud/bar)=0 80 82 84 86 88 90 92 94 96 98 100102104106108110112114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 M. Lafarga et al. www.eso.org/sci/software/pipelines/espresso/ espresso-pipe-recipes.html MNRAS 000, 1-20(2022) https://www.eso.org/observing/etc/skycalc MNRAS 000, 1-20 (2022) ACKNOWLEDGEMENTSThis work is based on observations made with ESO Telescopes at the La Silla Paranal Observatory under the programme ID 106.21EM. ML, HMC, and LD acknowledge funding from a UKRI Future Leader Fellowship, grant number MR/S035214/1. SG is grateful to Leiden Observatory at Leiden University for the award of the Oort Fellowship. RA is a Trottier Postdoctoral Fellow and acknowledges support from the Trottier Family Foundation. This work was supported in part through a grant from FRQNT. MLe acknowledges support of the Swiss National Science Foundation under grant number PCEFP2_194576. The contribution of MLe has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. We thank G. Frame for useful discussions of this work. We thank the anonymous referee for the very helpful and thorough review of the article. This work made use of numpy(Harris et al. 2020), scipy(Virtanen et al. 2020), astropy (Astropy Collaboration et al. 2013, 2018, and matplotlib(Hunter 2007).DATA AVAILABILITYThe ESPRESSO data used in this work is publicly available from the ESO archive under programme ID 106.21EM.APPENDIX A: PCA IMPLEMENTATION DETAILSBefore applying the PCA, the observations of each slice are cleaned from bad pixels. We corrected for flux anomalies caused by cosmic rays. To do this, we first identified outliers by performing a sigmaclip on values deviating more than +3 and −6 times the standard deviation of the slice flux (values tailored for these data), and then . R Allart, C Lovis, L Pino, A Wyttenbach, D Ehrenreich, F Pepe, 10.1051/0004-6361/201730814A&A. 606144Allart R., Lovis C., Pino L., Wyttenbach A., Ehrenreich D., Pepe F., 2017, A&A, 606, A144 . 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{'abstract': "With high-resolution spectroscopy we can study exoplanet atmospheres and learn about their chemical composition, temperature profiles, and presence of clouds and winds, mainly in hot, giant planets. State-of-the-art instrumentation is pushing these studies towards smaller exoplanets. Of special interest are the few planets in the 'Neptune desert', a lack of Neptune-size planets in close orbits around their hosts. Here, we assess the presence of water in one such planet, the bloated super-Neptune WASP-166 b, which orbits an F9-type star in a short orbit of 5.4 days. Despite its close-in orbit, WASP-166 b preserved its atmosphere, making it a benchmark target for exoplanet atmosphere studies in the desert. We analyse two transits observed in the visible with ESPRESSO. We clean the spectra from the Earth's telluric absorption via principal component analysis, which is crucial to the search for water in exoplanets. We use a cross-correlation-to-likelihood mapping to simultaneously estimate limits on the abundance of water and the altitude of a cloud layer, which points towards a low water abundance and/or high clouds. We tentatively detect a water signal blue-shifted ∼5 km s −1 from the planetary rest frame. Injection and retrieval of model spectra show that a solar-composition, cloud-free atmosphere would be detected at high significance. This is only possible in the visible due to the capabilities of ESPRESSO and the collecting power of the VLT. This work provides further insight on the Neptune desert planet WASP-166 b, which will be observed with JWST.", 'arxivid': '2302.04794', 'author': ['M Lafarga \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK\n', 'M Brogi \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom\n\nINAF -Osservatorio Astrofisico di Torino\nVia Osservatorio 2010025\n\nPino Torinese\nItaly\n\nDipartimento di Fisica\nUniversità degli Studi di Torino\nvia Pietro Giuria 1I-10125TorinoItaly\n', 'S Gandhi \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK\n\nLeiden Observatory\nLeiden University\n9513, 2300 RAPostbus, LeidenThe Netherlands\n', 'H M Cegla \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK\n', '† J V Seidel \nEuropean Southern Observatory\nAlonso de Córdova 3107VitacuraRegión MetropolitanaChile\n', 'L Doyle \nDepartment of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom\n\nCentre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK\n', 'R Allart \nDepartment of Physics\nTrottier Institute for Research on Exoplanets\nUniversité de Montréal\nH3T 1J4MontréalCanada\n', "N Buchschacher \nObservatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland\n", "M Lendl \nObservatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland\n", "C Lovis \nObservatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland\n", "D Sosnowska \nObservatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland\n"], 'authoraffiliation': ['Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom', 'Centre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK', 'Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom', 'INAF -Osservatorio Astrofisico di Torino\nVia Osservatorio 2010025', 'Pino Torinese\nItaly', 'Dipartimento di Fisica\nUniversità degli Studi di Torino\nvia Pietro Giuria 1I-10125TorinoItaly', 'Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom', 'Centre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK', 'Leiden Observatory\nLeiden University\n9513, 2300 RAPostbus, LeidenThe Netherlands', 'Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom', 'Centre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK', 'European Southern Observatory\nAlonso de Córdova 3107VitacuraRegión MetropolitanaChile', 'Department of Physics\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUnited Kingdom', 'Centre for Exoplanets and Habitability\nUniversity of Warwick\nCV4 7ALCoventryUK', 'Department of Physics\nTrottier Institute for Research on Exoplanets\nUniversité de Montréal\nH3T 1J4MontréalCanada', "Observatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland", "Observatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland", "Observatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland", "Observatoire Astronomique de l'Université de Genève\nChemin Pegasi 51bCH-1290VersoixSwitzerland"], 'corpusid': 256697560, 'doi': '10.1093/mnras/stad480', 'github_urls': [], 'n_tokens_mistral': 34128, 'n_tokens_neox': 28293, 'n_words': 18190, 'pdfsha': 'be17f5add20925459e6677071c62063e56db1611', 'pdfurls': ['https://export.arxiv.org/pdf/2302.04794v1.pdf'], 'title': ['The hot Neptune WASP-166 b with ESPRESSO III: A blue-shifted tentative water signal constrains the presence of clouds', 'The hot Neptune WASP-166 b with ESPRESSO III: A blue-shifted tentative water signal constrains the presence of clouds'], 'venue': ['MNRAS']}

arxiv

AN UNCERTAINTY PRINCIPLE, WEGNER ESTIMATES AND LOCALIZATION NEAR FLUCTUATION BOUNDARIES 18 May 2009 Anne Boutet De Monvel ⋆ Daniel Lenz Peter Stollmann AN UNCERTAINTY PRINCIPLE, WEGNER ESTIMATES AND LOCALIZATION NEAR FLUCTUATION BOUNDARIES 18 May 2009 We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven. Introduction Starting point of the present paper was the lamentable fact that for certain random models with possibly quite small and irregular support there was a proof of localization via fractional moment techniques (at least for d ≤ 3) but no proof of Wegner estimates necessary for multiscale analysis. The classes of models include models with surface type random potentials as well as Anderson models with displacement (see [1]) but actually much more classes of examples could be seen in the framework established there which was labelled "fluctuation boundaries". Actually, the big issue in the treatment of random perturbations with small or irregular support is the question, whether the spectrum at low energies really feels the random perturbation. This is exactly what is formalized in the fluctuation boundary framework. In the present paper we stablish the necessary Wegner estimates by using the method from Combes, Hislop, and Klopp [6] so that we get the correct volume factor and the modulus of continuity of the random variables. One of the main ideas we borrow from the last mentioned work is to show that spectral projectors are "spread out", a property we call "uncertainty principle". The solution to the above mentioned problem is now quite simple in fact. In an abstract framework we show that such an uncertainty principle of the form P I (H 0 )W P I (H 0 ) ≥ κP I (H 0 ), (0.1) where W ≥ 0 is bounded and P I (H 0 ) denotes the spectral projection, is in a sense equivalent to the mobility of λ(t) := inf σ(H 0 + tW ). (0.2) This is done in Section 1. That fits perfectly with the fluctuation boundary concept and gives the appropriate Wegner estimates. Actually, if the integrated density of states exists, it then must be continuous, provided the distribution of the random variables has a common modulus of continuity. We will prove this in Section 2. Finally, in Section 3 we show how to exploit these Wegner estimates for a proof of localization. It lies in the nature of these different methods that we thus get localization under Date: 21.4.2009. less restrictive conditions than what was needed in [1]. One main point is the dimension restriction of the latter paper, d ≤ 3, which certainly is not essential but is essential for a proof of digestable length. Clearly, the estimates one gets via the fractional moment method are more powerful. Acknowledgement. Fruitful discussions with G. Stolz are gratefully acknowledged. Part of this work was done at a stay of D.L. and P.S. at Paris, financial support by the DFG (German Science Foundation) and the University Paris Diderot Paris 7 are gratefully acknowledged. 1. An uncertainty principle and mobility of the ground state energy In this section we fix a rather abstract setting: H is a Hilbert space, H 0 is a selfadjoint operator in H with λ(0) := inf σ(H 0 ) > −∞. (1.1) Moreover, W is assumed to be bounded and nonnegative. The uncertainty principle we want to study is the existence of a positive κ such that P I W P I ≥ κP I (⋆) where I ⊂ R is some compact interval, I = [min I, max I] and P I = P I (H 0 ) = χ I (H 0 ) is the corresponding spectral projection. It is reasonable to call (⋆) an uncertainty principle as a state in the range of P I cannot be "concentrated where W vanishes". In our main application, H 0 will be a Schrödinger operator so that (⋆) is in fact a variant of the usual uncertainty principle, at least for H 0 = −∆. The use of (⋆) for the proof of Wegner estimates is due to Combes, Hislop, and Klopp, see [5,6]. Its importance lies in the fact that it takes care of random potentials with small support. Our purpose here is to prove a simple criterion that implies (⋆) and can be checked rather easily. 1.1. Theorem. Let for t ≥ 0 λ(t) := inf σ(H 0 + tW ) (1.2) and assume that λ(t 0 ) > max I for some t 0 > 0. Then P I W P I ≥ sup t>0 λ(t) − max I t P I . (1.3) Of course, the assumption in the theorem is merely there to guarantee that the square bracket is positive! Proof. Assume that (⋆) does not hold for some κ > 0. Then we find g ∈ Ran P I with g = 1 and W g, g = P I W P I g, g < κ. Since H 0 g, g ≤ max I, by the functional calculus, we get, for any t > 0, λ(t) ≤ (H 0 + tW )g, g < max I + t κ, which implies κ > λ(t) − max I t . By contraposition, we get the assertion. Remarks. (1) One particularly nice aspect of the above result is that the important constant is controlled in a simple way. (2) Once the ground state energy is pushed up by W we get an uncertainty principle (⋆) at least for intervals I near λ(0). (3) The corresponding uncertainty result in [5] for periodic Schrödinger operators does not follow from the preceding theorem. There is a kind of converse to Theorem 1.1. 1.2. Lemma. If (⋆) holds for I with min I = λ(0) = inf σ(H 0 ) and max I > min I, then λ(t) > λ(0) for all t > 0. Proof. We only need to consider small t > 0 since W ≥ 0. For f ∈ D(H 0 ), f = 1, let f 1 := P I f and f 2 := P I c f so that f 1 2 + f 2 2 = 1. We consider (H 0 + tW )f, f = H 0 f 1 , f 1 + H 0 f 2 , f 2 + t W f, f ≥ (max I) f 2 2 + λ(0) f 1 2 + tκ f 1 2 − 2t W f 1 f 2 ≥ λ(0) f 2 + (max I − λ(0)) f 2 2 − 2t f 1 f 2 W + tκ f 1 2 . A knowing smile at the last quadratic (!) expression in t reveals that it is strictly larger than λ(0) for t small enough. Continuity of the IDS near weak fluctuation boundaries The main result here is, in fact, rather an "optimal" Wegner estimate meaning that we recover at least the modulus of continuity of the random variables in the Wegner estimate as well as the correct volume factor. The models we consider needn't have a homogeneous background so that the integrated density of states, IDS need not exist. See [12] for a recent survey on how to prove the existence of the IDS in various different settings. We show that a straightforward application of Theorem 1.1 above gives the necessary input to perform the analysis of [6] in a rather general setting which we are going to introduce now. (A1) The background potential V 0 ∈ L p loc,unif (R d ) with p = 2 if d ≤ 3, and p > d 2 if d > 3. (A2) The set I ⊂ R d , where the random impurities are located, is uniformly discrete, i.e., inf α,β∈I α =β |α − β| =: r I > 0. ( A3) For the probability measure P on Ω = α∈I [0, η max ] we use conditional probabilities to define the following uniform bound s(ε) = sup α ess sup E∈R ess sup (ω β ) β =α P ω α ∈ [E, E + ε] | (ω β ) β =α . ( A4) Let E 0 := inf σ(H 0 ) and let H F := H 0 + η max α∈I U α the subscript F standing for "full coupling". The single site potentials U α , α ∈ I are measurable functions on R d that satisfy c U χ Λr U (α) ≤ U α ≤ C U χ Λ R U (α) for all α ∈ I, with c U , C U , r U , R U > 0 independent of α. Here, Λ s (β) denotes the box with sidelength 2s and center β. V ω (x) = α∈I ω α U α (x) and H := H(ω) := H 0 + V ω in L 2 (R d ). Assume that E 0 is a weak fluctuation boundary in the sense that E F := inf σ(H F ) > E 0 . Remarks. (1) In [1] (A3) and (A4) are stronger than their counterparts ( A3) (which actually isn't an assumption at all) and ( A4) here. (2) The modulus of continuity s( · ) from (A3) also appears in [6], where, however, the variables appearing in the conditional probabilities are not displayed correctly. For a detailed discussion of regular conditional probabilities see, e.g., [9]. Wegner estimates, named after Wegner's original work [13], are an important tool in random operator theory. They give a bound on the probability that the eigenvalues of a local Hamiltonian come close to a given energy. For a list of some recent papers, see [2,3,4,5,6,8,10] and the account in the recent Lecture Notes Volume [12]. We consider a box Λ ⊂ R d and denote by H Λ (ω) the restriction of H(ω) to L 2 (Λ) with Dirichlet boundary conditions and with H Λ 0 the restriction of H 0 to L 2 (Λ) with Dirichlet boundary conditions. Here comes our main application of Theorem 1.1: 2.1. Theorem. Assume (A1)-(A2) and ( A3)-( A4). Then, for every δ > 0 there exists a constant C W = C W (δ) such that for any interval I = [E 0 , E F − δ] we have: P σ H Λ (ω) ∩ I = ∅ ≤ E tr P I H Λ (ω) ≤ C W · |Λ| · s(ε). Clearly, any application will need some further assumptions on s( · ) for which we a priori just know that 0 ≤ s(ε) ≤ 1 for all ε > 0. Proof. We rely on the analysis from [6]. The main point is to find an estimate P I (H Λ 0 )W Λ P I (H Λ 0 ) ≥ κP I (H Λ 0 ) (⋆⋆) with a constant κ independent of Λ and I (as long as I ⊂ [E 0 , E F − δ]), and W Λ := α∈I U α · χ Λ . Once (⋆⋆) is established, the proof of [6, Theorem 1.3] takes over, with minor modifications of notation. But (⋆⋆) follows easily from Theorem 1.1 and ( A4): As Dirichlet boundary conditions shift the spectrum up, for any t ≥ η max : λ(t) = inf σ(H Λ 0 + tW Λ ) ≥ inf σ(H 0 + tW ) ≥ E F . For I ⊂ [E 0 , E F − δ] we see that λ(η max ) − max I > δ so that we get an uncertainty inequality with κ = δ η max . Remark. We should point out that the input from [6] is rather substantial. While the uncertainty principle is important to deal with possibly small support, there is also the issue of the correct volume factor which is settled in [6]. Like in the latter paper, if we assume on top that the IDS N ( · ) of the random operator H exists, then the preceding theorem implies that N ( · ) is as continuous as s( · ) is. ( · ) on [E 0 , E F ) such that N (E + ε) − N (E) ≤ c W (E) · s(ε) for ε small enough. In particular, N ( · ) is continuous on [E 0 , E F ), whenever s(ε) → 0 as ε → 0. Localization near fluctuation boundaries As mentioned already in the introduction, the validity of a Wegner estimate was missing for a proof of localization via multiscale analysis. Due to Theorem 2.1, this is now resolved. The assumptions we need to make now are weaker than what is found in [1] but stronger than what we needed in the preceding section. (A3) The random variables η α : Ω → R, ω → ω α are independent, supported in [0, η max ] and the modulus of continuity s(ε) := sup α∈I sup E∈R P{η α ∈ [E, E + ε]} satisfies s(ε) ≤ (− ln ε) −α for some α > 4d 2−m , where m ∈ (0, 2) is as in (A4). (A4) Additionally to ( A4) assume that there exists m ∈ (0, 2) and L * such that for some ξ > 0, all L ≥ L * and all x ∈ Z d : P σ H Λ L (x) (ω) ∩ [E 0 , E 0 + L −m ] = ∅ ≤ L −ξ . Remark. Cleary, the assumptions (A1)-(A4) from [1] imply (A1)-(A2) and (A3)-(A4) so that the localization result below extends the localization result from the latter paper. E |X| p η H(ω) · χ K < ∞ for every compact K ⊂ R d . Remarks. (1) Maybe one can strengthen the estimate of Theorem 3.1 in the sense of [7]. Note, however, that in the latter paper a stronger Wegner estimate is supposed to hold. (2) The theorem provides an extension to d ≥ 4 of the main result of [1]. Moreover, there is no technique at the moment to include single site distributions as singular as the ones allowed here in the fractional moment methods. In these aspects, our result considerably extends the main result of [1]. (3) At the same time, the estimates that come out of our analysis are weaker than those in the latter paper. Sketch of the proof. We use the multiscale setup from [11]. By now it is quite well understood that homogeneity doesn't play a major role so that multiscale analysis goes through without much alterations if we can verify the necessary input, i.e., Wegner estimates and initial length scale estimates. Let us begin with the latter: Combes-Thomas estimates give that (A4) implies an initial estimate of the form G(I, ℓ, γ, ξ) with ξ from (A4), I ℓ = [E 0 , E 0 + 1 2 ℓ −m ], γ ℓ = ℓ − m 2 so that the exponent is of the form γ ℓ = ℓ β−1 with β = β m = 2−m 2 . We have to check that an appropriate Wegner estimate is valid as well, i.e., that, for some q > d, θ < β 2 we have that P dist σ H Λ (ω) , E ≤ exp(−L θ ) ≤ L −q for L large enough. We check that 2. 2 . 2Corollary. Assume (A1)-(A2) and ( A3)-( A4), and, additionally that the IDS N ( · ) of H exists. Then there exists a locally bounded c W P dist σ H Λ (ω) , E ≤ exp(−L θ ) ≤ s 2 exp(−L θ ) have chosen θ = β 2 − x = 2−m 4 − κ with positive κ. Then, the Wegner estimate is fulfilled for q = d + κ 4d 2−m . The appropriate p in the strong dynamical localization estimate can be chosen at most inf κ 4d 2−m , ξ with ξ from (A3). An appeal to [11, Theorems 3.2.2 and 3.4.1] gives the result. 3.1. Theorem. Assume (A1)-(A2) and (A3)-(A4). Then there is a δ > 0 such that in [E 0 , E 0 + δ] the spectrum of H(ω) is pure point P-a.s. Moreover, for p small enough and η ∈ L ∞ with supp η ⊂ [E 0 , E 0 + δ] it follows that Localization near fluctuation boundaries via fractional moments and applications. A Boutet De Monvel, S Naboko, P Stollmann, G Stolz, J. Anal. Math. 100A. Boutet de Monvel, S. Naboko, P. Stollmann, and G. Stolz. Localization near fluctuation boundaries via fractional moments and applications. J. Anal. Math., 100:83-116, 2006. Wegner-Stollmann type estimates for some quantum lattice systems. V Chulaevsky, Adventures in mathematical physics. Providence, RIAmer. Math. Soc447V. Chulaevsky. Wegner-Stollmann type estimates for some quantum lattice systems. In Adventures in mathematical physics, volume 447 of Contemp. Math., pages 17-28. Amer. Math. Soc., Providence, RI, 2007. A Wegner-type estimate for correlated potentials. V Chulaevsky, Math. Phys. Anal. Geom. 112V. Chulaevsky. A Wegner-type estimate for correlated potentials. Math. Phys. Anal. Geom. 11(2): 117-129, 2008. Wegner bounds for a two-particle tight binding model. V Chulaevsky, Y Suhov, Comm. Math. Phys. 2832V. Chulaevsky and Y. Suhov. Wegner bounds for a two-particle tight binding model. Comm. Math. Phys. 283(2): 479-489, 2008. Hölder continuity of the integrated density of states for some random operators at all energies. J.-M Combes, P D Hislop, F Klopp, Int. Math. Res. Not. 4J.-M. Combes, P. D. Hislop, and F. Klopp. Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Not., 2003(4):179-209, 2003. An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. J.-M Combes, P D Hislop, F Klopp, Duke Math. J. 1403J.-M. Combes, P. D. Hislop, and F. Klopp. An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J., 140(3):469- 498, 2007. Bootstrap multiscale analysis and localization in random media. F Germinet, A Klein, Comm. Math. Phys. 2222F. Germinet and A. Klein. Bootstrap multiscale analysis and localization in random media. Comm. Math. Phys. 222(2): 415-448, 2001. Bounds on the spectral shift function and the density of states. D Hundertmark, R Killip, S Nakamura, P Stollmann, I Veselić, Comm. Math. Phys. 2622D. Hundertmark, R. Killip, S. Nakamura, P. Stollmann and I. Veselić. Bounds on the spectral shift function and the density of states. Comm. Math. Phys. 262(2): 489-503, 2006. Probability theory. A Klenke, Springer-Verlag London LtdLondonUniversitextA comprehensive course, Translated from the 2006 German originalA. Klenke. Probability theory. Universitext. Springer-Verlag London Ltd., London, 2008. A comprehensive course, Translated from the 2006 German original. Continuity of integrated density of states-independent randomness. M Krishna, Proc. Indian Acad. Sci. Math. Sci. 1173M. Krishna. Continuity of integrated density of states-independent randomness. Proc. Indian Acad. Sci. Math. Sci., 117(3):401-410, 2007. Caught by disorder. P Stollmann, Progress in Mathematical Physics. Birkhäuser Boston Inc. 20P. Stollmann. Caught by disorder, volume 20 of Progress in Mathematical Physics. Birkhäuser Boston Inc., Boston, MA, 2001. Existence and regularity properties of the integrated density of states of random Schrödinger operators. I Veselić, Lecture Notes in Mathematics. 1917Springer-VerlagI. Veselić. Existence and regularity properties of the integrated density of states of random Schrödinger operators, volume 1917 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008. Bounds on the density of states in disordered systems. F Wegner, Z. Phys. B. 441-2Institut de Mathématiques de Jussieu. Université Paris Diderot Paris 7, 175 rue du ChevaleretF. Wegner. Bounds on the density of states in disordered systems. Z. Phys. B, 44(1-2):9-15, 1981. ⋆ Institut de Mathématiques de Jussieu, Université Paris Diderot Paris 7, 175 rue du Chevaleret, France E Paris, $^{\star}[email protected] URL. Paris, France E-mail address: $^{\star}[email protected] URL: http://www.math.jussieu.fr/ ∼ aboutet/

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{'abstract': 'We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven.', 'arxivid': '0905.2845', 'author': ['Anne Boutet De Monvel ⋆ ', 'Daniel Lenz ', 'Peter Stollmann '], 'authoraffiliation': [], 'corpusid': 13997064, 'doi': '10.1007/s00209-010-0756-8', 'github_urls': [], 'n_tokens_mistral': 5727, 'n_tokens_neox': 5032, 'n_words': 3144, 'pdfsha': 'ac852cf53b3e5dc829e0a98a8aafadeb092fda50', 'pdfurls': ['https://arxiv.org/pdf/0905.2845v1.pdf'], 'title': ['AN UNCERTAINTY PRINCIPLE, WEGNER ESTIMATES AND LOCALIZATION NEAR FLUCTUATION BOUNDARIES', 'AN UNCERTAINTY PRINCIPLE, WEGNER ESTIMATES AND LOCALIZATION NEAR FLUCTUATION BOUNDARIES'], 'venue': []}

arxiv

Causalainer: Causal Explainer for Automatic Video Summarization Jia-Hong Huang [email protected] University of Amsterdam Georgia Institute of Technology IBM Research AI NVIDIA University of Amsterdam Chao-Han Huck University of Amsterdam Georgia Institute of Technology IBM Research AI NVIDIA University of Amsterdam Yang University of Amsterdam Georgia Institute of Technology IBM Research AI NVIDIA University of Amsterdam Pin-Yu Chen [email protected] University of Amsterdam Georgia Institute of Technology IBM Research AI NVIDIA University of Amsterdam Min-Hung Chen [email protected] University of Amsterdam Georgia Institute of Technology IBM Research AI NVIDIA University of Amsterdam Marcel Worring [email protected] University of Amsterdam Georgia Institute of Technology IBM Research AI NVIDIA University of Amsterdam Causalainer: Causal Explainer for Automatic Video Summarization The goal of video summarization is to automatically shorten videos such that it conveys the overall story without losing relevant information. In many application scenarios, improper video summarization can have a large impact. For example in forensics, the quality of the generated video summary will affect an investigator's judgment while in journalism it might yield undesired bias. Because of this, modeling explainability is a key concern. One of the best ways to address the explainability challenge is to uncover the causal relations that steer the process and lead to the result. Current machine learning-based video summarization algorithms learn optimal parameters but do not uncover causal relationships. Hence, they suffer from a relative lack of explainability. In this work, a Causal Explainer, dubbed Causalainer, is proposed to address this issue. Multiple meaningful random variables and their joint distributions are introduced to characterize the behaviors of key components in the problem of video summarization. In addition, helper distributions are introduced to enhance the effectiveness of model training. In visual-textual input scenarios, the extra input can decrease the model performance. A causal semantics extractor is designed to tackle this issue by effectively distilling the mutual information from the visual and textual inputs. Experimental results on commonly used benchmarks demonstrate that the proposed method achieves state-of-the-art performance while being more explainable. Introduction Video summarization is the process of automatically generating a concise video clip that conveys the primary message or story in the original video. Various automatic video summarization algorithms have been proposed in recent years to tackle this task using different supervision schemes. These include fully-supervised methods that utilize visual input alone [10,13,35,36,[68][69][70][71] or multi-modal input [26,27,42,43,46,51,54,56,58,65,72], as well as † Work done during an internship at Microsoft Research in Cambridge, UK and Amsterdam, NL. ‡ ex-Microsoft. 19 29 34 Figure 1. Visualization of human-annotated and machine-predicted frame-level scores for creating a video summary. Comparing the human-annotated video summary score pattern to the one generated by existing state-of-the-art video summarization methods, e.g., [26,27,56], we observe these methods are capable of learning visual consecutiveness and diversity which are some key factors considered by humans for creating a good video summary. These methods mainly focus on capturing the visual cues to achieve such a purpose. Red bars denote discarded frames and grey bars indicate selected frames used to form a summary. The video has 199 frames and the numbers, except for 199, denote the indices of frames. weakly-supervised methods [4,6,17,38,47,60]. According to [10,13,14,27,55,56], when human experts perform the task of video summary generation, they will not only consider concrete/visual factors, e.g., visual consecutiveness and visual diversity, but also abstract/nonvisual factors, such as interestingness, representativeness, and storyline smoothness. Hence, a human-generated video summary is based on many confounding factors. These factors/causes result in the video summary. Existing works do not, or in a very limited way, consider abstract factors and mainly focus on proposing various methods to exploit concrete visual cues to perform video summarization. See the illustration in Figure 1. This leads to limited modeling explainability of automatic video summarization [27,37]. Machine learning (ML) models can be made more explainable through causation modeling based on Bayesian probability [39,48,49,66,67]. In this work, we propose a novel method for improving the inherent explainability of video summarization models called Causalainer, which is based on causation modeling. Our approach aims to address the challenge of model explainability in video summarization by leveraging the insights gained from Bayesian probability and causation modeling. See Figure 2 for the method flowchart of the proposed Causalainer. To model the problem of video summarization and increase the explainability, four meaningful random variables are introduced to characterize the behaviors of the data intervention, the model's prediction, observed potential confounders, and unobserved confounders, respectively. Note that data intervention is a way to help a model learn the causal relations that lead to the result [5,11,12,45,52,53]. A prior joint distribution and its posterior approximation can be built on top of those four random variables. The proposed method is trained based on minimizing the distance between the prior distribution and the posterior approximation. We identify that predicting the behaviors of the data intervention and model's outcome can be challenging in practice due to various factors, e.g., video noise, lens or motion blur. We address this issue by introducing helper distributions for them. The helper distributions form a new loss term to guide the model learning. Furthermore, when multi-modal inputs are available, we identify that the extra input sometimes can harm the model performance most likely due to the interactions between different modalities being ineffective. We address this challenge by introducing a causal semantics extractor to effectively distill the mutual information between multi-modal inputs. These novel design choices have been instrumental in improving the explainability and performance of video summarization models. The extensive experimentation on commonly used video summarization datasets verifies that the proposed method outperforms existing state-of-the-art while also providing greater explainability. By leveraging causal learning techniques, our approach represents a promising attempt to reinforce the causal inference ability and explainability of an ML-based video summarization model. Methodology We now present the details of the proposed Causal Explainer method for automatic video summarization, dubbed Causalainer. First, the assumptions of causal modeling are described in detail. Secondly, we introduce four random variables y, t, X, and Z to characterize the behaviors of the model's prediction, the data intervention, observed potential confounders, and unobserved confounders, respectively. Finally, the derivation of our training objective with helper distributions and the proposed causal semantics extractor are presented. Causalainer consists of prior and posterior probabilistic networks. See Figure 2 for an overview. Assumptions In general, causal learning for real-world observational studies is complicated [1,2,7,[19][20][21][22][23][24][25][28][29][30][31][32][33][34]44,45,59,[61][62][63]. With the established efforts [45,49,66] on causal learning under noisy interventions, two assumptions are imposed when modeling the problem of video summarization. First, the information of having visual/textual intervention t or not is binary. Second, the observations (X, t, y) from a deep neural network (DNN) are sufficient to approximately recover the joint distribution p(Z, X, t, y) of the unobserved/latent confounding variable Z, the observed confounding variable X, the intervention t, and the outcome y. The proposed Causalainer method is built on top of multiple probability distributions as described in the following subsections. Causal Explainer for Video Summarization In the proposed Causalainer, x i denotes an input video and an optional text-based query indexed by i, z i indicates the latent confounder, t i ∈ {0, 1} denotes the intervention assignment, and y i indicates the outcome. Prior Probability Distributions. The prior network is conditioning on the latent variable z i and mainly consists of the following components: (i) The latent confounder distribution: p(z i ) = z∈zi N (z|µ = 0, σ 2 = 1), where N (z|µ, σ 2 ) denotes a Gaussian distribution with a random variable z, z is an element of z i , and the mean µ and variance σ 2 follow the settings in [41], i.e., µ = 0 σ 2 = 1. (ii) The conditional data distribution: p(x i |z i ) = x∈ xi p(x|z i ), where p(x|z i ) is an appropriate probability distribution with a random variable x, the distribution is conditioning on z i , and x is an element of x i . (iii) The conditional in- tervention distribution: p(t i |z i ) = Bernoulli(σ(f θ1 (z i ))), where σ(·) is a logistic function, Bernoulli(·) indicates a Bernoulli distribution for a discrete outcome, and f θ1 (·) denotes a neural network parameterized by the parameter θ 1 . (iv) The conditional outcome distribution: p(y i |z i , t i ) = σ(t i f θ2 (z i ) + (1 − t i )f θ3 (z i )), where f θ2 (·) and f θ3 (·) are neural networks parameterized by the parameters θ 2 and θ 3 , respectively. In this work, y i is tailored for a categorical classification problem, i.e., frame-based importance score classification in video summarization. Posterior Probability Distribution. Since a priori knowledge on the latent confounder does not exist, we have to marginalize over it in order to learn the model parameters, θ 1 , θ 2 , and θ 3 in (iii) and (iv). The non-linear neural network functions make inference intractable. Hence, variational inference [41] along with the posterior network is employed. These neural networks output the parameters of a fixed form posterior approximation over the latent variable z, given the observed variables. Similar to [45,50], in this work, the proposed posterior network is conditioning on observations. Also, the true posterior over Z depends on X, t and y. Hence, the posterior approximation defined below is employed to build the posterior network. q(z i |x i , y i , t i ) = z∈zi N (z|µ i , σ 2 i ), where µ i = t i µ t=1,i + (1 − t i )µ t=0,i , σ 2 i = t i σ 2 t=1,i + (1 − t i )σ 2 t=0,i , µ t=0,i = g φ1 • g φ0 (x i , y i ), σ 2 t=0,i = σ(g φ2 • g φ0 (x i , y i )), µ t=1,i = g φ3 • g φ0 (x i , y i ), σ 2 t=1,i = σ(g φ4 • g φ0 (x i , y i )) , g φ k (·) denotes a neural network with variational parameters φ k for k = 0, 1, 2, 3, 4, and g φ0 (x i , y i ) is a shared representation. Note that a feature map is multiplied with the approximated posterior q(y i |x i , t i ) without logistic function σ(·) to get g φ0 (x i , y i ). Training Objective with Helper Distributions In practice, various factors, e.g., video noise, motion blur, or lens blur, make the prediction of the behaviors of the data intervention and the model's outcome challenging. Therefore, two helper distributions are introduced to alleviate this issue. We have to know the intervention assignment t along with its outcome y before inferring the distribution over Z. Hence, the helper distribution q(t i |x i ) = Bernoulli(σ(g φ5 (x i ))) is introduced for the intervention assignment t i , and the other helper distribution q(y i |x i , t i ) = σ(t i g φ6 (x i ) + (1 − t i )g φ7 (x i )) is introduced for the outcome y i , where g φ k (·) indicates a neural network with variational parameters φ k for k = 5, 6, 7. The introduced helper distributions benefit the prediction of t i and y i for new samples. To estimate the variational parameters of the distributions q(t i |x i ) and q(y i |x i , t i ), a helper objective function L helper = N i=1 [log q(t i = t * i |x * i ) + log q(y i = y * i |x * i , t * i )] is introduced to the final training objective over N data samples, where x * i , t * i and y * i are the observed values in the training set. The overall training objective L causal for the proposed method is defined below. L causal = L helper + N i=1 E q(zi|xi,ti,yi) [log p(x i , t i |z i ) + log p(y i |t i , z i ) + log p(z i ) − log q(z i |x i , t i , y i )]. Causal Semantics Extractor Existing commonly used video summarization datasets, e.g., TVSum [55] and QueryVS [27], provide visual and textual inputs. Since the textual input cannot always help the model performance because of the ineffective extraction of mutual information from the visual and textual inputs, a causal semantics extractor is introduced to alleviate this issue. The proposed extractor is built on top of transformer blocks [57]. Vanilla transformers exploit all of the tokens in each layer for attention computation. However, the design philosophy of the proposed causal semantics extractor, dubbed causal attention, is effectively using fewer but relatively informative tokens to compute attention maps, instead of using the total number of tokens. According to [57], the computation of the vanilla attention matrix A ∈ R n×n is based on the dot-product. It is defined as A = softmax QK √ d ; Q = TW q , K = TW k , where the query matrix Q ∈ R n×d and key matrix K ∈ R n×d are generated by the linear projection of the input token matrix T ∈ R n×dm based on the learnable weights matrices W q ∈ R dm×d and W k ∈ R dm×d . n indicates the total number of input tokens. d represents the embedding dimension and d m denotes the dimension of an input token. The new value matrix V new ∈ R n×d can be obtained via V new = A V; V = TW v , where the value matrix V ∈ R n×d and W v ∈ R dm×d . In [57], the vanilla attention matrix is based on the calculation of all the query-key pairs. However, in the proposed Causal Semantics Extractor, only the top κ most similar keys and values for each query are used to compute the causal attention matrix. Similar to [57], all the queries and keys are calculated by the dot-product. Then, the row-wise top κ elements are used for the softmax calculation. In the proposed Causal Semantics Extractor, the value matrix V κ ∈ R n×d is defined as V κ = softmax (τ κ (A )) V new = softmax τ κ QK √ d V new , where τ κ (·) denotes an operator for the row-wise top κ elements selection. τ κ (·) is defined as [τ κ (A )] ij = A ij , A ij ∈ top κ factors at row i −∞ , otherwise. Then, V κ can be further used to generate X mul , i.e., an output of the proposed Causal Semantics Extractor. The procedure for calculating X mul is defined below. Z ta = TextAtten(FFN(LayerNorm(V κ )), where LayerNorm(·) denotes a layer normalization, FFN(·) indicates a feed forward network, and TextAtten(·) denotes an element-wise multiplication-based textual attention mechanism. Z va = VisualAtten(C3D(I)), where I denotes an input video, C3D(·) indicates an operation of the spatialtemporal feature extraction, e.g., 3D version of ResNet-34 [15,16], for the input video, and VisualAtten(·) indicates a visual attention mechanism based on the element-wise multiplication. X mul = FC(Z ta Z va ), where denotes the operation of feature concatenation and FC(·) indicates a fully connected layer. Note that the Causal Semantics Extractor's output X mul is an input of the proposed posterior network based on the scheme of using multi-modal inputs. Similar to the final step of video summary generation in [27], after the end-to-end training of the proposed causal video summarization model is complete, the trained model can be used for video summary generation. Finally, based on the generated score labels, a set of video frames is selected from the original input video to form a final video summary. Note that the summary budget is considered as a user-defined hyper-parameter in multi-modal video summarization [27]. Experiments Experimental Setup and Datasets Preparation Experimental Setup. We consider three scenarios: 1) fullysupervised training with human-defined frame-level labels, 2) fully-supervised training with multi-modal input including text-based query, and 3) weakly-supervised learning with two-second segment-level scores, which can be considered as a form of weak label [3,4,6]. Note that [55] empirically finds that a two-second segment length is appropriate for capturing video local context with good visual coherence. Hence, in this work, a video segment-level score is produced per two seconds based on given frame-level scores. Video Summarization Datasets. In the experiments, three commonly used video summarization datasets, i.e., TV-Sum [55], QueryVS [27], and SumMe [13], are exploited to evaluate the proposed method. The TVSum dataset contains 50 videos. The length of the video in TVSum is ranging from 2 to 10 minutes. The human expert frame-level importance score label in TVSum is ranging from 1 to 5. The QueryVS dataset contains 190 videos. The video length in QueryVS is ranging from 2 to 3 minutes. The human expert frame-level importance score label in QueryVS is ranging from 0 to 3. Every video is retrieved based on a given text-based query. The SumMe dataset contains 25 videos. The video duration in SumMe is ranging from 1 to 6 minutes. In SumMe, the importance score annotated by human experts ranges from 0 to 1. Note that SumMe is not used for multi-modal video summarization. Hence, we do not have textual input when a model is evaluated on this dataset. Videos from these datasets are sampled at Table 1. Comparison with fully-supervised state-of-the-art methods. The proposed method performs the best on both datasets. Note that textual query input is not used in this experiment. [40], with hyper-parameters set as = 1e − 8, β 1 = 0.9, and β 2 = 0.999. Causal Learning Dataset. When we observe people's writing behaviors, we notice some of them happen very often, such as synonym replacement, accidentally missing some words in a sentence, and so on. Motivated by the above, we randomly pick up one of the behaviors, e.g., accidentally missing some words in a sentence, and write a textual intervention function to simulate it. Similarly, we know that when people make videos in their daily life, some visual disturbances may exist, e.g., salt and pepper noise, image masking, blurring, and so on. We also randomly pick up some of them, e.g., blur and salt and pepper noise, and make a visual intervention function to do the simulation. Based on the visual and textual simulation functions, we can make our causal video summarization dataset with visual and textual interventions. The dataset is made based on the following steps. First, 50% of the (video, query) data pairs are randomly selected from the original training, validation, and testing sets. Secondly, for each selected video, 0 or 1 intervention labels are randomly assigned to 30% of the video frames and the corresponding queries. Note that in realworld scenarios, there are various disturbances beyond the previously mentioned visual and textual interventions that could be utilized in the proposed method. Evaluation and Analysis Evaluation protocol. Following existing works [13,26,27,37,55], we evaluate the proposed method under the same setting. TVSum, QueryVS, and SumMe datasets are randomly divided into five splits, respectively. For each of them, Table 3. Comparison with weakly-supervised state-of-the-art methods. The performance of the proposed approach is better than the existing weakly-supervised method. Figure 3. Causal graph in video summarization. t is an intervention, e.g., visual or textual perturbation. y is an outcome, e.g., an importance score of a video frame or a relevance score between the input text query and video. Z is an unobserved confounder, e.g., representativeness, interestingness, or storyline smoothness. X is noisy views on the hidden confounder Z, say the input text query and video. The causality graph of video summarization leads to more explainable modeling. 80% of the dataset is used for training, and the remaining for evaluation. F 1 -score [13,18,37,55] is adopted to measure the matching degree of the generated video summaries S i and the ground-truth video summariesŜ i for video i. State-of-the-art comparisons. The proposed method outperforms existing state-of-the-art (SOTA) models based on different supervision schemes, as shown in Table 1, Table 2, and Table 3. This is because the introduced causal modeling strengthens the causal inference ability of a video summarization model by uncovering the causal relations that guide the process and result. Effectiveness analysis of the proposed causal modeling. The proposed approach differs from existing methods by introducing causal modeling. Hence, the results in Tables 1, 2, and 3, demonstrate the effectiveness of this approach and serve as an ablation study of causal learning. An auxiliary task/distribution is a key component of the proposed approach, helping the model learn to diagnose input to make correct inferences for the main task, i.e., video summary inference, During training, a binary causation label is provided to teach the model to perform well regardless of intervention. This implies the model has the ability to analyze input and perform well in the main task, making it more robust. Explainability improvement analysis. The Causalainer method benefits modeling explainability with its associated causal graph of video summarization. Latent factors affecting video summary generation are treated as the causal effect in the proposed causal modeling. A causal graphical model is used to approach the video summarization problem, and the modeling explainability is illustrated in Figure 3. Conclusion ML-based decision-making systems, like video summarization, suffer from a lack of explainability, resulting in mistrust. To improve modeling explainability, we propose a new Causalainer method that achieves state-of-the-art F 1score performance in video summarization. Figure 2 . 2Flowchart of the proposed Causal Explainer (Causalainer) method for video summarization. The proposed method is mainly composed of a prior network, a posterior network, helper distributions, and a causal semantics extractor. ⊗ denotes element-wise multiplication and × indicates matrix multiplication. "Token + PE" denotes the operations of token embedding and positional encoding. Fully-supervised Method TVSum SumMeSASUM [58] 53.9 40.6 dppLSTM [68] 54.7 38.6 ActionRanking [8] 56.3 40.1 H-RNN [70] 57.7 41.1 CRSum [64] 58.0 47.3 M-AVS [36] 61.0 44.4 VASNet [9] 61.4 49.7 iPTNet [37] 63.4 54.5 DASP [35] 63.6 45.5 Causalainer 67.5 52.4 Table 2. Comparison with the multi-modal state-of-the-art. The proposed method outperforms the existing multi-modal approaches. '-' denotes unavailability from previous work. Multi-modal Method TVSum QueryVS DSSE [65] 57.0 - QueryVS [27] - 41.4 DQSN [72] 58.6 - GPT2MVS [26] - 54.8 Causalainer 68.2 55.5 1 frame per second (fps). The input image size is 224 by 224 with RGB channels. 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In Proceedings Proceedings of the 2nd International Workshop on Multimedia Content Analysis in Sports, pages 16-24, 2019. 1 Adjusting for errors in classification and measurement in the analysis of partly and purely categorical data. Jan Selén, Journal of the American Statistical Association. 81393Jan Selén. Adjusting for errors in classification and mea- surement in the analysis of partly and purely categorical data. Journal of the American Statistical Association, 81(393):75- 81, 1986. 2 Estimating individual treatment effect: generalization bounds and algorithms. Uri Shalit, D Fredrik, David Johansson, Sontag, PMLRInternational Conference on Machine Learning. Uri Shalit, Fredrik D Johansson, and David Sontag. Esti- mating individual treatment effect: generalization bounds and algorithms. In International Conference on Machine Learning, pages 3076-3085. PMLR, 2017. 2 Category driven deep recurrent neural network for video summarization. Xinhui Song, Ke Chen, Jie Lei, Li Sun, Zhiyuan Wang, Lei Xie, Mingli Song, 2016 IEEE International Conference on Multimedia & Expo Workshops (ICMEW). IEEEXinhui Song, Ke Chen, Jie Lei, Li Sun, Zhiyuan Wang, Lei Xie, and Mingli Song. Category driven deep recurrent neural network for video summarization. In 2016 IEEE International Conference on Multimedia & Expo Workshops (ICMEW), pages 1-6. IEEE, 2016. 1 Tvsum: Summarizing web videos using titles. Yale Song, Jordi Vallmitjana, Amanda Stent, Alejandro Jaimes, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognition14Yale Song, Jordi Vallmitjana, Amanda Stent, and Alejandro Jaimes. Tvsum: Summarizing web videos using titles. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5179-5187, 2015. 1, 3, 4 Query-adaptive video summarization via quality-aware relevance estimation. Michael Arun Balajee Vasudevan, Anna Gygli, Luc Volokitin, Van Gool, Proceedings of the 25th ACM international conference on Multimedia. the 25th ACM international conference on MultimediaArun Balajee Vasudevan, Michael Gygli, Anna Volokitin, and Luc Van Gool. Query-adaptive video summarization via quality-aware relevance estimation. In Proceedings of the 25th ACM international conference on Multimedia, pages 582-590, 2017. 1 Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, Illia Polosukhin, arXiv:1706.03762Attention is all you need. arXiv preprintAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Il- lia Polosukhin. Attention is all you need. arXiv preprint arXiv:1706.03762, 2017. 3 Video summarization via semantic attended networks. Huawei Wei, Bingbing Ni, Yichao Yan, Huanyu Yu, Xiaokang Yang, Chen Yao, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence324Huawei Wei, Bingbing Ni, Yichao Yan, Huanyu Yu, Xiaokang Yang, and Chen Yao. Video summarization via semantic attended networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018. 1, 4 Expert-defined keywords improve interpretability of retinal image captioning. Ting-Wei Wu, Jia-Hong Huang, Joseph Lin, Marcel Worring, Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision. the IEEE/CVF Winter Conference on Applications of Computer VisionTing-Wei Wu, Jia-Hong Huang, Joseph Lin, and Marcel Wor- ring. Expert-defined keywords improve interpretability of retinal image captioning. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1859-1868, 2023. 2 Self-supervised learning to detect key frames in videos. Xiang Yan, Mingtao Syed Zulqarnain Gilani, Liang Feng, Hanlin Zhang, Ajmal Qin, Mian, Sensors. 20236941Xiang Yan, Syed Zulqarnain Gilani, Mingtao Feng, Liang Zhang, Hanlin Qin, and Ajmal Mian. Self-supervised learning to detect key frames in videos. Sensors, 20(23):6941, 2020. 1 A novel hybrid machine learning model for auto-classification of retinal diseases. C-H Huck Yang, Jia-Hong Huang, Fangyu Liu, Fang-Yi Chiu, Mengya Gao, Weifeng Lyu, Jesper Tegner, Workshop on Computational Biology. 2ICMLC-H Huck Yang, Jia-Hong Huang, Fangyu Liu, Fang-Yi Chiu, Mengya Gao, Weifeng Lyu, Jesper Tegner, et al. A novel hy- brid machine learning model for auto-classification of retinal diseases. Workshop on Computational Biology, ICML, 2018. 2 Chao-Han Huck Yang, Te Hung, Yi Danny, Pin-Yu Ouyang, Chen, arXiv:2102.09677Causal inference q-network: Toward resilient reinforcement learning. arXiv preprintChao-Han Huck Yang, I Hung, Te Danny, Yi Ouyang, and Pin-Yu Chen. Causal inference q-network: Toward resilient reinforcement learning. arXiv preprint arXiv:2102.09677, 2021. 2 Treatment learning causal transformer for noisy image classification. Chao-Han Huck Yang, I-Te Hung, Yi-Chieh Liu, Pin-Yu Chen, Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision. the IEEE/CVF Winter Conference on Applications of Computer VisionChao-Han Huck Yang, I-Te Hung, Yi-Chieh Liu, and Pin- Yu Chen. Treatment learning causal transformer for noisy image classification. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 6139- 6150, 2023. 2 Spatiotemporal modeling for video summarization using convolutional recurrent neural network. Yuan Yuan, Haopeng Li, Qi Wang, IEEE Access. 74Yuan Yuan, Haopeng Li, and Qi Wang. Spatiotemporal mod- eling for video summarization using convolutional recurrent neural network. IEEE Access, 7:64676-64685, 2019. 4 Video summarization by learning deep side semantic embedding. Yitian Yuan, Tao Mei, Peng Cui, Wenwu Zhu, IEEE Transactions on Circuits and Systems for Video Technology. 294Yitian Yuan, Tao Mei, Peng Cui, and Wenwu Zhu. Video sum- marization by learning deep side semantic embedding. IEEE Transactions on Circuits and Systems for Video Technology, 29(1):226-237, 2017. 1, 4 Advances in variational inference. Cheng Zhang, Judith Bütepage, Hedvig Kjellström, Stephan Mandt, IEEE transactions on pattern analysis and machine intelligence. 41Cheng Zhang, Judith Bütepage, Hedvig Kjellström, and Stephan Mandt. Advances in variational inference. IEEE transactions on pattern analysis and machine intelligence, 41(8):2008-2026, 2018. 1, 2 Xian-Sheng Hua, and Qianru Sun. Causal intervention for weakly-supervised semantic segmentation. Dong Zhang, Hanwang Zhang, Jinhui Tang, Advances in Neural Information Processing Systems. 33Dong Zhang, Hanwang Zhang, Jinhui Tang, Xian-Sheng Hua, and Qianru Sun. Causal intervention for weakly-supervised semantic segmentation. Advances in Neural Information Processing Systems, 33, 2020. 1 Video summarization with long short-term memory. Ke Zhang, Wei-Lun Chao, Fei Sha, Kristen Grauman, ECCV. Springer14Ke Zhang, Wei-Lun Chao, Fei Sha, and Kristen Grauman. Video summarization with long short-term memory. In ECCV, pages 766-782. Springer, 2016. 1, 4 Dtr-gan: Dilated temporal relational adversarial network for video summarization. Yujia Zhang, Michael Kampffmeyer, Xiaoguang Zhao, Min Tan, Proceedings of the ACM Turing Celebration Conference-China. the ACM Turing Celebration Conference-ChinaYujia Zhang, Michael Kampffmeyer, Xiaoguang Zhao, and Min Tan. Dtr-gan: Dilated temporal relational adversarial network for video summarization. In Proceedings of the ACM Turing Celebration Conference-China, pages 1-6, 2019. 1 Hierarchical recurrent neural network for video summarization. Bin Zhao, Xuelong Li, Xiaoqiang Lu, Proceedings of the 25th ACM international conference on Multimedia. the 25th ACM international conference on Multimedia14Bin Zhao, Xuelong Li, and Xiaoqiang Lu. Hierarchical recur- rent neural network for video summarization. In Proceedings of the 25th ACM international conference on Multimedia, pages 863-871, 2017. 1, 4 Hsa-rnn: Hierarchical structure-adaptive rnn for video summarization. Bin Zhao, Xuelong Li, Xiaoqiang Lu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionBin Zhao, Xuelong Li, and Xiaoqiang Lu. Hsa-rnn: Hier- archical structure-adaptive rnn for video summarization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7405-7414, 2018. 1 Video summarisation by classification with deep reinforcement learning. Kaiyang Zhou, Tao Xiang, Andrea Cavallaro, arXiv:1807.0308914arXiv preprintKaiyang Zhou, Tao Xiang, and Andrea Cavallaro. Video sum- marisation by classification with deep reinforcement learning. arXiv preprint arXiv:1807.03089, 2018. 1, 4

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{'abstract': "The goal of video summarization is to automatically shorten videos such that it conveys the overall story without losing relevant information. In many application scenarios, improper video summarization can have a large impact. For example in forensics, the quality of the generated video summary will affect an investigator's judgment while in journalism it might yield undesired bias. Because of this, modeling explainability is a key concern. One of the best ways to address the explainability challenge is to uncover the causal relations that steer the process and lead to the result. Current machine learning-based video summarization algorithms learn optimal parameters but do not uncover causal relationships. Hence, they suffer from a relative lack of explainability. In this work, a Causal Explainer, dubbed Causalainer, is proposed to address this issue. Multiple meaningful random variables and their joint distributions are introduced to characterize the behaviors of key components in the problem of video summarization. In addition, helper distributions are introduced to enhance the effectiveness of model training. In visual-textual input scenarios, the extra input can decrease the model performance. A causal semantics extractor is designed to tackle this issue by effectively distilling the mutual information from the visual and textual inputs. Experimental results on commonly used benchmarks demonstrate that the proposed method achieves state-of-the-art performance while being more explainable.", 'arxivid': '2305.00455', 'author': ['Jia-Hong Huang [email protected] \nUniversity of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n\n', 'Chao-Han Huck \nUniversity of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n\n', 'Yang \nUniversity of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n\n', 'Pin-Yu Chen [email protected] \nUniversity of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n\n', 'Min-Hung Chen [email protected] \nUniversity of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n\n', 'Marcel Worring [email protected] \nUniversity of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n\n'], 'authoraffiliation': ['University of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n', 'University of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n', 'University of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n', 'University of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n', 'University of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n', 'University of Amsterdam\nGeorgia Institute of Technology\nIBM Research AI\nNVIDIA\nUniversity of Amsterdam\n'], 'corpusid': 258426833, 'doi': '10.48550/arxiv.2305.00455', 'github_urls': [], 'n_tokens_mistral': 15869, 'n_tokens_neox': 13425, 'n_words': 7511, 'pdfsha': 'd25e5a1950d58c773fdb9641c65d429316346943', 'pdfurls': ['https://export.arxiv.org/pdf/2305.00455v1.pdf'], 'title': ['Causalainer: Causal Explainer for Automatic Video Summarization', 'Causalainer: Causal Explainer for Automatic Video Summarization'], 'venue': []}

arxiv

On the use of a non-extensive distribution in the solar plasma 23 Oct 2000 A Lavagno P Quarati Dipartimento di Fisica Politecnico di Torino C.so Duca degli Abruzzi 24I-10129TorinoItaly Sezioni di Torino e di Cagliari Istituto Nazionale di Fisica Nucleare On the use of a non-extensive distribution in the solar plasma 23 Oct 2000 We describe the physical motivations why we are urged to look for an ionic equilibrium distribution function slightly different from the Maxwellian to calculate nuclear reaction rates in the solar core plasma answering, at the same time, to a recent paper by Bahcall et al. (astro-ph/0010055). In a recent paper, Bahcall et al. [1] discuss motivations why Salpeter formula for screening of nuclear reactions in the Sun is correct (in contrast with conclusions contained in several other papers) and show five different derivations of the Salpeter screening factor f 0 . A paper of us [2] is also included in this critical discussion (although our work is not concerned directly on the factorizable screening factor f 0 but, rather, on the evaluation of the reaction rates) because of the use we make of an ionic equilibrium distribution which differs very slightly from the Maxwellian distribution. The reason of using this equilibrium distribution is related to the need of including many-body effects. It was already proposed ad hoc by Clayton [3] and we can see, a posteriori, that coincides to the equilibrium distribution of the non-extensive thermostatistics developed by Tsallis [4] and used recently in many different physical applications [5], as for instance, gravitational [6] and high energy problems [7]. In our calculations the screening factor f 0 is taken to be unity because its effect on the reaction rates is negligible if compared to the important depletions (or enhancements) of the standard rates due to other effects (electromagnetic fluctuations in a plasma) responsible of the non-extensive equilibrium distribution function. Also, by following the derivation of f 0 in Ref. [8] and using non-extensive distribution, we can calculate for the solar core about the same value of the screening factor. For comments based on Molecular Dynamics Model (MDM) on the validity of Salpeter screening formula see Ref. [9]. Several authors are actually working on MDM within non-extensive Tsallis statistics [10]. It seems to us obvious that, if all the derivations of the Salpeter formula are based on the validity of the assumptions needed for the Boltzmann-Gibbs statistics, the equilibrium distribution must be, of course, the Maxwellian distribution. However the motivations that deviations from these assumptions are not correct should be explicitly given. Results within non-extensive statistics go toward a better agreement with the experimental results of the calculated neutrino fluxes. In spite of this achievement, we never excluded the possibility of neutrino oscillation or new physics because only experiments will give the correct answer on this problem. At the same time, we have also shown that modifications of the reaction rates do not affect bulk properties of the gravitationally stabilized solar core such as sound speed or hydrostatic equilibrium, depending on mean values obtained averaging over Maxwellian distribution function [11]. We confirm that Tsallis distribution is an equilibrium distribution and that the argument that there is enough time to arrive at equilibrium after a reaction is fully accomplished by the Tsallis equilibrium distribution that can be derived in, at least, three different ways containing not alternative reasons. In one approach we exploit the knowledge that the distribution in the solar interior cannot be too much different from the Maxwellian one and add small corrections (higher-order terms in a derivative expansion) to the coefficients of the standard Fokker-Plank equation. Tsallis' distribution is immediately generated [12]. A second approach focuses on the electric microfields that have been shown to exists in the solar plasma: the time-spatial fluctuations in the particles positions produce specific fluctuations of microscopic electric field in a given point of the plasma. We have shown that such electric microfields in the solar plasma (with energy density of the order of 10 −16 MeV/fm 3 ) implies a deviation from the Maxwellian distribution; the entity depending on the value of the plasma parameter and on the correlation among ions [2,13]. The third approach has been just started and aims to connect the distribution of collective variables to memory effects and long-time correlations between velocities [14,15]. There should exist solutions compatible with the Tsallis' distribution and/or other non-Maxwellian distributions. These three approaches are not exhaustive and not necessary alternative. Nevertheless, it is suggestive that all of them point in the same direction: the Maxwell-Boltzmann distribution of velocity should have small but nonnegligible corrections in the solar plasma and the Tsallis' distribution could provide a better description. Normal stellar matter, such as the one in the Sun, is non-degenerate, i.e., quantum effects are small (in fact, they are small for electrons and completely negligible for ions), it is nonrelativistic, and it is in good thermodynamical equilibrium. On this ground, the particle velocity distribution is almost universally taken to be a Maxwell-Boltzmann distribution, without much questioning. However, derivations of the ubiquitous Maxwell-Boltzmann distribution are based on several assumptions [12]. In a kinetical approach, one assumes (1) that the collision time be much smaller than the mean time between collisions, (2) that the interaction be sufficiently local, (3) that the velocities of two particles at the same point are not correlated (Boltzmann's Stosszahlansatz), and (4) that energy is locally conserved when using only the degrees of freedom of the colliding particles (no significant amount of energy is transferred to collective variables and fields). In the equilibrium statistical mechanics approach, one uses the assumption that the velocity probabilities of different particles are independent, corresponding to (3), and that the total energy of the system could be expressed as the sum of a term quadratic in the momentum of the particle and independent of the other variables, and a term independent of momentum, but if (1) and (2) are not valid the resulting effective two-body interaction is not local and depends on the momentum and energy of the particles. Finally, even when the one-particle distribution is Maxwellian, additional assumptions about correlations between particles are necessary to deduce that the relative-velocity distribution, which is the relevant quantity for rate calculations, is also Maxwellian. At least in one limit the Maxwell-Boltzmann distribution can be rigorously derived: systems that are dilute in the appropriate variables, whose residual interaction is small compared to the one-body energies. In spite of the fact that the effects of the residual interaction cannot be neglected, as a good first approximation the solar interior can be studied in this dilute limit; therefore, it is reasonable to suppose that the velocity distribution in the Sun is not too far from the Maxwellian one. Likewise, one often assumes that the solar core could be treated as an ideal (Debye) plasma. However, there are physical conditions and/or specific applications that needs higher accuracy for which becomes necessary to take into account modifications of the standard plasma theory. When Γ ≈ 0.05 ÷ 0.1, the mean Coulomb energy potential is not much smaller of the thermal kinetic energy and the screening length R D ≈ a. It is not possible to clearly separate individual and collective degrees of freedom. The presence of at least two different scales of energies of the same rough size produces deviations from the standard statistics which describe the system in terms of a single scale, kT . The reaction time necessary to build up screening after a hard collision can be estimated from the inverse solar plasma frequency t pl = ω −1 pl = m/(4πne 2 ) ≈ 10 −17 sec, and it is comparable to the collision time t coll = σvn −1 ≈ 10 −17 sec. Therefore, several collisions are likely necessary before the particle looses memory of the initial state and the scattering process can not be considered Markovian. In addition, screening starts to become dynamical: the time necessary to build up again the screening after hard collisions is not negligible any more. Many remarks we have just made are reported in well known books on statistical mechanics, kinetic and plasma theory and in works by astrophysicist as, for instance, Paquette et al. [16] and Cox et al. [17]. Let us report, specifically, from Cox et al.: "one usually assumes (1) negligible radiative forces, (2) complete ionization, (3) Maxwellian velocity distributions and the same kinetic temperature for all ions and electrons, (4) diffusion velocities which are much smaller than mean thermal velocities, (5) no magnetic fields, (6) collisions dominated by "classical" interactions between two point particles, and (7) plasma which can be considered a diluite gas, i.e., the ideal gas equation of state applies. [..] When the number of particles in a Debye sphere around an ion is not ≫ 1, assumption (6) and (7) above are inappropriate. Ions are no longer effectively screened from their surrounding, and multiple particle collisions and collective effects become important. The "classical" approach to calculating transport properties of ions using Boltzmann's equation becomes invalid. In the Sun, at number for particles per Debye sphere is only about 1.4 at the base of the convection zone, and 2.2 at the core, so the "diluite gas" approximation fails." Finally, let us recall that the introduced equilibrium distribution contains the factor e −(1−q)/2(E/kT ) 2 where q is the Tsallis parameter. In the solar core plasma we have found for it the value |(1 − q)/2| ≈ 0.003 (q = 1 corresponds to the Boltzmann-Gibbs statistics and Maxwellian distribution). In spite of the very small deviation of the behavior at high energy of the Tsallis distribution respect to the standard curve the effect on the reaction rates (depletion or enhancement) is quite important [2,12,13]. . J N Bahcall, S Brown, A Gruzinov, R F Sawyer, astro-ph/0010055J.N. Bahcall, S. Brown, A. Gruzinov, R.F. Sawyer, astro-ph/0010055. . A Lavagno, P Quarati, Nucl. Phys. B (PS). 87209A. Lavagno, P. Quarati, Nucl. Phys. B (PS) 87 (2000) 209. . D D Clayton, ApJ. 199494D.D. Clayton et al., ApJ. 199 (1975) 494. . C Tsallis, J. Stat. Phys. 52479C. Tsallis, J. Stat. Phys. 52 (1988) 479. . A Lavagno, G Kaniadakis, M Rego Monteiro, P Quarati, C Tsallis, Astrophys. Lett. Comm. 35449A. Lavagno, G. Kaniadakis, M. Rego Monteiro, P. Quarati, C. Tsallis, Astrophys. Lett. Comm. 35 (1998) 449. . W M Alberico, A Lavagno, P Quarati, Eur. Phys. J. C. 12499W.M. Alberico, A. Lavagno, P. Quarati, Eur. Phys. J. C 12 (2000) 499. . J N Bahcall, X Chen, M Kamionkowski, Phys. Rev. C. 572756J.N. Bahcall, X. Chen, M. Kamionkowski, Phys. Rev. C 57 (1998) 2756. . G Shaviv, astro-ph/0010152G. Shaviv, astro-ph/0010152. . C Tsallis, cond-mat/0010150C. Tsallis, cond-mat/0010150. . S Degl&apos;innocenti, Phys. Lett B. 449291S. Degl'Innocenti et al., Phys. Lett B 449 (1998) 291. . G Kaniadakis, A Lavagno, M Lissia, P Quarati, Physica A. 261359G. Kaniadakis, A. Lavagno, M. Lissia, and P. Quarati, Physica A 261 (1998) 359. . A Lavagno, P Quarati, Oct Preprint, submitted for publicationA. Lavagno, P. Quarati, preprint Oct. 2000, submitted for publication. . A Valuev, A Kaklyugin, G Norman, Journ. of Exp. Theor. Phys. 86480A. Valuev, A. Kaklyugin, G. Norman, Journ. of Exp. Theor. Phys. 86 (1998) 480. . M Coraddu, Braz. J. Phys. 29153M. Coraddu et al., Braz. J. Phys. 29 (1999) 153. . C Paquette, C Pelletier, G Fontaine, G Michaud, ApJ. Suppl. 61177C. Paquette, C. Pelletier, G. Fontaine and G. Michaud, ApJ. Suppl. 61 (1986) 177. . A N Cox, J A Guzik, R B Kidman, ApJ. 3421187A.N. Cox, J.A. Guzik, R.B. Kidman, ApJ. 342 (1989) 1187.

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{'abstract': 'We describe the physical motivations why we are urged to look for an ionic equilibrium distribution function slightly different from the Maxwellian to calculate nuclear reaction rates in the solar core plasma answering, at the same time, to a recent paper by Bahcall et al. (astro-ph/0010055).', 'arxivid': 'astro-ph/0010452', 'author': ['A Lavagno ', 'P Quarati ', '\nDipartimento di Fisica\nPolitecnico di Torino\nC.so Duca degli Abruzzi 24I-10129TorinoItaly\n', '\nSezioni di Torino e di Cagliari\nIstituto Nazionale di Fisica Nucleare\n\n'], 'authoraffiliation': ['Dipartimento di Fisica\nPolitecnico di Torino\nC.so Duca degli Abruzzi 24I-10129TorinoItaly', 'Sezioni di Torino e di Cagliari\nIstituto Nazionale di Fisica Nucleare\n'], 'corpusid': 17800340, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3370, 'n_tokens_neox': 2948, 'n_words': 1903, 'pdfsha': 'e12a31bbfc7eed2ceccc813e4f60859bf4e80a9a', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0010452v1.pdf'], 'title': ['On the use of a non-extensive distribution in the solar plasma', 'On the use of a non-extensive distribution in the solar plasma'], 'venue': []}

arxiv

Freeze the Discriminator: a Simple Baseline for Fine-Tuning GANs Sangwoo Mo [email protected] Jinwoo Shin KAIST Kaist Jinwoo Shin KAIST POSTECHMinsu Cho [email protected] Jinwoo Shin KAIST Freeze the Discriminator: a Simple Baseline for Fine-Tuning GANs Generative adversarial networks (GANs) have shown outstanding performance on a wide range of problems in computer vision, graphics, and machine learning, but often require numerous training data and heavy computational resources. To tackle this issue, several methods introduce a transfer learning technique in GAN training. They, however, are either prone to overfitting or limited to learning small distribution shifts. In this paper, we show that simple fine-tuning of GANs with frozen lower layers of the discriminator performs surprisingly well. This simple baseline, FreezeD, significantly outperforms previous techniques used in both unconditional and conditional GANs. We demonstrate the consistent effect using StyleGAN and SNGAN-projection architectures on several datasets of Animal Face, Anime Face, Oxford Flower, CUB-200-2011, and Caltech-256 datasets. The code and results are available at https://github.com/sangwoomo/FreezeD. Introduction Generative adversarial networks (GANs) [11] have shown a remarkable success across a broad range of applications in computer vision, graphics, and machine learning, e.g., image generation [5,20,21], image-to-image translation [28,33,8], and video-to-video synthesis [43,3,6]. Current state-of-the-art GANs, however, often require a large amount of training data and heavy computational resources, which thus limits the applicability of GANs in practical scenarios. Numerous techniques have been proposed to overcome this limitation, e.g., transferring knowledge of a welltrained source model [45,32,44], learning meta-knowledge for quick adaptation to a target domain [24,47,42], using an auxiliary task to facilitate training [7,26,48,49], improving an inference procedure of suboptimal models [2,39,29,38], using an expressive prior distribution [13], actively choosing samples to give supervision for conditional generation [29], or actively sampling mini-batches for training [37]. Among the approaches, transfer learning [46] is arguably the most promising way to training models under limited [15] scores of fine-tuning and our proposed baseline, FreezeD, on 'Dog' class in the Animal Face [36] dataset. While fine-tuning suffers from overfitting, FreezeD shows consistent stability in training GANs. data and resources. Indeed, most of the recent success in deep learning is built upon strong backbones pre-trained on large datasets in supervised [9] or self-supervised [10,14] ways. Following the success of transferring classifiers in recognition tasks, one can also consider utilizing welltrained GAN backbones for downstream generation tasks. While several methods propose such transfer-learning approaches to training GANs [45,32,44], they are often prone to overfitting with limited training data [45] or not robust in learning a significant distribution shift [32,44]. In this paper, we propose a simple yet effective baseline for transfer learning of GANs. In particular, we show that simple fine-tuning of GANs (both generator and discriminator) with frozen lower layers of the discriminator performs surprisingly well (see Figure 1). Intuitively, the lower layers of the discriminator learn generic features of images while the upper layers learn to classify whether the image is real or fake based on the extracted features. We remark that this dichotomous view of a feature extractor and a classifier (and freezing the feature extractor for fine-tuning) is not new; it has been widely used for training classifiers [46]. We confirm that this view is also useful for GANs, and set its proper baseline for transfer learning of GANs. We demonstrate the effectiveness of the simple baseline, dubbed FreezeD, using various architectures and datasets. For unconditional GANs, we fine-tune the StyleGAN [20] architecture, which is pre-trained on FFHQ [20], onto Animal Face [36] and Anime Face [30] datasets, and for conditional GANs, fine-tune the SNGAN-projection [27] architecture, which is pre-trained on ImageNet [9], onto Oxford Flower [31], CUB-200-2011 [40], and Caltech-256 [12] datasets. FreezeD outperforms previous techniques for all experiment settings, e.g., improving the FID [15] score from 64.28 of fine-tuning to 61.46 (-4.4%) on 'Dog' class of Animal Face dataset. Methods The goal of GANs [11] is to learn a generator (and a corresponding discriminator) to match with a target data distribution. In transfer learning, we assume one can utilize a pre-trained source generator (and a corresponding discriminator) trained on the source data distribution to improve the target generator. See [25,22] for the survey of GANs. We first briefly review previous methods for transfer learning of GANs. • Fine-tuning [45]: The most intuitive and effective way to transferring knowledge is fine-tuning; initialize the parameters of target models as the pre-trained weights of the source models. The authors report that finetuning both the generator and the discriminator indeed shows the best performance. 1 However, fine-tuning often suffer from overfitting; hence one needs a proper regularization. • Scale/shift [32]: Since naïve fine-tuning is prone to overfitting, scale/shift suggest to update the normalization layers only (e.g., batch normalization (BN) [17]) while fixing all other weights. However, it often shows inferior results due to its restriction, especially when there is a significant shift between the source and the target distribution. • Generative latent optimization (GLO) [32,4]: Since GAN loss is given by the discriminator, which can be unreliable for limited data, GLO suggests fine-tuning the generator with supervised learning, where the loss is given by the sum of the L1 loss and the perceptual loss [19]. Here, GLO jointly optimizes the generator and the latent codes to avoid overfitting; one latent code (and its corresponding generated sample) matches one real sample; hence, the generator can generalize samples by interpolation. While GLO improves the stability, it tends to produce blurry images due to the lack of adversarial loss (and prior knowledge of the source discriminator). 1 It is more crucial for our case, as we use stronger source models. • MineGAN [44]: To avoid overfitting of the generator, MineGAN suggests to fix the generator and modify the latent codes. To this end, MineGAN train a miner network that transforms the latent code to another latent code. While this importance-sampling-like approach can be effective when the source distribution and the target distributions share support, it may not be generalized when their supports are disjointed. We now introduce a simple baseline, FreezeD, which outperforms the previous methods despite its simplicity, and suggest two other methods for possible future directions, which may give further improvement. We remark that our goal is not to advocate the state-of-the-art but to set a simple and effective baseline. By doing so, we hope to encourage new techniques that outperform the proposed baseline. • FreezeD (our proposed baseline): We find that simply freezing the lower layers of the discriminator and only fine-tune the upper layers performs surprisingly well. We call this simple yet effective baseline as FreezeD, and will demonstrate its consistent gain over the previous methods in the experimental section. • L2-SP [23]: In addition to the prior methods, we test L2-SP, which is known to be effective for the classifiers. Built upon to the fine-tuning, L2-SP regularizes the target models not to move far from the source models. In particular, it regularizes the L2-norm of the parameters of source models and target models. In our experiments, we applied L2-SP to the generator, discriminator, and both, but the results were not satisfactory. However, since freezing layers can be viewed as giving the infinite weight of L2-SP for the chosen layers and 0 for the other layers, using proper weights for each layer may perform better. • Feature distillation [16,35]: We also test feature distillation, one of the most popular approaches to transfer learning of classifiers. Among the variants, we simply distill the activations of the source models and target models (initialized to the source models). We find that feature distillation shows comparable results to FreezeD while takes twice computation. Investigating more advanced techniques (e.g., [1,18,34]) would be an interesting and promising future direction. 2 Experiments In this section, we demonstrate the effectiveness of the simple yet effective baseline, FreezeD. We conduct extensive experiments for both unconditional GANs and conditional GANs in Section 3.1 and Section 3.2, respectively. [20] dataset, and fine-tune it on Animal Face [36] and Anime Face [30] datasets. We use full 20 classes of the Animal Face dataset, and the first 10 classes among the total 1,000 classes of the Anime Face dataset. Each class contains around 100 samples. We use the public pre-trained model 3 of resolution 256×256 and fine-tune the models following the original training scheme for 50,000 iterations. We remark that the training performed successfully without progressive training by utilizing the source models. Figure 2 visualizes the generated samples using the orig-3 https://github.com/rosinality/ style-based-gan-pytorch inal weights and the fine-tuned weights on 'Cat' and 'Dog' classes in the Animal Face dataset. Notably, the same latent code shares the same semantics even after fine-tuning. See Appendix D for more qualitative results. We also evaluate the FID [15] scores of the vanilla fine-tuning and FreezeD under Animal Face and Anime Face datasets in Table 1 and Table 2, respectively. We freeze the discriminator until layer 4. See Appendix A for the ablation study on different layers. FreezeD improves both the best performance and the stability as shown by the best and final FID scores. We finally compare FreezeD with several previous methods, including scale/shift, GLO, MineGAN, L2-SP, and feature distillation (FD). We choose the weights of L2-SP and FD from {0.1, 1, 10} and simply use 1 for all experiments. We follow the hyperparameters of [32] for GLO, and use 2-layer MLP with ReLU activation for the Miner network. Conditional GAN We also demonstrate the results for conditional GANs. We use the SNGAN-projection [27] architecture pre-trained on ImageNet [9] dataset, and fine-tune it on Oxford Flower [31], CUB-200-2011 [40], and Caltech-256 [12] datasets. Each dataset contains 102, 200, and 256 classes, respectively, where each class has 50-100 samples. We use the public pre-trained model 4 of resolution 128×128 and finetune the networks following the original training scheme for 20,000 iterations. SNGAN-projection has a larger variance than StyleGAN, but still the trend is similar. Figure 3 visualizes the samples generated using the model trained by fine-tuning and FreezeD. FreezeD generates more class-consistent samples than fine-tuning as shown in the 2nd and 8th rows. See Appendix E for more qualitative results. We also evaluate the FID [15] scores of the vanilla fine-tuning and FreezeD in Table 4. We freeze the discriminator until {3, 2, 1} layers for {Oxford Flower, CUB-200-2011, Caltech-256 datasets}, respectively, as the FreezeD improves both the performance and stability for most cases, but harms the stability for Oxford Flower. We find that feature distillation shows more stable results in our experiments. We leave this investigation for future work. Conclusion We have introduced a simple yet effective baseline, FreezeD, for transfer learning of GANs. FreezeD splits the discriminator into a feature extractor and a classifier and then fine-tune the classifier only. We demonstrate that this simple baseline clearly outperforms most of the previous methods using various architectures and datasets. Our observation raises two questions. First, the transferability of the feature extractor of the discriminator could be applied for the universal detector of generated images [41]. Second, one can design a more sophisticated method that outperforms our proposed baseline. We hypothesize that the advanced version of feature distillation [16,35] could be a promising direction. Supplementary Material: Freeze the Discriminator: a Simple Baseline for Fine-Tuning GANs A. Ablation Study on Freezing Layers We study the effect of freezing layers of the discriminator for StyleGAN and SNGAN-projection in Table 5 and Table 6, respectively. In StyleGAN, layer 4 consistently shows the best performance. However, in SNGAN-projection, layer {3, 2, 1} were the best for Oxford Flower, CUB-200-2011, and Caltech-256 datasets, respectively. It is since Caltech-256 is harder to learn compared to Oxford Flower (i.e., distribution shift is larger). Intuitively, one should less restrict the model to adapt to the large distribution shift. One can also see that FreezeD is less stable than fine-tuning for the Oxford Flower dataset. We observe that feature distillation shows better stability while showing a similar best performance in our early experiments. Investigating a more sophisticated method would be an interesting research direction. B. Comparison to Feature Distillation We compare FreezeD with feature distillation. We linearize the activations of the i-th layer of the discriminator, and match the activations of the source and target discriminators. Since the activation has a different size for each layer, we use the L2-norm normalized by the feature dimension. We simply use 1 for the weight of the regularizer regardless of the layer. Table 7 presents the comparison results. Feature distillation and FreezeD shows comparable results, while feature distillation is twice slower. Hence, we choose to FreezeD as the baseline for this paper. C. Qualitative Results for Prior Methods We visualize the samples generated by the prior methods in Figure 4. Scale/shift and L2-SP generates reasonable samples, but have less diversity as measured by FID scores. GLO generates blurry images due to the lack of adversarial loss and the knowledge of source discriminator. In our experiments, MineGAN totally fails to adapt to the target distribution. Note that MineGAN assumes the source distribution covers (or at least close to) the target distribution (e.g., adult faces to child faces as in the original paper [44]), but cannot be applied if the distributions have disjoint support (e.g., human faces to dog faces). Figure 1 : 1Trends of FID Figure 2 : 2Samples generated by StyleGAN of (a) original weights, and trained by FreezeD under (b) 'Cat', and (c) 'Dog' classes in the Animal Face dataset. Each entry indicates the same latent code. Same latent code shares the same semantics even after fine-tuning, e.g., the background color and hair color are preserved. See Appendix D for more qualitative results. Figure 3 : 3Samples generated by SNGAN-projection trained by (a) fine-tuning and (b) FreezeD under the Oxford Flower dataset. Each row indicates the same class. FreezeD generates more class-consistent samples than fine-tuning, e.g., finetuning generates some abnormal samples for row 2 and 8. See Appendix E for more qualitative results. Figure 4 : 4Samples generated by prior methods under 'Dog' class in the Animal Face dataset. Table 1 : 1FID scores under Animal Face dataset. Left and right values indicate the best and final FID scores.Bear Cat Chicken Cow Deer Dog Duck Eagle Elephant Human Fine-tuning 82.82/84.38 71.76/73.47 88.10/88.83 87.07/87.46 82.11/84.04 64.28/67.42 92.54/92.54 85.52/86.88 84.10/84.33 76.62/76.72 FreezeD 78.77/78.77 69.64/69.97 86.20/86.53 84.32/84.39 78.67/79.73 61.46/61.67 88.82/89.14 82.15/82.62 80.00/80.24 73.51/73.89 Lion Monkey Mouse Panda Pigeon Pig Rabbit Sheep Tiger Wolf Fine-tuning 76.86/78.36 86.70/87.30 84.95/85.61 74.29/76.07 81.24/81.36 85.31/86.08 89.11/89.82 86.98/87.89 73.21/75.06 79.97/81.37 FreezeD 73.49/73.59 82.31/82.61 81.72/82.30 72.19/72.62 77.79/78.07 83.22/83.31 85.65/85.65 84.33/84.55 71.26/71.54 76.47/76.47 Table 2 : 2FID scores under the Anime Face dataset. Left and right values indicate the best and final FID scores.Miku Sakura Haruhi Fate Nanoha Lelouch Mio Yuki Shana Reimu Fine-tuning 95.54/98.44 66.94/67.43 76.34/77.44 79.81/83.94 71.03/72.04 83.58/84.11 86.14/88.24 81.38/83.12 79.05/79.79 80.82/82.44 FreezeD 93.37/95.63 65.40/65.91 74.50/74.56 77.76/78.80 68.41/68.41 80.20/82.31 81.55/85.90 79.65/79.83 77.39/77.39 79.27/79.31 Table 3 : 3Comparison of various methods under 'Cat' and 'Dog' classes in the Animal Face dataset. Left and right values indicate the best and final FID scores. † indicates the model is trained by GLO loss, otherwise by GAN loss. Dog 64.28/67.42 75.19/75.45 64.12/67.79 79.08/79.91 79.05/79.23 79.11/79.20 64.18/67.14 64.28/66.68 64.25/66.06 61.46/61.67 3.1. Unconditional GAN We first demonstrate results for unconditional GANs. We use the StyleGAN [20] architecture pre-trained on FFHQFine-tuning Fine-tuning † Scale/shift Scale/shift † MineGAN MineGAN † L2-SP (G) L2-SP (D) L2-SP (G,D) FreezeD Cat 71.76/73.47 78.21/78.32 71.99/73.42 80.63/80.63 82.67/82.67 82.68/82.95 71.77/73.78 71.54/72.67 71.70/73.47 69.64/69.97 Table 3 3presents the FID scores of each method. Feature distillation and qualitative results are in Appendix B and C, respectively. Scale/shift and L2-SP are too restrictive and thus harms diversity. GLO produces blurry images while MineGAN fails to learn the distribution shift. Table 4 : 4FID scores under SNGAN-projection architecture. Left and right values indicate the best and final FID scores.Oxford Flower CUB-200-2011 Caltech-256 Fine-tuning 27.05/32.51 32.29/32.60 62.20/63.37 FreezeD 24.80/52.92 26.37/27.63 60.53/60.53 distribution shift goes larger. See Appendix A for details. Table 5 : 5Ablation study on freezing layers of D on StyleGAN architecture under 'Cat' and 'Dog' classes in the Animal Face dataset. Layer i indicates that the first i layers of the discriminator are frozen. Layer 4 performs the best. Dog 64.28/67.42 63.63/66.69 63.18/64.88 61.85/62.65 61.46/61.67 62.42/62.86 63.51/64.15 76.52/87.86Fine-tuning Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7 Cat 71.76/73.47 71.44/72.57 70.81/71.86 70.03/70.46 69.64/69.97 70.12/70.40 70.93/75.92 79.41/85.26 Table 6 : 6Ablation study on freezing layers of D on SNGAN-projection architecture under Oxford Flower, CUB-200-2011, Caltech-256 datasets. Layer i indicates that the first i layers of the discriminator are frozen. CUB-200-2011 32.29/32.60 28.80/31.80 26.37/27.63 28.48/28.48 26.87/29.29Fine-tuning Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Oxford Flower 27.05/32.51 27.65/42.14 25.85/42.31 24.80/52.92 25.41/87.60 25.35/104.07 29.92/34.08 Calteth-256 62.20/63.37 60.53/60.53 61.59/61.94 61.29/61.95 61.92/62.88 62.90/62.90 Table 7 : 7Comparison of FreezeD and feature distillation (FD) on StyleGAN architecture under 'Bear', 'Cat', and 'Dog' classes in the Animal Face dataset. FM (layer i) indicates the activations after layer i are matched. Feature distillation shows comparable results to FreezeD while it is twice slower. 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Transferring gans: generating images from limited data. Y Wang, C Wu, L Herranz, J Van De Weijer, A Gonzalez-Garcia, B Raducanu, ECCV. Y. Wang, C. Wu, L. Herranz, J. van de Weijer, A. Gonzalez- Garcia, and B. Raducanu. Transferring gans: generating im- ages from limited data. In ECCV, 2018. How transferable are features in deep neural networks? In NeurIPS. J Yosinski, J Clune, Y Bengio, H Lipson, J. Yosinski, J. Clune, Y. Bengio, and H. Lipson. How trans- ferable are features in deep neural networks? In NeurIPS, 2014. Few-shot adversarial learning of realistic neural talking head models. E Zakharov, A Shysheya, E Burkov, V Lempitsky, arXiv:1905.08233arXiv preprintE. Zakharov, A. Shysheya, E. Burkov, and V. Lempitsky. Few-shot adversarial learning of realistic neural talking head models. arXiv preprint arXiv:1905.08233, 2019. Consistency regularization for generative adversarial networks. H Zhang, Z Zhang, A Odena, H Lee, ICLR. H. Zhang, Z. Zhang, A. Odena, and H. Lee. Consistency regularization for generative adversarial networks. In ICLR, 2020. Z Zhao, S Singh, H Lee, Z Zhang, A Odena, H Zhang, arXiv:2002.04724,2020.(a)CUB-200-2011(fine-tuning)(b)CUB-200-2011Improved consistency regularization for gans. FreezeDarXiv preprintZ. Zhao, S. Singh, H. Lee, Z. Zhang, A. Odena, and H. Zhang. Improved consistency regularization for gans. arXiv preprint arXiv:2002.04724, 2020. (a) CUB-200-2011 (fine-tuning) (b) CUB-200-2011 (FreezeD) Caltech-256 (fine-tuning) (d) Caltech-256 (FreezeD). Caltech-256 (fine-tuning) (d) Caltech-256 (FreezeD)

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{'abstract': 'Generative adversarial networks (GANs) have shown outstanding performance on a wide range of problems in computer vision, graphics, and machine learning, but often require numerous training data and heavy computational resources. To tackle this issue, several methods introduce a transfer learning technique in GAN training. They, however, are either prone to overfitting or limited to learning small distribution shifts. In this paper, we show that simple fine-tuning of GANs with frozen lower layers of the discriminator performs surprisingly well. This simple baseline, FreezeD, significantly outperforms previous techniques used in both unconditional and conditional GANs. We demonstrate the consistent effect using StyleGAN and SNGAN-projection architectures on several datasets of Animal Face, Anime Face, Oxford Flower, CUB-200-2011, and Caltech-256 datasets. The code and results are available at https://github.com/sangwoomo/FreezeD.', 'arxivid': '2002.10964', 'author': ['Sangwoo Mo [email protected] \nJinwoo Shin KAIST\n\n', 'Kaist \nJinwoo Shin KAIST\n\n', 'POSTECHMinsu Cho [email protected] \nJinwoo Shin KAIST\n\n'], 'authoraffiliation': ['Jinwoo Shin KAIST\n', 'Jinwoo Shin KAIST\n', 'Jinwoo Shin KAIST\n'], 'corpusid': 211296588, 'doi': None, 'github_urls': ['https://github.com/sangwoomo/FreezeD.', 'https://github.com/rosinality/', 'https://github.com/pfnet-research/sngan_'], 'n_tokens_mistral': 11133, 'n_tokens_neox': 9291, 'n_words': 4649, 'pdfsha': '0b10d741f6c2cf946604105bd85707af65c7ccf6', 'pdfurls': ['https://arxiv.org/pdf/2002.10964v2.pdf'], 'title': ['Freeze the Discriminator: a Simple Baseline for Fine-Tuning GANs', 'Freeze the Discriminator: a Simple Baseline for Fine-Tuning GANs'], 'venue': []}

arxiv

UprightNet: Geometry-Aware Camera Orientation Estimation from Single Images Wenqi Xian Cornell Tech Cornell University Zhengqi Li Cornell Tech Cornell University Matthew Fisher Adobe Research Jonathan Eisenmann Adobe Research Eli Shechtman Adobe Research Noah Snavely Cornell Tech Cornell University UprightNet: Geometry-Aware Camera Orientation Estimation from Single Images We introduce UprightNet, a learning-based approach for estimating 2DoF camera orientation from a single RGB image of an indoor scene. Unlike recent methods that leverage deep learning to perform black-box regression from image to orientation parameters, we propose an end-to-end framework that incorporates explicit geometric reasoning. In particular, we design a network that predicts two representations of scene geometry, in both the local camera and global reference coordinate systems, and solves for the camera orientation as the rotation that best aligns these two predictions via a differentiable least squares module. This network can be trained end-to-end, and can be supervised with both ground truth camera poses and intermediate representations of surface geometry. We evaluate UprightNet on the single-image camera orientation task on synthetic and real datasets, and show significant improvements over prior state-of-the-art approaches. Introduction We consider the problem of estimating camera orientation from a single RGB photograph. This is an important problem with applications in robotics, image editing, and augmented reality. Classical approaches often rely on 2D projective geometric cues such as vanishing points [26]. However, more recent methods have sought to leverage the power of deep learning to directly regress from an image to extrinsic calibration parameters, by training on images with known ground truth orientation information [48,19]. But these methods typically do not explicitly leverage the knowledge of projective geometry, treating the problem as a black-box regression or classification. In this work, we introduce UprightNet, a novel deep network model for extrinsic camera calibration that incorporates explicit geometric principles. We hypothesize that injecting geometry will help achieve better performance and better * indicates equal contribution Figure 1: UprightNet overview. UprightNet takes a single RGB image and predicts surface geometry in both local camera and global upright coordinate systems. The camera orientation is then computed as the alignment between these two predictions, solved for by a differentiable least squares module, and weighted using predicted weight maps. generalization to a broader class of images, because geometry affords generally applicable principles, and because geometric representations provide a structured intermediary in the otherwise highly non-linear relationship between raw pixels and camera orientation. In particular, we define and use surface frames as an intermediate geometric representation. The orthogonal basis of the surface frames include the surface normal and two vectors that span the tangent plane of the surface. Surface frames allow us to capture useful geometric features-for instance, predicted surface normals on the ground will point directly in the up direction, and horizontal lines in the scene will point perpendicular to the up direction. However, it is not enough to know normals and other salient vectors in camera coordinates, we also need to know which normals are on the ground, etc. Therefore, our insight is to predict surface geometry not only in local camera coordinates, but also in global upright coordinates, as shown in Figure 1. Such a global prediction is consistent across different camera views and is highly related to the semantic task of predicting which pixels are horizontal surfaces (floors and ceilings), and which are vertical (walls). The camera orientation can then be estimated as the rotation that best aligns these two representations of surface geometry. This overall approach is illustrated in Figure 1. This alignment problem can be solved as a constrained least squares problem. We show in this paper that such an approach is end-to-end differentiable, allowing us to train the entire network using both supervisions on the intermediate representation of surface geometry, as well as on the final estimated orientation. We evaluate UprightNet on both synthetic and real-world RGBD datasets, and compare it against classical geometrybased and learning-based methods. Our method shows significant improvements over prior methods, suggesting that the geometry guidance we provide to the network is important for learning better camera orientation. Related Work Single image camera calibration is a longstanding problem in computer vision. Classical geometry-based methods highly rely on low-level image cues. When only a single image is available, parallel and mutually orthogonal line segments detected in the images can be used to estimate vanishing points and the horizon lines [22,36,11,26,35,6,51,5,27,47,50,3]. Other techniques based on the shape of objects such as coplanar circles [9] and repeated patterns [40] have also been proposed. When an RGB-D camera is available, one can solve for the upright orientation by assuming an ideal Manhattan world [15,42,43,24]. When the scene in question has been mapped in 3D, one can also solve for the camera pose by re-localizing the cameras with respect to the 3D maps [29,7,39,8,23]. On the other hand, using machine learning methods to estimate camera orientation from a single image has gained attention. Earlier work proposes to detect and segment skylines of the images in order to estimate horizon lines [13,2]. More recently, CNN-based techniques have been developed for horizon estimation from a single image [52,19,48]. Most of these methods formulate the problem as either regression or classification and impose a strong prior on the location of features correlated with the visible horizon and of corresponding camera parameters. Single-image surface normal prediction powered by deep networks [46,12] can provide a supervision signal for many 3D vision tasks such as planar reconstruction [33], depth completion [53] and 2D-3D alignment [4]. Recently, surface normal was used for single-image camera estimation by directly estimating a ground plane from the depth and normal estimates of segmented ground regions [34]. Unfortunately, such method assumes the ground plane is always visible in the images, and only applies to vehicle-control use cases. In addition, there are recent work making use of local surface frame representation for a variety of 3D scene understanding tasks [20,21]. However, our method extends beyond these ideas by estimating both local and global aligned surface geometry from single images and use such correspondences to estimate camera orientation. Suwajanakorn et al. [44] shows that the supervision on relative pose could automatically discover consistent keypoints of 3D objects, and our method is inspired by this work on end-to-end learning of intermediate representation via pose supervision. Approach Man-made indoor scenes typically consist of prominently structured surfaces such as walls, floors, and ceilings, as well as lines and junctions arising from intersections between these structures. Prominent lines also arise from other scene features, such as oriented textures. In indoor images, such geometric cues provides rich information about camera orientation. We propose to exploit such geometric features by explicitly predicting surface geometry as an intermediate step in estimating camera orientation. To understand the benefits of such an approach, imagine that we take an image and predict per-pixel surface normals, in camera coordinates. How do these relate to camera orientation? Surface normals on the ground and other horizontal surfaces point in the same direction as the cameras up vector-exactly the vector we wish to estimate. Similarly, surface normals on walls and other vertical surfaces are perpendicular to the up vector. Hence, finding the camera orientation can be posed as finding the vector that is most parallel to ground normals, and at the same time most perpendicular to wall normals. However, such an approach assumes that we know which pixels are ground and which are walls. Thus, we propose to also predict normals in global scene coordinates. This approach is in contrast to most work that predicts surface normals, which usually predict them only in the camera reference frame (e.g., if the camera is rolled 45 degrees about its axis, the predicted normals should be rotated accordingly). Given surface geometry predicted in both camera and scene coordinates, the camera orientation can be found as the rotation that best aligns these two frames. This approach is suitable for learning. If the alignment procedure is differentiable, then we can train a method endto-end to predict orientation by comparing to ground-truth orientations. A key advantage is that we can also apply supervision to the intermediate geometric predictions if we have ground truth. What kind of surface geometry should we estimate? Normals are useful as described above, but do not capture inplane features such as junctions and other lines. Hence, we propose to estimate a full orthonormal coordinate frame at each pixel, comprised of a normal and two tangent vectors. We predict these frames as a dual surface geometry representation in two coordinate systems: Figure 2: Visualization of surface geometry. From left to right: (a) image, (b-d) local camera surface frames F c , (e) the third row of global upright surface frames F g . (a) Image (b) n c (c) t c (d) b c (e) f g z • F c : the surface geometry in local camera coordinates. • F g : the surface geometry in global upright coordinates. Surface frames. To represent surface geometry, we define a surface frame F(i) at every pixel location i as a 3 × 3 matrix formed by three mutually orthogonal unit vectors F(i) = [n(i) t(i) b(i)] in either local camera coordinates or global upright coordinates: • n c , n g : surface normal in camera and upright coordinates, respectively. • t c , b c , t g , b g : mutually orthogonal unit vectors that span the tangent plane of the corresponding surface normal, in camera and upright coordinates respectively. (t stands for tangent, and b for bitangent.) As usual, we define the camera coordinate system as a viewdependent local coordinate system, and the global upright coordinate system as the one whose camera up vector aligns with global scene up vector. Which tangent vectors should we choose for t and b? For curved surfaces, these tangent vectors are often defined in terms of local curvature. However, for man-made indoor scenes, many surfaces are planar and hence lack curvature. Instead, we define these vectors to align with the upright orientation of the scene. In particular, we define the tangent vector t as a unit vector derived from the cross product between the surface normal n and the camera y-axis (pointing rightward in our case). The bitangent vector b is then b = n × t. This definition of tangent vectors has a degeneracy when the surface normal is parallel to the camera y-axis, in which case we instead compute t to align with the up vector. However, an advantage of this choice of tangents is that this degeneracy is rare in practice. We find this choice leads to the best performance in our experiments. However, other surface frame representations could also be used [20,21]. Camera orientation. Let R be the 3 × 3 rotation matrix transforming local camera coordinates to global upright coordinates. R maps an upright surface frame F g (i) to its corresponding camera surface frame F c (i) as follows: F g (i) = RF c (i)(1) Note that there is no natural reference for determining the camera heading (yaw angle) from a single image, and moreover we are most interested in determining camera roll and pitch because they are useful for graphics applications. Therefore, our problem is equivalent to finding the scene up vector in the camera coordinate system, which we denote u, and which happens to be the same as the third row of R. The scene up vector u encodes both roll and pitch, but not yaw. It also relates the two surface frames as follows: f g z (i) = u T F c (i)(2) where we define f g z (i) = [n g z (i) t g z (i) b g z (i)] ∈ R 3 as the third row of F g (i) and it has unit length by definition. The last column of Figure 2 shows the vectors f g z (i). Note that f g z (i) is consistent in the same supporting surfaces across images, and hence we refer to it as a scene layout vector. For example, n g z for ground, wall and ceiling pixels is always fixed, to 1, 0 and -1, respectively, across all images, while they can differ in camera coordinates for different images according to camera orientation. Therefore, a beneficial property of the global upright frame representation is that it is similar in spirit to performing a semantic segmentation of the ground, ceiling, and other supporting structures. To estimate 3DoF camera orientation, we could predict both camera and upright surface frames for an image, then estimate a rotation matrix that best aligns these frames. However, since we only estimate 2DoF camera orientation, it is sufficient to predict F c and f g z . Figure 1 shows an overview of our approach. Given a single RGB image, our network predicts per-pixel local camera surface frames F c and scene layout vectors f g z . Using corresponding local/upright frames, we can formulate computing the best up vector as a constrained least squares problem. We show how this problem can be solved in a differentiable manner (Sec. 3.1), allowing us to train a network end-to-end by supervising it with ground truth camera orientations. Predicting weights. A key challenge in our problem formulation is the varying uncertainty of surface geometry predictions in different image regions. We solve orientation estimation via rigid alignment as a least squares problem, which is sensitive to outliers in the predicted surface frames. To address this problem, at each pixel i, we propose to additionally predict separate weights w n (i), w t (i), w b (i) for each of the n, t and b maps, and integrate these weights into the least squares solver. We have no ground truth weights available for supervision, but because we can train our system end-to-end, the network can learn by itself to focus on only the most reliable predicted regions. Hence during training, our model jointly optimizes for surface frames, weights maps, and camera orientation. Up vector from surface frame correspondences Differentiable constrained least squares. Given local surface frames F c and the corresponding f g z , our goal is to find the up vector u that best aligns them. Given Eq. 2, we can write the following constrained minimization problem: min u N i=1 u T F c (i) − f g z (i) 2 2 subject to u 2 = 1 (3) where N is the number of pixels. Eq. 3 can be rewritten in matrix form as: min u Au − b 2 2 , subject to u 2 = 1(4) where the matrix A ∈ R 3N ×3 can be formed by vertically stacking matrices F c (i) for each pixel i, and similarly vector b ∈ R 3N can be formed by stacking the vectors f g z (i). If there were no unit-norm constraint, this problem would be a standard least squares problem. Similarly, if b = 0, the problem becomes a homogeneous least squares problem that can be solved in closed form using SVD [17]. In our case, b is not necessarily a zero vector, preventing us using such standard approaches. However, we show that Eq. 4 can be solved analytically, allowing us to use it to compute a loss in an end-to-end training pipeline. In particular, we can write the Lagrangian of Eq. 4 as L = (Au − b) T (Au − b) − λ(u T u − 1)(5) where λ is a Lagrange multiplier. The KarushKuhnTucker condition of Eq. 5 leads to the following equations: (A T A − λI)u = A T b, u T u = 1(6) To solve for λ and u from Eq. 6 analytically, we use the techniques proposed in [14]. Specifically, we have following theorem [14]: Theorem 1 Eq. 6 can be reduced to a quadratic eigenvalue problem (QEP): Iλ 2 − 2Hλ + H 2 − gg T = 0(7) where H = A T A, g = A T b, and Eq. 7 has a solution for λ. Further, the solution λ and u = (H − λI) −1 g satisfies (H − λI)u = g and u T u = 1. We refer readers to the supplementary material and to [14] for the proof. Fortunately, to solve this QEP, we can reduce it to an ordinary eigenvalue problem [14]: H −I −gg T H γ µ = λ γ µ(8) where γ = (H − λI) −2 g and µ = (H − λI)γ. Since the block matrix on the left hand side of Eq. 8 is not necessarily symmetric, the optimal λ corresponds to its minimum real eigenvalue. The derivative of this eigenvalue can be found in closed form [45], and so the solver is fully differentiable. Weighted least squares. To improve the robustness of the least squares solver, we weight each correspondence in Eq. 3: min u N i=1 W(i) u T F c (i) − f g z (i) T 2 2 subject to u 2 = 1(9) and corresponding Lagrangian can be similarly modified as L = (Au − b) T W T W(Au − b) − λ(u T u − 1) (10) where W ∈ R 3N ×3N is a diagonal matrix, and each 3 × 3 block, denoted as W(i), is diag([w n (i) w t (i) w b (i)]). Hence, we can use the technique described above to solve for λ and u. In our experiments, we show that the predicted weights not only help to reduce the overall estimation error in the presence of noisy predictions, but also focus on supporting structures, as shown in Figure 4 and Figure 5. Loss functions UprightNet jointly optimizes for surface frames, weights, and camera orientation in an end-to-end fashion. Our overall loss function is the weighted sum of terms: L total = L o + α F L F + α ∇ L ∇(11) In contrast to prior approaches that directly perform regression or classification on the ground-truth camera orientation, our method explicitly makes use of geometric reasoning over the entire scene, and we can train a network end-to-end with two primary objectives: • A camera orientation loss that measures the error between recovered up vector and the ground-truth. • A surface geometry loss that measures errors between predicted surface frames and ground-truth surface frames in both local camera and global upright coordinate systems. Camera orientation loss L o . The camera orientation loss is applied to the up vector estimated by the surface frame correspondences and weights using our proposed constrained weighted least squares solver. Specifically, the loss is defined as the angular distance between the estimated up vectorû and the ground-truth one u: L o = arccos (û · u)(12) Note that bothû and u are unit vectors. We can backpropagate through our differentiable constrained weighted least squares solver to minimize this loss directly. A numerical difficulty is that the gradient of arccos(x) reaches infinity when x = 1. To avoid exploding gradients, our loss automatically switches to 1 −û T u whenû · u is greater than 1 − . In our experiments, we set = 10 −6 and find that this strategy leads to faster training convergence and better performance compared to alternatives. Surface frames loss L F . We also introduce a supervised loss L F over predicted surface frames in both coordinate systems to encourage the network to learn a consistent surface geometry representation. In particular, we compute the cosine similarity between each column ofF c and the corresponding column of the ground-truth F c . We also compute the cosine similarity betweenf g z and the ground-truth f g z , yielding the following loss: LF = 2 − 1 3N N i=1 f ∈{n,t,b}f c (i) · f c (i) − 1 N N i=1f g z (i) · f g z (i)(13) Gradient consistency loss L ∇ . Finally, to encourage piecewise constant predictions on flat surfaces and sharp discontinuities, we include a gradient consistency loss across multiple scales, similar to prior work [30,31,32]. The gradient consistency loss L ∇ measures the 1 error between the gradients of the prediction and the corresponding ground truth: L ∇ = S s=1 1 3N s Ns i=1 f ∈{n,t,b} ||∇f c (i) − ∇f c (i)|| 1 + S s=1 1 N s Ns i=1 ||∇f g z (i) − ∇f g z (i)|| 1 (14) where S is the number of scales, and N s is the number of pixels in each scale. In our experiments, we set S = 4 and use nearest neighborhood downsampling to create image pyramids for both the prediction and ground-truth. Network architecture We adopt a U-Net-style network architecture [16,37] for UprightNet. Our network consists of one encoder and three separate decoders forF c (9 channels),f g z (3 channels) and weight maps (3 channels), respectively. We adopt an ImageNet [38] pretrained ResNet-50 [18] as the backbone encoder. Each decoder layer is composed of a 3 × 3 convolutional layer followed by bilinear upsampling, and skip connections are also applied. We normalize each column of F c andf g z to unit length. For the weight maps, we add a sigmoid function at the end of the weight stream and normalize predicted weight maps by dividing them by their mean. Experiments To validate the effectiveness of UprightNet, we train and test on synthetic images from the InteriorNet dataset [28] and real data from ScanNet [10], and compare with several prior single-image orientation estimation methods. Furthermore, to test generalization ability, we directly apply all methods trained on ScanNet to images drawn from the SUN360 [49] dataset without fine-tuning. For all datasets, we show both qualitative and quantitative results, as well as comparisons to other baselines. Datasets InteriorNet [28] is a large, photo-realistic indoor scene dataset of high-quality rendered images with associated ground-truth surface normals and camera poses. We use a pre-release subset of around 34k images. Each scene includes 3 images randomly sampled from a rendered videos. Compared to other synthetic dataset such as SUNCG [41], InteriorNet has a much larger variation in camera pitch and roll. In our experiments, we randomly split InteriorNet into training, validation, and test sets using ratios of 77%, 3%, and 20% based on different scene IDs. During training, we generate randomly cropped images with a vertical field of view (FoV) varying between 35 and 45 degrees. ScanNet [10] is an RGB-D video dataset containing indoor scenes with 3D camera poses and dense surface reconstructions. We use around 50K image frames, sampled approximately every second, for training and evaluation. In addition, during training we use rendered surface normals produced by Zhang and Funkhouser [53] for ground truth supervision, and we use the official train/val/test split based on scene id. We also use the same technique we use with InteriorNet to randomly crop images. SUN360. We also construct a test set of rectilinear crops from the SUN360 panorama dataset [49] for use as a crossdataset test set. Specifically, we extract six rectified images from each indoor panorama, with output camera parameters uniformly sampled from the ranges present in the training set of ScanNet. For all datasets and methods, we resize images to 384x288 before feeding them to the networks. Training details We implement UprightNet in PyTorch [1]. For all experiments, we train using Adam [25], starting from an ImageNetpretrained encoder with initial learning rate 0.0004 and minibatch size of 8. During training, we halve the learning rate Comparisons to baseline methods We compare UprightNet with four baseline methods: • A regression baseline: a CNN that predicts roll and pitch angles, trained with an 1 loss. • DeepHorizon [48], a CNN classification approach to predicting horizon offsets and slopes. • Hold-Geoffroy et al. [19]: a CNN classification method for predicting horizon lines and fields of view. • Lee et al. [26]: a state-of-the-art classical geometrybased method. To facilitate fair comparison, we re-implemented [48] and [19] based on their published descriptions. For all learningbased methods, we adopt the same network architecture and pretrained weights as our method, and train and evaluate them on the same datasets using the same training strategy. To evaluate estimated camera orientations, we compute the mean and median angular error between the predicted and ground truth up vectors, as in Equation 12, as well as absolute errors in both pitch and roll angles. Table 1 shows quantitative results on InteriorNet, and the top row of Figure 3 shows a visualization of estimated horizon lines superimposed on the input images for qualitative comparisons. Our proposed method achieve a significant improvement (30%) compared to prior CNN-based and geometry-based methods for all error metrics. Table 1 for descriptions. Table 2 shows quantitative results on ScanNet, and the second row of Figure 3 shows predicted horizon lines on that dataset. ScanNet is more challenging for image calibration compared to InteriorNet due to motion blur, extreme pitch angles, and more complex scene structures. Nevertheless, our method still improves calibration accuracy by a relative 20% in angular and pitch error, and is slightly better than the best baseline methods in terms of roll angle. Figure 4 shows visualizations of our predicted n c , f g z , and weights for InteriorNet and ScanNet test images. For visualization, in Figure 4(d) we combine the three weights maps by adding the normalizing them, and overlay them on the original input images. Interestingly, the network learns weight maps that tend to focus on the vicinity of vertical/horizontal surface line junctions. Such lines are very strong cues for vanishing points. For instance, t c in the vicinity of vertical junctions and b c in the vicinity of horizontal junctions both represent 3D directions of their respective junction lines, and also point towards 2D vanishing points [17]. It is interesting that the network "discovers" such features automatically Ablation analysis We now explore how the weight maps and different configurations of surface geometry supervision affect the performance of UprightNet through validation on InteriorNet. Impact of weights. We wanted to see if the network learns to correctly assign large weights to regions with smaller alignment error. We compute an alignment score map by using the ground-truth up vector u to align each column of F c with its corresponding scalar inf c z . We define alignment scores for surface normals as S n = exp(−10|u Tnc −n g z |) and similarly for the two tangent vectors. Figure 5 visualizes these alignment scores, and compares them with the predicted weights. The network indeed tends to predict large weights where the alignment error is small. This suggests that, while our surface geometry predictions are not always accurate, the weights can capture which image regions are more likely to have correct predictions, thus leading to better estimation of camera orientation. (a) Image (b) n c (c) f g z (d) weights Impact of camera orientation loss. As shown in Table 3, we evaluate our models using different configurations to analyze their influence on the performance. Comparing with direct estimation of camera pose from the predicted surface frame correspondences, end-to-end optimization with camera orientation loss boosts performance significantly. We observe an additional increase in performance by incorporating predicted weights into orientation estimation. Since the supervision of surface frames in both coordinate systems is not required during training, we can also train a model using supervision only from local camera frames, or only from global upright frames. The ablations shown in Table 3 suggest that using both local camera and global upright surface geometry as supervision leads to the best performance. Impact of surface geometry representation. We also ex- plore the influence of different surface geometry representations on camera orientation estimation. In particular, we compare our full surface geometry representation to (1) a single vector representation, i.e., just one of (n, t, b), and (2) a combination of any two of them. As shown in Table 4, our proposed full surface frame representation achieves the best performance. This suggests that each basis vector of our surface frames captures complementary geometry cues in indoor scenes. In particular, n c of grounds/ceilings and t c of vertical lines directly represent the scene up vectors we seek, while b c of horizontal lines on supporting surfaces could correspond to major vanishing points in the scene. (a) Image (b) n c (c) wn (d) Sn (a) Image (b) t c (c) wt (d) St (a) Image (b) b c (c) w b (d) S b Generalization to SUN360 We explore the ability of different learning-based calibration methods to generalize across datasets by taking models trained on ScanNet and testing them on crops from the SUN360 indoor panorama dataset. We summarize the results in Table 5 predictions of our model in Figure 6. While the errors are naturally higher on this unseen dataset, UprightNet still outperforms prior methods by large margins in all error metrics. This suggests that, compared with implicit features learned from direct regression or classification, using an intermediate geometric representation can help a model attain better generalization to new distributions of indoor imagery. We also demonstrate the use of UprightNet for an application in virtual object insertion in Figure 7. We orient a 3D object with the camera's pitch and roll estimated by Upright-Net. The 2D translation of the object along the ground plane is manually chosen by the artist. Finally, a 2D render of the object is composited on top of the image to produce a final results, shown in the bottom row of Figure 7. Limitations and future work The last row of Figure 6 demonstrate a common failure mode of our approach, where the image lacks sufficient supporting structure for the network to reason about geometry, resulting in inaccurate surface geometry and camera orientation predictions. In future work, other explicit 2D geometric priors, such as vanishing points, or camera intrinsics (if known) can also be integrated to our framework. Conclusion We introduced UprightNet, a new method for predicting 2DoF camera orientation from a single indoor image. Our key insight is to leverage surface geometry information from both the camera and upright coordinate systems, and pose camera orientation prediction as an alignment problem between these two frames. In particular, we showed how a network can be trained to predict camera-centric and global surface frames, and combine them with weights to estimate the camera orientation. Our evaluations demonstrated not only more accurate orientation estimation, but also better generalization to unseen datasets, compared with prior stateof-the-art methods. Figure 3 : 3Qualitative comparison of horizon line predictions. From top row to bottom row: InteriorNet, ScanNet and SUN360. Our trained model outperforms other baselines in terms of accuracy on all three datasets. every 5 epochs. More details on hyperparameter settings are included in the supplemental material. Figure 4 : 4Visualizations of predictions in InteriorNet (top 3 rows) and ScanNet (bottom 3 rows). In (d), we overlay weight maps (combined weights for n, t and b) over input images. Blue=small weights, red=large weights. Figure 6 : 6Visualizations of predictions on the SUN360 testset. The last row shows a failure case. Figure 7 : 7Application. Virtual insertion of 3D objects in images from SUN360 dataset using the camera orientation estimated by UprightNet. Table 2 : 2Quantitative comparisons on the ScanNet testset. See Figure 5: Comparisons between predicted weights and alignment scores. From left to right (a) input image, (b) predict camera surface frames, (c) predicted weights, (d) alignment scores S. In (c) and (d), white indicates large weights and high alignment score (i.e. low alignment error). angular error ( • ) pitch error ( • ) roll error ( • )Method avg. med. avg. med. avg. med. Ours (w/o weight) 1.83 1.15 1.22 0.81 0.90 0.57 Ours (w/o L o ) 2.71 1.74 1.21 2.83 1.37 0.93 Ours (w/o F c loss) 1.70 1.11 1.31 0.99 0.76 0.57 Ours (w/o F u loss) 1.82 1.18 1.32 0.98 0.79 0.55 Ours (full) 1.17 0.52 0.99 0.47 0.44 0.11 Table 3 : 3Ablation studies for different configurations on the InteriorNet test set. Ours (full) indicate our proposed full method, which achieves the best performance among all tested configurations. , and visualize surface normal and weight Method angular error ( • ) pitch error ( • ) roll error ( • )avg. med. avg. med. avg. med. n 1.81 1.13 1.35 0.85 0.90 0.45 t 2.11 1.28 1.60 0.96 1.06 0.47 b 1.82 1.09 1.39 0.84 0.81 0.47 nt 1.52 0.97 1.27 0.83 0.58 0.29 nb 1.17 0.89 1.33 0.77 0.78 0.29 tb 1.56 0.66 1.29 0.62 0.66 0.20 ntb 1.17 0.52 0.99 0.47 0.44 0.11 Table 4 : 4Ablation studies for different surface geometry representations on the InteriorNet test set. ntb indicates our full surface geometry representation. Method angular error ( • ) pitch error ( • ) roll error ( • ) Hold-Geoffroy et al.[19] 10.41avg. med. avg. med. avg. med. CNN Regression 10.43 6.99 9.57 6.10 3.10 2.17 DeepHorizon [48] 9.53 6.28 8.68 5.50 2.98 1.89 6.92 9.57 6.09 3.11 2.20 Ours 7.81 5.53 7.59 4.94 2.30 1.53 Table 5 : 5Generalization performance on the SUN360 test set. 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{'abstract': 'We introduce UprightNet, a learning-based approach for estimating 2DoF camera orientation from a single RGB image of an indoor scene. Unlike recent methods that leverage deep learning to perform black-box regression from image to orientation parameters, we propose an end-to-end framework that incorporates explicit geometric reasoning. In particular, we design a network that predicts two representations of scene geometry, in both the local camera and global reference coordinate systems, and solves for the camera orientation as the rotation that best aligns these two predictions via a differentiable least squares module. This network can be trained end-to-end, and can be supervised with both ground truth camera poses and intermediate representations of surface geometry. We evaluate UprightNet on the single-image camera orientation task on synthetic and real datasets, and show significant improvements over prior state-of-the-art approaches.', 'arxivid': '1908.07070', 'author': ['Wenqi Xian \nCornell Tech\nCornell University\n\n', 'Zhengqi Li \nCornell Tech\nCornell University\n\n', 'Matthew Fisher \nAdobe Research\n\n', 'Jonathan Eisenmann \nAdobe Research\n\n', 'Eli Shechtman \nAdobe Research\n\n', 'Noah Snavely \nCornell Tech\nCornell University\n\n'], 'authoraffiliation': ['Cornell Tech\nCornell University\n', 'Cornell Tech\nCornell University\n', 'Adobe Research\n', 'Adobe Research\n', 'Adobe Research\n', 'Cornell Tech\nCornell University\n'], 'corpusid': 201107189, 'doi': '10.1109/iccv.2019.01007', 'github_urls': [], 'n_tokens_mistral': 15269, 'n_tokens_neox': 13390, 'n_words': 8313, 'pdfsha': 'd384dbcda90fddf9c345f091a45e0d730d3c1033', 'pdfurls': ['https://arxiv.org/pdf/1908.07070v1.pdf'], 'title': ['UprightNet: Geometry-Aware Camera Orientation Estimation from Single Images', 'UprightNet: Geometry-Aware Camera Orientation Estimation from Single Images'], 'venue': []}

arxiv

Talk given at the first symposium of FCNC February 19-21, 1997 Yuval Grossman Stanford Linear Accelerator Center Stanford University 94309StanfordCA Talk given at the first symposium of FCNC Santa Monica, CAFebruary 19-21, 1997arXiv:hep-ph/9704208v1 3 The KL → π 0 νν decay is analyzed in a model independent way. When lepton flavor is conserved, this decay mode is a manifestation of CP violating interference between mixing and decay. Consequently, a theoretically clean relation between the measured rate and electroweak parameters holds in any given model. K L → π 0 νν is unique among K decays in several aspects: (a) It is theoretically very clean; (b) it is purely CP violating 1,2 ; and (c) it can be measured in the near future 3 even if the rate is as small as the Standard Model prediction. In the Standard Model a measurement of Γ(K L → π 0 νν) provides a clean determination of the Wolfenstein CP violating parameter η or, equivalently, of the Jarlskog measure of CP violation J and, together with a measurement of Γ(K + → π + νν), of the angle β of the unitarity triangle 2 . Here we explain what can be learned from the K → πνν decay in a model independent way 4 . We define λ ≡ q pĀ A ,(1) where p and q are the components of interaction eigenstates in mass eigenstates, |K L,S = p|K 0 ∓ q|K 0 , and A(Ā) is the K 0 (K 0 ) → π 0 νν decay amplitude. Then, the ratio between the K L and K S decay rates is 4 Γ(K L → π 0 νν) Γ(K S → π 0 νν) = 1 + |λ| 2 − 2Reλ 1 + |λ| 2 + 2Reλ .(2) In general, a three body final state does not have a definite CP parity. However, if the light neutrinos are purely left-handed, and if lepton flavor is conserved, the final state is CP even (to an excellent approximation) 4 . If lepton flavor is violated, the final state in K L → π 0 νν is not necessarily a CP eigenstate; specifically, K L → π 0 ν iνj with i = j is allowed. Here, we concentrate on the case where the above two conditions are satisfied, so that the final state is purely CP even. The contributions to the K L → π 0 νν decay from CP violation in mixing (|q/p| = 1) and from CP violation in decay (|Ā/A| = 1) are negligibly small. The deviation of |q/p| from unity is experimentally measured (by the CP asymmetry in K L → πℓν) and is O(10 −3 ). The deviation of |Ā/A| from unity is expected to be even smaller 4 . Therefore, |λ| = 1 + O(10 −3 ), and the leading CP violating effect is Imλ = 0, namely interference between mixing and decay. This puts the ratio of decay rates (2) in the same class as CP asymmetries in various B decays to final CP eigenstates, e.g. B → ψK S , where a very clean theoretical analysis is possible 5 . As a result of this cleanliness, the CP violating phase can be extracted almost without any hadronic uncertainty, even if this phase comes from New Physics. Defining θ to be the relative phase between the K −K mixing amplitude and the s → dνν decay amplitude, namely λ = e 2iθ , we get from eq. (2) Γ(K L → π 0 νν) Γ(K S → π 0 νν) = 1 − cos 2θ 1 + cos 2θ = tan 2 θ.(3) In reality, however, it will be impossible to measure Γ(K S → π 0 νν). We can use the isospin relation, A(K 0 → π 0 νν)/A(K + → π + νν) = 1/ √ 2, to replace the denominator by the charged kaon decay mode: a CP ≡ r is Γ(K L → π 0 νν) Γ(K + → π + νν) = 1 − cos 2θ 2 = sin 2 θ,(4) where r is = 0.954 is the isospin breaking factor 6 . The ratio (4) may be experimentally measurable as the relevant branching ratios are O(10 −10 ) in the Standard Model 2 and even larger in some of its extensions. Eq. (4) implies that a measurement of a CP will allow us to determine the CP violating phase θ without any information about the magnitude of the decay amplitudes. Also, using sin 2 θ ≤ 1 and τ KL /τ K + = 4.17, we get the model independent bound BR(K L → π 0 νν) < 1.1 × 10 −8 BR(K + → π + νν) 2.4 × 10 −9 .(5) This bound is much stronger than the direct experimental upper bound 7 BR(K L → π 0 νν) < 5.8 × 10 −5 . New Physics can modify both the mixing and the decay amplitudes. ε = O(10 −3 ) implies that any new contribution to the mixing amplitude carries almost the same phase as the Standard Model one. On the other hand, the upper bound 8 BR(K + → π + νν) < 2.4 × 10 −9 , which is much larger than the Standard Model prediction 2 , allows New Physics to dominate the decay amplitude (with an arbitrary phase). We conclude that a significant modification of a CP can only come from New Physics in the decay amplitude. For example, in models with extra quarks, the decay amplitudes can be dominated by tree level Z-mediated diagrams 4 . In superweak models, all CP violating effects appear in the mixing amplitudes. Then, CP violation in K L → π 0 νν should be similar in magnitude to that in K L → ππ. In models of approximate CP symmetry, all CP violating effects are small. Both scenarios predict then a CP = O(10 −3 ), in contrast to the Standard Model prediction, a CP = O(1). In other words, a measurement of a CP ≫ 10 −3 (and, in particular, BR(K L → π 0 νν) > ∼ O(10 −11 )) will exclude these two scenarios of New Physics in CP violation. In the Standard Model there are two clean ways to determine the unitarity triangle: (1) CP asymmetries in B 0 decays 5 ; and (2) the combination of BR(K L → π 0 νν) and BR(K + → π + νν) 2 . In general, New Physics will affect both determinations. Moreover, it is very unlikely that the modification of the two methods will be the same. Consequently, a comparison between these two clean determinations will be a very powerful tool to probe CP violation beyond the Standard Model. Because of the very small theoretical uncertainties in both methods even a small new physics effect can be detected. In practice, we will be limited only by the experimental sensitivity. In conclusion: a measurement of BR(K L → π 0 νν) is guaranteed to provide us with valuable information. It will either give a new clean measurement of CP violation or indicate lepton flavor violation. given at the first symposium of FCNC, February 19-21, 1997, Santa Monica, CA. arXiv:hep-ph/9704208v1 3 Apr 1997 SLAC-PUB-7443 hep-ph/9704208 Acknowledgments. I thank Yossi Nir for collaboration on this work. Y.G. is supported by the Department of Energy under contract DE-AC03-76SF00515. . L S Littenberg, Phys. Rev. D. 393322L.S. Littenberg, Phys. Rev. D 39 (1989) 3322. . G Buchalla, A J Buras, Nucl. Phys. B. 400225G. Buchalla and A.J. Buras, Nucl. Phys. B 400 (1993) 225; . Phys. Rev. D. 546782Phys. Rev. D 54 (1996) 6782; . A J Buras, Phys. Lett. B. 333476A.J. Buras, Phys. Lett. B 333 (1994) 476; . G Buchalla, these proceedingsG. Buchalla, these proceedings. . T Inagaki, these proceedingsT. Inagaki, these proceedings; . D Bryman, these proceedingsD. Bryman, these proceedings; . K Arisaka, these proceedingsK. Arisaka, these proceedings. . Y Grossman, Y Nir, hep-ph/9701313Phys. Lett. B. to appear inY. Grossman, and Y. Nir, hep-ph/9701313, to appear in Phys. Lett. B . . E G Y See, H R Nir, Quinn, Ann. Rev. Nucl. Part. Sci. 42211See, e.g. Y. Nir and H.R. Quinn, Ann. Rev. Nucl. Part. Sci. 42 (1992) 211. . W Marciano, Z Parsa, Phys. Rev. D. 531W. Marciano and Z. Parsa, Phys. Rev. D 53 (1996) 1. . M Weaver, E799 CollaborationPhys. Rev. Lett. 723758M. Weaver et al., E799 Collaboration, Phys. Rev. Lett. 72 (1994) 3758. . S Adler, BNL 787 CollaborationPhys. Rev. Lett. 761421S. Adler et al., BNL 787 Collaboration, Phys. Rev. Lett. 76 (1996) 1421.

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{'abstract': 'The KL → π 0 νν decay is analyzed in a model independent way. When lepton flavor is conserved, this decay mode is a manifestation of CP violating interference between mixing and decay. Consequently, a theoretically clean relation between the measured rate and electroweak parameters holds in any given model.', 'arxivid': 'hep-ph/9704208', 'author': ['Yuval Grossman \nStanford Linear Accelerator Center Stanford University\n94309StanfordCA\n'], 'authoraffiliation': ['Stanford Linear Accelerator Center Stanford University\n94309StanfordCA'], 'corpusid': 118612611, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2593, 'n_tokens_neox': 2145, 'n_words': 1356, 'pdfsha': '57b38ce9dd05850651ede85e6872e85e78db68ca', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/9704208v1.pdf'], 'title': ['Talk given at the first symposium of FCNC', 'Talk given at the first symposium of FCNC'], 'venue': []}

arxiv

A Portal for High-Precision Atomic Data and Computation: Design and Best Practices Parinaz Barakhshan Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Department of Computer and Information Sciences Department of Electrical and Computer Engineering Department of Physics and Astronomy Perimeter Institute for Theoretical Physics and Guru Nanak Dev University University of Delaware University of Delaware University of Delaware University of Delaware University of Delaware Akshay Bhosale Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Department of Computer and Information Sciences Department of Electrical and Computer Engineering Department of Physics and Astronomy Perimeter Institute for Theoretical Physics and Guru Nanak Dev University University of Delaware University of Delaware University of Delaware University of Delaware University of Delaware Amani Kiruga Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Department of Computer and Information Sciences Department of Electrical and Computer Engineering Department of Physics and Astronomy Perimeter Institute for Theoretical Physics and Guru Nanak Dev University University of Delaware University of Delaware University of Delaware University of Delaware University of Delaware Rudolf Eigenmann Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Department of Computer and Information Sciences Department of Electrical and Computer Engineering Department of Physics and Astronomy Perimeter Institute for Theoretical Physics and Guru Nanak Dev University University of Delaware University of Delaware University of Delaware University of Delaware University of Delaware Marianna S Safronova Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Department of Computer and Information Sciences Department of Electrical and Computer Engineering Department of Physics and Astronomy Perimeter Institute for Theoretical Physics and Guru Nanak Dev University University of Delaware University of Delaware University of Delaware University of Delaware University of Delaware Bindiya Arora Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Department of Computer and Information Sciences Department of Electrical and Computer Engineering Department of Physics and Astronomy Perimeter Institute for Theoretical Physics and Guru Nanak Dev University University of Delaware University of Delaware University of Delaware University of Delaware University of Delaware A Portal for High-Precision Atomic Data and Computation: Design and Best Practices This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. The atom portal [1], udel.edu/atom, provides the scientific community with easily accessible high-quality data about properties of atoms and ions, such as energies, transition matrix elements, transition rates, radiative lifetimes, branching ratios, polarizabilities, and hyperfine constants. The data are calculated using a high-precision state-of-the-art linearized coupled-cluster method, high-precision experimental values are used where available. All values include estimated uncertainties. Where available, experimental results are provided with references. This paper provides an overview of the portal and describes the design as well as applied software engineering practices. INTRODUCTION In a number of present applications, ranging from studies of fundamental interactions to the development of future technologies, accurate atomic theory is indispensable to the design and interpretation of experiments, with direct experimental measurement of relevant parameters being impossible or infeasible. These data are also in high demand by broader atomic, plasma, astrophysics, and nuclear physics communities. The need for high-precision atomic modeling has increased significantly in recent years with the development of atomic-based quantum technologies for a wide range of fundamental and practical applications. Further rapid advances will require accurate knowledge of basic atomic properties, most of which remain highly uncertain and difficult to measure experimentally. The lack of a comprehensive atomic database for modern research applications created a bottleneck in experimental design and analysis as well as in-dustry applications which motivated the creation of the portal for high-precision atomic data and computation. This paper reports the release of Version 2.0 of an online portal that provides high-precision atomic data derived from computationally intensive calculations performed on a growing number of atoms and ions. This project fills the need for high-quality atomic data and software in several scientific communities. The latest version of the portal, Version 2.0, provides over half a dozen properties of atoms and ions, including highly charged ions. As of November 2022, over 2600 users have accessed the website since it was published in April 2021. Users are connecting from 93 countries. The top ten countries that have accessed the website were the United States (36%), Germany (11%), India (10%), United Kingdom (6%), China (5%), Canada (3%), Switzerland (3%), Singapore (3%), France (2%), and Russia (2%). There have been 5,567 downloads and prints from the website. The latest version of the portal has added new properties and elements as well as enhanced the user interface based on community feedback. The majority of web pages are generated in an automated manner from the data supplied by the physicists. In this way, human error and repetitive work for different elements and properties can be reduced and the creation of pages for newly introduced elements is made easy for non-experts. The project is an interdisciplinary effort where physicists have produced the data and computer scientists have developed the website for presenting the data. Software engineering approaches have been applied to facilitate this joint portal development, resulting in savings of time and effort. The remainder of the paper is organized as follows. The portal overview section provides a brief overview of the portal. A description of how the data is generated is provided in the generation of the Physics data section. The section on portal design challenges describes the challenges we encountered during the development of the portal. The practices employed in this project are described in the section entitled applied software engineering practices, followed by conclusions. Portal Overview The current version of the portal provides users with pre-computed data on atomic properties through an interactive interface, allowing them to view and download the provided information. This information includes the properties transition matrix elements, transition rates, radiative lifetimes, branching ratios, hyperfine constants, quadrupole moments, scalar and dynamic polarizabilities, and energies of atoms and ions, including Li, Be + , Na, Mg + , K, Ca + , Rb, Sr + , Cs, Ba + , Fr, and Ra + . Furthermore, 13 highly charged ions have been added to this version, including Cs 6+ , Ba 7+ , Ce 9+ , Pr 10+ , Nd 11+ , Nd 12+ , Nd 13+ , Sm 13+ , Sm 14+ , Sm 15+ , Eu 14+ , Cf 15+ , and Cf 17+ . Figure 1 illustrates the home page of the Atom portal, which provides instant access to information about atoms and ions based on precomputed properties. An example property calculated for these atoms is their polarizability, which describes their response to an external electric field. Figure 2 illustrates the graphical interface designed for easy interaction by the user. Data for 12 elements are calculated using a state-of-the-art linearized coupled-cluster method that will be discussed in more detail in the next section. High-precision experimental values are used where available with references. Data for highly-charged ions is taken from the literature and references to the original source. The goal of the portal is to provide a recommended value that users should use for any given property, i.e. this is the most accurate value given the state-of-the-art knowledge in atomic physics. Generation of the Physics Data This section explains how physicists generate data. Data provided by the Atom portal for Li, Be + , Na, Mg + , K, Ca + , Rb, Sr + , Cs, Ba + , Fr, and Ra + are computed and evaluated for accuracy based on a long history of Physics research in developing accurate models for the computation of atomic properties and corresponding computational codes using a linearized coupled-cluster method reviewed in [2]. The polarizability computations are described in [3]. The computations of the quadrupole moments are described in [4]. This method is also referred to as the allorder method in literature as it involves summing series of dominant many-body perturbation terms to all orders. In the single-double (SD) all-order approach, single and double excitations of the Dirac-Fock orbitals are included. The single, double, particle triple (SDpT) all-order approach also includes classes of the triple excitations. Omitted higher excitations are estimated by the scaling procedure which can be started from either SD or SDpT approximations. We carry out four allorder computations for each of the electric-dipole matrix elements, ab initio SD and SDpT and scaled SD and SDpT. Either SD or SD scaled data are taken as final values based on the comparison of different contributions to the matrix elements. An algorithm is used to determine the uncertainties of the electric-dipole matrix elements based on the spread of the four results, size of the correlation correction, and comparison of different contributions to the matrix elements [5]. A detailed description of the SD and SDpT allorder approaches with all formulas is given in the same document. The computational codes have been made efficient through parallel computation using OpenMP pragmas embedded in the Fortran 90 code and running on multi-core machines. Due to the high resource demand for the code runs, computations are performed on compute nodes of the Caviness computational cluster [7] at the University of Delaware (UD). These code runs calculate the broad range of atomic properties mentioned in the introduction. Energies provide information about the electronic energy level relative to the ground state energy for each of the elements. The wavelength refers to photon wavelength for the transition between two electronic energy levels. Transition matrix elements quantify the strength of electronic transition between two levels. Transition rates give the probability of an electron making the transition. Radiative lifetimes are the average times that an excited atom can remain in that state before emitting a photon. Branching ratios describe the relative probabilities of all possible transitions. Hyperfine constants describe the interactions between the nucleus and the electrons and are used to quantify the resulting splitting of energy levels due to this effect. Quadrupole moments give information about the shape of an atoms' electron cloud. The scalar and dynamic polarizabilities describe the atoms' response to an external electric field; dynamic polarizability is used to understand the response of an atom to a laser light of a given frequency. The data for Cs 6+ , Ba 7+ , Ce 9+ , Pr 10+ , Nd 11+ , Nd 12+ , Nd 13+ , Sm 13+ , Sm 14+ , Sm 15+ , Eu 14+ , Cf 15+ , and Cf 17+ highly-charged ions have been computed by the University of Delaware team and collaborators using various methods, including the all-order method described above, and other methods that combine configuration interaction and coupled cluster approaches [6]. The use of more complicated methods is required due to the complicated electronic structures of some of the ions evaluated. As these computations are recent and represent the state-of-the-art, we use the generated data for the portal and provide references to original papers. Portal Design Challenges This section describes the challenges faced in creating a portal that is easy to populate with data for physicists and easy to navigate for the user community. Automated portal generation from physics data The portal interface provides functions that allow the user to select the data, present it in a user-oriented way, and print/download the viewed information. Users can select a particular atom or ion, choose one or several electronic states, and obtain data for a particular property they want to view. The original data underlying the portal is provided by physicists in well-formatted files that are placed in a source directory. The names and format of all data files are standardized for automated input. One design goal was to avoid manual portal changes when physicists provide or update new data. Data updates could stem from improvements of the computational codes or from high-precision physics experiments conducted by any researcher in the community. Another goal was to create the user interface functionality without the need for dynamic web content support, making the website portable to a large range of potential web servers. We addressed these challenges through the development of a portal-generation algorithm, implemented through Python scripts. The algorithm produces static web pages with the physics data directly embedded, and with JavaScript methods providing the needed user functionality. Data interface for physics developers We aimed to make it easy for physicists, both those involved in the project and external researchers providing experimental data, to supply their information without needing to understand web technology. To this end, we defined appropriate data formats for expressing elements, ions, and their properties. The data is in part generated automatically by the computational codes. Manual assistance is needed for such tasks as ensuring data validity, providing data from physics experiments, and uploading the data to the portal's source directory. The chosen format uses CSV (comma-separated values), which can also be generated from an excel spreadsheet. Integrating data from different sources The provision of data from two sourcesthe computational codes and from external experimental measurements -poses challenges for both the user interface and the implementation. For the user interface, we chose a display that marks data from sources other than our computational codes with a tag. Clicking on this tag opens a dialog box providing a reference and a link to the paper describing the experiment. The presence of experimental data is signaled to the portal-generation algorithm by the presence of a corresponding CSV file. The algorithm reads this file and replaces the computed properties by the experimentally provided values in the display, adding the tags. Presentation of complex state designations Physicists use complex notations for electron states, which involve superscripts, subscripts, and fractions. An example of such a complex notation is 5f 6p 2 2 F 5/2 . In general, web page designers can use HTML notation for displaying such expressions. Alternatively, LaTeX libraries support the same. The state designations also pose challenges for the formats of physics input and download files, which use plain text CSV. While there are Unicode representations of superscripts and subscripts for plain text, many viewing tools only support ASCII code; they display special Unicode characters as "unknown", such as question marks or blank squares. We discarded this solution, as we do not control the users' viewer tools. The variance in the first part of the designation requires formatting decisions (consider 6s6p, 6s 2 6p, 6s 2 6p 2 , and 5d 3 6s6p all of which appear for some atoms). Our solution is to employ a plain-text, ASCII representation of the states. We refer to this representation as the dot notation. For example, in dot notation, the example state mentioned above is represented as 5f.6p2 2F5/2, which indicates there is one 5f electron and two 6f electrons in this electronic state designation, allowing for automated input and conversion to the HTML. The second part of the designation always has a format of a superscript, an upper case letter, and a subscript, simplifying automation. Such formatting allows to describe complicated state designations and is already in use by the physics community [11]. An alternative would be the use of a LaTeX math notation, which is also an ASCII. The LaTeX representation of the above example would be 5f6pˆ2\;ˆ2Fˆ0_{5/2}. We preferred dot notation for its simplicity and the fact that it is already in use by physicists. Furthermore, for internal processing, the states are enumerated. A search function, for example, can look up the state's number, instead of performing a string search operation. Eliminated redundancy in menu navigation In our initial version of the portal, it was possible to navigate to certain information pages on more than one path. For example, to see the transition rates for atom K, one could choose the Transition Rate tab on the front page and then select the atom. Alternatively, the user could choose K in the "Transition Rates" menu, which was available on all pages. While convenient, this form of redundant path navigation was confusing to some users. We eliminated the redundancy in Version 2. Links to related external websites The information provided by the Atom portal is complimentary to the data provided by other websites. In atomic structure, the most commonly used database is NIST's ASD (Atomic Spectra Database) [11]. It has the largest collection of measured atomic energy levels and transition rates. However, while the accuracy of experimental energies is higher, the accuracy of experimental transition rates is generally low, with a few exceptions for most commonly studied elements. The database only focuses on these properties. To avoid redundancy, we use ASD energy data where available, creating references to the ASD website instead of duplicating ASD data. While redundant data could add convenience of having all data in the same page, for example -users would like to see energy levels next to transition rates -pointing the user to the ASD website better acknowledges their contribution and improves maintainability, in case NIST updates ASD information. The transition rates on our website are expected to be more accurate than on the ASD website, and we do not use ASD data. We also provide energy levels that are omitted on the ASD website due to lack of experimental data. Data consistency checks Ensuring that the data shown on the Atom portal is always correct is a non-trivial task. Currently, substantial manual work is involved in verifying correctness, while certain checks are automated already. For example, the display format for uncertainty is provided in the physics input files and also checked by a method in our portalgeneration algorithm. To eliminate the need for manual intervention, it is vital to automate the data consistency checks, and is being addressed in our ongoing work. User feedback function Websites can only reach maturity with substantial user feedback. A prominent user-feedback button serves to solicit suggestions for improvements from portal users. The same function allows users to see suggestions made by others, including the implementation status. Users can also view the features that are forthcoming in the immediate and future releases. Tracking Portal Use The Google Analytics tool allows website administrators to track and analyze users who visit their websites. A Google Analytics script has been embedded in each page to collect data, such as the number of users, session statistics, approximate geographical location, browser type, device information, and the number of downloads and prints. This information allows us to understand the portal's use and make data-driven design decisions. Applied Software Engineering Practices Careful attention to software engineering principles not only increases maintainability and sustainability; it is also a requirement of our project sponsor, the National Science Foundation (NSF). We describe the practices, as defined in an effort to develop best practices for multidisciplinary projects, in which research software engineering teams support, and collaborate with, domain science groups [8]. We briefly explain each of the recommended practices, followed by the specific use in our project and illustrating examples. The recommendations identify two categories: (i) best practices for project collaborations and (ii) best practices for software development. For reference, we use the same titles of the practices; we also use the term Xperts for computer and data experts supporting domain scientists (physicists, in our project). Diversity of Xpert Backgrounds Software development teams often include experts with diverse backgrounds. As a result, training should be planned so that less-familiar practices can be acquired as needed [9]. In the Atom project, developers from different backgrounds joined the development team at various stages of the development process. To be able to contribute to the project, each participant had to undergo a learning process. For example, to automate the creation of web pages, developers from both computer science and physics departments were added to the project for a limited time. The on-boarding process included steps, such as becoming familiar with the code, gaining the skills needed to make modifications, and asking questions. Breadth of Xpert Skills Needed Computational and Data-intensive (CDI) applications today incorporate a broad range of technologies, including computing paradigms, programming languages, architectures, and algorithms. The technologies to be used in the project and the skills required for the development team were determined from the project requirements. We then engaged project members with the necessary expertise. Collaborative Assistance Between Xperts and Domain scientists A recommended form of interaction between Xperts and domain scientists is through collaborative assistance, where Xperts work side by side (physically or virtually) with the domain scientists whose projects they support. Throughout the joint work, the collaborators tend to pick up sufficient knowledge of each others' skills and terminology. Supporting and facilitating collaboration between the members of the project is crucial. Joint, periodic reviews of milestones will reduce issues that may impede progress. The project engaged both physicists and computer scientists in close collaboration. The collaboration involved weekly meetings to discuss the progress, provide updates, possibly revise decisions, assign new development tasks, and discuss next steps. Overcoming the Terminology Gap Computer science and domain science use different jargon. This issue may complicate the understanding of the problem domain and project requirements since domain scientists tend to use rich vocabulary. The key to successful collaboration is keeping the vocabulary to the essentials and investing time in explaining new terms. Using many examples and frequent feedback from both sides will help bridge this gap. At first, in this project, it was difficult to understand the domain requirements, due to terminology gaps. While many terms were familiar to domain scientists, getting a clear understanding of their exact meaning was challenging for computer scientists. As a result, there were numerous questions asked by both parties so as to understand the terminologies. Detailed written explanation of physics terminology with examples were provided by physicists and discussed during the meetings, so they can be used for future references as well as by new team members. Additionally, the development team requested that the domain scientists provide multiple examples of the functionalities expected from the portal. Prototypes were used to arrive at a common understanding. Understanding the Domain Problem and Developing a Project Plan Xperts must devote substantial time to understand the domain problem, turning possibly vague ideas into a feasible approach, and developing a clear project plan. At the start of the development of Version 2, a detailed document describing new functionalities with examples was developed by the combined team and thoroughly discussed to guide future work. Having a clear understanding of the requirements for each version of the project is crucial and impacts many of the decisions that should be made along the way. Project requirements need to be well thought out, balanced, and clearly understood by all involved. It is important that the identified requirements are not dropped or compromised halfway through the project. In this project, the requirements of different versions of the project were identified, documented, discussed, and sometimes prototyped to make sure they were feasible to be implemented and reflected what the team had envisioned and then implemented. The implementation was subjected to multiple revisions to ensure that it met the requirements of not only the physicists but also of the user community. Many examples were provided by both parties to ensure that the requirements were intelligible to team members. Prototyping helped team members visualize how the final pages would look like. It also helped in highlighting unanticipated design or technical challenges. By giving everyone a clear vision of the project, the prototype made everyone more involved in the process of building the pages. Prioritize Functional Requirements A dilemma that is often encountered in the process of planning any project is the existence of many desirable features but only a short project duration. Essential features should be differentiated from desirable requirements and should be strictly prioritized. Keep in mind that the requirements can change as new insights into a research project emerge, so re-assessments and re-prioritizing the requirements are necessary. As part of this project, there was a discussion on how users can interact with the portal. One option was to display a periodic table to the user, with elements for which data were available being active and others greyed out. We decided, however, to prioritize elements and ions for which data was ready. We began with a simpler visual design of those elements only and created pages for each of their available properties. Source Code Management and Version Control Source code management and version control systems are used to track the evolution of an application. Source code management tools greatly facilitate collaborative software development. Such tools enable software roll-back to a previous, well-defined state and help developers start a new branch of the software. The latter can be important if two collaborators want to add their own separate feature sets. Because we only wanted team members to have access to the repository, we used GitHub's private repository as the version control tool. Documentation There are many advantages to a welldocumented application. Here, we will enlist those advantages that apply specifically to our project. Well-documented programs are easier to maintain and extend. Documentation makes it easier for new developers to reuse and extend code, which is a great concern for research teams where students graduate, leave, and are replaced by new students. Furthermore, documentation substantially enhances reproducibility. For this purpose, we published documentation to be used exclusively by the project team on GitHub Pages. Along with Github repositories of project files, the use of GitHub pages to share instructions for generating the web pages greatly improved collaboration between team members. To evaluate the usefulness of the documentation to new users, testers from departments other than computer science were added to the project and were asked to follow instructions provided in the documentation to accomplish specific tasks. Their feedback helped us improve the documentation. Maintainability & Sustainability Sustainability [10] ensures that the software will continue to be available in the future, on new platforms, meeting new needs. A requirement of this project was maintaining the portal for a long period of time without significant additional web development, even if funding were to be discontinued. It is important for physicists to have the ability to add new elements and properties to the portal without extending the code. To fulfill this requirement, the process of creating web pages had to be automated. To this end, we standardized the file naming conventions and file content formats that were provided by the physics collaborators. A script to create templates for different properties was developed. Us-ing these templates, HTML files were generated based on the contents of specific property files. A Python script can process and modify the files that are read in and generate HTML files or even graphical representations. The script has the ability to run on all elements and property files associated with those elements. This feature is added in case the template itself is changed, and we want all web pages created for that property to have the same look. In addition, the script allows the user to pass the name of an individual element for which the property pages should be created. This option is used to add a new element or to update a specific data file. Creating web pages is now as simple as dragging and dropping new data files into the repository and then running the script in one of its two available forms in order to generate the relevant web pages. User Community Engagement and Exchange It is vital to engage the user community in the design and development of the user interface. Prototypes help obtain better feedback from the user community. Giving presentations and talks on initial versions of this portal was a way to collect feedback. Moving forward, we will hold workshops to further engage the user community. Conclusion The Atom Portal provides the user community with high-quality atomic data. The current version offers access to the properties of 25 atoms and ions. We have described the design challenges in creating the portal as well as the practices that have been applied in the development process. The practices have led to considerable time savings in generating and updating the Atom Portal as well as in enhancing maintainability and extensibility. Future releases will cover up to 100 atoms and ions. The implementation will be capable of retrieving data from a database as opposed to the current implementation of embedding the data directly into the web pages. Using dynamic web development, the creation of portal pages will be fully automated. Additionally, future updates will include the release of atomic computation codes, allowing researchers to reproduce some of the data currently provided through the portal. Figure 1 . 1Home page of the atom portal at udel.edu/atom Figure 2 . 2Display of polarizability for element Na Portal for highprecision atomic data and computation (version 2.0)," 2022. P Barakhshan, A Marrs, A Bhosale, B Arora, R Eigenmann, M S Safronova, P. Barakhshan, A. Marrs, A. Bhosale, B. Arora, R. Eigenmann, and M. S. Safronova, "Portal for high- precision atomic data and computation (version 2.0)," 2022. [Online]. Available: https://www.udel.edu/atom All-Order Methods for Relativistic Atomic Structure Calculations. M S Safronova, W R Johnson, 10.1016/S1049-250X(07)55004-4Advances in Atomic Molecular and Optical Physics. 55M. S. Safronova and W. R. Johnson, "All-Order Methods for Relativistic Atomic Structure Calculations," Advances in Atomic Molecular and Optical Physics, vol. 55, pp. 191-233, Jan. 2008. Available: https: //doi.org/10.1016/S1049-250X(07)55004-4. Magic wavelengths for the np-ns transitions in alkali-metal atoms. B Arora, M S Safronova, C W Clark, 10.1103/PhysRevA.76.052509Phys. Rev. A. 76552509B. Arora, M. S. Safronova, and C. W. Clark, "Magic wavelengths for the np-ns transitions in alkali-metal atoms," Phys. Rev. A, vol. 76, no. 5, p. 052509, Nov. 2007. Available: https://doi.org/10.1103/PhysRevA.76. 052509. Electric quadrupole moments of metastable states of Ca + , Sr + , and Ba +. D Jiang, B Arora, M S Safronova, 10.1103/PhysRevA.78.022514Phys. Rev. A. 78222514D. Jiang, B. Arora, and M. S. Safronova, "Electric quadrupole moments of metastable states of Ca + , Sr + , and Ba + ," Phys. Rev. A, vol. 78, no. 2, p. 022514, Aug. 2008. Available: https://doi.org/10.1103/PhysRevA.78. 022514 High-precision calculations of atomic properties and parity nonconservation in systems with one valence electron. M S Safronova, University of Notre DameM. S. Safronova, High-precision calculations of atomic properties and parity nonconservation in systems with one valence electron. University of Notre Dame, 2001. . S G Porsev, U I Safronova, M S Safronova, P , S. G. Porsev, U. I. Safronova, M. S. Safronova, P. O. Optical clocks based on the Cf 15+ and Cf 17+ ions. A I Schmidt, M G Bondarev, I I Kozlov, Tupitsyn, 10.1103/PhysRevA.102.012802Phys. Rev. A. 10212802Schmidt, A. I. Bondarev, M. G. Kozlov, I. I. Tupitsyn, Optical clocks based on the Cf 15+ and Cf 17+ ions, Phys. Rev. A, vol. 102, p. 012802, July 2020. Available: https://doi.org/10.1103/PhysRevA.102.012802 CAVINESS, Supporting Researchers At University Of Delaware. On-7. "CAVINESS, Supporting Researchers At Uni- versity Of Delaware ," 2022. [On- Exchanging Best Practices for Supporting Computational and Data-Intensive Research, The Xpert Network. P Barakhshan, R Eigenmann, 10.1145/3491418.3530293Practice And Experience In Advanced Research Computing. Barakhshan, P. & Eigenmann, R. Exchanging Best Practices for Supporting Computational and Data- Intensive Research, The Xpert Network. Practice And Experience In Advanced Research Computing. (2022), https://doi.org/10.1145/3491418.3530293 The Xpert Network, workshop on best practices and tools for computational and data-intensive research. R Eigenmann, P Barakhshan, University Of DelawareReport P-29R. Eigenmann and P. Barakhshan, "The Xpert Network, workshop on best practices and tools for computational and data-intensive research," University Of Delaware, Report P-29, 2019. [Online]. Available: https://cpb-us-w2.wpmucdn.com/sites.udel.edu/dist/6/ 8980/files/2019/08/ICS-workshop-report-RE4.pdf Defining software sustainability. D S Katz, D. S. Katz, "Defining software sustainability," 2016. [Online]. Available: https://danielskatzblog.wordpress. com/2016/09/13/defining-software-sustainability/ . A Kramida, Yu Ralchenko, J Reader, Team, NIST Atomic Spectra Database. ver. 5.8)," 2021. [OnlineA. Kramida, Yu. Ralchenko, J. Reader and NIST ASD Team, "NIST Atomic Spectra Database (ver. 5.8)," 2021. [Online]. Available: https://physics.nist.gov/asd

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{'abstract': 'The atom portal [1], udel.edu/atom, provides the scientific community with easily accessible high-quality data about properties of atoms and ions, such as energies, transition matrix elements, transition rates, radiative lifetimes, branching ratios, polarizabilities, and hyperfine constants. The data are calculated using a high-precision state-of-the-art linearized coupled-cluster method, high-precision experimental values are used where available. All values include estimated uncertainties. Where available, experimental results are provided with references. This paper provides an overview of the portal and describes the design as well as applied software engineering practices.', 'arxivid': '2212.10665', 'author': ['Parinaz Barakhshan \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n\n', 'Akshay Bhosale \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n\n', 'Amani Kiruga \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n\n', 'Rudolf Eigenmann \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n\n', 'Marianna S Safronova \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n\n', 'Bindiya Arora \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n\n'], 'authoraffiliation': ['Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n', 'Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n', 'Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n', 'Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n', 'Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n', 'Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer and Information Sciences\nDepartment of Electrical and Computer Engineering\nDepartment of Physics and Astronomy\nPerimeter Institute for Theoretical Physics and Guru Nanak Dev University\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\nUniversity of Delaware\n'], 'corpusid': 254926677, 'doi': '10.48550/arxiv.2212.10665', 'github_urls': [], 'n_tokens_mistral': 8188, 'n_tokens_neox': 7228, 'n_words': 5213, 'pdfsha': '89d93b2c7571b764c87f319ee5b08e499b84f22f', 'pdfurls': ['https://export.arxiv.org/pdf/2212.10665v1.pdf'], 'title': ['A Portal for High-Precision Atomic Data and Computation: Design and Best Practices', 'A Portal for High-Precision Atomic Data and Computation: Design and Best Practices'], 'venue': []}

arxiv

Equivalence of Order and Algebraic Properties in Ordered * -Algebras 11 Jun 2020 June 2020 Matthias Schötz Département de Mathématiques Université libre de Bruxelles Equivalence of Order and Algebraic Properties in Ordered * -Algebras 11 Jun 2020 June 2020 The aim of this article is to describe a class of * -algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered * -algebras ( * -algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a C * -norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered * -algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su * -algebras as uniformly complete Archimedean ordered * -algebras which have all these equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su * -algebras generalize some important properties of C * -algebras to algebras of unbounded operators. Similarly, they generalize some properties of Φ-algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered * -algebras. It is also shown that the order on Su * -algebra is uniquely determined, so Su * -algebras are indeed just a class of well-behaved * -algebras. As an example, Su * -algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as * -algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system. * Boursier de l'ULB, [email protected]. This work was supported by the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk Onderzoek -Vlaaderen (FWO) under EOS Project n 0 30950721. Introduction Many important examples of * -algebras, especially * -algebras of complex-valued functions or * -algebras of adjointable endomorphisms, carry a partial order on their Hermitian elements that is compatible with the algebraic structure: In the former case, this is the order by pointwise comparison of realvalued functions, in the latter it is the usual order on Hermitian operators. From a more abstract point of view, it has long been known that there exists an intrinsic partial order on the Hermitian elements of a C * -algebra, which can be defined in many equivalent ways (e.g. by declaring squares of Hermitian elements to be the positive ones, or elements with non-negative real spectrum). This is of course not surprising as C * -algebras can always be represented as * -algebras of bounded operators, and in the commutative case even as * -algebras of continuous functions. However, a generalization of this approach seems to be difficult, at least in the realm of topological * -algebras: Already Banach * -algebras can have extremely pathological order properties. Because of this, examining * -algebras having in some sense "unbounded" elements by means of locally convex * -algebras is a rather hard task. However It will be shown that every Archimedean ordered * -algebra carries a metrizable, translation-invariant topology. In the bounded case, this topology comes from a C * -norm as in [10], but there is no need for restriction to this special case: Theorem 7.1 shows that for complete Archimedean ordered * -algebras, the first properties mentioned above (from existence of suprema and infima to existence of inverses) are all equivalent and then imply the others (automatic continuity, uniqueness of the order and compatibility or equivalence of some further order-theoretic and algebraic concepts). Those algebras where these equivalent properties are fulfilled will be called Su * -algebras. They include C * -algebras as well as (complexifications of) complete Φ-algebras as special cases. In the end, examples of Su * -algebras of unbounded operators, which are neither C * -nor Φ-algebras, will be constructed. This way, Su * -algebras allow to examine * -algebras of unbounded operators independently of concrete representations. The article is organized as follows: The next Section 2 explains the notation and gives some basic and well-known facts especially about ordered vector spaces. Section 3 contains the definition of (quasi-) ordered * -algebras as well as some important examples, both well-behaved and ill-behaved ones, and discusses the construction of the uniform metric. After that, Section 4 describes radical ordered * -algebras which fulfil an additional compatibility between multiplication and order, and it is shown that especially all symmetric ordered * -algebras (those in which "strictly" positive elements have a multiplicative inverse) are radical. The operations ∨ and ∧, which describe especially well-behaved suprema and infima of two commuting Hermitian elements, are discussed in Section 5. This leads to the definition of Φ * -algebras which are essentially non-commutative generalizations of Φ-algebras. Square roots, which allow the construction of absolute values and thus of suprema and infima, are examined in Section 6. All this then leads to the main Theorem 7.1 in Section 7, which essentially states that in the uniformly complete case, the existence of suprema, infima, absolute values, square roots and inverses are equivalent, and motivates Definition 7.2 of Su * -algebras as those complete ordered * -algebras where these equivalent conditions are fulfilled. Moreover, all the results obtained in the previous sections (like uniqueness of the order or automatic continuity of unital * -homomorphisms) then apply especially to these Su * -algebras. Finally, in Section 8, examples of Su * -algebras of unbounded operators on a Hilbert space are constructed. Preliminaries The natural numbers are AE = {1, 2, 3, . . . }, AE 0 := AE∪{0} and the sets of real and complex numbers are denoted by Ê and , respectively. If X is a set, then id X : X → X is x → id X (x) := x. A quasi-order on X is a reflexive and transitive relation, hence a partial order is a quasi-order that is additionally anti-symmetric. If X and Y are both endowed with a quasi-order , then a map Ψ : X → Y is called increasing if Ψ(x) Ψ(x) for all x,x ∈ X with x x. If Ψ is injective and increasing and if conversely also x x holds for all x,x ∈ X with Ψ(x) Ψ(x), then Ψ is called an order embedding. A quasi-ordered vector space is a real vector space V endowed with a quasi-order such that u + w v + w and λu λv hold for all u, v, w ∈ V with u v and all λ ∈ [0, ∞[. An ordered vector space is a quasi-ordered vector space whose order is even a partial order, which is then typically denoted by ≤ instead of . For every quasi-ordered vector space V , the convex cone (non-empty subset of a real vector space closed under addition and scalar multiplication with non-negative reals) of positive elements is V + := { v ∈ V | v 0 }, and one can check that this describes a one-to-one correspondence between convex cones in V and orders on V that turn V into a quasi-ordered vector space. From this point of view, V is even an ordered vector space if and only if V + ∩ (−V + ) = {0}. A quasi-ordered vector space V is called Archimedean if it has the following property: Whenever v ǫw holds for fixed v ∈ V and w ∈ V and all ǫ ∈ ]0, ∞[, then v 0. The real vector space L(V, W ) of all linear maps Ψ : V → W between two quasi-ordered vector spaces is again a quasi-ordered vector space by declaring the positive elements to be precisely the increasing linear maps. Because of this, the increasing linear maps are called positive. Note that a linear map Ψ : V → W is increasing if and only if Ψ(v) ∈ W + for all v ∈ V + . In ordered vector spaces it makes sense to discuss suprema and infima of arbitrary non-empty subsets. A Riesz space (or vector lattice) is an ordered vector space R in which suprema and infima of all pairs of elements exist. It is well-known that this is already the case if sup{r, −r} exists for all r ∈ R. The usual notations ∨ and ∧ for suprema and infima in Riesz spaces will be avoided and will be reserved for a similar concept that is introduced later for ordered * -algebras. Endowing Riesz spaces with an additional algebraic structure leads to e.g. the concept of Φ-algebras, which are Archimedean Riesz spaces R endowed with a multiplication that turns R into a real unital associative algebra such that rs ∈ R + for all r, s ∈ R + and inf{rt, s} = inf{tr, s} = 0 for all r, s, t ∈ R + with inf{r, s} = 0. Note that this property, applied twice with t = r and t = s, especially implies that rs = inf{rs, rs} = 0 for all r, s ∈ R + with inf{r, s} = 0, and from 0 = inf sup{r, −r} − r, sup{r, −r} + r for all r ∈ R it follows that (sup{r, −r}) 2 = r 2 . One remarkable result about Φ-algebras is a representation theorem as algebras of extended real-valued functions on compact Hausdorff spaces [5], and especially that Φ-algebras are automatically commutative. This shows that Φ-algebras are a good abstraction of lattice-ordered algebras of real-valued functions. There are also many similar notions of Riesz spaces with multiplication that have been studied extensively, most notably (almost) f -algebras. One essential takeaway is that a multiplication on an Archimedean Riesz space is automatically commutative under very mild assumptions of compatibility with the order [1,3,6,9]. This unfortunately means that such algebras are not suitable for the description of reasonably well-behaved non-commutative algebras of operators, which are the usual non-commutative analog of algebras of functions. A * -vector space is a complex vector space V endowed with an antilinear involution · * : V → V . An element v of a * -vector space V is called Hermitian if v = v * and the real linear subspace of Hermitian elements in V is denoted by V H . Then V = V H ⊕ iV H as a real vector space, and this decomposition can explicitly be described as v = Re(v)+i Im(v) with Re(v) = 1 2 (v+v * ) and Im(v) = 1 2i (v−v * ) for all v ∈ V . The most obvious example of a * -vector space is of course given by with complex conjugation · as * -involution. The complex vector space L(V, W ) of all linear maps Ψ : V → W between two * -vector spaces is again endowed with an antilinear involution defined by Ψ * (v) := Ψ(v * ) * for all Ψ ∈ L(V, W ) and all v ∈ V . A linear map Ψ : V → W thus is Hermitian if and only if Ψ(v * ) = Ψ(v) * holds for all v ∈ V , or equivalently, if and only if Ψ(v) ∈ W H for all v ∈ V H . A * -algebra is a unital associative complex algebra A which is also a * -vector space such that (ab) * = b * a * holds for all a, b ∈ A. Its unit is denoted by ½ or, more explicitly, by ½ A , and is automatically Hermitian. Moreover, a unital * -homomorphism between two * -algebras is a unital homomorphism of algebras which is additionally Hermitian, and a unital * -subalgebra of a * -algebra is a unital subalgebra that is stable under · * . It is not explicitly required that 0 = ½, but the only case in which this is not fulfilled is the not very interesting algebra {0}. For a subset S ⊆ A of a * -algebra A, the commutant S ′ := a ∈ A ∀ s∈S : sa = as is a unital subalgebra, and even a unital * -subalgebra if S is stable under · * . If S is commutative, then the bicommutant S ′′ is again commutative and S ⊆ S ′′ ⊆ S ′ . For example, the multiplicative inverse a −1 of an invertible a ∈ A is in the bicommutant of a. A C * -(semi)norm on a * -algebra A is a (semi)norm · for which ab ≤ a b and a * a = a 2 hold for all a, b ∈ A, hence especially a * = a , and a C * -algebra is a * -algebra that is complete with respect to the topology of a C * -norm. A (quasi-)ordered * -vector space is a * -vector space V whose real linear subspace of Hermitian elements V H is endowed with an order that turns it into a (quasi-)ordered vector space. The properties of ordered vector spaces and linear functions between them, like being Archimedean or positive, apply to ordered * -vector spaces in the obvious way, i.e. they refer to the order on the Hermitian elements. 3 Archimedean Ordered * -Algebras (Quasi-)ordered * -algebras are defined analogously to (quasi-)ordered * -vector spaces, and have already been studied in e.g. [8,11] as " * -algebras that are equipped with an admissible wedge" in the context of * -algebras of unbounded operators and, especially in the commutative case and up to complexification, as "rings equipped with a quadratic module" in real algebraic geometry, see e.g. [10] for a survey. However, it is important to point out that with respect to quadratic modules, the term "Archimedean" unfortunately is used in a different way than with respect to ordered vector spaces. Definition 3.1 A quasi-ordered * -algebra is a * -algebra A whose real linear subspace A H is endowed with a quasi-order such that a + c b + c , d * a d d * b d and 0 ½ (3.1) hold for all a, b, c ∈ A H with a b and all d ∈ A. An ordered * -algebra is a quasi-ordered * -algebra A for which A H is partially ordered. As * -algebras are required to have a unit, these axioms especially imply that every (quasi-)ordered * -algebra is a (quasi-)ordered * -vector space. Thus, a quasi-ordered * -algebra A will be called Archimedean if A H is Archimedean as a quasi-ordered vector space and we will especially be interested in positive Hermitian linear maps and positive unital * -homomorphisms between quasi-ordered * -algebras. Note that the set A + H of positive Hermitian elements of A generates A H as a real vector space because 4a = (a + ½) 2 − (a − ½) 2 holds for all a ∈ A H and because (a ± ½) 2 ∈ A + H . Moreover, one easily checks that λa + µb ∈ A + H , d * a d ∈ A + H and ½ ∈ A + H (3.2) hold for all a, b ∈ A + H , d ∈ A and scalars λ, µ ∈ [0, ∞[. Conversely, if A is a * -algebra and A + H an arbitrary subset of A H that fulfils these three conditions (3.2), then there is a unique order on A H such that A becomes a quasi-ordered * -algebra whose set of positive Hermitian elements is precisely this set A + H . This order is given for a, b ∈ A H by a b iff b − a ∈ A + H . Again, the most obvious example of an ordered * -algebra is with the usual order on H ∼ = Ê. More interesting ones are: Example 3.2 Let X be a non-empty set and X the unital * -algebra of all complex-valued functions on X with the pointwise operations. Then X with the pointwise order on its Hermitian elements, i.e. f ≤ g if and only if f (x) ≤ g(x) for all x ∈ X, is an Archimedean ordered * -algebra. Consequently, all unital * -subalgebras of X with this pointwise order are Archimedean ordered * -algebras as well. Special cases of such ordered * -algebras of functions are of course those of continuous functions, denoted by C (X) if X is a topological space. Another special case are polynomials, which demonstrate that there can be, in general, many possible orders on the same * -algebra: Example 3.4 Let D be a complex pre-Hilbert space with inner product · | · (antilinear in the first, linear in the second argument), then a linear endomorphism a : D → D is said to be adjointable if there exists a (necessarily unique) linear map a * : D → D such that a * (ξ) | η = ξ | a(η) holds for all ξ, η ∈ D. In this case, a * is called the adjoint endomorphism. The set of all adjointable linear endomorphisms on D is a * -algebra and becomes an Archimedean ordered * -algebra, denoted L * (D), with the usual order of Hermitian operators on D, i.e. a ≤ b if and only if ξ | a(ξ) ≤ ξ | b(ξ) for all ξ ∈ D. Consequently, all unital * -subalgebras of L * (D) are Archimedean ordered * -algebras as well. These are the O * -algebras on D, see e.g. the monograph [11] for more details. In these first examples, the order is essentially determined by positive Hermitian linear functionals, namely the evaluation functionals at points of X and S in Examples 3.2 and 3.3, or the vector functionals Then setting A + H := ⟪ G ⟫ pos turns A into a quasi-ordered * -algebra. This order on A will be called the order generated by G, and ⟪ G ⟫ pos is the smallest (with respect to inclusion) choice of positive Hermitian elements that contains G and with which A becomes a quasi-ordered * -algebra. Especially L * (D) ∋ a → ξ | a(ξ) ∈ infor G = ∅ we write A ++ H := ⟪ ∅ ⟫ pos = N n=1 a * n a n N ∈ AE; a 1 , . . . , a N ∈ A Positivstellensatz of Krivine and Stengle for polynomials but also newer results for non-commutative * -algebras like [13,14]. Choosing the order on a * -algebra A for which A + H = A ++ H yields a canonical way to construct a quasi-ordered * -algebra out of any * -algebra. For example, the canonical order on C * -algebras can be described like this. There is also another canonical (yet pathological) choice, namely A + H = A H . For a general quasi-ordered * -algebra A one clearly has A ++ H ⊆ A + H ⊆ A H , but it is well-known that these extreme cases may coincide: Example 3.6 Let Ë := { z ∈ | |z| = 1 } and let A be the unital associative algebra A := C (Ë), but endowed with the * -involution f * := · • f • τ for all f ∈ C (Ë) (instead of the usual pointwise one f * := · • f ), where τ : Ë → Ë is z → τ (z) := −z. This way C (Ë) indeed becomes a * -algebra. The usual norm f max := max z∈Ë |f (z)| turns C (Ë) into a Banach space and makes multiplication and * -involution continuous. However, id Ë describes a function in C (Ë) for which −½ Ë = (id Ë ) * id Ë ∈ A ++ H holds, thus A H = A ++ H − A ++ H = A ++ H . Finally, there is a standard example of a non-Archimedean ordered * -algebra: Example 3.7 The commutative unital subalgebra A := M a,b := a b 0 a ∈ 2×2 a, b ∈ (3.5) of the matrix algebra 2×2 with elementwise complex conjugation as * -involution becomes a * -algebra. Its algebraically positive elements are A ++ H = M a,b a, b ∈ Ê with a > 0 or a = b = 0 (3.6) and A with the algebraic order is an ordered * -algebra, but not Archimedean because M 0,1 ≤ ǫM 1,0 for all ǫ ∈ ]0, ∞[. Note especially that M 0,1 is a non-zero Hermitian element that squares to 0. Examples 3.6 and 3.7 already indicate that one should not expect to be able to prove many strong results for quasi-ordered * -algebras without any additional assumptions. Nevertheless, there is at least a possibility to characterize the pathological elements and in many cases one can eventually get rid of them by taking a suitable quotient. This way it will become clear that an order on a * -algebra can be seen as a generalization of a C * -norm. This follows essentially [4], but caution is advised because of the different usage of the term "Archimedean" there: Lemma 3.8 Let A be a quasi-ordered * -algebra, a ∈ A H and λ ∈ ]0, ∞[, then a 2 λ 2 ½ if and only if −λ½ a λ½. If A isλ½ ± a = λ 2 ½ − a 2 + (λ½ ± a) 2 2λ and λ 2 ½ − a 2 = (λ½ + a)(λ½ − a)(λ½ + a) + (λ½ − a)(λ½ + a)(λ½ − a) 2λ . So a 2 0 implies −ǫ½ a ǫ½ and 0 a 0 implies a 2 ǫ 2 ½ for all ǫ ∈ ]0, ∞[. If A is Archimedean, then this shows that a 2 0 and 0 a 0 are also equivalent. Proposition 3.9 Let A be an Archimedean ordered * -algebra and a ∈ A H nilpotent, then a = 0. Proof: Let n ∈ AE be the minimal exponent for which a n = 0. Then n is odd because otherwise 0 ≤ a n/2 ≤ 0 by the previous Lemma 3.8, which contradicts minimality of n. But a n+1 = 0 now implies 0 ≤ a (n+1)/2 ≤ 0, so n = 1 by minimality of n. In Example 3.7 we have seen that nilpotent Hermitian elements can indeed occur in non-Archimedean ordered * -algebras. Like in [4] we define: Definition 3.10 Let A be a quasi-ordered * -algebra, then define the map · ∞ : A → [0, ∞], a → a ∞ := inf λ ∈ ]0, ∞[ a * a λ 2 ½ ,(3. 7) where it is understood that the infimum of the empty set is ∞. An element a ∈ A is called uniformly bounded if a ∞ < ∞ and the set of all uniformly bounded elements in A is denoted by A bd . The algebra A itself is called uniformly bounded if A = A bd . Lemma 3.8 immediately gives an alternative description of · ∞ on Hermitian elements: Proposition 3.11 Let A be a quasi-ordered * -algebra and a ∈ A H , then a ∞ = inf λ ∈ ]0, ∞[ − λ½ a λ½ , (3.8) where again the infimum of the empty set is ∞. In the Archimedean case, these infima are even minima: Proposition 3.12 Let A be an Archimedean quasi-ordered * -algebra and a ∈ A bd , then a * a a 2 ∞ ½ . (3.9) If even a ∈ (A bd ) H , then also − a ∞ ½ a a ∞ ½ . (3.10) Proof: From the definition of a ∞ one sees that a * a a 2 ∞ + ǫ ½ for all ǫ ∈ ]0, ∞[, hence a * a a 2 ∞ ½ as A is Archimedean. If a is even Hermitian, then this implies − a ∞ ½ a a ∞ ½ by Lemma 3.8 again. The crucial property of · ∞ is that it yields a C * -(semi)norm on the uniformly bounded elements. Recall that a * -ideal of a * -algebra A is a linear subspace I ⊆ A that is stable under the * -involution and fulfils ba ∈ I for all a ∈ A and all b ∈ I (thus also ab = (b * a * ) * ∈ I for all a ∈ A and all b ∈ I). Proposition 3.13 Let A be a quasi-ordered * -algebra, then A bd is a unital * -subalgebra of A, and the restriction of · ∞ to A bd is a C * -seminorm. Its kernel K := a ∈ A a ∞ = 0 is a * -ideal of A bd . If A is Archimedean, then K is even a * -ideal of whole A, and if A is an Archimedean ordered * -algebra, then K = {0} so that · ∞ is a C * -norm on A bd . Proof: The claims for general, not necessarily Archimedean A have been proven in [4,Thm. 3.2]. For convenience of the reader, the details are given here as well: From the definition of · ∞ it is clear that ½ ∈ A bd with ½ ∞ ≤ 1 and that αa ∈ A bd with αa ∞ = |α| a ∞ for all a ∈ A bd and all α ∈ \{0}, as well as for α = 0 because clearly 0 ∞ = 0. Now given a, b ∈ A bd and λ, µ ∈ ]0, ∞[ such that a * a λ 2 ½ and b * b µ 2 ½, then (λ + µ) 2 ½−(a + b) * (a + b) = = 1 + µ λ λ 2 ½ − a * a + 1 + λ µ µ 2 ½ − b * b + √ λ √ µ b − √ µ √ λ a * √ λ √ µ b − √ µ √ λ a is positive, so a + b ∞ ≤ λ + µ. Moreover, (ab) * (ab) = b * a * a b λ 2 b * b (λµ) 2 ½ shows that ab ∞ ≤ λµ. Thus a + b ∞ ≤ a ∞ + b ∞ and ab ∞ ≤ a ∞ b ∞ , and especially a + b, ab ∈ A bd . So A bd is a unital subalgebra of A and · ∞ a submultiplicative seminorm on A bd . In order to show that A bd is stable under · * , let a ∈ A bd be given as well as λ ∈ ]0, ∞[ such that a * a λ 2 ½. Then 0 (λ 2 ½ − aa * ) 2 λ 2 = λ 2 ½ − 2aa * + a a * a λ 2 a * λ 2 ½ − aa * shows that a * ∞ ≤ λ. It follows that A bd is a unital * -subalgebra of A and a * ∞ = a ∞ for all a ∈ A because · * is an involution. The C * -property also is fulfilled: The increasing and continuous map · 2 : [0, ∞[ → [0, ∞[ commutes with infima so that Definition 3.10 yields a 2 ∞ = inf λ 2 ∈ ]0, ∞[ a * a λ 2 ½ for all a ∈ A bd , which coincides with a * a ∞ by Proposition 3.11. So · ∞ is indeed a C * -seminorm on A bd and its kernel K is a * -ideal of A bd . Now given a, b ∈ A such that a ∞ = 0, then (ab) * (ab) = b * a * a b ǫ 2 b * b holds for all ǫ ∈ ]0, ∞[, which implies (ab) * (ab) 0 if A is additionally Archimedean. In this case ab ∞ = 0 so that K is even a * -ideal of A. Finally, assume that A is even an Archimedean ordered * -algebra and let a ∈ K be given. If a is Hermitian, then a = 0 by Proposition 3.12. Otherwise a can be expressed as the linear combination a = Re(a) + i Im(a) of Hermitian elements Re(a), Im(a) ∈ K, which are both 0 so that again a = 0. So we see that uniformly bounded Archimedean ordered * -algebras with the norm · ∞ are pre-C * -algebras (i.e. * -algebras endowed with a C * -norm). Using some standard results about C * -algebras, e.g. the possibility to represent every C * -algebra as a * -algebra of bounded operators on a Hilbert space by the Gelfand-Naimark theorem, one can also show that the converse is true as well: every pre-C * -algebra with the canonical order inherited from its completion to a C * -algebra is a uniformly bounded Archimedean ordered * -algebra. It will be interesting to extend the concept of completeness of a C * -algebra to general Archimedean ordered * -algebras. While · ∞ is finite only on the uniformly bounded elements, and thus does not describe a norm on all Archimedean ordered * -algebras, it still allows to construct a translation-invariant metric: Definition 3.14 Let A be an Archimedean ordered * -algebra, then the uniform metric on A is defined as the map d ∞ : A × A → [0, ∞[, (a, b) → d ∞ (a, b) := min a − b ∞ , 1 . (3.11) All metric notions will always refer to this uniform metric, and A is especially called uniformly complete if it is complete with respect to d ∞ . Note that it is easy to check that d ∞ is indeed a translation-invariant metric. In this language, C * -algebras are the uniformly bounded and uniformly complete Archimedean ordered * -algebras. However, neither the product, nor the left or right multiplication with a fixed element are continuous in the general case: Consider the * -algebra [x] of polynomials in one Hermitian element x like in Example 3.3 with the Ê-pointwise ordering. Then lim n→∞ ½/n = 0 but the sequence AE ∋ n → x/n ∈ [x] H does not converge. Nevertheless, this metric is still sufficiently well-behaved for some purposes. For example, it is easy to see that every positive unital * -homomorphism between Archimedean ordered * -algebras is automatically continuous with respect to the uniform metric, because this metric is induced by the order. Moreover: Lemma 3.15 Let A be a quasi-ordered * -algebra and a, b ∈ A, then a * b + b * a χ −2 a * a + χ 2 b * b (3.12) holds for all χ ∈ ]0, ∞[. Proof: This follows from 0 (χ −1 a − χb) * (χ −1 a − χb) = χ −2 a * a − a * b − b * a + χ 2 b * b.= Im −1 ({0}) is closed. Now consider a sequence (a n ) n∈AE in A H that converges against someâ := lim n→∞ a n ∈ A H and let ǫ ∈ ]0, ∞[ be given, then there exists an n ∈ AE such that â − a n ∞ ≤ ǫ, i.e. −ǫ½ ≤â − a n ≤ ǫ½ by Proposition 3.12. If all a n with n ∈ AE are uniformly bounded, then this shows thatâ is also uniformly bounded, so A bd ∩ A H and A bd = Re −1 (A bd ∩ A H ) ∩ Im −1 (A bd ∩ A H ) are closed in A. Moreover, let s ∈ A H be given. If a n ≤ s for n ∈ AE, thenâ ≤ a n + ǫ½ ≤ s + ǫ½, and if a n ≥ s for n ∈ AE, thenâ ≥ a n − ǫ½ ≥ s − ǫ½. If a n ∈ {s} ′ for n ∈ AE, then i(âs − sâ) = (â − a n )(is) + (−is)(â − a n ) ≤ ǫ −1 (â − a n ) 2 + ǫs 2 ≤ ǫ(½ + s 2 ) by Lemma 3.15 with χ = √ ǫ and Lemma 3.8. As {−s} ′ = {s} ′ , the same estimate holds with −s in place of s, so −ǫ(½ + s 2 ) ≤ i(âs − sâ) ≤ ǫ(½ + s 2 ). Using that A is Archimedean, these estimates show that { a ∈ A H | a ≤ s }, { a ∈ A H | a ≥ s }, and {s} ′ ∩ A H are closed in A. As {s} ′ = Re −1 ({s} ′ ∩ A H ) ∩ Im −1 ({s} ′ ∩ A H ) also {s} ′ is closed in A. Consequently, intersections of such sets, and especially { a ∈ A H | a ≤ s for all s ∈ S }, { a ∈ A H | a ≥ s for all s ∈ S } and S ′ are closed. From S ⊆ A H it follows that S ′ is stable under · * , and thus S ′′ = (S ′ ∩ A H ) ′ is also closed in A. Radical and Symmetric Ordered * -Algebras The only compatibility between order and multiplication that has been discussed so far is the axiom of quasi-ordered * -algebras A that b * a b ∈ A + H for all a ∈ A + H and all b ∈ A. If the order is sufficiently nice (especially antisymmetric and Archimedean), then this is indeed enough to guarantee that the elements of A essentially behave like adjointable endomorphisms on a pre-Hilbert space, which can be made rigorous by a representation theorem like in [15]. However, it is well-known that such * -algebras of (unbounded) adjointable endomorphisms can still exhibit some unexpected behaviour because Hermitian endomorphisms need not be essentially self-adjoint. Because of this, it will be necessary to introduce another compatibility between order and multiplication that gurantees that commuting elements essentially behave like complex-valued functions (see again [15]): Definition 4.1 Let A be an ordered * -algebra, then an element a ∈ A H is called coercive if there exists an ǫ ∈ ]0, ∞[ such that a ≥ ǫ½. An ordered * -algebra A is called radical if the following is fulfilled: Whenever a, b ∈ A H are two commuting elements such that a is coercive and ab ≥ 0, then b ≥ 0. One obvious example of radical Archimedean ordered * -algebras are function algebras like in Example 3.2. Non-commutative examples will be constructed later on. Some basic observations about radical Archimedean ordered * -algebras are: Proposition 4.2 Let A be a radical Archimedean ordered * -algebra, a, b ∈ A H commuting and a ≥ 0. Then b 2 ≤ a 2 is equivalent to −a ≤ b ≤ a. Proof: One argues like in the proof of Lemma 3.8: First assume that b 2 ≤ a 2 , then also b 2 ≤ (a + ǫ½) 2 for all ǫ ∈ ]0, ∞[, so 2(a + ǫ½)(a + ǫ½ ± b) = (a + ǫ½) 2 − b 2 + (a + ǫ½ ± b) 2 ≥ 0. As A is radical, this shows that a + ǫ½ ± b ≥ 0, and then a ± b ≥ 0 because A is also Archimedean; so −a ≤ b ≤ a. Conversely, if −a ≤ b ≤ a, then also −(a + ǫ½) ≤ b ≤ a + ǫ½ for all ǫ ∈ ]0, 1], so 2(a + ǫ½) (a + ǫ½) 2 − b 2 = (a + ǫ½ + b)(a + ǫ½ − b)(a + ǫ½ + b) + (a + ǫ½ − b)(a + ǫ½ + b)(a + ǫ½ − b) ≥ 0. As A is radical, this shows that (a + ǫ½) 2 − b 2 ≥ 0, so b 2 ≤ (a + ǫ½) 2 ≤ a 2 + ǫ(2a + ½). It follows that b 2 ≤ a 2 because A is Archimedean. Corollary 4.3 If A is a radical Archimedean ordered * -algebra and a, b ∈ A + H commute, then ab ≥ 0. Proof: From −(a + b) ≤ a − b ≤ a + b we get (a − b) 2 ≤ (a + b) 2 , so 4ab = (a + b) 2 − (a − b) 2 ≥ 0. Note that even in the matrix * -algebra 2×2 with the composition of complex conjugation and transposition as * -involution and the usual order on the Hermitian matrices, which is a C * -algebra and certainly should be regarded as one of the most well-behaved ordered * -algebras, there exist Hermitian such that (1 + r 1 )q = 1 + r 2 (this version of the Positivstellensatz can be obtained from the more traditional formulation r 1 q = 1 + r 2 by the well-known trick of multiplying with q and adding the identities, which yields (1 + r 1 + r 2 )q = 1 + r 2 + r 1 q 2 ). This shows that ½ + r 1 (a 1 , . . . , a N ) q(a 1 , . . . , a N ) = ½ + r 2 (a 1 , . . . , a N ) and from the previous Corollary 4.3 it follows that r 1 (a 1 , . . . , a N ), r 2 (a 1 , . . . , a N ) ∈ A + H , so q(a 1 , . . . , a N ) ∈ A + H because A is radical. For general q ∈ [x 1 , . . . , x N ] H which is S-pointwise positive, this shows that q(a 1 , . . . , a N ) + ǫ½ = (q + ǫ)(a 1 , . . . , a N ) ∈ A + H for all ǫ ∈ ]0, ∞[, and thus q(a 1 , . . . , a N ) ∈ A + H because A is Archimedean. From this proof it also becomes clear that [x 1 , . . . , x N ] with the algebraic order is not radical for N ≥ 2, because there exist real polynomials that are (strictly) pointwise positive on whole Ê N but not sums of squares, hence not algebraically positive. The first paragraph thus fails for this algebra and a n := x n , M := 1, p 1 := 1. In order to construct radical Archimedean ordered * -algebras, it will be helpful to discuss algebras in which many elements are invertible. Recall that a * -algebra A is called symmetric if a ± i½ has a multiplicative inverse for all a ∈ A H , or equivalently if ½ + a 2 is invertible for all a ∈ A H . However, there are also similar, but non-equivalent notions where one demands that e.g. ½ + a * a is invertible for all a ∈ A or that ½ + N n=1 a * n a n is invertible for all a 1 , . . . , a N ∈ A with N ∈ AE, see [7, Chap. 9.8] for a comparison. In ordered * -algebras, there is another, even stronger possibility: Definition 4.5 An ordered * -algebra A is called symmetric if every coercive element of A H has a multiplicative inverse. In order to prove that every symmetric Archimedean ordered * -algebra is radical, we need some preliminary lemmas: Lemma 4.6 Let A be an ordered * -algebra, a ∈ A H coercive and ǫ ∈ ]0, ∞[ such that a ≥ ǫ½, then a −1 is Hermitian, positive and uniformly bounded with a −1 ∞ ≤ ǫ −1 . Proof: Let u := a ∞ + 1 so that 0 ≤ a ≤ u½ by Proposition 3.11. By the (Stone-)Weierstraß theorem, applied to the continuous function Proof: We have a −1 = (a a −1 ) * a −1 = (a −1 ) * a a −1 ∈ A + H , and a = ǫ −1 a 2 − ǫ −1 (a − ǫ½) 2 − (a − ǫ½) ≤ ǫ −1 a 2 implies a −1 = a −1 a a −1 ≤ ǫ −1 a −1 a 2 a −1 = ǫ −1 ½ so that a −1 ∞ ≤ ǫ −1 by Proposition 3.11.√ · : [0, u] → Ê, there exists for every n ∈ AE a polynomial p ′ n ∈ [x] H such that | √ t − p ′ n (t)| ≤ 1/(4n( √ u + 1) ) holds for all t ∈ [0, u]. Define p n := p ′ n + 1/(4n( √ u + 1)) and q n := p 2 n −x. Then the estimate 0 ≤ √ t ≤ p n (t) ≤ √ t+1/(2n( √ u+1)) and thus 0 ≤ p 2 n (t)−t ≤ 1/n hold for all t ∈ [0, u]. Using the fundamental theorem of algebra, one can show that every polynomial r ∈ [x] H which is pointwise positive on [0, u] is an element of ⟪ {x, u − x} ⟫ pos , see e.g. [12,Prop. 3.3], hence r(a) ∈ A + H . The pointwise estimates for p n and q n thus imply that 0 ≤ p n (a) and 0 ≤ q n (a) ≤ ½/n, and the identity a + q n (a) = p n (a) 2 is fulfilled by construction. Proposition 4.8 Every symmetric Archimedean ordered * -algebra is radical. Proof: Let two commuting elements a, b ∈ A H and ǫ ∈ ]0, ∞[ be given such that a is coercive with a ≥ ǫ½ and ab ≥ 0. Using the previous Lemmas 4.6 and 4.7 one can construct sequences of polynomials (p n ) n∈AE and (q n ) n∈AE such that a −1 + q n (a −1 ) = p n (a −1 ) 2 with 0 ≤ q n (a −1 ) ≤ ½/n for all n ∈ AE, so 0 ≤ p n (a −1 ) ab p n (a −1 ) = b+q n (a −1 ) ab. Using Lemma 3.15 with χ = √ n and that q n (a −1 ) 2 ≤ ½/n 2 by Lemma 3.8, it follows that 2 q n (a −1 ) ab ≤ χ −2 a 2 b 2 + χ 2 q n (a −1 ) 2 ≤ (a 2 b 2 + ½)/n. As A is Archimedean it follows that 0 ≤ b, so A is radical. In the uniformly complete case, we will also see that the various notions of symmetric * -algebras that were mentioned before are actually equivalent: Lemma 4.9 Let A be an ordered * -algebra, a, b ∈ A and d ∈ A + H , then it follows that a * c b + b * c a ≤ a * d a + b * d b (4.2) holds for all c ∈ A H fulfilling −d ≤ c ≤ d. Proof: Given c ∈ A H with −d ≤ c ≤ d,a * c (+) b + b * c (+) a ≤ a * c (+) a + b * c (+) b and − a * c (−) b − b * c (−) a ≤ a * c (−) a + b * c (−) b hold. Adding these two estimates yields (4.2). Lemma 4.10 Let A be an Archimedean ordered * -algebra,â ∈ A H and (a n ) n∈AE a sequence in A H of invertible elements such that the sequence of their inverses converges with respect to the uniform metric against some limit e := lim n→∞ a −1 n ∈ A. Moreover, assume that there exist elements c, d ∈ A + H such that a 2 n ≤ c for all n ∈ AE and such that for all ǫ ∈ ]0, ∞[ there exists an N ∈ AE for which −ǫd ≤â−a n ≤ ǫd is fulfilled for all n ∈ AE with n ≥ N . Thenâ is invertible andâ −1 = e = lim n→∞ a −1 n . Proof: As all a n with n ∈ AE and thus also their inverses a −1 n are Hermitian, e is Hermitian by Proposition 3.16. Therefore it is sufficient to show thatâe = ½, which then also implies eâ = (âe) * = ½. So let ǫ ∈ ]0, ∞[ be given, then there exists an n ∈ AE such that −ǫd ≤â−a n ≤ ǫd and e−a −1 n ∞ ≤ ǫ hold, thus also (e − a −1 n ) 2 ≤ ǫ 2 ½ by Proposition 3.12. Using the previous Lemma 4.9 and Lemma 3.15 with χ := 1/ √ ǫ one finds that (i k ½) * (â − a n )e + e(â − a n )(i k ½) ≤ǫ(d+ede) + (i k a n ) * (e − a −1 n ) + (e − a −1 n )(i k a n ) ≤ǫ(c+½) ≤ ǫ(½ + c + d + ede) holds for all k ∈ {0, 1, 2, 3}, or equivalently, −ǫ(½ + c + d + ede) ≤ 2 Re(âe − ½) ≤ ǫ(½ + c + d + ede) and −ǫ(½ + c + d + ede) ≤ 2 Im(âe − ½) ≤ ǫ(½ + c + d + ede) becauseâe − ½ = (â − a n )e + a n (e − a −1 n ). Proposition 4.11 Let A be a uniformly complete Archimedean ordered * -algebra and a, b ∈ A + H commuting such that a is coercive, b invertible and a ≤ b 2 . Then a is also invertible. Proof: It is sufficient to show thatâ := bab has an inverse, then a is also invertible with a −1 = bâ −1 b. There is an ǫ ∈ ]0, ∞[ such that ǫ½ ≤ a, and consequently ǫ½ ≤ b 2 and ǫ 2 ½ ≤ ǫb 2 ≤â ≤ b 4 hold. Define a n :=â + b 4 /n for all n ∈ AE, c :=â 2 + 3b 8 and d := b 4 , then a 2 n =â 2 + 2b 2â b 2 /n + b 8 /n 2 ≤ c and −d/n =â − a n ≤ d/n. In order to apply the previous Lemma 4.10 it only remains to show that all a n with n ∈ AE are invertible and that the sequence of their inverses is a Cauchy sequence. Consider b −2 a n b −2 = b −2â b −2 + ½/n. Then ½/n ≤ b −2 a n b −2 ≤ (1 + 1/n)½, so b −2 a n b −2 is a coercive element of A bd . From Proposition 3.16 it follows that A bd is uniformly complete itself, hence a C * -algebra, so b −2 a n b −2 is invertible in A bd (the inverse can be constructed explicitly e.g. using a Neumann series). Consequently, a n is also invertible with a −1 n = b −2 (b −2 a n b −2 ) −1 b −2 . Moreover, using −|m −1 − n −1 |b 4 ≤ a n − a m ≤ |m −1 − n −1 |b 4 and Lemma 4.9, one obtains the estimate a −1 m − a −1 n = a −1 m (a n − a m )a −1 n + a −1 n (a n − a m )a −1 m 2 ≤ 1 m − 1 n a −1 m b 4 a −1 m + a −1 n b 4 a −1 n 2 for all m, n ∈ AE. From ǫb 2 ≤â it follows that ǫ½ ≤ b −1â b −1 and thus ǫ 2 b 4 ≤ (â − ǫb 2 ) 2 + ǫ 2 b 4 =â 2 − 2ǫb(â − ǫb 2 )b ≤â 2 ≤â 2 + 2b 2â b 2 /n + b 8 /n 2 = a 2 n for all n ∈ AE. The combination of these estimates yields a −1 m − a −1 n ≤ ǫ −2 |m −1 − n −1 |½ for all m, n ∈ AE, so (a −1 n ) n∈AE is indeed a Cauchy sequence. Corollary 4.12 Let A be a uniformly complete Archimedean ordered * -algebra in which ½ + a 2 is invertible for all a ∈ A H , then A is symmetric. Proof: Given a coercive a ∈ A H , then one can apply the previous Proposition 4.11 with b := ½ + a 2 because a ≤ 2a = ½ + a 2 − (½ − a) 2 ≤ ½ + a 2 ≤ (½ + a 2 ) 2 . Φ * -Algebras It has already been mentioned in Section 2 that Riesz spaces which carry a non-commutative multiplication have rather pathological properties. Because of this, a well-behaved non-commutative generalization of the notion of Φ-algebras must necessarily deal with some restrictions to the infima and suprema. Moreover, like in Φ-algebras, there should also be a compatibility between suprema, infima and the product, but it might not be immediately clear what exactly this compatibility should be. The following observation, which gives a mostly algebraic characterization of suprema and infima, might serve as a motivation (recall that · ′ denotes the commutant): Proposition 5.1 Let A be a radical Archimedean ordered * -algebra, a, b ∈ A H commuting and let x ∈ {a, b} ′′ ∩ A H be such that x 2 + ab = x(a + b), then the following holds: • If 2x ≥ a + b, then x is the supremum of a and b in {a, b} ′ ∩ A H . • If 2x ≤ a + b, then x is the infimum of a and b in {a, b} ′ ∩ A H . Proof: First assume that 2x ≥ a + b, then (2x − a − b) 2 = 4x 2 − 4x(a + b) + 4ab + (a − b) 2 = (a − b) 2 implies −(2x − a − b) ≤ a − b ≤ 2x − a − bb) 2 = 4y 2 − 4y(a + b) + 4ab + (a − b) 2 ≥ (a − b) 2 = (2x − a − b) 2 . Proposition 4.2 now shows that 2y − a − b ≥ 2x − a − b, so y ≥ x and x is indeed the supremum of a and b in {a, b} ′ ∩ A H . If 2x ≤ a + b, then one can apply the above argument to −x, −a and −b. The following definition thus makes sense and describes suprema and infima that fulfil additional algebraic conditions: Definition 5.2 Let A be a radical Archimedean ordered * -algebra and a, b ∈ A H commuting. Then a ∨ b is (if it exists) the element in {a, b} ′′ ∩ A H which fulfils 2(a ∨ b) ≥ a + b and (a ∨ b) 2 + ab = (a ∨ b)(a + b). (5.1) Similarly, a ∧ b is (if it exists) the element in {a, b} ′′ ∩ A H which fulfils 2(a ∧ b) ≤ a + b and (a ∧ b) 2 + ab = (a ∧ b)(a + b). (5.2) A radical Archimedean ordered * -algebra A in which a ∨ b and a ∧ b exist for all commuting a, b ∈ A H will be called a Φ * -algebra. There are some basic results about these suprema and infima which are not very surprising as they mostly mimic the rules in ordered vector spaces, and which can easily be checked: (a ∨ b) + (a ∧ b) = a + b and (a ∨ b)(a ∧ b) = ab. (5.3) Moreover, if one, hence both of a ∨ b and a ∧ b exist, then the following holds: i.) b ∨ a = a ∨ b and b ∧ a = a ∧ b exist. ii.) (λa) ∨ (λb) = λ(a ∨ b) and (λa) ∧ (λb) = λ(a ∧ b) exist for all λ ∈ [0, ∞[. iii.) (−a) ∧ (−b) = −(a ∨ b) and (−a) ∨ (−b) = −(a ∧ b) exist. iv.) (a + c) ∨ (b + c) = (a ∨ b) + c exists for all c ∈ {a, b} ′ ∩ A H with (a ∨ b) + c ∈ {a + c, b + c} ′′ , and (a + c) ∧ (b + c) = (a ∧ b) + c exists for all c ∈ {a, b} ′ ∩ A H with (a ∧ b) + c ∈ {a + c, b + c} ′′ . Proof: If x ∈ {a, b} ′′ ∩ A H fulfils x 2 + ab = x(a + b), then also y := a + b − x ∈ {a, b} ′′ ∩ A H fulfils y 2 + ab = y(a + b). Especially using x = a ∨ b and x = a ∧ b it follows that a ∨ b exists if and only if a ∧ b exists and that (a ∨ b) + (a ∧ b) = a + b. As a consequence, (a ∨ b)(a ∧ b) = (a ∨ b)(a + b) − (a ∨ b) 2 = ab. Checking that i.), ii.) and iii.) hold is easy and for part iv.) one essentially only needs to verify that (a ∨ b) + c 2 + (a + c)(b + c) = (a ∨ b) 2 + ab + 2c(a ∨ b) + (a + b + 2c)c = (a + b)(a ∨ b) + 2c(a ∨ b) + (a + b + 2c)c = (a + b + 2c) (a ∨ b) + c . With respect to part iv.) we note that the conditions (a ∨ b) + c, (a ∧ b) + c ∈ {a + c, b + c} ′′ are superfluous if it is a priori known that (a + c) ∨ (b + c) and (a + c) ∧ (b + c) exist, i.e. especially if A is a Φ * -a ∨ b = a + b + |a − b| 2 and a ∧ b = a + b − |a − b| 2 . (5.4) Proof: This is just an application of Proposition 5.3 using that |a − b| ∈ {a, b} ′′ . As immediate consequences of Propositions 5.3 and 5.5 we obtain: Corollary 5.6 Let A be a radical Archimedean ordered * -algebra and a ∈ A H such that |a| exists. Then a∨0 = a + ∈ A + H and (−a)∨0 = −(a∧0) = a − ∈ A + H exist and fulfil a + +a − = a and a − a + = a + a − = 0. Corollary 5.7 Let A be a radical Archimedean ordered * -algebra, then A is a Φ * -algebra if and only if |a| exists for all a ∈ A H . One motivation to study Φ * -algebras is that they indeed are a non-commutative generalization of Φ-algebras: Proposition 5.8 Let R be a Φ-algebra, then its complexification A := R ⊗ with * -involution and multiplication defined by (r ⊗ λ) * := r ⊗ λ and (r ⊗ λ)(s ⊗ µ) := rs ⊗ λµ for all r, s ∈ R and all λ, µ ∈ is a commutative Φ * -algebra. Conversely, if A is a commutative Φ * -algebra, then its real unital subalgebra A H is a Φ-algebra. Proof: First let R be a Φ-algebra and A := R ⊗ . Then it is clear that A is a commutative Archimedean ordered * -algebra, and it is also radical: Given two commuting a, b ∈ A H ∼ = R such that Proof: Let a ∈ A H with Ψ(a) ≥ 0 be given. Then a = a + − a − with a + , a − ∈ A + H and a + a − = 0 by Corollary 5.6, so on the one hand (a − ) 3 ≥ 0 implies Ψ(a − ) 3 ≥ 0, and on the other, (a − ) 3 = − a − aa − implies Ψ(a − ) 3 = −Ψ(a − ) Ψ(a) Ψ(a − ) ≤ 0, so Ψ(a − ) 3 = 0. As Ψ is injective, it follows that (a − ) 3 = 0 and thus a − = 0 by Proposition 3.9. So a = a + ≥ 0 and Ψ is an order embedding. b = b + −b − and b + b − = 0. Consequently, 0 ≤ b − ab = −b − ab − ≤ −ǫ(b − ) 2 ≤ 0, so (b − ) 2 = 0. Moreover, positive unital * -homomorphisms between Φ * -algebras are compatible with ∨, ∧ and the absolute value: Proof: As ∨ and ∧ can be expressed using the absolute value, it is enough to show that |Ψ(a)| = Ψ(|a|) holds. It is easy to check that Ψ(|a|) ∈ {Ψ(a)} ′ ∩ B + H and that Ψ(|a|) 2 = Ψ(a) 2 . As it is already known that |Ψ(a)| ∈ {Ψ(a)} ′′ ∩ B + H exists, it follows from the previous Lemma 5.10 that |Ψ(a)| = Ψ(|a|). As a last result we note that in the uniformly complete case, the existence of infima is helpful for the construction of a multiplicative inverse: Lemma 5.12 Let A be a radical Archimedean ordered * -algebra, a ∈ A H and λ ∈ ]0, ∞[. If a ∧ λ½ exists, then it fulfils the estimate a ≤ (a ∧ λ½) + a 2 /(4λ). Proof: As a and a ∧ λ½ commute, a(a ∧ λ½) = a(a ∧ λ½) + (a ∧ λ½)a /2 ≤ a 2 /4 + (a ∧ λ½) 2 holds by Lemma 3.15 with χ := √ 2. From (a ∧ λ½) 2 + λa = (a + λ½)(a ∧ λ½) it now follows that (a ∧ λ½) 2 + λa ≤ a 2 /4 + (a ∧ λ½) 2 + λ(a ∧ λ½). Lemma 5.13 Let A be a radical Archimedean ordered * -algebra and a, b ∈ A + H with a ≤ b commuting and invertible, then a −1 ≥ b −1 . Proof: As b−a, a −1 , b −1 ∈ {a, b} ′′ ∩A + H are pairwise commuting, their product a −1 (b−a)b −1 = a −1 −b −1 is positive by Corollary 4.3. Proposition 5.14 Let A be a radical and uniformly complete Archimedean ordered * -algebra and let a ∈ A H be a coercive element for which a ∧ n½ exists for all n ∈ AE, then a is invertible. Proof: Proposition 5.1 shows that a ∧ n½ is, for every n ∈ AE, the infimum of a and n½ in the commutative real unital subalgebra {a} ′′ ∩ A H of A. As a is coercive, there exists ǫ ∈ ]0, 1] such that ǫ½ ≤ a, and then ǫ½ ≤ a∧ n½ ≤ n½ shows that a∧ n½ is a coercive element of A bd . By Proposition 3.16, A bd is uniformly complete itself, hence a C * -algebra, and thus a ∧ n½ is invertible. For fixed m, n ∈ AE with n ≤ m, the estimate a ∧ n½ ≤ a ∧ m½ yields (a ∧ m½) −1 ≤ (a ∧ n½) −1 by the previous Lemma 5. 13. Moreover, let b := (a ∧ m½) −1 + ½/n ∈ {a} ′′ ∩ A H , then b is also a coercive element of A bd , hence invertible. From ½/n ≤ b and (a ∧ m½) −1 ≤ b it follows that b −1 ≤ n½ and b −1 ≤ a ∧ m½ by the previous Lemma 5. 13. So b −1 ≤ a ∧ n½ and therefore (a ∧ n½) −1 ≤ b by Lemma 5.13 again. Altogether, this shows that (a ∧ m½) −1 ≤ (a ∧ n½) −1 ≤ (a ∧ m½) −1 + ½/n for all m, n ∈ AE with n ≤ m, so AE ∋ n → (a ∧ n½) −1 ∈ A H is a Cauchy sequence with respect to the uniform metric, and thus converges. From 0 ≤ (a ∧ n½) ≤ a it follows that (a ∧ n½) 2 ≤ a 2 by Proposition 4.2 and that 0 ≤ a − (a ∧ n½), and Lemma 5.12 shows that a − (a ∧ n½) ≤ a 2 /(4n). So one can apply Lemma 4.10 to the sequence (a∧n½) n∈AE with c := a 2 and d := a 2 /4, which shows that a is invertible and a −1 = lim n→∞ (a∧n½) −1 . Square Roots The usual way to construct absolute values is via square roots of the square. In order to guarantee the uniqueness of the square roots, it makes sense to discuss square roots only in radical Archimedean ordered * -algebras, in which Lemma 5.10 applies: exists, then |a| exists and is given by |a| = √ a 2 . Proof: Note that {a} ′ ⊆ {a 2 } ′ , therefore {a 2 } ′′ ⊆ {a} ′′ . So √ a 2 ∈ {a 2 } ′′ ∩ A + H ⊆ {a} ′′ ∩ A + H and √ a 2 2 = a 2 show that |a| = √ a 2 exists. Especially if √ a exists for all positive Hermitian elements a of a radical Archimedean ordered * -algebra A, then A is a Φ * -algebra and A + H = A ++ H , thus every unital * -homomorphism Ψ : A → B to another ordered * -algebra B is automatically positive as Ψ(a) = Ψ( √ a) 2 ∈ B + H . On such algebras, the order is even uniquely determined, a result that generalizes the uniqueness of the norm of C * -algebras: Proposition 6.3 Let A be a radical Archimedean ordered * -algebra in which √ a exists for all a ∈ A + H , then the order on A H is uniquely determined in the following sense: Denote the order on A H by ≤, as always. If is any order on A H such that A with is an ordered * -algebra, then ≤ and coincide. Proof: Consider the injective unital * -homomorphism id A as a map from A with ≤ to A with . Then id A is automatically positive due to the existence of square roots, and as A with ≤ is a Φ * -algebra by the previous Proposition 6.2 and Corollary 5.7, Proposition 5.9 applies and shows that id A is even an order embedding, i.e. that ≤ and coincide. Moreover, unital * -homomorphisms between such algebras are compatible with square roots: In the uniformly complete case, square roots can oftentimes be explicitly constructed: Lemma 6.5 Let A be a radical and uniformly complete Archimedean ordered * -algebra,â ∈ A + H and (a n ) n∈AE a sequence in {â} ′′ ∩ A + H with limitâ. If a sequence (b n ) n∈AE in {â} ′′ ∩ A + H fulfils b 2 n = a n for all n ∈ AE and is bounded from above by some c ∈ A + H , then √â exists and √â = lim n→∞ b n . Proof: Given ǫ ∈ ]0, 1], then there exists an N ∈ AE such that −ǫ 2 ½ ≤ a n −a N ≤ ǫ 2 ½ holds for all n ∈ AE It only remains to show thatb 2 =â, then √â =b exists. For all ǫ ∈ ]0, 1] there exists an N ∈ AE such that −ǫ½ ≤ b n −b ≤ ǫ½ and −ǫ½ ≤ a n −â ≤ ǫ½ hold for all n ∈ AE with n ≥ N . The first estimate givesb ≤ b n + ǫ½ and b n ≤b + ǫ½, and using Proposition 4.2 one obtainsb 2 ≤ (b n + ǫ½) 2 ≤ b 2 n + ǫ(2b n + ½) ≤ a n + ǫ(2c + ½) and a n = b 2 n ≤ (b + ǫ½) 2 ≤b 2 + ǫ(2b + ½). Together with the second estimate this yieldsb 2 −â =b 2 − a n + a n −â ≤ ǫ(2c + 2½) andâ −b 2 =â − a n + a n −b 2 ≤ ǫ(2b + 2½), soâ =b 2 because A is Archimedean. with n ≥ N , hence b 2 N ≤ b 2 n + ǫ 2 ½ and b 2 n ≤ b 2 N + ǫ 2 ½. This implies b 2 N ≤ (b n + ǫ½) 2 and b 2 n ≤ (b N + ǫ½) 2 , so b N ≤ b n + ǫ½ and b n ≤ b N + ǫ½ by Proposition 6.6 Let A be a radical and uniformly complete Archimedean ordered * -algebra and let a ∈ A + H . If additionally a is uniformly bounded, or a + ½/n invertible for all n ∈ AE, then √ a exists. Proof: As A bd is a C * -algebra due to the completeness of A, it is clear that the square root of a exists in the uniformly bounded case. This can also be obtained directy by combining Lemmas 4.7 and the previous Lemma 6.5. If a + ½/n is invertible for all n ∈ AE, then (a + ½/n) −1 ∈ {a} ′′ ∩ A + H ∩ A bd for every n ∈ AE by Lemma 4. holds and let a ∈ A H be given. Then ½ + a 2 is coercive and (½ + a 2 ) ∧ (n½) = n(((½ + a 2 )/n) ∧ ½) exists for all n ∈ AE by Proposition 5.3, so Proposition 5.14 applies and shows that ½ + a 2 has a multiplicative inverse. But then (a ± i½) −1 = (½ + a 2 ) −1 (a ∓ i½) exists as well. Definition 7.2 A Su * -algebra is a uniformly complete Archimedean ordered * -algebra that has one, hence all of the equivalent additional properties of the above Theorem 7.1 It is obvious that "Su" refers to "symmetric and uniformly complete". Besides the equivalent characterizations given by Theorem 7.1, a Su * -algebra A also has some other interesting properties: Because of the existence of square roots, the order on A is simply the algebraic one, i.e. A + H = A ++ H , and every unital * -homomorphism Ψ : A → B into an ordered * -algebra B is automatically positive, hence continuous for Archimedean B. If Ψ is in addition injective, then it is already an order embedding by Proposition 5.9. Proposition 6.3 shows that the order on A H is the unique one with which A becomes an ordered * -algebra. Unital * -homomorphisms between Su * -algebras are not only compatible with the algebraic operations and positive, they are also compatible with ∨ and ∧, absolute values and square roots by Propositions 5.11 and 6.4. From this point of view, Su * -algebras can be seen just as wellbehaved * -algebra, not necessarily as * -algebra with an additional structure, because the order is not subject to any choice and because unital * -homomorphisms between them fulfil all compatibilities one would reasonably expect. All these properties are typical for C * -algebras and complete Φ-algebras, which are important examples of Su * -algebras: The uniformly bounded Su * -algebras are precisely the C * -algebras, because · ∞ in this case is a C * -norm by Proposition 3.13, and because conversely, C * -algebras carry a natural ordering with respect to which they are well-known to be uniformly bounded and uniformly complete Archimedean ordered * -algebras, and also symmetric due to their well-behaved spectral theory. The commutative Su * -algebras are the complexifications of complete Φ-algebras by Proposition 5.8. As a consequence, the representation theorems for C * -and Φ-algebras partly apply: All uniformly bounded and closed unital * -subalgebras of a Su * -algebra A (especially A bd ) are isomorphic to a C * -algebra of bounded operators on a Hilbert space. Similarly, all commutative and closed unital * -subalgebras of A, that also contain the inverses of all coercive elements (especially bicommutants S ′′ of commutative subsets S ⊆ A H ), are isomorphic to the complexification of a complete Φ-algebra of continuous functions on a compact Hausdorff space with values in the extended real line by [5]. Similar representation theorems specifically adapted to ordered * -algebras are developped in [15]. With respect to a well-behaved functional calculus we note that already the well-behaved polynomial calculus described in Proposition 4.4 is far from being trivial, but still remains to be extended to a continuous calculus for Su * -algebras. Important existing results in this direction are of course the continuous calculus for C * -algebras, which also applies to uniformly bounded elements of Su * -algebras, and the continuous calculus for Φ-algebras from [2]. Q ↓ := a ∈ Q ′ ∃ q∈Q : a * a q * q and a a * q * q . (8.1) Lemma 8.2 Let A be a quasi-ordered * -algebra and q, r ∈ A commuting and with the property that q * q and r * r are coercive. Then λ 2 q * r * r q is coercive for all λ ∈ ]0, ∞[ and there exists a λ ∈ [1, ∞[ such that q * q + r * r λ 2 q * r * r q holds. Proof: Let ǫ ∈ ]0, 2] be given such that q * q ǫ½ and r * r ǫ½, then λ 2 q * r * r q is coercive for all λ ∈ ]0, ∞[ because λ 2 q * r * r q λ 2 ǫ q * q λ 2 ǫ 2 ½. Moreover, (2/ǫ) q * r * r q = q * (r * r/ǫ − ½) q + r * (q * q/ǫ − ½) r + q * q + r * r q * q + r * r holds. So q * q + r * r λ 2 q * r * r q if one chooses λ := 2/ǫ ∈ [1, ∞[. Proposition 8.3 Let A be a quasi-ordered * -algebra and Q a dominant subset of A, then Q ↓ is a unital * -subalgebra of Q ′ , hence of A, and Q ⊆ Q ↓ . Proof: Clearly ½ ∈ Q ⊆ Q ↓ , λa ∈ Q ↓ for all λ ∈ if a ∈ Q ↓ and Q ↓ is stable under · * . Let a, b ∈ Q ↓ be given, then there are q, r ∈ Q that fulfil a * a q * q, a a * q * q and b * b r * r, b b * r * r. Then b * a * a b b * q * q b = q * b * b q q * r * r q and similarly also a b b * a * r * q * q r = q * r * r q show that ab ∈ Q ↓ . Moreover, by the previous Lemma 8.2, there exists a λ ∈ [1, ∞[ for which (a + b) * (a + b) (a + b) * (a + b) + (a − b) * (a − b) = 2(a * a + b * b) 2(q * q + r * r) 2λ 2 r * q * q r and similarly also (a + b) (a + b) * 2λ 2 r * q * q r hold, so a + b ∈ Q ↓ . for all ξ, η ∈ D. The induced seminorm is denoted by ξ a := ξ | ξ a for all ξ ∈ D. Recall that the graph topology (see e.g. can check that this set of seminorms · b with b ∈ A + H is even cofinal in the set of all seminorms on D that are continuous with respect to the graph topology, i.e. for every such continuous seminorm p there exists a b ∈ A + H such that p ≤ · b . Proposition 8.5 Let D be a pre-Hilbert space and Q ⊆ L * (D) a dominant subset. Then the graph topology induced by the O * -algebra Q ↓ ⊆ L * (D) on D is the locally convex topology defined by all the seminorms · q * q with q ∈ Q, and the set of seminorms · q * q with q ∈ Q is cofinal in the set of all seminorms on D that are continuous with respect to the graph topology of Q ↓ . Moreover, for all a ∈ Q ′ the following is equivalent: i.) a ∈ Q ↓ , ii.) a and a * are both continuous as maps from D with the graph topology of Q ↓ to D with the · -topology, iii.) a and a * are both continuous as maps from D with the graph topology of Q ↓ to itself. Proof: The locally convex topology on D defined by all the seminorms · q * q with q ∈ Q is clearly weaker than the graph topology of Q ↓ . Conversely, given b ∈ (Q ↓ ) + H , then also ½ + b ∈ (Q ↓ ) + H and so there exists a q ∈ Q such that (½ + b) 2 ≤ q * q. Consequently · b ≤ · (½+b) 2 ≤ · q * q , and it follows that the locally convex topology on D defined by all the seminorms · q * q and the graph topology of Q ↓ coincide and that the set of seminorms · q * q with q ∈ Q is cofinal in the set of all seminorms on D that are continuous with respect to the graph topology of Q ↓ . Now let a ∈ Q ↓ be given. Then also a * ∈ Q ↓ and a(ξ) = ξ a * a ≤ ξ ½+a * a and a * (ξ) = ξ aa * ≤ ξ ½+aa * hold for all ξ ∈ D, which shows that a and a * are both continuous as maps from D with the graph topology of Q ↓ to D with the · -topology, i.e. i.) implies ii.). Next assume that some a ∈ Q ′ is continuous as a map from D with the graph topology of Q ↓ to D with the · -topology. Then there is a q ∈ Q such that a(ξ) ≤ ξ q * q holds for all ξ ∈ D, and thus a(ξ) r * r = r a(ξ) = a r(ξ) ≤ r(ξ) q * q = ξ r * q * q r holds for all ξ ∈ D and all r ∈ Q because a and r commute. This shows that a is continuous as a map from D with the graph topology of Q ↓ to itself. So ii.) implies iii.). Finally, assume that a ∈ Q ′ is such that a and a * are both continuous as maps from D with the graph topology of Q ↓ to itself. Then there especially exist q, r ∈ Q such that a(ξ) ≤ ξ q * q = q(ξ) and a * (ξ) ≤ ξ r * r = r(ξ) hold for all ξ ∈ D, hence a * a ≤ q * q ≤ q * q + r * r and a a * ≤ r * r ≤ q * q + r * r. By Lemma 8.2, there exists a λ ∈ [1, ∞[ such that a * a ≤ t * t and a a * ≤ t * t hold for t := λqr ∈ Q. We conclude that iii.) implies i.). Such dominated * -algebras of operators are especially interesting if they are closed: Recall that an O * -algebra A ⊆ L * (D) on a pre-Hilbert space D is closed if D is complete with respect to the graph topology of A. holds for all a ∈ A. Proof: The supremum on the right hand side of (8.3) is by definition the minimum of the set of λ ∈ [0, ∞] for which a(ξ) ≤ λ holds for all ξ ∈ D with ξ = 1, or equivalently, for which a * a ≤ λ 2 ½ holds (where a * a ≤ ∞ 2 ½ is defined to be always true). By Definition 3.10 and Proposition 3.12, this minimum is just a ∞ . Proposition 8.7 Let D be a pre-Hilbert space, Q ⊆ L * (D) a dominant subset and such that Q ↓ is a closed O * -algebra. Then Q ↓ is a uniformly complete Archimedean ordered * -algebra. Proof: Let (a n ) n∈AE be a Cauchy sequence in Q ↓ . Then for every ǫ ∈ ]0, ∞[ there exists an N ∈ AE such that a n − a N ∞ ≤ ǫ holds for all n ∈ AE with n ≥ N , and due to the previous Lemma 8. 6, this implies that the estimate a n (ξ) − a N (ξ) q * q = (a n − a N ) q(ξ) ≤ ǫ q(ξ) holds for all ξ ∈ D, all q ∈ Q and all n ∈ AE with n ≥ N . So for every ξ ∈ D, this together with Proposition 8.5 shows that a n (ξ) n∈AE is a Cauchy sequence with respect to the graph topology of Q ↓ on D, and thus converges against a limitâ(ξ) ∈ D. The resulting mapâ : D → D, ξ →â(ξ) is the pointwise limit of the sequence (a n ) n∈AE and is easily seen to be a linear endomorphism of D. The convergence is even uniform in the sense that â − a N (ξ) ≤ â − a n (ξ) + a n − a N (ξ) ≤ 2ǫ ξ holds for all ξ ∈ D if n ∈ AE with n ≥ N is chosen sufficiently large. As the * -involution is continuous with respect to d ∞ on Q ↓ , also (a * n ) n∈AE is a Cauchy sequence in Q ↓ and yields a pointwise limitã : D → D. The inner product · | · : D × D → is ·continuous as a consequence of the Cauchy Schwarz inequality. Using this it is easily seen thatã is the adjoint endomorphism ofâ, soâ ∈ L * (D). The previous Lemma 8.6 together with the above uniform convergence estimate imply thatâ is the limit of the sequence (a n ) n∈AE with respect to d ∞ on L * (D). It only remains to show thatâ ∈ Q ↓ : Proposition 3.16 already shows thatâ ∈ Q ′ . Moreover, there exists an n ∈ AE with â − a n ∞ ≤ 1, i.e. â − a n (ξ) ≤ ξ for all ξ ∈ D, and a q ∈ Q fulfilling a * n a n ≤ q * q and a n a * n ≤ q * q, i.e. a n (ξ) ≤ q(ξ) = ξ q * q and a * n (ξ) ≤ ξ q * q for all ξ ∈ D. So â(ξ) ≤ â − a n (ξ) + a n (ξ) ≤ ξ + ξ q * q for all ξ ∈ D, which shows thatâ is continuous as a map from D with the graph topology of Q ↓ to D with the · -topology. The same is also true forâ * , soâ ∈ Q ↓ by Proposition 8.5. This shows that uniform completeness of certain O * -algebras can be guaranteed essentially by a completeness condition on the domain. For the existence of inverses of coercive elements one can then make use of Proposition 4.11: Theorem 8.8 Let D be a pre-Hilbert space and Q ⊆ L * (D) a dominant subset such that Q ↓ is a closed O * -algebra. Then Q ↓ is a Su * -algebra if and only if all q ∈ Q are invertible. Proof: First assume that Q ↓ is a Su * -algebra and let q ∈ Q be given. Then q * q is coercive, hence has an inverse, and as q * q = q q * it follows that q is also invertible with q −1 = (q * q) −1 q * . Conversely, if all q ∈ Q are invertible, then also all q * q with q ∈ Q. The previous Proposition 8.7 already shows that Q ↓ is uniformly complete. Given a coercive a ∈ A + H then there exists a q ∈ Q such that (½ + a) 2 ≤ q * q, and then ½ ≤ (½ + a) 2 ≤ q * q implies q * q ≤ q * (q * q) q = (q * q) 2 and therefore a ≤ (½ + a) 2 ≤ q * q ≤ (q * q) 2 . It follows from Proposition 4.11 that a is invertible, so Q ↓ is a Su * -algebra. Example 8.9 Let H be a Hilbert space and h a self-adjoint (not necessarily bounded) operator on H which is coercive in the sense that there exists an ǫ ∈ ]0, ∞[ such that ξ | h(ξ) ≥ ǫ ξ | ξ holds for all vectors ξ in the domain of h. Let D be the dense linear subspace of H consisting of all smooth vectors of h, i.e. the intersection of the domains of the operators h n for all n ∈ AE, and write q ∈ L * (D) for the endomorphism described by the restriction of h to D, which is coercive in the sense of Definition 4.1. Then D is complete with respect to the locally convex topology defined by all the seminorms · q 2n with n ∈ AE 0 . Moreover, q is invertible in L * (D) because the inverse operator h −1 ∈ L * (H) of the self-adjoint coercive h restricts to an endomorphism of D as well. One can check that the set Q := λq n λ ∈ [1, ∞[, n ∈ AE 0 is a dominant subset of L * (D), and as q and hence all elements of Q are invertible, the previous Theorem 8.8 applies and shows that Q ↓ is a Su * -algebra. One important application of the above Example 8.9 is the case where h is the Hamiltonian operator of a quantum mechanical system. Then the Su * -algebra Q ↓ is essentially the algebra of all symmetries of this system that are bounded by a power of h. Note that choosing h = ½ L * (H) produces the C * -algebra Q ↓ = L * (H), so the construction of this Example 8.9 is sufficiently general to cover (up to taking suitable * -subalgebras) at least all C * -algebras, and clearly many more. , in the commutative case, * -algebras are just the complexifications of real associative algebras. So the theory of ordered real algebras, especially of lattice ordered ones like (almost) f -algebras and Φ-algebras, immediately carries over and yields examples of well-behaved ordered * -algebras even beyond the scope of C * -algebras. The representation theorem [5, Thm. 2.3] for Φ-algebras as algebras of functions on a compact Hausdorff space with values in the extended real numbers further exemplifies the close relation between commutative C * -algebras and (complexifications of) Φ-algebras. The aim of the present article is to examine ordered * -algebras and ultimately to determine a class of very well-behaved ordered * -algebras that generalize important properties of C * -algebras to the unbounded case, as well as properties of Φ-algebras to the non-commutative case. This includes the existence of suprema and infima of finitely many commuting Hermitian elements, of absolute values, square roots of positive elements and inverses of elements that are coercive (i.e. "strictly" positive), as well as automatic continuity of unital * -homomorphisms and the uniqueness of the order. Special attention is given to situations where order-theoretic and algebraic concepts are equivalent. The most obvious example for this are absolute values: The absolute value |a| of a Hermitian element a should be, from the purely order-theoretic point of view, the supremum of a and −a. But from a more algebraic point of view, |a| should be the (positive) square root of a 2 . This raises the question whether, or under which circumstances, the two descriptions are equivalent. Example 3. 3 3Let [x 1 , . . . , x N ] with N ∈ AE be the * -algebra of complex polynomials in N Hermitian variables x 1 , . . . , x N , i.e. the * -involution is given by complex conjugation of all coefficients. For every subset S ⊆ Ê N , the S-pointwise order on the Hermitian polynomials, i.e. p ≤ q if and only if p(s 1 , . . . , s N ) ≤ q(s 1 , . . . , s N ) for all (s 1 , . . . , s N ) ∈ S, turns [x 1 , . . . , x N ] into an Archimedean quasiordered * -algebra, which is even an Archimedean ordered * -algebra e.g. if S has non-empty interior. Non-commutative examples are provided by * -algebras of operators, i.e. O * -algebras: Example 3.4. But there are also other types of examples: Example 3.5 Let A be a unital * -algebra, G ⊆ A H and define ⟪ G ⟫ pos as ⟪ G ⟫ pos := N n=1a * n g n a n N ∈ AE; g 1 , . . . , g N ∈ G ∪ {½}, a 1 , . . . , a N ∈ A . the elements of A ++ H algebraically positive. There are many strong results (called "Positivstellensätze") in real algebraic geometry that link orders on * -algebras that are induced by positive Hermitian linear functionals like in Examples 3.2, 3.3 and 3.4 to orders generated by a set of positive elements like in Example 3.5. These include the classical ( but not commuting) matrices a and b with 0 ≤ a ≤ b which do not fulfil a 2 ≤ b 2 .Because of this, one should not expect an analog of Proposition 4.2 to be fulfilled for non-commutating Hermitian elements in well-behaved examples. Using results from real algebraic geometry, the aboveCorollary 4.3 can be improved significantly, which demonstrates the importance of the radical-property: Proposition 4.4 Let A be a radical Archimedean ordered * -algebra, a 1 , . . . , a N ∈ A H with N ∈ AE pairwise commuting and p 1 , . . . , p M ∈ Ê[x 1 , . . . , x N ] ∼ = [x 1 , . . . , x N ] H with M ∈ AE polynomials fulfilling p m (a 1 , . . . , a N ) ∈ A + H for all m ∈ {1, . . . , M }. Define the associated semialgebraic set S := s ∈ Ê N p m (s 1 , . . . , s N ) ≥ 0 for all m ∈ {1, . . . , M } . (4. 1 ) 1Then the unital * -homomorphism [x 1 , . . . , x N ] ∋ q → q(a 1 , . . . , a N ) ∈ A is positive with respect to the S-pointwise order on [x 1 , . . . , x N ].Proof: First let q ∈ [x 1 , . . . , x N ] H be given such that q(s 1 , . . . , s N ) > 0 for all s ∈ S. By the Positivstellensatz of Krivine and Stengle, there exist two polynomials Lemma 4. 7 7Let A be an ordered * -algebra and a ∈ (A bd ) + H , then there exist two sequences of polynomials (p n ) n∈AE and (q n ) n∈AE in Ê[x] ∼ = [x] H such that the identity a + q n (a) = p 2 n (a) and the estimates 0 ≤ p n (a) and 0 ≤ q n (a) ≤ ½/n hold. then write c (+) := (d + c)/2 ∈ A + H and c (−) := (d − c)/2 ∈ A + H . Note that c = c (+) − c (−) and d = c (+) + c (−) . From 0 ≤ (a − b) * c (+) (a − b) and 0 ≤ (a + b) * c (−) (a + b) by Proposition 4.2, so x ≥ a and x ≥ b. Moreover, if some y ∈ {a, b} ′ ∩ A H also fulfils y ≥ a and y ≥ b, then 0 ≤ (y − a)(y − b) = y 2 − y(a + b) + ab by Corollary 4.3 and thus (2y − a − Proposition 5.1 especially guarantees that a ∨ b and a ∧ b (if they exist) are uniquely determined as certain suprema and infima. In a Φ * -algebra A, Proposition 5.1 also has a rather trivial, but noteworthy converse: If a, b ∈ A H commute and x ∈ {a, b} ′ ∩ A H is the supremum or infimum of a and b in somereal linear subspace V of A H such that {a, b} ′′ ∩ A H ⊆ V ⊆ {a, b} ′ ∩ A H , then x coincides with a ∨ bor a ∧ b, respectively, due to the uniqueness of the suprema and infima; so especially x ∈ {a, b} ′′ ∩ A H and x 2 + ab = x(a + b). Proposition 5. 3 3Let A be a radical Archimedean ordered * -algebra and a, b ∈ A H commuting. Then a ∨ b exists if and only if a ∧ b exists, and the two are related by algebra. This is due to the observation that (a ∨ b) + c and (a ∧ b) + c with c ∈ {a, b} ′ ∩ A H are the supremum and infimum, respectively, of a + c and b + c in V := {a, b} ′ ∩ {a + c, b + c} ′ ∩ A H , so the discussion under Definition 5.2 applies. One important special case of these suprema in ordered * -algebras are absolute values: Definition 5.4 Let A be a radical Archimedean ordered * -algebra and a ∈ A H , then the absolute value of a is defined (if it exists) as the element |a| := a ∨ (−a). If the absolute value exists, then one also defines the positive part a + := 1 2 (|a| + a) and the negative part a − := 1 2 (|a| − a) of a. Clearly, |−a| = |a| if |a| exists. By definition, the absolute value of an element a ∈ A H is (if it exists) the element |a| ∈ {a} ′′ ∩ A + H that fulfils |a| 2 = a 2 . The earlier results about suprema and infima now show that, like for Riesz spaces, the existence of all absolute values already implies the existence of all suprema and infima of commuting Hermitian elements:Proposition 5.5 Let A be a radical Archimedean ordered * -algebra and a, b ∈ A H commuting and such that |a − b| exists. Then a ∨ b and a ∧ b exist and are given by a ≥ ǫ½ for some ǫ ∈ ]0, ∞[ and 0 ≤ ab, then write b + := sup{b, 0} and b − := sup{−b, 0}. Note that it is not yet clear that b + and b − are the positive and negative part of b like in Definition 5.4, but it follows from the general calculation rules in Riesz spaces and Φ-algebras that Proposition 3.9 now shows that b − = 0 and therefore b = b + ≥ 0. As the order-theoretic absolute value |a| := sup{a, −a} ∈ A + H of any a ∈ A H indeed fulfils |a| 2 = a 2 by the calculation rules in Φ-algebras, it also describes the absolute value as in Definition 5.4 and therefore A is a Φ * -algebra by the previous Corollary 5.7. Now let A be an arbitrary commutative Φ * -algebra. Then A H is a real commutative unital associative algebra and a Riesz space by Proposition 5.1. Corollary 4.3 shows that ab ∈ A + H for all a, b ∈ A + H . Given a, b, c ∈ A + H with inf{a, b} = 0, then a ∧ b = inf{a, b} = 0 and thus ab = 0. It follows that 0 ≤ (ac ∧ b) 2 ≤ (ac ∧ b)(ac ∨ b) = acb = 0 holds by Corollary 4.3 and Proposition 5.3, so (ac ∧ b) 2 = 0. Proposition 3.9 now shows that ac ∧ b = 0 and thus A H is a Φ-algebra.Non-commutative examples of Φ * -algebras will be described in Sections 7 and 8.Another interesting observation about Φ * -algebras is that injective positive unital * -homomorphisms between them are automatically order embeddings. This is roughly similar to the case of * -algebrasendowed with a Fréchet-topology, where surjective continuous linear maps are automatically open by the open mapping theorem: Proposition 5.9 Let Ψ : A → B be an injective positive unital * -homomorphism from a Φ * -algebra A to an ordered * -algebra B, then Ψ is automatically an order embedding. Lemma 5 . 510 Let A be a radical Archimedean ordered * -algebra and a ∈ A +H , b ∈ {a} ′′ ∩ A + H and c ∈ {a} ′ ∩ A + H such that b 2 = c 2 ,then b = c and especially c ∈ {a} ′′ . Proof: Note that b and c commute, so b 2 = c 2 implies b = c by Proposition 4.2. Proposition 5.11 Let Ψ : A → B be a positive unital * -homomorphism between two Φ * -algebras A, B and a,ã ∈ A H commuting. Then Ψ(a) ∨ Ψ(ã) = Ψ(a ∨ã), Ψ(a) ∧ Ψ(ã) = Ψ(a ∧ã) and |Ψ(a)| = Ψ(|a|) hold. Proposition 6. 4 4Let A and B be two radical Archimedean ordered * -algebras and assume that the square roots of all positive Hermitian elements in A and B exist. Moreover, let Ψ : A → B be an (automatically positive) unital * -homomorphism. Then Ψ( √ a) = Ψ(a) holds for all a ∈ A + H . Proof: Like Proposition 5.11, this follows from Lemma 5.10 because Ψ( √ a) ∈ {Ψ(a)} ′ ∩ B + H fulfils Ψ( √ a) 2 = Ψ(a) and because it is already known that Ψ(a) ∈ {Ψ(a)} ′′ ∩ B + H exists. Proposition 4.2, or equivalently −ǫ½ ≤ b n − b N ≤ ǫ½. The sequence (b n ) n∈AE thus is a Cauchy sequence and has a limitb := lim n→∞ b n ∈ {â} ′′ ∩ A + H as {â} ′′ ∩ A + H is a closed subset of the complete metric space A by Propositions 3.16. 6 and (a + ½/n) −1 ∈ {(a + ½/n) −1 } ′′ ∩ A + H exists by the first part, and one can thus construct b n := (a + ½/n) (a + ½/n) −1 . By Corollary 4.3, b n ≥ 0 and one can easily check that even b n ∈ {a} ′′ ∩ A + H and b 2 n = a + ½/n. So the previous Lemma 6.5 applies and shows that √ a exists. 7 Su * -Algebras Essentially all of the previous results hold for a class of very well-behaved ordered * -algebras: Theorem 7.1 Let A be a uniformly complete Archimedean ordered * -algebra, then the following additional properties are all equivalent: i.) All elements of the form a + i½ and a − i½ with a ∈ A H are invertible. ii.) All coercive elements in A H are invertible, i.e. A is symmetric. iii.) A is radical and √ a exists for all a ∈ A + H . iv.) A is radical and |a| exists for all a ∈ A H . v.) A is radical and both a ∨ b and a ∧ b exist for all commuting a, b ∈ A H , i.e. A is a Φ * -algebra. vi.) A is radical and a ∧ ½ exists for all coercive a ∈ A + H . Proof: Implication i.) =⇒ ii.) follows from Corollary 4.12 because ½+a 2 = (a+i½)(a−i½), and for the implication ii.) =⇒ iii.) one uses that every symmetric Archimedean ordered * -algebra is automatically radical by Proposition 4.8 and thus one can apply Proposition 6.6. iii.) =⇒ iv.) is Proposition 6.2, iv.) =⇒ v.) is Corollary 5.7 and v.) =⇒ vi.) is trivial. Finally, for vi.) =⇒ i.), assume that vi.) 8 Su * -Algebras of Unbounded Operators While Su * -algebras are unbounded generalizations of C * -algebras and non-commutative generalizations of Φ-algebras, it still remains to give examples of Su * -algebras that are not of one of these already wellunderstood types. Clearly, such examples should especially be provided by * -algebras of unbounded operators, i.e. by O * -algebras like in Example 3.4. In the general case, when A ⊆ L * (D) is an arbitraryO * -algebra on a pre-Hilbert space D, it is clear that A is an Archimedean ordered * -algebra. It remains to find sufficient conditions for A to be symmetric and uniformly complete. Recall that an element a of a * -algebra is called normal if a * a = a a * . For example, if D = H is a Hilbert space, then Q := { λ½ | λ ∈ [1, ∞[ } is a dominant subset of L * (H) and Q ↓ = L * (H) is the * -algebra of all bounded, i.e. of all · -continuous linear operators on H. Similar examples generated by a coercive Hermitian, but not necessarily bounded operator in L * (D) on a general pre-Hilbert space D can be constructed analogously. The characterization as a * -algebra of continuous adjointable operators carries over as well: Definition 8.4 Let D be a pre-Hilbert space and a ∈ L * (D) + H . Then define the positive Hermitian sesquilinear form · | · a : D × D → as ξ η a := ξ a(η) (8.2) [ 11 , 11Def. 2.1.1]) induced by an O * -algebra A ⊆ L * (D) on the pre-Hilbert space D is the locally convex topology defined by all the seminorms · ½+a * a with a ∈ A, or equivalently by all the seminorms · b with b ∈ A + H because b ≤ ½ + (½ + b) 2 for all b ∈ A + H . One Lemma 8. 6 6Let D be a pre-Hilbert space and A ⊆ L * (D) an O * -algebra on D, then a ∞ = sup ξ∈D, ξ =1 a(ξ) ∈ [0, ∞] (8.3) Archimedean, then this equivalence also holds for λ = 0.Proof: For λ ∈ ]0, ∞[ this is essentially [4, Lemma 3.1], an immediate consequence of the identities Proposition 3.16 Let A be an Archimedean ordered * -algebra and S ⊆ A H , then the space A H of Hermitian elements, the space A bd of uniformly bounded elements, the commutant S ′ , the bicommutant S ′′ and the sets { a ∈ A H | a ≤ s for all s ∈ S } and { a ∈ A H | a ≥ s for all s ∈ S } are closed in A with respect to the uniform metric.Proof: Using that Re(a) ∞ ≤ a ∞ and Im(a) ∞ ≤ a ∞ hold for all a ∈ A bd , it is easy to check that the Ê-linear projectors Re, Im : A → A are continuous and thus A H Definition 8.1 Let A be a quasi-ordered * -algebra. Then a dominant subset of A is a subset Q ⊆ A of normal and pairwise commuting elements, stable under · * , with ½ ∈ Q and such that q * q is coercive and λq ∈ Q as well as qr ∈ Q hold for all λ ∈ [1, ∞[ and all q, r ∈ Q. For such a dominant subset define the subset Q ↓ of the commutant of Q in A as Almost f-algebras and d-algebras. S J Bernau, C B Huijsmans, Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press107S. J. Bernau and C. B. Huijsmans. Almost f-algebras and d-algebras. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 107, pages 287-308. Cambridge University Press, 1990. 4 Functional calculus on riesz spaces. G Buskes, B De Pagter, A Van Rooij, Indagationes Mathematicae. 2421G. Buskes, B. de Pagter, and A. van Rooij. Functional calculus on riesz spaces. Indagationes Mathematicae, 2(4):423-436, 1991. 21 Almost f-algebras: Commutativity and the cauchy-schwarz inequality. G Buskes, A Van Rooij, Positivity. 43G. Buskes and A. van Rooij. Almost f-algebras: Commutativity and the cauchy-schwarz inequality. Positivity, 4(3):227-231, 2000. 4 A representation theorem for archimedean quadratic modules on *-rings. J Cimprič, Canadian Mathematical Bulletin. 521J. Cimprič. A representation theorem for archimedean quadratic modules on *-rings. Canadian Mathematical Bulletin, 52(1):39-52, 2009. 7, 8 On the structure of a class of archimedean lattice-ordered algebras. M Henriksen, D Johnson, Fundamenta Mathematicae. 501M. Henriksen and D. Johnson. On the structure of a class of archimedean lattice-ordered algebras. Fundamenta Mathematicae, 50(1):73-94, 1961. 2, 4, 21 Commutativity of some archimedean ordered algebras. N Kouki, M A Toumi, N Toumi, Positivity18N. Kouki, M. A. Toumi, and N. Toumi. Commutativity of some archimedean ordered algebras. Positivity, 18(4):805-821, 2014. 4 Banach Algebras and the General Theory of *-Algebras. T W Palmer, of Encyclopedia of Mathematics and its Applications. Cambridge University Press212T. W. Palmer. Banach Algebras and the General Theory of *-Algebras, volume 2 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994. 12 Self-adjoint algebras of unbounded operators ii. R T Powers, Transactions of the American Mathematical Society. 1874R. T. Powers. Self-adjoint algebras of unbounded operators ii. Transactions of the American Mathematical Society, 187:261-293, 1974. 4 . E Scheffold, 177ff-banachverbandsalgebren. Mathematische ZeitschriftE. Scheffold. ff-banachverbandsalgebren. Mathematische Zeitschrift, 177(2):193-205, 1981. 4 Positivity and sums of squares: A guide to recent results. C Scheiderer, Emerging Applications of Algebraic Geometry. Mihai Putinar and Seth SullivantNew York, New York, NYSpringer25C. Scheiderer. Positivity and sums of squares: A guide to recent results. In Mihai Putinar and Seth Sullivant, editors, Emerging Applications of Algebraic Geometry, pages 271-324. Springer New York, New York, NY, 2009. 2, 5 Unbounded Operator Algebras and Representation Theory. K Schmüdgen, 422Birkhäuser, BaselK. Schmüdgen. Unbounded Operator Algebras and Representation Theory. Birkhäuser, Basel, 1990. 4, 6, 22 The Moment Problem. K Schmüdgen, SpringerK. Schmüdgen. The Moment Problem. Springer, 2017. 13 A strict positivstellensatz for the weyl algebra. K Schmüdgen, Mathematische Annalen. 3314K. Schmüdgen. A strict positivstellensatz for the weyl algebra. Mathematische Annalen, 331(4):779-794, 2005. 6 A strict positivstellensatz for enveloping algebras. K Schmüdgen, Mathematische Zeitschrift. 2543K. Schmüdgen. A strict positivstellensatz for enveloping algebras. Mathematische Zeitschrift, 254(3):641-653, 2006. 6 M Schötz, arXiv:1906.08752Gelfand-Naimark Theorems for Ordered *-Algebras. arXiv e-prints. 1021M. Schötz. Gelfand-Naimark Theorems for Ordered *-Algebras. arXiv e-prints, page arXiv:1906.08752, 2019. 10, 21

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{'abstract': "The aim of this article is to describe a class of * -algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered * -algebras ( * -algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a C * -norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered * -algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su * -algebras as uniformly complete Archimedean ordered * -algebras which have all these equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su * -algebras generalize some important properties of C * -algebras to algebras of unbounded operators. Similarly, they generalize some properties of Φ-algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered * -algebras. It is also shown that the order on Su * -algebra is uniquely determined, so Su * -algebras are indeed just a class of well-behaved * -algebras. As an example, Su * -algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as * -algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system. * Boursier de l'ULB, [email protected]. This work was supported by the Fonds de la Recherche Scientifique (FNRS) and the Fonds Wetenschappelijk Onderzoek -Vlaaderen (FWO) under EOS Project n 0 30950721.", 'arxivid': '1811.04878', 'author': ['Matthias Schötz \nDépartement de Mathématiques\nUniversité libre de Bruxelles\n\n'], 'authoraffiliation': ['Département de Mathématiques\nUniversité libre de Bruxelles\n'], 'corpusid': 119142342, 'doi': '10.1007/s11117-020-00792-4', 'github_urls': [], 'n_tokens_mistral': 28737, 'n_tokens_neox': 25515, 'n_words': 15846, 'pdfsha': '0f54a48b3ea24bb08d07926e2e1e564995ff4e64', 'pdfurls': ['https://arxiv.org/pdf/1811.04878v3.pdf'], 'title': ['Equivalence of Order and Algebraic Properties in Ordered * -Algebras', 'Equivalence of Order and Algebraic Properties in Ordered * -Algebras'], 'venue': []}

arxiv

Massive Dirac particles on the background of charged de-Sitter black hole manifolds 19 Jun 2009 (Dated: June 19, 2009) F Belgiorno Dipartimento di Fisica F.N Dipartimento di Fisica e Matematica Università di Milano Università dell'Insubria20133, 22100Milano, sezione di Milano, ComoItaly, and I.N.F.N., sezione di MilanoItaly, and I.N., Italy, Italy S L Cacciatori Dipartimento di Fisica F.N Dipartimento di Fisica e Matematica Università di Milano Università dell'Insubria20133, 22100Milano, sezione di Milano, ComoItaly, and I.N.F.N., sezione di MilanoItaly, and I.N., Italy, Italy Massive Dirac particles on the background of charged de-Sitter black hole manifolds 19 Jun 2009 (Dated: June 19, 2009) We consider the behavior of massive Dirac fields on the background of a charged de-Sitter black hole. All black hole geometries are taken into account, including the Reissner-Nordström-de-Sitter one, the Nariai case and the ultracold case. Our focus is at first on the existence of bound quantum mechanical states for the Dirac Hamiltonian on the given backgrounds. In this respect, we show that in all cases no bound state is allowed, which amounts also to the non-existence of normalizable time-periodic solutions of the Dirac equation. This quantum result is in contrast to classical physics, and it is shown to hold true even for extremal cases. Furthermore, we shift our attention on the very interesting problem of the quantum discharge of the black holes. Following Damour-Deruelle-Ruffini approach, we show that the existence of level-crossing between positive and negative continuous energy states is a signal of the quantum instability leading to the discharge of the black hole, and in the cases of the Nariai geometry and of the ultracold geometries we also calculate in WKB approximation the transmission coefficient related to the discharge process. * I. INTRODUCTION Black holes in de Sitter space are an interesting subject of investigation, both on the theoretical side and on the experimental one. On one hand, the contextual presence of a black hole event horizon and of a cosmological event horizon, to be associated with the corresponding quantum emission of thermal radiation [1] is a feature which enriches the framework of black hole thermodynamics in itself and also because of the possibility to obtain a true nonequilibrium situation when two different temperatures coexist on the same manifold. On the other hand, black hole physics in spacetimes with positive cosmological constant appear to be of direct physical interest, because the presentday measurements of cosmological parameters confirm the presence of a small positive cosmological constant, which implies that dS backgrounds are the real black hole backgrounds to be taken into account for physical considerations. In this paper, we consider some relevant aspects of the physics of massive quantum Dirac particles on dS black hole backgrounds. We first show that, as expected, the Dirac Hamiltonian is well behaved in the sense that its self-adjointness can be ensured without imposing any boundary condition. We also determine, by means of spectral analysis, two relevant physical properties: there is no mass gap in the spectrum, even if the particles are massive, and there exist no quantum bound state for charged particles around a charged black hole, in contrast to classical physics. The latter property amounts to the absence of normalizable and time-periodic solutions of the Dirac equation on the background of a non-extremal Reissner-Nordström-dS black hole, in full agreement with the recent literature on this topic [2,3,4,5,6]. Furthermore, we show that this holds true also in the extremal case, due to the prominent role of the cosmological event horizon, as well as in the so-called Nariai case and in the ultracold ones. In the second part of the paper, we also take into account the problem of pair-creation by a charged black hole. This is a long-standing topic in the framework of quantum effects in the field of a black hole, as old as the Hawking effect but still different in its origin [7,8]. The latter can be brought back to vacuum instability in presence of an external field (see e.g. [9,10] and, in the recent literature [11,12,13]). It is shown that the presence of level crossing, i.e. of overlap between positive continuum energy states and negative continuum energy ones, according to a criterion introduced by Ruffini, Damour and Deruelle [14,15,16,17], recently extended to include the Reissner-Nordström-AdS case [18], is a still valid tool for investigating pair-creation of charged Dirac particles even in presence of a positive cosmological constant. We point out that this method is equivalent to the ones commonly exploited in order to investigate instability properties of the vacuum [11,13], even if the criterion of level-crossing seems to be specific of the above references [14,15,16,17]. A special attention is focused on special cases, like the Nariai and the ultracold ones, for which an estimate in WKB approximation of the transmission coefficient related to the process of pair-creation is provided. This work, together with the analogous one concerning the Dirac equation on the background of a Reissner-Nordström-AdS black hole, completes the analysis of the process of pair-creation by a charged black hole in presence of a cosmological constant, and in this sense it also extends the analysis on the background of a Reissner-Nordström black hole [7,8]. We recall that, in spite of the fact that dS and AdS differ for a change of sign in the cosmological constant, very different manifolds and very different physics occur on these backgrounds. We mention for example the occurrence of closed timelike curves in the AdS case, a problem which can be overcome by passing to the universal covering, but at the price to deal with the lack of global hyperbolicity [19]. On the quantum level, self-adjointness of the wave operators cannot be ensured in general (see e.g. [18] for RN-AdS black holes), and boundary conditions have to be introduced for some cases, because of a boundary-like behavior of the AdS asymptotic region. In particular, for µ 3 |Λ| < 1 2 , where µ is the Dirac particle mass and Λ is the (negative) cosmological constant, several boundary conditions can be chosen (see e.g. [20] for explicit choices of boundary conditions for the Dirac Hamiltonian on pure AdS), and then physics is not uniquely defined. In the dS cases we discuss herein, no such features arise. Moreover, black holes are characterized by a single event horizon in the AdS case and by two event horizons in the dS one. This fact is shown to be at the root of the fact that in the de Sitter case there is always level-crossing, which is in contrast not only to the AdS case but also with the standard RN case (Λ = 0). This feature is then peculiar of these solutions; notwithstanding, the actual presence of pair-creation is to be associated with further conditions, to be related with the actual largeness of the forbidden region separating positive energy states from negative energy ones. For completeness, we recall that charged Dirac fields in the more general Kerr-Newman-de Sitter background have been studied with the aim to determine their quasinormal modes in [21]. II. DIRAC HAMILTONIAN IN THE CASE r+ < rc: REISSNER-NORDSTRÖM-DS BLACK HOLES We first define the one-particle Hamiltonian for Dirac massive particles on the Reissner-Nordström-dS black hole geometry (RN-dS black hole in the following). We use natural units = c = G = 1 and unrationalized electric units. The metric of the RN-dS black hole manifold (t ∈ R; r ∈ (r + , r c ); Ω ∈ S 2 ) is ds 2 = −f (r)dt 2 + 1 f (r) dr 2 + r 2 dΩ 2 f (r) = 1 − 2M r + Q 2 r 2 − Λ 3 r 2 ;(1) M is the mass and Q is the electric charge of the black hole, and Λ > 0 is the cosmological constant; let us define L = 3 Λ ; the equation f (r) = 0 is assumed to admit solutions r c > r + ≥ r − > r 0 ; then one obtains f (r) = 1 L 2 r 2 (r c − r)(r − r + )(r − r − )(r − r 0 ).(2) r c is the radius of the cosmological horizon, r + is the radius of the black hole event horizon, r − is the radius of the Cauchy horizon. Moreover, due to the actual lack of a term proportional to r 3 , one has r 0 = −(r c + r + + r − ). The above reparameterization of the metric amounts to implementing the following relations between r c , r + , r − and M, Q, L: L 2 = r c (r + + r − + r c ) + r 2 + + r 2 − + r + r − 2L 2 M = r 2 c r + + r c r 2 + + 2r c r + r − + r 2 c r − + r c r 2 − + r 2 + r − + r + r 2 − L 2 Q 2 = r c r + r − (r c + r + + r − ). It is not difficult to show that four real zeroes of f (r) = 0 exist (and three are positive) if and only if the following conditions are implemented: 0 < Q 2 < L 2 12 (3) M extr ≤ M < M max ,(4) where M extr = L 3 √ 6 1 − 1 − 12 Q 2 L 2 2 + 1 − 12 Q 2 L 2 (5) is the mass of the extremal black hole with r − = r + , and M max = L 3 √ 6 1 + 1 − 12 Q 2 L 2 2 − 1 − 12 Q 2 L 2(6) is the mass of the black hole with r + = r c (see sect. IV). See [22] for the analysis of the more general Kerr-Newman-dS case. The vector potential associated with the RN-dS solution is A µ = (−Q/r, 0, 0, 0). Spherical symmetry, as usual, allows to separate variables [23,24,25,26] and to obtain the following reduced Hamiltonian H red = − √ f µ + e Q r f ∂ r + k √ f r −f ∂ r + k √ f r √ f µ + e Q r(7) where f (r) is the same as in (1), k = ±(j + 1/2) ∈ Z − {0} is the angular momentum eigenvalue and µ is the mass of the Dirac particle. The Hilbert space in which H red is formally defined is the Hilbert space L 2 [(r + , r c ), 1/f (r) dr] 2 of the two-dimensional vector functions g ≡ g 1 g 2 such that rc r+ dr f (r) (|g 1 (r)| 2 + |g 2 (r)| 2 ) < ∞. As a domain for the minimal operator associated with H red we can choose the following subset of L 2 [(r + , r c ), 1/f (r) dr] 2 : the set C ∞ 0 (r + , r c ) 2 of the two-dimensional vector functions g whose components are smooth and of compact support [27]. It is useful to define a new tortoise-like variable y dy dr = 1 f (r) (8) and then one obtains y ∈ R, with y → ∞ ⇔ r → r − c and y → −∞ ⇔ r → r + + . The reduced Hamiltonian becomes H red = D 0 + V (y)(9) where D 0 = 0 ∂ y −∂ y 0 and V (r(y)) = − √ f µ + e Q r k √ f r k √ f r √ f µ + e Q r . The Hilbert space of interest for the Hamiltonian (9) is L 2 [R, dy] 2 . We check if the one-particle Hamiltonian is welldefined in the sense that no boundary conditions are required in order to obtain a self-adjoint operator. This means that we have to check if the reduced Hamiltonian is essentially self-adjoint; with this aim, we check if the solutions of the equation H red g = λ g(10) are square integrable in a right neighborhood of y = −∞ and in a left neighborhood of y = +∞. The so called Weyl alternative generalized to a system of first order ordinary differential equations ( [27], theorem 5.6) can be applied, in particular if in a right neighborhood of y = −∞ at least one solution not square integrable exists for every λ ∈ C, then no boundary condition is required and the so-called limit point case (LPC) is verified; if instead for every λ ∈ C all the solutions of (H red − λ) g = 0 lie in L 2 [(−∞, c), dy] 2 , with −∞ < c < ∞, the so-called limit circle case (LCC) occurs (and boundary conditions are required). Analogously one studies the behavior of solutions in a left neighborhood of y = ∞. The Hamiltonian operator is essentially self-adjoint if the LPC is verified both at y = −∞ and at y = ∞ (cf. [27], theorem 5.7). In the case at hand, we can refer to corollary to theorem 6.8 (p. 99) of [27], both for y → −∞ and for y → ∞. Thus, the Dirac operator defined on C ∞ 0 (r + , r c ) 2 is essentially self-adjoint on the RN-dS black hole background. We first show that the essential spectrum of the unique self-adjoint extension of the Dirac Hamiltonian [still indicated with H red ] coincides with R both in the case of non-extremal black holes and in the extremal case. This feature is expected in presence of a black hole horizon and is well-known in the case of scalar particles [28], and also verified in the case of Dirac particles on Kerr-Newman black hole manifold (see e.g. [29,30]). We confirm that it is verified also in the present cases. From a physical point of view, it also implies that there is no room for isolated eigenvalues, and then that there is no "standard" bound state (in the sense that a charged particle with charge opposite to the charge of the black hole cannot form a bound state which is analogous to the bound state an electron forms around an atomic nucleus). Moreover, a finer analysis allows also to conclude that, both in the non-extremal case and in the extremal one, the point spectrum is empty, and then no quantum bound state, i.e. no possibility to obtain a normalizable time-periodic solution of the Dirac equation exists. A. Essential spectrum One expects that, in presence of an event horizon, i.e. of a so-called ergosurface, the mass gap vanishes and that the continuous spectrum includes the whole real line. We recall that qualitative spectral methods for the Dirac equation (see e.g. [26,27]) have been applied to Dirac fields on a black hole background in [4,29]. In order to verify this property, we adopt the decomposition method [27]. We split the interval (r + , r c ) at a inner point r 1 and then consider the formal differential expression (7) restricted to the sub-intervals (r + , r 1 ] and [r 1 , r c ). Roughly speaking, we refer to the aforementioned expressions as to the "restriction of the Hamiltonian H red to the interval (r + , r 1 ] and to the interval [r 1 , r c )" and write e.g. H red | [r1,rc) for the latter. We limit ourselves to consider the latter restriction, which is relative to the novel feature of space-time, with respect to previously discussed cases, represented by the cosmological horizon. In the tortoise-like coordinate y one finds a potential P such that P = − √ f µ + e Q r k √ f r k √ f r √ f µ + e Q r and it holds lim y→∞ P (y) = P 0 = Φ c 0 0 Φ c which is in diagonal form and whose eigenvalues coincide. We apply theorem 16.6 p. 249 of Ref. [27], which implies that, if ν − , ν + , with ν − ≤ ν + , are the eigenvalues of the matrix P 0 , then {R − (ν − , ν + )} ⊂ σ e (H red | [y(r1),∞) ) if lim y→∞ 1 y y ǫ0 dt|P (t) − P 0 | = 0,(11) where | · | stays for any norm in the set of 2 × 2 matrices (we choose the Euclidean norm). In our case one has to find the limit as y → ∞ for the following expression: 1 y r(y) r1 dr 1 h(r) 1 √ r c − r 2µ 2 h(r) + 2 Φ 2 + (r c − r) + k 2 h(r) 1 r 2 ,(12) where we put h(r) = f (r) rc−r . Both in the non-extremal case and in the extremal one, the above integral is finite as r → r c i.e. as y → ∞, and then the limit is zero (we recall that the difference between non-extremal case and extremal one from this point occurs when studying the limit as r → r + , i.e. as y → −∞. In the extremal case r + = r − the corresponding integral diverges but a trivial use of the l'Hospital's rule allows to find that the aforementioned limit is still zero). As a consequence, we can state that σ e (H red ) = R.(13) A completely analogous conclusion can be stated for the restriction to (−∞, r 1 ), and again the essential spectrum contribution one finds is R both in the non-extremal case and in the extremal one. B. Absence of states of the point spectrum Qualitative spectral analysis of the reduced Hamiltonian in the non-extremal case can be implemented by means of theorems in [31] or also in [27]. In [22] a proof was given for the more general case of the Dirac equation in a Kerr-Newman-de Sitter black hole background, again in the case r + < r c . For the sake of completeness, we sketch the strategy and also provide some details involving some differences with respect to [22]. We note that, given a decomposition point r 1 ∈ (r + , r c ), we can introduce the following self-adjoint operators H hor and H c on the respective domains D(H hor ) = { g ∈ L 2 [(r + , r 1 ), 1/f (r) dr] 2 , g is locally absolutely continuous; g 1 (r 1 ) = 0; H hor g ∈ L 2 [(r + , r 1 ), 1/f (r) dr] 2 }, and analogously D(H c ) = { g ∈ L 2 [(r 1 , r c ), 1/f (r) dr] 2 , g is locally absolutely continuous; g 1 (r 1 ) = 0; H c g ∈ L 2 [(r 1 , r c ), 1/f (r) dr] 2 }. According to the decomposition method applied to the absolutely continuous spectrum, one has σ ac (H red ) = σ ac (H hor ) ∪ σ ac (H c ) (cf. e.g. [22]). Theorem 16.7 in [27] allows to conclude that, in the non-extremal case, H hor has absolutely continuous spectrum in R − Φ + , where Φ+ e is the electrostatic potential at the black hole event horizon, and that H c has absolutely continuous spectrum in R − Φ c , where Φc e is the electrostatic potential at the cosmological horizon. Moreover, for eQ > 0 it has to hold Φ c < Φ + , and Φ c > Φ + for eQ < 0, due to the inequality r c > r + . In any case, Φ c = Φ + occurs, and this is an interesting fact in the light of the study of the pair-creation process, as we shall see in the following section. As to the spectral properties of the reduced Hamiltonian, one can easily infer that the spectrum is absolutely continuous in R (indeed, the above analysis allows to conclude that the spectrum is absolutely continuous in R − {{Φ c } ∩ {Φ + }} but of course the latter set coincides with R). As to the extremal case, one can again refer to theorem 16.7 in [27] for H c and to theorem 1 in [31] for H hor to conclude that the spectrum is absolutely continuous in R − {{Φ c } ∩ {Φ + }}. Again, the latter set is R. IV. NARIAI SOLUTION We take into consideration the special case of the so-called charged Nariai solution [32,33], which is a black hole solution with r − < r + = r c . As known, the metric (1) is no more valid and a suitable transformation is necessary. It can be shown that the manifold can be described by ds 2 = 1 A (− sin 2 (χ)dψ 2 + dχ 2 ) + 1 B (dθ 2 + sin 2 (θ)dφ 2 )(14) with ψ ∈ R, χ ∈ (0, π) and where B = 1 [32,33]. We note that there is no warping factor in the metric between the "radial" part and the S 2 part. For an electrically charged black hole we can choose A i = −Q B A cos(χ)δ 0 i . We study the Dirac equation as in [23,24]. With the same notation as in [23], we introduce the so-called generalized Dirac matrices such that {γ i , γ j } = 2g ij : 2Q 2 1 − 1 − 12 Q 2 L 2 and A = 6 L 2 − B are constants such that A B < 1γ 0 = sin(χ) √ Aγ 0 γ 0 = − √ A sin(χ)γ 0 γ 1 = 1 √ Aγ 1 γ 1 = √ Aγ 1 γ 2 = 1 √ Bγ 2 γ 2 = √ Bγ 2 γ 3 = sin(θ) √ Bγ 3 γ 3 = √ B sin(θ)γ 3 ,(15) whereγ i , i = 0, 1, 2, 3 are the usual Dirac matrices in Minkowski space. The Dirac equation is [γ k (∂ k − Γ k ) − µ]Ψ = 0,(16) with Γ k = − 1 4 γ j (∂ k γ j − γ l Γ l jk ) + ieA k .(17) One finds the following non-vanishing Christoffel symbols Γ 0 01 = cot(χ), Γ 1 00 = sin(χ) cos(χ), Γ 2 33 = − sin(θ) cos(θ), Γ 3 23 = cot(θ). Then, due to our choice for A i , we get Γ 0 = − 1 2 cos(χ)γ 0γ1 + ieA 0 , Γ 1 = 0, Γ 2 = 0, Γ 3 = 1 2 cos(θ)γ 2γ3 . Then the Dirac equation becomes − √ A sin(χ)γ 0 (∂ ψ − ieA 0 ) + √ Aγ 1 (∂ χ + 1 2 cot(χ))+ √ B γ 2 ∂ θ + 1 2 cot(θ) +γ 3 1 sin(θ) ∂ φ − µ Ψ = 0.(18) By posing Ψ = (sin(χ)) −1/2 (sin(θ)) −1/2 ζ, we eliminate the terms proportional to cot(θ) and to cot(χ) in the previous equation. We now consider a static solution with ζ = exp(−iωψ)η(χ, θ, φ). Then a trivial manipulation of the Dirac equation leads to the following eigenvalue equation: Hη = ωη,(19) with H = −iγ 0γ1 sin(χ)∂ χ − eA 0 I 4 + B A sin(χ)γ 1 K + iγ 0 µ √ A sin(χ).(20) I 4 stays for the 4 × 4 identity matrix and K is the following operator: K = −iγ 1γ0γ2 ∂ θ − iγ 1γ0γ3 1 sin(θ) ∂ φ(21) which commutes with H and whose eigenvalues are k ∈ Z − 0 [23,24]. By restricting H to eigenspaces of K and by choosingγ 0 = iI 2 O 2 O 2 −iI 2 ,γ 1 = O 2 I 2 I 2 O 2 (I 2 is the 2 × 2 identity matrix, O 2 is the 2 × 2 zero matrix) , we obtain the reduced Hamiltonian H k = − sin(χ)∂ χ O 2 −I 2 I 2 O 2 + eQ B A cos(χ) I 2 O 2 O 2 I 2 + B A sin(χ)k O 2 I 2 I 2 O 2 − µ √ A sin(χ) I 2 O 2 O 2 −I 2 = h k ⊗ I 2 ,(22) where h k =   eQ B A cos(χ) − µ √ A sin(χ) sin(χ)∂ χ + B A sin(χ)k − sin(χ)∂ χ + B A sin(χ)k eQ B A cos(χ) + µ √ A sin(χ)  (23) The coordinate transformation x = log(tan( χ 2 )) ←→ χ = 2 arctan(exp(x))(24) is such that x ∈ R and, furthermore, h k becomes h k = 0 ∂ x −∂ x 0 + P (χ(x)),(25) where P (χ) =   eQ B A cos(χ) − µ √ A sin(χ) B A sin(χ)k B A sin(χ)k eQ B A cos(χ) + µ √ A sin(χ)   .(26) h k is formally self-adjoint in L 2 [R, dx] 2 and it is essentially self-adjoint in C ∞ 0 (R) 2 , as follows from corollary to theorem 6.8 (p. 99) of [27] (the limit point case occurs both at x = −∞ and at x = ∞). It is easy to show that the essential spectrum of h k coincides with R and the same is true for the absolutely continuous spectrum. The latter claim can be checked by following the ideas displayed in sect. III B. See Appendix A for more details. V. ULTRACOLD CASE There is still a sub-case to be taken into account. It corresponds to the so-called ultracold case [32], where the three horizons coincide: r − = r + = r c . Also in this case the metric (1) is no more valid, and a suitable limit has to be considered [32]. Actually, one can introduce two different metrics for the ultracold case. As a consequence, also our analysis is split into two parts. A. ultracold I A first metric [32] is ds 2 = −χ 2 dψ 2 + dχ 2 + 1 2Λ (dθ 2 + sin 2 (θ)dφ 2 ),(27) with χ ∈ (0, ∞) and ψ ∈ R. One gets Γ 0 01 = 1 χ , Γ 1 00 = χ, Γ 2 33 = − sin(θ) cos(θ), Γ 3 23 = cot(θ). The electromagnetic field strength is F = √ Λχdχ ∧ dψ, and we can choose A 0 = √ Λ 2 χ 2 and A j = 0, j = 1, 2, 3. We introduce γ 0 = χγ 0 γ 0 = − 1 χγ 0 γ 1 =γ 1 γ 1 =γ 1 γ 2 = 1 √ 2Λγ 2 γ 2 = √ 2Λγ 2 γ 3 = sin(θ) √ 2Λγ 3 γ 3 = √ 2Λ sin(θ)γ 3 ,(28) and then we obtain Γ 0 = − 1 2γ 0γ1 + ieA 0 , Γ 1 = 0, Γ 2 = 0, Γ 3 = 1 2 cos(θ)γ 2γ3 . Calculations which are strictly analogous to the ones performed in the Nariai case (with Ψ = exp(−iωψ) 1 √ χ √ sin(θ) ζ) and the variable change χ = exp(x), x ∈ R, lead to the following reduced Hamiltonian: h k = − e √ Λ 2 exp(2x) − µ exp(x) ∂ x + √ 2Λk exp(x) −∂ x + √ 2Λk exp(x) − e √ Λ 2 exp(2x) + µ exp(x) .(29) As in the Nariai case, h k is formally self-adjoint in L 2 [R, dx] 2 and it is essentially self-adjoint in C ∞ 0 (R) 2 , as follows from corollary to theorem 6.8 (p. 99) of [27]. The analysis of the spectrum can be pursued by means of the decomposition method applied to the absolutely continuous spectrum, and one can again conclude that the absolutely continuous spectrum of the self-adjoint extension of h k on R coincides with the whole real line. See Appendix B for details. B. ultracold II The second allowed metric [32] for the ultracold case is ds 2 = −dψ 2 + dx 2 + 1 2Λ (dθ 2 + sin 2 (θ)dφ 2 ),(30) with x ∈ R and ψ ∈ R. One gets Γ 2 33 = − sin(θ) cos(θ), Γ 3 23 = cot(θ). The electromagnetic field strength is F = − √ Λdψ ∧ dx, and we can choose A 0 = √ Λx and A j = 0, j = 1, 2, 3. We introduce γ 0 =γ 0 γ 0 = −γ 0 γ 1 =γ 1 γ 1 =γ 1 γ 2 = 1 √ 2Λγ 2 γ 2 = √ 2Λγ 2 γ 3 = sin(θ) √ 2Λγ 3 γ 3 = √ 2Λ sin(θ)γ 3 ,(31) and then we obtain Γ 0 = ieA 0 , Γ 1 = 0, Γ 2 = 0, Γ 3 = 1 2 cos(θ)γ 2γ3 . Again calculations as above (with Ψ = exp(−iωψ) 1 √ sin(θ) ζ) lead to the following reduced Hamiltonian: h k = −e √ Λx − µ ∂ x + √ 2Λk −∂ x + √ 2Λk −e √ Λx + µ .(32) Also in this case, h k is formally self-adjoint in L 2 [R, dx] 2 and it is essentially self-adjoint in C ∞ 0 (R) 2 . As in the previous case, the decomposition method applied to the absolutely continuous spectrum allows to draw the conclusion that the spectrum of the self-adjoint extension of the Hamiltonian (32) is absolutely continuous and coincides with R. See Appendix C for details. VI. PAIR CREATION AND LEVEL-CROSSING We follow the Ruffini-Damour-Deruelle [14,15,16,17] approach, which was summarized in a previous paper [18]. Herein, we limit ourselves to recall some very basic properties, focusing on the RN-dS case (the other cases can be dealt with analogously). In this approach one introduces effective potentials E ± 0 (r) for the positive and negative energy states respectively; they represent the classical turning points for the particle motion and lead to the definition of the so-called effective ergosphere. These potentials enter the Hamilton-Jacobi (HJ) equation for a classical particle. They can be interpreted also at the quantum level, as in [17]. In particular, they indicate the regions of level-crossing between positive and negative energy states [15,16]. In the case of the Dirac equation, it is known that the HJ equation corresponds to a WKB approximation to the Dirac equation at the lowest order [34,35]. Variable separation in the quantum case allows to obtain an obvious improvement of the semi-classical formulas, amounting in replacing the classical value of the angular momentum with the quantum eigenvalues of the corresponding quantum operator [18]. We limit ourselves to recall one of the main point of our analysis of the Dirac Hamiltonian in [18]. The key-observation resides in the following fact: if one consider the potential term in the Dirac Hamiltonian V (r) = p 11 (r) p 12 (r) p 21 (r) p 22 (r) , and formally calculates the eigenvalues of the above matrix, which are found by solving (p 11 (r) − λ)(p 22 (r) − λ) − p 12 (r)p 21 (r) = 0; then, defining S(r) ≡ (p 11 (r) + p 22 (r)) 2 − 4p 11 (r)p 22 (r) + 4p 12 (r)p 21 (r) one finds λ ± (r) = 1 2 (p 11 (r) + p 22 (r) ± S(r)) , and moreover one gets λ ± (r) = E ± 0 (r),(35) i.e., the semi-classical potentials introduced by Damour-Deruelle-Ruffini coincide with the eigenvalues of the matrix potential term in the Dirac Hamiltonian. Moreover, the square of the classical angular momentum term is replaced by the square of the eigenvalues k = ±(j + 1/2) for the quantum angular momentum, and one obtains E ± 0 (r) = eQ r ± f (r)(µ 2 + k 2 r 2 ).(36) A. Level crossing in the RN-dS case Level-crossing amounts to the presence of overlap between the range of E + 0 and the range of E − 0 , signalling the possibility of a tunneling between positive energy states and negative energy ones; the latter phenomenon is in turn interpreted as pair-creation at the barrier potential, and is strictly related to the Klein paradox (incidentally, it could be called for this reason also Klein effect [29]). There is a peculiar property in the case of RN-dS black holes: due to the inequality Φ c = Φ + , an overlap is always present, indeed one gets lim r→r+ E ± 0 (r) = Φ + ,(37) and lim r→rc E ± 0 (r) = Φ c . Assuming for definiteness eQ > 0, one finds E − 0 (r + ) = Φ + > Φ c = E + 0 (r c ),(39) which proves the above claim. Moreover, it is easy to realize that level crossing occurs for energy ω such that Φ c = min E + 0 (r) ≤ ω ≤ max E − 0 (r) = Φ + .(40) For eQ < 0 the overlap still exists, being E − 0 (r c ) = Φ c > Φ + = E + 0 (r + ), and (40) changes accordingly: Φ + ≤ ω ≤ Φ c . It is to be immediately pointed out that this overlap is not enough to conclude that a sensible pair-creation process occurs in the given background. Indeed, the potential barrier to be overcome by the negative energy particle can be as large as the whole external region of the spacetime. As a consequence, no efficient process can be expected on this ground in general, and further conditions taking into account the effective largeness of the barrier have to be looked for. Notice also that E + 0 , as a function of µ 2 , is increasing, and the same is true for its dependence on k 2 ; both these properties are reversed in the case of E − 0 , which is in fact decreasing. As a consequence, one expects a more difficult level-crossing for increasing values of µ 2 or k 2 . We start giving sample-examples for a non-extremal black hole manifold. In figure 1, we keep fixed the geometric background parameters L, M, Q and the charge e of the Dirac particle, and consider two sample values of the mass. On the left, we find the former phenomenon described above, i.e. a level-crossing with a barrier as large as ∼ r c − r + . On the right-hand case, which is obtained by considering a lower fermion mass, we obtain instead a level-crossing associated with a much smaller extent of the barrier. The latter case is expected to be involved in a effective phenomenon of pair creation at the barrier. In figures 2 and 3 a further non-extremal case is displayed, with rc r+ ∼ 1.01 (to be compared with rc r+ ∼ 10.4 of the example displayed in figure 1). In figure 3 we show the details of the potentials near the horizons. In figure 4, an extremal case is also shown. In figure 5 we display the so-called lukewarm case [32,36], which is such that the same temperature occurs in the case of the cosmological horizon and of the black hole event horizon, still with r + < r c (this happens for Q = M ). It is displayed for the sake of completeness, even if from the point of view of the given phenomenon no peculiar behavior is expected with respect to the cases explored in figures 1 and 2. Notice that, from a physical point of view, the phenomenon of pair creation by a charged black hole has been related to the Schwinger calculation of pair creation by a homogeneous electric field [7]. The highest value of the electrostatic potential, which corresponds to the highest intensity of the electrostatic field, occurs near the black hole event horizon, hence one could naively expect that the standard condition Φ 2 + > µ 2 , i.e. Q r + 2 > µ 2 e 2 ,(41) which is enough for the standard Reissner-Nordström case [7,8,15], is also qualitatively relevant in the present one, at least in some approximation. We recall that, in the case of the lightest known charged particle, i.e. the electron, one has µe e ∼ 10 −21 . The above condition is not necessary for the existence of a sensible level-crossing. A proof of its sufficiency in the extremal case can be given under the hypothesis k 2 r 2 + ≪ 1, which allows to neglect the angular momentum contribution in the potentials, and also for r + ≪ r c . In the extremal case, it is easy to show that the condition to be satisfied is that dE + 0 dr (r + ) dE − 0 dr (r + ) > 0, which means that both potentials are increasing or decreasing near r = r + . One has to impose dE + 0 dr (r + ) dE − 0 dr (r + ) = 1 r 2 + Φ 2 + − r 2 c L 2 1 − r + r c 1 + 3 r + r c µ 2 + k 2 r 2 + > 0,(42) and then, under the above hypotheses k 2 r 2 + ≪ 1 and r + ≪ r c , one obtains dE + 0 dr (r + ) dE − 0 dr (r + ) ≃ 1 r 2 + Φ 2 + − µ 2 r 2 c L 2 > 0,(43) for which, being rc L < 1, the aforementioned condition (41) is sufficient. In figure 6 we show an example of extremal black hole where a significant level crossing occurs but condition (41) is not satisfied. Explicit evaluations of the transmission coefficient which is related to the pair-creation phenomenon (discharge) can be given e.g. in a WKB approximation, as pointed out in the original literature [7,8,15]. See also [18] for a short summary. We do not delve into quantitative evaluation herein in the RN-dS case but limit ourselves to some estimates in the cases which will be analyzed in the following subsections. The upper straight line represents eQ/r+, the lower is eQ/rc. The upper potential is E + 0 (r), the lower one is E − 0 (r). Level-crossing occurs, but the potential barrier is as large as the whole spacetime region at hand. On the right, the only change with respect to the previous figure stays in the smaller value µ = 0.01 of the fermion mass. The upper straight line represents eQ/r+, the lower is eQ/rc. The upper potential is E + 0 (r), the lower one is E − 0 (r). Level-crossing occurs in this case with a much smaller extent of the potential barrier with respect to rc − r+. √ 190/ √ 31791, which are such that rc = 100, r+ = 99 and r− = 10. Moreover, we choose e = 1, k = 1, and the same sign for the black hole charge and for the particle charge. The upper straight line represents eQ/r+, the lower is eQ/rc. The figure on the right displays the potentials for µ = 1, and shows that a very large potential barrier occurs. In the figure on the right one has µ = 0.01 and a very narrower potential barrier. Note that in the latter case eQ/r+ > µ holds, whereas in the former the opposite inequality is implemented. B. the Nariai case The potentials E ± 0 (χ) in the Nariai case are E ± 0 (χ) = eQ B A cos(χ) ± µ 2 A + k 2 B A sin(χ).(44) Also in this case level-crossing is always present, being E + 0 (π) < E − 0 (0) for eQ > 0 and E + 0 (0) < E − 0 (π) for eQ < 0. Level-crossing occurs for energies ω such that E + 0 (π) ≤ ω ≤ E − 0 (0) in the former case and for E + 0 (0) ≤ ω ≤ E − 0 (π) in the latter one. Again, the largeness of the potential barrier depends on the choice of the parameters. See figure 7. An estimation of the transmission coefficient can be given in the WKB approximation; one obtains [15] |T W KB (5)). Particle parameters e = 1, k = 1 are kept fixed, whereas it holds µ = 1 on the left and µ = 0.01 on the right. Level-crossing is more effective in the latter case, and it occurs without any bump near the black hole event horizon. where x stays for the coordinate defined in (24)) and where we have stressed the dependence on the energy ω of T W KB ω and of ω | 2 = exp(−2 barrier dx Z ω ),(45)Z ω = B A k 2 + µ 2 A sin 2 (χ(x)) − (ω − eQ B A cos(χ(x))) 2 .(46) Let us introduce: µ 2 k = B A k 2 + µ 2 A ,(47) and also E m = |Q| B A ,(48) which corresponds to 1 A times the maximum value for the modulus of the electrostatic field. Note that positivity of Z ω requires ω 2 < µ 2 k + e 2 E 2 m . We obtain |T W KB ω | 2 = exp −2π|e|E m ( 1 + µ 2 k e 2 E 2 m − 1) ,(49) which does not depend on ω. See Appendix D for more details. At the leading order as µ 2 k ≪ e 2 E 2 m one is also able to recover the approximation |T W KB ω | 2 ∼ exp −π µ 2 k |e|E m ,(50) which shares a nice resemblance with the WKB estimate of the transmission coefficient related to the pair creation process in a uniform constant electrostatic field. Cf. e.g. [15]. C. the ultracold cases We obtain E ± I (χ) = − e √ Λ 2 χ 2 ± 2Λk 2 + µ 2 χ(51) in the case of the first kind of ultracold solution (27), and level-crossing requires that ω ≤ 0 for e > 0 and ω ≥ 0 for e < 0. In the case (30) one finds E ± II (x) = −e √ Λx ± 2Λk 2 + µ 2 ,(52) It is evident that for any ω ∈ R one obtains level-crossing. In figures 8 and 9 level-crossing is displayed for both the ultracold I and the ultracold II cases. We choose Λ = 0.01, k = 1, e = 1 in both cases and µ = 1 on the left and µ = 0.01 on the right. We display only a part of the full plot (but qualitatively all relevant information is given). The potentials converge both to zero as χ → 0 and to −∞ as χ → +∞. As to a WKB approximation for |T ω | 2 , we get in both cases |T W KB ω | 2 = exp − π(µ 2 + 2Λk 2 ) |e| √ Λ ,(53) which again is independent from ω. Notice that, by keeping into account that for the electrostatic field one finds E = √ Λ, and with the replacement µ 2 → µ 2 k = µ 2 + 2Λk 2 , one obtains again a formula which is very similar to the one which is associated with the description of the pair creation in a uniform constant electric field in flat space-time in the same approximation, and this time no requirement about the smallness of the ratio between µ 2 k and |e|E is imposed. A deeper analysis for the special cases ultracold II, ultracold I and Nariai is in progress [37]. Some considerations about the problem of the choice of the quantum state playing the role of vacuum are addressed. If one were to assume that the positive and negative frequencies associated with the Hamiltonian define the vacuum, one would end up with the so-called Boulware vacuum, which is viable as the real vacuum only in the ultracold II case, where the background temperature is zero [36]. For a Reissner-Nordström-de-Sitter black hole background, a further difficulty arises due to the presence of both a cosmological temperature and of a black hole temperature in the non-extremal and non-lukewarm cases, involving a true non-equilibrium situation. A simpler case is the extremal one, because of the occurrence of a single temperature for the given manifold, and the same considerations can be made in the case of the lukewarm solution and in the Nariai case. We are not aware of a rigorous construction for quantum field theory on the given backgrounds. One could expect that, in presence of a single non-zero temperature, suitable analyticity requirements for the fields on the extended manifold can lead to the thermal state as in [7], and that "heating up" the Boulware vacuum (as it can be rigorously done in the case of a scalar field on a Schwarzschild black hole background [38]), taking into account the complication of the level-crossing displayed above, could be a viable solution. The instability associated with the pair-creation process induced by the presence of the electrostatic field generated by the black hole still remains, and gives rise to the process of discharge we have taken into account. Thermality of the physical state modifies such a pair-creation process but the transmission coefficient |T | we have calculated for a vacuum situation still plays a role, as it is shown e.g. in Ref. [39] for the case of quantum electrodynamics in flat spacetime (see also [40]). One obtains that for an initial thermal state pair-creation is still proportional to |T | 2 with a multiplicative factor depending on the temperature. We shall come back to this topic in [37]. The general RN-dS case is evidently more tricky and challenging, and requires a non-equilibrium framework. VII. CONCLUSIONS We have shown that, on the background of a charged black hole in de-Sitter space, massive and charged Dirac particles are described by an Hamiltonian operator which is well-behaved both on the cosmological horizon and on the black hole horizon. We have also inferred that in all cases the point spectrum of the Hamiltonian is empty, and then there is no bound state and no normalizable time-periodic solution of the Dirac equation. The presence of two different horizons allows a simpler analysis even in the extremal case. Moreover, the same occurrence of two event horizons involving different values of the electrostatic potential is at the root of the presence in any case of level-crossing between positive energy states and negative energy ones. This fact per se is not enough for claiming that a sensible pair creation effect is present on the given manifold, due to the fact that a priori the potential barrier to be overcome can be very large (even large as almost the whole external manifold in the RN-dS case and in the Nariai one) and then the effect is expected to be very suppressed. Nevertheless, in all cases examples can be found where the barrier is of much more reduced extent, in such a way to allow a physical ground to the pair-creation phenomenon. Some estimates in WKB approximation have been given for the transmission coefficient which is related to the pair-creation process [7,8,15] in the case of the Nariai geometry and in the ultracold ones. We introduce a decomposition pointd ∈ R and also the following self-adjoint operators H −∞ and H ∞ on the respective domains D(H −∞ ) = { g ∈ L 2 [(−∞,d], dx] 2 , g is locally absolutely continuous; g 1 (d) = 0; H −∞ g ∈ L 2 [(−∞,d], dx] 2 }, and analogously D(H ∞ ) = { g ∈ L 2 [[d, ∞), dx] 2 , g is locally absolutely continuous; g 1 (d) = 0; H ∞ g ∈ L 2 [[d, ∞), dx] 2 }. We define P − := lim x→−∞ P (χ(x)) and P + := lim x→−∞ P (χ(x)), where the P (χ(x)) is the potential (26), and write P = P ∓ + (P − P ∓ ). (A1) The first term on the right hand side of eq. (A1) is obviously of bounded variation, whereas the latter term is such that |P − P + | ∈ L 1 [[d, ∞), dx] and |P − P − | ∈ L 1 [(−∞,d], dx] respectively. Moreover, notice that P ∓ = ±eQ B A 0 0 ±eQ B A .(A2) As a consequence, in both cases the hypotheses of Theorem 16.7 in [27] are implemented, and one is allowed to conclude that H −∞ has absolutely continuous spectrum in R − {eQ B A }, and that H ∞ has absolutely continuous spectrum in R − {−eQ B A }. Then the absolutely continuous spectrum of the self-adjoint extension of the Hamiltonian operator (25) is R. APPENDIX B: ABSOLUTELY CONTINUOUS SPECTRUM IN THE ULTRACOLD I CASE Let us introduce a self-adjoint extension H −∞ of the formal differential expression (29) on the interval (−∞, 0] (0 is the decomposition point). Notice that the potential term in (29) is P (x) = − e √ Λ 2 exp(2x) − µ exp(x) √ 2Λk exp(x) √ 2Λk exp(x) − e √ Λ 2 exp(2x) + µ exp(x) (B1) and it is such that lim x→−∞ P (x) = O. Moreover, it is easy to show that |P (x)| ∈ L 1 [(−∞, 0], dx]. As a consequence, Theorem 16.7 of [27] can be applied and the given self-adjoint extension has absolutely continuous spectrum in R − {0}. It is also true that 0 is not an eigenvalue for H −∞ , because no normalizable solution exists as a consequence of Levinson theorem (whose applicability is related to the property that each entry in P (x) is integrable near x = −∞; cf. [41], p.8). Cf. also [4] for the Kerr-Newman case. Thus σ ac (H −∞ ) = R. As a consequence (cf. e.g. [22]), also the absolutely continuous spectrum of the self-adjoint extension of h k on R coincides with the whole real line. APPENDIX C: ABSOLUTELY CONTINUOUS SPECTRUM IN THE ULTRACOLD II CASE Let us notice that the equation h k g = λ g, by putting g(x) = w 1 (x)u 1 (x) 1 w1(x) u 2 (x) , where w 1 (x) = exp(−2 √ 2Λkx), is equivalent to the following equation: d dx u(x) = 0 (−e √ Λx − α)w 1 (x) (e √ Λx + β) 1 w1(x) 0 u(x),(C1) where α = λ − µ and β = λ + µ. |e|x is such that lim x→+∞ ∆ = 0 and d∆ dx ∈ L 1 [[c, +∞), dx]. Moreover, α1 p11 , α2 p21 , p12 p11 , p22 p21 are long-range (in the sense that they vanish as x → +∞ and their derivative is in L 1 [[c, +∞), dx]). Then the hypotheses of Theorem 2 in [31] are implemented, which means that the absolutely continuous spectrum is R. where Z ω is given in (46) and χ(x) = 2 arctan e x (cf. (24)). Using cos χ(x) = − tanh x, sin χ(x) = 1 cosh x , we can rewrite barrier Z ω dx = barrier B A k 2 + µ 2 A − ω cosh x + eQ B A sinh x 2 dx cosh x = barrier B A k 2 + µ 2 A e −2x − 1 4 [ω + eQ B A ] + [ω − eQ B A ]e −2x 2 2e 2x 1 + e 2x dx.(D2) If we define µ 2 k as in (47) and ω ± = ω ± eQ B A , and change variable to z = e 2x we get FIG. 1 : 1On the left, we display the level-crossing in the case of a non-extremal RN-dS black hole, with L = 1000, M = 50, Q = 30, µ = 1, e = 1, k = 1. The particle and the black hole have charges with the same sign. One finds r+ ∼ 90.842, rc ∼ 946.214, r− ∼ 9.999 and r0 ∼ −1047.056. FIG. 2 : 2Level-crossing in the case of a non-extremal RN-dS black hole having L = √ 31791, M = 1193005/31791 and Q = 330 FIG. 3 : 3Level-crossing in the case of a non-extremal RN-dS black hole having L = √ 31791, M = 1193005/31791 and Q = 330 √ 190/ √ 31791, as in the previous figure, with e = 1, k = 1. We display on the left the presence of a bump near the black hole horizon in the case of E + 0 . Analogously, on the right we show the behavior of the potentials very near the cosmological horizon. FIG. 4 : 4Level-crossing in the case of an extremal RN-dS black hole, with L = 1000, Q = 10 and then M ≃ 9.999499 (see eq. FIG. 5 :FIG. 6 :FIG. 7 : 567Level-crossing in the case of a lukewarm RN-dS black hole, with L = 1000, M = 50, Q = 50, e = 1, k = 1, and with µ = 1 on the left and µ = 0.01 on the right. One finds r+ ∼ 52.786 and rc ∼ 947.214. Level-crossing is qualitatively similar to the one displayed in figure 1. Level-crossing in the case of an extremal RN-dS black hole, with L = 50, Q = 14.4336845607765725, which are such that M = 13.608225263871805121, r+ ≃ 20.380115 and rc ≃ 20.476977. With µ = 0.01, e = 0.01, k = 1 one gets Φ+ ≃ 0.708µ, which violates (41). Level-crossing in the case of a Nariai solution.The potentials E ± 0 (χ) are plotted, with L = 1000, Q = 80. Particle parameters e = 1, k = 1 are kept fixed; particle mass is chosen to be µ = 1 on the left and µ = 0.01 on the right. Also in the latter case, a bump is present for E + 0 near χ = 0 and a hollow occurs for E − 0 near χ = π. FIG. 8 : 8Level-crossing in the case of the ultracold I metric. FIG. 9 : 9Level-crossing in the case of the ultracold II metric. We choose Λ = 0.01, k = 1, e = 1 in both cases and µ = 1 on the left and µ = 0.01 on the right. We display only a part of the full plot, which shows of course a linear behavior. III. QUALITATIVE SPECTRAL PROPERTIES AND TIME-PERIODIC SOLUTIONS IN THE CASE r+ < rc. APPENDIX A: ABSOLUTELY CONTINUOUS SPECTRUM IN THE NARIAI CASE Then, by restricting our attention to the interval [c, ∞), with c > 0, and by applying Theorem 2 in [31] the result follows. Let us define, according to the notations of [31], p 1 = (−e √ Λx + µ) exp(−2 √ 2Λkx) =: p 11 + p 12 , p 2 = (−e √ Λx − µ) exp(2 √ 2Λkx) =: p 21 + p 22 , α 1 = exp(−2 One obtains p 11 p 21 = e 2 Λx 2 > 0 and +∞ c dx √ p 11 p 21 = +∞. Moreover both it is sufficient that one of them diverges [31]). Moreover, if η = ( p21 p11 ) 1/4 , then ∆ := dη dx 1 η √ p11p21 =√ 2Λkx) and α 2 = exp(2 √ 2Λkx); p 11 := −e √ Λx exp(−2 √ 2Λkx) and p 21 := −e √ Λx exp(2 √ 2Λkx). +∞ c dxα 1 p21 p11 and +∞ c dxα 2 p11 p21 diverge (√ 2Λ|k| APPENDIX D: EVALUATION OF THE NARIAI TRANSMISSION INTEGRALWe need to evaluatebarrier Z ω dx (D1) AcknowledgmentsThis research was supported in part by Perimeter Institute for Theoretical Physics. We thank Francesco Dalla Piazza for his help in drawing figures 10 and 11.where 0 ≤ z − < z + are the two real solutions ofNote that such solutions exist if the discriminant of the polynomial is positive. This givesThe integral is indeed well defined for − µ 2 k + e 2 Q 2 B 2 A 2 < ω < µ 2 k + e 2 Q 2 B 2 A 2 . Note however that, only the region of level crossing −|eQB/A| ≤ ω ≤ |eQB/A|, corresponding to ω + ω − ≤ 0, is relevant for computing the transmission coefficient. Let us first distinguish the "generic case" ω + ω − = 0 from the "particular case" ω + ω − = 0. In order to compute the integral in the generic case, the residue method is used. Let us cut the complex plane C along the segment [z − , z + ] ⊂ R + (note that z − > 0 in this case), so defining a Riemann sheet for the square rootIn particular we choose the phase of (z + − z)(z − z − )ω 2 + /4 to be 0 (modulo 4π) along the lower border of the cut, so that it will be 2π (modulo 4π) along the upper border. Thus, it makes sense to take the closed path Γ = s ↓ ∪ c + ∪ s ↑ ∪ c − as infigure 10. When the radii of the two circles c ± are set to zero, Γ approaches the cut without crossing any singularity so that the value of the integraldoes not change. In this limit, the contributions from the circles vanish, whereasand being the phase of f equal to 0 on the lower cut and to π on the upper cut, we see thatTo compute the integral I, let us blow up Γ, without crossing the poles of the integrand (which are z = 0, −1, ∞) like infigure 11.FIG. 11: Blow up of the path ΓWe see that (cf. figure(11))where we used c r (z) to indicate the counterclockwise oriented circle with center z and radius r. Note that if we take the change of variable z = 1/t in the last integral, we have [42] c R (0) −→ −c 1/R (0) so thatTo compute the square roots we need to specify their phases. This is easily done looking at the Riemann sheet. 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{'abstract': 'We consider the behavior of massive Dirac fields on the background of a charged de-Sitter black hole. All black hole geometries are taken into account, including the Reissner-Nordström-de-Sitter one, the Nariai case and the ultracold case. Our focus is at first on the existence of bound quantum mechanical states for the Dirac Hamiltonian on the given backgrounds. In this respect, we show that in all cases no bound state is allowed, which amounts also to the non-existence of normalizable time-periodic solutions of the Dirac equation. This quantum result is in contrast to classical physics, and it is shown to hold true even for extremal cases. Furthermore, we shift our attention on the very interesting problem of the quantum discharge of the black holes. Following Damour-Deruelle-Ruffini approach, we show that the existence of level-crossing between positive and negative continuous energy states is a signal of the quantum instability leading to the discharge of the black hole, and in the cases of the Nariai geometry and of the ultracold geometries we also calculate in WKB approximation the transmission coefficient related to the discharge process. *', 'arxivid': '0810.1642', 'author': ["F Belgiorno \nDipartimento di Fisica\nF.N\nDipartimento di Fisica e Matematica\nUniversità di Milano\nUniversità dell'Insubria20133, 22100Milano, sezione di Milano, ComoItaly, and I.N.F.N., sezione di MilanoItaly, and I.N., Italy, Italy\n", "S L Cacciatori \nDipartimento di Fisica\nF.N\nDipartimento di Fisica e Matematica\nUniversità di Milano\nUniversità dell'Insubria20133, 22100Milano, sezione di Milano, ComoItaly, and I.N.F.N., sezione di MilanoItaly, and I.N., Italy, Italy\n"], 'authoraffiliation': ["Dipartimento di Fisica\nF.N\nDipartimento di Fisica e Matematica\nUniversità di Milano\nUniversità dell'Insubria20133, 22100Milano, sezione di Milano, ComoItaly, and I.N.F.N., sezione di MilanoItaly, and I.N., Italy, Italy", "Dipartimento di Fisica\nF.N\nDipartimento di Fisica e Matematica\nUniversità di Milano\nUniversità dell'Insubria20133, 22100Milano, sezione di Milano, ComoItaly, and I.N.F.N., sezione di MilanoItaly, and I.N., Italy, Italy"], 'corpusid': 53597106, 'doi': '10.1103/physrevd.79.124024', 'github_urls': [], 'n_tokens_mistral': 19213, 'n_tokens_neox': 16590, 'n_words': 10348, 'pdfsha': '30d67f807f82641abf130b913180c62cdeffcce0', 'pdfurls': ['https://arxiv.org/pdf/0810.1642v2.pdf'], 'title': ['Massive Dirac particles on the background of charged de-Sitter black hole manifolds', 'Massive Dirac particles on the background of charged de-Sitter black hole manifolds'], 'venue': []}

arxiv

Symmetry and geometry in generalized Higgs effective field theory -Finiteness of oblique corrections v.s. perturbative unitarity 16 Apr 2019 Ryo Nagai [email protected]†e-mail:[email protected] Institute for Cosmic Ray Research (ICRR) The University of Tokyo 277-8582KashiwaChibaJapan Department of Physics Tohoku University 980-8578SendaiMiyagiJapan Masaharu Tanabashi Department of Physics Nagoya University 464-8602NagoyaJapan Nagoya University 464-8602NagoyaJapan Koji Tsumura Department of Physics Kyoto University 606-8502KyotoJapan Yoshiki Uchida Department of Physics Nagoya University 464-8602NagoyaJapan Nagoya University 464-8602NagoyaJapan Symmetry and geometry in generalized Higgs effective field theory -Finiteness of oblique corrections v.s. perturbative unitarity 16 Apr 2019(Dated: April 17, 2019) 14 Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, 2 We formulate a generalization of Higgs effective field theory (HEFT) including arbitrary number of extra neutral and charged Higgs bosons (generalized HEFT, GHEFT) to describe non-minimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The one-loop divergences in the oblique correction parameters S and U can also be written in terms of the Killing vectors (symmetry) and the Riemann curvature tensor (geometry). It is found that perturbative unitarity of the scattering amplitudes involving the Higgs bosons and the longitudinal gauge bosons demands the flatness of the scalar manifold. The relationship between the finiteness of the electroweak oblique corrections and perturbative unitarity of the scattering amplitudes is also clarified in this language: we verify that once the tree-level unitarity is ensured, then the one-loop finiteness of the oblique correction parameters S and U is automatically guaranteed. * I. INTRODUCTION What is the origin of the electroweak symmetry breaking (EWSB)? In the standard model (SM) of particle physics, the EWSB is caused by a vacuum expectation value of a complex scalar field (SM Higgs field), which linearly transforms under the SU(2) W × U(1) Y electroweak gauge symmetry. The Higgs sector of the SM is constructed to be minimal, as it includes only a scalar boson (SM Higgs boson) and three would-be Nambu-Goldstone bosons eaten by massive gauge bosons after the EWSB. There are no cousin particles of Higgs in the SM. The scalar particle discovered by the ATLAS and CMS experiments in 2012 with the mass of 125 GeV [1,2] can now be successfully interpreted as the SM(-like) Higgs boson. The Higgs sector in the SM, however, does not ensure the stability of the EWSB scale against quantum corrections. In other words, the SM itself cannot explain why the EWSB scale is an order of 100 GeV, much smaller than its cutoff scale such as Planck (or Grand Unification) scale. The SM Higgs sector is therefore inherently incomplete. It should be extended. Many extensions/generalizations of the SM Higgs sector, such as Two Higgs Doublet Model , Composite Higgs Models [25][26][27][28][29][30][31][32][33][34], Georgi-Machacek Model [35][36][37][38], etc., have been proposed. The 125GeV Higgs boson accompanies extra Higgs particles in these scenarios. The Effective Field Theory (EFT) approach is widely used to study these beyond-SM (BSM) physics in a model independent manner. The physics below 1TeV can be described by the Standard Model Effective Field Theory (SMEFT) , which parametrizes the BSM contributions using the coefficients of SM field higher dimensional operators. The SMEFT is successful if the BSM particles are much heavier than 1TeV and they decouple from the low energy physics. The SMEFT cannot be applied, however, if the heavy BSM particles do not decouple from the low energy physics. The Higgs Effective Field Theory (HEFT) [67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83] should be applied instead. These existing EFTs cannot be applied if there exist BSM particles lighter than 1TeV. We should include these BSM particles explicitly in the EFT approach. In this paper, we propose a generalization of HEFT (GHEFT) for this purpose. As in the HEFT, GHEFT is based on the electroweak chiral perturbation theory (EWChPT) [84][85][86][87][88][89]. In GHEFT, the BSM particles, as well as the 125GeV Higgs boson, are introduced as matter particles in the Callan-Coleman-Wess-Zumino (CCWZ) construction [90][91][92] of EWChPT. Note that the longitudinal gauge boson scattering amplitudes exceed perturbative unitarity limits at high energy in the EWChPT. The GHEFT couplings should satisfy special conditions, known as the unitarity sum rules [93][94][95], to keep the amplitudes perturbative in the high energy scatterings, if the model is considered to be ultraviolet (UV) complete. We also note that the EWChPT is not renormalizable. The UV completed GHEFT couplings should satisfy the finiteness conditions in order to cancel these UV divergences. The GHEFT can also be described in a geometrical language using the scalar manifold metric, as discussed in Refs. [96,97] in the HEFT context. We point out that both the scalar scattering amplitudes and the one-loop UV divergences in the electroweak oblique correction parameters S and U [98] are described by using the Riemann curvature tensor (geometry) and the Killing vectors (symmetry) of the scalar manifold. Therefore, both the unitarity sum rules and the oblique correction finiteness conditions are described in terms of the geometry and the symmetry. We find that the perturbative unitarity is ensured by the flatness of the scalar manifold (vanishing Riemann curvature). We also find that the divergences in the oblique correction parameters (S and U parameters) are canceled if a subset of the perturbative unitarity conditions and the SU(2) W × U(1) Y gauge symmetry are satisfied. These findings generalize our previous observation [99] which relates the perturbative unitarity to the one-loop finiteness of the oblique correction parameters 1 . This paper is organized as follows: in §. II we introduce the GHEFT Lagrangian at its lowest order (O(p 2 )). We investigate the scalar boson scattering amplitudes in §. III. §. IV and §. V are for one-loop computations with and without the gauge boson contributions. The relationship between the perturbative unitarity and the one-loop finiteness of the oblique correction parameters is clarified in §. VI. We conclude in §. VII. II. GENERALIZED HEFT LAGRANGIAN OF SU (2) W × U (1) Y → U (1) em The electroweak chiral perturbation theory (EWChPT) [84][85][86][87][88][89] provides a systematic framework to describe the low energy phenomenologies of the electroweak symmetry breaking physics. It utilizes the electroweak chiral Lagrangian method for parametrizing the 1 Possible relations between the unitarity and the renormalizabilty have also been investigated in gravity models. See Refs. [100][101][102][103][104]. non-decoupling corrections, which appear ubiquitously in models with strongly interacting electroweak symmetry breaking sector. Although the original version of the EWChPT was constructed to be a Higgsless theory [94,95,[105][106][107][108][109][110][111][112][113][114], after the discovery of the 125GeV Higgs particle, the EWChPT is extended to the Higgs Effective Field Theory (HEFT), incorporating the 125 GeV Higgs particle h as a neutral spin-0 matter particle in the electroweak chiral Lagrangian [67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83]. Introducing functions F (h) and V (h), which parametrize the phenomenological properties of the 125GeV Higgs, the HEFT provides a systematic description for a neutral spin-0 particle in the electroweak symmetry breaking sector, including the one-loop radiative corrections [72][73][74][75][76][77][78][79][80][81][82][83]. It can parametrize the low energy properties of the 125GeV Higgs particle in the strongly interacting model context, as well as weakly interacting model context. We need to generalize the HEFT further (generalized HEFT, GHEFT), if we want to introduce extra Higgs particles other than the discovered 125GeV Higgs particle. It is not trivial to introduce non-singlet extra particles in the EWChPT, however, since the electroweak gauge symmetry SU(2) W × U(1) Y is realized nonlinearly in the EWChPT. The interaction Lagrangian needs to be arranged carefully to make the theory invariant under the electroweak gauge symmetry SU(2) W × U(1) Y . These extra non-singlet Higgs particles can be regarded as matter particles in the EWChPT Lagrangian context. The Callan-Coleman-Wess-Zumino (CCWZ) formulation [90][91][92] provides an ideal framework for the concrete construction of the matter particle interaction Lagrangian in a manner consistent with the nonlinear sigma model symmetry structure. In this section, we apply the CCWZ formulation for the construction of the GHEFT Lagrangian. A. Electroweak chiral Lagrangian For simplicity, in this subsection, we consider the EWChPT Lagrangian in the gaugeless limit, i.e., g W = g Y = 0. The couplings with the electroweak gauge fields will be re-introduced in §. II C. ξ W (x) = exp i a=1,2 π a (x) τ a 2 ,(1)ξ Y (x) = exp iπ 3 (x) τ 3 2 ,(2) with τ a (a = 1, 2, 3) being Pauli spin matrices. Under the G = [SU(2) W × U(1) Y ] transfor- mation, g W ∈ SU(2) W , g Y ∈ U(1) Y ,(3) these NG boson fields transform as ξ W (x) → ξ ′ W (x) = g W ξ W (x) h † (π, g W , g Y ) ,(4)ξ Y (x) → ξ ′ Y (x) = h(π, g W , g Y ) ξ Y (x) g † Y .(5) Here h(π, g W , g Y ) is an element of the unbroken group H, which is determined to pullback the coset space coordinates to their original forms (1) and (2). Note that the H transformation h(π, g W , g Y ) depends not only on the SU(2) W and U(1) Y elements g W and g Y , but also on the NG boson fields π(x). The NG boson fields π a (a = 1, 2, 3) therefore transform nonlinearly under the G symmetry. It is useful to introduce objects called Maurer-Cartan (MC) one-forms α a ⊥µ (a = 1, 2, 3) defined as α a ⊥µ = tr 1 i ξ † W (∂ µ ξ W )τ a , (a = 1, 2)(6) and α 3 ⊥µ = tr 1 i ξ † W (∂ µ ξ W )τ 3 + tr 1 i (∂ µ ξ Y )ξ † Y τ 3 .(7) Although the NG boson fields π transform nonlinearly, these MC one-forms transform homogeneously, i.e., a=1,2 α a ⊥µ τ a 2 → h(π, g W , g Y ) a=1,2 α a ⊥µ τ a 2 h † (π, g W , g Y ) ,(8)α 3 ⊥µ τ 3 2 → h(π, g W , g Y ) α 3 ⊥µ τ 3 2 h † (π, g W , g Y ) ,(9) under the G symmetry. We see that the MC one-forms transform as α a ⊥µ → [ρ α (h)] a b α b ⊥µ ,(10) with ρ α (h) being a 3 × 3 matrix ρ α (h) = exp iθ h (π, g W , g Y ) Q α , h = exp iθ h (π, g W , g Y ) τ 3 2 .(11) In the expression (10) and hereafter, summation b=1,2,3 is implied whenever an index b is repeated in a product. Here the NG boson charge matrix Q α is defined by Q α =   −σ 2 0   ,(12) with σ 2 being the Pauli spin matrix σ 2 =   0 −i +i 0   .(13) It is now straightforward to construct the lowest order (O(p 2 )) G invariant Lagrangian of the NG bosons: L π = 1 2 G (0) ab α a ⊥µ α bµ ⊥(14) with G (0) ab = 1 4      v 2 v 2 v 2 Z      .(15) The Lagrangian can be rewritten as L π = v 2 4 tr (∂ µ U † ) (∂ µ U) − v 2 Z − v 2 8 tr U † (∂ µ U)τ 3 tr U † (∂ µ U)τ 3 ,(16) with U := ξ W ξ Y .(17) It should be emphasized here that v and v Z (decay constants of π 1,2 and π 3 ) are independently adjustable parameters in the EWChPT on the G/H = [SU(2) W ×U(1) Y ]/U(1) em coset space. Phenomenologically preferred relation ρ := v 2 v 2 Z ≃ 1(18) is realized only by a parameter tuning v ≃ v Z in this setup. B. Matter particles coupled with the electroweak chiral Lagrangian Thanks to the homogeneous transformation properties of the MC one-forms (10), matter particles can be introduced easily in the CCWZ formulation of the EWChPT Lagrangian (16). We consider a set of real scalar matter fields φ I , which transforms homogeneously as φ I → [ρ φ (h)] I J φ J ,(19) under the unbroken group H. Here ρ φ (h) stands for a representation matrix ρ φ (h) = exp iθ h Q φ , h = exp iθ h τ 3 2 ,(20) with Q φ being a hermitian matrix. Note here that the h transformation depends on the NG boson fields π(x). It therefore is a local transformation depending on the space-time point x. If the set of scalar matter particles consists of n N species of neutral particles and n C species of charged particles, the matrix Q φ can be expressed as a (2n C + n N ) × (2n C + n N ) matrix Q φ =              −q 1 σ 2 . . . −q n C σ 2 0 . . . 0              .(21) Here q i (i = 1, 2, · · · n C ) are the charges of the scalar matter particles. Since h is a local transformation, ∂ µ φ I transforms non-homogeneously under h. In order to write a kinetic term for the matter field φ I , we therefore introduce a covariant derivative of the matter field φ I : (D µ φ) I = ∂ µ φ I + iV 3 µ [Q φ ] I J φ J , (I, J = 1, 2, · · · , 2n C + n N ) .(22) We take the connection V 3 µ as V 3 µ = −tr 1 i (∂ µ ξ Y )ξ † Y τ 3 + cα 3 ⊥µ ,(23) with c being an arbitrary constant. Hereafter we take c = 0 for simplicity. The covariant derivative (22) transforms homogeneously (D µ φ) I → [ρ φ (h)] I J (D µ φ) J ,(24) as we designed so in Eq. (22). It is now straightforward to write down an O(p 2 ) EWChPT Lagrangian including additional scalar bosons with arbitrary charges: L = 1 2 G ab α a ⊥µ α bµ ⊥ + G aI α a ⊥µ (D µ φ) I + 1 2 G IJ (D µ φ) I (D µ φ) J − V .(25) Here G ab , G aI , G IJ and V are functions of the scalar fields φ I . Also, G ab , G aI and G IJ transform homogeneously as multiplets of corresponding representations. They satisfy G ab φ=0 = G (0) ab , G aI φ=0 = 0 , G IJ φ=0 = δ IJ ,(26) and Lagrangian (14) in the absence of Higgs particles φ I . The stability around the vacuum φ = 0 is guaranteed by the conditions (27). ∂ ∂φ I V φ=0 = 0 , ∂ ∂φ J ∂ ∂φ I V φ=0 = M 2 I δ IJ ,(27) C. Electroweak gauge fields It is easy to reintroduce the electroweak gauge fields W a µ (a = 1, 2, 3) and B µ in our EWChPT Lagrangian (25). When the gauge coupling is switched on, we just need to replace the derivatives ∂ µ ξ W and ∂ µ ξ Y by the covariant derivatives: D µ ξ W = ∂ µ ξ W − ig W W a µ τ a 2 ξ W ,(28)D µ ξ Y = ∂ µ ξ Y + ig Y ξ Y B µ τ 3 2 ,(29) with g W and g Y being the SU(2) W and U(1) Y gauge coupling strengths, respectively. The lowest order (O(p 2 )) GHEFT Lagrangian is therefore L = 1 2 G abα a ⊥µα bµ ⊥ + G aIα a ⊥µ (D µ φ) I + 1 2 G IJ (D µ φ) I (D µ φ) J − V − 1 4 W a µν W aµν − 1 4 B µν B µν ,(30) witĥ α a ⊥µ = tr 1 i ξ † W (∂ µ ξ W )τ a − g W tr ξ † W W b µ τ b 2 ξ W τ a , (a = 1, 2)(31)andα 3 ⊥µ = tr 1 i ξ † W (∂ µ ξ W )τ 3 + tr 1 i (∂ µ ξ Y )ξ † τ 3 − g W tr ξ † W W b µ τ b 2 ξ W τ 3 + g Y B µ .(32) We define the covariant derivative of the matter fields (D µ φ) I (D µ φ) I = ∂ µ φ I + iV 3 µ [Q φ ] I J φ J ,(33)withV 3 µ = −tr 1 i (∂ µ ξ Y )ξ † Y τ 3 − g Y B µ .(34) It should be noted that the GHEFT Lagrangian (30) reproduces HEFT Lagrangian [67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83] for n N = 1 and n C = 0. Here φ I=h stands for the 125 GeV Higgs boson field. In the HEFT, G aI and G IJ are taken as G ah = 0 , G hh = 1 .(35) G ab is tuned to be G ab = v 2 4 F (h)δ ab .(36) D. Geometrical form of the O(p 2 ) GHEFT Lagrangian The lowest order (O(p 2 )) GHEFT Lagrangian (30) can also be expressed in a geometrical form: L = 1 2 g ij (φ)D µ φ i D µ φ j − V (φ) − 1 4 W a µν W aµν − 1 4 B µν B µν ,(37) where φ i stands a scalar field multiplet containing both Higgs bosons φ I and the NG bosons π a as its component, i.e., {φ i } = {π a , φ I } .(38) The geometrical form of the GHEFT Lagrangian (37) can be understood as a gauged nonlinear sigma model on a scalar manifold. The scalar manifold (internal space) is coordinated by the scalar multiplet φ i . Both the metric g ij (φ) and the potential V (φ) are functions of φ i . They should be invariant under the SU(2) W × U(1) Y transformation: 0 = w k a g ij,k + (w k a ) ,i g kj + (w k a ) ,j g ik ,(39)0 = y k g ij,k + (y k ) ,i g kj + (y k ) ,j g ik ,(40)0 = w k a V ,k ,(41)0 = y k V ,k ,(42) with g ij,k := ∂ ∂φ k g ij , V ,k := ∂ ∂φ k V , (w k a ) ,i := ∂ ∂φ i w k a , (y k ) ,i := ∂ ∂φ i y k .(43) The SU(2) W and U(1) Y Killing vectors are denoted by w k a (a = 1, 2, 3) and y k , respectively, in Eqs. (39)- (42). The GHEFT Lagrangian (30) provides the most general and systematic method to construct the geometrical form of the Lagrangian (37) having these symmetry properties (39)- (42). The translation dictionary from the GHEFT Lagrangian (30) to the geometrical form (37) will be published elsewhere. The SU(2) W × U(1) Y gauge interactions are introduced in the scalar sector through the covariant derivative D µ φ i = ∂ µ φ i + g W W a µ w i a (φ) + g Y B µ y i (φ) .(44) It should be noted that the gauge fields interact with the scalar sector through the SU(2) W × U(1) Y Killing vectors w i a and y i . The scalar potential V (φ) should be minimized at the vacuum, φ i =φ i .(45) Note that, since the electroweak symmetry is spontaneously broken at the vacuum, the vacuum φ i =φ i cannot be a fixed point of the SU(2) W × U(1) Y transformation, i.e., w i a (φ) = 0 , y i (φ) = 0 .(46) It should be a fixed point of the U(1) em transformation, w i 3 (φ) + y i (φ) = 0,(47) however. The electroweak gauge bosons (W and Z) acquire their masses M 2 W ∝ g 2 W g ij (φ) w i 1 (φ) w j 1 (φ) , M 2 Z ∝ (g 2 W + g 2 Y )g ij (φ) w i 3 (φ) w j 3 (φ) .(48) The Killing vectors at the vacuum (46) therefore play the role of the Higgs vacuum expectation value in the SM. It should be emphasized that the vanishing scalar vacuum expectation valueφ i = 0 does not imply the electroweak symmetry recovery in the GHEFT Lagrangian. Actually, in the GHEFT coordinate (38), even though the vacuum expectation values of the scalar fields are all vanishingφ i = 0, the electroweak symmetry is still spontaneously broken by the non-vanishing Killing vectors at the vacuum (46). The dynamical excitation fields ϕ i are obtained after the expansion around the vacuum, φ i =φ i + ϕ i .(49) The scalar manifold metric g ij is expanded as g ij =ḡ ij +ḡ ij,k ϕ k + 1 2ḡ ij,kl ϕ k ϕ l + · · · ,(50) with g ij := g ij (φ),ḡ ij,k := ∂ ∂φ k g ij (φ) φ=φ ,ḡ ij,kl := ∂ ∂φ l ∂ ∂φ k g ij (φ) φ=φ , · · · . (51) In a similar manner, the potential term is expanded as V (φ) =V +V ,i ϕ i + 1 2V ,ij ϕ i ϕ j + 1 3!V ,ijk ϕ i ϕ j ϕ k + 1 4!V ,ijkl ϕ i ϕ j ϕ k ϕ l + · · · ,(52) withV := V φ=φ ,V ,i := ∂ ∂φ i V φ=φ ,V ,ij := ∂ ∂φ j ∂ ∂φ i V φ=φ , · · · .(53) Since the potential V is minimized at the vacuum, the potential should satisfȳ V ,i = 0 .(54) The scalar manifold is coordinated by the scalar field multiplet φ i . Hereafter, we normal- ize/diagonalize the coordinate φ i asḡ ij = δ ij ,(55)andV ,ij = δ ij m 2 i ,(56) so that the excitation fields ϕ i are canonically normalized and diagonalized. III. SCALAR SCATTERING AMPLITUDES AND PERTURBATIVE UNITAR-ITY We next consider implications of the perturbative unitarity in the GHEFT framework. It is well known that, in the effective field theory framework, the longitudinally polarized electroweak (EW) gauge boson scattering amplitudes grow in the high energy and tend to cause violations of the perturbative unitarity [115,116]. The effective field theory coupling constants need to be arranged to keep the perturbative unitarity in the high energy gauge boson scattering amplitudes. For such a purpose, we use the equivalence theorem between the longitudinally polarized gauge boson scattering amplitudes and the corresponding would-be NG boson amplitudes [117][118][119][120][121]. The equivalence theorem allows us to estimate the longitudinally polarized gauge boson high energy scattering amplitudes by using the NG boson amplitudes in the gaugeless limit i.e., g W = g Y = 0 with uncertainty of O(M 2 W /E 2 ) . The computation of the amplitudes is simplified greatly in the gaugeless limit. Note that the energy growing behavior in the longitudinal polarized gauge bosons amplitudes is exactly canceled in the SM [117,[122][123][124]. The energy growing behavior coming from the EW gauge boson exchange and contact interaction diagrams is exactly canceled by the Higgs exchange diagram in the SM. The Higgs boson plays an essential role to keep the perturbative unitarity in the SM. On the other hand, it is highly non-trivial whether the cancellation of the energy growing terms does work or not in the GHEFT. In fact, in order to ensure the cancellation, the coupling strengths between the Higgs boson(s) and the EW gauge bosons should satisfy special conditions known as the "unitarity sum rules" [93][94][95]. The unitarity sum rules provide a guiding principle to investigate the extended Higgs scenarios in a model-independent manner. Model-independent studies on extended EWSB scenarios have been done based on the unitarity argument [70,93,99,125,126]. We estimate the amplitudes of EW gauge boson scattering by the NG boson scattering with the help of the equivalence theorem. In subsequent subsections, we explicitly calculate the on-shell amplitudes among the scalar fields ϕ i in the gaugeless limit, and express the unitarity sum rules in terms of the scalar manifold's geometry. A. Scalar scattering amplitudes We consider here an N-point on-shell scalar scattering amplitude at the tree-level, iM(123 · · · N) := iM(ϕ i 1 (p 1 ), ϕ i 2 (p 2 ), ϕ i 3 (p 3 ), · · · , ϕ i N (p N )) ,(57) with p n and i n (n = 1, 2, · · · , N) being outgoing momenta and the particle species, respectively. We define s n 1 := p 2 n 1 , s n 1 n 2 := (p n 1 + p n 2 ) 2 , s n 1 n 2 n 3 := (p n 1 + p n 2 + p n 3 ) 2 , · · · .(58) External momenta p n are taken on-shell, s n = m 2 in .(59) We note s n 1 n 2 n 3 = s n 1 n 2 + s n 2 n 3 + s n 1 n 3 − m 2 in 1 − m 2 in 2 − m 2 in 3 , · · · .(60) The N-point amplitude (57) can thus be written as a function of the scalar particle masses m 2 i and the generalized Mandelstam variables s n 1 n 2 . As we will show explicitly below, the three-and four-point on-shell scattering amplitudes are described in terms of the geometry of the scalar manifold, iM(123) = −iV ;(i 1 i 2 i 3 ) ,(61)iM(1234) = iM(1234) + iM(125) [D(s 12 )] i 5 i 6 iM(346) + iM(135) [D(s 13 )] i 5 i 6 iM(246) + iM(145) [D(s 14 )] i 5 i 6 iM(236) ,(62) with iM(1234) = −iV ;(i 1 i 2 i 3 i 4 ) − i 3 R i 1 i 3 i 4 i 2 +R i 1 i 4 i 3 i 2 s 12 − i 3 R i 1 i 2 i 4 i 3 +R i 1 i 4 i 2 i 3 s 13 − i 3 R i 1 i 2 i 3 i 4 +R i 1 i 3 i 2 i 4 s 14 ,(63) and [D(s)] ij := i s − m 2 iḡ ij .(64)HereV ;(i 1 i 2 i 3 ) ,V ;(i 1 i 2 i 3 i 4 ) andR i 1 i 2 i 3 i 4L 3 = 1 2ḡ ij,k ϕ k (∂ µ ϕ i )(∂ µ ϕ j ) − 1 3!V ,ijk ϕ i ϕ j ϕ k ,(65) at the tree-level. It is straightforward to evaluate the on-shell three-point amplitude iM(123) = i 2 (ḡ i 1 i 2 ,i 3 +ḡ i 2 i 1 ,i 3 )(−p 1 · p 2 ) + i 2 (ḡ i 2 i 3 ,i 1 +ḡ i 3 i 2 ,i 1 )(−p 2 · p 3 ) + i 2 (ḡ i 3 i 1 ,i 2 +ḡ i 1 i 3 ,i 2 )(−p 3 · p 1 ) − iV ,i 1 i 2 i 3 = i 2ḡ i 1 i 2 ,i 3 m 2 i 1 + m 2 i 2 − s 12 + i 2ḡ i 2 i 3 ,i 1 m 2 i 2 + m 2 i 3 − s 23 + i 2ḡ i 3 i 1 ,i 2 m 2 i 3 + m 2 i 1 − s 31 − iV ,i 1 i 2 i 3 ,(66) from the vertices in (65). The conservation of the total momentum p 1 + p 2 + p 3 = 0 , implies s 12 = (p 1 + p 2 ) 2 = p 2 3 = m 2 i 3 , and similarly s 23 = m 2 i 1 , s 31 = m 2 i 2 . The on-shell three-point amplitude (66) can therefore be expressed as iM(123) = i 2ḡ i 1 i 2 ,i 3 m 2 i 1 + m 2 i 2 − m 2 i 3 + i 2ḡ i 2 i 3 ,i 1 m 2 i 2 + m 2 i 3 − m 2 i 1 + i 2ḡ i 3 i 1 ,i 2 m 2 i 3 + m 2 i 1 − m 2 i 2 − iV ,i 1 i 2 i 3 = i 2 m 2 i 1 ḡ i 1 i 2 ,i 3 +ḡ i 1 i 3 ,i 2 −ḡ i 2 i 3 ,i 1 + i 2 m 2 i 2 ḡ i 2 i 3 ,i 1 +ḡ i 2 i 1 ,i 3 −ḡ i 3 i 1 ,i 2 + i 2 m 2 i 3 ḡ i 3 i 1 ,i 2 +ḡ i 3 i 2 ,i 1 −ḡ i 1 i 2 ,i 3 − iV ,i 1 i 2 i 3 .(67) Note that the m 2 i 1 , m 2 i 2 and m 2 i 3 are related with the second derivative of the potential V ,ij by (56). The first derivative of the metric tensor in the interaction vertex (65) is related with the the Affine connection Γ l jk g il Γ l jk := 1 2 [g ij,k + g ki,j − g jk,i ] .(68) The amplitude (67) can then be rewritten as iM(123) = iV ,i 1 lΓ l i 2 i 3 + iV ,i 2 lΓ l i 3 i 1 + iV ,i 3 lΓ l i 1 i 2 − iV ,i 1 i 2 i 3 ,(69) withΓ l jk being the Affine connection at the vacuum Γ l jk := Γ l jk φ=φ .(70) Our final task is to rewrite the amplitude (69) in terms of the covariant derivatives of the potential V . It is straightforward to show V ;ijk = V ,ijk − (Γ l ij ) ,k V ,l − Γ l ij V ,lk − Γ l ik V ,lj − Γ l jk V ,li + Γ l ik Γ m lj V ,m + Γ l jk Γ m li V ,m .(71) Since the first derivative of the potential vanishes at the vacuum, we obtain V ;ijk =V ,ijk −Γ l ijV ,lk −Γ l ikV ,lj −Γ l jkV ,li .(72) Moreover, as we see in (72),V ;ijk is symmetric under the i ↔ j, i ↔ k and j ↔ k exchanges. We therefore obtainV ;(ijk) := 1 3! V ;ijk +V ;jki +V ;kij +V ;ikj +V ;kji +V ;jik = V ;ijk .(73) It is now easy to obtain a geometrical formula for the three-point amplitude iM(123) = −iV ;(i 1 i 2 i 3 ) .(74) We next consider the four-point amplitude iM(1234) = iM 0 (1234) + iM(12[5])[D(s 12 )] i 5 i 6 iM(34[6]) + iM(13[5])[D(s 13 )] i 5 i 6 iM(24[6]) + iM(14[5])[D(s 14 )] i 5 i 6 iM(23[6]) ,(75) where the first line comes from the four-point contact interaction vertices, while the second, the third and the fourth lines are from the i 5 -particle exchange diagrams in the s 12 , s 13 and s 14 channels, respectively. The three-point amplitude M(ij[k]) is for on-shell i and j, allowing off-shell k-particle. We first study M(12 [5]), M(12[5]) = −V ;(125) +ḡ i 5 i ′ 5Γ i ′ 5 i 1 i 2 (s 12 − m 2 i 5 ) ,(76) which can be related with the on-shell three-point amplitude M(125) as M(12[5]) = M(125) +ḡ i 5 i ′ 5Γ i ′ 5 i 1 i 2 (s 12 − m 2 i 5 ) .(77) It is easy to rewrite the amplitude (75) as iM(1234) = iM(1234) +iM(125)[D(s 12 )] i 5 i 6 iM(346) +iM(135)[D(s 13 )] i 5 i 6 iM(246) +iM(145)[D(s 14 )] i 5 i 6 iM(236) ,(78) with M(1234) being M(1234) = M 0 (1234) −ḡ i 5 i 6Γ i 5 i 1 i 2Γ i 6 i 3 i 4 (s 12 − m 2 i 5 ) −ḡ i 5 i 6Γ i 5 i 1 i 3Γ i 6 i 2 i 4 (s 13 − m 2 i 5 ) −ḡ i 5 i 6Γ i 5 i 1 i 4Γ i 6 i 2 i 3 (s 14 − m 2 i 5 ) +V ;(i 1 i 2 i 5 )Γ i 5 i 3 i 4 +V ;(i 1 i 3 i 5 )Γ i 5 i 2 i 4 +V ;(i 1 i 4 i 5 )Γ i 5 i 2 i 3 +V ;(i 2 i 3 i 5 )Γ i 5 i 1 i 4 +V ;(i 2 i 4 i 5 )Γ i 5 i 1 i 3 +V ;(i 3 i 4 i 5 )Γ i 5 i 1 i 2 .(79) The evaluation of the four-point contact interaction contribution is a bit tedious but straightforward. We obtain iM 0 (1234) = −iV ,i 1 i 2 i 3 i 4 + i 2 − (ḡ i 1 i 2 ,i 3 i 4 +ḡ i 3 i 4 ,i 1 i 2 ) s 12 − (ḡ i 1 i 3 ,i 2 i 4 +ḡ i 2 i 4 ,i 1 i 3 ) s 13 − (ḡ i 1 i 4 ,i 2 i 3 +ḡ i 2 i 3 ,i 1 i 4 ) s 14 + (ḡ i 1 i 2 ,i 3 i 4 +ḡ i 1 i 3 ,i 2 i 4 +ḡ i 1 i 4 ,i 2 i 3 ) m 2 i 1 + (ḡ i 2 i 1 ,i 3 i 4 +ḡ i 2 i 3 ,i 1 i 4 +ḡ i 2 i 4 ,i 1 i 3 ) m 2 i 2 + (ḡ i 3 i 1 ,i 2 i 4 +ḡ i 3 i 2 ,i 1 i 4 +ḡ i 3 i 4 ,i 1 i 2 ) m 2 i 3 + (ḡ i 4 i 1 ,i 2 i 3 +ḡ i 4 i 2 ,i 1 i 3 +ḡ i 4 i 3 ,i 1 i 2 ) m 2 i 4 .(80) Combining these results, we obtain a geometrical formula for the on-shell four-point ampli- tude iM(1234) = −iV ;(i 1 i 2 i 3 i 4 ) − i 3 R i 1 i 3 i 4 i 2 +R i 1 i 4 i 3 i 2 s 12 − i 3 R i 1 i 2 i 4 i 3 +R i 1 i 4 i 2 i 3 s 13 − i 3 R i 1 i 2 i 3 i 4 +R i 1 i 3 i 2 i 4 s 14 .(81) We used the on-shell condition s 12 + s 13 + s 14 = m 2 i 1 + m 2 i 2 + m 2 i 3 + m 2 i 4 ,(82) in the computation above. HereR ijkl andV ;(ijkl) denote the Riemann curvature tensor and the totally symmetrized covariant derivatives of the potential at the vacuum: R ijkl := R ijkl φ=φ ,V ;(ijkl) := V ;(ijkl) φ=φ .(83) We here give formulas to computeR ijkl andV ;(ijkl) from the metric tensor g ij and the potential V :R ijkl = 1 2 (ḡ il,jk +ḡ jk,il −ḡ ik,jl −ḡ jl,ik ) +ḡ mn Γ m ilΓ n jk −Γ m ikΓ n jl ,(84) andV (37). In order to keep the perturbative unitarity in the high energy limit, the GHEFT Lagrangian should satisfy special conditions known as the unitarity sum rules [93][94][95]. We here give a geometrical interpretation for the unitarity sum rules. ;(ijkl) =V ,ijkl −V ,ijmΓ m kl −V ,klmΓ m ij −V ,ikmΓ m jl −V ,jlmΓ m ik −V ,ilmΓ m jk −V ,jkmΓ m il +V ,mn Γ m ijΓ n kl +Γ m ikΓ n jl +Γ m ilΓ n jk + A ijkl + A jikl + A kijl + A lijk ,(85) Applying the on-shell condition s 12 + s 13 + s 14 = m 2 i 1 + m 2 i 2 + m 2 i 3 + m 2 i 4(87) in the four-point amplitude (81) we obtain iM(1234) = − i 3 (R i 1 i 3 i 4 i 2 +R i 1 i 4 i 3 i 2 −R i 1 i 2 i 3 i 4 −R i 1 i 3 i 2 i 4 )s 12 − i 3 (R i 1 i 2 i 4 i 3 +R i 1 i 4 i 2 i 3 −R i 1 i 2 i 3 i 4 −R i 1 i 3 i 2 i 4 )s 13 + O(E 0 ).(88) Therefore, the unitarity sum rules can be summarized in the geometrical language as R i 1 i 3 i 4 i 2 +R i 1 i 4 i 3 i 2 −R i 1 i 2 i 3 i 4 −R i 1 i 3 i 2 i 4 = 0 ,(89)R i 1 i 2 i 4 i 3 +R i 1 i 4 i 2 i 3 −R i 1 i 2 i 3 i 4 −R i 1 i 3 i 2 i 4 = 0 .(90) Note that the Riemann curvature tensor R ijkl is antisymmetric under the k ↔ l exchange: R ijkl ≡ −R ijlk .(91) The unitarity sum rules (89) can thus be rewritten as 2R i 1 i 3 i 4 i 2 −R i 1 i 4 i 2 i 3 −R i 1 i 2 i 3 i 4 = 0 .(92) The Bianchi identity R ijkl + R iklj + R iljk ≡ 0(93) can be expressed as −R i 1 i 4 i 2 i 3 −R i 1 i 2 i 3 i 4 ≡R i 1 i 3 i 4 i 2 ,(94) which enables us to simplify the unitarity sum rules (92) further. We obtain the sum rules (89) can be expressed in a simple form: 3R i 1 i 3 i 4 i 2 = 0 .(95) The both of the unitarity sum rules (89) and (90) can be expressed in a compact form: R ijkl = 0 .(96) Note that the unitarity sum rules (96) imply the flatness of the scalar manifold only at the vacuum. The unitarity conditions (96) is lifted to R ijkl = 0 ,(97) i.e., the complete flatness of the entire scalar manifold at least in the vicinity of the vacuum, by imposing the perturbative unitarity in the arbitrary N-point amplitudes. See appendix. A for details. The perturbative unitarity is violated at the certain high energy scale in an extended Higgs scenario with a curved scalar manifold. For instance, if we consider the HEFT with F (h) = (1 + (κ V h)/v) 2 and take κ V < 1, the corresponding scalar manifold has non-zero curvature proportional to 1 −κ 2 V [96,97]. The model causes the violation of the perturbative unitarity at Λ ∼ 4πv/(1 − κ 2 V ) 1/2 . In that case, we need to introduce new particles lighter than Λ and/or to consider non-perturbative effects for ensuring the unitarity in the model. IV. ONE-LOOP DIVERGENCES IN THE GAUGELESS LIMIT As we have shown in the previous section, the tree-level perturbative unitarity requires the GHEFT scalar manifold should be flat at the vacuum. What does this imply at the loop level, then? Refs. [96,97] investigated the structure of the one-loop divergences in the nonlinear sigma model Lagrangian (37). They found the logarithmic divergences in the scalar one-loop integral are described in the gaugeless limit by ∆L ϕ−loop div = 1 (4π) 2 ǫ 1 12 tr(Y µν Y µν ) + 1 2 tr(X 2 ) .(98) Here ǫ is defined as ǫ := 4 − D,(99) with D being the spacetime dimension. Y µν and X are defined as [Y µν ] i j = R i jkl (D µ φ) k (D ν φ) l + W a µν (w i a ) ;j + B µν (y i ) ;j ,(100)[X] i k = R i jkl (D µ φ) j (D µ φ) l + g ij V ;jk ,(101) with (w i a ) ;j = ∂ ∂φ j w i a + Γ i lj w l a , (y i ) ;j = ∂ ∂φ j y i + Γ i lj y l ,(102) and R i jkl = g im R mjkl . Remember that the perturbative unitarity implies the flatness at the vacuum, R ijkl = 0 .(103) It is easy to see that the unitarity condition (103) The divergence structure in the operators proportional to W a µν W bµν , W a µν B µν , B µν B µν ,(104) is not manifest, however. Note that the oblique correction parameters S and U [98] are related with the gauge-kinetic-type operators listed in (104). There is no obvious reason to ensure the absence of the one-loop divergences in the S and U parameters even in the perturbatively unitary models. Moreover, the one-loop divergence formula (98) does not include quantum corrections arising from the gauge-boson loop diagrams, which should be evaluated to deduce the conclusion on the divergence structure for the oblique correction parameters. V. OBLIQUE CORRECTIONS AND FINITENESS CONDITIONS A. Vacuum polarization functions at one-loop The electroweak oblique correction parameters S and U are defined as S := 16π(Π ′ 33 (0) − Π ′ 3Q (0)) ,(105)U := 16π(Π ′ 11 (0) − Π ′ 33 (0)) ,(106)with Π ′ A (0) being Π ′ A (0) := d dp 2 Π A (p 2 ) p 2 =0 .(107) Here Π A (p 2 ) stands for the non-SM contribution to the gauge boson vacuum polarization function in the A-channel. Π 11 (p 2 ) and Π 33 (p 2 ) are charged and neutral weak SU(2) W current correlators at momentum p, respectively. Π 3Q (p 2 ) is the correlator between the neutral weak SU(2) W current and the electromagnetic current. Note that, in the GHEFT, a number of scalar particles other than the 125GeV Higgs contribute to Π A (p 2 ) at loop. The oblique correction parameter T is related with Veltman's ρ parameter [127], with v 0 and v Z0 being the "bare" parameters corresponding to the charged and neutral wouldbe NG boson decay constants v and v Z . The GHEFT Lagrangian loses its predictability on αT := ρ − 1 , ρ = v 2 0 4 + Π 11 (0) v 2 Z0 4 + Π 33 (0) ,(108) the T -parameter, if we allow to introduce independent counter terms for v and v Z . On the other hand, if we assume the counter terms for v and v Z are related with each other, v 2 0 = v 2 (1 + δ v ) , v 2 Z0 = v 2 Z (1 + δ v ) ,(109) the ρ is calculated as ρ = v 2 v 2 Z 1 + δ v + 4 v 2 Π 11 (0) 1 + δ v + 4 v 2 Z Π 33 (0)(110) and we regain a counter-term independent predictability on the ρ parameter ρ = v 2 v Z 1 + αT ,(111) with αT := 4 1 v 2 Π 11 (0) − 1 v 2 Z Π 33 (0) .(112) In what follows, we calculate Π 11 , Π 33 , and Π 3Q at one-loop level in the GHEFT and derive the required conditions for ensuring the UV finiteness of Eqs. (105), (106) and (112). We apply a background field method [128][129][130][131][132][133] to calculate the vacuum polarization functions to keep the gauge invariance. See Appendix B for the details of the calculation. Although there exist UV divergences in Π 11 (0) and Π 33 (0) associated with tadpole diagrams as shown in Figure 1, we assume these UV divergences are canceled by the corresponding tadpole counter terms. FIG. 2: Feynman diagrams for Π ϕϕ 11 , Π ϕϕ 33 , and Π ϕϕ 3Q . The internal lines correspond to ϕ fields. Scalar loop Let us start with the scalar loop corrections to the vacuum polarization functions. The relevant Feynman diagrams are shown in Figure 2, which are evaluated to be Π ϕϕ 3Q (p 2 ) = 1 (4π) 2 −2 i,j (w i 3 ) ;j (w j 3 ) ;i + (ȳ j ) ;i B 22 (p 2 , m 2 i , m 2 j ) + i,j (w i 3 ) ;j (w j 3 ) ;i + (ȳ j ) ;i A(m 2 i ) ,(113) and Π ϕϕ bc (p 2 ) = 1 (4π) 2 −2 i,j (w i b ) ;j (w j c ) ;i B 22 (p 2 , m 2 i , m 2 j ) + i,j (w i b ) ;j (w j c ) ;i +ḡ ij (w k b ) (w l c )R kilj A(m 2 i ) ,(114) A and B 22 are loop functions defined as i (4π) 2 A(m 2 ) = d 4 k (2π) 4 1 k 2 − m 2 ,(116)i (4π) 2 B 22 (p 2 ; m 2 1 , m 2 2 ) = d 4 k (2π) 4 k µ k ν (k 2 − m 2 1 ) {(k + p) 2 − m 2 2 } gµν .(117) Scalar-Gauge loop We next calculate the Feynman diagrams shown in Figure 3. In the 't Hooft-Feynman gauge, we obtain Π ϕV 3Q (p 2 ) = 0 ,(118)2 ) = − 4 (4π) 2 a=1,2 i,jḡ ij (G W a ) i b (G W a ) j c B 0 (p 2 , M 2 W , m 2 i ) + i,jḡ ij (G Z ) i b (G Z ) j c B 0 (p 2 , M 2 Z , m 2 i ) + i,jḡ ij (G A ) i b (G A ) j c B 0 (p 2 , 0, m 2 i ) ,(119)for b, c = 1, 2, 3. Here (G W a ) i b , (G Z ) i b , (G A ) i b are defined as (G W a ) i b := g W (w i a ) ;jw j b , (a = 1, 2)(120)(G Z ) i b := 1 g 2 W + g 2 Y g 2 W (w i 3 ) ;j − g 2 Y (ȳ i ) ;j w j b ,(121)(G A ) i b := g W g Y g 2 W + g 2 Y (w i 3 ) ;j + (ȳ i ) ;j w j b(122) and M 2 W = g 2 W 4 v 2 , M 2 Z = g 2 W + g 2 Y 4 v 2 Z .(123) B 0 is defined as i (4π) 2 B 0 (p 2 ; m 2 1 , m 2 2 ) = d 4 k (2π) 4 1 (k 2 − m 2 1 ) {(k + p) 2 − m 2 2 } .(124) Gauge and Faddeev-Popov (FP) ghost loop Finally, we calculate the contributions which are independent of the scalar interactions. The relevant Feynman diagrams are depicted in Figure 4. In the 't Hooft-Feynman gauge, = 4 (4π) 2 −A(M 2 W ) − c 2 W A(M 2 Z ) − s 2 W A(0) + 2p 2 c 2 W B 0 (p 2 ; M 2 Z , M 2 W ) + s 2 W B 0 (p 2 ; 0, M 2 W ) + 4 c 2 W B 22 (p 2 ; M 2 Z , M 2 W ) + s 2 W B 22 (p 2 ; 0, M 2 W ) ,(125)Π Gauge 33 (p 2 ) = 8 (4π) 2 p 2 B 0 (p 2 ; M 2 W , M 2 W ) + 2B 22 (p 2 ; M 2 W , M 2 W ) − A(M 2 W ) ,(126)Π Gauge 3Q (p 2 ) = 8 (4π) 2 p 2 B 0 (p 2 ; M 2 W , M 2 W ) + 2B 22 (p 2 ; M 2 W , M 2 W ) − A(M 2 W ) ,(127) and Faddeev-Popov (FP) ghost contributions are calculated as Π cc 11 (p 2 ) = Π cc 22 (p 2 ) = 2 (4π) 2 A(M 2 W ) + c 2 W A(M 2 Z ) + s 2 W A(0) − 4 c 2 W B 22 (p 2 ; M 2 Z , M 2 W ) + s 2 W B 22 (p 2 ; 0, M 2 W ) ,(128)Π cc 33 (p 2 ) = − 4 (4π) 2 2B 22 (p 2 ; M 2 W , M 2 W ) − A(M 2 W ) ,(129)Π cc 3Q (p 2 ) = − 4 (4π) 2 2B 22 (p 2 ; M 2 W , M 2 W ) − A(M 2 W ) .(130) Here s W and c W are s W = g Y g Z , c W = g W g Z , g Z = g 2 W + g 2 Y .(131) B. Finiteness of the oblique corrections We are now ready to derive the UV finiteness conditions for the oblique correction parameters at the one-loop level, i.e., the finiteness of Eqs. (105), (106) and (112). For the estimation of the UV cutoff dependence, we regularize A, B 0 , and B 22 functions as A(m 2 ) = −Λ 2 + m 2 ln Λ 2 µ 2 − (4π) 2 A r (m),(132)B 0 (p 2 , m 2 1 , m 2 2 ) = ln Λ 2 µ 2 + (4π) 2 B r (m 1 , m 2 , p 2 ),(133)B 22 (p 2 , m 2 1 , m 2 2 ) = − 1 2 Λ 2 + 1 4 m 2 1 + m 2 2 − p 2 3 ln Λ 2 µ 2 + 1 4 (4π) 2 B 0r (m 1 , m 2 , p 2 ),(134) where Λ and µ denote UV cutoff and renormalization scale, respectively. A r , B r , and B 0r are Λ-independent (µ-dependent) functions. The explicit expressions of the Λ-independent functions are given in Ref. [99]. S and U parameter Let us focus on the UV divergences in Eqs. (105) and (106). Combining the results derived in subsection V A and Eqs. (132)-(134), we find that the UV divergent parts of S and U are given as S div = − 1 12π (w i 3 ) ;j (ȳ j ) ;i ln Λ 2 µ 2 ,(135)U div = 1 12π (w i 1 ) ;j (w j 1 ) ;i − (w i 3 ) ;j (w j 3 ) ;i ln Λ 2 µ 2 .(136) The gauge boson loops do not contribute to the one-loop divergences in S and U parameters. These results are thus identical with the results computed in the gaugeless limit [96,97]. The UV divergences in Eq. (112) other than the tadpole contributions can also be extracted using Eqs. (132)- (134). We obtain 1 v 2 Π 11 (0) − 1 v 2 Z Π 33 (0) div = 1 v 2 Π 11 (0) − 1 v 2 Z Π 33 (0) Λ 2 + 1 v 2 Π 11 (0) − 1 v 2 Z Π 33 (0) ln Λ 2 ,(137) where 1 v 2 Π 11 (0) − 1 v 2 Z Π 33 (0) Λ 2 = − 1 (4π) 2 1 v 2 (w i 1 )(w j 1 ) − 1 v 2 Z (w i 3 )(w j 3 ) R ikjlḡ kl Λ 2 ,(138) and 1 v 2 Π 11 (0) − 1 v 2 Z Π 33 (0) ln Λ 2 = 1 (4π) 2 1 v 2 (w i 1 )(w j 1 ) − 1 v 2 Z (w i 3 )(w j 3 ) × × −4g 2 W (w k a ) ;i (w l a ) ;jḡkl − 4g 2 Y (ȳ k ) ;i (ȳ l ) ;jḡkl +R ikjl ( M 2 ) kl ln Λ 2 µ 2 ,(139) with ( M 2 ) kl being the scalar boson mass matrix in the 't Hooft-Feynman gauge: ( M 2 ) ij :=ḡ ikḡjlV ;kl + g 2 W (w i a )(w j a ) + g 2 Y (ȳ i )(ȳ j ) ,V ;ij := V ;ij φ=φ . (140) VI. PERTURBATIVE UNITARITY VS. FINITENESS CONDITIONS We are now ready to discuss the implications of the perturbative unitarity to the oneloop finiteness of the oblique correction parameters. We first concentrate ourselves on the S-parameter, the UV-divergence of which is given by Eq. (135). As we stressed in §. IV, since there are no obvious connections between the Riemann curvature tensor (geometry) R ijkl and the SU(2) W × U(1) Y Killing vectors (symmetry) w i a and y i , the relation between the perturbative unitarityR ijkl = 0 and the one-loop finiteness of the S-parameter is not evidently understood in Eq. (135). We note, however, that the scalar manifold should be invariant under the SU(2) W ×U(1) Y transformations, and thus the Killing vectors should satisfy the Killing equations, 0 = (w k a )g ij,k + (w k a ) ,i g kj + (w k a ) ,j g ik , 0 = (y k )g ij,k + (y k ) ,i g kj + (y k ) ,j g ik .(141) There do exit connections between the geometry (R ijkl ) and the symmetry (w i a and y i ) embedded in the Killing equations Eqs. (141). Moreover, the Killing vectors w i a and y i should obey the SU(2) W × U(1) Y Lie algebra, [w a , w b ] = ε abc w c , [w a , y] = 0 ,(142) with w a := w i a ∂ ∂φ i , y := y i ∂ ∂φ i .(143) The connections can be studied most easily if we take the Riemann Normal Coordinate (RNC) around the vacuumφ, in which the metric tensor g ij (φ) can be expressed in a Taylorexpanded form aroundφ as, g ij (φ) = δ ij − 1 3R ikjl ϕ k ϕ l + · · · ,(144) with δ ij =ḡ ij = g ij (φ) φ=φ ,R ijkl = R ijkl φ=φ .(145) Solving the Killing equations (141) in terms of the Taylor expansion around the vacuum, w i a =w i a + (w i a ) ,j ϕ j + 1 2! (w i a ) ,jk ϕ j ϕ k + · · · ,(146)y i =ȳ i + (ȳ i ) ,j ϕ j + 1 2! (ȳ i ) ,jk ϕ j ϕ k + · · · ,(147) we find the Taylor expansion coefficients satisfy 0 =ḡ ik (w k a ) ,j +ḡ jk (w k a ) ,i ,(148)0 =ḡ ik (ȳ k ) ,j +ḡ jk (ȳ k ) ,i ,(149)(w i a ) ,jk = 1 3 R i jkl +R i kjl w l a ,(150)(ȳ i ) ,jk = 1 3 R i jkl +R i kjl ȳ l ,(151) . . . There certainly exist connections between the geometry R ijkl and the symmetry w i a and y i in Eqs. (150) and (151). However, Eqs. (150) and (151) are not enough to clarify the relation between the perturbative unitarity and the S-parameter coefficient in (135). Note that the S-parameter coefficient is written in terms of the first derivative of the Killing vectors (w i a ) ;j and (ȳ i ) ;j . We need physical principles to relate (w i a ) ;j and (ȳ i ) ;j with the second derivatives (w i a ) ,jk and (ȳ i ) ,jk . Actually, the SU(2) W × U(1) Y Lie algebra (symmetry) (142) plays the role. Plugging Eqs. (150) and (151) into Eq. (142), we obtain (T a ) j i = 1 2 ε abc ([T b , T c ]) j i + 1 2 ε abc (w k b ) (w l c )R i jkl ,(152)0 = ([T a , T Y ]) j i + (w k a ) (ȳ l )R i jkl ,(153) with T a and T Y being matrices denoting the first derivatives of the SU(2) W × U(1) Y Killing vectors at the vacuum, (T a ) j i := (w i a ) ,j , (T Y ) j i := (ȳ i ) ,j .(154) It is now easy to show tr(T 3 T Y ) = 1 2 ε 3bc tr([T b , T c ]T Y ) + 1 2 ε 3bc (w k b ) (w l c )R i jkl (T Y ) i j = 1 2 ε 3bc tr([T c , T Y ]T b ) + 1 2 ε 3bc (w k b ) (w l c )R i jkl (T Y ) i j = − 1 2 ε 3bc (w k c ) (ȳ l )R i jkl (T b ) i j + 1 2 ε 3bc (w k b ) (w l c )R i jkl (T Y ) i j = 1 2 ε 3bc (w k c ) (w l 3 )R i jkl (T b ) i j + 1 2 ε 3bc (w k b ) (w l c )R i jkl (T Y ) i j .(155) In the last line of Eq. (155), we used the fact that U(1) em is unbroken at the vacuum Eq. (47), i.e., 0 =w i 3 +ȳ i .(156) Eq. (155) can be rewritten in a covariant form (w i 3 ) ;j (ȳ j ) ;i = 1 2 ε 3bc (w k c ) (w l 3 )R i jkl (w j b ) ;i + ε 3bc (w k b ) (w l c )R i jkl (ȳ j ) ;i .(157) In a similar manner, we obtain the divergent coefficient in the U-parameter (136), (w i 1 ) ;j (w j 1 ) ;i − (w i 3 ) ;j (w j 3 ) ;i = 1 2 ε 1bc (w k b ) (w l c )R i jkl (w j 1 ) ;i − ε 3bc (w k b ) (w l c )R i jkl (w j 3 ) ;i .(158) Combining Eqs. (135), (136), (157), and (158), we find S div = − 1 12π ε 3bc (w k c ) (w l 3 )R i jkl (w j b ) ;i + ε 3bc (w k b ) (w l c )R i jkl (ȳ j ) ;i ln Λ 2 µ 2 ,(159)U div = 1 12π ε 1bc (w k b ) (w l c )R i jkl (w j 1 ) ;i − ε 3bc (w k b ) (w l c )R i jkl (w j 3 ) ;i ln Λ 2 µ 2 .(160) The relation between the symmetry and the geometry hidden in the expressions (135) and (136) is now unveiled in the expressions (159) and (160). The one-loop divergences of both S and U are proportional to the Riemann curvature tensorR ijkl at the vacuum. Once the four-point tree-level unitarity is ensured, i.e.,R ijkl = 0, then the one-loop finiteness of S and U is automatically guaranteed in Eqs. (159) and (160). The physical implications of the S and U parameter formulas (159) and (160) can be studied more closely. Note that both of them vanishes when (w k b ) (w l c )R ijkl = 0 ,(161) even if there might exist non-vanishingR ijkl . What does the condition (161) imply, then? Combining the equivalence theorem and the results presented in §. III, we see that the condition (161) ensures the tree-level unitarity of the high energy p-wave scattering amplitude in the V b L V c L → ϕ i ϕ j(162) channel. Here V a L stands for the longitudinally polarized massive gauge bosons, V 1,2 L = W 1,2 L and V 3 L = Z L . The one-loop finiteness of the S and U parameters does not require a completely flat scalar manifold. Once the p-wave tree-level unitarity in the channel (162) is somehow ensured, it is potentially possible to construct strongly interacting EWSB models without violating the one-loop finiteness of the S and U parameters. Moreover, as we see in Appendix C, the covariant derivative of the Killing vector (w i c ) ;j is related with the light-fermion scattering amplitudes ff → ϕ i ϕ j .(163) Here f (andf ) stands for light quarks or leptons (light anti-quarks or anti-leptons). Since the coefficients in front of the logarithmic divergences in Eqs. (159) and (160) can be expressed in a form (w k b ) (w l c )R i jkl (w j a ) ;i ,(164) the precise measurements of the S and U parameters can be used to constrain the high energy scattering amplitudes in V b L V c L → ϕ i ϕ j , ff → ϕ i ϕ j(165) channels, which can be tested in future collider experiments. Finally we make a comment on the UV finiteness condition of (137). We find that the UV finiteness of (137) is not ensured solely by the flatness of the scalar manifold. For an example, even if we assume that the scalar manifold is completely flat and v = v Z at the tree-level, an extra condition (w i 1 )(w j 1 ) − (w i 3 )(w j 3 ) g 2 W (w k a ) ;i (w l a ) ;jḡkl + g 2 Y (ȳ k ) ;i (ȳ l ) ;jḡkl = 0 .(166) is required to ensure the finiteness of the one-loop T parameter correction. The Georigi- Machacek model [35][36][37][38] is one of examples where the condition (166) is not satisfied. We need to introduce independent counter terms for v and v Z in these models. VII. SUMMARY We have formulated a generalized Higgs effective field theory (GHEFT), which includes extra Higgs particles other than the 125 GeV Higgs boson as a low energy effective field theory describing the electroweak symmetry breaking. The scalar scattering amplitudes are expressed by the geometry (Riemann curvature) and the symmetry (Killing vectors) of the scalar manifold in the GHEFT. The one-loop radiative corrections to electroweak oblique corrections are also expressed in terms of geometry and symmetry of the scalar manifold. By using the results, we have clarified the relationship between the perturbative unitarity and the UV finiteness of oblique corrections in the GHEFT. Especially, we have shown that once the tree-level unitarity is ensured, then the S and We also found connections between the coefficients of S and U parameter divergences and the particle scattering amplitudes which can be measured in future collider experiments. We emphasize that future precision measurements of the discovered Higgs couplings, cross section, and oblique parameters are quite important for investigating the geometry and symmetry of the scalar manifold in the generalized Higgs sector. Combining collider/precision experimental data with our effective theoretical approach, we should be able to obtain new prospects of the physics beyond the SM. The one-particle-irreducible on-shell N-point amplitude M(12 · · · N) can thus be expressed 2 as iM(12 · · · N) = − i 2 m<n s mnḠ(i min)(i1i2···ǐm···ǐn···iN ) ,(A8) in the gaugeless flat-potential (V = 0) scalar model. Scalar particles are all massless in this model. The indices inside parentheses are understood to be totally symmetrized. The check symbols on top ofǐ m andǐ n in the sequence i 1 i 2 · · ·ǐ m · · ·ǐ n · · · i N denote the absence of the corresponding indices, i.e., i 1 i 2 · · ·ǐ m · · ·ǐ n · · · i N = i 1 i 2 · · · i m−1 i m+1 · · · i n−1 i n+1 · · · i N .(A9) We show, in this appendix, that the perturbative unitarity up to the N-point amplitudes requires R i 1 i 2 i 3 i 4 = 0 ,R i 1 i 2 i 3 i 4 ;i 5 = 0 ,R i 1 i 2 i 3 i 4 ;i 5 i 6 = 0 , · · ·R i 1 i 2 i 3 i 4 ;i 5 ···i N = 0 . (A10) The scalar manifold needs to be completely flat at least in the vicinity of the vacuum. It Here we introduce an abbreviation for the Riemann curvature tensor {12|34} :=R i 1 i 3 i 4 i 2 .(A15) The indices inside parentheses are, again, understood to be totally symmetrized. Considering the amplitude (A13) for large s, we see the perturbative unitarity requires A(1234) = 0 . (A16) Using the Riemann curvature tesor symmetry {12|34} = −{32|14} = −{14|32} = {34|12} = {21|43} ,(A17) and the first Bianchi identity {12|34} + {13|42} + {14|23} = 0 ,(A18) the coefficient A(1234) can be computed as A(1234) = 2{(12)|(34)} − 2{(13)|(24)} = {12|34} + {12|43} − {13|24} − {13|42} = {12|34} − {13|42} + {14|23} − {13|42} = −3{13|42} .(A19) It is now easy to see that the perturbative unitarity requires the vanishing Riemann curvature tensor at the vacuum,R i 1 i 4 i 2 i 3 = 0 .(A20) Taking the external lines i 1 , · · · , i 4 arbitrary, the result (A20) requiresR ijkl = 0, which is enough to guarantee the perturbative unitarity in the arbitrary four-point amplitudes given in the form of Eq. (A8). The considerations in the limit (A12) thus provide necessary and sufficient conditions for the perturbative unitarity in the four-point amplitudes. Note that the fifth particle is considered to be very soft. We introduce an abbreviation for the covariant derivative of the Riemann curvature tensor, {12|34; 5} :=R i 1 i 3 i 4 i 2 ;i 5 .(A22) The five-point amplitude in the limit behaves as M(12345) ∝ sA(12345) ,(A23) We obtain A(12345) = −{13|42; 5} − 1 2 {24|35; 1} + {25|31; 4} − 1 2 {13|45; 2} + {15|42; 3} = −{13|42; 5} + 1 2 {21|34; 5} + 1 2 {12|43; 5} = −2{13|42; 5} .(A29) The perturbative unitarity in the five-point amplitude thus requires R i 1 i 3 i 4 i 2 ;i 5 = 0 . (A30) It is easy to see that Eq. (A30) gives necessary and sufficient conditions for the perturbative unitarity in the five-point amplitudes. N = 6 It is now straightforward to derive the perturbative unitarity conditions for the six-point amplitude M(123456). It will be turned out considerations in the limit s := s 12 = s 34 = −s 13 = −s 24 = 0 ,(A31) are enough. Generalized Mandelstam variables other than s 12 ,s 34 ,s 13 and s 24 are taken to be zero. Note that the fifth-particle and the sixth-particle are both considered to be very soft in this limit. Note also this choice of the Mandelstam variables is consistent with the momentum conservation constraints and the Gram determinant constraints. We already know the Riemann curvature tensorR ijkl vanishes at the vacuum thanks to the perturabative unitarity of the four-point amplitude. We therefore concentrate ourselves to theR ijkl;mn term in (A6). The six-point amplitude coming from theR ijkl;mn term in (A6) behaves as we obtain M(123456) ∝ s A(123456) ,(A32)A(123456) := A 1 + A 2 + A 3 + A 4 ,(A36) Here we used the fact that the covariant derivatives are commutable, justified by the vanishing curvature tensorR ijkl = 0 at the vacuum. The A 1 term can be computed easily by using the result of A(1234). The A 2 and A 3 terms can be computed in a manner similar to the computations of A(12345). We obtain A 1 = A 2 = A 3 = − 1 2 {13|42; 56} .(A41) The A 4 term can be computed as The second Bianchi identity is used in the first-and fourth-lines in the above calculation. Combining these results, we find the six-point amplitude can be expressed in a simple form, A(123456) = − 5 3 {13|42; 56} .(A43) The perturbative unitarity condition in the six-point amplitude A(123456) = 0 can now be written in terms of the covariant derivative of the Riemann curvaturē R i 1 i 4 i 2 i 3 ;i 5 i 6 = 0 .(A44) It is straightforward to generalize the calculation presented above to the perturbative unitarity conditions in the N-point amplitude, R i 1 i 4 i 2 i 3 ;i 5 i 6 ···i N = 0 .(A45) Since the Taylor expansion coefficients of R ijkl (φ) are required to vanish at any order, the N-point perturbative unitarity requires the Riemann curvature to be R ijkl (φ) = 0 ,(A46) at least in the vicinity of the vacuum. There may exist non-perturbative essential singularity type corrections to (A46), though. Appendix B: Background field method In this appendix, we briefly summarize the interaction terms used in the calculation of the vacuum polarization functions in the background field method at the one-loop level. The background field method is reviewed in Refs. [128][129][130][131][132][133]. We start with the lowest order (O(p 2 )) gauged nonlinear sigma model Lagrangian (37). Let us first decompose φ i , W a µ and B µ into the background fields and the fluctuation fields as φ i :=φ i + ξ i − 1 2Γ i jk ξ j ξ k + · · · , (B1) W a µ :=W a µ + W a µ ,(B2)B µ :=B µ + B µ ,(B3) whereφ i ,W a µ , andB µ are the background fields. The dynamical fluctuation fields are denoted by ξ i , W a µ , and B µ .Γ i jk represents the Christoffel symbols for the metric g ij at φ =φ. The metric tensor and the Killing vector fields are expanded as g ij =g ij + 1 3R iklj ξ k ξ l + · · · ,(B4)w i a =w i a + (w i a ) ;j ξ j + 1 3R i kljw j a ξ k ξ l + · · · ,(B5)y i =ỹ i + (ỹ i ) ;j ξ j + 1 3R i kljỹ j ξ k ξ l + · · · ,(B6) whereg ij , andR ikjl denote the metric, and the Riemann curvature tensor evaluated at φ =φ.w i a andỹ i are the SU(2) W and U(1) Y Killing vectors, while (w i a ) ;j and (ỹ i ) ;j are the covariant derivatives of the Killing vectors evaluated at φ =φ. The Lagrangian (37) is expanded as L = L (0) + L (1) + L (2) + · · · ,(B7) where L (n) is of order n in the fluctuation fields. The quadratic terms L (2) is given as L (2) = − 1 2 W a µ −D 2 δ ab η µν +D νDµ δ ab − g 2 Wg ijw i aw j b η µν − g WW cµν ε abc W b ν − 1 2 B µ −∂ 2 η µν + ∂ ν ∂ µ − g 2 Yg ijỹ iỹj η µν B ν + g W g Y W a µ g ijw i aỹ j η µν B ν + 1 2 ξ i −D 2g ij −D µφ kDµφlR kilj −Ṽ ;ij ξ j + 2g W W a µ g jk (w k a ) ;iD µφj ξ i + 2g Y B µ g jk (ỹ k ) ;iD µφj ξ i − g Wgjiw j a (D µ W a µ )ξ i − g Ygjiỹ j (∂ µ B µ )ξ i ,(B8) with η µν being the space-time metric. Here we definẽ D µ W a µ := ∂ µ W a µ − g W ε abcW b µ W c µ ,(B9)D µφ i := ∂ µφ i + g WW a µw i a + g YBµỹ i ,(B10) D µ ξ i := ∂ µ ξ i +Γ i kj (∂ µφ ) j ξ k + g WW a µ (w i a ) ;j ξ j + g YBµ (ỹ i ) ;j ξ j , (B11) V ;ij := V ;ij φ=φ .(B12) In order to compute radiative corrections, we introduce the gauge fixing action, L GF := − 1 2α W G a W G a W − 1 2α Y G Y G Y ,(B13) where G a W :=D µ W a µ − g W α Wgijw i a ξ j ,(B14)G Y := ∂ µ B µ − g Y α Ygijỹ i ξ j .(B15) with α W and α Y being gauge fixing parameters for SU(2) W and U(1) Y symmetry, respectively. In the one-loop calculation performed in §. V A, we take α W = α Y = 1 ('t Hooft Feynman gauge). We then obtain L (2) + L GF = − 1 2 W a µ −D 2 δ ab η µν + 1 − 1 α W D µDν δ ab − g 2 Wg ijw i aw j b η µν − 2g WW cµν ε abc W b ν − 1 2 B µ −∂ 2 η µν + 1 − 1 α Y ∂ µ ∂ ν − g 2 Yg ijỹ iỹj η µν B ν + g W g Y W a µ g ijw i aỹ j η µν B ν + 1 2 ξ i −D 2g ij −D µφ kDµφlR kilj − α W g 2 Wg ljgkiw l aw k a − α Y g 2 Yg ljgkiỹ lỹk −Ṽ ;ij ξ j + 2g W W a µ g jk (w k a ) ;iD µφj ξ i + 2g Y B µ g jk (ỹ k ) ;iD µφj ξ i . We also need to introduce the Faddeev-Popov (FP) action L FP := ig Wc a W δG a W δθ b W c b W + ig YcY δG Y δθ Y c Y + ig Yc a W δG a W δθ Y c Y + ig WcY δG Y δθ b W c b W ,(B17)δG Y δθ b W := −g Y α Ygijỹ iwj b + O(ξ), (B20) δG Y δθ Y := − 1 g Y ∂ 2 + g 2 Y α Ygijỹ iỹj + O(ξ).(B21) The L FP is expanded as L FP = i (D µca W )D µ c a W − g 2 W α Wgijw i aw j bc a W c b W + i (∂ µc Y )∂ µ c Y − g 2 Y α Ygijỹ iỹjc Y c Y − ig W g Y α Wc a Wg ijw i aỹ j c Y − ig W g Y α YcYgijỹ iwj a c a W + · · · ,(B22) whereD µ c a W := ∂ µ c a W − g W ε abcW b µ c c W , through its covariant derivative D µ f := ∂ µ f + g V V µ T (f ) V f ,(C6) with T (f ) V being the charge matrix of the fermion multiplet f . Note that, in order to keep the Lagrangian gauge invariant, the fermion current J µ V :=f γ µ T (f ) V f (C7) must be conserved 0 = ∂ µ J µ V .(C8) In order to calculate the ff → V µ → ϕ i ϕ j amplitude, we consider the gauge interaction Lagrangian L V φ = g V V µ g ij (φ) (∂ µ φ i ) v j (φ) ,(C9) which can be derived from the nonlinear sigma model kinetic term, 1 2 g ij (φ) (D µ φ) i (D µ φ) j ∈ L .(C10) Expanding the scalar manifold metric g ij (φ) and the Killing vector v j (φ) by the dynamical excitation field ϕ i , we obtain g ij (φ) =ḡ ij + ϕ kḡ ij,k + · · · , (C11) v j (φ) =v j + ϕ k (v j ) ,k + · · · , (C12) withḡ ij,k := ∂ ∂φ k g ij φ=φ , (v j ) ,k := ∂ ∂φ k v j φ=φ ,(C13) and φ i =φ i + ϕ i .(C14) The interaction Lagrangian (C9) can be expanded as L V φ = g V V µḡij (∂ µ ϕ i ) (v j ) + g V V µ (∂ µ ϕ i )ϕ k ḡ ij (v j ) ,k +ḡ ij,k (v j ) + · · · .(C15) Note that on-shell amplitudes are not affected by total derivative terms in the Lagrangian. The interaction Lagrangian (C15) can thus be replaced by L ′ V φ = L V φ − 1 2 ∂ µ g V V µ ϕ i ϕ k (ḡ ij (v j ) ,k +ḡ ij,k (v j )) = g V V µḡij (∂ µ ϕ i )(v j ) − g V (∂ µ V µ )ϕ i ϕ k (ḡ ij (v j ) ,k +ḡ ij,k (v j )) + 1 2 g V V µ (∂ µ ϕ i )ϕ k ḡ ij (v j ) ,k +ḡ ij,k (v j ) −ḡ kj (v j ) ,i −ḡ kj,i (v j ) + · · · .(C16) On the other hand, it is straightforward to show g ij (v j ) ;k − g kj (v j ) ;i = g ij (v j ) ,k + g ij Γ j kl v l − g kj (v j ) ,i + g kj Γ j il v l = g ij (v j ) ,k + 1 2 [g il,k + g ki,l − g kl,i ] v l −g kj (v j ) ,i − 1 2 [g kl,i + g ik,l − g il,k ] v l = g ij (v j ) ,k + g il,k (v l ) − g kj (v j ) ,i − g kl,i (v l ) .(C17) Here the Affine connection Γ j kl is defined by Γ j kl = 1 2 g jm (g ml,k + g km,l − g kl,m ) . It is now easy to see L ′ V φ = g V V µḡij (∂ µ ϕ i )(v j ) − g V (∂ µ V µ )ϕ i ϕ k (ḡ ij (v j ) ,k +ḡ ij,k (v j )) + 1 2 g V V µ (∂ µ ϕ i )ϕ k ḡ ij (v j ) ;k −ḡ kj (v j ) ;i + · · · .(C19) Thanks to the fermion current conservation, the term proportional to ∂ µ V µ does not contribute to the ff → V µ → ϕ i ϕ j amplitude. It is now easy to show M(ff → V µ → ϕ i ϕ j ) ∝ ḡ ik (v k ) ;j −ḡ jk (v k ) ;i g 2 V δ ab s − M 2 V . (C20) The first covariant derivative of the Killing vector,ḡ ik (v k ) ;j , thus plays the role of the V µϕ i -ϕ j interaction vertex in the ff → V µ → ϕ i ϕ j amplitude. with M I being the φ I boson mass. The second and the third conditions in Eq. (26) can be achieved by redefining the scalar field φ I in the Lagrangian. The first condition in Eq. (26) ensures that the extended Lagrangian (25) reproduces the lowest order EWChPT stand for the totally symmetrized covariant derivatives of the potential and the Riemann curvature tensor of the scalar manifold at the vacuum. Let us start with the three-point scalar scattering amplitude M(123). The interaction vertices relevant for this amplitude are [ḡ jk,nl +ḡ kl,nj +ḡ jl,nk − 2(ḡ nj,kl +ḡ nk,jl +ḡ nl,jk )] we have shown in Eq.(81), the scalar four-point amplitude M(1234) contains the energy-squared terms proportional to s 12 , s 13 and s 14 . This implies that the perturbative unitarity of the scattering amplitude is generally violated at certain high energy scale in the GHEFT is enough to guarantee the absence of the divergences in the (∂ µ φ) 4 type operators, which affect the scalar boson high energy fourpoint scattering amplitudes. The flatness of the scalar manifold at the vacuum (103) also automatically guarantees the absence of the divergences in the anomalous triple gauge boson operators. FIG. 1 : 1Feynman diagrams for tadpole contributions to Π 11 (0) and Π 33 (0) and their counter terms. for b, c = 1, 2, 3. Here (w i a ) ;j and (ȳ i ) ;j denote the covariant derivatives of the Killing vectors at the vacuum, (w i a ) ;j := (w i a ) ;j φ=φ , (ȳ i ) ;j := (y i ) ;j φ=φ . FIG. 3 : 3Feynman diagrams for Π ϕV 11 , Π ϕV 33 , and Π ϕV 3Q . The internal lines correspond to ϕ and gauge fields. FIG. 4 :. 4Feynman The wavy and dotted lines correspond to gauge fields and Faddeev-Popov ghost fields, respectively.we find the gauge bosons contributions are U parameters' one-loop finiteness is automatically guaranteed. The tree-level perturbative unitarity in the scalar amplitudes requires the complete flatness of the scalar manifold at the vacuum. On the other hand, the one-loop finiteness of electroweak oblique correction does not require the complete flatness. The findings enable us to verify that tree-level unitarity condition is stronger than the one-loop UV finiteness condition in extended Higgs scenarios. Interestingly, the findings also indicate possibilities of extended Higgs scenarios where perturbative unitarity is broken at certain energy scale but keeps the consistency with the electroweak precision measurements. The construction of a concrete model is deferred to our future work. should be stressed here, even though we already have a compact expression for the N-point amplitude (A8), it is nontrivial to obtain the unitarity condition (A10), since the generalized Mandelstam variables s mn need to satisfy the momentum conservation conditionsN n=1 s mn = 0 ,(A11)and the conditions coming from the four-dimensional space-time (Gram determinant conditions)[137]. We need to make full use of the Riemann tensor symmetry in order to deduce our conclusions (A10).N = 4 Let us start with the analysis on the four-point scattering amplitude. We compute the amplitude in the limit s := s 12 = s 34 = −s 13 = −s 24 = 0 , s 14 = s 23 = 0 . (A12) Clearly the momentum conservation conditions (A11) are satisfied in (A12). The Gram determinant conditions do not give extra conditions in N = 4. In the limit above, the four-point on-shell amplitude behaves as 1234) := {(12)|(34)} + {(34)|(12)} − {(13)|(24)} − {(24)|(13)} . consider the five-point scattering amplitude. Again, we consider the amplitude in the limit s := s 12 = s 34 = −s 13 = −s 24 = 0 , s 14 = s 23 = s 15 = s 25 = s 35 = s 45 = 0 . with A(12345) := {(12)|(34; 5)} + {(34)|(12; 5)} − {(13)|(24; 5)} − {(24)|(13; 5)} . 12)|(34); 5} + {(34)|(12); 5} − {(13)|(24); 5} − {(24)|(13); 12)|(35); 4} − {(13)|(25); 4} . (A26) The first line in (A26) can be computed easily using the result on the four-point amplitude. The second and the third lines can also be computed in a manner similar to Eq. (A19). We find A(12345) = −{13|42; 5} − 1 2 {42|53; 1} − 1 2 {31|54; 2} − 1 2 {24|51; 3} − 1 2 {13|52; 4} . (A27) Eq. (A27) can be simplifed further with the help of the second Bianchi identity {12|34; 5} + {14|35; 2} + {15|32; 4} = 0 . with A(123456) := {(12)|(34; 56)} + {(34)|(12; 56)} − {(13)|(24; 56)} − {(24)|(13; 56)} . (A33) Here we introduce an abbreviation {12|34; 56} :=R i 1 i 3 i 4 i 2 ;i 5 i 6 . ; 56} + {(34)|(12); 56} − {(13)|(24); 56} − {(24)|(13); 56} , 12)|(35); 46} + {(12)|(45); 36} + {(34)|(15); 26} + {(34)|(25); 16} − {(13)|(25); 46} − {(13)|(45); 26} − {(24)|(15); 36} − {(24)|(35); 16} , (A38) 12)|(36); 45} + {(12)|(46); 35} + {(34)|(16); 25} + {(34)|(26); 15} − {(13)|(26); 45} − {(13)|(46); 25} − {(24)|(16); 35} − {(24)|(36); 15} , (A39) 12)|(56); 34} + {(34)|(56); 12} − {(13)|(56); 24} − {(24)|(56); 13} . . (B22), we only show the quadratic terms of the fluctuation fields.The one-loop vacuum polarizations among the electroweak gauge boson can be evaluated by using the quadratic Lagrangian, L (2) + L GF + L FP . In §. V A, we calculate the one-loop diagrams where the internal lines are the fluctuation fields or FP ghosts. The electroweak symmetry G = [SU(2) W × U(1) Y ] is broken spontaneously to the H = U(1) em symmetry in the SM Higgs sector. The most general scalar sector Lagrangian consistent with the symmetry breaking structure G/H = [SU(2) W × U(1) Y ]/U(1) em can be constructed as the CCWZ nonlinear sigma model Lagrangian on the coset space G/H. The coset manifold G/H = [SU(2) W × U(1) Y ]/U(1) em iscoordinated by the Nambu-Goldstone (NG) boson fields π a (a = 1, 2, 3) as 34} + {12|65; 34} + {34|56; 12} + {34|65; 12} − {13|56; 24} − {13|65; 24} − {24|56; 13} − {24|65; 13} 34} + {12|65; 34} + {43|65; 12} + {43|56; 12} + {16|53; 24} + {15|63; 24} + {45|62; 13} + {46|52; 13} {43|65; 12} + {45|62; 13}) + ({43|56; 12} + {46|52; 13}) . Applying the second Bianchi identity, it can be simplified further A 4 = − 1 12 {13|52; 64} + {13|62; 54} + {42|63; 15} + {42|53; 16} 46} + {26|31; 45} + {24|36; 15} + {35|24; 16}A 4 = 1 12 {12|56; = 1 12 {12|56; = 1 12 ({12|56; 34} + {16|53; 24}) + ({12|65; 34} + {15|63; 24}) + (= − 1 12 {31|25; = − 1 12 ({31|25; 46} + {35|24; 16}) + ({26|31; 45} + {24|36; 15}) = 1 12 {34|21; 56} + {21|34; 65} = − 1 6 {13|42; 56} . associated with the gauge fixing term where (∂ µ δ ad − g W ε abdW bµ )(∂ µ δ dc − g W ε decW e µ ) + g 2 W α WgijwδG a W δθ c W := − 1 g W i aw j c + O(ξ) (B18) δG a W δθ Y := −g W α Wgijw i aỹ j + O(ξ), (B19) Eq. (A8) can be regarded as a geometrical manifestation of Weinberg's soft-theorem in on-shell amplitudes. See Ref.[136], for an exampe, for a recent review on the computational techniques of various on-shell amplitudes including nonlinear sigma models. Appendix A: N -point amplitudeLet us consider the Taylor expansion of the scalar manifold metric tensor g ij (φ) around the vacuum pointφ i ,with φ i =φ i + ϕ i . The Taylor coefficients can be expressed in terms of the covariant derivatives of the Riemann curvature tensor in RNC. They are[134,135]. . . When we consider a gauged nonlinear sigma model, the derivative ∂ µ φ i is replaced by thewith V µ and v i (φ) being a gauge field and its corresponding Killing vector. The gauge coupling strength is denoted by g V in (C1). If the Killing vector v i (φ) does not vanish at the vacuumvit implies that the gauge symmetry is spontaneously broken, and the gauge boson V µ acquires its massThe magnitude of the Killing vector at the vacuum,ḡ ij (v i ) (v j ), can therefore be determined by the gauge boson mass measurement.How can we measure the first covariant derivative of the Killing vectorfrom experimental observables in the gauged nonlinear sigma model, then? 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{'abstract': 'We formulate a generalization of Higgs effective field theory (HEFT) including arbitrary number of extra neutral and charged Higgs bosons (generalized HEFT, GHEFT) to describe non-minimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The one-loop divergences in the oblique correction parameters S and U can also be written in terms of the Killing vectors (symmetry) and the Riemann curvature tensor (geometry). It is found that perturbative unitarity of the scattering amplitudes involving the Higgs bosons and the longitudinal gauge bosons demands the flatness of the scalar manifold. The relationship between the finiteness of the electroweak oblique corrections and perturbative unitarity of the scattering amplitudes is also clarified in this language: we verify that once the tree-level unitarity is ensured, then the one-loop finiteness of the oblique correction parameters S and U is automatically guaranteed. *', 'arxivid': '1904.07618', 'author': ['Ryo Nagai [email protected]†e-mail:[email protected] \nInstitute for Cosmic Ray Research (ICRR)\nThe University of Tokyo\n277-8582KashiwaChibaJapan\n\nDepartment of Physics\nTohoku University\n980-8578SendaiMiyagiJapan\n', 'Masaharu Tanabashi \nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan\n\nNagoya University\n464-8602NagoyaJapan\n', 'Koji Tsumura \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n', 'Yoshiki Uchida \nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan\n\nNagoya University\n464-8602NagoyaJapan\n'], 'authoraffiliation': ['Institute for Cosmic Ray Research (ICRR)\nThe University of Tokyo\n277-8582KashiwaChibaJapan', 'Department of Physics\nTohoku University\n980-8578SendaiMiyagiJapan', 'Department of Physics\nNagoya University\n464-8602NagoyaJapan', 'Nagoya University\n464-8602NagoyaJapan', 'Department of Physics\nKyoto University\n606-8502KyotoJapan', 'Department of Physics\nNagoya University\n464-8602NagoyaJapan', 'Nagoya University\n464-8602NagoyaJapan'], 'corpusid': 118713811, 'doi': '10.1103/physrevd.100.075020', 'github_urls': [], 'n_tokens_mistral': 44731, 'n_tokens_neox': 34832, 'n_words': 17596, 'pdfsha': '6006f007a129e6666fc7df07553e58b8817635b0', 'pdfurls': ['https://arxiv.org/pdf/1904.07618v1.pdf'], 'title': ['Symmetry and geometry in generalized Higgs effective field theory -Finiteness of oblique corrections v.s. perturbative unitarity', 'Symmetry and geometry in generalized Higgs effective field theory -Finiteness of oblique corrections v.s. perturbative unitarity'], 'venue': []}

arxiv

18 Mar 1998 March 16, 1998 Salah-Eldin A Mohammed Michael K R Scheutzow 18 Mar 1998 March 16, 1998arXiv:math/9803160v1 [math.PR] THE STABLE MANIFOLD THEOREM FOR STOCHASTIC DIFFERENTIAL EQUATIONS * We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory. Introduction. Consider the following Stratonovich and Itô stochastic differential equations (sde's) on R d : dφ(t) = • F (•dt, φ(t)), t > s φ(s) = x (S) dφ(t) = F (dt, φ(t)), t > s φ(s) = x (I) defined on a filtered probability space (Ω, F , (F t s ) s≤t , P ). Equation (S) is driven by a continuous forward-backward spatial semimartingale F : R × R d × Ω → R d (Kunita [Ku]). Both • F and F have stationary ergodic increments. It is known that, under suitable regularity conditions on the driving spatial semimartingale • F , the sde (S) admits a continuous (forward) stochastic flow φ s,t : [Ku]). The inverse flow is denoted by φ t,s := φ −1 s,t : R d × Ω → R d , −∞ < s ≤ t < ∞. This flow is generated by Kunita's backward Stratonovich sde R d × Ω → R d , −∞ < s ≤ t < ∞ (dφ(s) = − • F (•ds, φ(s)), s < t φ(t) = x. (S − ) Similarly, the inverse flow φ t,s := φ −1 s,t : R d × Ω → R d , −∞ < s ≤ t < ∞ of the Itô equation (I) solves a backward Itô sde with a suitable correction term ( [Ku],p. 117). In this article, we prove a local stable-manifold theorem for the sde's (S) and (I) under the condition that the driving semimartingales • F and F have stationary ergodic increments. Our main result is Theorems 3.1. It gives a random flowinvariant local splitting of R d into stable and unstable differentiable submanifolds in the neighborhood of each hyperbolic stationary solution. The method we use to establish these results is based on a non-linear discrete-time multiplicative ergodic theorem due to Ruelle ([Ru.1], cf. [Ru.2]). Although the article is largely selfcontained, familiarity with the arguments in [Ru.1] will sometimes be needed. Key ingredients of this approach are Ruelle-Oseledec integrability conditions which we prove in Lemma 3.1. The proof of this lemma is in turn based on spatial estimates on the flow and its derivatives ( [Ku], [M-S.2]). These estimates are stated in Theorem 2.1 for easy reference. During the past few years, several authors have contributed to the development of the stable-manifold theorem for non-linear sde's . The first successful attempt was carried out by Carverhill for sde's on compact manifolds ( [C]). In [C], a stable manifold theorem is obtained in the globally asymptotically stable case where the Lyapunov exponents of the linearized flow are all negative. The general hyperbolic case with positive Lyapunov exponents is not treated in [C]. The work by Boxler ([Bo]) focuses on the existence of a (global) center manifold under small (white) noise. Wanner ([Wa]) deals essentially with the real-noise case. In this paper, we are able to handle a much more general type of noise with a simple and transparent approach. In addition, we expect the method of construction of the unstable manifold to work if the state space is replaced by a finite-dimensional Riemannian manifold, cf. [C]. The multiplicative ergodic theory of linear finite-dimensional systems was initiated by Oseledec in his fundamental work ( [O]). An infinite-dimensional stablemanifold theorem for linear stochastic delay equations was developed by Mohammed [Mo.1] in the white-noise case, and by Mohammed and Scheutzow for general semimartingales with stationary ergodic increments ([M-S.1]). Basic setting and preliminary results. Let (Ω, F , P ) be a probability space. Let θ : R × Ω → Ω be a P -preserving flow on Ω, viz. (i) θ is jointly measurable, (ii) θ(t + s, ·) = θ(t, ·) • θ(s, ·), s, t ∈ R, (iii) θ(0, ·) = I Ω , the identity map on Ω, (iv) P • θ(t, ·) −1 = P, t ∈ R. Denote byF the P -completion of F . Let (F t s : ∞ < s ≤ t < ∞) be a family of sub-σ-algebras ofF satisfying the following conditions: (i) θ(−r, ·)(F t s ) = F t+r s+r for all r ∈ R, −∞ < s ≤ t < ∞. (ii) For each s ∈ R, both (Ω,F, (F s+u s ) u≥0 , P ) and (Ω,F, (F s s−u ) u≥0 , P ) are filtered probability spaces satisfying the usual conditions ([Pr.2]). A random field F : R × R d × Ω → R d is called a (continuous forward) spatial semimartingale helix if it satisfies the following: (i) For every s ∈ R, there exists a sure event Ω s ∈ F such that F (t + s, x, ω) = F (t, x, θ(s, ω)) + F (s, x, ω) for all t ∈ R, all ω ∈ Ω s and all x ∈ R d . (ii) For almost all ω ∈ Ω, the mapping R × R d ∋ (t, x) → F (t, x, ω) ∈ R d is continuous. (iii) For any fixed s ∈ R and x ∈ R d , the process F (s + t, x, ω) − F (s, x, ω), t ≥ 0, is an (F s+t s ) t≥0 -semimartingale. Similarly, a random field F : R × R d × Ω → R d is called a continuous backward spatial semimartingale helix if it satisfies (i) and (ii) and has the property that for fixed s ∈ R and x ∈ R d , the process F (s − t, x, ω) − F (s, x, ω), t ≥ 0, is an (F s s−t ) t≥0 -semimartingale. In (iii) above, it is enough to require that the semimartingale property holds for some fixed s (e.g. s = 0); then it will hold automatically for every s ∈ R ([A-S], Theorem 14). Note that a semimartingale helix F always satisfies F (0, x, ω) = 0 for a.a. ω ∈ Ω and all x ∈ R d . It is also possible to select a suitable perfect version of F such that the helix property (i) holds for every ω ∈ Ω. See [A-S] for further details, and [Pr.1] for other general properties of semimartingale helices. Suppose that the continuous forward semimartingale helix F : R ×R d ×Ω → R d is decomposed as F (t, x) = V (t, x) + M (t, x), t ∈ R + , x ∈ R d , where V (·, x) := (V 1 (·, x), · · · , V d (·, x)) is a continuous bounded variation process, M (·, x) := (M 1 (·, x), · · · , M d (·, x)) is a continuous local martingale with respect to (F t 0 ) t≥0 , and V (0, x) = M (0, x) = 0 for each x ∈ R d . Let M i (·, x), M j (·, y) be the joint quadratic variation of M i (·, x), M j (·, y), for x, y ∈ R d , 1 ≤ i, j ≤ d. Throughout this paper, assume that F |[0, ∞) has forward local characteristics (a(t, x, y), b(t, x)) that satisfy the relations M i (·, x), M j (·, y) (t) = t 0 a i,j (u, x, y) du, V i (t, x) = t 0 b i (u, x) du for all 1 ≤ i, j ≤ d, 0 ≤ t ≤ T , and where a(t, x, y) := (a i,j (t, x, y)) i,j=1,··· ,d , b(t, x) := (b 1 (t, x), · · · , b d (t, x)). Further measurability properties of the local characteristics are given in ( [Ku],. Note that the local characteristics are uniquely determined by F up to null sets. In what follows, let △ denote the diagonal △ : = {(x, x) : x ∈ R d } in R d × R d , and let △ c be its complement. The space R d carries the usual Euclidean norm | · |. We shall use the notation α := (α 1 , α 2 , · · · , α d ), D α x := ∂ |α| (∂x 1 ) α 1 · · · (∂x d ) α d , |α| := d i=1 α i , for α i non-negative integers, i = 1, · · · , d. Following [Ku], we shall say that the spatial forward semimartingale F has forward local characteristics of class (B m,δ ub , B k,δ ub ) for non-negative integers m, k and δ ∈ (0, 1], if for all T > 0, its characteristics satisfy ess sup ω∈Ω sup 0≤t≤T [ a(t)˜ m+δ + b(t) k+δ ] < ∞, where a(t)˜ m+δ := sup x,y∈R d |a(t, x, y)| (1 + |x|)(1 + |y|) + 1≤|α|≤m sup x,y∈R d |D α x D α y a(t, x, y)| + |α|=m D α x D α y a(t, ·, ·)ˆ δ , b(t) k+δ := sup x∈R d |b(t, x)| (1 + |x|) + 1≤|α|≤k sup x∈R d |D α x b(t, x)| + |α|=k sup (x,y)∈△ c |D α x b(t, x) − D α y b(t, y)| |x − y| δ , and fˆ δ := sup |f (x, y) − f (x ′ , y) − f (x, y ′ ) + f (x ′ , y ′ )| |x − x ′ | δ |y − y ′ | δ : (x, x ′ ), (y, y ′ ) ∈ △ c for any δ-Hölder continuous function f : R d × R d → R d . Similar definitions hold for the backward local characteristics of a backward spatial semimartingale. Now consider the Stratonovich and Itô stochastic differential equations dφ(t) = • F (•dt, φ(t)), t > s, φ(s) = x,(S)dφ(t) = F (dt, φ(t)), t > s, φ(s) = x.(I) The sde (S) is driven by a continuous forward(-backward) spatial helix semimartin- gale • F (t, x) := ( • F 1 (t, x), · · · , • F d (t, x)), x ∈ R d . In the sde (I), F denotes a spatial continuous forward helix semimartingale. It is known that, under suitable regularity hypotheses on the local characteristics of • F (or F ), the sde's (S) and (I) generate the same stochastic flow. Throughout this article, these flows will denoted by the same symbol {φ s,t : s, t ∈ R}. More precisely, we will need the following hypotheses: Hypothesis (ST(k, δ)). • F is a continuous spatial helix forward semimartingale with forward local characteristics of class (B k+1,δ ub , B k,δ ub ). The function [0, ∞) × R d ∋ (t, x) → d j=1 ∂a ·,j (t, x, y) ∂x j y=x ∈ R d belongs to B k,δ ub . Hypothesis (ST − (k, δ)). • F is a continuous helix backward semimartingale with backward local characteristics of class (B k+1,δ ub , B k,δ ub ). Hypothesis (IT(k, δ)). F : R × R d × Ω → R d is a continuous spatial helix forward semimartingale with forward local characteristics of class (B k,δ ub , B k,δ ub ). The following proposition establishes a relationship between the sde's (S) and (I). Proposition 2.1. Suppose the helix semimartingale • F satisfies Hypothesis (ST(k, δ)) for some positive integer k and δ ∈ (0, 1]. Let the following relation hold: F (t, x, ω) := • F (t, x, ω) + 1 2 t 0 d j=1 ∂a ·,j (u, x, y) ∂x j y=x du, t ∈ R, x ∈ R d . Then F is a helix semimartingale which satisfies Hypothesis (IT(k, δ)). In this case, the sde's (S) and (I) generate the same stochastic flow φ s,t , s, t ∈ R, on R d . Proof. The assertion of the proposition follows from Theorem 3.4.7 in [Ku], except for the helix property. The helix property of F follows from that of • F and the fact that the R d×d -valued process < • F (·, x), • F (·, y) > (t) =< F (·, x), F (·, y) > (t), t ∈ R, is a helix for any x, y ∈ R d ([Pr.1]). Proposition 2.1 shows that for a given k, δ, Hypothesis (ST(k, δ)) is stronger than (IT(k, δ)). Although our results will cover both the Stratonovich and Itô cases, the reader may note that the Stratonovich sde (S) allows for a complete and more aesthetically pleasing dynamic characterization of the stochastic flow φ s,t and its inverse. Indeed, under (ST(k, δ)) and (ST − (k, δ)), φ −1 s,t solves the backward Stratonovich sde based on • F and hence provides a natural dynamical representation of the local unstable manifold in terms of trajectories of the backward Stratonovich sde. Such a dynamical characterization is not available for the Itô sde (I). See Section 3. From now on, we will implicitly assume that the spatial semimartingales • F and F are related by the formula in Proposition 2.1. In this context, all our results will be derived under both sets of hypotheses (ST(k, δ)) and (IT(k, δ)), although the conclusions pertain invariably to the generated flow φ s,t . The following proposition is elementary. Its proof is an easy induction argument using the chain rule. Proposition 2.2. Let f := (f 1 , f 2 , · · · , f d ) : R d → R d be a C k diffeomorphism for some integer k ≥ 1. Then, for each α with 1 ≤ |α| ≤ k, D α x (f ) −1 i (x) = p α,i (f −1 (x)) | det[Df (f −1 (x))]| n α , i = 1, · · · , d,(1) for all x ∈ R d , and some integer n α ≥ 1. In the above identity, p α,i (y) is a polynomial in the partial derivatives of f of order up to |α| evaluated at y ∈ R d . Proof. We use induction on α. For |α| = 1, the chain rule gives Df −1 (x) = [Df (f −1 (x))] −1 . By Cramer's rule, this implies (1) with n α = 1. Assume by induction that for some integer 1 ≤ n < k, (1) holds for all α such that |α| ≤ n and all i = 1, · · · , d. Take α such that |α| = n, and fix i, j ∈ {1, 2, · · · , d}. Taking partial derivatives with respect to x j in both sides of (1) shows that the right-hand-side of the resulting equation is again of the same form with α replaced byα := (α 1 ,α 2 , · · · ,α d ), whereα i := α i + δ i,j . This completes the proof of the proposition. The next proposition allows the selection of sure θ(t, ·)-invariant events in F from corresponding ones inF . Proposition 2.3. Let Ω 1 ∈F be a sure event such that θ(t, ·)(Ω 1 ) ⊆ Ω 1 for all t ≥ 0. Then there is a sure event Ω 2 ∈ F such that Ω 2 ⊆ Ω 1 and θ(t, ·)(Ω 2 ) = Ω 2 for all t ∈ R. Proof. DefineΩ 1 := ∞ k=0 θ(k, ·)(Ω 1 ). ThenΩ 1 is a sure event,Ω 1 ⊆ Ω 1 and θ(t, ·)(Ω 1 ) = Ω 1 for all t ∈ R. SinceF is the completion of F , we may pick a sure event Ω 0 ⊆Ω 1 such that Ω 0 ∈ F . Define Ω 2 := {ω : ω ∈ Ω, θ(t, ω) ∈ Ω 0 for Lebesgue-a.e t ∈ R}. Using Fubini's theorem and the P -preserving property of θ, it is easy to check that Ω 2 satisfies all the conclusions of the proposition. Theorem 2.1. Let • F satisfy Hypothesis (ST(k, δ)) (resp. F satisfies (IT(k, δ))) for some k ≥ 1 and δ ∈ (0, 1]. Then there exists a jointly measurable modification of the trajectory random field of (S) (resp. (I)) also denoted by {φ s,t (x) : −∞ < s, t < ∞, x ∈ R d }, with the following properties: If φ : R × R d × Ω → R d is defined by φ(t, x, ω) := φ 0,t (x, ω), x ∈ R d , ω ∈ Ω, t ∈ R, then the following is true for all ω ∈ Ω: (i) For each x ∈ R d , and s, t ∈ R, φ s,t (x, ω) = φ(t − s, x, θ(s, ω)). (ii) (φ, θ) is a perfect cocycle: φ(t + s, ·, ω) = φ(t, ·, θ(s, ω)) • φ(s, ·, ω), for all s, t ∈ R. (iii) For each t ∈ R, φ(t, ·, ω) : R d → R d is a C k diffeomorphism. (iv) The mapping R 2 ∋ (s, t) → φ s,t (·, ω) ∈ Diff k (R d ) is continuous, where Diff k (R d ) denotes the group of all C k diffeomorphisms of R d , given the C k -topology. (v) For every ǫ ∈ (0, δ), γ, ρ, K, T > 0, and 1 ≤ |α| ≤ k, the quantities sup 0≤s,t≤T, x∈R d |φ s,t (x, ω)| [1 + |x|(log + |x|) γ ] , sup 0≤s,t≤T, x∈R d |D α x φ s,t (x, ω)| (1 + |x| γ ) , sup x∈R d sup 0≤s,t≤T, 0<|x ′ −x|≤ρ |D α x φ s,t (x, ω) − D α x φ s,t (x ′ , ω)| |x − x ′ | ǫ (1 + |x|) γ , are finite. Furthermore, the random variables defined by the above expressions have p-th moments for all p ≥ 1. Proof. The cocycle property stated in (ii) is proved in [I-W] for the white-noise case using an approximation argument (cf. [Mo.1], [Mo.2]). Assertions (iii) and (iv) are well-known to hold for a.a. ω ∈ Ω ( [Ku], Theorem 4.6.5). A perfect version of φ s,t satisfying (i)-(iv) for all ω ∈ Ω, is established in [A-S]. The arguments in [A-S] use perfection techniques and Theorem 4.6.5 of [Ku] (cf. also [M-S.1]). Assume that for every γ, T > 0 the two random variables in (v) have finite moments of all orders. Let Ω T,γ be the set of all ω ∈ Ω for which all random variables in (v) are finite. Define the set Ω 0 by Ω 0 := T ∈N n∈N s∈R θ(s, ·)(Ω T,1/n ). Then θ(s, ·)(Ω 0 ) = Ω 0 for all s ∈ R. Furthermore, it is not hard to see that T ∈N n∈N m∈Z θ(mT, ·)(Ω 2T,1/n ) ⊆ Ω 0 . Therefore Ω 0 is a sure event inF. By Proposition 2.3, Ω 0 contains a sure invariant event Ω ′ 0 ∈ F . Hence we can redefine φ s,t (·, ω) and φ(t, ·, ω) to be the identity map R d → R d for all ω ∈ Ω\Ω ′ 0 . This can be done without violating properties (i)-(iv). By Proposition 2.2, Theorem 1 in [M-S.2] and the remark following its proof, it follows that the two random variables X 1 := sup 0≤s≤t≤T, x∈R d |φ s,t (x, ·)| [1 + |x|(log + |x|) γ ] , X 2 := sup 0≤s≤t≤T, x∈R d |x| [1 + |φ s,t (x, ·)|(log + |x|) γ ] have p-th moments for all p ≥ 1. To complete the proof of the first assertion in (v), it is sufficient to show that the random variablê X 1 := sup 0≤s≤t≤T, x∈R d |φ t,s (x, ·)| [1 + |x|(log + |x|) γ ] has p-th moments for all p ≥ 1. To do this, assume (without loss of generality) that γ ∈ (0, 1). From the definition of X 2 , we have |y| ≤ X 2 [1 + |φ s,t (y, ·)|(log + |y|) γ ] for all 0 ≤ s ≤ t ≤ T, y ∈ R d . Use the substitution y = φ t,s (x, ω) = φ −1 s,t (x, ω), φ s,t (y, ω) = x, 0 ≤ s ≤ t ≤ T, ω ∈ Ω, x ∈ R d , to rewrite the above inequality in the form |y| ≤ X 2 [1 + |x|(log + |y|) γ ]. By an elementary computation, the above inequality may be solved for log + |y|. This gives a positive non-random constant K 1 (possibly dependent on ǫ and T ) such that |y| ≤ K 1 X 2 [1 + |x|{1 + (log + |X 2 |) γ + (log + |x|) γ }]. Since X 2 has moments of all orders, the above inequality implies thatX 1 also has p-th moments for all p ≥ 1. We now prove the second assertion in (v). First, note that the following two random variables X 3 := sup 0≤s≤t≤T, x∈R d |D α x φ s,t (x, ·)| (1 + |x| γ ) , |α| ≤ k, X 4 := sup 0≤s≤t≤T, x∈R d |[Dφ s,t (x, ·)] −1 | (1 + |x| γ ) have p-th moments for all p ≥ 1 ( [Ku], Ex. 4.6.9, p. 176; [M-S.2], Remark (i) following Theorem 2). We must show that the random variableŝ X 3 := sup 0≤s≤t≤T, x∈R d |D α x φ −1 s,t (x, ·)| (1 + |x| γ ) , 1 ≤ |α| ≤ k, have p-th moments for all p ≥ 1. Note that there is a positive constant C such that for any non-singular matrix A, one has |(det A) −1 | = | det(A −1 )| ≤ C A −1 d . Using this fact and applying Proposition 2.2 with f := φ s,t , 1 ≤ s ≤ t ≤ T , shows that for every δ ′ > 0, any i ∈ {1, 2, · · · , d} and any 1 ≤ |α| ≤ k, there exists a random variable K δ ′ ∈ p≥1 L p (Ω, R) such that |D α x (φ −1 s,t ) i (x)| ≤ K δ ′ (1 + |x| δ ′ ) m α,i for all x ∈ R d and some positive integer m α,i . Now for any given ǫ > 0, choose δ ′ = γ m α,i to obtain |D α x (φ −1 s,t ) i (x)| ≤ K δ ′ (1 + |x| γ/m α,i ) m α,i ≤ 2 m α,i K δ ′ (1 + |x| γ ) for all x ∈ R d . This shows thatX 3 has p-th moments for all p ≥ 1. The last estimate in (v) follows from a somewhat lengthy argument. We will only sketch it. First note that for every p ≥ 1, there exists a constant c ≥ 0 such that Ku], Theorem 4.6.4, pp. 172-173). Using the above estimate, we can employ the inequality of Garsia-Rodemich-Rumsey in its majorising measure version in order to show that the expression E(|D α x φ s,t (x) − D α x φ s ′ ,t ′ (x ′ )| 2p ) ≤ c(|x − x ′ | 2pδ + |s − s ′ | p + |t − t ′ | p ) uniformly for all x, x ′ ∈ R d , 0 ≤ s ≤ t ≤ T ([sup x∈R d sup 0≤s≤t≤T, 0<|x ′ −x|≤ρ |D α x φ s,t (x, ω) − D α x φ s,t (x ′ , ω)| |x − x ′ | ǫ (1 + |x|) γ has moments of all orders. The argument used to show this is similar to the one used in [I-S]. The application of the Garsia-Rodemich-Rumsey inequality is effected using the following metric on the space [0, T ] × [0, T ] × R d : d((s, t, x), (s ′ , t ′ , x ′ )) := |x − x ′ | δ + |s − s ′ | 1/2 + |t − t ′ | 1/2 . Finally, we extend the estimate to cover the sup over all (s, t) ∈ [0, T ] × [0, T ] by appealing to Proposition 2.2 and the argument used above to establish the existence of p-th moments ofX 3 . This completes the proof of the theorem. The local stable manifold theorem. In this section, we shall maintain the general setting and hypotheses of Section 2. Furthermore, we shall assume from now on that the P -preserving flow θ : R × Ω → Ω is ergodic. For any ρ > 0 and x ∈ R d denote by B(x, ρ) the open ball with center x and radius ρ in R d . Denote byB(x, ρ) the corresponding closed ball. Recall that (φ, θ) is the perfect cocycle associated with the trajectories φ s,t (x) of (S) or (I) (Theorem 2.1). Definition 3.1. Say that the cocycle φ has a stationary trajectory if there exists an F -measurable random variable Y : Ω → R d such that φ(t, Y (ω), ω) = Y (θ(t, ω))(1) for all t ∈ R and every ω ∈ Ω. In the sequel, we will always refer to the stationary trajectory (1) by φ(t, Y ). If (1) is known to hold on a sure event Ω t that may depend on t, then there are "perfect" versions of the stationary random variable Y and of the flow φ such that (1) and the conclusions of Theorem 2.1 hold for all ω ∈ Ω (under the hypotheses therein)( [Sc]). We may replace ω in (1) by θ(s, ω), s ∈ R, to get φ(t, Y (θ(s, ω)), θ(s, ω)) = Y (θ(t + s, ω))(2) for all s, t ∈ R and every ω ∈ Ω. To illustrate the concept of a stationary trajectory, we give a few simple examples. Examples. (i) Consider the SDE: dφ(t) = h(φ(t)) dt + m i=1 g i (φ(t)) • dW i (t) with vector fields h, g i : R d → R d , i = 1, · · · , m, in C ∞ b and globally bounded. Suppose h(x 0 ) = g i (x 0 ) = 0, 1 ≤ i ≤ m for some fixed x 0 ∈ R d . Take Y (ω) = x 0 for all ω ∈ Ω. Then Y is a stationary trajectory of the above SDE. (ii) Consider the affine linear one-dimensional SDE: dφ(t) = λφ(t) dt + dW (t) where λ > 0 is fixed, and W (t) ∈ R is one-dimensional Brownian motion. Take Y (ω) := − ∞ 0 e −λu dW (u), θ(t, ω)(s) = ω(t + s) − ω(t). Using integration by parts and variation of parameters, the reader may check that there is a version of Y such that φ(t, Y (ω), ω) = Y (θ(t, ω)) for all (t, ω) ∈ R × Ω. (iii) Consider a 2-dimensional affine linear SDE in R 2 : dφ(t) = Aφ(t) dt + GdW (t) with A a fixed hyperbolic (2 × 2)-diagonal matrix A := λ 1 0 0 λ 2 where λ 2 < 0 < λ 1 . G is a constant matrix, e.g. G := g 1 g 2 g 3 g 4 where g i ∈ R, i = 1, 2, 3, 4. W := W 1 W 2 is 2-dimensional Brownian motion. Set Y := Y 1 Y 2 where Y 1 := −g 1 ∞ 0 e −λ 1 u dW 1 (u) − g 2 ∞ 0 e −λ 1 u dW 2 (u) and Y 2 := g 3 0 −∞ e −λ 2 u dW 1 (u) + g 4 0 −∞ e −λ 2 u dW 2 (u) . Using variation of parameters and integration by parts (as in (ii)), it is easy to see that Y has a measurable version Y : Ω → R 2 which gives a stationary trajectory of the SDE that satisfies (1). (iv) By Itô's formula, non-linear transforms of the SDE in (iii) under a fixed global diffeomorphism of R 2 , immediately yield a stationary trajectory of the transformed SDE. Lemma 3.1. Let the conditions of Theorem 2.1 hold. Assume also that log + |Y (·)| is integrable. Then the cocycle φ satisfies Ω log + sup −T ≤t 1 ,t 2 ≤T φ(t 2 , Y (θ(t 1 , ω)) + (·), θ(t 1 , ω)) k,ǫ dP (ω) < ∞(3) for any fixed 0 < T, ρ < ∞ and any ǫ ∈ (0, δ). The symbol · k,ǫ denotes the C k,ǫ -norm on C k,ǫ mappingsB(0, ρ) → R d . Furthermore, the linearized flow (D 2 φ(t, Y (ω), ω), θ(t, ω), t ≥ 0) is an L(R d )-valued perfect cocycle and Ω log + sup −T ≤t 1 ,t 2 ≤T D 2 φ(t 2 , Y (θ(t 1 , ω)), θ(t 1 , ω)) L(R d ) dP (ω) < ∞(4) for any fixed 0 < T < ∞. The forward cocycle (D 2 φ(t, Y (ω), ω), θ(t, ω), t > 0), has a non-random finite Lyapunov spectrum {λ m < · · · < λ i+1 < λ i < · · · < λ 2 < λ 1 }. Each Lyapunov exponent λ i has a non-random (finite) multiplicity q i , 1 ≤ i ≤ m, and m i=1 q i = d. The backward linearized cocycle (D 2 φ(t, Y (ω), ω), θ(t, ω), t < 0), admits a "backward" non-random finite Lyapunov spectrum defined by lim t→−∞ 1 t log |D 2 φ(t, Y (ω), ω)(v)|, v ∈ R d , and taking values in {−λ i } m i=1 with non-random (finite) multiplicities q i , 1 ≤ i ≤ m, and m i=1 q i = d. Note that Lemma 3.1 stipulates regularity only on the forward characteristics of • F and F . Proof of Lemma 3.1. We first prove (4). Start with the perfect cocycle property for (φ, θ): φ(t 1 + t 2 , ·, ω) = φ(t 2 , ·, θ(t 1 , ω)) • φ(t 1 , ·, ω),(5) for all t 1 , t 2 ∈ R and all ω ∈ Ω. The perfect cocycle property for (D 2 φ(t, Y (ω), ω), θ(t, ω)) follows directly by taking Fréchet derivatives at Y (ω) on both sides of (5); viz. D 2 φ(t 1 + t 2 , Y (ω), ω) = D 2 φ(t 2 , φ(t 1 , Y (ω), ω), θ(t 1 , ω)) • D 2 φ(t 1 , Y (ω), ω) = D 2 φ(t 2 , Y (θ(t 1 , ω)), θ(t 1 , ω)) • D 2 φ(t 1 , Y (ω), ω)(6) for all ω ∈ Ω 0 , t 1 , t 2 ∈ R. The existence of a fixed discrete spectrum for the linearized cocycle follows the analysis in [Mo.1] and [M-S.1]. This analysis uses the integrability property (4) and the ergodicity of θ. Although (4) is an easy consequence of (6) and Theorem 2.1 (v), it is clear that (3) implies (4). Therefore it is sufficient to establish (3). In view of (1) and the identity φ t 1 ,t 1 +t 2 (x, ω) = φ(t 2 , x, θ(t 1 , ω)), x ∈ R d , t 1 , t 2 ∈ R, (Theorem 2.1(i)), (3) will follow if we show that the following integrals are finite for 0 ≤ |α| ≤ k: Ω log + sup 0≤s,t≤T, |x ′ |≤ρ |D α x φ s,t (φ 0,s (Y (ω), ω) + x ′ , ω)| dP (ω),(7)Ω log + sup 0≤s,t≤T, |x,x ′ ∈B(0,ρ),x =x ′ |D α x φ s,t (φ 0,s (Y (ω), ω) + x, ω) − D α x φ s,t (φ 0,s (Y (ω), ω) + x ′ , ω)| |x − x ′ | ǫ dP (ω), (7 ′ ) For simplicity of notation, we shall denote random constants by the letters K i , i = 1, 2, 3, 4. Each K i , i = 1, 2, 3, 4, has p-th moments for all p ≥ 1 and may depend on ρ and T . The following string of inequalities follows easily from Theorem 2.1 (v). log + sup s,t∈[0,T ], |x ′ |≤ρ |D α x φ s,t (φ 0,s (Y (ω), ω) + x ′ , ω)| ≤ log + sup s∈[0,T ] K 1 (ω)[1 + (ρ + |φ 0,s (Y (ω), ω)|) 2 ] ≤ log + K 2 (ω) + log + [1 + 2ρ 2 + K 3 (ω)(1 + |Y (ω)| 4 )] ≤ log + K 4 (ω) + log [1 + 2ρ 2 ] + 4 log + |Y (ω)|(8) for all ω ∈ Ω. Now (8) and the integrability hypothesis on Y imply that the integral (7) is finite. The finiteness of (7 ′ ) follows in a similar manner using Theorem 2.1 (v). This completes the proof of the lemma. Definition 3.2. A stationary trajectory φ(t, Y ) of φ is said to be hyperbolic if E log + |Y (·)| < ∞ and the linearized cocycle (D 2 φ(t, Y (ω), ω), θ(t, ω), t ≥ 0) has a Lyapunov spectrum {λ m < · · · < λ i+1 < λ i < · · · < λ 2 < λ 1 } which does not contain 0. Let {U (ω), S(ω) : ω ∈ Ω} denote the unstable and stable subspaces for the linearized cocycle (D 2 φ(t, Y (·), ·), θ(t, ·)) as given by Theorem 5.3 in [M-S.1]. See also [Mo.1] . This requires the integrability property (4). The following discussion is devoted to the Stratonovich sde (S) and the linearization of the stochastic flow around a stationary trajectory. The Linearization. In (S), suppose • F is a forward-backward semimartingale helix satisfying Hypotheses (ST(k, δ)) and (ST − (k, δ)) for some k ≥ 2 and δ ∈ (0, 1]. Then it follows from Theorem 4.2(i) that the (possibly anticipating) process φ(t, Y (ω), ω) is a trajectory of the anticipating Stratonovich sde dφ(t, Y ) = • F (•dt, φ(t, Y )), t > 0 φ(0, Y ) = Y.(SII) In the above sde, the Stratonovich differential • F (•dt, ·) is defined as in Section 4 (Definition 4.1, cf. [Ku], p. 86). The above sde follows immediately by substituting x = Y (ω) in dφ(t, x) = • F (•dt, φ(t, x)), t > 0 φ(0) = x ∈ R d    (SI) (Theorem 4.2 (i)). This substitution works in spite of the anticipating nature of φ(t, Y (ω), ω) = Y (θ(t, ω)), because the Stratonovich integral is stable under random anticipating substitutions (Theorem 4.1). Furthermore, we can linearize the sde (S) along the stationary trajectory and then match the solution of the linearized equation with the linearized cocycle D 2 φ(t, Y (ω), ω). That is to say, the (possibly non-adapted) process y(t) := D 2 φ(t, Y (ω), ω), t ≥ 0, satisfies the associated Stratonovich linearized sde dy(t) = D 2 • F (•dt, Y (θ(t)))y(t), t > 0 y(0) = I ∈ L(R d ).    (SIII) In (SIII), the symbol D 2 denotes the spatial (Fréchet) derivative of the driving semimartingale along the stationary trajectory φ(t, Y (ω), ω) = Y (θ(t, ω)) (Theorem 4.2(ii)). In view of Hypothesis (ST − (k, δ)) for k ≥ 2, δ ∈ (0, 1), and Theorem 4.2(iii),(iv), it follows that the backward trajectories φ(t, Y ),ŷ(t) := D 2 φ(t, Y, ·), t < 0, satisfy the backward sde's dφ(t, Y ) = − • F (•dt, φ(t, Y )), t < 0 φ(0, Y ) = Y, (SII − ) dŷ(t) = −D 2 • F (•dt, φ(t, Y ))ŷ(t), t < 0, y(0, Y ) = I ∈ L(R d ).    (SIII − ) Note however that the significance of (SIII) is to provide a direct link between the linearized flow D 2 φ(t, Y (ω), ω) and the linearized sde. The Stratonovich equation (SII) does not play a direct role in the construction of the stable and unstable manifolds (cf. [Wa], Section 4.2). On the other hand, (SII) and (SII − ) provide a dynamic characterization of the stable and unstable manifolds in Theorem 3.1 (a), (d). In order to apply Ruelle's discrete theorem ([Ru.1], Theorem 5.1, p. 292), we will introduce the following auxiliary cocycle Z : R × R d × Ω → R d , which is essentially a "centering" of the flow φ about the stationary solution: Z(t, x, ω) := φ(t, x + Y (ω), ω) − Y (θ(t, ω))(9) for t ∈ R, x ∈ R d , ω ∈ Ω. Lemma 3.2. Assume the hypotheses of Theorem 2.1. Then (Z, θ) is a perfect cocycle on R d and Z(t, 0, ω) = 0 for all t ∈ R, and all ω ∈ Ω. Proof. Let t 1 , t 2 ∈ R, ω ∈ Ω, x ∈ R d . Then by the cocycle property for φ and Definition 3.1, we have Z(t 2 , Z(t 1 , x, ω), θ(t 1 , ω)) = φ(t 2 , Z(t 1 , x, ω) + Y (θ(t 1 , ω)), θ(t 1 , ω)) − Y (θ(t 2 , θ(t 1 , ω))) = φ(t 2 , φ(t 1 , x + Y (ω), ω), θ(t 1 , ω)) − Y (θ(t 2 + t 1 , ω)) = Z(t 1 + t 2 , x, ω). The assertion Z(t, 0, ω) = 0, t ∈ R, ω ∈ Ω, follows directly from the definition of Z and Definition 3.1. The next lemma will be needed in order to construct the shift-invariant sure events appearing in the statement of the local stable manifold theorem. The lemma essentially gives "perfect versions" of the ergodic theorem and Kingman's subadditive ergodic theorem. Lemma 3.3. (i) Let h : Ω → R + be F -measurable and such that Ω sup 0≤u≤1 h(θ(u, ω)) dP (ω) < ∞. Then there is a sure event Ω 1 ∈ F such that θ(t, ·)(Ω 1 ) = Ω 1 for all t ∈ R, and lim t→∞ 1 t h(θ(t, ω)) = 0 for all ω ∈ Ω 1 . (ii) Suppose f : R + × Ω → R ∪ {−∞} is a measurable process on (Ω, F , P ) satisfying the following conditions (a) Ω sup 0≤u≤1 f + (u, ω) dP (ω) < ∞, Ω sup 0≤u≤1 f + (1 − u, θ(u, ω)) dP (ω) < ∞ (b) f (t 1 + t 2 , ω) ≤ f (t 1 , ω) + f (t 2 , θ(t 1 , ω)) for all t 1 , t 2 ≥ 0 and all ω ∈ Ω. Then there is sure event Ω 2 ∈ F such that θ(t, ·)(Ω 2 ) = Ω 2 for all t ∈ R, and a fixed number f * ∈ R ∪ {−∞} such that lim t→∞ 1 t f (t, ω) = f * for all ω ∈ Ω 2 . Proof. A proof of (i) is given in [Mo.1] , Lemma 5 (iii) with a sure eventΩ 1 ∈F such that θ(t, ·)(Ω 1 ) ⊆Ω 1 for all t ≥ 0. Proposition 2.3 now gives a sure event Ω 1 ⊆Ω 1 such that Ω 1 ∈ F and satisfies assertion (i) of the lemma. Assertion (ii) follow from [Mo.1] , Lemma 7, and Proposition 2.3. The proof of the local stable-manifold theorem (Theorem 3.1) uses a discretization argument that requires the following lemma. Lemma 3.4. Assume the hypotheses of Lemma 3.2, and suppose that log + |Y (·)| is integrable. Then there is a sure event Ω 3 ∈ F with the following properties: (i) θ(t, ·)(Ω 3 ) = Ω 3 for all t ∈ R, (ii) For every ω ∈ Ω 3 and any x ∈ R d , the statement lim sup n→∞ 1 n log |Z(n, x, ω)| < 0 (10) implies lim sup t→∞ 1 t log |Z(t, x, ω)| = lim sup n→∞ 1 n log |Z(n, x, ω)|.(11) Proof. The integrability condition (3) of Lemma 3.1 implies that Ω log + sup 0≤t 1 ,t 2 ≤1, x * ∈B(0,1) D 2 Z(t 1 , x * , θ(t 2 , ω)) L(R d ) dP (ω) < ∞.(12) Therefore by (the perfect version of) the ergodic theorem (Lemma 3.3(i)), there is a sure event Ω 3 ∈ F such that θ(t, ·)(Ω 3 ) = Ω 3 for all t ∈ R, and lim t→∞ 1 t log + sup 0≤u≤1, x * ∈B(0,1) D 2 Z(u, x * , θ(t, ω)) L(R d ) = 0(13) for all ω ∈ Ω 3 . Let ω ∈ Ω 3 and suppose x ∈ R d satisfies (10). Then (10) implies that there exists a positive integer N 0 (x, ω) such that Z(n, x, ω) ∈B(0, 1) for all n ≥ N 0 . Let n ≤ t < n + 1 where n ≥ N 0 . Then by the cocycle property for (Z, θ) and the Mean Value Theorem, we have is obvious. Hence (11) holds, and the proof of the lemma is complete. sup n≤t≤n+1 1 t log |Z(t, x, ω)| ≤ 1 n log + sup 0≤u≤1, x * ∈B(0,1) D 2 Z(u, x * , θ(n, ω)) L(R d ) + n (n + 1) 1 n log |Z(n, x, ω)|. In order to formulate the measurability properties of the stable and unstable manifolds, we will consider the class C(R d ) of all non-empty compact subsets of R d . Give C(R d ) the Hausdorff metric d * : d * (A 1 , A 2 ) := sup{d(x, A 1 ) : x ∈ A 2 } ∨ sup{d(y, A 2 ) : y ∈ A 1 } where A 1 , A 2 ∈ C(R d ), and d(x, A i ) := inf{|x − y| : y ∈ A i }, x ∈ R d , i = 1, 2. Denote by B(C(R d )) the Borel σ-algebra on C(R d ) with respect to the metric d * . Then (C(R d ), d * ) is a complete separable metric space. Morevover, it is not hard to see that finite non-empty intersections are jointly measurable and translations are jointly continuous on C(R d ). These facts are used in the proof of Theorem 3.1 (h). We now state the local stable manifold theorem for the sde's (S) and (I) around a hyperbolic stationary solution. Theorem 3.1. (Local Stable and Unstable Manifolds) Assume that • F satisfies Hypothesis (ST(k, δ)) (resp. F satisfies (IT(k, δ))) for some k ≥ 1 and δ ∈ (0, 1]. Suppose φ(t, Y ) is a hyperbolic stationary trajectory of (S) (resp. (I)) with E log + |Y | < ∞. Suppose the linearized cocycle (D 2 φ(t, Y (ω), ω), θ(t, ω), t ≥ 0) has a Lyapunov spectrum {λ m < · · · < λ i+1 < λ i < · · · < λ 2 < λ 1 }. Define λ i 0 := max{λ i : λ i < 0} if at least one λ i < 0. If all λ i > 0, set λ i 0 = −∞. (This implies that λ i 0 −1 is the smallest positive Lyapunov exponent of the linearized flow, if at least one λ i > 0; in case all λ i are negative, set λ i 0 −1 = ∞.) Fix ǫ 1 ∈ (0, −λ i 0 ) and ǫ 2 ∈ (0, λ i 0 −1 ). Then there exist (i) a sure event Ω * ∈ F with θ(t, ·)(Ω * ) = Ω * for all t ∈ R, (ii) F -measurable random variables ρ i , β i : Ω * → [0, ∞), β i > ρ i > 0, i = 1, 2, such that for each ω ∈ Ω * , the following is true: There are C k,ǫ (ǫ ∈ (0, δ)) submanifoldsS(ω),Ũ(ω) ofB(Y (ω), ρ 1 (ω)) and B(Y (ω), ρ 2 (ω)) (resp.) with the following properties: (a)S(ω) is the set of all x ∈B(Y (ω), ρ 1 (ω)) such that |φ(n, x, ω) − Y (θ(n, ω))| ≤ β 1 (ω) e (λ i 0 +ǫ 1 )n for all integers n ≥ 0. Furthermore, lim sup t→∞ 1 t log |φ(t, x, ω) − Y (θ(t, ω))| ≤ λ i 0(14) for all x ∈S(ω). Each stable subspace S(ω) of the linearized flow D 2 φ is tangent at Y (ω) to the submanifoldS(ω), viz. T Y (ω)S (ω) = S(ω). In particular, dimS(ω) = dim S(ω) and is non-random. (b) lim sup t→∞ 1 t log sup |φ(t, x 1 , ω) − φ(t, x 2 , ω)| |x 1 − x 2 | : x 1 = x 2 , x 1 , x 2 ∈S(ω) ≤ λ i 0 . (c) (Cocycle-invariance of the stable manifolds): There exists τ 1 (ω) ≥ 0 such that φ(t, ·, ω)(S(ω)) ⊆S(θ(t, ω)), t ≥ τ 1 (ω).(15) Also D 2 φ(t, Y (ω), ω)(S(ω)) = S(θ(t, ω)), t ≥ 0. (16) (d)Ũ(ω) is the set of all x ∈B(Y (ω), ρ 2 (ω)) with the property that |φ(−n, x, ω) − Y (θ(−n, ω))| ≤ β 2 (ω) e (−λ i 0 −1 +ǫ 2 )n(17) for all integers n ≥ 0. Also lim sup t→∞ 1 t log |φ(−t, x, ω) − Y (θ(−t, ω))| ≤ −λ i 0 −1 .(18) for all x ∈Ũ(ω). Furthermore, U(ω) is the tangent space toŨ (ω) at Y (ω). In particular, dimŨ(ω) = dim U(ω) and is non-random. (e) lim sup t→∞ 1 t log sup |φ(−t, x 1 , ω) − φ(−t, x 2 , ω)| |x 1 − x 2 | : x 1 = x 2 , x 1 , x 2 ∈Ũ(ω) ≤ −λ i 0 −1 . (f ) (Cocycle-invariance of the unstable manifolds): There exists τ 2 (ω) ≥ 0 such that φ(−t, ·, ω)(Ũ(ω)) ⊆Ũ(θ(−t, ω)), t ≥ τ 2 (ω). Also D 2 φ(−t, Y (ω), ω)(U(ω)) = U(θ(−t, ω)), t ≥ 0. (g) The submanifoldsŨ (ω) andS(ω) are transversal, viz. R d = T Y (ω)Ũ (ω) ⊕ T Y (ω)S (ω).(21) (h) The mappings Ω → C(R d ), Ω → C(R d ), ω →S(ω) ω →Ũ(ω) are (F , B(C(R d )))-measurable. Assume, in addition, that • F satisfies Hypothesis (ST(k, δ)) (resp. F satisfies (IT(k, δ))) for every k ≥ 1 and δ ∈ (0, 1]. Then the local stable and unstable manifoldsS(ω),Ũ(ω) are C ∞ . The following corollary follows from Theorem 3. dx(t) = h(x(t)) dt + m i=1 g i (x(t))dW i (t) (V ) Suppose that for some k ≥ 1, δ ∈ (0, 1), h, g i , 1 ≤ i ≤ m, are C k,δ b vector fields on R d , and W := (W 1 , · · · , W m ) is m-dimensional Brownian motion on Wiener space (Ω, F , P ). Let θ : R × Ω → Ω denote the canonical Brownian shift θ(t, ω)(s) := ω(t + s) − ω(t), t, s ∈ R, ω ∈ Ω.(22) Suppose φ(t, Y ) is a hyperbolic stationary trajectory of (V) with E log + |Y | < ∞. Then the conclusions of Theorem 3.1 hold. Furthermore, if the vector fields h, g i , 1 ≤ i ≤ m, are C ∞ b , then the conclusions of Theorem 3.1 hold, whereS(ω),Ũ(ω) are C ∞ manifolds. Remarks. (i) A similar statement to that of Corollary 3.1.1 holds for the corresponding Stratonovich sde driven by finite-dimensional Brownian motion: dx(t) = h(x(t)) dt + m i=1 g i (x(t)) • dW i (t)(SIV ) However, in this case one needs stronger conditions to ensure that Hypothesis (ST(k, δ)) hold for (SIV). In fact, such hypotheses will hold if we assume that the functions R d ∋ x → m l=1 ∂g i l (x) ∂x j g j l (x) ∈ R are in C k,δ b for each 1 ≤ i, j ≤ d and some k ≥ 1, δ ∈ (0, 1). Cf. the conditions expressed in [A-I]. For example this holds if for some k ≥ 1, δ ∈ (0, 1), the vector field h is of class C k,δ b and g i , 1 ≤ i ≤ m, are globally bounded and of class C k+1,δ b . We conjecture that the the global boundedness condition is not needed. This conjecture is not hard to check if the vector fields g i , 1 ≤ i ≤ m are C ∞ b and generate a finite-dimensional solvable Lie algebra. See [Ku], Theorem 4.9.10, p. (ii) Recall that if • F is a forward-backward semimartingale helix satisfying Hypotheses (ST(k, δ)) and (ST − (k, δ)) for some k ≥ 2 and δ ∈ (0, 1), then the inverse φ(t, ·, θ(−t, ω)) −1 (x) = φ(−t, x, ω), t > 0, corresponds to a solution of the sde (S − ). Furthermore, φ(−t, Y ) and D 2 φ(−t, Y ), t > 0, satisfy the anticipating sde's (SII − ) and (SIII − ), respectively. See Theorem 4.2 (iii), (iv), of Section 4. (iii) We may replace the stationary random variable Y by its invariant distribution µ and then formulate all our results with respect to the product measure µ⊗P and the underlying skew-product flow. This would give stable and unstable manifolds that are defined a.e.(µ ⊗ P ); cf. [C] for the globally asymptotically stable case on a compact manifold. (iv) In Corollary 3.1.1, one can allow for infinitely many Brownian motions (cf. [Ku], p. 106-107). Details are left to the reader. Proof of Theorem 3.1. Assume the hypotheses of the theorem. Consider the cocycle (Z, θ) defined by (9). Define the family of maps F ω : R d → R d by F ω (x) := Z(1, x, ω) for all ω ∈ Ω and x ∈ R d . Let τ := θ(1, ·) : Ω → Ω. Following Ruelle ([Ru.1], p. 292), define F n ω := F τ n−1 (ω) • · · · • F τ (ω) • F ω . Then by the cocycle property for Z, we get F n ω = Z(n, ·, ω) for each n ≥ 1. Clearly, each F ω is C k,ǫ (ǫ ∈ (0, δ)) and (DF ω )(0) = D 2 φ(1, Y (ω), ω). By measurability of the flow φ, it follows that the map ω → (DF ω )(0) is F -measurable. By (4) of Lemma 3.1, it is clear that the map ω → log + D 2 φ(1, Y (ω), ω) L(R d ) is integrable. Furthermore, the discrete cocycle ((DF n ω )(0), θ(n, ω), n ≥ 0) has a non-random Lyapunov spectrum which coincides with that of the linearized continuous cocycle (D 2 φ(t, Y (ω), ω), θ(t, ω), t ≥ 0), viz. {λ m < · · · < λ i+1 < λ i < · · · < λ 2 < λ 1 }, where each λ i has fixed multiplicity q i , 1 ≤ i ≤ m (Lemma 3.1). Note that λ i 0 (and λ i 0 −1 ) are well-defined by hyperbolicity of the stationary trajectory. If λ i > 0 for all 1 ≤ i ≤ m, then takẽ S(ω) := {Y (ω)} for all ω ∈ Ω. The assertions of the theorem are trivial in this case. From now on suppose that at least one λ i < 0. We use Theorem 5.1 of Ruelle ([Ru.1], p. 292) and its proof to obtain a sure event Ω * 1 ∈ F such that θ(t, ·)(Ω * 1 ) = Ω * 1 for all t ∈ R, F -measurable positive random variables ρ 1 , β 1 : Ω * 1 → (0, ∞), ρ 1 < β 1 , and a random family of C k,ǫ (ǫ ∈ (0, δ)) submanifolds ofB(0, ρ 1 (ω)) denoted byS d (ω), ω ∈ Ω * 1 , and satisfying the following properties for each ω ∈ Ω * 1 : S d (ω) = {x ∈B(0, ρ 1 (ω)) : |Z(n, x, ω)| ≤ β 1 (ω)e (λ i 0 +ǫ 1 )n for all integers n ≥ 0}. (23) EachS d (ω) is tangent at 0 to the stable subspace S(ω) of the linearized flow D 2 φ, viz. T 0Sd (ω) = S(ω). In particular, dimS d (ω) is non-random by the ergodicity of θ. Furthermore, lim sup n→∞ 1 n log sup x 1 =x 2 , x 1 ,x 2 ∈S d (ω) |Z(n, x 1 , ω) − Z(n, x 2 , ω)| |x 1 − x 2 | ≤ λ i 0 .(24) Before we proceed with the proof, we will indicate how one may arrive at the above θ(t, ·)-invariant sure event Ω * 1 ∈ F from Ruelle's proof. Consider the proof of Theorem 5.1 in [Ru.1], p. 293. In the notation of [Ru.1], set T t ω := D 2 Z(t, 0, ω), T n (ω) := D 2 Z(1, 0, θ(n − 1, ω)), τ t (ω) := θ(t, ω), for t ∈ R + , n = 1, 2, 3, · · · . By the integrability condition (4) (Lemma 3.1) and Lemma 3.3 (i),(ii), there is a sure event Ω * 1 ∈ F such that θ(t, ·)(Ω * 1 ) = Ω * 1 for all t ∈ R, with the property that continuous-time analogues of equations (5.2), (5.3), (5.4) in ([Ru.1], p. 45) hold. In particular, lim t→∞ {[D 2 Z(t, 0, ω)] * [D 2 Z(t, 0, ω)]} 1/(2t) = Λ(ω), lim t→∞ 1 t log + Z(1, ·, θ(t, ω)) 1,ǫ = 0,     (25) for all ω ∈ Ω * 1 , ǫ ∈ (0, δ). See Theorem ( For each ω ∈ Ω * 1 , letS(ω) be the set defined in part (a) of the theorem. Then it is easy to see from (23) and the definition of Z that S(ω) =S d (ω) + Y (ω). (26) SinceS d (ω) is a C k,ǫ (ǫ ∈ (0, δ)) submanifold ofB(0, ρ 1 (ω)), it follows from (26) thatS(ω) is a C k,ǫ (ǫ ∈ (0, δ)) submanifold ofB(Y (ω), ρ 1 (ω)). Furthermore, T Y (ω)S (ω) = T 0Sd (ω) = S(ω). In particular, dimS(ω) = dim S(ω) = m i=i 0 q i , and is non-random. Now (24) implies that lim sup n→∞ 1 n log |Z(n, x, ω)| ≤ λ i 0(27) for all ω in the shift-invariant sure event Ω * 1 and all x ∈S d (ω). Therefore by Lemma 3.4, there is a sure event Ω * 2 ⊆ Ω * 1 such that θ(t, ·)(Ω * 2 ) = Ω * 2 for all t ∈ R, and lim sup t→∞ 1 t log |Z(t, x, ω)| ≤ λ i 0(28) for all ω ∈ Ω * 2 and all x ∈S d (ω). This immediately implies assertion (14) of the theorem. To prove assertion (b) of the theorem, let ω ∈ Ω * 1 . By (24), there is a positive integer N 0 := N 0 (ω) (independent of x ∈S d (ω)) such that Z(n, x, ω) ∈B(0, 1) for all n ≥ N 0 . Let Ω * 4 := Ω * 2 ∩ Ω 3 , where Ω 3 is the shift-invariant sure event defined in the proof of Lemma 3.4. Then Ω * 4 is a sure event and θ(t, ·)(Ω * 4 ) = Ω * 4 for all t ∈ R. Using an argument similar to the one used in the proof of Lemma 3.4, it follows that sup n≤t≤n+1 1 t log sup x 1 =x 2 , x 1 ,x 2 ∈S(ω) |φ(t, x 1 , ω) − φ(t, x 2 , ω)| |x 1 − x 2 | = sup n≤t≤n+1 1 t log sup x 1 =x 2 , x 1 ,x 2 ∈S d (ω) |Z(t, x 1 , ω) − Z(t, x 2 , ω)| |x 1 − x 2 | ≤ 1 n log + sup 0≤u≤1, x * ∈B(0,1) D 2 Z(u, x * , θ(n, ω)) L(R d ) + n (n + 1) 1 n log sup x 1 =x 2 , x 1 ,x 2 ∈S d (ω) |Z(n, x 1 , ω) − Z(n, x 2 , ω)| |x 1 − x 2 | for all ω ∈ Ω * 4 , all n ≥ N 0 (ω) and sufficiently large. Taking lim sup n→∞ in the above inequality and using (24), immediately gives assertion (b) of the theorem. To prove the invariance property (16), we apply the Oseledec theorem to the linearized cocycle (D 2 φ(t, Y (ω), ω), θ(t, ω)) ([Mo.1], Theorem 4, Corollary 2). This gives a sure θ(t, ·)-invariant event, also denoted by Ω * 1 , such that D 2 φ(t, Y (ω), ω)(S(ω)) ⊆ S(θ(t, ω)) for all t ≥ 0 and all ω ∈ Ω * 1 . Equality holds because D 2 φ(t, Y (ω), ω) is injective and dim S(ω) = dim S(θ(t, ω)) for all t ≥ 0 and all ω ∈ Ω * 1 . To prove the asymptotic invariance property (15), we will need to take a closer look at the proofs of Theorems 5.1 and 4.1 in [Ru.1], pp. 285-297. We will first show that ρ 1 , β 1 and a sure event (also denoted by) Ω * 1 may be chosen such that θ(t, ·)(Ω * 1 ) = Ω * 1 for all t ∈ R, and with the property that for any ǫ ∈ (0, ǫ 1 ) and every ω ∈ Ω * 1 , there exists a positive K ǫ 1 (ω) for which the inequalities ρ 1 (θ(t, ω)) ≥ K ǫ 1 (ω)ρ 1 (ω)e (λ i 0 +ǫ)t , β 1 (θ(t, ω)) ≥ K ǫ 1 (ω)β 1 (ω)e (λ i 0 +ǫ)t(29) hold for all t ≥ 0. The above inequalities hold in the discrete case (when t = n, a positive integer) from Ruelle's theorem ([Ru.1], Remark (c), p. 297, following the proof of Theorem 5.1). We claim that the relations (29) hold also for continuous time. To see this, we will use the method of proof of Theorems 5.1 and 4.1 in [Ru.1]. In the notation of the proof of Theorem 5.1 ([Ru.1], p. 293), observe that the random variable G in (5.5) may be replaced by the larger onẽ G(ω) := sup t≥0 F τ t ω 1,θ e (−tη−λθ) < +∞, θ ∈ (0, 1](30) for 0 < η < −(λ i 0 + ǫ)/4, and Ruelle's λ corresponds to λ i 0 + ǫ 1 in our notation. Now β 1 may be chosen using δ, A from Theorem (4. , where B ′ ǫ is given by (4.5) in Theorem (4.1) of ([Ru.1], p. 285). Therefore, given any fixed ω ∈ Ω * 1 , we need to determine how the choices of Ruelle's constants δ, A and B ′ ǫ are affected if ω is replaced by θ(l, ω) = τ l (ω) where l is any positive real number. Since T n (τ l (ω)) = D 2 Z(1, 0, θ(n − 1, θ(l, ω))) = T n+l (ω), for all positive integers n, it is sufficient to apply Theorem 4.1 ([Ru.1]) to the sequence {T n+l (ω)} ∞ n=1 . Hence we may follow the discussion in Section (4.7) ([Ru.1], pp. 291-292). We claim that the argument therein still works for positive real l. We will indicate the reasoning for δ and leave the rest of the details to the reader. Consider the definition of δ in (4.15) in the proof of Lemma (4.2) ([Ru.1], p. 288). Set δ(ω) := δ, D(ω) := D, C(ω) := C given by (4.15), (4.11), (4.13), respectively. Redefine D and C by larger constants which we will denote by the same symbols: D(ω) := sup t≥0 e −tη (ξ (t) ) −1 < ∞,(31)C(ω) := sup 0<s<t<∞ 1≤h,k≤m T t ξ (0) h T s ξ (0) k T s ξ (0) h T t ξ (0) k exp{[λ (r(k)) − λ (r(h)) ](t − s)} < ∞,(32) where ξ (t) := (ξ (t) 1 , · · · , ξ (t) m ), ξ (t) k := T t ξ (0) k T t ξ (0) k , 1 ≤ k ≤ m. The λ (r(k)) are the eigenvalues of log Λ(ω) with multiplicities. Observe that D(ω) is finite because the following continuous-time version of (4.9)([Ru.1], p. 287) lim t→∞ 1 t log T t ξ (0) 1 ∧ · · · ∧ T t ξ (0) m = m k=1 λ (r(k))(33) holds everywhere on a θ(t, ·)-invariant sure event in F also denoted by Ω * 1 . This is an immediate consequence of Lemma 3.3 (ii). Cf. [Ru.1], p. 287 and p. 303. The constant C(ω) satisfies the inequality (4.13) of [Ru.1], p. 288, because (by choice of (t (n) k ) * := t (n) k e [λ (r(µ)) −λ (r(k)) ] ) one has N k=1 (t (n) k ) * = T N ξ (0) k e N[λ (r(µ)) −λ (r(k)) ] , for all positive integers N and 1 ≤ k ≤ µ ≤ m. Indeed, C(ω) is finite because lim t→∞ 1 t log T t ξ (0) k = λ (r(k)) on a θ(t, ·)-invariant sure event in F (also denoted by Ω * 1 ). Now replace ω in (31) and (32) by θ(l, ω). This changes ξ (t) to ξ (t+l) , and T t ξ (0) h to T t+l ξ (0) h . Hence we get positive constants K ǫ 2 (ω), K ǫ 3 (ω) such that D(θ(l, ω)) ≤ K ǫ 2 (ω)e l(λ i 0 +ǫ) D(ω), C(θ(l, ω)) ≤ K ǫ 2 (ω)e l(λ i 0 +ǫ) C(ω) for sufficiently small ǫ ∈ (0, ǫ 1 ) and all sufficiently large l. From (4.5) in ([Ru.1], p. 288), we get a positive constant K ǫ 4 (ω) such that δ(θ(l, ω)) ≥ K ǫ 4 (ω)e l(λ i 0 +ǫ) δ(ω) for all sufficiently large l ∈ R + and all sufficiently small ǫ. The behavior of the constants A and B ′ ǫ in Theorem 4.1 ([Ru.1], p. 285) can be analysed in a similar fashion. See [Ru.1], Section (4.7). This yields the inequalities (29). We now proceed to prove (15). Use (b) to obtain a sure event Ω * 5 ⊆ Ω * 4 such that θ(t, ·)(Ω * 5 ) = Ω * 5 for all t ∈ R, and for any 0 < ǫ < ǫ 1 and ω ∈ Ω * 4 , there exists β ǫ (ω) > 0 (independent of x) with |φ(t, x, ω) − Y (θ(t, ω))| ≤ β ǫ (ω)e (λ i 0 +ǫ)t(34) for all x ∈S(ω), t ≥ 0. Fix any real t ≥ 0, ω ∈ Ω * 5 and x ∈S(ω). Let n be a non-negative integer. Then the cocycle property and (34) imply that |φ(n, φ(t, x, ω), θ(t, ω)) − Y (θ(n, θ(t, ω)))| = |φ(n + t, x, ω) − Y (θ(n + t, ω))| ≤ β ǫ (ω)e (λ i 0 +ǫ)(n+t) ≤ β ǫ (ω)e (λ i 0 +ǫ)t e (λ i 0 +ǫ 1 )n .(35) If ω ∈ Ω * 5 , then it follows from (29),(34), (35) and the definition ofS(θ(t, ω)) that there exists τ 1 (ω) > 0 such that φ(t, x, ω) ∈S(θ(t, ω)) for all t ≥ τ 1 (ω). This proves (15) and completes the proof of assertion (c) of the theorem. Note that assertions (a), (b) and (c) still hold for all ω ∈ Ω * 5 . We now prove assertion (d) of the theorem, regarding the existence of the local unstable manifoldsŨ(ω). We do this by running both the flow φ and the shift θ backward in time. Definẽ φ(t, x, ω) := φ(−t, x, ω),Z(t, x, ω) := Z(−t, x, ω),θ(t, ω) := θ(−t, ω) for all t ≥ 0 and all ω ∈ Ω. Clearly (Z(t, ·, ω),θ(t, ω), t ≥ 0) is a smooth cocycle, withZ(t, 0, ω) = 0 for all t ≥ 0. By the hypothesis on F and Y , it follows that the linearized flow (D 2φ (t, Y (ω), ω),θ(t, ω), t ≥ 0) is an L(R d )-valued perfect cocycle with a non-random finite Lyapunov spectrum {−λ 1 < −λ 2 < · · · < −λ i < −λ i+1 < · · · < −λ m } where {λ m < · · · < λ i+1 < λ i < · · · < λ 2 < λ 1 } is the Lyapunov spectrum of the forward linearized flow (D 2 φ(t, Y (ω), ω), θ(t, ω), t ≥ 0). Now apply the first part of the proof of this theorem. This gives stable manifolds for the backward flowφ satisfying assertions (a), (b), (c). This immediately translates into the existence of unstable manifolds for the original flow φ, and assertions (d), (e), (f) automatically hold. In particular, we get a sure event Ω * 6 ∈ F such that θ(−t, ·)(Ω * 6 ) = Ω * 6 for all t ∈ R, and with the property that assertions (d), (e) and (f) hold for all ω ∈ Ω * 6 . Define the sure event Ω * := Ω * 6 ∩ Ω * 5 . Then θ(t, ·)(Ω * ) = Ω * for all t ∈ R. Furthermore, assertions (a)-(f) hold for all ω ∈ Ω * . Assertion (g) follows directly from the following facts T Y (ω)Ũ (ω) = U(ω), T Y (ω)S (ω) = S(ω), R d = U(ω) ⊕ S(ω) for all ω ∈ Ω * . We shall now prove assertion (h). Recall that by (26), S(ω) = T (Y (ω),S d (ω))(36) for all ω ∈ Ω * 1 , where T : R d × C(R d ) → C(R d ) denotes the translation map T (x, A) := x + A, x ∈ R d , A ∈ C(R d ). Hence, by joint continuity of T and measurability of Y , the F -measurability of the mapping Ω ∋ ω →S(ω) ∈ C(R d ) would follow from (36) if we can show that the map Ω ∋ ω →S d (ω) ∈ C(R d ) is F -measurable. The rest of the argument will demonstrate this. Define the sequence of random diffeomorphisms f n (x, ω) := β 1 (ω) −1 e −(λ i 0 +ǫ 1 )n Z(n, x, ω), x ∈ R d , ω ∈ Ω * 1 , for all integers n ≥ 0. Let Hom(R d ) be the topological group of all homeomorphisms of R d onto itself. Hom(R d ) carries the topology of uniform convergence of sequences of maps and their inverses on compacta. The joint measurability of f n implies that for each positive integer n, the map Ω ∋ ω → f n (·, ω) ∈ Hom(R d ) is measurable into the Borel field of Hom(R d ). Using (23),S d (ω) can be expressed in the formS d (ω) = lim m→∞B (0, ρ 1 (ω)) ∩ m i=1 f i (·, ω) −1 (B(0, 1))(37) for all ω ∈ Ω * 1 . In (37), the limit is taken in the metric d * on C(R d ). The Fmeasurability of the map ω →S d (ω) follows directly from (37), the measurability of f i , ρ 1 , that of finite intersections and the continuity of the maps R + ∋ r →B(0, r) ∈ C(R d ). Hom(R d ) ∋ f → f −1 (B(0, 1)) ∈ C(R d ). Hence the mapping Ω ∋ ω →S(ω) ∈ C(R d ) is (F , B(C(R d )))-measurable. A similar argument yields the measurability of Ω ∋ ω →Ũ(ω) ∈ C(R d ). This completes the proof of assertion (h) of the theorem. If • F (resp. F ) satisfy Hypothesis (ST(k, δ)) (resp. (IT(k, δ))) for every k ≥ 1 and δ ∈ (0, 1], then a simple adaptation of the argument in [Ru.1], Section (5.3) (p. 297) gives a sure event in F , also denoted by Ω * such thatS(ω),Ũ(ω) are C ∞ for all ω ∈ Ω * . This completes the proof of Theorem 3.1. Global Stable and Unstable Sets. We will conclude this section by a discussion of global stable and unstable sets for the sde's (S) and (I). Assume all the conditions of Theorem 3.1. Define the set S g (ω) := {x ∈ R d : lim sup t→∞ 1 t log |φ(t, x, ω) − Y (θ(t, ω))| ≤ λ i 0 } for each ω ∈ Ω * . The familyS g (ω), ω ∈ Ω * , is clearly invariant under φ; that is φ(t, ·, ω)(S g (ω)) =S g (θ(t, ω)) for all t ∈ R and all ω ∈ Ω * . Using induction, we may define the family {S n (ω)} ∞ n=0 of C k,ǫ stable submanifolds as follows: S 0 (ω) :=S(ω) S n (ω) := φ(−n, ·, ω)(S(θ(n, ω)), ifS n−1 (ω) ⊆ φ(−n, ·, ω)(S(θ(n, ω))) S n−1 (ω), otherwise for n ≥ 1. In the above definition,S(ω) refers to the stable manifolds constructed in the proof of Theorem 3.1. Note thatS n (ω) ⊆S n+1 (ω) for all n ≥ 0. Furthermore, the global stable setS g (ω) is given bỹ S g (ω) = ∞ n=1S n (ω), ω ∈ Ω * .(38) We will indicate a proof of (38). Fix any ω ∈ Ω * . Then by asymptotic cocycle invariance of the stable manifolds, there is an a positive l 0 := l 0 (ω) such that φ(l, ·, ω)(S(ω)) ⊆S(θ(l, ω)) for all integers l ≥ l 0 . The inclusion (39) follows from Remark (5.2)(c) in ([Ru.1], p. 297). In particular, and by the definition ofS n (ω), it follows thatS n (ω) = φ(−n, ·, ω)(S(θ(n, ω)) for infinitely many integers n > 0. Now let x ∈S g (ω). Then it is easy to see that φ(k, x, ω) ∈S(θ(k, ω)) for sufficiently large k. Fix such a k and call it k 0 . Then there exists l ≥ k 0 such that φ(l, x, ω) ∈S(θ(l, ω)) andS l (ω) = φ(−l, ·, ω)(S(θ(l, ω)). Hence x ∈S l (ω) and therefore x ∈ ∞ n=1S n (ω). Conversely, let x belong to the set on the right-hand-side of (38). Then by definition of thẽ S n (ω), there exists k such that x ∈ φ(−k, ·, ω)(S(θ(k, ω)). By Theorem 3.1(a), this implies that lim sup t→∞ 1 t log |φ(t, x, ω) − Y (θ(t, ω))| ≤ λ i 0 . Hence x ∈S g (ω) , and the proof of (38) is complete. Similar remarks hold for the global unstable set U g (ω) := {x ∈ R d : lim sup t→∞ 1 t log |φ(−t, x, ω) − Y (θ(−t, ω))| ≤ λ i 0 −1 }. The above considerations also show thatS g andŨ g are C k,ǫ manifolds which are immersed (but not in general imbedded) in R d . Appendix. The Substitution Rule. In this appendix, we will establish some results that are aimed towards showing that certain extensions of the Itô integral and the Stratonovich integral are stable under random substitutions. Throughout this appendix, F : R × R l × Ω → R d is a continuous spatial semimartingale based on a filtered probability space such that F (0, x) = 0 for all x ∈ R l . (Note that the helix property is not needed in this section.) We shall use the notation in Sections 1 and 2. Decompose F as F (t, x) = V (t, x) + M (t, x), t ≥ 0, x ∈ R l(1) where V (·, x) := (V 1 (·, x), · · · , V d (·, x)) is a continuous bounded variation process and M (·, x) := (M 1 (·, x), · · · , M d (·, x)), x ∈ R l , is a continuous spatial local martingale such that M (0, x) = V (0, x) = 0 for all x ∈ R l and all ω ∈ Ω. We now introduce a definition of the Itô and the Stratonovich integral with respect to integrands that are possibly anticipating. Let f : [0, ∞) × Ω → R l be a measurable process with continuous sample paths. Take any sequence of partitions π n := {0 = t n 0 < t n 1 < · · · < t n n } of [0, ∞). Suppose lim n→∞ t n n = ∞ and lim n→∞ max{t n i − t n i−1 , i = 1, · · · , n} = 0. Define the sequence I n (t) := n−1 k=0 [M (t n k+1 ∧ t, f (t n k )) − M (t n k ∧ t, f (t n k ))], n ≥ 1, t ≥ 0.(2) If, in addition, M is a C 1 spatial local martingale, then define S n (t) := I n (t) + C n (t), n ≥ 1, t ≥ 0, where C n (t) := 1 2 n−1 k=0 [D 2 M (t n k+1 ∧ t, f (t n k )) − D 2 M (t n k ∧ t, f (t n k ))][f (t n k+1 ∧ t) − f (t n k ∧ t)]. when the limit exists uniformly on compact subsets of [0, ∞) in probability for any sequence of partitions as above. (iii) If F is a spatial semimartingale given by (1), define the Stratonovich integral of f with respect to F by T 0 F (•dt, f (t)) := T 0 V (dt, f (t)) + T 0 M (•dt, f (t)).(6) provided the right side of (6) is defined. The Itô integral is defined analogously (without the circle). Note that our definitions of the Itô and the Stratonovich integral agree with the classical ones when the integrand process f is a continuous semimartingale. For the Stratonovich integral this follows from ( [Ku], Theorem 3.2.5, p. 86). As will be clear from the sequel, the computations become simpler under Definition 4.1 than if we had directly extended Kunita's definition to the non-adapted case. We remark that our definition of the Itô integral does not always coincide with the well-known Skorohod integral even if both are defined. In the following theorem, B k,δ c denotes the class of all C k spatial semimartingales such that for any T > 0, any p ≥ 1 and any compact subset K of R l (or R m ) the p-th moment of the (k + δ)-norms of the characteristics restricted to K are uniformly bounded on [0, T ]. Observe that B k,δ ub ⊂ B k,δ c . The flow generated by an Itô equation driven by F ∈ B 0,1 ub is always in B 0,1 c but generally not in B 0,1 ub (see Theorem 4.2). We now state the substitution rule. Theorem 4.1. Fix δ ∈ (0, 1) and let F (t, y) = M (t, y) + V (t, y) ∈ R d , y ∈ R l , be a spatial semimartingale of class B 0,δ c such that M (0, y) = V (0, y) = 0 for all y ∈ R l . Further let f : [0, ∞) × R m × Ω → R l be a continuous spatial semimartingale such that for any compact subset K of R m , any T > 0 and any p > 1, there exists a constant c such that E(|f (t, x) − f (t, y)| p ) ≤ c|x − y| δp for all x, y ∈ K and all 0 ≤ t ≤ T . Then there is a modification of the Itô integral such that for any F -measurable random variable Y : Ω → R m , one has a.s. T 0 F (dt, f (t, x)) x=Y = T 0 F (dt, f (t, Y ))(7) for all T > 0. If, moreover, M is of class B 1,1 c and f ∈ B 0,δ c , then we also have T 0 F (•dt, f (t, x)) x=Y = T 0 F (•dt, f (t, Y )) a.s.(8) for all T > 0. For Brownian linear integrators, a similar result is given in ([N-P], Propositions 7.7, 7.8), ( [A-I], Theorem 2, Corollary 1) and ( [Nu] (Theorem 5.3.3). In order to prove Theorem 4.1, we will adopt the approach in ( [Nu]). The essence of the argument is to replace f by f (t, x) on the right-hand-side of (2), substitute x = Y (ω) in each finite sum in (2), and then pass to the limit in probability in order to get (7) and (8). Note that the substitution rule holds trivially in the bounded variation integral on the right-hand-side of (6). Hence in all subsequent computations, we can and will assume that V ≡ 0 and F = M . The proof of Theorem 4.1 turns on the following lemma. Lemma 4.1. Let {S n (x), x ∈ R m }, n ≥ 1 be a sequence of (jointly) measurable random fields taking values in a complete separable metric space (E, ρ) such that lim n→∞ S n (x) = S(x) in probability, where {S(x), x ∈ R m } is a random field. Assume that there exist positive constants p ≥ 1, α > m, C = C(T, K, p) such that whenever K > 0, and |x|, |x ′ | ≤ K, one has E[ρ(S n (x), S n (x ′ )) p ] ≤ C|x − x ′ | α for all n ≥ 1. Then the random fields S, S n , n ≥ 1, have continuous modifications (denoted by the same symbols). For any such modifications and any random variable Y : Ω → R m , one has lim n→∞ S n (Y ) = S(Y ) in probability. The proof is given in ( [Nu], Lemma 5.3.1) in the special case where E = R d but the argument therein carries over to our case without change. Observe that the conditions of the Lemma imply that S n (·) converges to S(·) uniformly on compact subsets of R m in probability (which is the reason why the substitution property holds). Proof of Theorem 4.1. Assume, without loss of generality, that F = M , a local martingale. Let us first assume in addition that M and f are bounded uniformly in (t, x, ω) on compact subsets of [0, ∞) × R l (resp. [0, ∞) × R m ). For a given sequence of partitions π n := {0 = t n 0 < t n 1 < · · · < t n n } of [0, ∞) as in Definition 4.1 define the sequence of random fields I n (t, x), x ∈ R m , n ≥ 1, by I n (t, x) := n−1 k=0 [M (t n k+1 ∧ t, f (t n k , x)) − M (t n k ∧ t, f (t n k , x))], n ≥ 1, t ≥ 0. (9) We want to check the assumptions of Lemma 4.1 for the sequence I n taking values in the space of R d -valued continuous functions on [0, ∞). Write I n (t, x) := (I 1 n (t, x), · · · , I d n (t, x)). It is enough to show that for every compact subset K of R m , every T > 0 and every j ∈ {1, 2, · · · , d}, we have E[ sup 0≤s≤T |I j n (s, x) − I j n (s, y)| p ] ≤ c|x − y| α for some p > 1, some α > m, some c > 0 and all x, y ∈ K. Fix a compact subset K of R m , T > 0, j ∈ {1, · · · , d}, p > 1, and abbreviate u k = t n k ∧ T . Using the Burkholder-Davis-Gundy inequality, we find constants c 1 , c 2 and c 3 (independent of n) such that for all x, y ∈ K, one has E[ sup 0≤s≤T |I j n (s, x) − I j n (s, y)| p ] ≤ c 1 E n−1 k=0 M j (u k+1 , f (u k , x)) − M j (u k+1 , f (u k , y))− + −M j (u k , f (u k , x)) + M j (u k , f (u k , y)) 2 p/2 ≤ c 1 n−1 k=0 E|M j (u k+1 , f (u k , x)) − M j (u k+1 , f (u k , y)) + −M j (u k , f (u k , x)) + M j (u k , f (u k , y))| p 2/p p/2 ≤ c 2 n−1 k=0 E( u k+1 u k a jj (s, f (u k , x), f (u k , x)) − 2a jj (s, f (u k , x), f (u k , y)) + a jj (s, f (u k , y), f (u k , y))ds) p/2 2/p p/2 ≤ c 3 n−1 k=0 (u k+1 − u k )(E(|f (u k , x) − f (u k , y)| 2δp )) 1/p p/2 , where we have used Hölder's inequality, the boundedness of f and the fact that M ∈ B 0,δ c to obtain the last inequality. Inserting the moment estimate on f in Theorem 4.1, we see that for each p ≥ 1, T > 0 and each compact subset K of R m , there exists a constant c 4 such that E[ sup 0≤s≤T |I n (s, x) − I n (s, y)| p ] ≤ c 4 |x − y| δ 2 p . for all n and all x, y ∈ K. Now take p sufficiently large so that δ 2 p > m. Therefore the substitution formula follows from Lemma 4.1 in the Itô case under the additional constraint that M and f are bounded. For general M and f , we get the uniform convergence I n (t, x) → I(t, x) in probability on compacts of [0, ∞) × R m , by a straightforward localization argument. To show (8) we assume first that M , D 2 M and f are uniformly bounded in (t, x, ω) on compact subsets. Let C n (t, x) = 1 2 n−1 k=0 [D 2 M (t n k+1 ∧t, f (t n k , x))−D 2 M (t n k ∧t, f (t n k , x))][f (t n k+1 ∧t, x)−f (t n k ∧t, x)]. To apply Lemma 4.1, we will show that for every compact subset K of R m and every T > 0 there exist p > 1, α > m and c 5 > 0 such that E sup 0≤s≤T |C n (s, x) − C n (s, y)| p ≤ c 5 |x − y| α for all n ∈ N and all x, y ∈ K. We will use the following abbreviations (suppressing the dependence on n): A k (t, x) = D 2 M (t n k+1 ∧ t, f (t n k , x)) − D 2 M (t n k ∧ t, f (t n k , x)) and B k (t, x) = f (t n k+1 ∧ t, x) − f (t n k ∧ t, x). Then we get for p ≥ 1 and all x, y ∈ K E sup 0≤t≤T |2C n (t, x) − 2C n (t, y)| p (10) ≤ 2 p (E sup 0≤t≤T | n−1 k=0 (A k (t, x) − A k (t, y))B k (t, x)| p + E sup 0≤t≤T | n−1 k=0 A k (t, y)(B k (t, x) − B k (t, y))| p ). Using Hölder's inequality, we get E sup 0≤t≤T | n−1 k=0 (A k (t, x) − A k (t, y))B k (t, x)| p ≤ ( n−1 k=0 (E(( sup 0≤t≤T |A k (t, x) − A k (t, y)| p )( sup 0≤t≤T |B k (t, x)| p ))) 1/p ) p ≤ ( n−1 k=0 (E sup 0≤t≤T |A k (t, x) − A k (t, y)| 2p ) 1/2p )(E sup 0≤t≤T |B k (t, x)| 2p ) 1/2p ) p .(11) Using the inequality of Burkholder, Davis and Gundy and [Ku], Theorem 3.1.2 which allows us to interchange spatial derivatives and the quadratic variation there exist constants c 6 , c 7 and c 8 (independent of k and n) such that (E sup 0≤t≤T |A k (t, x) − A k (t, y)| 2p ) 1/2p ≤ c 6 (u k+1 − u k ) 1/2 |x − y| δ 2 (this is derived just like the first part of the proof) and (E sup 0≤t≤T (|B k (t, x)| 2p )) 1/2p ≤ c 7 (u k+1 − u k ) 1/2 + c 8 (u k+1 − u k ). Since (u k+1 −u k ) ≤ T 1/2 (u k+1 −u k ) 1/2 we can in fact delete the term c 8 (u k+1 −u k ) in (13) by increasing c 7 accordingly. Similarly, for p ≥ 1 there exists some constant c 9 such that for all x, y ∈ K, we have E sup 0≤t≤T | n−1 k=0 A k (t, y)(B k (t, x) − B k (t, y))| p ≤ n−1 k=0 (E sup 0≤t≤T |A k (t, y)| 2p ) 1/2p )(E sup 0≤t≤T |B k (t, x) − B k (t, y)| 2p ) 1/2p p ≤ c 9 ( n−1 k=0 ((u k+1 − u k ) 1/2 |x − y| δ (u k+1 − u k ) 1/2 )) p ≤ c 9 T p |x − y| δp .(14) Inserting (12) and (13) into (11) and then (11) and (14) into (10) we see that for all p ≥ 1 there exists a constant c 5 such that for all x, y ∈ K we have E sup 0≤t≤T |2C n (t, x) − 2C n (t, y)| p ≤ c 5 |x − y| δp as desired. The case of general M and f is again easily obtained by localization. This proves Theorem 4.1. The next result allows us to start the solution of a sde and its linearization at any random (possibly anticipating) initial state. In all parts of the following theorem the stochastic flow associated with an Itô equation driven by F or a Stratonovich equation driven by • F will be denoted by φ. Recall that φ(t, x) := φ 0t (x). Theorem 4.2. Suppose Y : Ω → R d is any F -measurable random variable and let δ ∈ (0, 1]. (i) Let F be a spatial (forward) semimartingale in B 0,1 ub . Then φ(t, Y ), t ≥ 0, is a solution of the anticipating Itô sde dφ(t, Y ) = F (dt, φ(t, Y )), t > 0 φ(0, Y ) = Y. (I ′ ) If the spatial (forward) semimartingale • F is in (B 2,δ ub , B 1,δ ub ), then φ(t, Y ), t ≥ 0, is a solution of the anticipating Stratonovich sde dφ(t, Y ) = • F (•dt, φ(t, Y )), t > 0 φ(0, Y ) = Y.(SII) (ii) Assume that • F is a (forward) spatial semimartingale of class (B 3,δ ub , B 2,δ ub ). Then the (possibly non-adapted) process y(t, ω) := D 2 φ(t, Y (ω), ω), t ≥ 0, satisfies the Stratonovich linearized sde dy(t) = D 2 • F (•dt, φ(t, Y ))y(t), t > 0, y(0) = I ∈ L(R d ).    (SIII) A similar result is true in the Itô case. (iii) Let • F be a spatial backward semimartingale of class (B 2,δ ub , B 1,δ ub ). Then φ(t, Y ), t ≤ 0, is a solution of the backward Stratonovich sde dφ(t, Y ) = − • F (•dt, φ(t, Y )), t < 0 φ(0, Y ) = Y. (SII − ) (iv) Assume that • F is a spatial backward semimartingale of class (B 3,δ ub , B 2,δ ub ). Then the process y(t, ω) := D 2 φ(t, Y (ω), ω), t ≤ 0, satisfies the backward Stratonovich linearized sde dŷ(t) = −D 2 • F (•dt, φ(t, Y ))ŷ(t), t < 0, y(0) = I ∈ L(R d ).    (SIII − ) Proof. Let F be in B 0,1 ub and define f (t, x) := φ(t, x). Then the moment estimate for f in Theorem 4.1 is satisfied (with δ = 1), thanks to [Ku], Lemma 4.5.6. Therefore (I ′ ) follows. Next suppose that • F is in (B 2,δ ub , B 1,δ ub ), so in particular the local martingale part is in B 1,1 ub . By [Ku], Theorem 3.4.7 (or our Proposition 2.1) we know that φ is also generated by an Itô equation which is driven by a semimartingale F with local characteristics of class (B 2,δ ub , B 1,δ ub ). Observe that f := φ 0· ∈ B 0,1 c because, for every compact subset K ⊂ R d , every T > 0 and every p ≥ 1, we have sup 0≤s≤T E sup x∈K |φ 0s (x)| p < ∞, and sup 0≤s≤T E sup x,y∈K,x =y |φ 0s (x) − φ 0s (y)| |x − y| p < ∞. The estimates (15) and (16) follow from Theorem 2.1(v). This proves part (i). We next proceed to prove assertion (ii) of the theorem. To do this we will reduce the problem to a system of sde's which satisfies the hypotheses of part (i) of the theorem. Define the spatial semimartingales z(t, x, v) := (φ(t, x), D 2 φ(t, x)(v)) G(t, x, v) := ( • F (t, x), D 2 • F (t, x)(v)) for all (x, v) ∈ R d × R d , t > 0. Then the sde's (S) and its linearization d[D 2 φ(t, x)(v)] = D 2 • F (•dt, φ(t, x))D 2 φ(t, x)(v), t > 0, D 2 φ(0, x)(v) = v ∈ R d viewed as a coupled pair, are equivalent to the sde dz(t, x, v) = G(•dt, z(t, x, v)), t > 0 z(0, x, v) = (x, v) ∈ R d × R d .(SIV ) By hypothesis, • F has local characteristics of class (B 3,δ ub , B 2,δ ub ). We claim that G has local characteristics of class (B 2,δ ub , B 1,δ ub ) (in R 2d ). To see this, use coordinates x := (x 1 , · · · , x d ), x ′ := (x ′ 1 , · · · , x ′ d ), v := (v 1 , · · · , v d ), v ′ := (v ′ 1 , · · · , v ′ d ), and observe that the semimartingale {D 2 • F (t, x)(v) : (x, v) ∈ R d × R d , t ≥ 0} has local characteristics {ã k,l (t, (x, v), (x ′ , v ′ )) : (x, v), (x ′ , v ′ ) ∈ R d × R d , t ≥ 0, 1 ≤ k, l ≤ d}, {b k (t, (x, v)) : (x, v) ∈ R d × R d , t ≥ 0, 1 ≤ k ≤ d} given bỹ a k,l (t, (x, v), (x ′ , v ′ )) = d i,j=1 ∂ 2 ∂x i ∂x ′ j a k,l (t, x, x ′ )v i v ′ j b k (t, (x, v)) = d i=1 ∂ ∂x i b k (t, x)v i . From these relations, our claim follows. By the first part of the proof, we can substitute x = Y (ω) in (SIV) and keep v ∈ R d arbitrary but fixed (non-random). This gives (SIII) (and (SII)). Hence assertion (ii) of the theorem holds. The proofs of assertions (iii) and (iv) are similar to those of (i) and (ii). This completes the proof of the theorem. In the case of Brownian linear integrators, a version of Theorem 4.2 (i) is given in ([M-N-S], Theorem 3.1, p. 1920) under somewhat more restrictive hypotheses. In this case too, similar results to Theorem 4.2 (i), (ii) appear in ([A-I], Theorems 4,5) and ( [Nu], Theorems 5.3.4, 6.1.1). •F : R × R d × Ω → R d , and equation (I) is driven by a continuous forward spatial semimartingale 1991 Mathematics Subject Classification. Primary 60H10, 60H20; secondary 60H25, 60H05.. Mohammed's research is supported in part by NSF Grants DMS-9503702, DMS-9703852 and by MSRI, Berkeley, California. Scheutzow's research is supported in part by MSRI, Berkeley, California. Research at MSRI is supported in part by NSF grant DMS-9701755. log |Z(n, x, ω)| ≤ lim sup t→∞ 1 t log |Z(t, x, ω)|, B.3) ([Ru.1], p. 304). The rest of the proof of Theorem 5.1 works for a fixed choice of ω ∈ Ω * 1 . In particular, (the proof of) the "perturbation theorem" ([Ru.1], Theorem 4.1) does not affect the choice of the sure event Ω * 1 because it works pointwise in ω ∈ Ω * 1 , and hence does not involve the selection of a sure event ([Ru.1], pp. 285-292). Definition 4. 1 . 1(i) Define the Itô integral of f with respect to the continuous spatial local martingale limit in probability exists uniformly on compact subsets of [0, ∞) for any sequence of partitions as above. (ii) Define the Stratonovich integral of f with respect to the C 1 local martingale M by T 0 M (•dt, f (t)) := lim n→∞ S n (T ) Acknowledgement.The authors are grateful for comments made by Ludwig Arnold on an earlier version of the manuscript. Random Dynamical Systems. L Arnold, Springer-VerlagTo appear). [A-SArnold, L., Random Dynamical Systems, Springer-Verlag (To appear). [A-S] Perfect cocycles through stochastic differential equations. L Arnold, M K R Scheutzow, Probab. Th. Rel. Fields. 101Arnold, L., and Scheutzow, M. K. R., Perfect cocycles through stochastic differ- ential equations, Probab. Th. Rel. Fields, 101, (1995), 65-88. Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory. L Arnold, P Imkeller, Stoch. Proc. Appl. 62Arnold, L., and Imkeller, P., Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stoch. Proc. Appl. 62 (1996), 19-54. A stochastic version of center manifold theory. P Boxler, Probab. Th. Rel. Fields. 83Boxler, P., A stochastic version of center manifold theory, Probab. Th. Rel. Fields, 83 (1989), 509-545. Flows of stochastic dynamical systems. A Carverhill, Ergodic theory, Stochastics. 14Carverhill, A., Flows of stochastic dynamical systems: Ergodic theory, Stochas- tics, 14 (1985), 273-317. On the spatial asymptotic behaviour of stochastic flows in Euclidean space. P Imkeller, M K R Scheutzow, preprintImkeller, P., and Scheutzow, M. K. R., On the spatial asymptotic behaviour of stochastic flows in Euclidean space (preprint)(1997). N Ikeda, S Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland-KodanshaSecond EditionIkeda, N., and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland-Kodansha (1989). H Kunita, Stochastic Flows and Stochastic Differential Equations. Cambridge, New York, Melbourne, SydneyCambridge University PressKunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, New York, Melbourne, Sydney (1990). The Lyapunov spectrum and stable manifolds for stochastic linear delay equations. S.-E A Mohammed, Stochastics and Stochastic Reports. 29Mohammed, S.-E. A., The Lyapunov spectrum and stable manifolds for stochas- tic linear delay equations, Stochastics and Stochastic Reports, Vol. 29 (1990), 89-131. Lyapunov exponents and stochastic flows of linear and affine hereditary Systems. S.-E A Mohammed, Diffusion Processes and Related Problems in Analysis. Mark Pinsky and Volker WihstutzBirkhauserIIM-N-SMohammed, S.-E. A., Lyapunov exponents and stochastic flows of linear and affine hereditary Systems, Diffusion Processes and Related Problems in Analysis, Vol. II, edited by Mark Pinsky and Volker Wihstutz, Birkhauser (1992), 141-169. [M-N-S] Large deviations for a class of anticipating stochastic differential equations. A Millet, D Nualart, M Sanz, The Annals of Probability. 20Millet, A., Nualart, D., and Sanz, M., Large deviations for a class of anticipating stochastic differential equations, The Annals of Probability, 20 (1992), 1902-1931. Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales, Part I: The multiplicative ergodic theory. S.-E A Mohammed, M K R Scheutzow, Probabilités et Statistiques. 32143Ann. Inst. Henri PoincaréMohammed, S.-E. A., and Scheutzow, M. K. R., Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales, Part I: The multiplicative ergodic theory, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, Vol. 32, 1, (1996), 69-105. pp. 43. Spatial estimates for stochastic flows in Euclidean space. S.-E A Mohammed, M K R Scheutzow, to appear in The Annals of ProbabilityMohammed, S.-E. A., and Scheutzow, M. K. R., Spatial estimates for stochastic flows in Euclidean space, to appear in The Annals of Probability. Analysis on Wiener space and anticipating stochastic calculus. D Nualart, St. Flour Notes. to appear inNualart, D., Analysis on Wiener space and anticipating stochastic calculus (to appear in) St. Flour Notes. Stochastic calculus with anticipating integrands, Analysis on Wiener space and anticipating stochastic calculus. N-P] Nualart, D Pardoux, E , Probab. Th. Rel. Fields. 78[N-P] Nualart, D., and Pardoux, E., Stochastic calculus with anticipating integrands, Analysis on Wiener space and anticipating stochastic calculus, Probab. Th. Rel. Fields, 78 (1988), 535-581. Lyapunov characteristic numbers for dynamical systems. V I Oseledec, English transl. Trans. Moscow Math. Soc. 19Trudy Moskov. Mat. Obšč.Oseledec, V. I., A multiplicative ergodic theorem. Lyapunov characteristic num- bers for dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179-210. English transl. Trans. Moscow Math. Soc. 19 (1968), 197-221. Application to a class of anticipating stochastic differential equations. D Ocone, E Pardoux, Probabilités et Statistiques. 251Ann. Inst. Henri PoincaréOcone, D., and Pardoux, E., A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, Vol. 25, no. 1 (1989), 39-71. Semimartingales and measure-preserving flows. Ph E Protter, Probabilités et Statistiques. 22Ann. Inst. Henri PoincaréProtter, Ph. E., Semimartingales and measure-preserving flows, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, vol. 22, (1986), 127-147. Ph E Protter, Stochastic Integration and Stochastic Differential Equations: A New Approach. SpringerProtter, Ph. E., Stochastic Integration and Stochastic Differential Equations: A New Approach, Springer (1990). Ergodic theory of differentiable dynamical systems. D Ruelle, Publ. Math. Inst. Hautes Etud. Sci. Ruelle, D., Ergodic theory of differentiable dynamical systems, Publ. Math. Inst. Hautes Etud. Sci. (1979), 275-306. Characteristic exponents and invariant manifolds in Hilbert space. D Ruelle, Annals of Mathematics. 115Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Annals of Mathematics 115 (1982), 243-290. On the perfection of crude cocycles. M K R Scheutzow, Random and Computational Dynamics. 4Scheutzow, M. K. R., On the perfection of crude cocycles, Random and Compu- tational Dynamics, 4, (1996), 235-255. Linearization of random dynamical systems. T Wanner, Dynamics Reported. U. Kirchgraber and H.O. Walther4SpringerWanner, T., Linearization of random dynamical systems, in Dynamics Reported, vol. 4, edited by U. Kirchgraber and H.O. Walther, Springer (1995), 203-269.

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{'abstract': "We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory.", 'arxivid': 'math/9803160', 'author': ['Salah-Eldin A Mohammed ', 'Michael K R Scheutzow '], 'authoraffiliation': [], 'corpusid': 16141382, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 32098, 'n_tokens_neox': 28089, 'n_words': 15672, 'pdfsha': '44a06093a09413296794942365aa8b2a4a812c0e', 'pdfurls': ['https://export.arxiv.org/pdf/math/9803160v1.pdf'], 'title': [], 'venue': []}

arxiv

Title: Selective generation of reactive oxygen species in plasma activated water using CO 2 plasma Vikas Rathore Institute for Plasma Research (IPR) Atmospheric Plasma Division 382428GandhinagarGujaratIndia Homi Bhabha National Institute Training School Complex 400094AnushaktinagarMumbaiIndia Sudhir Kumar Nema Institute for Plasma Research (IPR) Atmospheric Plasma Division 382428GandhinagarGujaratIndia Homi Bhabha National Institute Training School Complex 400094AnushaktinagarMumbaiIndia Title: Selective generation of reactive oxygen species in plasma activated water using CO 2 plasma 1 *Email: [email protected] activated waterCO 2 plasmaCO 2 emission spectrareactive oxygen- nitrogen species In the present work, a process of a selective generation of reactive oxygen species (ROS) such as H 2 O 2 and dissolved O 3 in plasma-activated water (PAW) is discussed. For selective ROS generation, pure CO 2 was used as a plasma-forming gas. The gases species present in plasma and properties of PAW are compared in details when CO 2 and air are used as plasma forming gas. The results reveal that PAW (CO 2 ) has significantly higher pH, and low oxidizing potential and electrical conductivity compared to PAW (air). The formed species in PAW (CO 2 ) due to CO 2 plasma-water interaction are dissolved O 3 , H 2 O 2 , dissolved CO 2 , and CO 3 2ˉ ions, etc. In addition, no detectable concentration of NO 2ˉ and NO 3ˉ ions is observed in PAW (CO 2 ). PAW (CO 2 ) has a substantially higher concentration of H 2 O 2 compared to PAW (air). Moreover, increasing plasma treatment time with water significantly increases H 2 O 2 and dissolved O 3 concentration in PAW (CO 2 ). However, PAW (air) showed a rise and fall in H 2 O 2 and dissolved 2 O 3 concentration with time. In conclusion, selective generation of ROS in PAW is possible using CO 2 as plasma-forming gas. Introduction The plasma-activated water (PAW) technology is one of the fastest growing novel technologies in cold plasma field. This is due to its continuously evolving applications in the field of plasma medicine, plasma agriculture, plasma food science and technology, etc. (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11). These applications of PAW are possible due to the presence of various stable reactive oxygennitrogen species (RONS) in it. These RONS in PAW are formed as a resultant product of plasma-water interaction. Moreover, it brings a physicochemical change in PAW properties such as pH, oxidizing potential, and electrical conductivity, etc. (4,(9)(10)(11)(12)(13)(14)(15)(16) Different RONS have different significance in their applications (2,4,5,17). Such as species like H 2 O 2 , dissolved O 3 , HO·, and ONOOˉ have applications in microbial inactivation, selective killing of cancer cells, enhancing the shelf life of various food products such as fruits, vegetables, meat, seafood, and dairy products, etc. (1, 3-5, 8-11, 17, 18) Moreover, reactive nitrogen species can be used as a nitrogen replacement source for numerous agriculture applications (2,7,15,(19)(20)(21). The PAW produced due to plasma-water exposure contains both the reactive oxygen species (H 2 O 2 , dissolved O 3 , HO·, etc.) and reactive nitrogen species (NO 2ˉ ions, NO 3ˉ ions, etc.) (4,13,22). This is due to conventionally used plasma forming gases during PAW production which contain oxygen and nitrogen molecules or ionization of surrounding air by noble gas. The frequently used plasma-forming gases at atmospheric pressure during PAW production are air, nitrogen (N 2 ), oxygen (O 2 ), argon (Ar), Helium (He), and their mixture in 3 different compositions (4,7,9,11,13,18,23). Working with 100% O 2 plasma at atmospheric pressure is not recommended. Since, O 2 is highly oxidizing and can ignite flammable material rapidly that can cause explosions at atmospheric pressure. The use of inert gases (Ar or He) for plasma production mainly discharges the atmospheric air resulting in the generation of various RONS during plasma-water exposure (23). Hence, the production of plasma-activated water with a selective generation of ROS (free from nitrogen species) at atmospheric pressure is still an open challenge to be overcome. The presence of RNS forms nitrous and nitric acid (strong acid) in PAW due to which the pH of PAW decreased substantially. Therefore, PAW cannot be used in applications that do not prefer low pH solutions. Due to this applicability of PAW is restricted and also one of the main disadvantages of PAW. As per the authors' knowledge, no work has been reported that emphasizes the selective generation of reactive oxygen species (ROS). The ROS have applications in microbial and biofilm inactivation, medicine, food preservation, and enhancing seeds germination, etc (2-4, 8, 9, 24). This research gap tries to be fulfilled in present work. The conventionally used oxidizing and inert gases plasma formed nitrogen species in water as discussed above (4,7,9,11,13,18,23). Moreover, nitrogen free gases such as phosphine (PH 3 ), hydrogen sulphide (H 2 S), arsine (AsH 3 ) and sulphur dioxide (SO 2 ), etc. have environmental hazards (highly toxic) at atmospheric pressure, hence not recommended. Therefore, the present work uses CO 2 as plasma forming gas for selective generation of reactive oxygen species in plasma-activated water. Moreover, the PAW produced using CO 2 plasma is also compared with PAW produced using air plasma. The comparison was performed based on plasma characterization, formation of plasma reactive species/radicals, and properties of PAW. Material and Methods Experimental Setup 4 The experimental schematic of characterization of air and CO 2 plasma and production of plasma-activated water (PAW) is shown in figure 1. The air and CO 2 plasma were produced in a co-axial cylindrical pencil plasma jet (PPJ) (25). The PPJ setup based on the principle of dielectric barrier discharge and the schematic is shown in figure 1. In which the central ground electrode was made using a 1.6 mm tungsten rod. The high voltage electrode was made from copper in cylindrical form with an inner diameter of 6 mm in which dielectric quartz tube (outer diameter × inner diameter -6 mm × 4 mm) was tightly fixed. The discharge gap between the dielectric surface and the ground electrode was 1.2 mm. To measure the voltage drop across the PPJ setup a 1000x voltage probe (Tektronix P6015A) and a 100 MHz bandwidth, 2 Gs s -1 sampling rate, and a 4-CH oscilloscope (Tektronix TDS2014C) was used (26). A 10x (Tektronix TPP0201) voltage probe was used to measure the current and transported charge. This probe measured the voltage drop across the resistor (R -30 Ω) and capacitor (C -100 µF) connected in series with the ground as shown in For plasma-activated water production, 50 ml of ultrapure milli-Q water was taken in 600 ml of a glass beaker. This water was treated with air and CO 2 plasma as shown in figure 1. The air and CO 2 gas flow rate were controlled using a flow controller and the flow rate was fixed at 3 l min -1 . To enhance the solubility of plasma produced reactive species in water and escalate the reaction between gases reactive species and water, a continuous stirring of water and cooling of water were performed. For water stirring, a mortarless magnetic stirrer was used and for cooling of water, ice-cooled water was placed in contact with a glass beaker in which PAW was kept during plasma-water interaction. 5 Figure 1. Schematic of production of plasma activated water using CO 2 and air plasma and these gases plasma electrical and optical emission characterization. Equipments used for measurement of physicochemical properties of PAW The physicochemical properties of PAW (air or CO 2 ) such as pH, oxidation-reduction potential (ORP), total dissolved solids (TDS), and electrical conductivity (EC) were measured for PAW characterization. A Hanna Instruments pH meter (HI98121), HM digital ORP meter (ORP-200), HM digital TDS meter (AP-1), and Contech Instruments Ltd. EC meter (CC-01) were used to measure the pH, ORP, TDS, and EC of PAW. Measurement of Reactive Oxygen-Nitrogen Species Concentration The reactive oxygen-nitrogen species (RONS) form in PAW (air or CO 2 ) due to air plasma or CO 2 plasma water interaction was determined semi-quantitatively and quantitatively. The quantitative estimation of RONS concentration present in PAW was measured spectrophotometrically. The NO 3ˉ ions concentration was measured using the ultraviolet screening method (27), and the standard curve of NO 3ˉ ions was made using NaNO 3 solution with molar attenuation coefficient 0.0602 (mg L -1 ) -1 (25). In acidic region, NO 2ˉ ions present in the solution when react with the reaction mixture of N-(1-naphthyl) ethylenediamine dihydrochloride and sulfanilamide give reddish purple azo dye (λ max = 540 nm)(27). This characterstic of NO 2ˉ ions was utilize to determine its unknown concentration. The standard curve of NO 2ˉ ions was made using NaNO 2 solution with a molar attenuation coefficient of 0.0009 (µg L -1 ) -1 (25). The unknown concentration of H 2 O 2 in PAW was determined spectrophotometrically using the titanium sulfate method (13). The standard curve of H 2 O 2 was made of 30% H 2 O 2 solution (molar attenuation coefficient 0.4857 mM -1 (25)). The unknown concentration of dissolved O 3 in PAW was determined using the indigo colorimetric method (27). The titratable acidity and dissolved CO 2 concentration in PAW (CO 2 ) were determined using the titration method (8,28). The titratable acidity of PAW (CO 2 ) was determined using 0.1 M sodium hydroxide (NaOH) solution and a freshly prepared phenolphthalein indicator. In addition, the dissolved CO 2 concentration in PAW was determined using 0.02 N sodium carbonate (Na 2 CO 3 ) solution and freshly prepared phenolphthalein as an indicator (28). The dissolved carbonate ions (CO 3 2ˉ) concentration in PAW (CO 2 ) was determined using the UV screening method (29,30) with CO 3 2ˉ molar attenuation coefficient 0.0008 (mg L -1 ) -1 . 7 Data analysis All the experimental were performed at least three times (n ≥ 3) in the present investigation. The results were shown as µ ± σ (mean ± standard deviation (Error)). The statistically significant difference with a significant level of 95% (p = 0.05) among the groups mean were calculated using one-way analysis of variance followed by a post-hoc test (Fischer Least Significant Difference (LSD)). Results and discussion Voltage current waveform The air and CO 2 plasma voltage current waveform produced in dielectric barrier discharge (DBD) pencil plasma jet (PPJ) is shown in figure 2. The air and CO 2 plasma current waveform showed nanosecond (ns) current filaments peaks (~ 100 ns, shown in figure S1 of supplementary material) over each negative and positive current half cycle. The cluster of these nanosecond current filaments lies in the microsecond region. Therefore, these current discharges are known as filamentary DBD micro discharges (26). The discharge current peaks observed in CO 2 plasma were higher than air plasma for the same applied voltage. This signifies the radicals and species produced in CO 2 plasma had higher current carrying affinity compared to air plasma. Alongside, the other possible region is the generation of high concentration of plasma radicals and species in CO 2 plasma compared to air plasma. Due to which high discharge current was observed in CO 2 plasma for the same process parameter. The plasma discharge power consumed during the air and CO 2 plasma generation was measured using voltage-transported charge Lissajous figure. Initially, the energy consumed during discharge was calculated using the integral of voltage over the charge domain. For power calculation, a product of energy consumed with frequency (40 kHz) was performed. To 8 compare the properties of PAW produced using air and CO 2 plasma, the energy and power kept in the range of 0.0125 mJ to 0.015 mJ and 0.5 to 0.6 W. The optical emission spectra of air and CO 2 plasma in the afterglow region are shown in figure 3. The overlay plot of air (solid line) and CO 2 (dotted lines) plasma showed the deviation between the emission bands peaks lines of air and CO 2 plasma. These emission bands peaks lines are formed due to electronic transition radiative decay of the upper vibration state to the lower vibration state of ions or molecules. The emission spectrum of air plasma consists of strong emission band peaks of N 2 second positive system (C 3 Π u → B 3 Π g ) along with weak emission intensity band peaks of N 2 + first negative system (B 2 Σ u + → X 2 Σ g + ) (31,32). The reactions associated with this transition are shown in equations (1-4). Moreover, the CO 2 plasma afterglow region showed strong intensity emission band peaks of CO 2 + first negative system (A 2 Π g → X 2 Π u ). Along with that weak intensity emission band peaks of CO + (A 2 Π 9 → X 2 Σ), CH (A 2 Δ → X 2 Π), CO (d 3 Δ → a 3 Π), and C 2 (A 3 Π g → X 3 Π u ) also observed in CO 2 plasma afterglow region (33). The formation of these upper state ions and molecules in the air and CO 2 plasma region occurs due to the collision of gas (air or CO 2 ) with high-energy electrons, photons, ions, and neutral particles. These collisions result in the excited/upper vibration state of the above molecules. The most common reactions involved in these collisions were excitation, ionization, and dissociation (33)(34)(35)(36). The details of air and CO 2 emission band peaks lines observed in the air and CO 2 plasma afterglow region are shown in Table S1 of supplementary material. For air emission spectrum: N 2 (X 1 Σ g + ) + eˉ → N 2 (C 3 Π u ) + eˉ (electron impact excitation) (1) N 2 (C 3 Π u ) → N 2 (B 3 Π g ) + hυ (radiative decay)(2)C 2 (A 3 Π g ) → C 2 (X 3 Π u ) (radiative decay)(13) Figure 3. Air and CO 2 plasma emission spectra recorded in afterglow region Physicochemical properties of plasma-activated water The plasma-activated water (PAW) produced using air and CO 2 plasma is colorless in appearance. However, PAW (CO 2 ) and PAW (air) can be differentiated based on odor. PAW (air) was odorless, but PAW (CO 2 ) has a smoky and unpleasant odor. The physicochemical properties of PAW produced using air and CO 2 plasma is shown in figure 4. The pH of PAW produced using air and CO 2 plasma showed a continuous decrease in its value with increasing plasma treatment time ( figure 4 (a)). This signifies increasing plasma-water treatment time results in more formation of acidic radicals/species in water (17,22,25). Moreover, the PAW produced using CO 2 plasma was considerably higher (82.14% higher after 60 minutes of treatment) than PAW produced using air plasma. Hence, the generated species in water when exposed to air plasma were more acidic compared to CO 2 plasma. Ma et al. The oxidizing potential of PAW when produced using air and CO 2 plasma is shown in figure 4 (b). The oxidizing potential of PAW gives information regarding net combinations of oxidizing and reducing species formed in water after plasma exposure (12,23,25). The oxidation-reduction potential (ORP) of PAW prepared using air plasma was higher compared to PAW prepared using CO 2 plasma. This was due to the formation of a high concentration of oxidizing species in PAW when produced using air plasma compared to CO 2 plasma. For 60 minutes of plasma treatment, the ORP of PAW prepared using air plasma was 27.1% higher RONS concentration in plasma-activated water The above-discussed variation in physicochemical properties of water after plasma treatment occurs due to the formation of numerous reactive species in water (1, 3, 4, 6-8, 14, 20, 22). The mechanism of formation of these reactive species in PAW is shown in equations (14-25) (1,4,12,13,16,18,22,25). 13 Formation of reactive oxygen species (ROS) in plasma-activated water: 2 ( ) → 2 ( ) (14) 2 ( ) + ( ) → 3 ( ) . → ( . ) (15) 2 ( ) → · ( ) + · ( ) . → + ( . ) + − ( . ) + · ( . )(16)2 · ( . ) → ( . )(17)( . ) + ( . ) → · ( . ) + 2 · ( . ) + 2 ( . )(18)2 2 · ( . ) → ( . ) + 2 ( . )(19) Formation of reactive nitrogen species (RNS) in plasma-activated water: Figure 5 showed the identified and measured concentration of RONS (reactive oxygen-nitrogen species) present in PAW when prepared using air and CO 2 plasma. Figure 5 (a, c, ( figure 5 (f, h) and figure 6). 2 ( ) → 2 ( )(20) The concentration of NO 3ˉ and NO 2ˉ ions present in PAW prepared using air and CO 2 plasma is shown in figure 5 (a-d). A continuous increase in NO 3ˉ and NO 2ˉ ions concentration with plasma treatment time observed in PAW prepared using air plasma ( figure 5 (a, c)). The figure 5 (g, h)). The acidic species concentration formed in PAW (CO 2 ) was measured by measuring titratable acidity. Increasing plasma treatment time with water continuously and significantly (p < 0.05) increases the titratable acidity, dissolved CO 2 , and CO 3 2ˉ ions concentration. The uniform increase in titratable acidity, dissolved CO 2 , and CO 3 2ˉ ions concentration signifies continuous production of reactive species in PAW (CO 2 ) with increasing plasma-water treatment time. The CO 3 2ˉ ions exist in the form of carbonic acid in PAW (CO 2 ). The dissolved CO 2 and CO 3 2ˉ ions (carbonic acid) are weak acids due to which the pH of PAW (CO 2 ) decreased. However, this decreases in pH of PAW (CO 2 ) significantly (p < 0.05) low compared to PAW (air). Formation of carbonic acid in PAW: Figure 6. (a) Titratable acidity and dissolved CO 2 , and (b) CO 3 2ˉ ions concentration in plasmaactivated water produced using CO 2 plasma. Statistically significant (p < 0.05) difference between the group mean ± standard deviation (µ ± σ) is shown by a different lowercase letter. ( . ) + 2 ( . ) → ( . )(26)( . ) + ( . ) → ( . )(27) Hence, the above results and discussion showed the higher discharge current filaments in CO 2 plasma compared to air plasma. Moreover, the emission spectrum of CO 2 plasma is free from nitrogen containing species. As a results, formation of reactive nitrogen species (RNS) is not occurring in PAW (CO 2 ). Hence, selective generation of reactive oxygen species (ROS) occurs in PAW (CO 2 ). Moreover, due to the use of CO 2 gas plasma for PAW preparation. The carbonic acid, dissolved CO 2 , CO 3 2ˉ ions also occurs in PAW due to which pH of PAW (CO 2 ) is decreased. However, the pH of PAW (CO 2 ) is significantly lower than PAW (air). Conclusion The present work compares the properties of PAW produced using air and CO 2 plasma. The acidity of PAW (air) is significantly higher than PAW (CO 2 ). This is due to the dissolution of strong acids (nitric acid) in PAW (air) compared to weak acids (carbonic acid) of PAW (CO 2 ). In addition, the oxidizing potential, total dissolved solids, and electrical conductivity of PAW (air) are significantly higher than PAW (CO 2 ). This is due to PAW (air) has high concentration of strong ionic species in the form of HNO 3 compared to weak H 2 CO 3 species of PAW (CO 2 ). The PAW prepared using CO 2 plasma does not contain any reactive nitrogen species. This is due to the emission spectra of CO 2 plasma not containing any N 2 emission band peaks. Hence, CO 2 plasma-water interaction does not form any reactive nitrogen species in PAW (CO 2 ). Hence, selective production of reactive oxygen species can be achieved without the interference of reactive nitrogen species. Therefore, the concentration of dissolved H 2 O 2 in PAW (CO 2 ) is higher than PAW (air). In conclusion, selective production of reactive oxygen species in plasma-activated water is possible by using CO 2 as a plasma-forming gas. The presence of reactive oxygen species in PAW (CO 2 ) makes it a useful antimicrobial agent. Moreover, it can also be used in numerous applications where conventional PAW could not be used due to its low pH (such as low pH PAW could not be used for surface disinfection of metal objects since it oxidizes its surface and damage it). figure 1 . 1The air and CO 2 plasma emission spectra were measured by capturing the afterglow light photons using optical fiber and a spectrometer (Plasma and Vacuum Solution (PVS), model UVH-1) as shown in figure 1. Figure 2 . 2Voltage current waveform of (a) air and (b) CO 2 plasma produced in pencil plasma jet Optical emission spectra of air and CO 2 plasma ( 9 ) 9Subramanian et al. (1), El Shaer et al. (11), and Lu et al. (16) also showed a decrease in pH of PAW with increasing plasma treatment with water. compared to CO 2 plasma. The increase in ORP of PAW with increasing plasma-water treatment is also shown in the work reported by Guo et al. (3), Xiang et al. (10), and Ma et al. (9), etc.The rough estimation of conducting ions were measured by measuring total dissolved solids (TDS) and electrical conductivity (EC). The TDS and EC give the information regarding conducting ions formed in water due to plasma-water interaction. The observed TDS and EC of PAW, when prepared using air and CO 2 plasma are shown infigure 4 (c, d). Increasing plasma treatment with water increased the TDS and EC of PAW for both the air and CO 2 plasma. The observed TDS and EC of PAW prepared using air plasma were substantially high compared to CO 2 plasma (937.0% and 987.3% higher after 60 minutes of treatment). Hence, the concentration of inorganic ions formed in water after air plasma exposure was extremely higher compared to CO 2 plasma. The increase in EC with plasma treatment was also supported12 by results of Subramanian et al. (1), Zhang et al. (37), and Sivachandiran et al. (15), and Lu et al. (16), etc. Figure 4 . 4The variation in physicochemical properties of plasma activated water prepared using air and CO 2 plasma with plasma treatment time. (a) pH, (b) oxidation-reduction potential (ORP), (c) total dissolved solids (TDS), and (d) electrical conductivity (EC). Statistically significant (p < 0.05) difference between the group mean ± standard deviation (µ ± σ) is shown by a different lowercase letter. observed maximum concentration of NO 3ˉ and NO 2ˉ ions in PAW (air) were given as 4.0 mg L -1 and 401.5 mg L -1 , respectively. The NO 3ˉ and NO 2ˉ ions form nitric and nitrous acid in PAW (air) (equations (20-25))(1,6,8,12,22). As nitric acid is a strong acid, therefore the lowest pH value of PAW (air) was given as 2.8. The increasing concentration of NO 2ˉ and NO 3ˉ ions with activation time was also reported by Subramanian et al.(1)and Xiang et al.(10) in PAW prepared in an air atmosphere. However, the PAW (CO 2 ) did not contain any observable concentration of NO 3ˉ and NO 2ˉ ions as shown infigure 5 (b, d). As discussed in equations(1)(2)(3)(4)(20)(21)(22)(23), the formation of RNS (reactive nitrogen species) in PAW required excited nitrogen species(12,18,22,25) that were not observed in emission spectra of CO 2 plasma (figure 3). Hence, the possible RNS present in PAW (CO 2 ) such as (NO 2ˉ and NO 3ˉ ions) were beyond the detection limit of the present investigation. Figure 5 . 5The variation in reactive oxygen-nitrogen species (RONS) concentration of plasmaactivated water prepared using air and CO 2 plasma with plasma treatment time. (a, b) NO 3ˉ ions, (c, d) NO 2ˉ ions, (e, f) H 2 O 2 concentration, and (g, h) Dissolved O 3 . Statistically significant (p < 0.05) difference between the group mean ± standard deviation (µ ± σ) is shown by a different lowercase letter. The variation in titratable acidity, dissolved CO 2 , and CO 3 2ˉ ions with plasma treatment time in PAW (CO 2 ) is shown in figure 6. The excited carbon oxides (CO x ) and carbon oxide ions (CO x + ), etc. observed in emission spectra of CO 2 plasma (figure 3) when comes in contact with water enhances the solubility of CO 2 and formed carbonic acid (H 2 CO 3 ), etc. in water (equations (26-27)(38)). Due to which physicochemical properties of PAW (CO 2 ) changed. Strip test and colorimetry test kits were used to determine the initial RONS concentration in PAW and plasma (VISOCOLOR alpha (MACHEREY-NAGEL item no. 935065) nitrate (NO 3ˉ) ions colorimetry test kit, dissolved O 3 test kit (Hanna Instruments item no. HI38054), H 2 O 2 determination test strips (QUANTOFIX Peroxide 25, MACHEREY-NAGEL item no. 91319), NO 2ˉ ions determination test strips (QUANTOFIX Nitrite, MACHEREY-NAGEL item no. 91311), gases O 3 determination test strips (Ozone Test for Ozone in air, MACHEREY-NAGEL item no. 90736)).6 results in a monotonous increase in H 2 O 2 and dissolved O 3 concentration with increasing plasma treatment time with water (Moreover, the concentration of H 2 O 2 present in PAW (CO 2 ) was 316.7% higher than PAW (air). This was due to no interference of NO 2ˉ ions in H 2 O 2 determination in PAW (CO 2 ). The NO 2ˉ ions present in PAW (air) react with H 2 O 2 to give more stable NO 3ˉ ions (equation 25) (6, 18, 25). Therefore, interfere with the H 2 O 2 determination in PAW (air). The interference of NO 2ˉ ions in H 2 O 2 concentration and variation can be seen in figure 5 (e). Initially (t = 0 minutes), no H 2 O 2 was present in PAW (air), as the plasma treatment increased to 30 minutes, a continuous increase in the H 2 O 2 concentration was observed. Increasing plasma treatment time to 45 minutes results in a decrease in H 2 O 2 concentration due to the reaction of NO 2ˉ ions with H 2 O 2 to give more stable NO 3ˉ ions. Further increasing plasma treatment time to 60 15 minutes results in H 2 O 2 concentration enhancement. This showed saturation of NO 2ˉ ions and H 2 O 2 reaction in PAW (air) and the unreacted H 2 O 2 shown by enhanced H 2 O 2 in PAW (air). Similar behavior as H 2 O 2 was observed in the concentration of dissolved O 3 in PAW (air). Since, NO 2ˉ ions present in PAW (air) also reacts with dissolved O 3 to give more stable NO 3ˉ ions by following equation (24). This rise and fall in H 2 O 2 concentration in PAW (air) with increasing plasma treatment time also was observed in work reported by Subramanian et al.(1) and Sivachandiran et al. (15). However, this rise and fall in H 2 O 2 and dissolved O 3 concentration in PAW prepared using CO 2 plasma was not observed due to the absence of NO 2ˉ ions (figure 5 (b, d, g, h)). As no N 2 emission peaks bands were observed in the CO 2 plasma (figure 3) that confirms the absence of nitrogen species in CO 2 plasma. Hence, no interference of NO 2ˉ ions in PAW (CO 2 ) Acknowledgments19This work was supported by the Department of Atomic Energy (Government of India) doctrate fellowship scheme (DDFS).Data availability statementThe data that support the findings of this study are available upon reasonable request from the authors.Conflict of interestsThe authors declare that there are no conflicts of interests.Authors' contributionsBoth authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Vikas Rathore. 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Vacuum ultraviolet photoabsorption spectroscopy of space-related ices: formation and destruction of solid carbonic acid upon 1 keV electron irradiation. Astronomy Astrophysics. 2021;646:A172.

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{'abstract': 'In the present work, a process of a selective generation of reactive oxygen species (ROS) such as H 2 O 2 and dissolved O 3 in plasma-activated water (PAW) is discussed. For selective ROS generation, pure CO 2 was used as a plasma-forming gas. The gases species present in plasma and properties of PAW are compared in details when CO 2 and air are used as plasma forming gas. The results reveal that PAW (CO 2 ) has significantly higher pH, and low oxidizing potential and electrical conductivity compared to PAW (air). The formed species in PAW (CO 2 ) due to CO 2 plasma-water interaction are dissolved O 3 , H 2 O 2 , dissolved CO 2 , and CO 3 2ˉ ions, etc. In addition, no detectable concentration of NO 2ˉ and NO 3ˉ ions is observed in PAW (CO 2 ). PAW (CO 2 ) has a substantially higher concentration of H 2 O 2 compared to PAW (air). Moreover, increasing plasma treatment time with water significantly increases H 2 O 2 and dissolved O 3 concentration in PAW (CO 2 ). However, PAW (air) showed a rise and fall in H 2 O 2 and dissolved 2 O 3 concentration with time. In conclusion, selective generation of ROS in PAW is possible using CO 2 as plasma-forming gas.', 'arxivid': '2301.05051', 'author': ['Vikas Rathore \nInstitute for Plasma Research (IPR)\nAtmospheric Plasma Division\n382428GandhinagarGujaratIndia\n\nHomi Bhabha National Institute\nTraining School Complex\n400094AnushaktinagarMumbaiIndia\n', 'Sudhir Kumar Nema \nInstitute for Plasma Research (IPR)\nAtmospheric Plasma Division\n382428GandhinagarGujaratIndia\n\nHomi Bhabha National Institute\nTraining School Complex\n400094AnushaktinagarMumbaiIndia\n'], 'authoraffiliation': ['Institute for Plasma Research (IPR)\nAtmospheric Plasma Division\n382428GandhinagarGujaratIndia', 'Homi Bhabha National Institute\nTraining School Complex\n400094AnushaktinagarMumbaiIndia', 'Institute for Plasma Research (IPR)\nAtmospheric Plasma Division\n382428GandhinagarGujaratIndia', 'Homi Bhabha National Institute\nTraining School Complex\n400094AnushaktinagarMumbaiIndia'], 'corpusid': 255749469, 'doi': '10.1116/6.0002460', 'github_urls': [], 'n_tokens_mistral': 12933, 'n_tokens_neox': 10638, 'n_words': 6684, 'pdfsha': '9d99cd06099bc366eff7c5e842dc5d7486b46ec6', 'pdfurls': ['https://export.arxiv.org/pdf/2301.05051v1.pdf'], 'title': ['Title: Selective generation of reactive oxygen species in plasma activated water using CO 2 plasma', 'Title: Selective generation of reactive oxygen species in plasma activated water using CO 2 plasma'], 'venue': []}

arxiv

Seeing Invisible Poses: Estimating 3D Body Pose from Egocentric Video Hao Jiang Computer Science Department Boston College USA Kristen Grauman Department of Computer Science University of Texas at Austin USA Seeing Invisible Poses: Estimating 3D Body Pose from Egocentric Video Egocentric videohuman pose predictiondiscrete optimization Understanding the camera wearer's activity is central to egocentric vision, yet one key facet of that activity is inherently invisible to the camerathe wearer's body pose. Prior work focuses on estimating the pose of hands and arms when they come into view, but this 1) gives an incomplete view of the full body posture, and 2) prevents any pose estimate at all in many frames, since the hands are only visible in a fraction of daily life activities. We propose to infer the "invisible pose" of a person behind the egocentric camera. Given a single video, our efficient learning-based approach returns the full body 3D joint positions for each frame. Our method exploits cues from the dynamic motion signatures of the surrounding scene-which changes predictably as a function of body pose-as well as static scene structures that reveal the viewpoint (e.g., sitting vs. standing). We further introduce a novel energy minimization scheme to infer the pose sequence. It uses soft predictions of the poses per time instant together with a nonparametric model of human pose dynamics over longer windows. Our method outperforms an array of possible alternatives, including deep learning approaches for direct pose regression from images. Introduction Wearable "egocentric" cameras are steadily gaining traction-thanks not only to smaller devices, but also the increasing promise of vision and learning technology to transform applications. Head-or chest-mounted cameras, initially perceived as the purview of hard-core life loggers, are now valuable tools for many others. Last year, President Obama authorized $20M for law enforcement agencies across the US for purchasing bodycams in an effort to promote transparency with the public. Psychologists leverage wearable cameras on infants to gain insights into motor and linguistic development [23]. In healthcare, egocentric vision could move daily-living activity monitoring required for motor rehabilitation from the hospital to the home [14,17]. For many applications, the important vision problems center around inferring the camera wearer's behavior, i.e., his activity and interactions with people and objects. As such, the ability to infer the camera wearer's 3D body pose is of great interest. However, doing so is challenging because most body parts are invisible to the egocentric camera! Existing work estimates a person's pose by analyzing the body parts visible in his first-person camera. Naturally, this makes them restricted to the arms and hands [3,9,10] arXiv:1603.07763v1 [cs.CV] 24 Mar 2016 [11,16]. However, from the view of a chest-mounted wide-angle camera, arms and legs are often not visible in daily life activity. For example, in our ground truth videos in which people perform normal activities in public places such as labs and offices, the chance to view arms and legs is less than 10%. To estimate full body pose, one creative approach [1] is to fasten multiple cameras to all the person's joints, then use structure from motion (SfM) to localize the cameras and hence the joints. However, this comes with the disadvantages of requiring 1) obtrusive multi-camera equipment not amenable to everyday casual use and 2) intensive computational requirements (hours to days of processing to infer pose for a minute of video [1]). We ask the question: Is it possible to estimate the "invisible" human body pose behind a single egocentric view? See Fig. 1. Despite the fact that we cannot see the person behind the body-mounted camera, the video seen from his point of view provides clues that may well be learnable. In particular, we expect clues from two sources: dynamic motion signatures and static scene structure. First, there exist scene-independent motion signatures for pose changes. For example, the act of standing up has a certain motion pattern as seen by the ego-camera, no matter if he stands up from a chair in a restaurant or a bench at the park. In fact, first-person games use these effects to guide the virtual camera, giving gamers the impression they are moving the same way as the virtual character. Second, static scene structure sets the context and offers a prior on likely poses. For example, the pose of typing on a keyboard occurs in similar views showing a monitor or laptop, even though the hands need not be visible. Or, if we see a table in front of us with a specific distance and angle, we can predict whether we are standing or sitting in front of the table. Of course, not all poses are distinguishable from egocentric video; some will be aliased, meaning different poses can produce the same visual signal. Our intent is to leverage the typical structure linking how the scene changes to how the body is posed. We introduce a novel approach to predict first-person body pose, given an egocentric video sequence. As training data, our approach takes videos from a wearable camera, where each frame is labeled with ground truth pose parameters. The pose is parameterized by 25 3D joint positions, i.e., a "stick figure" representation, and is obtained with Kinect during training. At test time, we are given a novel egocentric video from a new user, and must infer the sequence of 3D body poses based on the single wearable camera video alone. Our learning approach capitalizes on the clues described above, while also incorporating longer term pose dynamics. First, classifiers based on dynamic and static cues estimate the probability of each of a (large) set of quantized poses per frame. Then, we jointly infer poses for a longer sequence (1 to 3 minutes) based on those initial predictions together with a non-parametric model of pose dynamics. The latter is used to identify a least-cost "pose path" through exemplar training video. This step regularizes the initial estimates with priors about how people can move, and is efficiently optimized with dynamic programming. The whole approach is fast-about 0.5 seconds per frame. We validate our method quantitatively on videos from ten camera wearers performing daily activity poses, as well as qualitatively on challenging videos in unconstrained environments. The experiments show the proposed method gives robust results. It greatly outperforms several alternative methods, including a CNN regression method modeled after the third-person DeepPose [4] approach retrained for our setting. In summary, our contributions are: (1) We tackle a new problem that estimates the wearer's "invisible" pose from a single ego-centric video; (2) We propose a novel global optimization method that leverages both learned dynamic and scene classifiers and the pose coupling over a long time span; and (3) We benchmark several methods, including hand crafted features and CNN learned features, for our task. Related work We deal with a new problem of predicting invisible human poses from a single egocentric video stream. To put our idea in context, we review related work on third-person pose, egocentric hand/arm estimation, and egocentric activity analysis. Third-person pose Pose estimation from images and video has been studied for decades [5]. Existing work tackles pose estimation from a third-person viewpoint, where the person is entirely visible. In contrast, we consider estimating the body pose of the person behind the camera; his/her body parts are rarely visible, if at all. As such, existing pose estimation methods are not applicable to our scenario. Some third-person pose methods use regression to map from images to pose parameters (e.g., [4,6,7]), including the recent DeepPose work using convolutional neural networks [4]. At a glance, a direct regression approach seems like a possible solution for our problem. Even though the body is not visible, we want to learn the connection between what the person sees and how his body is posed. However, a naive application of that idea is inadequate, since 1) even large training sets cannot fully capture the possible variation in environments, poses, and movements, and 2) the relevant egocentric visual signals are inherently temporal. The proposed method learns the connection between pose dynamic and static cues from snippets of video, and enforces long term constraints between estimated poses. Our experiments show this yields superior results to a DeepPose-like scheme applied to our task. First-person pose Limited research explores ways to infer the body pose of an egocentric camera wearer [3,1,11,9,10]. Given interest in understanding handled objects, some methods are dedicated to estimating pixel-wise 2D maps of the camera wearer's hands [11,9,10]. Recent work also investigates how depth data from an egocentric RGBD camera can help estimate shoulder, arm, and hand poses in 3D. Both lines of work assume the body parts are visible in the egocentric view. In contrast, we aim to estimate the full body pose of the person (e.g., 25 joint positions), and we do so even when the body is entirely out of view of the egocentric camera. In this sense, our goal is more related to the "inside-out" mocap approach of [1]. In that work, 16 or more body-mounted cameras are placed on a person's joints, and then each camera's 3D location is recovered via structure from motion (SfM). There are important differences with our technical approach and motivation. First, rather than 16+ cameras attached at joints worn expressly for the purpose of a mocap session [1], we employ a single chest-mounted camera-the sort typical wearable-computer-users may wear anyway while going about daily activities. Thus, the SfM approach cannot be directly applied to our setting, and our system requirements are more lightweight and flexible. Secondly, our approach is novel. Whereas the mocap method employs a geometric solution to localize the joints, we devise a learning solution that discovers the connection between how the ego-centered scene changes as a function of body pose. We also note that the mocap method requires substantial computational resources-about 1.5 days for a minute of capture [1] due to SIFT matching expenses-whereas our method requires only 15 minutes. The possible disadvantage of our method relative to [1] is our need for representative training data, though the data is relatively easy to collect, given that it requires no manual annotations (see Sec. 3.1). Egocentric activity analysis Most recent egocentric vision work studies activity recognition [12,15,17,18,19,20,13] or object recognition [16,11]. Once again, the focus is largely on visible activity happening in front of the camera-particularly hand-object manipulation activities. However, some work shows that ego-actions (like riding a bus, snowboarding, etc.) are detectable from the scene video [19,15], and the walking style of the camera wearer can even aid person identification [8]. We consider whether egovideo can go further to reveal full 3D body pose. While we also use movement information, our method does not infer action classes. For instance, rather than recognize the current action as "walking", our approach will produce the detailed pose across the walking cycle. Thus, our method provides a mid-level representation-explicit posewhich could be further used in high-level activity recognition or other applications. Method We estimate 3D human poses from the video of a chest-mounted camera. Predicting human poses from egocentric video is essentially a regression problem: from the input video, we estimate the 3D position of each body joint in the wear's local frame. In the following sections, we give details about instantaneous pose estimation using local features and full sequence estimation using the pose path method. Pose parameterization and data collection We use a Kinect V2 sensor to capture the ground truth human poses. Pose is represented as the 3D positions of 25 body joints defined in the MS Kinect SDK. The predicted 3D pose is positioned in a local coordinate system. The first axis is parallel to the ground and points to the wearer's facing direction. The second one is parallel to the ground and in the same plane as the shoulder line. The third axis is perpendicular to the ground. The joint coordinates are normalized by five times the shoulder length of the subject. In data capture, the subject wears a chest-mounted camera to record egocentric video. We choose chest-mounted (vs. head-mounted) because they provide a stable view unaffected by constant head bobbles. The frame rate of both the Kinect sensor and the ego-camera is 30Hz. The two are synchronized using time stamps. We capture a total of 18 ground truth videos, in which 3 videos are for training and the rest for testing. Ten subjects with different height, body shape, and gender are involved in data collection. They are instructed to perform normal daily activities in public places such as offices, labs, and libraries. Our ground truth dataset is collected indoors due to the limitation of Kinect V2 sensor. However, our approach is general, and we demonstrate outdoor tests as well. With more advanced motion capture setups, our method can be trained in even broader action domains. Instantaneous pose estimation We construct a function f (v, p) that gives the probability of video segment v corresponding to pose p ∈ P , where P is the set of all possible poses. In this paper, P includes all the poses in the train sequence. Here v is a mini-sequence of egocentric video frames, e.g. a one-second clip. In the following, we also use v to represent the feature vector extracted from a video segment. Due to the large number of possible poses, directly constructing f is difficult. We therefore introduce pose clusters as an intermediate pose representation. We cluster the normalized poses using k-means with L 2 norm to obtain K pose clusters. For everyday movement, K = 300 is sufficient. Then we train a classifier to obtain the function g(v, c) to extract the probability of video segment v matching the pose cluster c. The mapping f is approximated as f (v, p) = g(v, c(p)), where c(p) is the pose cluster identity of a pose p. Dynamic clues Egocentric video has specific motion patterns for different human movements. Human poses thus have strong correlation with the scene dynamics in the egocentric view. This is more so for the transient poses. A human observer can often infer the wearer's pose from the global scene motion. To construct a feature that is scene invariant, we extract the sequence of homographies between successive video frames. Strictly speaking, the homography is scene invariant only when the camera is purely rotating. However, the egocentric camera translates very little between successive frames if the frame rate is high, making the camera rotation dominant and the representation close to scene invariant. This useful property allows us to use very few training data to obtain good classifiers (as opposed to attempting to learn appearance-specific cues, which would be overly restrictive to a given training environment.) The feature is still related to the camera's intrinsic matrices. If the camera matrix K is known, we can normalize the result by computing the approximate rotation matrix R = K −1 HK, where H is the homography. To compute a homography between frames, we use optical flow to find the point correspondence. A least squares method is used to estimate the homographies, which can be implemented using SVD. The elements in each homography are then normalized by the top-left corner element. The stack of normalized homographies over a fixed time interval (one second), is used to represent the camera movement. Fig. 2 illustrates how the proposed feature helps differentiate poses of the wearer. In Fig. 2, the homographies are vectorized and combined into a matrix in each one-second time interval. Using the above feature, we train a random forest to predict the probability of the pose at each instant of the input video belonging to each of the K pose clusters. We build 100 random trees with arbitrary depth. The dynamic feature classifier gives reasonable results. However, the result is ambiguous when there is little motion in the egocentric video. To resolve this issue, we also use static scene structure, as defined next. Static scene structure clues In everyday life, two static poses are most common: standing and sitting. Many dynamic poses are often similar to these broad categories, e.g., walking is standing-like and kneeling is sitting-like. Indeed, in the dataset in [15], roughly 95% of frames can be classified as standing-like or sitting-like. 3 Given image n from the egocentric view, we compute h n , the probability the corresponding pose is sitting-like; its probability to be standing-like is 1 − h n . We collect a training dataset containing 5,530 standing images and 2,946 sitting images in different indoor environments. Fig. 3 shows sample images from the dataset. We train a CNN classifier by fine tuning the last three layers of the fully connected network in the AlexNet [22]; the learning rates of other layers are set to be zero. The two-class classifier generalizes well. On our ground truth dataset with 71,623 egocentric video frames and poses from Kinect V2, the sitting-like and standing-like image classification accuracy is 65.09% and 77.97%, respectively. The dataset is composed of 79.71% images with standing-like poses. Local cost of pose estimation Thus far we have provided two ways to estimate pose for each frame, using dynamic and static cues. These instantaneous estimates are not the final output of our system, however. As we will explain in Sec. 3.3, errors can be corrected in a global optimization stage where we infer the entire pose path over the entire sequence. In particular, the two classification outputs above serve as unary terms of an energy function for the longer sequence of surrounding frames (1-3 minutes per clip in our dataset). Let x i,n be an indicator variable, which is 1 if at time n the pose i is predicted. Here i is the id of a pose in P . Let e i,n be the cost of predicting pose i at time n. The overall unary cost term is U = n=1..N,i∈P e i,n x i,n , where N is the number of frames. Here P is also used to represent the ids of all the possible poses. We define the cost e i,n = 1 − g(v n , c(p i )) + d i,n , where g is the probability of dynamic feature v n being classified to pose cluster c(p i ), and d i,n is determined by both the static scene classifier and the dynamic scene classifier. We use d i,n to penalize the selection of pose i at time n if there is large chance that the estimations from the dynamic features and the static features mismatch. Specifically, we define d as: d i,n = δ if h n > τ andĝ(v n ) is standing, or h n < 1 − τ andĝ(v n ) is sitting, and otherwise 0. Recall that h n is the probability of sitting from the static scene feature at time n. Theĝ is derived from g to classify the current video frame as sitting or standing using the dynamic features and random forest, based on the known listing of which pose clusters are sitting/standinglike. Simply optimizing the local cost is not sufficient. Without considering the interframe pose constraints the pose predictions can be noisy. Another issue is the resolution. Since the local pose cost is estimated from the probability of quantized poses, it tends to be a staircase function over time. In the following, we show how to solve both of these problems by optimizing poses simultaneously over a long time span. Non-parametric prior on pose dynamics Next we show how we optimize the final sequence of pose estimates based on the local costs and a non-parametric prior on pose dynamics. First we define the prior, then we introduce an efficient optimization approach. Pose paths in an implicit motion graph To infer a likely sequence of poses over time, our method constructs an implicit motion graph that controls the possible transitions between poses in the exemplar training videos. The graph nodes correspond to poses in exemplar videos. The edges indicate possible transitions from one pose to another. The optimal pose sequence corresponds to the optimal pose path on an exemplar pose sequence. The pose path is composed of a sequence of "steps", each of which represents a transition from one pose to the next. We enforce that each step can only move from a pose cluster to the same pose cluster or a direct neighbor pose cluster. Each pose in the exemplar pose sequence belongs to a pose cluster. We define pose clusters as direct neighbors if we can find two poses that are drawn from each of the two pose clusters and are adjacent in time in the exemplar pose sequence. Since the same pose cluster may appear at different times in the exemplar pose sequence, the above rule allows large jumps. To further regularize the pose path, we constrain the step sizes, uniformity of the step sizes, and control the stationary steps on the pose path (see below). Therefore when determining where a step should lead to, we also have to consider previous decisions on the pose path. Thus, the transition costs dynamically change with the traversal history. This graph is reminiscent of motion-graphs used for motion synthesis in computer graphics [21,2]. However, whereas motion synthesis aims to generate convincing movements within an annotated mocap database based on a few user-specified anchor poses, our task is to jointly infer the sequence of poses in a novel egocentric video. Furthermore, unlike traditional motion graphs, edge weights in our graph dynamically change to allow the regularizers mentioned above. We have used P to denote the set of all the poses in the training dataset. Here we overload the notion; we also use P to represent the concatenation of all the training pose sequences from the training dataset. The poses in P thus preserve the original temporal order. Selecting a sequence of poses from P is equivalent to find a path on P so that the following energy function is minimized: min X {U (X) + T (X) + V (X) + S(X)} s.t. The assignment of X represents a sequence of poses drawn from P. Here X is the matrix [x i,n ], where n is the time index and i is the index of poses in P and recall that x i,n is a binary variable to indicate whether pose i is selected at time n. To represent a path, at each time instant n, we have i x i,n = 1. Here U (.) is the unary term defined in the previous section. T (.), V (.), S(.) are terms that control coupling between poses in the whole sequence. T (.) constrains the step size between successive footprints on the path, V (.) controls the speed of the pose transition, and S(.) restricts stationary steps. The step size term T We order the pose ids in P according to their temporal sequence in the training video. If we choose pose l ∈ P at instant n − 1, we say we step on point l at time n − 1. At time n, we may step to l + k, where k is the step size from time n − 1 to n. Since the original exemplar video is continuous, the smaller the k the smoother the pose transition is likely to be. If the step size is 0, we keep the same pose in the time interval. The stationary step can be used to infer a slower movement in the testing video. If the step size is 1, the movement has the same speed in the training and testing video. For k > 1, the movement in the testing video is faster than the exemplar sequence. In the energy function, we prefer the step size to be small and at the same time we allow occasional large jumps from one point to the other. In particular, T = i,j,n w j,i x j,n−1 x i,n , where w j,i = 0 if i − j ≤ 2, i ≥ j and otherwise w j,i = δ, where δ is a positive constant penalizing backward steps and steps that are great than two. Apart from the step size constraint, we also constrain that if c(p i ) = c(p j ) and c(p i ) and c(p j ) are not consecutive in the training video w j,i = +∞. Here c(p i ) is the pose cluster of pose i. This prohibits the path from going from one pose to another with too much difference or using a transition of pose clusters not seen in the exemplars. However, it does allow long jumps from one pose cluster to the same pose cluster or one that is a direct neighbor to the cluster. However, such long jumps do have a penalty. So, we prefer that steps on the path move to a directly adjacent frame if possible. We allow the path to go forward or backward. The speed smoothness of the path V The above step size term roughly enforces a first order constraint on the path: small steps are taken when possible. However, the path may still have a non-uniform speed of steps in a short time span, which is undesirable because within a time of 1 or 2 seconds human body motion is usually uniform. We thus introduce a second order term to penalize the speed changes: V = i,j,n q(|s j,n−1 − (i − j)|)x j,n−1 x i,n , where s j,n−1 is the speed at time n − 1, for step j. Here q is a truncated linear function: q(x) = µx if x < γ and otherwise q(x) = µγ, where γ and µ are constant parameters. This term encourages the path to maintain a constant speed. The stationary step penalty S in the path Simply minimizing the first order and second order smoothness of the path is not enough. Recall that the local cost in short time intervals tends to be constant. The steps in the pose path thus tend to be stationary because the first and second order smoothness terms will be zero. The step size penalty helps but is not sufficient. We thus penalize stationary steps: S = i,j,n r(u(j, n − 1), i)x j,n−1 x i,n , where r(u(j, n − 1), i) = 0 if i = j, otherwise r(u(j, n − 1), i) = t(u(j, n − 1) + 1). We therefore count the number of stationary steps and penalize the pose stop changing for a long time. Here, u(i, n) is the number of stationary steps accumulated at time n if the current pose is i; u(j, n − 1) is similarly defined. Similar to q, t(.) is a truncated linear function.The stationary step penalty term thus makes the path less likely to stay at one point and helps resolve the temporal resolution loss problem. Optimizing the pose path using DP We can rewrite the problem into a recursion: H(i, n) = ei,n + min j∈S i {H(j, n − 1) + wj,i + q(|s(j, n − 1) − (i − j)|) + r(u(j, n − 1), i)} u(i, n) = u(j * , n − 1) + 1, if j * = i and otherwise u(i, n) = 0 s(i, n) = i − j * , p(i, n) = j * , where j * = arg minj∈S i {H(j, n − 1) + wj,i + q(|s(j, n − 1) − (i − j)|) + r(u(j, n − 1), i)}. S i is the set of poses that can transform to i. Here, H(i, n) is the optimal energy of pose path if the path ends at a specific pose i at time n. u(i, n), s(i, n), p(i, n) are the stationary step number, speed of steps and previous optimal pose selection of the optimal pose path ending at pose i at time n. We initialize H(i, 1) = e i,1 , u(i, 1) = 0, s(i, 1) = 0, ∀i ∈ P . All the other H are initialized to be +∞, and p to be −1. We can verify that solving the recursion is equivalent to optimizing min X {U (X) + T (X) + V (X) + S(X)}, where the solution of X represents the pose path. The recursion can be efficiently solved using dynamic programming (DP). It helps to visualize the optimization in a trellis. The trellis contains M columns and N rows, where M is the number of possible poses in P and N is the number of input video frames. Fig. 4 illustrates the edge connection from layer (n − 1) to node i in layer n. Each edge corresponds to one possible step in the path. Each node has a cost e i,n , where i is the column and n is the row of the trellis. Each edge has a weight w j,i + q(|s j,n−1 − (i − j)|) + r(u(j, n − 1), i). The DP finds a minimum cost path in the trellis. Solving the DP involves updating the state variables H, s, u, p in each node. Since only the nodes inside the same or neighboring cluster are connected by each stage of the trellis, the complexity is much lower than O (M 2 N ). Moreover, we can use the local pose probability to prune impossible nodes from the trellis. In fact, most of the poses have near zero probability from the random forest classifier. If we only keep nodes that correspond to poses that have probability greater than 0.01, the trellis becomes very sparse and the corresponding DP can be quickly completed (typically contributing 0.01 seconds per frame for our whole system). Experimentation We evaluate the performance of the proposed method on both a ground truth dataset and challenging videos in unconstrained environments. (See videos at www.cs.bc. edu/˜hjiang/egopose/index.html). In our ground truth data, the 3D human poses are captured from the Kinect V2 for ten human subjects. The synchronized egocentric video is from a chest-mounted GoPro camera. Below we consider two settings. In the first setting, training and testing videos are from the same human subject, but taken in disjoint indoor environments such as lab, office, hallway and living room. In the second setting, the training and testing videos are from different human subjects and recorded at different locations. There are in total 71,623 test video frames (about 40 minutes) in the ground truth experiments, consisting of clips ranging from 1-3 minutes each. We also test about 15 minutes of video from unconstrained video, which lacks ground truth for evaluation. Implementation details For the unary term U , we set δ = 0.1, τ = 0.99. We thus include a penalty δ only when the confidence of the sitting-standing classifier is above 99%. For the truncated linear functions q and t, we fix γ = 10, µ = 0.01 and γ = 5, µ = 0.02, respectively. All parameters were set based on manual inspection of a few examples during method development, then fixed for all experiments. With sufficient labeled data, their values could be set with DP to minimize pose errors. Baselines No prior work predicts body pose from egocentric video. We therefore devise a series of informative baselines to gauge the impact of our method, including methods inspired by today's best image-based third-person pose estimators: -CNN-Regression: an adaption of the DeepPose [4] method to our task. Our problem is still a regression problem, even though the camera wearer is not visible from the egocentric view. We use the same network structure as the DeepPose except that our input is a stack of grayscale images in every one-second video clip and output is the 25 body joints defined by the Kinect SDK. We scale each image to 100 × 100. -KdTree: simple nearest neighbor approach using Kd-trees. It finds the "closest" video segments in the training data and then takes the corresponding 3D poses as the prediction result. The stacked homography in every 30 frames is used as the feature, and the L 2 norm is the distance metric. Other norms give similar results. deep-trained features in place of our hand crafted homography features. We train a deep neural network to classify each sequence of 30 frames to one of the 300 pose clusters. We use AlexNet [22] due to its good results in many applications. In the first setting (Path-CNN), we rescale each input video frame to 100 × 100 and retrain the network from scratch. In the second setting (Path-CNN-Refine), we fine-tune on the modified AlexNet with depth 30. The fine-tuning is only on the first convolution layer and the last three fully connected layers. We compute the local pose cost as one minus the class probability from the CNN output. The proposed global optimization is then applied to obtain the final result. -AlwaysStanding and AlwaysSitting: simple guessing methods that exploit the prior that poses are typically somewhere near a standing or sitting pose (hence much stronger than a truly random guess). We compute the standing and sitting poses by the average over training subjects. Figs. 5 and 6 show qualitatively that the proposed method indeed gives better results than the DeepPose adaptation (CNN-Regression) and nearest neighbors (KdTree), neither of which considers long term pose coupling. Our method also gives better results than the three variations of the proposed method as shown in Figs. 7 and 8. Fig. 7 shows that if we directly use the the estimated pose cluster centers as the predicted poses, the results have lower temporal resolution than the proposed method. Refining the pose selection in each estimated pose cluster is inferior to the proposed approach because the errors in the first stage cannot be undone. The predicted pose sequence is also not as smooth as the proposed method. Path-Cluster is essentially an interpolation method that smooths the cluster centers estimated in the first step. Note that a simpler linear interpolation method is not directly usable because it does not always give valid poses. Fig. 8 shows qualitatively that using deep neural networks to train the dynamic and scene structure features does not give better results. Neither training from scratch nor fine-tuning improves the result. Neural network approaches need a large dataset to capture different variations of the scene and human poses. Our method is able to train on a small dataset and achieve good performance. Now we present the quantitative comparisons with all baselines. We analyze the errors of the joints with highest variance in everyday activity: head, elbows, wrists, knees, and ankles. In the wearer's local frame joints such as shoulders and hips do not vary much in normal daily activities. Fingers and toes are also not included because they are not accurately estimated by Kinect; they mostly follow the wrists and ankles. We quantify error by the distance between the predicted 3D joints and the ground truth, after rotating each joint point cloud so that the shoulder is parallel to the yz plane and Path (Ours) Path-Cluster Path-CNN Path-CNN-R KdTree CNN-Regr AwaysStanding AwaysSitting Head 15.8(0.08) 16.5(0.08) 21.6(0.14) 22.9(0.14) 18. the body center is at the origin. Recall that the predicted coordinates are already in normalized coordinates according to the shoulder length of the subject. We convert raw errors to centimeters based on a reference shoulder joint distance of 30 cm. Tables 1 and 2 show the results, for the two settings defined above. Overall the proposed method gives smaller errors than all the competing methods. The tables show our method outperforms the DeepPose-like CNN-Regression, presenting the value in our scene-invariant dynamic homography features. It is also better than all the variants of our method we tested. While AlwaysStanding is a reasonable prior for most test frames, our method still makes noticeable gains on it, showing our ability to make fine-grained estimates (e.g., 6 cm better on average for the ankles and knees). AlwaysSitting has much larger errors than any method, in line with the distribution of the test data. Finally, between Table 1 Table 2. Average joint errors (cm) and standard errors when training and testing on disjoint people and environments. The training sequence has 10,000 frames from two subjects. There are 8 test videos with a total of 46,428 frames. See Table 1 for the definition of NAvgAll and NAvg(W+A). 10. Experiments on data without ground truth. There are three subjects (S1, S2, and S3). Row 1-2: Classroom (S1). Row 3: Classroom (S2). Row 4-5: Lab (S3). Row 6: Library (S1). Row 7-8: Library (S2). Row 9-10: Art gallery (S1). Row 11: Outdoor (S1). Row 12-13: Hallway (S1). Each result contains three columns: egocentric view, side view (unseen by our method) and pose prediction. see that absolute error is lower for all methods with the benefit of observing the same subject during training. Fig. 9 shows some failure cases by our method. Failures are mostly due to the ambiguity of the input. Arm poses are not always predictable, if they do not affect the motion or the viewing angle of the egocentric camera. Other errors are due to the misclassification between the standing and sitting poses. Improving the features and instantaneous pose estimation accuracy may further improve the results. Finally, we test our method on 8 video sequences with no ground truth, captured in varying environments and with 3 subjects. 4 The training dataset is the same as above. Fig. 10 shows sample results. For each example, we display the frame from the egocentric camera as well as one from a side camera viewing the subject. Note that the side view is for display only, and never used by our method. The 3D pose is estimated using only the egocentric video. Our method works well on this data, including for outdoor test sequences despite all training taking place indoors. Please see the Supp video. Conclusion We tackle a new problem in computer vision: predicting human poses from egocentric video. The proposed global optimization method is able to give accurate pose predictions in both same-person and cross-person tests. Our experiments show our method gives superior results to a number of alternative approaches. We believe our method will be useful for many different applications including egocentric video logging, summarization, and information retrieval, and it could facilitate action and movement understanding. Fig. 1 . 1Our goal is to infer the full 3D body pose of a person using the video captured from a single chest-mounted camera. (a): One person with a chest-mounted camera. (b): The egocentric view. (c): The predicted pose of the person using only video from view (b). Fig. 2 . 2Example poses and the corresponding dynamic features for the surrounding 1-second video segment. Similar poses often have similar dynamic features (see first two examples), and distinct poses have different features. Fig. 3 . 3Samples from the training dataset of sitting (Rows 1-2) and standing (Rows 3-4). Fig. 4 . 4(a): Connection between two trellis layers. Colors indicate pose clusters. We only allow pose transitions to the same or neighboring cluster. In this example, the blue cluster's neighbors have colors: light green, dark green, light blue and yellow. (b): State variables in each node. Fig. 5 . 5Comparison with the DeepPose [4] method retrained for our task. GT: ground truth. Path: proposed method. CNNR: CNN-Regression baseline. Fig. 6 .Fig. 7 . 67Comparing with the Kd-tree baseline. Row 1: Sample frames. Row 2: Ground truth poses. Row 3: Our result in left box and Kd-tree result in right box. The x, y, z coordinates of ground truth body joints (left), versus the results from proposed method (middle) and pose cluster centers of Path-Cluster (right) on a 50-frame video. - Path-Cluster: a variant of the proposed method. Instead of directly optimizing the poses, this method first finds the pose clusters and then refines the pose estimates using dynamic programming. The refinement is similar to the proposed pose path optimization, except that the pose candidates at each instant can only come from the pose clusters estimated in the first stage. -Path-CNN and Path-CNN-Refine: a variant of the proposed method that uses Fig. 8 . 8Comparison with methods using deeply learned features. GT: ground truth. Path: proposed method. PNN: Path-CNN. PNR: Path-CNN-Refine. Fig. Fig. 10. Experiments on data without ground truth. There are three subjects (S1, S2, and S3). Row 1-2: Classroom (S1). Row 3: Classroom (S2). Row 4-5: Lab (S3). Row 6: Library (S1). Row 7-8: Library (S2). Row 9-10: Art gallery (S1). Row 11: Outdoor (S1). Row 12-13: Hallway (S1). Each result contains three columns: egocentric view, side view (unseen by our method) and pose prediction. Table 1. Average joint error (cm) and standard errors, when training and testing on same subject but in different environments. The training sequence has 6,950 frames. There are 7 test videos with a total of 25,195 frames. We compute the mean error normalized by the standard error for the nine joints denoted NAvgAll, and for the wrists and ankles denoted NAvg(W+A).Fig. 9. Failure cases. See text for details.1(0.11) 16.2(0.10) 15.1(0.08) 32.5(0.09) Elbow 14.4(0.07) 15.4(0.07) 18.6(0.12) 19.4(0.12) 15.8(0.10) 14.4(0.09) 14.5(0.08) 20.7(0.08) Wrist 19.1(0.09) 20.6(0.10) 26.5(0.17) 27.1(0.17) 21.3(0.13) 22.0(0.14) 22.9(0.12) 21.3(0.08) Knee 15.4(0.09) 17.2(0.09) 27.3(0.17) 26.2(0.17) 22.0(0.14) 21.3(0.13) 21.2(0.11) 40.0(0.11) Ankle 20.7(0.10) 22.9(0.10) 33.8(0.21) 33.3(0.21) 28.4(0.18) 26.4(0.17) 26.7(0.13) 37.9(0.09) NAvgAll 17.2 19.1 48.1 48.7 32.8 29.7 24.6 31.9 NAvg(W+A) 19.9 22.6 60.0 60.2 40.8 38.7 32.4 27.1 and 2, as expected we Path (Ours) Path-Cluster Path-CNN Path-CNN-R KdTree CNN-Regr AwaysStanding AwaysSittingHead 16.6(0.07) 18.0(0.07) 19.4(0.09) 21.3(0.10) 20.1(0.09) 15.8(0.07) 14.3(0.07) 29.1(0.07) Elbow 15.3(0.06) 16.9(0.06) 19.1(0.09) 19.5(0.09) 18.0(0.08) 15.8(0.07) 14.9(0.06) 20.9(0.06) Wrist 22.2(0.08) 24.2(0.08) 29.7(0.14) 29.4(0.14) 24.9(0.12) 24.3(0.11) 23.8(0.09) 22.9(0.07) Knee 18.9(0.07) 24.4(0.09) 21.6(0.10) 21.8(0.10) 31.9(0.15) 27.6(0.13) 21.7(0.08) 45.7(0.09) Ankle 24.9(0.09) 29.9(0.10) 29.2(0.14) 29.2(0.14) 38.1(0.18) 33.3(0.15) 28.2(0.10) 43.0(0.09) NAvgAll 19.9 24.6 35.4 36.4 44.5 34.6 22.4 32.9 NAvg(W+A) 23.6 28.4 46.6 46.3 53.3 44.6 28.9 30.7 We do not include lying down because it does not happen often in the day. It is easy to capture egocentric video in arbitrary environments for test data; it is the Kinect ground truth capture for training that places restrictions. AcknowledgmentsThis research is supported in part by U.S. NSF 1018641 (HJ) and ONR PECASE N00014-15-1-2291 and a gift from Intel (KG). Motion Capture from Body-Mounted Cameras. T Shiratori, H S Park, L Sigal, Y Sheikh, J K Hodgins, ACM Transactions on Graphics. 304Shiratori T., Park H.S., Sigal L., Sheikh Y., Hodgins J.K.: Motion Capture from Body- Mounted Cameras. ACM Transactions on Graphics, Vol. 30, No. 4, July 2011. 2, 4 Motion Synthesis from Annotations. O Arikan, D A Forsyth, J F Obrien, 8Arikan O., Forsyth D.A., OBrien J.F.: Motion Synthesis from Annotations. Siggraph 2003. 8 First-person Pose Recognition using Egocentric Workspaces. G Rogez, J J Supancic, D Ramanan, 14Rogez G., Supancic J.J., Ramanan D.: First-person Pose Recognition using Egocentric Workspaces. CVPR 2015. 1, 4 A Toshev, C Szegedy, DeepPose: Human Pose Estimation via Deep Neural Networks. CVPR. 312Toshev A., Szegedy C.: DeepPose: Human Pose Estimation via Deep Neural Networks. CVPR 2014. 3, 11, 12 A Survey of Human Pose Estimation: The Body Parts Parsing based Methods. Z Liu, J Zhu, J Bu, C Chen, 1019. 3Journal of Visual Communication and Image Representation. 32Liu Z., Zhu J., Bu J., Chen C.: A Survey of Human Pose Estimation: The Body Parts Parsing based Methods. Journal of Visual Communication and Image Representation, Volume 32, October 2015, Pages 1019. 3 3D Human Pose from Silhouettes by Relevance Vector Regression. A Agarwal, B Triggs, CVPR 2004. 3Agarwal A., Triggs B.: 3D Human Pose from Silhouettes by Relevance Vector Regression. CVPR 2004. 3 G Shakhnarovich, P A Viola, T Darrell, Fast Pose Estimation with Parameter-Sensitive Hashing. ICCV. 3Shakhnarovich G., Viola P.A., Darrell T.: Fast Pose Estimation with Parameter-Sensitive Hashing. ICCV 2003. 3 An Egocentric Look at Video Photographer Identity. Y Hoshen, S Peleg, arXiv:1411.75911To appear in CVPRHoshen Y., Peleg S.: An Egocentric Look at Video Photographer Identity. arXiv:1411.7591, v1-Nov 2014, v2-Nov 2015. To appear in CVPR, June 2016. 4 Model Recommendation with Virtual Probes for Ego-Centric Hand Detection. C Li, K M Kitani, 14Li C., Kitani K.M.: Model Recommendation with Virtual Probes for Ego-Centric Hand De- tection. ICCV 2013. 1, 4 Pixel-level Hand Detection for Ego-centric Videos. C Li, K M Kitani, 14Li C.and Kitani K.M.: Pixel-level Hand Detection for Ego-centric Videos. CVPR 2013. 1, 4 X Ren, C Gu, Figure-ground Segmentation Improves Handled Object Recognition in Egocentric Video. CVPR 2010. 24Ren X. and Gu C.: Figure-ground Segmentation Improves Handled Object Recognition in Egocentric Video. CVPR 2010. 2, 4 Understanding Egocentric Activities. A Fathi, A Farhadi, Rehg J , 2011. 4Fathi A., Farhadi A., and Rehg J.: Understanding Egocentric Activities. ICCV 2011. 4 Delving into Egocentric Actions. Y Li, Z Ye, J Rehg, 2015. 4Li Y., Ye Z., Rehg J.: Delving into Egocentric Actions. CVPR 2015. 4 The Arm Motor Ability Test: reliability, validity, and sensitivity to change of an instrument for assessing disabilities in activities of daily living. Arch. of physical medicine and rehab. B Kopp, A Kunkel, H Flor, T Platz, U Rose, K Mauritz, K Gresser, K Mcculloch, E Taub, 1997. 178Kopp B., Kunkel A., Flor H., Platz T., Rose U., Mauritz K., Gresser K., McCulloch K., Taub E.: The Arm Motor Ability Test: reliability, validity, and sensitivity to change of an instrument for assessing disabilities in activities of daily living. Arch. of physical medicine and rehab, 78(6), 1997. 1 Y Poleg, C Arora, Peleg S , Temporal Segmentation of Egocentric Videos. CVPR. 46Poleg Y., Arora C., and Peleg S.: Temporal Segmentation of Egocentric Videos. CVPR 2014. 4, 6 You-Do, I-Learn: Discovering Task Relevant Objects and their Modes of Interaction from Multi-User Egocentric Video. D Damen, T Leelasawassuk, O Haines, A Calway, W W Mayol-Cuevas, BMVC. 24Damen D., Leelasawassuk T., Haines O., Calway A., Mayol-Cuevas W.W.: You-Do, I-Learn: Discovering Task Relevant Objects and their Modes of Interaction from Multi-User Egocentric Video. BMVC 2014 2, 4 Detecting Activities of Daily Living in First-Person Camera Views. H Pirsiavash, D Ramanan, 14Pirsiavash H. and Ramanan D.: Detecting Activities of Daily Living in First-Person Camera Views. CVPR 2012. 1, 4 Action Recognition in the Presence of One Egocentric and Multiple Static Cameras. B Soran, A Farhadi, L G Shapiro, 2014. 4Soran B., Farhadi A., Shapiro L.G.: Action Recognition in the Presence of One Egocentric and Multiple Static Cameras. ACCV 2014. 4 Fast Unsupervised Ego-Action Learning for First-Person Sports Video. K Kitani, T Okabe, Y Sato, A Sugimoto, CVPR 2011. 4Kitani K., Okabe T., Sato Y., and Sugimoto A.: Fast Unsupervised Ego-Action Learning for First-Person Sports Video. CVPR 2011. 4 E Spriggs, F D La Torre, Hebert M , Temporal Segmentation and Activity Classification from First-Person Sensing. CVPR Workshop on Egocentric Vision. Spriggs E., la Torre F.D., and Hebert M.: Temporal Segmentation and Activity Classification from First-Person Sensing. CVPR Workshop on Egocentric Vision, 2009. 4 L Kovar, M Gleicher, F Pighin, Motion Graphs. SIGGRAPH. Kovar L., Gleicher M., Pighin F.: Motion Graphs. SIGGRAPH 2002. 8 A Krizhevsky, I Sutskever, G E Hinton, ImageNet Classification with Deep Convolutional Neural Networks. NIPS 2012. 712Krizhevsky A., Sutskever I., Hinton G.E.: ImageNet Classification with Deep Convolutional Neural Networks. NIPS 2012. 7, 12 Not your mothers view: the dynamics of toddler visual experience. Linda B Smith, Chen Yu, Alfredo F Pereira, Developmental Science. 141Linda B. Smith, Chen Yu, and Alfredo F. Pereira. Not your mothers view: the dynamics of toddler visual experience. Developmental Science, Volume 14, No. 1, pp 917, 2011. 1

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{'abstract': 'Understanding the camera wearer\'s activity is central to egocentric vision, yet one key facet of that activity is inherently invisible to the camerathe wearer\'s body pose. Prior work focuses on estimating the pose of hands and arms when they come into view, but this 1) gives an incomplete view of the full body posture, and 2) prevents any pose estimate at all in many frames, since the hands are only visible in a fraction of daily life activities. We propose to infer the "invisible pose" of a person behind the egocentric camera. Given a single video, our efficient learning-based approach returns the full body 3D joint positions for each frame. Our method exploits cues from the dynamic motion signatures of the surrounding scene-which changes predictably as a function of body pose-as well as static scene structures that reveal the viewpoint (e.g., sitting vs. standing). We further introduce a novel energy minimization scheme to infer the pose sequence. It uses soft predictions of the poses per time instant together with a nonparametric model of human pose dynamics over longer windows. Our method outperforms an array of possible alternatives, including deep learning approaches for direct pose regression from images.', 'arxivid': '1603.07763', 'author': ['Hao Jiang \nComputer Science Department\nBoston College\nUSA\n', 'Kristen Grauman \nDepartment of Computer Science\nUniversity of Texas at Austin\nUSA\n'], 'authoraffiliation': ['Computer Science Department\nBoston College\nUSA', 'Department of Computer Science\nUniversity of Texas at Austin\nUSA'], 'corpusid': 206595684, 'doi': '10.1109/cvpr.2017.373', 'github_urls': [], 'n_tokens_mistral': 13231, 'n_tokens_neox': 11814, 'n_words': 7770, 'pdfsha': '445dc963f522f3a2885a98f3e4ac65e50e9110e2', 'pdfurls': ['https://arxiv.org/pdf/1603.07763v1.pdf'], 'title': ['Seeing Invisible Poses: Estimating 3D Body Pose from Egocentric Video', 'Seeing Invisible Poses: Estimating 3D Body Pose from Egocentric Video'], 'venue': []}

arxiv

A Ricci-flat metric on D-brane orbifolds 9803192v1 23 Mar 1998 Koushik Ray [email protected] Dipartimento di Fisica Ricerca Scientifica 00133RomaITALY Università Di Dipartimento di Fisica Ricerca Scientifica 00133RomaITALY Roma &quot; Tor Vergata Dipartimento di Fisica Ricerca Scientifica 00133RomaITALY &quot; I N F N Dipartimento di Fisica Ricerca Scientifica 00133RomaITALY -Sezione Di Dipartimento di Fisica Ricerca Scientifica 00133RomaITALY Roma &quot; Tor Vergata Dipartimento di Fisica Ricerca Scientifica 00133RomaITALY A Ricci-flat metric on D-brane orbifolds 9803192v1 23 Mar 1998arXiv:hep-th/ We study issues pertaining to the Ricci-flatness of metrics on orbifolds resolved by Dbranes. We find a Kähler metric on the three-dimensional orbifold C 3 /Z 3 , resolved by D-branes, following an approach due to Guillemin. This metric is not Ricci-flat for any finite value of the blow-up parameter. Conditions for the envisaged Ricci-flat metric for finite values of the blow-up parameter are formulated in terms of a correction to the Kähler potential. This leads to an explicit construction of a Ricci-flat Kähler metric on the resolved orbifold. The correction constitutes a part of the superspace-interaction in the corresponding gauged linear sigma-model. * I.N.F.N. Fellow. † Duality symmetries in Type-I and Type-II string theories necessitate incorporating Dbranes in these theories, even though the branes may be of dimensions other than unity, and are not "strings" in general. Apart from the fact that these objects fit nicely into the conformal field theoretic framework, it looms large that D-branes will play a crucial role in our final understanding of space-time, as they can be used to probe the topology and geometry of spacetime. In fact, D0-branes are deemed to be more efficient probes of space-time than strings in that they probe distance scales shorter than the string scale [1]. These considerations have made D-branes interesting objects to study in recent times.The world-volume theory of n parallel Dp-branes is described by a supersymmetric gauge theory, namely the ten-dimensional supersymmetric Yang-Mills theory, with gauge group U(n), dimensionally reduced to (p + 1) dimensions[2]. The moduli space of this reduced theory has been interpreted as the space-time sensed by D-branes[2]. In this sense space-time is a derived concept in the theory of D-branes. It has been found that it is possible to realize the moduli space of D-branes as orbifolds by properly truncating the reduced theory to sectors invariant under the action of some discrete group. The theory of D-branes on two(complex)-dimensional ALE-spaces has been studied in detail[3,4]. Such studies realize Krönheimer's construction of ALE-spaces in physical terms for the Kleinian subgroups of SU(2) acting on the two-dimensional complex space, C 2 . Fundamental strings on ALE-spaces have been considered earlier[5].More recently, it has been found that D-branes can also be used to resolve three-dimensional orbifold singularities. One can thus study the short-distance behavior of Calabi-Yau manifolds as sensed by a D-brane probe in the neighborhood of a resolved orbifold singularity. In order to study the theory of a single D-brane on C 3 /Γ, where Γ is a discrete group of order |Γ|, one starts with |Γ| number of D-branes on the covering space C 3 , arranged in the regular representation of Γ. The world-volume theory of |Γ| D-branes on C 3 is a four-dimensional gauge theory with N = 4 supersymmetry and gauge group U(|Γ|). The complex positions of the D-branes on C 3 are given by the scalars in the theory. Restricting to the sector of the D-brane Lagrangian invariant under the discrete group yields a gauged linear sigma-model. The nexus between the two-dimensional gauged linear sigma-model and the world-volume theory of a D1-brane probe has been exploited earlier[6,7]. The moduli space of this linear sigma-model is interpreted as the sub-stringy space-time. This moduli space can be thought of as an internal space on which D-branes move as points, with the world-volume of the D-brane lying in directions transverse to this internal space. Now, if the discrete group Γ is chosen to be a subgroup of SU(3), then the resulting theory after truncation retains N = 1 supersymmetry. This construction has been realized for different choices of the discrete group Γ; for example, Γ = Z n for n = 3, 5, 7, 9, 11 [8, 9] and Γ = Z 2 × Z 2[10,11]. It should be mentioned that in these considerations one may start with an arbitrary number |Γ| of D-branes on the covering space because the space one considers is non-compact. The moduli space of the D-brane can be viewed as a local description of a Calabi-Yau manifold only near an orbifold point. Further generalization of these ideas to four dimensions has also been considered[12]. It has been found that toric geometry provides a convenient language for such considerations[8,13,14]. In the present paper we shall concentrate on the example of C 3 /Z 3 [8]. As mentioned above, the theory of a D-brane on C 3 /Z 3 is a fourdimensional N = 1 theory, which is equivalent to the two-dimensional N = 2 gauged linear sigma-model[15,16]upon dimensional reduction.Apart from the topological properties, some geometric, alias metric, properties of these resolved orbifolds have also been considered[17]. It has been found that, starting from the flat Kähler potential on the covering space C 3 , and then adding the Fayet-Iliopoulos terms, one can derive, by proper gauging, a Kähler metric on the three-dimensional resolved orbifold C 3 /Z 3 . However, this metric turns out to be a non-Ricci-flat one [17,18]. It has also been found that the same prescription leads to the Eguchi-Hanson metric on the two-dimensional resolved orbifold C 2 /Z 2 , which is Ricci-flat. This dichotomy has been attributed to the fact that, unlike a resolved C 2 /Z 2 , the resolved orbifold C 3 /Z 3 , although Kähler, is not hyper-Kähler. In particular, it has been shown that the metric derived in this fashion for C 3 /Z 3 resolved into O CP 2 (−3) assumes the form discovered by Calabi, where O CP 2 (−3) is looked upon as having a canonical fiber over a base manifold of constant curvature. This metric is Ricci-flat only for certain complex values of the Fayet-Iliopoulos parameters, which is forbidden both physically and mathematically. In this paper we shall follow an alternative approach to deriving Kähler metrics on toric varieties [19]. A novel feature of this approach is that it makes it possible to derive the metric starting from the toric data alone. This approach ensues from endowing a toric variety X with a symplectic structure arising from a Hamiltonian action of the torus T R on X [20]. It identifies the polytope ∆ corresponding to the toric variety -with ∆ required to satisfy certain non-singularity and non-degeneracy conditions (in particular, the polytope ∆ must be Delzant [19,21]) -with the moment polytope of the moment map of the Hamiltonian action of the torus on X by a homeomorphism X/T R −→ ∆, up to translation. This correspondence provides a prescription for obtaining a Kähler metric on the variety [19]. We shall refer to this metric as the canonical metric in the sequel. This metric is not Ricci-flat in general. However, starting from the canonical metric, one may go over to any other Kähler metric within the same Kähler class by adding a well-behaved function to the (Legendre) dual (to be defined below) of the Kähler potential [21]. We pursue this line of approach here. We first find out the canonical metric, following [19], on the resolved orbifold C 3 /Z 3 , starting from the toric data as obtained in [8]. We find that this metric is not Ricci-flat for any finite value of the blow-up parameter. We then add a function f to the (Legendre) dual Kähler potential and find out a differential equation for f , demanding that the new metric thus obtained is Ricci-flat. The solution of this equation leads to a Ricci-flat metric on the orbifold C 3 /Z 3 resolved by a D-brane. Moreover, this additional function f constitutes a part of the superspace-interaction in the linear sigma-model. In order to fix notations, let us start with some relevant features of the present construction [19]. Let (X, ω) be an effective Hamiltonian T n -space with a Kähler form ω, and let ϕ : X −→ R n denote the corresponding moment map associated to the Hamiltonian action of T n on X. Let ∆ denote the image of X on R n under the moment map, i.e. ∆ = ϕ(X) ⊂ R n . ∆ is referred to as the moment polytope and X ∆ = X is the toric variety associated to ∆. Conversely, one can associate a toric variety X ∆ with the above properties, to a Delzant polytope ∆ in R d , such that ∆ is the moment polytope of X ∆ . Let us recall that the polytope ∆ in R d is called Delzant if there are d edges meeting at each vertex p of ∆ and any edge meeting at p can be given the form p + sv i , for 0 ≤ s ≤ ∞, where {v i } is a basis of Z d [19,21]. The moment polytope ∆ can be described by a set of inequalities of the form y, u i ≥ λ i , i = 0, 2, · · · d − 1. Here u i denotes the inward-pointing normal to the i-th (n − 1)-dimensional face of ∆ and is a primitive element of the lattice Z n ⊂ R n ; , denotes the standard scalar product in R n and y is an n-dimensional real vector. We can thus define a set of linear maps, ℓ i : R n −→ R, ℓ i (y) = y, u i − λ i , i = 0, · · · d − 1.(1) Denoting the interior of ∆ by ∆ • , y ∈ ∆ • , if and only if ℓ i (y) > 0 for all i. On the open T n C -orbit in X ∆ , associated to a Delzant polytope ∆, the Kähler form ω can be written in terms of a potential F as [19]: ω = 2i∂∂F ,(2) with F = 1 2 ϕ ⋆ d−1 i=0 λ i ln ℓ i + ℓ ∞ ,(3) where ϕ ⋆ denotes the pull-back of the moment map on ∆ and we have defined ℓ ∞ = d−1 i=0 y, u i .(4) The Kähler form ω can be written as: ω = i 2 n−1 j,k=0 ∂ 2 F ∂x j ∂x k dz j ∧ dz k .(5) The restriction to R n = Re C n of the Kähler metric corresponding to ω is the Riemannian metric given by, ds 2 = n−1 j,k=0 ∂ 2 F ∂x j ∂x k dx j dx k .(6) Now, under the Legendre transform determined by the moment map ϕ, y i = ∂F ∂x i ,(7) the metric given by (6) is the pull-back of the metric given by, ds 2 = n−1 j,k=0 ∂ 2 G ∂y j ∂y k dy j dy k(8) on ∆ • , where G is given by: G = 1 2 d−1 k=0 ℓ k (y) ln ℓ k (y).(9) The inverse of the Legendre transform (7) is: x i = ∂G ∂y i + r i , i = 0, · · · , d − 1,(10) where r i are constants. This means that up to a linear term in the co-ordinates y i , G is the Kähler potential Legendre-dual to F . Moreover, the matrix G ij = ∂ 2 G ∂y i ∂y j ,(11) evaluated at (7), i.e. at y i = ∂F ∂x i , is the inverse of the matrix F ij = ∂ 2 F ∂x i ∂x j .(12) The Ricci-tensor for the metric (12) takes the following form: R ij = − 1 2 ∂ 2 ln det F ∂x i ∂x j (13) = − 1 2 n−1 k,l=0 G lj ∂ 2 G ik ∂y k ∂y l ,(14) where G ij denotes the inverse of G ij . Note that, F in (13) refers to the matrix F ij , and not the Kähler potential, unlike other places in the paper. The Ricci-scalar for this metric is then derived by multiplying (14) with G ij , which is the inverse of the metric F ij in the y co-ordinates and is given by [21], R = − 1 2 n−1 i,j=0 F ij ∂ 2 ln det F ∂x i ∂x j (15) = − 1 2 n−1 i,j=0 ∂ 2 G ij ∂y i ∂y j ,(16) where F ij denotes the inverse of F ij . For our purposes, it will be convenient to use the matrix G ij in the coordinates y. One can, in principle, rewrite all the relevant expressions in terms of the coordinates x i and the matrix F ij . With this introduction, let us now begin the study of metric properties of the orbifold C 3 /Z 3 resolved by D-branes. The description of this orbifold along with its topological properties and phase structure has been studied extensively [8,10]. We shall not repeat the construction here. Let us start from the toric data obtained in [8] for the resolved orbifold C 3 /Z 3 . The toric data is specified by the charges of the components of the chiral superfields in terms of a 3 × 4 matrix T as, T =   −1 1 0 0 −1 0 1 0 3 0 0 1   .(17) Triangulating on all points yields the blow-up of the orbifold C 3 /Z 3 . Given the toric data (17), one can find out the inequalities describing the corresponding Delzant polytope [20]. For this let us consider the fan in R 3 , corresponding to the toric data (17) whose one-dimensional cone-generators are given by the four three-vectors constituting the four columns of T : e 0 =   −1 −1 3   , e 1 =   1 0 0   , e 2 =   0 1 0   , e 3 =   0 0 1   .(18) Thus, this case corresponds to n = 3 and d = 4 of the above discussion. The Kernel of the matrix T is Ker T =     1 1 1 −3     .(19) Hence, the Kähler cone of the variety X ∆ associated to T is given by ξ ≡ a 0 + a 1 + a 2 − 3a 3 > 0,(20) where we have denoted the image of the support function [13,20] at e i as −a i for i = 0, · · · , 3. The moment map is given by |z 0 | 2 + |z 1 | 2 + |z 2 | 2 − 3|z 3 | 3 = ξ,(21) which corresponds to the D-flatness condition of a U(1) gauged linear sigma-model with four chiral superfields [8,9,15,16]. In (21), z i , i = 0, · · · , 3, denote the homogeneous coordinates on the toric variety. Let us also note that this correspondence allows us to relate the blow-up parameter ξ, which is proportional to the Kähler class of the variety, to the Fayet-Iliopoulos parameters of the linear sigma-model, using the charge matrix (17), and hence study the phase structure [8]. For all possible real values of the Fayet-Iliopoulos parameters, ξ remains positive in concordance with (20). Physically, this means that the D-branes abstain from entering the non-geometric phases of the linear sigma-model [8,22]. Using the one-dimensional cone generators e i , we can now write down the Delzant polytope ∆ in the shifted coordinates as the one defined by the inequalities: ξ − y 1 − y 2 + 3y 3 ≥ 0,(22)y 1 ≥ 0, (23) y 2 ≥ 0,(24)y 3 ≥ 0,(25) where ξ = a 0 + a 1 + a 2 − 3a 3 is positive as above, and we have applied a shift of a i to each co-ordinate y i , for i = 1, 2, 3. The above expressions are required to be strictly positive in the interior ∆ • of ∆ and equality holds in each of the above relations on the boundary of the polytope defined by two-dimensional faces. We thus have the following linear functions ℓ i , for i = 0, 1, 2, 3, ℓ 1 (y) = y 1 , ℓ 2 (y) = y 2 , ℓ 3 (y) = y 3 , ℓ 0 (y) = ξ − y 1 − y 2 + 3y 3 ,(26) whose positivity determine the Delzant polytope ∆ up to shifts, corresponding to the resolved orbifold C 3 /Z 3 . We can now write down the potential G from (26): G = 1 2 [y 1 ln y 1 + y 2 ln y 2 + y 3 ln y 3 + (ξ − y 1 − y 2 + 3y 3 ) ln(ξ − y 1 − y 2 + 3y 3 )] .(27) The inverse Legendre transform (10) expresses the co-ordinates x i in terms of the co-ordinates y i . The relations between these dual co-ordinates can be written in the following form: (ξ + 3y 3 ) 3 y 3 = (1 + e 2x 1 + e 2x 2 ) 3 e 2x 3 ,(28)y 1 y 2 = e 2(x 1 −x 2 ) ,(29)y 1 A = e 2x 1 ,(30) where we have defined, A = ξ − y 1 − y 2 + 3y 3 ,(31) and chosen the constants r i as r 1 = r 2 = 0 and r 3 = 2 for simplicity. Equations (28)-(30) can be solved explicitly to write down expressions for x 1 , x 2 and x 3 in terms of y 1 , y 2 and y 3 . One can now calculate the matrix G ij = ∂ 2 G ∂y i ∂y j , using the potential G obtained in (27). It takes the form: G ij = 1 2A   1 + A/y 1 1 −3 1 1 + A/y 2 −3 −3 −3 9 + A/y 3   .(32) Inverting this matrix to write G ij , and then differentiating twice with respect to the co-ordinates, we can compute the Ricci-scalar according to (16) as: R = 12(ξ 2 + 72y 2 3 ) (ξ + 12y 3 ) 3 .(33) Thus, the metric on the variety is not Ricci-flat for any finite value of the parameter ξ. However, the Ricci-scalar (33) vanishes as 1/ξ as ξ is taken to infinity. The metric in this limit is given by G ij =   2y 1 0 0 0 2y 2 0 0 0 2y 3   .(34) The Ricci-tensor (14) also vanishes for the metric (34). Thus, in the limit of infinite ξ, we have a Ricci-flat metric. However, let us note that using the formulation of [19], one can explicitly calculate the first Chern class of the variety under consideration and check that it vanishes. For this, let us note that the Kähler class of the canonical metric can be expressed in terms of the parameters λ i as [19]: [ω] 2π = − d−1 i=0 λ i α i ,(35) where α i , i = 0, · · · , d − 1 are constants. One can express the first Chern class of the variety X ∆ as a sum of the constants α i : c 1 (X ∆ ) = d−1 i=0 α i .(36) Physically, equation (36) expresses the first Chern class of the variety as a sum of the charges of the fields in the corresponding linear sigma-model. Using the expression (20) for ξ, which is proportional to the Kähler class of the variety, and noting that λ i = −a i , when the polytope ∆ is written in terms of the unshifted co-ordinates, we find that α 0 = α 1 = α 2 = 1, α 3 = −3,(37) up to a constant of proportionality multiplying each of α i . Hence, by (36), the first Chern class of the resolved orbifold C 3 /Z 3 vanishes. This is not surprising in view of the fact that the action of Z 3 on the chiral superfields were chosen in the beginning in such a way that the Z 3 was a subgroup of SU(3) [8]. Nonetheless, the present approach provides an explicit reconfirmation of the vanishing of the first Chern class of the resolved orbifold C 3 /Z 3 . Let us now go over to explore the possibility of obtaining a Ricci-flat metric for finite values of the blow-up parameter ξ. For this purpose, let us note [21] that one can add a function, which is smooth on some open subset of R n containing ∆, to the potential G, such that the Hessian of the new potential is positive definite on ∆ • , and derive a new Kähler metric in the same Kähler class as the canonical one. The variety X ∆ endowed with the two different Kähler forms are related by a T n -equivariant symplectomorphism by virtue of the function f being non-singular [21, Remark 3.1]. For example, the function one has to add to the potential G, in order to derive the extremal Kähler metric on CP 2 #CP 2 in Calabi's form, was determined utilizing this observation [21]. Following this approach, let us as add a function f to G, and define a new potential G = G + 1 2 f . Now the matrix G ij corresponding to the potential G assumes the form G ij = G ij + 1 2 ∂ 2 f ∂y i ∂y j .(38) One can then find out the Kähler metric by inverting G ij and hence the curvature for this new metric. This will also give rise to a new F corresponding to F and also new co-ordinates x. What we propose to do next is to write down the general form of the metric for a function f and then determine the function by demanding that the Ricci-tensor given by the formula (14) vanishes. This gives a differential equation for the function f . However, for practical purposes of obtaining a tractable equation for f , one needs to start with an ansätz for the function f . In view of the fact that the Ricci-scalar (33) is a function of y 3 alone, not depending on y 1 or y 2 , let us assume that f is a function of y 3 only, i.e. f = f (y 3 ). The matrix G ij is then given by G ij = 1 2A   1 + A/y 1 1 −3 1 1 + A/y 2 −3 −3 −3 9 + A/y 3 + Af ′′   ,(39) where A = ξ − y 1 − y 2 + 3y 3 , as before and a prime denotes a differentiation with respect to y 3 . The inverse of the matrix G ij is given by G ij =    2y 1 − 2 y 2 1 F φ −2 y 1 y 2 F φ 6 y 1 y 3 F −2 y 1 y 2 F φ 2y 2 − 2 y 2 2 F φ 6 y 2 y 3 F 6 y 1 y 3 F 6 y 2 y 3 F 2 y 3 (ξ+3y 3 ) F    ,(40) where we have defined F = ξ + 12y 3 + (ξ + 3y 3 )y 3 f ′′ and φ = F − 9y 3 ξ + 3y 3 .(41) The determinant of the matrix G ij is det G ij = F 8Ay 1 y 2 y 3 . Hence, throughout this paper we shall assume F to be nowhere-vanishing in order to keep G ij non-singular. For the metric (40), the surviving terms in the different components of R ij as evaluated from (14) take the following form: − 2R 11 = G 11 ∂ 2 G 11 ∂y 1 ∂y 1 + ∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 13 ∂y 1 ∂y 3 + G 13 ∂ 2 G 11 ∂y 1 ∂y 3 + ∂ 2 G 12 ∂y 2 ∂y 3 + ∂ 2 G 13 ∂y 3 ∂y 3 ,(42) − 2R 12 = G 12 ∂ 2 G 11 ∂y 1 ∂y 1 + ∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 13 ∂y 1 ∂y 3 + G 23 ∂ 2 G 11 ∂y 1 ∂y 3 + ∂ 2 G 12 ∂y 2 ∂y 3 + ∂ 2 G 13 ∂y 3 ∂y 3 ,(43) − 2R 13 = G 13 ∂ 2 G 11 ∂y 1 ∂y 1 + ∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 13 ∂y 1 ∂y 3 + G 33 ∂ 2 G 11 ∂y 1 ∂y 3 + ∂ 2 G 12 ∂y 2 ∂y 3 + ∂ 2 G 13 ∂y 3 ∂y 3 ,(44) − 2R 22 = G 22 ∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 22 ∂y 2 ∂y 2 + ∂ 2 G 23 ∂y 2 ∂y 3 + G 23 ∂ 2 G 12 ∂y 1 ∂y 3 + ∂ 2 G 22 ∂y 2 ∂y 3 + ∂ 2 G 23 ∂y 3 ∂y 3 ,(45) − 2R 23 = G 23 ∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 22 ∂y 2 ∂y 2 + ∂ 2 G 23 ∂y 2 ∂y 3 + G 33 ∂ 2 G 12 ∂y 1 ∂y 3 + ∂ 2 G 22 ∂y 2 ∂y 3 + ∂ 2 G 23 ∂y 3 ∂y 3 ,(46) − 2R 33 = G 33 ∂ 2 G 13 ∂y 1 ∂y 3 + ∂ 2 G 23 ∂y 2 ∂y 3 + ∂ 2 G 33 ∂y 3 ∂y 3 .(47) Thus, the Ricci-tensor vanishes if the following five equations are satisfied: ∂ 2 G 11 ∂y 1 ∂y 1 + ∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 13 ∂y 1 ∂y 3 = 0,(48)∂ 2 G 11 ∂y 1 ∂y 3 + ∂ 2 G 12 ∂y 2 ∂y 3 + ∂ 2 G 13 ∂y 3 ∂y 3 = 0,(49)∂ 2 G 12 ∂y 1 ∂y 2 + ∂ 2 G 22 ∂y 2 ∂y 2 + ∂ 2 G 23 ∂y 2 ∂y 3 = 0,(50)∂ 2 G 12 ∂y 1 ∂y 3 + ∂ 2 G 22 ∂y 2 ∂y 3 + ∂ 2 G 23 ∂y 3 ∂y 3 = 0,(51)∂ 2 G 13 ∂y 1 ∂y 3 + ∂ 2 G 23 ∂y 2 ∂y 3 + ∂ 2 G 33 ∂y 3 ∂y 3 = 0.(52) Let us note that adding the equations (48), (50) and (52) yields the equation of vanishing Ricci-scalar, as derived from (16). Using the explicit form of the matrix (40), one can now write down the equations for F corresponding to (48)-(52). The equations following from (48) and (50) are identical; those from (49) and (51) are also identical. We are thus left with three equations following from (48), (49) and (52). These are, respectively, F − y 3 F ′ − φF = 0 (53) 2y 3 (F ′ ) 2 − (φ ′ F − φF ′ )F − y 3 F F ′′ − 2F F ′ = 0 (54) 2y 3 (ξ + 3y 3 )(F ′ ) 2 − y 3 (ξ + 3y 3 )F F ′′ − 2(ξ + 9y 3 )F F ′ + 12F 2 = 0.(55) However, these three equations are not independent. Equations (54) and (55) can be solved by using only (53). Thus, finally, we need to solve only (53) in order to determine F , i.e. the function F is not over-determined by the condition of Ricci-flatness of the metric. Rewriting (53) by plugging in the expression of φ from (41), we derive the following equation, y 3 (ξ + 3y 3 )F ′ + (F − ξ − 12y 3 )F = 0.(56) As a check on consistency, let us note that (56) solves the equation for vanishing Ricci-scalar, which, by (16), can be written as y 3 (ξ + 3y 3 ) 2 F F ′′ + 2(ξ + 3y 3 )(ξ + 12y 3 )F F ′ − 2y 3 (ξ + 3y 3 ) 2 (F ′ ) 2 + 6(F − 3ξ − 18y 3 )F 2 = 0.(57) We shall linearize and solve (56) for F . Once F is determined, we can use (41) to determine f ′′ , and this yields the Ricci-flat metric, using f ′′ in (40). In order to solve for the Kähler potential one has to solve for f . Let us point out that using the definition of F from (41) in (56), we have the following relation between F and f : f ′′ = −F ′ /F.(58) In order to solve (56), we divide both sides of (56) by F , as F is nowhere-vanishing, to derive the first order linear equation, y 3 (ξ + 3y 3 )χ ′ + (ξ + 12y 3 )χ = 1,(59) where χ = 1/F . Equation (59) can be solved to derive the following expression for F : F = 9y 3 (ξ + 3y 3 ) 3 c + (ξ + 3y 3 ) 3 ,(60) where c is a constant of integration. This in turn yields the following expression for f ′′ : f ′′ = 9y 3 (ξ + 3y 3 ) 3 − (ξ + 12y 3 )[c + (ξ + 3y 3 ) 3 ] y 3 (ξ + 3y 3 ) .(61) Using this in the expression (40) for G ij , we have a Ricci-flat metric on the variety. Equation (61) can be integrated to solve for f and hence determine G. Thus, we finally have the explicit form of a Ricci-flat metric and the Kähler potential on the variety in a neighborhood of the resolved orbifold singularity C 3 /Z 3 . Finally, let us discuss the physical meaning of the function f . Let us recall that it is customary in the context of gauged linear sigma-models to work with the metric induced from the standard metric on CP n , in which one can embed the space described by the D-flatness condition. This metric, although Kähler, is not Ricci-flat [15]. It is believed that there exists a unique Ricci-flat Kähler metric for large values of the Fayet-Iliopoulos parameter, ξ, in the present case. Moreover, this Ricci-flat metric differs from the one induced from the Fubini-Study metric on CP n in such a way that the difference between the cohomologous Kähler forms corresponding to the two metrics is given by −i∂∂T . The difference between the corresponding the Kähler potentials, T , provides the superspace interaction term in the Lagrangian of the linear sigma-model, namely, d 2 xd 4 θT,(62) in the notation of [15]. In the limit of infinite coupling, the linear sigma-model [15] differs from the conformal invariant non-linear sigma-model by the term (62). Therefore, in order to learn about the conformal invariant model from the linear one of [15], one needs to know the form of T in (62). Now, let us recall the following results from [19]. It can be proved that the de Rham cohomology classes in H 2 (X, R) of the canonical Kähler form ω obtained through Guillemin's construction and the metric induced on C 3 /Z 3 from the Fubini-Study metric on CP N are the same. It can also be proved [19] that: there exists a smooth function, Q, on R n , such that ω FS = ω + i∂∂ϕ ⋆ Q.(63) Moreover, Q is unique up to an additive constant. Here ω FS denotes the pull-back from the Fubini-Study metric on CP N . An explicit form for Q is given in [19]. The Kähler classes of the forms corresponding to G and G are the same [21]. The Kähler form corresponding to G can be written as : ω = ω FS + i∂∂(f − ϕ ⋆ Q).(64) In other words, the expression (f − ϕ ⋆ Q) is the difference between the potentials corresponding to the induced Fubini-Study metric on the resolved orbifold and the Ricci-flat Kähler metric (40). Hence, in the limit of strong coupling this may be identified with T in (62), providing the "irrelevant" superspace interaction term needed to drive the linear sigma-model to the conformal invariant limit [15]. To conclude, in this paper we have followed a construction due to Guillemin to derive a Kähler metric on the orbifold C 3 /Z 3 resolved by D-branes, in the neighborhood of the blownup orbifold point. The metric derived following [19] is not Ricci-flat for any finite value of the blow-up parameter. Then, making use of the observation that one can add a well-behaved function to the Legendre-dual Kähler potential G [21], we determined a function f which can be added to the canonical potential to obtain a Ricci-flat metric. In this approach one derives an explicit form of the Ricci-flat metric, for all values of the blow-up parameter. The novelty of the construction of [19] is that it allows to derive the metric on the variety and the corresponding Kähler potential starting from the toric data alone, and hence avoids the complications due to the choice of seed metric [17]. The canonical metric itself can be thought of as the seed metric in this context. Since the resolution of the orbifold C 3 /Z 3 as effected by D-branes is encoded in the toric data T , which correspond to charges of the fields in the corresponding linear sigmamodel [8], the metric (40) with f ′′ given by (61) is the metric on the resolved orbifold sensed by D-branes in a neighborhood of the orbifold point. It will be interesting to check this result by direct world-sheet computations as suggested in [17]. In view of the result of Guillemin relating the Kähler potentials corresponding to the canonical metric and the metric induced on the resolved orbifold from the Fubini-Study metric on CP N , by a function Q, we find that the function f , combined with Q, gives the superspace interaction term in the strong coupling regime of the linear gauged sigma-model, which is required to drive the linear sigma model to the non-linear one with conformal symmetry. It will be interesting to see how this formulation can be used to drive the linear sigma-model to the limit of strong coupling with a large Fayet-Iliopoulos parameter. It is expected that this formulation will shed some light on the emergence of conformal symmetry in this context. It will be interesting to extend this formulation for other choices of ansätz for f . Another interesting point to note is that while Calabi's form of the metric is derived from the condition that the metric on the base of the resolution O CP 2 (−3) has a constant determinant, it is not so in the present case. Here we demand the vanishing of the Ricci-tensor R ij as given by (14), which is different from demanding constant determinant on the base. It will be interesting, at least mathematically, to find out an explicit relation between the two metrics. Finally, it should be possible, by following the same line of arguments as here, to generalize this formulation for other cases, for example, Z n and Z 2 × Z 2 . We hope to return to some of these issues in near future. Acknowledgements: I take this opportunity to thank M Bianchi and A Sagnotti for encouragement and enlightening discussions. I also thank D Polyakov for useful discussions. D-brane and short distances in string theory. 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{'abstract': 'We study issues pertaining to the Ricci-flatness of metrics on orbifolds resolved by Dbranes. We find a Kähler metric on the three-dimensional orbifold C 3 /Z 3 , resolved by D-branes, following an approach due to Guillemin. This metric is not Ricci-flat for any finite value of the blow-up parameter. Conditions for the envisaged Ricci-flat metric for finite values of the blow-up parameter are formulated in terms of a correction to the Kähler potential. This leads to an explicit construction of a Ricci-flat Kähler metric on the resolved orbifold. The correction constitutes a part of the superspace-interaction in the corresponding gauged linear sigma-model. * I.N.F.N. Fellow. † Duality symmetries in Type-I and Type-II string theories necessitate incorporating Dbranes in these theories, even though the branes may be of dimensions other than unity, and are not "strings" in general. Apart from the fact that these objects fit nicely into the conformal field theoretic framework, it looms large that D-branes will play a crucial role in our final understanding of space-time, as they can be used to probe the topology and geometry of spacetime. In fact, D0-branes are deemed to be more efficient probes of space-time than strings in that they probe distance scales shorter than the string scale [1]. These considerations have made D-branes interesting objects to study in recent times.The world-volume theory of n parallel Dp-branes is described by a supersymmetric gauge theory, namely the ten-dimensional supersymmetric Yang-Mills theory, with gauge group U(n), dimensionally reduced to (p + 1) dimensions[2]. The moduli space of this reduced theory has been interpreted as the space-time sensed by D-branes[2]. In this sense space-time is a derived concept in the theory of D-branes. It has been found that it is possible to realize the moduli space of D-branes as orbifolds by properly truncating the reduced theory to sectors invariant under the action of some discrete group. The theory of D-branes on two(complex)-dimensional ALE-spaces has been studied in detail[3,4]. Such studies realize Krönheimer\'s construction of ALE-spaces in physical terms for the Kleinian subgroups of SU(2) acting on the two-dimensional complex space, C 2 . Fundamental strings on ALE-spaces have been considered earlier[5].More recently, it has been found that D-branes can also be used to resolve three-dimensional orbifold singularities. One can thus study the short-distance behavior of Calabi-Yau manifolds as sensed by a D-brane probe in the neighborhood of a resolved orbifold singularity. In order to study the theory of a single D-brane on C 3 /Γ, where Γ is a discrete group of order |Γ|, one starts with |Γ| number of D-branes on the covering space C 3 , arranged in the regular representation of Γ. The world-volume theory of |Γ| D-branes on C 3 is a four-dimensional gauge theory with N = 4 supersymmetry and gauge group U(|Γ|). The complex positions of the D-branes on C 3 are given by the scalars in the theory. Restricting to the sector of the D-brane Lagrangian invariant under the discrete group yields a gauged linear sigma-model. The nexus between the two-dimensional gauged linear sigma-model and the world-volume theory of a D1-brane probe has been exploited earlier[6,7]. The moduli space of this linear sigma-model is interpreted as the sub-stringy space-time. This moduli space can be thought of as an internal space on which D-branes move as points, with the world-volume of the D-brane lying in directions transverse to this internal space. Now, if the discrete group Γ is chosen to be a subgroup of SU(3), then the resulting theory after truncation retains N = 1 supersymmetry. This construction has been realized for different choices of the discrete group Γ; for example, Γ = Z n for n = 3, 5, 7, 9, 11 [8, 9] and Γ = Z 2 × Z 2[10,11]. It should be mentioned that in these considerations one may start with an arbitrary number |Γ| of D-branes on the covering space because the space one considers is non-compact. The moduli space of the D-brane can be viewed as a local description of a Calabi-Yau manifold only near an orbifold point. Further generalization of these ideas to four dimensions has also been considered[12]. It has been found that toric geometry provides a convenient language for such considerations[8,13,14]. In the present paper we shall concentrate on the example of C 3 /Z 3 [8]. As mentioned above, the theory of a D-brane on C 3 /Z 3 is a fourdimensional N = 1 theory, which is equivalent to the two-dimensional N = 2 gauged linear sigma-model[15,16]upon dimensional reduction.Apart from the topological properties, some geometric, alias metric, properties of these resolved orbifolds have also been considered[17]. It has been found that, starting from the flat Kähler potential on the covering space C 3 , and then adding the Fayet-Iliopoulos terms,', 'arxivid': 'hep-th/9803192', 'author': ['Koushik Ray [email protected] \nDipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY\n', 'Università Di \nDipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY\n', 'Roma &quot; Tor Vergata \nDipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY\n', '&quot; I N F N \nDipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY\n', '-Sezione Di \nDipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY\n', 'Roma &quot; Tor Vergata \nDipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY\n'], 'authoraffiliation': ['Dipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY', 'Dipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY', 'Dipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY', 'Dipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY', 'Dipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY', 'Dipartimento di Fisica\nRicerca Scientifica\n00133RomaITALY'], 'corpusid': 14715871, 'doi': '10.1016/s0370-2693(98)00722-9', 'github_urls': [], 'n_tokens_mistral': 11596, 'n_tokens_neox': 9865, 'n_words': 6219, 'pdfsha': '11eefb7875618edd6b85c8b561c2315f30ba9381', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/9803192v1.pdf'], 'title': ['A Ricci-flat metric on D-brane orbifolds', 'A Ricci-flat metric on D-brane orbifolds'], 'venue': []}

arxiv

Topological Origins of Flexibility and Internal Stress in Sodium Aluminosilicate Glasses Ernest Ching Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab) University of California 90095-1593Los AngelesCAU.S.A Mathieu Bauchy [email protected] Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab) University of California 90095-1593Los AngelesCAU.S.A Topological Origins of Flexibility and Internal Stress in Sodium Aluminosilicate Glasses 1 * Corresponding author: Prof. Mathieu Bauchy, In the framework of topological constraint theory, network glasses are classified as flexible, stressed-rigid, or isostatic if the number of atomic constraints is smaller, larger, or equal to the number of atomic degrees of freedom. Here, based on molecular dynamics simulations, we show that sodium aluminosilicate glasses exhibit a flexible-to-stressed-rigid transition driven by their composition. This transition manifests itself by a loss of atomic mobility and an onset of internal atomic stress. Importantly, we find that the flexible-to-rigid (i.e., loss of internal flexibility) and unstressed-to-stressed transitions (i.e., onset of internal stress) do not occur at the same composition. This suggests that the isostatic state (i.e., rigid but unstressed) is achieved within a window rather than at a threshold composition. I. INTRODUCTION Sodium aluminosilicate glasses are well recognized for their outstanding mechanical properties and its ability to undergo ion-exchange [1-4] with larger alkali elements, such as potassium, which increases its surface compressive strength. As such, sodium aluminosilicate glasses find a wide range of commercial applications, including its use as smartphone display surfaces, e.g. Corning® Gorilla® Glass [5,6] and as substrates for organic electronics [7]. In order to better understand the post-ion exchange properties of aluminosilicate glasses, this paper examines the origins of flexibility and internal stress in sodium aluminosilicate glasses within the framework of topological constraint theory (TCT) [8][9][10][11][12][13][14][15]. TCT describes the rigidity of a glass network by modeling atoms as truss nodes, and chemical bonds as truss members [10]. The number of constraints per atom (nc) is the sum of the following two types of constraints: radial bond stretching constraints and angular bond bending constraints, which varies between different atomic types based on the geometry of local connectivity. TCT allows for three modes: (i) flexible (nc < 3), where the number of inter-atomic constraints is fewer than the number of atomic degrees of freedom, (ii) isostatic (nc = 3), where the number of constraints exactly balance the degrees of freedom, and (iii) stressed-rigid (nc > 3), where the network is over-constrained, leading to the development of internal stresses [16]. The atomic structure and physical properties, including density, stiffness, thermal expansion, and corrosion behavior of sodium aluminosilicate glasses have been well studied in previous experiments and simulations [17][18][19][20][21]. However, it should be noted that per-aluminous glasses with high alumina content have not been studied extensively in experiments due to decreased glass formation ability with increased alumina content [22]. Hence, this study relies on molecular dynamics simulations in order to simulate alumina-rich glasses-this study examines compositions of (Na2O)30(SiO2)70-x(Al2O3)x, with x = 0% to 70% at 5% increments. Moreover, although topological constraint models have been proposed for different glasses, e.g., sodium silicate [23], sodium phosphosilicate [24], and soda lime borate [25], no comprehensive topological constraint model currently exist for sodium aluminosilicate glasses. Hence, this paper aims to fill this gap of knowledge by proposing a topological constraint model (TCM) for sodium aluminosilicates, which is compared to and verified with signatures of flexibility and internal stress obtained from simulations. II. SIMULATION METHODOLOGY A. Preparation of the glasses Sodium aluminosilicate glasses with compositions of (Na2O)30(SiO2)70-x(Al2O3)x, with x = 0% to 70% at 5% increments are simulated by molecular dynamics using the LAMMPS [26] package. All simulations are performed using the well-established empirical potential parametrized by Teter [27], which has been verified to predict realistic structural, mechanical and dynamical properties for sodium aluminosilicate glasses [18,[28][29][30][31][32]. The short-range interactions are modeled by a Buckingham potential with a cutoff of 8.0 Å, and the Coulombic interactions are evaluated using the Ewald method, with a cutoff of 12.0 Å [28]. All glass compositions are composed of roughly 3000 atoms and the initial configuration is created by randomly placing atoms in a cubic simulation box while avoiding any unrealistic overlap. The cooled glasses are then formed by (i) creating the melts at 4000 K with a Gaussian distribution to lose the memory of the initial configuration, (ii) equilibrating the melts at 4000 K and 2 GPa for 100 ps, and then at 4000 K and zero pressure for an additional 100 ps to lose the memory of initial configurations, (iii) cooling from 4000 K to 300 K at 1 K/ps at zero pressure, and (iv) relaxing the cooled glass at 300 K and zero pressure for 200 ps. The glass forming procedure described above is performed in the NPT ensemble with a Nosé-Hoover thermostat and barostat [33,34]. A timestep of 1 fs is used for all simulations. All glass compositions are repeated for 6 trials by using different random velocities for the creation of glass melts. B. Structure characterization Determining the radial and angular constraint of different atomic sub-species is important in creating a realistic topological constraint model. To this end, each as-cooled glass composition is relaxed at 300 K for 0.1 ns in the NVT ensemble. An atomic trajectory data file is obtained during this relaxation, which is used to analyze the structure of the glass using MATLAB [35]. Each oxygen atom is classified as a non-bridging oxygen (NBO), bridging oxygen (BO) or a tricluster oxygen (TO) based on the number of network formers (silicon and aluminum) each oxygen atoms is connected to. Similarly, each aluminum atom is classified as a four-fold aluminum (Al IV ) or a five-gold aluminum (Al V ) based on the number of oxygen atoms each aluminum is connected to. A MATLAB script is used to perform the above classification, which, for each oxygen atom, calculates the distance between the oxygen atom and each silicon or aluminum atom, and adds up the number of silicon or aluminum atoms that are less than a specified cutoff distance from the oxygen. The O-Si/Al cutoff is determined by examining the O-Si and O-Al partial radial distribution functions (PDF) in OVITO [36], and choosing the first minimum in the PDF, to a precision of 0.05 Å, as the cutoff [37,38]. It is found that the cutoffs vary between 2.0 Å (for x = 0%) and 2.2 Å (for x = 70%) depending on the composition of the glass since O-Si/Al bond lengths increase as the percentage of alumina increases. Having classified each O and Al atom into their respective sub-species, the bond angle distribution for each sub-species of O and Al is obtained by calculating the angles between each pair of bonds within the cutoff of each central O or Al atom. C. Mean square displacement computation The mean square displacement of a glass after an energy bump can be used as a measure of flexibility. In this investigation, the as-cooled glasses were first cooled to from 1 K to 10 -7 K in 20 ps in the NVT ensemble. An energy minimization is performed using the Polak-Ribiere version of the conjugate gradient (CG) algorithm [39] with energy and force tolerances of 10 -8 and 10 -8 kcal/mol·Å, respectively. Then, the system is subjected to a 200 meV energy bumpapplied in the form of an increment in the kinetic energy of the atoms, following a Gaussian velocity distribution. Based on the equipartition theorem, half of this energy bump eventually results in an increase in the potential energy of the atoms, while the other half yields an increase in temperature. This energy bump corresponds to a final system temperature of 750 K, which is below the glass transition temperature of the simulated glasses [17,19] to ensure that the glasses remain in the solid phase. The glasses are then allowed to relax in the NVE ensemble for 2 ns, and the mean square displacement of all atoms is computed at every 1 ps. 6 trials are obtained for each composition by using different random velocities for the energy bump. D. Internal stress computation Stress is normally defined for macroscopic objects or ensembles of atoms as the energy per unit volume, but ill-defined for individual atoms due to the lack of a clear physical interpretation. Nevertheless, to quantify the magnitude of the local forces acting on individual atoms, the "stress per atom" framework formulated by Thompson et al. [40] is adopted in our investigation. The stress per atom (σi) is defined in the following equation [41]: 3 # # = # # ( + + , ,⃗ • + ,,⃗ (Eq. 1) where i denotes the atom of interest, and Vi, mi, vi, and + , ,⃗ denote the Voronoi volume, mass, velocity, and position of the atom, and + ,,⃗ is the sum of all forces applied on the atom by all the other atoms in the system. Several previous studies have utilized this approach to quantify local stresses in an atomic network [42][43][44][45][46]. The computed stresses are either positive (in tension) or negative (in compression), and the sum of all stresses per atom multiplied by their respective Voronoi volumes equal zero since there is no external pressure applied to the glass macroscopically. The stress per atom is calculated in LAMMPS after bringing the glass to 10 -7 K and performing the energy minimization procedure as described in Section II.C. The internal stress per silicon atom ( 01,13456378 ) is defined in Eq. 2 as the difference between the stress per silicon atom in the as-cooled glass ( 01,987:: ) and the stress per silicon atom in an isolated Q n cluster (an SiO4 tetrahedron unit connected to n bridging oxygens), henceforth termed "reference stress" ( 01,65;5653<5 ): 01,13456378 = 01,987:: − 01,65;5653<5 (Eq. 2) More details regarding this calculation can be found in Ref. [41]. III. RESULTS AND DISCUSSION A. Radial constraints In order to construct a realistic topological constraint model, the number of radial constraints per atom is modeled by examining the connectivity of the glass. In this section, the oxygen atoms are classified into 3 sub-species, each of which creates different numbers of constraints: non-bridging oxygen (NBO), bridging oxygen (BO) and tricluster oxygen (TO) [47]. The aluminum atoms are classified into 2 sub-species: four-fold aluminum (Al IV ) and five-fold aluminum (Al V ) [48]. For a pure sodium silicate (x = 0%), each Na acts as a network modifier by breaking one Si-BO covalent bond and creating an Na-NBO ionic bond. Hence, the theoretical number of NBO is equal to the number of Na. As the percentage of alumina increases from 0% to 30%, each four-fold coordinated Al (Al IV ) removes one Na-NBO bond by taking an Na into its local vicinity in order to balance the excess unit negative charge on the AlO4 tetrahedron. At x = 30%, the number of Na exactly equal the number of Al, and the structure of the glass is comprised of SiO4 and AlO4 tetrahedrons, with all oxygen atoms acting as BO as seen in FIG. 1. At x > 30%, as the ratio of Al/Na > 1, the newly added Al must rely on a different mechanism in order to maintain local charge neutrality. This is achieved by creating one TO (tricluster oxygen) for each non-sodium charge compensated Al, where each TO is connected to 3 network formers (Al or Si) [19]. The above model has been used in a number of studies [17,19,49], and the existence of TO has been verified in simulations and experiments [17,18,50]. Furthermore, simulated glasses in this study suggest that this model is fairly consistent with simulations; the model predicts the formation of NBO accurately for prealuminous glasses, but slightly under-predicts the fraction of TO for per-aluminous glasses. Since the fraction of Al V is hard to predict, this paper assumes a theoretical model where all Al are four-fold coordinated. In our simulations, the fraction of Al V is observed to increase as the percentage of alumina increases, up to a maximum of 10% at 70% alumina content as shown in FIG. 2. The existence of Al V has been observed in previous simulations and experiments [18,19,51], and are a consequence of increased network connectivity in aluminarich glasses. B. Angular constraints In addition to identifying the radial connectivity of each atomic sub-species, it is also important to assign the correct number of angular constraints to each sub-species. This can be achieved by examining the simulated bond angle distributions (BAD) of each sub-species. In general, distributions featuring sharp peaks suggest low angular bond mobility and wide distributions suggest high angular flexibility [52][53][54][55]. Upon examination of FIG. 3(a), it is observed that the NBO BAD has a wide distribution between 60° and 180°, which suggests that the NBO creates no angular constraints. On the contrary, the BO BAD has a concentrated peak at 150°. The TO BAD has two concentrated peaks at 90° and 120°, which corresponds to edge-sharing and trigonal planar geometries respectively. These bond angles agree with previous simulations and experiments [18,[56][57][58][59][60]. The narrow BADs of BO and TO suggest that BO and TO have low angular bond mobility. For the BADs of different Al sub-species as shown in FIG. 3(b), Al IV shows a narrow peak at 109°, which corresponds to the expected tetrahedron geometry. However, Al V shows a wide distribution between 75° and 180° with peaks at 90° and 180°, which corresponds to a distorted octahedral geometry. These bond angles agree with previous simulations [18,19,51,56]. Due to the sharpness of their respective BADs, the Al IV is theorized to create angular constraints while the Al V is thought to create no angular constraints. Typical radial bond stretching (BS) constraints, BS = n /2 [49], where n denotes the connectivity of the central atom, are assumed for all atomic types except for NaAl since a charge compensating NaAl does not create a directional "bond" [61]. Moreover, n for NaNBO is set to equal 1 instead of its coordination number (i.e., 6) [59]. Typical angular bond bending (BB) constraints, BB = 2n -3, are assumed for all atomic types except for NBO, Al V , NaNBO, and NaAl. As explained in Section III.B, no bond bending constraints are assumed for NBO and NaNBO because a NaNBO ionically bonded to an NBO is not thought to impose any angular constraints. Again, a NaAl in the vicinity of an aluminum atom does not create a directional "bond" and therefore does not impose any angular constraints. As shown in FIG. 3, NBO and Al V have wide BADs, which suggests that no angular constraints are imposed by NBO or Al V . The theoretical atomic fraction, BS, BB, and the total number of constraints per atom for each atomic sub-species is tabulated in TABLE 2. The number of constraints per atom (nc) as a function of percentage Al2O3 is shown in FIG. 4. Simulation data points are obtained by multiplying the simulated fractions of each atomic sub-species with their respective number of constraints per atom. The close agreement between the number of topological constraints predicted by theoretical atomic fractions and the number of topological constraints calculated by simulated atomic fractions suggest that the simulated fractions of each type of oxygen (NBO, BO and TO) and aluminum atoms (Al IV and Al V ) closely reflect the theoretical fractions. The nc is observed to increase as the percentage of alumina increases, and nc increases more steeply after 30% alumina due to the formation of TO. At 25% alumina, the nc is exactly balanced by the number of degrees of freedom, i.e. the glass is isostatic. The proposed topological constraint model can be indirectly verified by experiments since isostatic glasses were found to have the best glass forming abilities [8]. Indeed, alumina rich glasses are extremely difficult to form in the laboratory [62], which is why molecular dynamics simulations are relied upon to study per-aluminous aluminosilicates. The mean square displacement (MSD) of all atoms, a signature of network flexibility, after an energy bump of 194 meV and a duration of 2 ns is shown in FIG. 5. The effect of the energy bump is to provide the atoms with enough kinetic energy to move and displace permanently, but not to the extent of melting. This method has been used in previous studies [43,63]. Mean square displacements of up to 23 Å 2 are observed, which imply that atoms are permanently displaced. This occurs when atoms are provided with enough energy to jump over energy barriers, moving from one minimum in the enthalpy landscape to other minimums. MSD is observed to decrease slightly from 0% to 25% alumina, but a sudden drop in MSD is observed at 30% alumina. High MSD values prior to 30% alumina suggest that those glass compositions are flexible and have internal degrees of freedom. The sharp drop at 30% signifies a rigidity transition at the percolation threshold [64], at which long-range connectivity in the glassy network is achieved. Hence, this paper defines 30% alumina as the lower bound of rigidity transition. MSD is observed to decrease linearly from 30% to 70% alumina as the number of constraints continues to increase due to the addition of alumina. For a pure sodium silicate glass (x = 0%), although it is in the flexible regime, there exists some internal stress due to residual thermal stress from quenching [41,65]. In FIG. 6, the internal stress per silicon atom is observed to increase as the percentage of alumina increases, which corresponds to an increase in nc. Moreover, an increase in slope is observed at 40% alumina, which indicates the presence of additional internal stress due to the network becoming over-constrained. Hence, this paper defines 40% as the point of transition into the stressedrigid domain. IV. CONCLUSIONS Altogether, this study provides a topological model for sodium aluminosilicate glasses, which can be used to predict the composition wherein the glass is expected to exhibit a flexibleto-stressed-rigid transition. We find that the enumeration of the topological constraints is independently supported by tracking the onset of flexibility and internal stress within the atomic network. As a major outcome of this study, we find that, as the atomic connectivity increases, the glass first exhibits a flexible-to-rigid transition (i.e., a loss of atomic mobilities) and then an unstressed-to-stressed transition (i.e., an onset of internal stress). The fact that these two transitions do not occur at the same threshold composition effectively defines a range of compositions wherein the glass is isostatic, that is, rigid but free of internal stress [66][67][68][69]. Internal stress per Si atom (GPa) REFERENCES FIG. 1 : 1Theoretical and simulated fractions of non-bridging oxygen (NBO), bridging oxygen (BO), and tricluster oxygen (TO) as a function of composition. The lines indicate the theoretical fractions as presented in TABLE 2, and the symbols indicate simulated fractions. The grey area indicates the region where the glass is isostatic. FIG. 2 : 2Simulated fractions of four-fold aluminum (Al IV ) and five-fold aluminum (Al V ) as a function of composition. The lines serve as guides for the eyes. The grey area indicates the region where the glass is isostatic. FIG. 3 : 3(a) Simulated bond angle distribution for non-bridging oxygen (NBO) and bridging oxygen (BO) in (Na2O)30(SiO2)70(Al2O3)00, and for tricluster oxygen (TO) in (Na2O)30(SiO2)00(Al2O3)70. (b) Simulated bond angle distribution for four-fold aluminum (Al IV ) and five-fold aluminum (Al V ) in (Na2O)30(SiO2)00(Al2O3)70. FIG. 4 : 4Topological constraint model for (Na2O)30(SiO2)70-x(Al2O3)x. The solid line is constructed based on the number of constraints calculated using theoretical atomic fractions, and the squares are constructed based on the number of constraints (nc) calculated using simulated atomic fractions. The dotted line indicates where the nc is exactly balanced by the number of degrees of freedom. The grey area indicates the region where glass is isostatic. FIG. 5 : 5Mean square displacement of all atoms after an energy bump of 194 meV and a duration of 2 ns, as a function of percentage Al2O3. The line serves as a guide for the eye. The gray area indicates the region of rigidity transition. FIG. 6 : 6Internal stress per silicon atom as a function of percentage Al2O3. The lines serve as a guide for the eye. The gray area indicates the region of rigidity transition. TABLE 1 : 1Assumptions used to calculate the theoretical atomic fractions for the construction of the proposed topological constraint model, where x is the percentage of Al2O3, which varies from 0% to 70%. Percentage of atoms are expressed per unit of (Na2O)30(SiO2)70-x(Al2O3)x.x ≤ 30%: x > 30%: 1) No Al V are formed. 2) No TO are formed. 3) Each Al creates one charge compensating Na (NaAl). 4) Each remaining Na (NaNBO) creates one ionic bond to an NBO. 1) No Al V are formed. 2) No NBO are formed. 3) Every Na is used to charge compensate an Al IV , i.e. every Na is an NaAl. 4) Each Al IV , except those that are charge compensated by an NaAl, create one TO. TABLE 2 : 2Theoreticalatomic fractions, bond stretching constraints, and bond bending constraints behind the proposed topological constraint model, where x is the percentage of Al2O3, which varies from 0% to 70%. Percentage of atoms are expressed per unit of (Na2O)30(SiO2)70-x(Al2O3)x. Atomic Type Percentage (x ≤ 30%) Percentage (x > 30%) Bond Stretching Constraints Bond Bending Constraints Total # of Constraints NBO 60 -2x 0 2 /2 0 1 BO 110 + 3x 230 -x 2 /2 1 2 TO 0 2x -60 3 /2 3 9 /2 Si 70 -x 70 -x 4 /2 5 7 Al IV 2x 2x 4 /2 5 7 Al V 0 0 5 /2 0 5 /2 NaNBO 60 -2x 0 1 /2 0 1 /2 NaAl 2x 60 0 0 0 [1] D.K. Hale, Strengthening of Silicate Glasses by Ion Exchange, Nature. 217 (1968) 1115-1118. doi:10.1038/2171115a0. [2] A.R. Cooper, D.A. Krohn, Strengthening of Glass Fibers: 11, Ion Exchange, Journal of the American Ceramic Society. 52 (1969) 665-669. doi:10.1111/j.1151-2916.1969.tb16073.x. Chemical Strengthening of Glass: Lessons Learned and Yet To Be Learned. A K Varshneya, 10.1111/j.2041-1294.2010.00010.xInternational Journal of Applied Glass Science. 1A.K. Varshneya, Chemical Strengthening of Glass: Lessons Learned and Yet To Be Learned, International Journal of Applied Glass Science. 1 (2010) 131-142. doi:10.1111/j.2041-1294.2010.00010.x. The Physics of Chemical Strengthening of Glass: Room for a New View. A K Varshneya, 10.1016/j.jnoncrysol.2010.05.010Journal of Non-Crystalline Solids. 356A.K. Varshneya, The Physics of Chemical Strengthening of Glass: Room for a New View, Journal of Non-Crystalline Solids. 356 (2010) 2289-2294. doi:10.1016/j.jnoncrysol.2010.05.010. Dynamics of Glass Relaxation at Room Temperature. R C Welch, J R Smith, M Potuzak, X Guo, B F Bowden, T J Kiczenski, D C Allan, E A King, A J Ellison, J C Mauro, 10.1103/PhysRevLett.110.265901Physical Review Letters. 110R.C. Welch, J.R. Smith, M. Potuzak, X. Guo, B.F. Bowden, T.J. Kiczenski, D.C. Allan, E.A. King, A.J. Ellison, J.C. Mauro, Dynamics of Glass Relaxation at Room Temperature, Physical Review Letters. 110 (2013). doi:10.1103/PhysRevLett.110.265901. Glass Substrates for Liquid Crystal Displays. A Ellison, I A Cornejo, 10.1111/j.2041-1294.2010.00009.xInternational Journal of Applied Glass Science. 1A. Ellison, I.A. 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{'abstract': 'In the framework of topological constraint theory, network glasses are classified as flexible, stressed-rigid, or isostatic if the number of atomic constraints is smaller, larger, or equal to the number of atomic degrees of freedom. Here, based on molecular dynamics simulations, we show that sodium aluminosilicate glasses exhibit a flexible-to-stressed-rigid transition driven by their composition. This transition manifests itself by a loss of atomic mobility and an onset of internal atomic stress. Importantly, we find that the flexible-to-rigid (i.e., loss of internal flexibility) and unstressed-to-stressed transitions (i.e., onset of internal stress) do not occur at the same composition. This suggests that the isostatic state (i.e., rigid but unstressed) is achieved within a window rather than at a threshold composition.', 'arxivid': '1812.04791', 'author': ['Ernest Ching \nPhysics of AmoRphous and Inorganic Solids Laboratory (PARISlab)\nUniversity of California\n90095-1593Los AngelesCAU.S.A\n', 'Mathieu Bauchy [email protected] \nPhysics of AmoRphous and Inorganic Solids Laboratory (PARISlab)\nUniversity of California\n90095-1593Los AngelesCAU.S.A\n'], 'authoraffiliation': ['Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab)\nUniversity of California\n90095-1593Los AngelesCAU.S.A', 'Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab)\nUniversity of California\n90095-1593Los AngelesCAU.S.A'], 'corpusid': 119007847, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 17452, 'n_tokens_neox': 13811, 'n_words': 6831, 'pdfsha': '4d7c5a6129ca80168ef5fdd682c7138ca79019b8', 'pdfurls': ['https://arxiv.org/pdf/1812.04791v1.pdf'], 'title': ['Topological Origins of Flexibility and Internal Stress in Sodium Aluminosilicate Glasses', 'Topological Origins of Flexibility and Internal Stress in Sodium Aluminosilicate Glasses'], 'venue': []}

arxiv

From Zero to Production: Baltic-Ukrainian Machine Translation Systems to Aid Refugees 2022 Toms Bergmanis Vienības gatve 75ALV-1004Tilde, RigaLatvia Mārcis Pinnis Vienības gatve 75ALV-1004Tilde, RigaLatvia From Zero to Production: Baltic-Ukrainian Machine Translation Systems to Aid Refugees Baltic J. Modern Computing 002022low resource machine translationUkrainianLatvianLithuanianEstonian In this paper, we examine the development and usage of six low-resource machine translation systems translating between the Ukrainian language and each of the official languages of the Baltic states. We developed these systems in reaction to the escalating Ukrainian refugee crisis caused by the Russian military aggression in Ukraine in the hope that they might be helpful for refugees and public administrations. Now, two months after MT systems were made public, we analyze their usage patterns and statistics. Our findings show that the Latvian-Ukrainian and Lithuanian-Ukrainian systems are integrated into the public services of Baltic states, leading to more than 127 million translated sentences for the Lithuanian-Ukrainian system. Motivated by these findings, we further enhance our MT systems by better Ukrainian toponym translation and publish an improved version of the Lithuanian-Ukrainian system. Introduction On February 20, 2014, Russian Federation started military aggression against Ukraine (Cosgrove, 2020). Eight years later, on February 24, 2022, following a Russian military build-up on the Russia-Ukraine border, Russian aggression culminated in a fullscale invasion of Ukraine. 1 As of May 2022, more than 6.1 million refugees have fled Ukraine. 2 The majority of refugees have left Ukraine for one of the seven neighboring countries. Still, many seek shelter in other countries, including the Baltic states. 3 The influx of Ukrainian refugees poses a new challenge for communication between individuals and governmental bodies in the Baltic states. In this paper, we examine six low-resource machine translation (MT) systems translating between the Ukrainian language and each of the official languages of the Baltic states. Their development took place in the wake of the escalating Ukrainian refugee crisis shortly after the Russian invasion of Ukraine. Thus it was motivated by apprehension for the future rather than a clear vision of how they might be used. Now, after MT systems have been online for more than two months, we analyze their usage statistics and draw conclusions for what are the aspects of MT integration in the public services, which have led to more than 127 million translated sentences for the Lithuanian-Ukrainian system, while the Latvian-Ukrainian system has been used seemingly relatively little having translated only 138 thousand sentences. Machine Translation Systems Due to data scarcity for the language pairs involving Ukrainian and the languages of the Baltic states, we use two data augmentation methods -one that enables dynamic terminology integration and another that allows training MT models that are more robust to unknown tokens and rare words. For terminology integration, we prepare data with Target Lemma Annotations (TLA) (Bergmanis and Pinnis, 2021b), while for the robustness, we use synthetic data augmentation as proposed by Pinnis et al. (2017). For system training, we use the Marian neural machine translation (NMT) toolkit by (Junczys-Dowmunt et al., 2018). We train standard NMT systems that largely follow the Transformer (Vaswani et al., 2017) base model configuration. The only departures from the standard configuration are the changes necessary for TLA support during training and inference. For the Marian toolkit, they were described in Bergmanis and Pinnis (2021a). Specifically, we employ the source-side factors using factor embeddings of dimensionality of 8 and concatenate them with subword embeddings. We also increase the delay of updates for the optimiser 4 (from 16 to 24 batches) and set the maximum sequence length to 196 tokens. The increased sequence length accounts for longer input sequences caused by the additional TLA tokens. On the other hand, the increased optimizer delay negates the effect of the smaller effective batch size due to fewer sentences fitting in the workspace memory-based batch because of their increased maximum length. Furthermore, all models are trained using the guided alignment functionality of the Marian toolkit. To train MT systems, we mostly use publicly available parallel data from the Tatoeba Challenge (Tiedemann, 2020) corpus. This constitutes 69%, 70%, and 74% of all parallel data for Latvian-Ukrainian, Estonian-Ukrainian, and Lithuanian-Ukrainian respectively. The remaining data were acquired from proprietary data sources. Data statistics are depicted in Table 1. We filtered all parallel data using parallel data filtering methods by Pinnis (2018) and then performed pre-processing, which included the following steps: normalisation of whitespaces and punctuation (e.g., quotation marks, apostrophes, hyphens, etc.), identification of non-translatable entities (e.g., e-mails, file paths, complex identifiers are replaced with placeholders), tokenisation, truecasing, synthetic unknown data generation (Pinnis et al., 2017), byte-pair encoding (Sennrich et al., 2016b), and finally TLA. For validation of our MT systems during training and for evaluation, we use the dev and devtest splits of FLORES-101-an evaluation benchmark specially created for lowresource language pairs (Goyal et al., 2022). We use the standard splits, which consist of 997 and 1012 sentences large validations and evaluation sets respectively that are parallel across all four languages. Automatic Evaluation We compare our systems with Google Translate 5 and eTranslation 6 . We compare against Google Translate because, for many people, it is the go-to MT service provider when the amount of text to be translated is small. However, Google Translate is not free of charge when translation volumes exceed a certain limit. Thus we also compare against eTranslation -the MT service provider of the European Commission. eTranslation is free of charge for European small and medium-sized enterprises, employees of public administrations across the European Union and public sector service providers. Table 2 shows results of automatic evaluation using SacreBLEU (Post, 2018) implementation of three MT evaluation metrics-ChrF 7 (Popović, 2015), BLEU 8 (Papineni et al., 2002), and TER 9 (Snover et al., 2006). The automatic evaluation using ChrF, which is the most suitable metric for morphologically complex languages (Kocmi et al., 2021) such as the languages considered in this work, shows that Google Translate performs the best. Our systems compare to eTranslation in the range from marginally better for Ukrainian-Lithuanian and Lithuanian-Ukrainian directions to substantially worse for Ukrainian-Latvian. While these results do not favor our MT systems, they serve as a sanity check. Even though our systems are a one-shot attempt at developing MT systems for a set of low-resource language pairs, they are, to an extent, comparable to other publicly available alternatives. Usage of MT Systems We published our MT systems on March 11, 2022, which means that they have been online for more than two months at the time of writing the paper. In this section, we aim to analyze how our systems are used and who are their users. Figure 1 shows the number of translated sentences by each system. Due to its large translation volume, usage statistics for the Lithuanian-Ukrainian MT system are plotted separately in Figure 2. The graphs show that Estonian systems were used the least, having translated only about five thousand sentences from Ukrainian to Estonian and almost twice as much from Estonian to Ukrainian. Slightly higher usage numbers are evident for Latvian systems, which have processed more than 138 thousand and 132 thousand Latvian-Ukrainian and Ukrainian-Latvian translation sentences, respectively. Although the Ukrainian-Lithuanian system has translated only about 16 thousand sentences, the Lithuanian-Ukrainian system has had the highest demand as it has translated more than 127 million sentences. Analyzing through what channels our systems are accessed reveals that the Latvian systems are only one-quarter of the time used by our paid clients. However, they are most often used via the Latvian language technology platform hugo.lv, which is popular among freelance translators and governmental organizations. As for Lithuanian systems, the users translating from Ukrainian into Lithuanian have almost exclusively used our public translation platform translate.tilde.com, which allows for speculation that individual users made these translation requests, most likely translating text snippets from news and social media. The system for the opposite translation direction is translating from Lithuanian into Ukrainian and has almost entirely been used via Tilde Web Translation Widget. To understand the 127 million sentences large volume of translated sentences, we inspect the distribution of translated sentences by their source website (see Figure 3). All websites using this MT system are related to the Lithuanian government. The top websites are uzt.lt and ldb.lt, which are services of the employment agency of Lithuania, paslaugos.vilnius.lt, which is the Vilnius City Council services' page and socmin.lrv.lt, which is the Ministry of Social Security and Labor of the Republic of Lithuania. This analysis reveals that, at least as far as the usage of the Lithuanian-Ukrainian MT system is concerned, even if just a little, our work has helped the people in need to access help and social services. It is also important to note that the difference of the usage levels for Latvian and Lithuanian systems can be explained with how the systems are used in Latvia and Lithuania. In Lithuania, the Lithuanian-Ukrainian system is (mostly) used to translate governmental websites. Whenever a user (a citizen, a refugee, or a tourist) accesses a certain page in a website, its content is translated by the MT system. This generates high numbers of translation requests. However, this method allows to provide instant multilinguality in a website regardless of which page a user wants to see. In Latvia, the systems are mostly used by translators and public service officials in post-editing scenarios. This means that different from Lithuania where we can grasp a rough estimate of how many end-users consume the translations, in Latvia we only know how many unique sentences were translated to create content in a different language. We cannot estimate how many end-users might have consumed that content. However, the volume is still substantial for post-editing scenarios. Step Forward Lithuanian-Ukrainian MT System with Back-translated Data In Section 3, we established that the Lithuanian-Ukrainian MT system is used the most as it has translated 127 million sentences helping Ukrainians in Lithuania to find jobs Table 3. Results of automatic evaluation using the SacreBLEU implementation of ChrF, BLEU and TER metrics for Lithuanian-Ukrainian MT systems. This Work's BT System refers to the system developed using back-tanslated data, while This Work's Baseline refers to MT systems described in Section 4.1.2. * denotes that the result difference between this and the system trained on back-translated data is statistically significant according to SacreBLEU's paired bootstrap resampling test. and access social services. Besides, unlike the Latvian-Ukrainian system, which is primarily used in post-editing scenarios, the translations of the Lithuanian-Ukrainian system reach its users without the supervision of professional translators. Therefore, we aim to deliver better technology where the people use it the most and retrain our Lithuanian-Ukrainian MT system. Machine Translation Systems We use the Ukrainian-Lithuanian MT system to create synthetic parallel data by back-translation (Sennrich et al., 2016a) of monolingual Ukrainian data. For data sources, we use the 2008 to 2021 News Crawl 10 corpus provided by the Machine Translation Group at the University of Edinburgh and the RSS News, Newscrawl, and Wikipedia corpora 11 collected by the University of Leipzig (Goldhahn et al., 2012). We also use the Ukrainian side of the Ukrainian-English Wikimedia (Tiedemann, 2012), TED 2020 (Reimers and Gurevych, 2020), and OpenSubtitles v2018 (Lison and Tiedemann, 2016) corpora from Opus 12 (Tiedemann, 2012). Altogether these corpora amount to around 11.4 million sentences. As before, we continue by using the synthetic data augmentation (Pinnis et al., 2017), which nearly doubles the number of sentences to about 21 million. We then translate this data into Lithuanian and use parallel data filters by Pinnis (2018) to get rid of noisy and poor quality translations, which leaves us with around 19 million backtranslated sentences. Finally, we add this data to the data we used to train the baseline system (see Section 4.1.2) to obtain a total of about 37.8 million sentences. We then use the same configuration as described in Section 4.1.2 with an exception that we increase optimizer delay from 24 to 64. Table 3 shows a comparison of the baseline MT systems from Section and the newly created Lithuanian-Ukrainian MT system. The new system achieves the second best results, conceding only to Google Translate, which is still 0.8 ChrF points better. However, the results also show that using back-translated data helps to yield statistically significant improvements in translation quality over the other two baselines. Automatic Evaluation Ukrainian Toponym Translation Historically Ukrainian toponyms in the languages of the Baltic states have been introduced via Russian. Thus traditionally, Latvian and Lithuanian representations of Ukrainian toponyms have leaned on the conventions of Russian pronunciation. Traditions, however, are subject to cultural changes, as exemplified by the decommunization of Ukrainian toponymy after the collapse of the USSR and the proclamation of independent Ukraine (Demska and Levchuk, 2020). Likewise, shifts in geopolitical allegiances can also be a decisive factor in changing language customs. Here, the example is the departure from the Russian-based representations of Ukrainian toponyms in Latvian to favour Ukrainian-based pronunciation. Since 2014 when the Russian Federation started military aggression against Ukraine, the expert committee of the Latvian State Language Centre has twice pushed for changes in the Latvian language representations of Ukrainian city names -first in 2017 13 and then in 2019. 14 The final decision to offi- Table 5. Examples of Ukrainian-Latvian and Ukrainian-Lithuanian toponym translation with and without terminology integration (incorrect translations of toponyms are underlined). Ukrainian-Latvian Source: Київ, Харкiв, Одеса, Днiпро i Донецьк мають пять найбiльших українських мiст. Without terms: Kijeva, Harkova, Odesa, Dn , epra un Don , ecka ir piecas lielākās Ukrainas pilsētas. With terms: Kijiva, Harkiva, Odesa, Dnipro un Donecka ir piecas lielākās Ukrainas pilsētas. Ukrainian-Lithuanian Source: Ми доставляємо товари в Запорiжжя, Львiв, Чернiгiв та Тернопiль. Without terms: Mes pristatome prekes į Zaporožę, Lvovą,Černigą ir Ternopilį. With terms: Prekes pristatome į Zaporižę, Lvovą,Černyhivą ir Ternopilį. cially stop using Russian-based representations of Ukrainian city names was reached on March 9, 2022, 15 only two weeks after the Russian invasion of Ukraine. Although these swift decisions reflect the political climate and the sentiment of the people, these changes hardly have had time to reach the training data of data-driven natural language processing tools. So we take advantage of our MT system's dynamic terminology integration capability and approach the problem of Ukrainian toponym translation as a terminology integration task. Specifically, we prepare toponym glossaries (see Table 4) mapping both the new and obsolete terms to their new and preferred translations. Before translating, we compare the stemmed version of each word in the sentence against the stemmed Ukrainian toponyms in the glossary and annotate them with their preferred translation if we find one. Then, we pass the annotated sentence to a system that is trained with TLA and can use the annotations to translate and inflect the toponym according to the sentence context. For more details, refer to Bergmanis and Pinnis (2021b) and Bergmanis and Pinnis (2021a). Terminology integration using TLA applies soft constraints on an NMT model. Contrary to methods that apply hard constraints, e.g., constrained decoding (Post and Vilar, 2018), this enables the NMT model to have flexibility in how the annotations are used. The NMT system can freely decide on the most suitable inflected form for the given morphosyntactic context. However, this also means that in some cases, the NMT model can choose to ignore the annotations if there is a stronger internal signal for a different lexical choice. Table 5 shows two examples where Ukrainian toponyms are translated from Ukrainian into Latvian and Lithuanian. The example shows that terminology integration improves toponym translation quality for most cases except for one example,'Львiв', was translated using the obsolete variant. Nevertheless, we believe that soft constraints are more appropriate for morphologically rich languages. There is room for future work to reduce cases where the NMT model decides not to rely on the annotations. Conclusions and Discussion We examined the quality, usage patterns and translation volume of six low-resource MT systems translating between the Ukrainian language and each of the official languages of the Baltic states. Although the translation quality analysis revealed that our systems are no better than the other publicly available alternatives, the MT usage statistics showed that the general public nevertheless uses some of our MT systems. Meanwhile, other MT systems are integrated into Lithuania's governmental websites or used by government translators in Latvia. We found that the different approaches to MT integration in public services have led to vastly different volumes of translation requests. In Lithuania, whenever a user accesses a certain page on a website, the MT system translates its content, generating many translation requests. This method provides flexible and instant multilingualism on a website regardless of which page a user wants to access. In Latvia, the systems are used mainly by translators and public service officials in MT post-editing scenarios. While this approach generates fewer translation requests, it also limits what content users can access in their native language. Knowing which systems are used most actively, we revisit them to improve their quality. Because of the different ways systems are integrated, in contrast to the Latvian-Ukrainian system, the translations of the Lithuanian-Ukrainian system reach users without being checked by professional translators. Motivated by this finding, we retrained the Lithuanian-Ukrainian MT system using nearly twenty million sentences of backtranslated data, which allowed us to, as measured by automatic metrics, outperform eTranslation and close the gap with Google Translate. Finally, we cast the Ukrainian toponym translation as a terminology integration task and show how to dynamically solve the changing and divergent spelling of place names when systems are deployed. There are options for future work to improve the MT between the official languages of Baltic states and Ukrainian beyond the quality achieved within this work. One, evident from Table 1, is to obtain more high-quality data. Indeed, the amount of training data after filtering ranges from two to nearly five million parallel sentences, which is not much compared to other European language pairs. Another potential avenue for future work is to train multilingual MT models (Dabre et al., 2020) translating from many source languages to one target language. In such a setup, including one resourcerich language pair, such as English-Ukrainian, could help via means of transfer-learning (Kocmi, 2019), or at the very least, as a form of regularization (Neubig and Hu, 2018). While this work provides a novel analysis of the MT usage in Baltic states to address language barriers rising from a refugee crisis, there are other similar efforts to use language technology to aid people displaced by the Russian war in Ukraine. One such effort is ÚFAL for Ukraine by Charles University in Prague, which offers an MT system for Czech-Ukrainian, Charles Translator for Ukraine. Their MT system builds on the previous work on Czech-English MT (Popel et al., 2020) and can be accessed on the web, 16 via an android app, 17 as well as in the form of chatbots for Telegram and Messenger and other messaging services. 18 Fig. 1 . 1The number of sentences translated daily for MT systems translating between Ukrainian and languages of Baltic states, except for the Lithuanian-Ukrainian system for which data is plotted separately (seeFigure 2). Statistics are fromMarch 11, 2022, to May 19, 2022 Fig. 2 . 2The number of sentences translated daily for Lithuanian-Ukrainian MT system. Statistics are fromMarch 11, 2022, to May 19, 2022 Fig. 3 . 3The number of sentences translated by the top users of the Lithuanian-Ukrainian MT system. Statistics are fromMarch 11, 2022, to May 19, 2022 Table 1 . 1Parallel data statistics for each language pair before and after filtering as well as after synthetic data generation.Before filtering After filtering After unknown data generation After TLA UK-LV 4.0M 2.3M 4.5M 8.9M UK-ET 5.1M 4.0M 7.8M 15.5M UK-LT 5.9M 4.7M 9.3M 18.6M Table 2 . 2Results of automatic evaluation using SacreBLEU implementation of ChrF, BLEU and TER metrics.ChrF ↑ BLEU ↑ TER ↓ ChrF ↑ BLEU ↑ TER ↓UK-LV eTranslation 53.7 23.0 65.4 LV-UK eTranslation 51.4 21.2 69.8 Google 57.0 26.9 60.7 Google 53.0 22.6 66.0 This Work 52.2 21.2 67.3 This Work 47.6 17.5 73.0 UK-LT eTranslation 54.0 21.4 69.0 LT-UK eTranslation 49.1 19.1 70.7 Google 56.6 24.4 64.5 Google 50.8 21.0 67.9 This Work 54.5 21.7 67.7 This Work 49.4 18.8 71.0 UK-ET eTranslation 53.6 19.4 69.2 ET-UK eTranslation 50.6 20.8 68.6 Google 56.2 22.0 65.7 Google 52.4 23.0 65.4 This Work 53.5 19.4 69.2 This Work 49.5 19.5 70.5 ChrF ↑ BLEU ↑ TER ↓eTranslation 49.1* 19.1 70.7 Google Translate 50.8* 21.0 67.9 This Work's Baseline 49.4* 18.8 71.0 This Work's BT System 50.0 19.3 70.7 Table 4 . 4Examples of Ukrainian toponyms and their translations previously represented via their pronunciation in Russian.UK EN LV LT New Obsolete New Obsolete New Obsolete Київ Kyiv Kiev Kijiva Kijeva Kyjivas Kyjevas Харкiв Kharkiv Kharkov Harkiva Harkova Charkivas Charkovas Одеса Odessa Odesa Odesa - Odesa - Днiпро Dnipro Dnipropetrovsk Dnipro Dn , epro Dnipras Dniepras Донецьк Donetsk - Donecka Don , ecka Doneckas - Запорiжжя Zaporizhzhia - Zaporižja Zaporožje Zaporižė Zaparožė Львiв Lviv Lvov L , viva L , vova Lvivas Lvovas Кривий Рiг Kryvyi Rih - Krivijriha Krivojroga Kryvyj Rihas - Миколаїв Mykolaiv Nikolaev Mikolajiva Mikolajeva Mykolajevas - Луганськ Luhansk Lugansk Luhanska Luganska Luhanskas - Макiївка Makiivka Makiyivka Makijivka - Makijivka - Вiнниця Vinnytsia - Vinnica Vin , n , ica Vinica - Чернiгiв Chernihiv ChernigovČernihivaČern , igovaČernyhivasČernyhovas Чернiвцi Chernivtsi MakiyivkaČernivciČernovciČernivciai - Горлiвка Horlivka - Horlivka Gorlovka Horlivka - Кам'янське Kamianske - Kamjanska Kamjanske Kamianskė - Тернопiль Ternopil Tarnopol Ternopil , a Ternopole Ternopilis Ternopolis Accessed May 16, 2022 https://news.un.org/en/focus/ukraine 2 Accessed May 16, 2022 https://data2.unhcr.org/en/situations/ukraine 3 Accessed May 5, 2022 https://en.wikipedia.org/wiki/2022_Ukrainian_refugee_ crisis#Other_European_countries arXiv:2209.14142v1 [cs.CL] 28 Sep 2022 The delay of updates for the optimiser can be enabled in Marian using the -optimizer-delay parameter. Accessed May 16, 2022 https://translate.google.com 6 Accessed May 16, 2022 https://ec.europa.eu/info/resources-partners/ machine-translation-public-administrations-etranslation_en SacreBLEU hash: nrefs:1|case:mixed|eff:yes |nc:6|nw:0|space:no|version:2.0.0 8 SacreBLEU hash: nrefs:1|case:mixed|eff:no |tok:13a |smooth:exp |version:2.0.0 9 SacreBLEU hash: nrefs:1|case:lc|tok:tercom|norm:no |punct:yes |asian:no |version:2.0.0 https://data.statmt.org/news-crawl/uk 11 https://wortschatz.uni-leipzig.de/en/download/Ukrainian 12 https://opus.nlpl.eu/ Accessed May 5, 2022 https://www.vestnesis.lv/op/2017/208.18 14 Accessed May 5, 2022 https://www.vestnesis.lv/op/2019/42.37 Accessed May 5, 2022 https://www.vestnesis.lv/op/2022/67.4 AcknowledgementsThe research has been supported by the European Regional Development Fund within the research project "AI Assistant for Multilingual Meeting Management" No. 1.1.1.1/ 19/A/082. Dynamic terminology integration for COVID-19 and other emerging domains. 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{'abstract': 'In this paper, we examine the development and usage of six low-resource machine translation systems translating between the Ukrainian language and each of the official languages of the Baltic states. We developed these systems in reaction to the escalating Ukrainian refugee crisis caused by the Russian military aggression in Ukraine in the hope that they might be helpful for refugees and public administrations. Now, two months after MT systems were made public, we analyze their usage patterns and statistics. Our findings show that the Latvian-Ukrainian and Lithuanian-Ukrainian systems are integrated into the public services of Baltic states, leading to more than 127 million translated sentences for the Lithuanian-Ukrainian system. Motivated by these findings, we further enhance our MT systems by better Ukrainian toponym translation and publish an improved version of the Lithuanian-Ukrainian system.', 'arxivid': '2209.14142', 'author': ['Toms Bergmanis \nVienības gatve 75ALV-1004Tilde, RigaLatvia\n', 'Mārcis Pinnis \nVienības gatve 75ALV-1004Tilde, RigaLatvia\n'], 'authoraffiliation': ['Vienības gatve 75ALV-1004Tilde, RigaLatvia', 'Vienības gatve 75ALV-1004Tilde, RigaLatvia'], 'corpusid': 252567862, 'doi': '10.22364/bjmc.2022.10.3.01', 'github_urls': ['https://github.com/martin-majlis/'], 'n_tokens_mistral': 11166, 'n_tokens_neox': 9606, 'n_words': 5367, 'pdfsha': '9f6d2d5bcb8eeff279fcfbcb3e1322bf6beab243', 'pdfurls': ['https://export.arxiv.org/pdf/2209.14142v1.pdf'], 'title': ['From Zero to Production: Baltic-Ukrainian Machine Translation Systems to Aid Refugees', 'From Zero to Production: Baltic-Ukrainian Machine Translation Systems to Aid Refugees'], 'venue': ['Baltic J. Modern Computing']}

arxiv

Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions ⋆ 24 Oct 2005 Myunghyun Oh Kevin Zumbrun Department of Mathematics University of Kansas 1460 Jayhawk Blvd66047LawrenceKSUSA Department of Mathematics Indiana University 47450BloomingtonINUSA Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions ⋆ 24 Oct 2005 We generalize work of Oh & Zumbrun and Serre on spectral stability of spatially periodic traveling waves of systems of viscous conservation laws from the one-dimensional to the multi-dimensional setting. Specifically, we extend to multi-dimensions the connection observed by Serre between the linearized dispersion relation near zero frequency of the linearized equations about the wave and the homogenized system obtained by slow modulation (WKB) approximation. This may be regarded as partial justification of the WKB expansion; an immediate consequence is that hyperbolicity of the multi-dimensional homogenized system is a necessary condition for stability of the wave. As pointed out by Oh & Zumbrun in one dimension, description of the low-frequency dispersion relation is also a first step in the determination of time-asymptotic behavior. Introduction Nonclassical viscous conservation laws arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena, including homoclinic (pulse-type) and periodic solutions along with the standard heteroclinic (shock, or front-type) solutions. Here, we investigate stability of periodic traveling waves: specifically, the spectrum of the linearized operator about the wave. Our main result generalizes the works [OZ.1], [Se.1] about stability of periodic traveling waves of systems of viscous conservation laws from the one-dimensional to the multi-dimensional setting. Consider a system of conservation laws u t + j f j (u) x j = j, k (B jk (u)u x k ) x j , (1.1) X 0 u(y)dy, F j := 1 X X 0 f j (u) − d k=1 B jk (u)ων k ∂ y u dy when u is a periodic solution of (1.4). Since these quantities are translation invariant, we consider the set P of periodic functions u that are solutions of (1.4) for some triple (s, ν, q), and construct the quotient set P := P/R under the relation (u R v) ⇐⇒ (∃h ∈ R; v = u(· − h)). We thus have class functions: X = X(u), ω = Ω(u), s = S(u), ν = N (u), q = Q(u), M = M (u), F j = F j (u), whereu is the equivalence class of translates of different periodic functions. Note thatū is a nonconstant periodic solution. Without loss of generality, assume S(ū) = 0 and N (ū) = e 1 , so that (1.4) takes the form B 11 (ū)ū ′ = f 1 (ū) −q forq = Q(ū). LettingX = X(ū) andā =ū(0) = u 0 , the map (y, a, s, ν, q) → u(y; a, s, ν, q) − a is smooth and well-defined in a neighborhood of (X;ā, 0, e 1 ,q), and it vanishes at this special point. Here and elsewhere, e j denotes the jth standard Euclidean basis element. We assume: (H0) f j , B jk ∈ C 2 . (H1) Re σ( jk ν j ν k B jk ) ≥ θ > 0. (H2) The map H : R × U × R × S d−1 × R n → R n taking (X; a, s, ν, q) → u(X; a, s, ν, q) − a is a submersion at point (X;ā, 0, e 1 ,q). As a consequence of (H0), (H2), there is a smooth n+d dimensional manifold P of periodic solutionṡ u in the vicinity ofū, where d is the spatial dimension. On this set, one may obtain, rescaling by (x, t) → (ǫx, ǫt) and carrying out a formal WKB expansion as ǫ → 0 a closed system of n + d averaged, or homogenized, equations ∂ t M (u) + j ∂ x j (F j (u)) = 0, ∂ t (ΩN (u)) + ∇ x (ΩS(u)) = 0 (1.5) in the (n + d)-dimensional unknownu, expected to correspond to large time-space behavior. For details, see Section 4. The problem of stability ofū may heuristically be expected to be related to the linearized equations of (1.5) about the constant solutionu(x, t) ≡ u 0 , u 0 ∼ū, provided that the WKB expansion is justifiable by stability considerations. This leads to the homogeneous degree n + d linearized dispersion relation ∆(ξ, λ) := det   λ ∂(M, ΩN ) ∂u (u) + j iξ j ∂(F j , SΩe j ) ∂u (u)   = 0. (1.6) On the other hand, one may also pursue the direct course of linearizing PDE (1.1) about the stationary solutionū and studying the spectrum of the associated linearized operator L. Taking the Fourier transform in constant directions x j , j = 1, and following the general construction of [G], [OZ.1], we obtain an Evans function D(ξ, λ), ξ ∈ R d , λ ∈ C, of which the zero set (ξ, λ(ξ)) determines the linearized dispersion relation for (1.1), with λ(ξ) running over the spectrum of L as ξ runs over R d . For details, see Section 2. In particular, the low-frequency expansion of λ(ξ) near (ξ, λ) = (0, 0) may be expected to determine long-time asymptotic behavior, provided that spectrum away from λ = 0 has strictly negative real part, and this in turn may be expected to derive from the lowest order terms of the D(ξ, λ) = ∆ 1 (ξ, λ) + O(|ξ, λ| n+2 ), (1.7) where ∆ 1 is a homogeneous degree n + 1 polynomial expressed as the determinant of a rather complicated 2n × 2n matrix in (ξ, λ): in particular, not in the simple form det λN 0 + j iξ j N j of a first-order hyperbolic (n + 1) × (n + 1) dispersion relation, or an obvious tensor product thereof. Our main result is the following theorem relating these two expansions, generalizing the result of [Se.1] in the one-dimensional case. Define ∆(ξ, λ) := λ 1−d∆ (ξ, λ), (1.8) where∆ is defined as in (1.6). Theorem 1. Under assumptions (H0)-(H2), ∆ 1 = Γ 0 ∆, i.e., D(ξ, λ) = Γ 0 ∆(ξ, λ) + O(|ξ, λ| n+2 ) (1.9) Γ 0 = 0 constant, for |ξ, λ| sufficiently small. That is, up to an additional factor of λ d−1 , the dispersion relation (1.6) for the averaged system (1.5) indeed describes the low-frequency limit of the exact linearized dispersion relation D(ξ, λ) = 0. The discrepancy λ d−1 is an interesting and at first glance puzzling new phenomenon in the multidimensional case. However, it is easily explained by a closer look at the formal approximation procedure described in Section 4. For, in the derivation of (1.5), it was assumed that ΩN represent the gradient ∇ x φ of a certain phase function φ(x, t). In one dimension, this is no restriction, since we may always take φ(x, t) := x 0 ω(z)dz. However, in multidimensions, it imposes the additional constraint curl (ΩN ) ≡ 0, (1.10) which properly should be adjoined to the averaged system. Taking the curl of the second equation of (1.5), we obtain the simple equation ∂ t curl (ΩN ) = 0, revealing at once that constraint (1.10) is compatible with the time-evolution of the system, and that the unconstrained system possesses (d − 1) spurious zero characteristices λ(ξ) ≡ 0, corresponding to the (d − 1) Fourier modes in the range of the curl operatorf → ξ curlf , lying in (ξ/|ξ|) ⊥ . Thus, ∆(ξ, λ) = λ 1−d∆ (ξ, λ) = 0 is exactly the linearized dispersion relation for the constrained averaged system (1.5), (1.10) relevant to time-asymptotic behavior. Theorem 1 may be regarded as partial justification of the WKB expansion. Roughly speaking, it states that if perturbed periodic waves exhibit coherent behavior near the unperturbed waveū, then this behavior is well-described by the constrained averaged equations (1.5), (1.10). In one dimension, additional results of [OZ.2] case give rigorous sense to this statement in the form of detailed pointwise linear bounds under the assumption of spectral stability of the linearized operator about the wave. To establish a comparable long time result on behavior in multidimensions would be a very interesting direction for future investigation. Moreover, the new description of the solution given in [Se.1] by modulation expansion might give a sufficiently good nonlinear Ansatz to carry out a complete nonlinear analysis, which was not done even in the one-dimensional case. Thus, it would be interesting to revisit also the one-dimensional setting of [OZ.2] from this new perspective, in particular, to resolve certain puzzling issues in the general, non-quasi-Hamiltonian case. An equally interesting direction for future investigation would be to rigorously validate the WKB expansion of Section 4 for the closely related small viscosity problem As an immediate consequence of Theorem 1, we obtain the following two corollaries, yielding a necessary condition for low-frequency multi-dimensional spectral stability strengthening the one- then for λ, ξ sufficiently small, the zero-set of D(·, ·), corresponding to spectra of L, consists of n + 1 characteristic surfaces: u t + j f j (u) x j = ǫ j, k (B jk (u)u x k ) x j ,(1.λ j (ξ) = −ia j (ξ) + O(ξ), j = 1, . . . , n + 1, (1.13) where a j (ξ) denote the eigenvalues of A := j ξ j ∂(F j , SΩe j ) ∂(M, ΩN ) , (1.14) excluding (d − 1) identically zero eigenvalues associated with modes not satisfying constraint (1.10). Proof. Similarly as in as in the proof of the analogous Lemma 7.5 [ZS] in the shock wave case, assuming (1.12), we may easily deduce (1.13) from (1.9) using Rouchés Theorem. Defining D ρ,ξ (λ) := ρ −(n+1) D(ρξ, ρλ), (1.15) for (ρ,ξ,λ) ∈ R×S d−1 ×C, we obtain a d-parameter family of analytic maps, converging as ρ → 0 to D 0,ξ = ∆(ξ, ·). Under assumption (1.12), D 0,ξ = ∆(ξ, ·) ∼λ n+1 as |λ| → ∞, hence, for ρ sufficiently small, D ρ,ξ has n + 1 continuously varying roots λ =â j (ξ, ρ). Defining a j (ξ) := |ξ|â j (ξ/|ξ|, |ξ|), we obtain the result. Remark 1.1. Evidently, a j (ξ) are smooth in |ξ| for fixedξ, but in general have a conical singularity at ξ = 0 when considered as a function of ξ, i.e., ∂a j /∂ξ is discontinuous at ξ = 0. Corollary 1.2. Assuming (H0)-(H2) and the nondegeneracy condition (1.12), a necessary condition for low-frequency spectral stability ofū, defined as Re λ ≤ 0 for D(ξ, λ) = 0, ξ ∈ R d , and |ξ, λ| sufficiently small, is that the averaged system (1.5) be "weakly hyperbolic" in the sense that it possesses a full set of real characteristicsλ j (ξ) for each ξ ∈ R d , i.e., the eigenvalues of A = j ξ j ∂(F j , SΩe j ) ∂(M, ΩN ) are real. Remark 1.2. Condition (1.12), or equivalently (∂/∂λ) n+1 D(0, 0) = 0, is a necessary condition for one-dimensional linearized stability [OZ.2], while hyperbolicity is necessary for stability of the homogenized system linearized about a constant state. Thus, Corollaries 1.1 and 1.2 are analogous to results of [ZS] in the shock wave case, stating that, given one-dimensional stability, stability of the inviscid equations linearized about an ideal shock is necessary for multi-dimensional stability of a viscous shock wave. Finally, we mention that, though the averaged system may in some cases be hyperbolic Plan of the paper In Section 2, we recall the basic Evans function construction of [G]. In Section 3, we carry out the expansion (1.7), and in Section 4, the multi-dimensional WKB expansion (1.5). Finally, in Section 5, we carry out the proof of Theorem 1 by a calculation similar to the one used by Serre [Se.1] to treat the one-dimensional case. Preliminaries Without loss of generality taking S(ū) = 0, N (ū) = e 1 ,ū =ū(x 1 ) represents a stationary solution. Linearizing (1.1) aboutū(·), we obtain v t = Lv := (B jk v x k ) x j − (A j v) x j , (2.1) where coefficients B jk := B jk (ū), A j v := Df j (ū)v − (DB j1 (ū)v)ū x 1 (2.2) are now periodic functions of x 1 . Taking the Fourier transform in the transverse coordinatex = (x 2 , · · · , x d ), we obtain v t = Lξv = (B 11v x 1 ) x 1 − (A 1v ) x 1 + i( j =1 B j1 ξ j )v x 1 + i( k =1 B 1k ξ kv ) x 1 − i j =1 A j ξ jv − j =1,k =1 B jk ξ k ξ jv ,(2. 3) whereξ = (ξ 2 , · · · , ξ d ) is the transverse frequency vector. The Laplace transform in time t leads us to study the family of eigenvalue equations 0 = (Lξ − λ)w = (B 11 w ′ ) ′ − (A 1 w) ′ + i j =1 B j1 ξ j w ′ + i( k =1 B 1k ξ k w) ′ − i j =1 A j ξ j w − j =1,k =1 B jk ξ k ξ j w − λw, (2.4) associated with operators Lξ and frequency λ ∈ C, where ' ′ ' denotes ∂/∂x 1 . Clearly, a necessary condition for stability of (1.1) is that (2.4) have no L 2 solutions w forξ ∈ R d−1 and Re λ > 0. For solutions of (2.4) correspond to normal modesv(x, t) = e λt e iξ·x w(x 1 ) of (2.1). The difficulty of our problem is due to accumulation at the origin of the essential spectrum of the linearized operator L about the wave as in the one dimensional case. Multidimensional stability concerns the behavior of the perturbation of the top eigenvalue, λ = 0 under small perturbations inξ. For study this stability, we use Floquet's theory and an Evans function [G] which not only depends on λ but also on ξ 1 which corresponds to the phase shift andξ. To define the Evans function, we choose a basis {w 1 (x 1 ,ξ, λ), . . . , w 2n (x 1 ,ξ, λ)} of the kernel of Lξ − λ, which is analytic in (ξ, λ) and is real when λ is real, for details see [OZ.1,Se.1]. Now we can define the Evans function by D(λ, ξ 1 ,ξ) := w l (X,ξ, λ) − e iXξ 1 w l (0,ξ, λ) (w l ) ′ (X,ξ, λ) − e iXξ 1 (w l ) ′ (0,ξ, λ) 1≤l≤2n (2.5) where ξ 1 ∈ R. Note that Xξ 1 is exactly θ in [Se.1]. We remark that D is analytic everywhere, with associated analytic eigenfunction w l for 1 ≤ l ≤ 2n. A point λ is in the spectrum of Lξ if and only if D(λ, ξ) = 0 with ξ = (ξ 1 ,ξ). Example 2.1. In the constant-coefficient case B 11 w ′′ − A 1 w ′ + i j =1 B j1 ξ j w ′ + i k =1 B 1k ξ k w ′ − i j =1 A j ξ j w − j =1,k =1 B jk ξ k ξ j w − λw = 0, (2.6) an elementary computation yields D(λ, ξ) = Π 2n l=1 (e µ l (λ,ξ)T − e iξ 1 T ) where µ l , l = 1, . . . , 2n, denote the roots of the characteristic equation (µ 2 B 11 + µ(−A 1 + i j =1 B j1 ξ j + i k =1 B 1k ξ k ) − (i j =1 A j ξ j + j =1,k =1 B jk ξ k ξ j + λI))w = 0, (2.7) where w = e µx 1w . The zero set of D consists of all λ and ξ 1 such that µ l (λ,ξ) = iξ 1 (mod2πi/X) for some l. Setting µ = iξ 1 in (2.7), we obtain the dispersion relation det(−B ξ − iA ξ − λI) = 0 (2.8) where A ξ = j A j ξ j and B ξ = j,k B jk ξ k ξ j . Remark 2.1. If (λ,ξ) = (0, 0) then (2.7) reduces to µ((B 11 ) −1 A 1 − µ)w = 0 giving n nonzero roots µ = s j ,w = t j , where s j , t j are eigenvalues and eigenvectors for the matrix (B 11 ) −1 A 1 , and an n-fold root µ = 0. Thus, D(0, 0) = 0 in the above example. We shall see later that this holds also in the general variable-coefficient case. Remark 2.2. In the constant coefficient case, (2.8) yields expansions λ j (ξ) = 0 − ia j (ξ) + O(|ξ|), j = 1, . . . , n, (2.9) for the n roots bifurcating from λ(0) = 0, where a j denote the eigenvalues of A ξ . Thus we obtain the necessary stability condition of hyperbolicity, σ(A ξ ) real. Evans function calculations Motivated by the example, we now find linearized dispersion relations for the variable-coefficient Evans function in the low-frequency limit. From now on, coordinatize ν in the vicinity of e 1 by ν =: (1, δ 2 , . . . , δ d ) 1 + |δ| 2 , (3.1) δ = (δ 2 , . . . , δ d ) ∈ R d−1 . Note that differentiation of (3.1) yields ∂ν = (0, δ). Plug the Taylor expansion of w(x 1 ,ξ, λ) at the origin of (λ,ξ) w l (·,ξ, λ) = w l (·, 0, 0) + λw l λ (·, 0, 0) + j =1 w l ξ j (·, 0, 0)ξ j (3.4) + 1 2 (λ 2 w l λλ (·, 0, 0) + 2λ j =1 w l λξ j (·, 0, 0)ξ j + j =1,k =1 w l ξ j ξ k (·, 0, 0)ξ k ξ j ) + . . . into (2.4) to find the identies: (Lw l ) ′ = 0, (Lw l λ ) ′ = w l , (Lw l λλ ) ′ = 2w l λ (3.5) and (Lw l λξ j ) ′ = (iA j w l λ − iB j1 (w l λ ) ′ − i(B 1j w l λ ) ′ + w l ξ j ), (Lw l ξ j ξ k ) ′ = (iA j w l ξ k + iA k w l ξ j − iB j1 (w l ξ k ) ′ − i(B 1j w l ξ k ) ′ − iB k1 (w l ξ j ) ′ − i(B 1k w l ξ j ) ′ + 2B jk w l ), whereLw = B 11 w ′ − A 1 w, and also: (Lw l ξ j ) ′ = i(A j w l − B j1 (w l ) ′ − (B 1j w l ) ′ ), j = 1 (3.6) and (Lw 1 ξ j ) ′ = i(f j (ū) − B j1 (ū)ū ′ − B 1j (ū)ū ′ ) ′ , j = 1 (3.7) by using the definition of A j in (2.2). In the Laplacian case B jk = δ j k , the latter identity simplifies to (Lw 1 ξ j ) ′ = if j (ū) ′ .[Lw 1 λξ j ] = X 0 iA j w 1 λ − iB j1 (w l λ ) ′ − i(B 1j w 1 λ ) ′ + w 1 ξ j dx 1 , [Lw 1 ξ j ξ k ] = X 0 iA j w 1 ξ k + iA k w 1 ξ j − iB j1 (w 1 ξ k ) ′ − i(B 1j w 1 ξ k ) ′ −iB k1 (w 1 ξ j ) ′ − i(B 1k w 1 ξ j ) ′ + 2B jk w 1 dx 1 . (3.11) In the Laplacian case B jk = δ j k , the last two identities simplify considerably, to [Lw 1 λξ j ] = X 0 (iA j w 1 λ + w 1 ξ j )dx 1 , [Lw 1 ξ j ξ k ] = X 0 iA j w 1 ξ k + iA k w 1 ξ j dx 1 . (3.12) Connection to traveling-wave variations From (3.3), we find easily that w j (·, 0, 0) = ∂u/∂a j |ū, w n+j (·, 0, 0) = ∂u/∂q j |ū for j = 1, . . . , n. (3.13) For example, taking the variation of traveling wave equation (1.4) with respect to q j , we find that z = ∂u/∂q j satisfies B 11 z ′ − A 1 z = −e j with z(0) = 0, so that L 0 z = 0 and z ′ (0) = −(B 11 ) −1 e j as claimed. Further (see [OZ.1], [Se.1]), w 1 λ = −∂u/∂s + 2n n+1 α l w l for α ∈ R 2n , since L 0 (−∂u/∂s) = L 0 w 1 λ (·, 0, 0) =ū ′ and w 1 λ (0, 0, 0) = (∂u/∂s)(0) = 0, and, similarly, for j = 1, using L 0 w 1 ξ j (·, 0, 0) = L 0 (i∂u/∂δ j )(·, 0, 0) = if j (ū) ′ and w 1 ξ j (0, 0, 0) = ∂u/∂δ j (0) = 0, w 1 ξ j (·, 0, 0) = i∂u/∂δ j + 2n n+1 β l j w l . Alternatively,w 1 λ (·, 0, 0) = −∂u/∂s,w 1 ξ j (·, 0, 0) = i∂u/∂δ j (3.14) forw 1 := w 1 − λ 2n n+1 α l w l − 2n ℓ=n+1 j ξ j β l j w l ,(3.15) withw 1 (0, 0) still equal toū ′ . We hereafter substitutew 1 for w 1 everywhere it appears, as we are free to do. (Recall, w ℓ can be an arbitrary basis of the kernel of L.) Reduction of the leading part We rewrite the Evans function (2.5) as D(λ, ξ 1 ,ξ) := [w l (ξ, λ)] + (1 − e iXξ 1 )w l (0,ξ, λ) (w l ) ′ (ξ, λ) + (1 − e iXξ 1 )(w l ) ′ (0,ξ, λ) 1≤l≤2n (3.16) and then multiply the second row in (3.16) by B 11 and then subtract A 1 times the first one (det B 11 )D(λ, ξ 1 ,ξ) := [w l (ξ, λ)] + (1 − e iXξ 1 )w l (0,ξ, λ) L w l (ξ, λ) + (1 − e iXξ 1 )(Lw l )(0,ξ, λ) 1≤l≤2n . (3.17) At this point, we restrict for readibility to the Laplacian case B jk = δ j k . The general case goes similarly. Then the Evans function D(λ, ξ 1 , 0) becomes Γ 0 det c(ξ, λ) [w 2 ] . . . C(ξ, λ) C 2 (ξ, λ) . . . + O(|λ| n+2 + |ξ 1 | n+2 ) (3.18) with a nonzero number Γ 0 (for details, see [OZ.1,Se.1]), where c(ξ, λ) = λ[w 1 λ ] + j =1 ξ j [w 1 ξ j ] − iXξ 1ū ′ (0) = −λ[∂u/∂s] + j =1 iξ j [∂u/∂δ j ] − iXξ 1ū ′ (0) (3.19) (3.20) is a homogeneous degree one polynomial, C(ξ, λ) := 1 2 λ 2 [Lw 1 λλ ] + λ j =1 ξ j [Lw 1 λξ j ] + 1 2 j,k =1 ξ j ξ k [Lw 1 ξ j ξ k ] −iXξ 1 λ(Lw 1 λ )(0) − iXξ 1 j =1 ξ j (Lw 1 ξ j )(0) = −λ 2 X 0 (∂u/∂s)dx 1 + iλ j =1 ξ j X 0 (A j (−∂u/∂s) + (∂u/∂δ j ))dx 1 − j,k =1 ξ j ξ k X 0 A j (∂u/∂δ k )dx 1 +iXξ 1 λ(L∂u/∂s)(0) +Xξ 1 j =1 ξ j (L∂u/∂δ j )(0) = −λ 2 X 0 (∂u/∂s)dx 1 + iλ j =1 ξ j − (∂/∂s) X 0 f j (u)dx 1 + (∂/∂δ j ) X 0 udx 1 − j,k =1 ξ j ξ k (∂/∂δ k ) X 0 f j (u)dx 1 +iXξ 1 λ(L∂u/∂s)(0) +Xξ 1 j =1 ξ j (L∂u/∂δ j )(0) (3.21) is a homogeneous degree two polynomial, and C ℓ (ξ, λ) := λ[Lw l λ ] + j =1 ξ j [Lw l ξ j ] − iXξ 1L w l (0) (3.22) are homogeneous degree one polynomials given by (∂/∂a j ) λ X 0 u(x 1 )dx 1 + j =1 ξ j X 0 if j (u(x 1 ))dx 1 − iXξ 1 (u ′ (0) − f 1 (u(0)) (3.23) for ℓ = j = 2, . . . , n, and (∂/∂q j ) λ X 0 u(x 1 )dx 1 + j =1 ξ j X 0 if j (u(x 1 ))dx 1 − iXξ 1 (u ′ (0) − f 1 (u(0)) (3.24) for ℓ = n + j = n + 1, . . . , 2n. Thus, the leading order part of D near (ξ, λ) = (0, 0) is the homogeneous degree (n + 1) polynomial ∆ 1 (ξ, λ) := Γ 0 det c(ξ, λ) . . . C(ξ, λ) C 2 (ξ, λ) . . . with c, C, C ℓ defined as above. In particular, the Evans function has a zero of order n + 1 at (λ, ξ) = (0, 0). Slow modulation approximation Next, we carry out a multi-dimensional version of the slow modulation (WKB) expansion in [Se.1]. Rescale (x, t) → (ǫx, ǫt) in (1.1) to obtain u t + j f j (u) x j = ǫ j, k (B jk (u)u x k ) x j . (4.1) Let u ǫ (x, t) = u 0 x, t, φ(x, t) ǫ + ǫu 1 x, t, φ(x, t) ǫ + · · · , (4.2) where y → u 0 (x, t, y) is a periodic function with ∂ x φ = 0. We plug (4.2) into (4.1) and consider the equations obtained by equating coefficients at successive powers of ǫ. At order ǫ −1 , we have − s∂ y u 0 + j ων j ∂ y (f j (u 0 )) − ∂ y ( j,k ω 2 ν j ν k B jk (u 0 )∂ y u 0 ) = 0, with s := − ∂ t φ |∂ x φ| , ν := ∂ x φ |∂ x φ| , ω := |∂ x φ|,(4.3) which may be recognized as the traveling-profile equation after rescaling y → ωy. That is, u 0 (y) = u(ωy) for a periodic profile of period X = ω −1 , hence u 0 is periodic of period one, as described in [Se.1]. The quantities ω(x, t), s(x, t), ν(x, t) are the local frequency, speed, and direction of the modulated wave. At order ǫ 0 , we have ∂ t u 0 + d j=1 ∂ x j f j (u 0 ) − d k=1 B jk (u 0 )ων k ∂ y u 0 = ∂ y (. . .). Taking the average with repect to y, and rescaling with y := ωy, we obtain ∂ t M (u 0 ) + j=1 ∂ x j F j (u 0 ) = 0 (4.4) where F j (u 0 ) = 1 X X 0 f j (u 0 ) − d k=1 B jk (u 0 )ων k ∂ y u 0 dy (4.5) is the averaged flux along orbit u 0 (now rescaled to actual periodX), with j ν j F j = (SM + Q)(u 0 ), by the profile equation. In the Laplacian case B jk = δ j k , (4.5) simplifies to F j (u 0 ) = 1 X X 0 (f j (u 0 ) − ν j (u 0 )∂ y u 0 )dy. (4.6) We have an additional d equations ∂ t (ΩN )(u 0 ) + ∂ x (ΩS(u 0 )) = 0 (4.7) from the Schwarz identity ∂ t ∂ x φ = ∂ x ∂ t φ, where d is the dimension of the spatial variable x. (Note: (Ω, N ) may be regarded as polar coordinates for ΩN .) Combining, we obtain finally the closed homogenized system ∂ t (M, ΩN ) + j ∂ x j (F j , ΩSe j ) (4.8) of the introduction, consisting of n+d equations in n+d unknowns. As discussed in the introduction, this should be supplemented with the constraint curl (ΩN ) ≡ 0 (4.9) coming from the relation ΩN = ∇ x φ. Proof of the main theorem We now carry out the proof of Theorem 1, restricting for readibility to the Laplacian case B jk = δ j k . The general case follows similarly. We want to see that the leading order part ∆ 1 of D, defined in (3.25), is given by a (nonzero) constant multiple of λ 1−d timeŝ δ j ∂u ∂δ j + γū ′ (0). ∆(ξ, λ) = det   λ ∂(M, ΩN ) ∂u (u) + j iωXξ j ∂(F j , SΩe j ) ∂u (u)   ,(5. We relabel β 1 for β 0 since we will not use w 1 hereafter. Thus, Z(β, δ, γ) := β 1 ∂u ∂s + 2n 2 β l [w l ] + d j=2 δ j ∂u ∂δ j + γū ′ (0), β ∈ C 2n , δ ∈ C d−1 γ ∈ C. (5.2) We easily compute (see (3.3) for dQ) the differentials dX · (β, δ, γ) = γ = ∂X, dS · (β, δ, γ) = β 1 = ∂S, dN · (β, δ, γ) = (0, δ) T = ∂N, (5.3) dQ · (β, δ, γ) = (β n+1 , . . . , β 2n ) T = − 2n 2 β lL w l = ∂Q, dΩ · (β, δ, γ) = −ω 2 γ = ∂Ω, where, following [Se.1], we use the notation ∂G to indicate the extension to C 2n+d of a differential dG defined on the kernel of Z. (Note: this includes the extension from real to complex values, of which we shall later make important use in parametrizations (5.14) and (5.15).) Likewise, XM = where N 2 is the 2n×2n matrix corresponding to linear operator H 2 Z operating on (ρ, β 2 , . . . , β 2n ) through the compositions (5.14) and (5.15), and thuŝ ∆(ξ, λ) = C 1 λ d−1 det N 2 . (5.16) Alternatively, we may observe that H 1 is full rank whenever λ = 0. Observing also a posteriori that det N 2 is homogeneous degree n + 1, we may conclude that C(ξ, λ) = det H 1 | ker(H 1 ) ⊥ as the ratio of n + d and n + 1 degree homogeneous polynomials must be a constant times λ d−1 . This discussion repairs a minor omission in [Se.1], where the dependence of C 2 on (ξ, λ) is not explicitly discussed. It remains to compute, under the compositions (5.14), (5.15), the 2n × 2n determinant det N 2 , which, transposing first and second n-row blocks, may be expressed as det N 2 = det β 1 [∂u/∂s] + 2n 2 β l [w l ] + γū ′ (0) + d 2 δ j [∂u/∂δ j ] λ∂M + iωXξ 1 (M ∂S + ∂Q − j =1 δ jF j ) + j =1 iωXξ j ∂F j . (5.17) Substituting from (5.14)-(5.15) and the variational formulae (5.4)-(5.9), and expressing N 2 as a matrix taking (ρ, β 2 , . . . , β 2n ) → C 2n , we obtain, similarly as in [Se.1], that det N 2 =∆λ 1−d is ω n times the determinant ∆ 1 defined in (3.25), giving the desired relation ∆ 1 =ω −n λ 1−d∆ , and completing the proof. Namely, the first line of (5.17) becomes dimensional version obtained in [OZ.1], [Se.1]. Corollary 1.1. Assuming (H0)-(H2) and the nondegeneracy condition det ∂(M, ΩN ) , [w 2 ], . . . , [w 2n ],ū ′ (0)}. Proof. Immediate, using [w 1 ] = 0; see [Se.1] for the one-dimensional case. 0 − λ)w = (B 11 w ′ ) ′ − (A 1 w) ′ − λw = 0, (3.2) which is associated with the one dimensional stability problem studied in [OZ.1,Se.1]. Recall that u isX-periodic in x 1 and the functions w 1 (x 1 ,ξ, λ), . . . , w 2n (x 1 ,ξ, λ) are in the basis of the kernel of Lξ − λ. Following [OZ.1], we normalize w j (0,ξ, λ) = e j , (w j ) ′ (0,ξ, λ) = (B 11 ) −1 A 1 e j ; w n+j (0,ξ, λ) = 0, (w n+j ) ′ (0,ξ, λ) = −(B 11 ) −1 e j for j = 1, . . . , n and all (ξ, λ), giving in particular Lw j (0,ξ, λ) = 0,Lw n+j (0,ξ, λ) = e j for j = 1, . . . , n. (3.3) the functions w 1 (x 1 , 0, 0), . . . , w 2n (x 1 , 0, 0) are in the basis of the kernel of L 0 . We omit (·, 0, 0) hereabove and denote [w] = w(X) − w(0). We also have w 1 =ū ′ ,Lw 1 = 0, [w 1 ] 1)whereu denotes the orbit class ofū, withν = (1, 0, . . . , 0) = N (u). tangent space to P atu is the β-projection of the kernel of Z(β 0 , β 2 , . . . , β 2n , δ 2 , . . . , δ d , γ) := β XM ) · (β, δ, γ) = γū(0) j dy = ∂(XM ), (5.4)thus determining ∂M =ω(∂(XM ) −M ∂X). j (u) − ν j (u)∂ y u)dy = ∂(XF j )(u) (5.6) for j = 1 have the simple form ∂(XF j )(u) = ∂Xf j (u)periodicity of u, hence ∂F j (u) = ω(∂(XF j ) − (∂X)F j (u)) = ωγf j (u)(0) + ω X 0∂f j (u)dy − ωγF j (u). β ℓ free for ℓ = 1, determines a choice of basis for kerH 1 , for which C 2 (ξ, λ) has the simple form λ Taylor expansion of D. A tedious, but fairly straightforward calculation following [OZ.1], [Se.1] shows that [OZ.1], so far, only unstable periodic traveling-wave solutions have been found for viscous conservation laws. However, essentially only the single, 2 × 2 model of van der Waals gas dynamics with viscositycapillarity in one dimension has so far been considered in detail [OZ.1], [Se.1]-[Se.2], and we see no obvious reason why a stable wave should not exist for other models. It would be extremely interesting to either find such an example, with the associated rich behavior described by the modulation equations, or show that it can in no case exist. As suggested by Serre [Se.3], a useful starting point might be to consider whether the averaged system (1.5) might ever possess an entropy. We make the inessential change ρ → −ρ for convenience in later calculations. To find the variation for F 1 , note that, by the first-order traveling wave system (1.4), j ν j F j = M S + Q, so that j (∂ν j )F j + j ν j (∂F j ) = ∂(M S + Q), hence, for ν = (1, 0, . . . , 0), s = 0,(5.9)We may now compute the determinant (5.1), i.e., the determinant of the restriction to ker Z of the linear mapwhich can be evaluated using an ingenious trick of [Se.1] asis well-defined thanks to full rank of Z, assumption (H2) ′ , and independent of (ξ, λ), by the corresponding property of Z, but C 2 (ξ, λ) = det H 1 | ker(H 1 ) ⊥ andboth depend on the specific dependence on (ξ, λ) of the basis chosen for kerH 1 .Note that determinant (5.1) is in the first place defined only up to a constant factor depending on the parametrization of P, so that we need only take care of the (ξ, λ) dependence of C 2 . Rewriting The second line of (5.17) becomeswith (5.5) and other identities. Denoting by N the matrix in (3.25) for which ∆ 1 = det N , we find, comparing term by term, that the first n rows of N 2 are equal to the first n rows of N , while the last n rows of N 2 are equal toω times the last n rows of N . Thus, det N 2 =ω n det N =ω n ∆ 1 as claimed, and we are done. On the structure of the spectra of periodic traveling waves. R Gardner, J. Math. Pures Appl. 72R. Gardner, On the structure of the spectra of periodic traveling waves, J. Math. Pures Appl. 72 (1993), 415-439. Multidimensional viscous shocks I: degenerate symmetrizers and long time stability. O Gues, G Metivier, M Williams, K Zumbrun, Journal of the Amer. Math. Soc. 18Gues, O., Metivier, G., Williams, M., and Zumbrun, K., Multidimensional viscous shocks I: degen- erate symmetrizers and long time stability, Journal of the Amer. Math. Soc. 18. (2005), 61-120. Multidimensional viscous shocks II: the small viscosity problem. O Guès, G Métivier, M Williams, K Zumbrun, Comm. Pure Appl. Math. 57Guès, O., Métivier, G., Williams, M., and Zumbrun, K., Multidimensional viscous shocks II: the small viscosity problem, Comm. Pure Appl. Math. 57. (2004), 141-218. Existence and stability of multidimensional shock fronts in the vanishing viscosity limit. O Guès, G Métivier, M Williams, K Zumbrun, Arch. Rat. Mech. Anal. 175Guès, O., Métivier, G., Williams, M., and Zumbrun, K., Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Rat. Mech. Anal. 175. (2004), 151-244. Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. G Metivier, K Zumbrun, Mem. Amer. Math. Soc. 175826107Metivier, G. and Zumbrun, K., Large viscous boundary layers for noncharacteristic nonlinear hy- perbolic problems, Mem. Amer. Math. Soc. 175 (2005), no. 826, vi+107 pp. Stability of periodic solutions of viscous conservation laws with viscosity-1. Analysis of the Evans function. M Oh, K Zumbrun, Arch. Rational Mech. Anal. to appearM. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation laws with viscosity- 1. Analysis of the Evans function, to appear, Arch. Rational Mech. Anal. (2002). Stability of periodic solutions of viscous conservation laws with viscosity-Pointwise bounds on the Green function, to appear. M Oh, K Zumbrun, Arch. Rational Mech. Anal. M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation laws with viscosity- Pointwise bounds on the Green function, to appear, Arch. Rational Mech. Anal. (2002). Spectral stability of periodic solutions of viscous conservation laws: Large wavelength analysis. D Serre, PreprintD. Serre, Spectral stability of periodic solutions of viscous conservation laws: Large wavelength analysis, Preprint. Entropie du mélange liquide-vapeur d'un fluide thermo-capillaire. D Serre, Archy. Rational Mech. Anal. 128D. Serre, Entropie du mélange liquide-vapeur d'un fluide thermo-capillaire, Archy. Rational Mech. Anal., No. 128 (1994) 33-73. . D Serre, Private Communicaton, Z-kD. Serre, Private Communicaton. [Z-k] Multidimensional stability of planar viscous shock waves, TMR Summer School Lectures: Kochel am See. K Zumbrun, Birkhauser's Series: Progress in Nonlinear Differential Equations and their Applications. 207K. Zumbrun, Multidimensional stability of planar viscous shock waves, TMR Summer School Lec- tures: Kochel am See, May, 1999, Birkhauser's Series: Progress in Nonlinear Differential Equations and their Applications (2001), 207 pp. Viscous and inviscid stability of multidimensional planar shock fronts. K Zumbrun, D Serre, Indiana Univ. Math. J. 48K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999) 937-992.

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{'abstract': 'We generalize work of Oh & Zumbrun and Serre on spectral stability of spatially periodic traveling waves of systems of viscous conservation laws from the one-dimensional to the multi-dimensional setting. Specifically, we extend to multi-dimensions the connection observed by Serre between the linearized dispersion relation near zero frequency of the linearized equations about the wave and the homogenized system obtained by slow modulation (WKB) approximation. This may be regarded as partial justification of the WKB expansion; an immediate consequence is that hyperbolicity of the multi-dimensional homogenized system is a necessary condition for stability of the wave. As pointed out by Oh & Zumbrun in one dimension, description of the low-frequency dispersion relation is also a first step in the determination of time-asymptotic behavior.', 'arxivid': 'math/0510515', 'author': ['Myunghyun Oh ', 'Kevin Zumbrun ', '\nDepartment of Mathematics\nUniversity of Kansas\n1460 Jayhawk Blvd66047LawrenceKSUSA\n', '\nDepartment of Mathematics\nIndiana University\n47450BloomingtonINUSA\n'], 'authoraffiliation': ['Department of Mathematics\nUniversity of Kansas\n1460 Jayhawk Blvd66047LawrenceKSUSA', 'Department of Mathematics\nIndiana University\n47450BloomingtonINUSA'], 'corpusid': 14955051, 'doi': '10.4171/zaa/1275', 'github_urls': [], 'n_tokens_mistral': 12563, 'n_tokens_neox': 11267, 'n_words': 6165, 'pdfsha': 'fd0c2309685a8f3fa15a95ff866f6291a679617e', 'pdfurls': ['https://export.arxiv.org/pdf/math/0510515v1.pdf'], 'title': ['Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions ⋆', 'Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions ⋆'], 'venue': []}

arxiv

Pair production of neutral Higgs bosons from the left-right twin Higgs model at the ILC and LHC 21 May 2009 May 21, 2009 Wei Ma Department of Physics Liaoning Normal University 116029DalianP. R. China Chong-Xing Yue Department of Physics Liaoning Normal University 116029DalianP. R. China Yong-Zhi Wang Department of Physics Liaoning Normal University 116029DalianP. R. China Pair production of neutral Higgs bosons from the left-right twin Higgs model at the ILC and LHC 21 May 2009 May 21, 2009number: 1260Cn1480Cp1215Ji * cxyue@lnnueducn 1 In the framework of the left-right twin Higgs model, we study pair production of the neutral Higgs bosons at the International Linear Collider (ILC) and the CERN LHC. We find that the production cross section of the process e + e − → φ 0 h are at the level of several tens f b at the ILC, the production cross section of the φ 0 φ 0 pair and φ 0 h pair are at the level of several hundreds f b at the LHC. As long as the neutral Higgs boson φ 0 is not too heavy, we conclude that its pair production might be used to test for the left-right twin Higgs model at the LHC experiment or in the future ILC experiment. I. Introduction The Higgs mechanism is the heart of the standard model (SM) providing masses to gauge bosons via electroweak symmetry breaking (EW SB). However, the SM fails to explain the origin of the fermion mass and has naturalness problems. Many alternative new physics models with extended Higgs sectors are free from the above difficulties. The hunt for the Higgs bosons came to be one of the most important goals for present and future high energy collider experiments. Apart from the SM, neutral Higgs bosons appear in almost every scenario exploring new phenomena [1]. Pair production of neutral Higgs bosons at the CERN LHC, which will provide a way to test the Higgs boson self-coupling, may be sensitive to new physics [2,3]. Many works have contributed to studies of the neutral Higgs pair production at the hadron collider in model independent [4], in SM [5][6][7][8], and in new physics models beyond the SM, such as little Higgs models [9], Randall-Sundrum-like models [10], top condensation models [11], supersymmetric models (SUSY ) [12,13] and models of universal extra dimensions (UED) [14]. The SM has been proved by all existing precise experimental data with its theoretical predictions beyond one-loop level being coincident with experimental observations. But in the SM the Higgs boson mass suffers from an instability under radiative corrections, which is called "hierarchy problem" [15]. Recently, the twin Higgs mechanism has been proposed as a solution to the little hierarchy problem. The Higgs bosons emerge as pseudo-Goldstone bosons once the global symmetry is spontaneously broken. Gauge and Y ukawa interactions that break the global symmetry give masses to the Higgses. The twin Higgs mechanism can be implemented in left-right models with the additional discrete symmetry being identified with left-right symmetry [16,17]. The left-right twin Higgs (LRT H) [17][18][19][20][21]. In the context of the LRT H model, pair production of the charged Higgs bosons (φ + , φ − ) in the LRT H model at the ILC and LHC are studied in Ref. [21], but they did not consider production of the neutral Higgs bosons (φ 0 , h). As we know, so far production of the neutral Higgs pair at the LHC and the ILC in the LRT H model has not been considered, which is the main aim of this paper. Besides the SM-like Higgs boson h, there are two additional neutral Higgs bosons in the LRT H model, which areĥ 0 2 and φ 0 . The neutral Higgs bosonĥ 0 2 is a possible dark matter candidate that only couples to the gauge bosons (including the SM gauge bosons γ, Z, W , and the new gauge boson Z H ). The production cross section ofĥ 0 2 at the collider is very small and escapes the detector. Therefore, in this paper, we will not discuss the production ofĥ 0 2 at the ILC or LHC. The neutral Higgs boson φ 0 is a pseudoscalar that couples to both the SM fermions and gauge bosons. The neutral Higgs boson pair φ 0 h can be produced via the processes e + e − → Z(Z H ) → φ 0 h at the ILC, and via the partonic processes qq → φ 0 h(q = u, c, d, s, b), gg → φ 0 h at the LHC, respectively. While the neutral Higgs pair φ 0 φ 0 can only be produced via the partonic process gg → φ 0 φ 0 and the t-channel partonic process bb → φ 0 φ 0 at the LHC. We calculate all above these processes. Our numerical results denote that, for m h = 120GeV , 120GeV ≤ m φ 0 ≤ 180GeV and 500GeV ≤ f ≤ 1500GeV : (i)the production cross section of φ 0 h at the ILC with the center-of-mass (c.m.) energy √ s = 500GeV is in the range of 0.92f b ∼ 20f b; (ii)the production cross section of φ 0 h at the LHC with the c.m. energy √ s = 14T eV is in the range of 34f b − 306f b, and the main contribution comes from light quarks; (iii)the production cross section of φ 0 φ 0 at the LHC is in the range of 4f b − 122f b, and the main contribution comes from the top quark loop. This paper is organized as follows. In Sec. II, we briefly review the essential features of the LRT H model. The relevant couplings of the neutral Higgs bosons to other particles and the feature of the decay for the neutral Higgs bosons φ 0 are also discussed in this section. In Secs. III and IV, we give our numerical results for pair production of neutral Higgs bosons predicted by the LRT H model at the ILC and LHC, respectively. Our conclusions are given in Sec. V. II. The LRTH Model The LRT H model was first proposed in Ref. [16] and the details of the model as well as the particle spectrum, F eynman rules, and some phenomenology analysis have been studied in Ref. [17]. Here we will briefly review the essential features of the model and The fermion sector of the LRT H model is similar to that of the SM, with the righthanded quarks (u R , d R ) and leptons (l R , υ R ) form fundamental representations of SU(2) R . In order to give the top-quark mass of the order of the electroweak scale, a pair of vectorlike quarks Q L and Q R are introduced. The mass eigenstates, which contain one the SM top quark t and a heavy top partner T , are mixtures of the gauge eigenstates. Their masses are given by m 2 t = 1 2 (M 2 + y 2 f 2 − N t ), M 2 T = 1 2 (M 2 + y 2 f 2 + N t ).(1) where N t = (y 2 f 2 + M 2 ) 2 − y 4 f 4 sin 2 2x with x = ν/ √ 2f , in which ν = 246GeV is the scale of the EW SB. Provided M T ≤ f and that the parameter y is of order one, the top Y ukawa coupling will also be of order one. The parameter M is essential to the mixing between the SM top quark and its partner T . According the symmetry-breaking pattern discussed above, with certain reparametriza- The couplings expression forms which are related our calculation, are shown as [17] φ 0d 1,2,3 d 1,2,3 : im d i γ 5 /( √ 2f ); φ 0ū 1,2 u 1,2 :−im u i γ 5 /( √ 2f ) ; htt : − em t C L C R / (2m W S W ) ; φ 0t t :−iyS R S L γ 5 / √ 2 ; hTT : −y(S R S L − C L C R x)/ √ 2; φ 0T T :−iyC L C R γ 5 / √ 2; hφ 0 φ 0 :x(30p 2 ·p 3 +11p 1 ·p 1 )/(27 √ 2f); hφ 0 Z µ : iexp 3µ /(6C W S W ); Z Hµū1,2 u 1,2 :−eγ µ (2S 2 W P L +(1 − 7cos2θ W )P R )/(12C W S W cos2θ W ); Z Hµd1,2,3 d 1,2,3 : − e γ µ (S 2 W P L + (3 − 5S 2 W )P R )/(6C W S W cos2θ W ); hφ 0 Z Hµ : iex((14 − 17S 2 W )p 2µ − (4 − S 2 W )p 1µ )/(18S W C W cos2θ W ).(2) Where p 1 , p 2 , and p 3 refer to the incoming momentum of the first, second and third particles, respectively. u i and d i represent the upper-and down-type fermions, respectively. S W = sin θ W , C W = cosθ W , and θ W is the W einberg angle. At the leading order of 1/f , the sine values of the mixing angles α L and α R can be written as S L = sin α L ≃ M M T sin x, S R = sin α R ≃ M M T (1 + sin 2 x).(3) C L and C R are the cosine values of the mixing angles α L and α R , respectively. P L(R) = (1 ∓ γ 5 )/2 is the left (right)-handed projection operator. In the framework of the LRT H model, the mass of the neutral Higgs boson φ 0 can be anything below f here we consider another possibility, in which the mass is around 150GeV [17]. Similar to the SM Higgs boson, φ 0 can decay to γγ through the top-quark loop and heavy top-quark loop. But unlike the SM Higgs boson, in the LRT H model, the light neutral Higgs boson φ 0 is a pseudoscalar boson, due to its pseudoscalar nature, there is no φ 0 W W and φ 0 ZZ couplings at tree level. So, the one-loop SM gauge boson contribution to φ 0 γγ is zero. In general, the light neutral Higgs boson φ 0 decays into bb, cc, τ + τ − , gg and γγ. Now we discuss the branching ratios for the possible decay modes of φ 0 . The decay width of φ 0 → ff is proportional to the square of the corresponding Y ukawa coupling, with an additional suppression factor of ν 2 /(2f 2 ) comparing to that of the SM Higgs boson. The concrete expressions of the decay widths for the different decay channels are given as follows: Γ(φ 0 → bb) = 3G F m φ 0 ν 2 m 2 b 8 √ 2πf 2 (1 − 4m 2 b /m 2 φ 0 ) 3 2 , Γ(φ 0 → cc) = 3G F m φ 0 ν 2 m 2 c 8 √ 2πf 2 , Γ(φ 0 → τ + τ − ) = G F m φ 0 ν 2 m 2 τ 8 √ 2πf 2 , Γ(φ 0 → γγ) = G F α 2 m 3 φ 0 128 √ 2π 3 | f N f c Q 2 f A φ 0 f (τ f )| 2 , Γ(φ 0 → gg) = G 2 F α 2 s m 3 φ 0 48 √ 2π 3 | q A φ 0 q (τ q )| 2 .(4) Where m b , m c , and m τ are the masses of the SM fermions b, c and τ , respectively. The index f corresponds to q and l (q = quark, l = lepton). Where N f c = 1, 3 for f = l, q, respectively. Q f is the charge of the fermion f . Similar with Ref. [22], the function A φ 0 f can be written as: [22], the function f (τ f ) has two parts corresponding to the τ f ≥ 1 and τ f < 1 two conditions. In our numerical estimation, we have neglected the contributions of the light fermions. Therefore, in the LRT H model, there is τ t(T ) = 4m 2 t(T ) /m 2 φ 0 ≥ 1, and the function f (τ f ) is given by A φ 0 f = 2τ f [1 + (1 − τ f )f (τ f )]. (5) where τ f = 4m 2 f /m 2 φ 0 . In Ref.f (τ ) = arcsin 2 1 √ τ .(6) where A φ 0 q , τ q , and f (τ q ) in Eq.(4) are defined the same as A φ 0 f , τ f and f (τ f ), but only for quarks. Using above partial widths of the neutral Higgs boson φ 0 , its total width Γ can be approximately written as and m φ 0 = 120GeV . One can see from F ig.1 that the decay branching ratios of φ 0 are sensitive to the parameter f . If we assume that the parameter f is in the range of 500GeV ∼ 1500GeV , the value of the branching ratio Br(φ 0 → bb) is in the range of 14% − 55%, and the branching ratio Br(φ 0 → gg) is in the range of 38% − 85%. The values of Br(φ 0 → cc), Br(φ 0 → τ + τ − ) and Br(φ 0 → γγ) are much smaller than those of Br(φ 0 → bb) and Br(φ 0 → gg). Therefore, in order to see the trend clearly, in F ig.1 Γ = Γ bb + Γ cc + Γ τ + τ − + Γ γγ + Γ gg .(7) we have multiplied them by 10, 20 and 300, respectively. The real numerical results are Br(φ 0 → cc)= 0.9% − 3.8%, Br(φ 0 → τ + τ − )= 0.6% − 2.6%, and Br(φ 0 → γγ)= 0.09% − 0.2%. Our numerical results agree quite well with Ref. [17], in that the branching ratio Br(φ 0 → γγ) is roughly same as Br(h → γγ) for m h = m φ 0 . III. Pair production of neutral Higgs bosons at the ILC In many cases, the ILC can significantly improve the LHC measurements. If a Higgs boson is discovered, it will be crucial to determine its couplings with high accuracy, to understand the so-called mechanism of EW SB [24]. The high resolution profile determination of a light Higgs boson (mass, couplings, self-couplings, etc.) can be carried out at the ILC, where clear signals of Higgs events are expected with backgrounds that can be reduced to a magnitude level. With the LHC guidance, the ILC, which is currently being designed, will further improve our knowledge of the Higgs sector if that is how nature decided to create mass [24]. It was demonstrated in Ref. [25] that physics at the LHC and at the ILC will be complementary to each other in many respects. So far, many works have been contributed to studies of the neutral Higgs boson pair production at the ILC, in the SM [26][27][28] and in new physics beyond the SM [29][30][31][32]. From the discussions given in Sec. II, we can see that the neutral Higgs boson pair At the leading order, the production amplitude of the process can be written as M 1 = M Z + M Z H(8) with M Z = e 2 x(−1 + 4S 2 W ) 24C 2 W S 2 Wv e (p 2 ) p 12 / p 2 12 − m 2 Z u e (p 1 ) + e 2 x 24C 2 W S 2 Wv e (p 2 ) p 12 / p 2 12 − m 2 Z γ 5 u e (p 1 ), M Z H = −e 2 x(14 − 17S 2 W ) 36C 2 W cos2θ Wv e (p 2 ) p 3 / p 2 12 − m 2 Z H P L u e (p 1 ) + −e 2 x(14 − 17S 2 W )(1 − 3C 2 W ) 72C 2 W S 2 W cos2θ Wv e (p 2 ) p 3 / p 2 12 − m 2 Z H P R u e (p 1 ) + e 2 x(4 − S 2 W ) 36C 2 W cos2θ Wv e (p 2 ) p 4 / p 2 12 − m 2 Z H P L u e (p 1 ) + e 2 x(1 − 3C 2 W )(4 − S 2 W ) 72C 2 W S 2 W cos2θ Wv e (p 2 ) p 4 / p 2 12 − m 2 Z H P R u e (p 1 ). where p 12 is the momentum of the propagator, which is the sum of the incoming momentums p 1 and p 2 . With the above production amplitudes, we can obtain the production cross section directly. From the above discussions, we can see that, except for the SM input parameters e + e − → φ 0 h → bbbb.(9) The production rate of the bbbb final state in the LRT H model can be easily estimated 1 Thanks to the referees for offering this reference to us. using the formula σ s = σ × Br(φ 0 → bb) × Br(h → bb). If we assume the integrated luminosity £ int = 500f b −1 for the ILC with the c.m. energy √ s = 500GeV , then there will be 9 − 3.0 × 10 3 bbbb events to be generated at the ILC, which is significantly larger than that for the SM Higgs boson pair production process e + e − → hh → bbbb [26][27][28]. Therefore, we hope that by using very efficient µ-vertex detectors to tag the b quark jets, IV. Pair production of neutral Higgs bosons at the LHC The LHC has a good potential for discovery of a neutral Higgs boson. Now we look at pair production of the neutral Higgs bosons predicted by the LRT H model at the LHC. From the above discussions, we can see that both the φ 0 φ 0 pair and φ 0 h pair can be produced at the LHC. In this section, we will consider both of these cases. A. φ 0 φ 0 pair production In this paper, we calculate all production channels for the neutral Higgs boson pair φ 0 φ 0 at the LHC, as shown in F ig.4, including triangle diagrams, box diagrams and treelevel diagram. Each loop diagram is composed of some scalar loop functions, which are calculated by using LoopT ools [34]. The hadronic cross section at the LHC is obtained by convoluting the partonic cross sections with the parton distribution functions (P DF s). In our numerical calculation, we will use CTEQ6L P DF s for the gluon and quark P DF s [35]. The renormalization scale µ R and the factorization scale µ F are chosen to be µ R = µ F = 2m φ 0 . Because the calculation of the loop diagrams are too tedious and the analytical expression are lengthy, we will not present those here. For M = 100GeV , m φ 0 = 120GeV , and 500GeV ≤ f ≤ 150GeV , the value of the total production cross section is in the range of 4f b ∼ 122f b, and the value of the production cross section coming from the top-quark loop diagrams is in the range of 1.5f b − 105f b. g g h t(T ) (a) h t(T ) t(T ) φ 0 φ 0 g g t(T ) (b) t(T ) t(T ) t(T ) φ 0 φ 0 b b b φ 0 φ 0 (c) This is because the contributions of the box diagrams are generally much smaller than those of the triangle diagrams, and furthermore the coupling htt is much larger than the coupling hTT or the coupling φ 0 bb. If we assume the integrated luminosity £ int = 100f b −1 for the LHC with the c.m. energy √ s = 14T eV , then there will be 4 × 10 2 − 1.22 × 10 4 events to be generated at the LHC. Using the relevant F eynman rules, we can write the invariant amplitude for the par- tonic process q(p 1 )q(p 2 ) → φ 0 (p 3 )h(p 4 ) as M 2 (q) = M 21 (q), f or q = u, c M 2 (q) = M 22 (q), f or q = d, s q q φ 0 h Z, Z H (a) (b) b b φ 0 b h g g t(T ) (d) t(T ) t(T ) t(T ) φ 0 h g g φ 0 t(T ) (c) t(T ) t(T ) φ 0 h g g Z, Z H t(T ) (e) t(T ) t(T )M 21 (q) = ( −e 2 x 24S 2 W C 2 W + e 2 x 18C 2 W )v(p 2 ) p 12 / p 2 12 − m 2 Z P L u(p 1 ) + e 2 x 18C 2 Wv (p 2 ) p 12 / p 2 12 − m 2 Z P R u(p 1 ) + e 2 x(14 − 17S 2 W ) 108C 2 W cos2θ Wv (p 2 ) p 3 / p 2 12 − m 2 Z H P L u(p 1 ) + −e 2 x(4 − S 2 W ) 108C 2 W cos2θ Wv (p 2 ) p 3 / p 2 12 − m 2 Z H P R u(p 1 ) + e 2 x(1 − 3S 2 W )(14 − 17S 2 W ) 216S 2 W C 2 W cos2θ Wv (p 2 ) p 4 / p 2 12 − m 2 Z H P L u(p 1 ) + −e 2 x(4 − S 2 W )(1 − 3S 2 W ) 216S 2 W C 2 W cos2θ Wv (p 2 ) p 4 / p 2 12 − m 2 Z H P R u(p 1 ). For the s-channel partonic processes qq → Z(Z H ) → φ 0 h (q = d, s and b), the invariant amplitude can be written M 22 (q) = ( e 2 x 24S 2 W C 2 W − e 2 x 18C 2 W )v(p 2 ) p 12 / p 2 12 − m 2 Z P L u(p 1 ) + e 2 x 36C 2 Wv (p 2 ) p 12 / p 2 12 − m 2 Z P R u(p 1 ) + e 2 x(14 − 17S 2 W ) 108C 2 W cos2θ Wv (p 2 ) p 3 / p 2 12 − m 2 Z H P L u(p 1 ) + e 2 x(4 − S 2 W )(3 − 5S 2 W ) 108S 2 W C 2 W cos2θ Wv (p 2 ) p 3 / p 2 12 − m 2 Z H P R u(p 1 ) + −e 2 x(4 − 3S 2 W ) 108C 2 W cos2θ Wv (p 2 ) p 4 / p 2 12 − m 2 Z H P L u(p 1 ) + −e 2 x(4 − S 2 W )(3 − 5S 2 W ) 108S 2 W C 2 W cos2θ Wv (p 2 ) p 4 / p 2 12 − m 2 Z H P R u(p 1 ). For the t-channel partonic process bb → φ 0 h as shown in F ig.7b, the invariant amplitude can be written M 23 (q) = m 2 b √ 2vfv (p 2 ) p 13 / + m b p 2 13 − m 2 b γ 5 u(p 1 ). Where of m φ 0 in F ig.9. One can see from F ig.9 that the total cross section σ is sensitive to mass parameter m φ 0 . For f = 500GeV and 120GeV ≤ m φ 0 ≤ 180GeV , its value is in the range of 101f b − 306f b. From the above discussions, we can see that the decay features of φ 0 are similar to those of the SM-like neutral Higgs boson h, as far as decays into bb and γγ are concerned. Therefore, when we analyze the signatures of the neutral Higgs boson pairs from the LRT H model at the colliders, we will take the φ 0 φ 0 pair, for example. to bb and the other decays to γγ, then pair production of the neutral Higgs boson φ 0 at the LHC can give rise to the bbγγ final state, and the production rate of the bbγγ final state can be easily estimated using the formula σ s = σ × Br(φ 0 → bb) × Br(φ 0 → γγ). If we assume the integrated luminosity £ int = 100f b −1 for the LHC with the c.m. energy √ s = 14T eV , then there will be several hundreds of bbγγ events to be generated at the LHC. Furthermore, the narrow γγ peak can be reconstructed to distinguish the signal from the backgrounds. Detailed analysis of the signals and the relevant backgrounds about this kind of the final state has been given in Ref. [36]. V. Conclusions The twin Higgs mechanism provides an alternative method to solve the little hierarchy problem. The LRT H model is a concrete realization of the twin Higgs mechanism. In this paper, we discuss the possible decay modes of the neutral Higgs boson φ 0 predicted by the LRT H model and consider its pair production at the ILC and LHC via suitable mechanisms. At the ILC, we study production of the neutral Higgs boson pair φ 0 h via the processes e + e − → Z(Z H ) → φ 0 h. Our numerical results show that, for m φ 0 = m h = 120GeV and 500GeV ≤ f ≤ 1500GeV , the total production cross section of neutral Higgs boson pair φ 0 h at ILC is in the range of 0.92f b − 20f b. If we assume the integrated luminosity £ int = 500f b −1 for the ILC with the c.m. energy √ s = 500GeV , there will be 10 2 − 10 4 φ 0 h events to be generated at the ILC. If we assume that the neutral Higgs bosons φ 0 and h both decay to bb, then the process e + e − → φ 0 h can give rise to the bbbb final state. There will be 9 − 3.0 × 10 3 bbbb events to be generated at the ILC. Owing to the bbbb events, we might detect the possible signatures of the neutral Higgs boson φ 0 via the processes e + e − → Z(Z H ) → φ 0 h in the future ILC experiments. At the LHC, we study production of the neutral Higgs boson pairs φ 0 φ 0 and φ 0 h. First, we study production of the neutral Higgs boson pair φ 0 φ 0 via the processes gg → φ 0 φ 0 and qq → φ 0 φ 0 . Our numerical results show that, for M = 100GeV , m φ 0 = 120GeV and 500GeV ≤ f ≤ 1500GeV , the value of the hadronic cross section σ φ 0 φ 0 is in the range of 4f b − 122f b, which mainly comes from the contributions of the top-quark loop. Then we study production of the neutral Higgs boson pair φ 0 h via the processes qq → φ 0 h(q = u, c, d, s, b) and gg → φ 0 h. Our numerical results show that, for M = 150GeV , m φ 0 = m h = 120GeV and 500GeV ≤ f ≤ 1500GeV , the value of σ φ 0 h is in the range of 34f b − 306f b, of which about 91% of the contributions comes from light quarks u, d, c, s. If we assume the integrated luminosity £ int = 100f b −1 for the LHC with the c.m. energy √ s = 14T eV , then there will be 3.4 × 10 3 − 3.1 × 10 4 φ 0 h events to be generated at the LHC. If we assume that one of the neutral Higgs bosons φ 0 and h decays to bb and the other decays to γγ, then the processes pp → φ 0 φ 0 + X and pp → φ 0 h + X all can give rise to the bbγγ final state. There will be several hundreds and up to thousands of bbγγ events to be generated at the LHC with the c.m. energy √ s = 14T eV and £ int = 100f b −1 . model contains the U(4) 1 × U(4) 2 global symmetry as well as the gauged symmetry SU(2) L × SU(2) R × U(1) B−L . After Higgs obtained vacuum expectation values (f ,f ), the global symmetry U(4) 1 × U(4) 2 breaks down to U(3) 1 × U(3) 2 , and the gauge group SU(2) R × U(1) B−L breaks down to the SM U(1) Y . Thus, the LRT H model predicts the existence of the new particles, such as heavy gauge bosons, heavy scalars, and the top partner T , which can generate rich phenomenology at present and in future collider experiments focus our attention on the neutral Higgs bosons. The LRT H model is based on the global U(4) 1 × U(4) 2 symmetry with a locally gauged subgroup SU(2) L × SU(2) R × U(1) B−L . Two Higgs fields, H = (H L , H R ) and H = (Ĥ L ,Ĥ R ), are introduced and each transforms as (4, 1) and (1, 4), respectively, under the global symmetry. H L,R (Ĥ L,R ) are two component objects which are charged under SU(2) L and SU(2) R , respectively. For the gauge couplings g 2L and g 2R of SU(2) L and SU(2) R , the left-right symmetry implies that g 2L = g 2R = g 2 . The U(4) 1 [U(4) 2 ] group is spontaneously broken down to its subgroup U(3) 1 [U(3) 2 ] with nonzero vacuum expectation value (V EV ) < H > = (0, 0, 0, f ) [<Ĥ > = (0, 0, 0,f )]. The Higgs V EV s also break SU(2) R × U(1) B−L down to the SM U(1) Y . After spontaneous global symmetry breaking by f andf , three Goldstone bosons are eaten by the new gauge bosons W ± H and Z H . After the SM electroweak symmetry breaking, the three additional Goldstone bosons are eaten by the SM gauge bosons W ± and Z. tions of the fields, there are left with four Higgs bosons in the LRT H spectrum that couple to both the fermion sector and the gauge boson sector. They are one neutral Higgs bosons φ 0 , a pair of charged Higgs bosons φ ± , and the SM-like physical Higgs h. In addition, there is an SU(2) L doubletĥ = (ĥ + 1 , h 0 2 ) that couples to the gauge boson sector only (including the SM gauge bosons γ, Z, W , and the new gauge boson Z H ). The lightest particle inĥ, typically one of the neutral components, is stable, and therefore constitutes a good dark matter candidate. These neutral Higgs bosons can couple to each others, and also can couple to the ordinary fermions, ordinary gauge bosons, new top quark T, and new gauge boson Z H . Figure 1 : 1The branching ratios of the neutral Higgs boson φ 0 for different decay modes as functions of the free parameter f for M = 150GeV , m φ 0 = 120GeV . In order to see the trend clearly, we have multiplied Br(φ 0 → cc), Br(φ 0 → τ + τ − ), and Br(φ 0 → γγ) by the factors 10, 20, and 300, respectively.We summed up our numerical results of the branching ratios of the neutral Higgs boson φ 0 for different decay modes Br(φ 0 ) in F ig.1. To get the numerical results, the SM parameters involved are taken as m b = 4.8GeV , m c = 1.25GeV and m τ = 1.78GeV[23]. In F ig.1, we plot Br(φ 0 ) as a function of free parameter f for M = 150GeV φ 0 φ 0 Figure 2 : 02cannot be produced exclusively at the ILC because φ 0 φ 0 can not couple with gauge boson Z or Z H . However, the neutral Higgs boson pair φ 0 h can be produced via the processes e + e − → Z(Z H ) → φ 0 h at the ILC. The F eynman diagrams of the process e + (p 1 )e − (p 2 ) → φ 0 (p 3 )h(p 4 ) are shown in F ig.Feynman diagrams for the process e + e − → φ 0 h. α = 1 Figure 3 : 13/128.8, S W = √ 0.2315, m Z = 91.1876GeV , m h = 120GeV [23], the cross section σ of pair production for the neutral Higgs boson φ 0 h at the ILC is dependent on the model dependent parameters f and m φ 0 . In our numerical estimation, we will assume that the values of the free parameters f and m φ 0 are in the ranges of 500GeV − 1500GeV and 100GeV − 180GeV , respectively. In F ig.3, we plot the production cross section σ of the process e + e − → φ 0 h as a function of the scale parameter f for the c.m. energy √ s = 500GeV , m h = 120GeV and three values of m φ 0 . We can see that σ is sensitive to the scale parameter f and the mass parameter m φ 0 . For 500GeV ≤ f ≤ 1500GeV and 120GeV ≤ m φ 0 ≤ 180GeV , its value is in the range of 0.92f b−20f b. According to an update of parameter for ILC at 2006 [33] 1 , one can see that, an integrated luminosity of 500f b −1 should be achieved in the first four years of running after one year of commissioning. Therefore, if we assume the integrated luminosity for the ILC is 500f b −1 , there will be 10 2 − 10 4 φ 0 h events to be generated at the ILC. The production cross section σ of e + e − → φ 0 h as a function of the parameter f for three values of m φ 0 , m h = 120GeV , and the c.m. energy √ s = 500GeV . From the discussions given in Sec. II, we can see that the possible decay modes of the neutral Higgs boson φ 0 are bb, cc, τ + τ − , gg and γγ. The SM-like neutral Higgs boson h has similar decay features with those of φ 0 . Therefore, the signatures of neutral Higgs boson pair φ 0 h is similar to those of the neutral Higgs boson pair φ 0 φ 0 at the high energy colliders. From the numerical results given in Sec. II, one can see that, for the masses m φ 0 ≤ 180GeV , the possible signals of φ 0 h can be seen as four b quarks, we might detect the possible signatures of the neutral Higgs boson φ 0 via the process e + e − → φ 0 h in the future ILC experiments. Certainly, detailed confirmation of the observability of the signals generated by the process e + e − → Z(Z H ) → φ 0 h would require Monte-Carlo simulations of the signals and backgrounds, which is beyond the scope of this paper. First , we study production of the neutral Higgs boson pair φ 0 φ 0 at the LHC. At the LHC, the neutral Higgs boson pair φ 0 φ 0 can be produced through two mechanisms.One is loop-induced production via gluon fusion (gg → φ 0 φ 0 ) and the other is from the t-channel quark-antiquark annihilation (qq → φ 0 φ 0 ). The relevant F eynman diagrams are shown in F ig.4. Considering the couplings of the neutral Higgs boson φ 0 to the SM fermions are proportional to the factor of m q /f and the smallness masses of the quarks q = u, c, d, and s, we have neglected their contributions to production of the neutral Higgs boson pair φ 0 φ 0 . Figure 4 : 4One-loop F eynman diagrams for the subprocess gg → φ 0 φ 0 (a,b) and treelevel F eynman diagram for the subprocess bb → φ 0 φ 0 (c) in the LRT H model. The diagrams obtained by exchanging the two gluons or exchanging the two Higgs bosons are not shown here. It is obvious that the production cross section σ of the neutral Higgs boson pair φ 0 φ 0 at the LHC are dependent on the model dependent parameters f , m φ 0 , and M. Similar to the calculation at the ILC, we assume that the values of the free parameters f and m φ 0 are in the ranges of 500GeV − 1500GeV and 100GeV − 180GeV , respectively. Besides, we assume the mixing parameter M is in the range of 100GeV − 200GeV . Our numerical results are summarized in F igs.5 and 6.To see contributions of the different partonic processes to the total hadronic cross section, we plot the total and partial hadronic cross sections for different partonic processes as functions of the scale parameter f for the parameters M = 100GeV and m φ 0 = 120GeV in F ig.5. We see from F ig.5 that production of the neutral Higgs boson pair φ 0 φ 0 is dominated by the partonic process gg → φ 0 φ 0 induced by the top-quark loop diagrams. Figure 5 :Figure 6 : 56The total and partial hadronic cross sections for different partonic processes as functions of the free parameter f for the parameters M = 100GeV andm φ 0 = 120GeV .In order to see the effects of the mass parameter m φ 0 on the total cross section σ, we plot σ as a function of m φ 0 for f = 500GeV and three values of the mixing parameter M in F ig.6. One can see from F ig.6 that the total cross section σ is sensitive to the mass parameter m φ 0 , while is not sensitive to the mixing parameter M. This is because M is introduced to generate the mass mixing term Mq L q R , which is included in the gauge invariant top Y ukawa terms allowed by gauge invariance. From the relevant F eynman rules we can see that, the mixing parameter M does not influence the production crosssection σ of the neutral Higgs boson φ 0 too much. For f = 500GeV , M = 200GeV , and m φ 0 = 100GeV − 180GeV , the total cross section σ is in the range of 16f b − 253f b. The total production cross section σ as a function of free parameter m φ 0 for three values of mixing parameter M. B. φ 0 h pair production Now we consider production of the neutral Higgs boson pair φ 0 h at the LHC. At the LHC, the neutral Higgs boson pair φ 0 h can be mainly produced through two mechanisms: (i) qq → φ 0 h, where q = u, d, c, s, b; (ii) the loop-induced gluon fusion process gg → φ 0 h. The relevant F eynman diagrams are shown in F ig.7. Figure 7 : 7Tree-level F eynman diagrams for the process qq → φ 0 h(q = u, d, c, s, b) (a,b) and one-loop F eynman diagrams for gg → φ 0 h (c,d,e) in the LRT H model. For the s-channel partonic processes qq → Z(Z H ) → φ 0 h (q = u and c), the invariant amplitude can be written p 13 = p 1 − p 3 . Considering the couplings of the neutral Higgs boson φ 0 to the SM fermions are proportional to the factor of m q /f and the smallness masses of the quark q = u, c, d, and s, we have neglected their contributions to production cross section of the neutral Higgs boson pair φ 0 h via the t-channel process in our calculations. When we calculate the loop diagrams F igs.7(c)-7(d), and F ig.7e, we will use the same method with F igs.4(a) and 4(b). To see contributions of the different partonic processes to the total hadronic cross section, we plot the total and partial hadronic cross sections for different partonic processes as functions of the parameter f for m φ 0 = m h = 120GeV and M = 150GeV in F ig.8. We see that the production cross sections of the neutral Higgs bosons φ 0 h mainly come from the contributions of the light quarks (u, d, c, s) through the s-channel Z exchange and Z H exchange. Our numerical results show that, the contributions coming from the partonic processes gg → φ 0 h [including F igs.7(c)-7(e)] to total production cross section are at the orders of 10 −5 f b − 10 −1 f b, which are much smaller than those of the tree-level processes.This is because the Y ukawa couplings depend sensitively on the free parameters M and f . The parameter M is very smaller than the scale parameter f . So, although the gluon fusion get an enhancement due to large parton distribution functions, the contribution of the gluon fusion process is suppressed by the order of (M/f ) 4[21]. Thus, in F ig.8, we did not show the line corresponding to the value of the production cross section contributed by the gg fusion. The value of the production cross section of the neutral Higgs bosons φ 0 h is insensitive to the mixing parameter M. For m φ 0 = m h = 120GeVand 500GeV ≤ f ≤ 150GeV , its value is in the range of 34f b − 306f b, the partial value of the total production cross section coming from light quarks contributions is in the range of 31f b − 281f b. If we assume the integrated luminosity £ int = 100f b −1 for the LHC with the c.m. energy √ s = 14T eV , then there will be 3.4 × 10 3 − 3.1 × 10 4 φ 0 h events generated at the LHC. Figure 8 : 8The total and partial hadronic cross sections for different partonic processes as function of the parameter f for m φ 0 = m h = 120GeV and M = 150GeV . Similar to those of the discussions for neutral Higgs boson pair φ 0 φ 0 production, we plot σ as a function of free parameter f for m h = 120GeV , M = 150GeV and three values Figure 9 : 9The total production cross section as a function of free parameter f for m h = 120GeV , M = 150GeV and three values of m φ 0 . In most of the parameter space of the LRT H model, the main decay modes of φ 0 are gg and bb. 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{'abstract': 'In the framework of the left-right twin Higgs model, we study pair production of the neutral Higgs bosons at the International Linear Collider (ILC) and the CERN LHC. We find that the production cross section of the process e + e − → φ 0 h are at the level of several tens f b at the ILC, the production cross section of the φ 0 φ 0 pair and φ 0 h pair are at the level of several hundreds f b at the LHC. As long as the neutral Higgs boson φ 0 is not too heavy, we conclude that its pair production might be used to test for the left-right twin Higgs model at the LHC experiment or in the future ILC experiment.', 'arxivid': '0905.0597', 'author': ['Wei Ma \nDepartment of Physics\nLiaoning Normal University\n116029DalianP. R. China\n', 'Chong-Xing Yue \nDepartment of Physics\nLiaoning Normal University\n116029DalianP. R. China\n', 'Yong-Zhi Wang \nDepartment of Physics\nLiaoning Normal University\n116029DalianP. R. China\n'], 'authoraffiliation': ['Department of Physics\nLiaoning Normal University\n116029DalianP. R. China', 'Department of Physics\nLiaoning Normal University\n116029DalianP. R. China', 'Department of Physics\nLiaoning Normal University\n116029DalianP. R. China'], 'corpusid': 16819155, 'doi': '10.1103/physrevd.79.095010', 'github_urls': [], 'n_tokens_mistral': 15356, 'n_tokens_neox': 12397, 'n_words': 8044, 'pdfsha': 'ca649ff5ac5be118c23cdb751f57b09c1774dab9', 'pdfurls': ['https://arxiv.org/pdf/0905.0597v2.pdf'], 'title': ['Pair production of neutral Higgs bosons from the left-right twin Higgs model at the ILC and LHC', 'Pair production of neutral Higgs bosons from the left-right twin Higgs model at the ILC and LHC'], 'venue': []}

arxiv

IMBALANCED SEMI-SUPERVISED LEARNING WITH BIAS ADAPTIVE CLASSIFIER Renzhen Wang [email protected] Xi'an Jiaotong University Xixi Jia Xidian University Quanziang Wang Xi'an Jiaotong University Yichen Wu City University of Hong Kong Deyu Meng [email protected] Xi'an Jiaotong University Macau University of Science and Technology Peng Cheng Laboratory IMBALANCED SEMI-SUPERVISED LEARNING WITH BIAS ADAPTIVE CLASSIFIER Published as a conference paper at ICLR 2023 Pseudo-labeling has proven to be a promising semi-supervised learning (SSL) paradigm. Existing pseudo-labeling methods commonly assume that the class distributions of training data are balanced. However, such an assumption is far from realistic scenarios and thus severely limits the performance of current pseudolabeling methods under the context of class-imbalance. To alleviate this problem, we design a bias adaptive classifier that targets the imbalanced SSL setups. The core idea is to automatically assimilate the training bias caused by class imbalance via the bias adaptive classifier, which is composed of a novel bias attractor and the original linear classifier. The bias attractor is designed as a light-weight residual network and optimized through a bi-level learning framework. Such a learning strategy enables the bias adaptive classifier to fit imbalanced training data, while the linear classifier can provide unbiased label prediction for each class. We conduct extensive experiments under various imbalanced semi-supervised setups, and the results demonstrate that our method can be applied to different pseudo-labeling models and is superior to current state-of-the-art methods. Raffel. Mixmatch: A holistic approach to semi-supervised learning. NeurIPS, 32, 2019b. Paula Branco, Luís Torgo, and Rita P Ribeiro. A survey of predictive modeling on imbalanced domains. ACM Computing Surveys (CSUR), 49(2):1-50, 2016. Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance problem in convolutional neural networks. -supervised learning (chapelle, o. et al., eds.; 2006)[book reviews]. IEEE Transactions on Neural Networks, 20(3):542-542, 2009. . Smote: synthetic minority over-sampling technique. . Model-agnostic meta-learning for fast adaptation of deep networks. INTRODUCTION Semi-supervised learning (SSL) (Chapelle et al., 2009) has proven to be promising for exploiting unlabeled data to reduce the demand for labeled data. Among existing SSL methods, pseudo-labeling (Lee et al., 2013), using the model's class prediction as labels to train against, has attracted increasing attention in recent years. Despite the great success, pseudo-labeling methods are commonly based on a basic assumption that the distribution of labeled and/or unlabeled data are class-balanced. Such an assumption is too rigid to be satisfied for many practical applications, as realistic phenomena always follows skewed distributions. Recent works (Hyun et al., 2020;Kim et al., 2020a) have found that class-imbalance significantly degrades the performance of pseudo-labeling methods. The main reason is that pseudo-labeling usually involves pseudo-label prediction for unlabeled data, and an initial model trained on imbalanced data easily mislabels the minority class samples as the majority ones. This implies that the subsequent training with such biased pseudo-labels will aggravate the imbalance of training data and further bias the model training. To address the aforementioned issues, recent literature attempts to introduce pseudo-label rebalancing strategies into existing pseudo-labeling methods. Such a re-balancing strategy requires the class distribution of unlabeled data as prior knowledge (Wei et al., 2021;Lee et al., 2021) or needs to estimate the class distribution of the unlabeled data during training (Kim et al., 2020a;Lai et al., 2022). However, most of the data in imbalanced SSL are unlabeled and the pseudo-labels estimated by SSL algorithms are unreliable, which makes these methods sub-optimal in practice, especially when there are great class distribution mismatch between labeled and unlabeled data. In this paper, we investigate pseudo-labeling SSL methods in the context of class-imbalance, in which class distributions of labeled and unlabeled data may differ greatly. In such a general scenario, the current state-of-the-art FixMatch (Sohn et al., 2020) may suffer from performance degradation. To illustrate this, we design an experiment where the entire training data (labeled data + unlabeled Published as a conference paper at ICLR 2023 data) are balanced yet the labeled data are imbalanced, as shown in Fig. 1(a). Then, we can obtain an upper bound model by training the classification network on the whole training data (the underlying true labels of unlabeled data are given during training) and a lower bound model (trained with the labeled data only). As shown in Fig. 1(b)(c), the performance of FixMatch is much worse than the upper bound model, and degrades significantly from the majority classes to the tail classes , which instantiates the existence of imbalance bias. Meanwhile, for the last two tail classes, the performance of FixMatch is even worse than the lower bound model, which indicates the existence of pseudolabel bias brought by inaccurate pseudo-labels and it further deteriorates the model training. To address this problem, we propose a learning to adapt classifier (L2AC) framework to protect the linear classifier of deep classification network from the training bias. Specifically, we propose a bias adaptive classifier which equips the linear classifier with a bias attractor (parameterized by a residual transformation). The linear classifier aims to provide an unbiased label prediction and the bias attractor attempts to assimilate the training bias arising from class imbalance. To this end, we learn the L2AC with a bi-level learning framework: the lower-level optimization problem updates the modified network with bias adaptive classifier over both labeled and unlabeled data for better representation learning; the upper-level problem tunes the bias attractor over an online class-balanced set (re-sampled from the labeled training data) for making the linear classifier predict unbiased labels. As a result, the bias adaptive classifier can not only fit the biased training data but also make the linear classifier generalize well towards each class (i.e., tend to equal preference to each class). In Fig. 1(c), we show that the linear classifier learned by our L2AC can well approximate the predicted class distribution of the upper bound model, indicating that L2AC obtains an unbiased classifier. In summary, our contributions are mainly three-fold: (1) We propose to learn a bias adaptive classifier to assimilate online training bias arising from class imbalance and pseudo-labels. The proposed L2AC framework is model-agnostic and can be applied to various pseudo-labeling SSL methods; (2) We develop a bi-level learning paradigm to optimize the parameters involved in our method. This allows the online training bias to be decoupled from the linear classifier such that the resulting network can generalize well towards each class. (3) We conduct extensive experiments on various imbalanced SSL setups, and the results demonstrate the superiority of the proposed method. The source code is made publicly available at https://github.com/renzhenwang/bias-adaptive-classifier. RELATED WORK Class-imbalanced learning attempts to learn the models that generalize well to each classes from imbalanced data. Recent studies can be divided into three categories: Re-sampling (He & Garcia, 2009;Chawla et al., 2002;Buda et al., 2018;Byrd & Lipton, 2019) that samples the data to rearrange the class distribution of training data; Re-weighting (Khan et al., 2017;Cui et al., 2019;Cao et al., 2019;Lin et al., 2017;Ren et al., 2018;Shu et al., 2019;Tan et al., 2020;Jamal et al., 2020) that assigns weights for each class or even each sample to balance the training data; Transfer learning (Wang et al., 2017;Liu et al., 2019;Yin et al., 2019;Kim et al., 2020b;Chu et al., 2020;Liu et al., 2020;) that transfers knowledge from head classes to tail classes. Besides, most recent works tend to decouple the learning of representation and classifier (Kang et al., 2020;Zhou et al., 2020;Tang et al., 2020;Zhang et al., 2021b). However, it is difficult to directly extend these techniques to imbalanced SSL, as the distribution of unlabeled data is unknown and may be greatly different from that of labeled data. Semi-supervised learning targets to learn from both labeled and unlabeled data, which includes two main lines of researches, namely pseudo-labeling and consistency regularization. Pseudo-labeling (Lee et al., 2013;Xie et al., 2020a;b;Sohn et al., 2020;Zhang et al., 2021a) is evolved from entropy minimization (Grandvalet & Bengio, 2004) and commonly trains the model using labeled data together with unlabeled data whose labels are generated by the model itself. Consistency regularization (Sajjadi et al., 2016;Tarvainen & Valpola, 2017;Berthelot et al., 2019b;Miyato et al., 2018;Berthelot et al., 2019a) aims to impose classification invariance loss on unlabeled data upon perturbations. Despite their success, most of these methods are based on the assumption that the labeled and unlabeled data follow uniform label distribution. When used for class-imbalance, these methods suffer from significant performance degradation due to the imbalance bias and pseudo-label bias. Imbalanced semi-supervised learning has been drawing extensive attention recently. Yang & Xu (2020) Oh et al. (2022) proposed to blend the pseudolabels from the linear classifier with those from a similarity-based classifier. Guo & Li (2022) found a fixed threhold for pseudo-labeled sample selection biased towards head classes and in turn proposed to optimize an adaptive threhold for each class. Assumed that labeled and unlabeled data share the same distribution, Wei et al. (2021) proposed a re-sampling method to iteratively refine the model, and Lee et al. (2021) proposed an auxiliary classifier combined with re-sampling technique to mitigate class imbalance. Most recently, Wang et al. (2022) proposed to combine counterfactual reasoning and adaptive margins to remove the bias from the pseudo-labels. Different from these methods, this paper aims to learn an explicit bias attractor that could protect the linear classifier from the training bias and make it generalize well towards each class. METHODOLOGY PROBLEM SETUP AND BASELINES Imbalanced SSL involves a labeled dataset D l = {(x n , y n )} N n=1 and an unlabeled dataset D u = {x m } M m=1 , where x n is a training example and y n ∈ {0, 1} K is its corresponding label. We denote the number of training examples of class k within D l and D u as N k and M k , respectively. In a classimbalanced scenario, the class distribution of the training data is skewed, namely, the imbalance ratio γ l := max k N k min k N k 1 or γ u := max k M k min k M k 1 always holds. Note that the class distribution of D u , i.e., {M k } K k=1 is usually unknown in practice. Given D l and D u , our goal is to learn a classification model that is able to correctly predict the labels of test data. We denote a deep classification model Ψ = f cls φ • f ext θ with the feature extractor f ext θ and the linear classifier f cls φ , where θ and φ are the parameters of f ext θ and f cls φ , respectively, and • is function composition operator. With pseudo-labeling techniques, current state-of-the-art SSL methods (Xie et al., 2020b;Sohn et al., 2020;Zhang et al., 2021a) generate pseudo-labels for unlabeled data to augment the training dataset. For unlabeled sample x m , its pseudo-labelŷ m can be a 'hard' one-hot label (Lee et al., 2013;Sohn et al., 2020;Zhang et al., 2021a) or a sharpened 'soft' label (Xie et al., 2020a;Wang et al., 2021). The model is then trained on both labeled and pseudo-labeled samples. Such a learning scheme is typically formulated as an optimization problem with objective function L = L l + λ u L u , where λ u is a hyper-parameter for balancing labeled data loss L l and pseudo-labeled data loss L u . To be more specific, L l = 1 |D l | xn∈D l H (Ψ(x n ), y n ) , whereD l denotes a batch of labeled data sampled from D l , H is cross-entropy loss; L u = 1 |Du| xm∈Du 1(max(p m ) ≥ τ )H (Ψ(x m ),ŷ m ) , where p m = softmax(Ψ(x m )) represents the output probability, and τ is a predefined threshold for masking out inaccurate pseudo-labeled data. For simplicity, we reformulate L as L = 1 |D l | x i ∈D l H(Ψ(xi), yi) + 1 |Du| x i ∈Du λiH(Ψ(xi),ŷi),(1) where λ i = λ u 1(max(p i ) ≥ τ ). The pseudo-labeling framework has achieved remarkable success in standard SSL scenarios. However, under class-imbalanced setting, the model is easily biased during training, such that the generated pseudo-labels can be even more biased and severely degrades the performance of minority classes. Moreover, due to the confirmation bias issue (Tarvainen & Valpola, 2017;Arazo et al., 2020), the model itself is hard to rectify such a training bias. LEARNING TO ADAPT CLASSIFIER Our goal is to enhance the existing pseudo-labeling SSL methods by making full use of both labeled and unlabeled data, while protecting the linear classifier from the training bias (imbalance bias and pseudo-label bias). To this end, we design to learn a bias adaptive classifier that equips the linear classifier with a bias attractor in order to assimilate complicated training bias. The proposed bias adaptive classifier: As shown in Fig. 2. The bias adaptive classifier (denoted as F ) consists of two modules: the linear classifier f cls φ and a nonlinear network ∆f w (dubbed bias attractor). The bias attractor is implemented by imposing a residual transformation on the output of the linear classifier, i.e., plugging ∆f w after f cls φ and then bridging their outputs with a shortcut connection. Mathematically, the bias adaptive classifier F can be formulated as F ω,φ (z) = (I + ∆f ω ) • f cls φ (z),(2) where z = f ext θ (x) ∈ R d , I denotes identity mapping, and ∆f w with parameters ω is a multi-layer perceptron (MLP) with one hidden layer in this paper. We design such a bias adaptive classifier for the following two considerations. On the one hand, the bias attractor ∆f w adopts a nonlinear network which can assimilate complicated training bias in theory due to the universal approximation properties 1 . Since the whole bias adaptive classifier (i.e., classifier with the bias attractor) is required to fit the imbalanced training data, we hope the bias attractor could indeed help the linear classifier to learn the unbiased class conditional distribution (i.e., let the classifier less effected by the biases.). By contrast, in the original classification network, a single linear classifier is required to fit biased training data such that it is easily misled by class imbalance and biased pseudo-labels. On the other hand, the residual connection conveniently makes the bias attractor a plug-in module, i.e., assimilate the training bias during training and be removed in the test stage, and it also has been proven successful in easing the training of deep networks (He et al., 2016;Long et al., 2016). Learning bias adaptive classifier: With the proposed bias adaptive classifier F ω,φ , the modified classification network can be formulated asΨ = F ω,φ • f ext θ . To make full use of the whole training data (D l ∪ D u ) for better representation learning, we can minimize the following loss function: L = 1 |D l | x i ∈D l H(Ψ(xi), yi) + 1 |Du| x i ∈Du λiH(Ψ(xi),ŷi).(3) This involves an optimization problem with respect to three parts of parameters {θ, φ, ω}, which can be jointly optimized via the stochastic gradient decent (SGD) in an end-to-end manner. However, such a training strategy poses a critical challenge: there is no prior knowledge on f cls φ to predict unbiased label prediction and on ∆f w to assimilate the training bias. In other words, we cannot guarantee the training bias to be exactly decoupled from the linear classifier f cls φ . To address this problem, we take ω as hyper-parameters associated with φ and design a bi-level learning algorithm to jointly optimize the network parameters {θ, φ} and hyper-parameters ω. We illustrate the process in Fig. 2 and Algorithm 1. In each training iteration t, we update the network parameters {θ, φ} by gradient descent as (θ t+1 , φ t+1 (ω)) = (θ t , φ t ) − α∇ θ,φ L,(4) where α is the learning rate. Note that we herein assume that ω is only directly related to the linear classifier f cls φ via φ t+1 (ω), which implies that the subsequent optimization of ω will not affect the feature extractor f ext θ . We then tune the hyper-parameters ω to make the linear classifier φ t+1 (ω) generalize well towards each class, and we thus minimize the following loss function of the network Ψ = f cls φ t+1 (ω) •f ext θ t+1 over a class-balanced set (dynamically sampled from the labeled training set): L bal = 1 |B| xi∈B H(f cls φ t+1 (ω) • f ext θ t+1 (x i ), y i ),(5) where B ⊂ D l is a batch of class-balanced labeled samples, which can be implemented by classaware sampling (Shen et al., 2016). This loss function reflects the effect of the hyper-parameter ω on making the linear classifier generalize well towards each class, we thus optimize ω by ω t+1 = ω t − η∇ ω L bal ,(6) where η is the learning rate on ω. Note that in Eq. (6) we need to compute a second-order gradient ∇ ω L bal with respect to ω, which can be easily implemented through popular deep learning frameworks such as Pytorch (Paszke et al., 2019) in practice. In summary, the proposed bi-level learning framework ensures that 1) the linear classifier f cls ω can fit unbiased class-conditional distribution by minimizing the empirical risk over balanced data via Eq. (5) and 2) the bias attractor ∆f ω can handle the implicit training bias by training the bias adaptive classifier F ω,φ over the imbalanced training data by Eq. (3). As such, our proposed ∆f w can protect f cls w from the training bias. Additionally, our L2AC is more efficient than most existing bi-level learning methods (Finn et al., 2017;Ren et al., 2018;Shu et al., 2019). To illustrate this, we herein give a brief complexity analysis of our algorithm. Since our L2AC introduces a bi-level optimization problem, it requires one extra forward passes in Eq. (5) and backward pass in Eq. (6) compared to regular single-level optimization problem. However, in the backward pass, the second-order gradient of ω in Eq. (6) only requires to unroll the gradient graph of the linear classifier f cls φ . As a result, the backward-on-backward automatic differentiation in Eq. (6) demands a lightweight of overhead, i.e., approximately #Params(f cls φ ) #Params(Ψ) × training time of one full backward pass. THEORETICAL ANALYSIS Note that the update of the parameter ω aims to minimize the problem Eq. (5), we herein give a brief debiasing analysis of our proposed L2AC by showing how the value of bias attractor ∆f (·) change with the update of ω. We use notation ∂f ∂ω | ω t to denote the gradient operation of f at ω t and superscript T to denote the vector/matrix transpose. We have the following proposition. Proposition 3.1 Let p i denote the predicted probability of x i , then Eq. (6) can be rewritten as ω t+1 = ω t + ηα 1 n n i=1 G i ∂∆f i ∂ω | ω t ,(7) where G i = ∂(pi−yi) ∂φ | T φ t ( 1 m m j=1 ∂L bal j (θ,φ) ∂φ | φ t+1 ) , which represents the similarity between the gradient of the sample x i and the average gradient of the whole balanced set B. This shows that the update of the parameter ω affects the value of the bias attractor, i.e, the value of ∆f i is adjusted according to the interaction G i between x i and B. If G i > 0 then ∆f i,k is increased, otherwise ∆f i,k is decreased, indicating L2AC adaptively assimilate the training bias. Table 1: Comparison results on CIFAR-10 under two typical imbalanced SSL settings, i.e., γ = γ l = γ u and γ l = γ u (γ l = 100). The performance (bACC / GM) is reported in the form of mean ±std across three random runs. , 1997;Branco et al., 2016) 2 as the evaluation metrics. We evaluate our L2AC under two different settings: 1) both labeled and unlabeled data follow the same class distribution, i.e., γ := γ l = γ u ; 2) labeled and unlabeled data have different class distributions, i.e., γ l = γ u , where γ u is commonly unknown. Methods CIFAR-10 (γ l = γ u ) CIFAR-10 (γ l = γ u ) γ = 100 γ = 150 γ u = 1 (uniform) γ u = 100 (reversed)Vanilla RESULTS ON CIFAR-10 Dataset. We follow the same experiment protocols as Kim et al. (2020a). In detail, a labeled set and an unlabeled set are randomly sampled from the original training data, keeping the number of images for each class to be the same. Then both the two sets are tailored to be imbalanced by randomly discarding training images according to the predefined imbalance ratios γ l and γ u . We denote the number of the most majority class within labeled and unlabeled data as N 1 and M 1 , respectively, and we then have N k = N 1 · γ k l and M k = M 1 · γ k u , where k = k−1 K−1 . We initially set N 1 = 1500 and M 1 = 3000 following Kim et al. Results under γ l = γ u . We evaluate the proposed L2AC based on two widely-used SSL methods: MixMatch (Berthelot et al., 2019b) and FixMatch (Sohn et al., 2020), and compare it with the following methods: 1) The Vanilla model merely trained with labeled data; 2) Recent re-balancing methods that are trained with labeled data by considering class imbalance, including: Re-sampling (Oh et al., 2022). Please refer to Appendix C for more details about these methods. The main results are shown in Table 1. It can be observed that L2AC significantly improves MixMatch and FixMatch at least 9% absolute gain on bACC and at least 14% Table 2: Comparison results on CIFAR-100 and STL-10 under two different imbalance ratios. The performance (bACC / GM) is reported in the form of mean ±std across three random runs. Methods CIFAR-100 (γ l = γ u ) STL-10 (γ u =N/A) γ l = 10 γ l = 20 γ l = 10 γ l = 20 on GM for all settings. This implies that our L2AC benefits the two baselines by learning an unbiased linear classifier. Moreover, our L2AC consistently surpasses all the comparison methods over both evaluation metrics. Take the extremely imbalanced case of γ = 150 for example, compared with the second best comparison method, our L2AC achieves up to 3.6% bACC gain and 4.9% GM gain upon MixMatch, and around 3.0% bACC gain and 3.6% GM gain upon FixMatch. Results under γ l = γ u . The class distribution of labeled and unlabeled data can be arguably different in practice. We herein simulate two typical scenarios following Oh et al. (2022), i.e., the unlabeled set follows an uniform class distribution (γ u = 1) and a reversed long-tailed class distribution against the labeled data (γ u = 100 (reversed)). Note that CReST+ (Wei et al., 2021) fails in this case as they require the class distribution of unlabeled data as prior knowledge for training. As shown in Table 1, L2AC can consistently improve both MixMatch and FixMatch by a large margin. An interesting observation is that the baselines MixMach and FixMatch under γ u = 1 perform much worse than that under γ u = 100, even if more unlabeled data are added for γ u = 1. This is mainly because the models under imbalanced SSL setting have a strong bias to generate incorrect labels for the tail classes, which will impair the entire learning process. As a result, more unlabeled tail class samples under γ u = 1 lead to more severe performance degradation. On the contrary, our L2AC can eliminate the influence of the training bias and predict high-quality pseudolabels for unlabeled data, such that it achieves significant performance gain in both settings. RESULTS ON CIFAR-100 AND STL-10 Dataset: To make a more comprehensive comparison, we further evaluate L2AC on CIFAR-100 (Krizhevsky et al., 2009) andSTL-10 (Coates et al., 2011). For CIFAR-100, we create labeled and unlabeled sets in the same way as described in Sec. 4.1 and set N 1 = 150 and M 1 = 300. STL-10 is a more realistic SSL task that has no distribution information for unlabeled data. In our experiments, we set N 1 = 450 to construct the imbalanced labeled set and adopt the whole unknown unlabeled set (i.e., M = 100k). It is worth noting that the unlabeled set of STL-10 is noisy as it contains samples that do not belong to any of the classes in the labeled set. Results: As shown in Table 2, L2AC achieves the best performance over GM and competitive performance over bACC compared to current state-of-the-art ABC (Lee et al., 2021) and DASO (Oh et al., 2022). This implies that our approach obtains a relatively balanced classification performance towards all the classes. While for SLT-10, a more realistic noisy dataset without distribution information for unlabeled data, L2AC significantly outperforms ABC and DASO on both bACC and GM, which demonstrates that it has greater potential to be applied in the practical SSL scenarios. RESULTS ON LARGE-SCALE SUN-397 Dataset: SUN397 (Xiao et al., 2010) is an imbalanced real-world scene classification dataset, which originally consists of 108,754 RGB images with 397 classes. Following the experimental setups in Results: The experimental results are summarized in Table 3. Compared to the baseline FixMatch (Sohn et al., 2020), our proposed L2AC results in about 4% performance gain over all evaluation metrics, and outperforms all the SOTA methods. This further verifies the efficacy of our proposed method toward the real-world imbalanced SSL applications. DISCUSSION What about the performance under various label ratios? To answer this question, we vary the ratios of labeled data (denoted as β) on CIFAR-10 and STL-10 to evaluate the proposed method. For CIFAR-10, we define β = N 1 /(N 1 + M 1 ) and set the imbalance ratio γ = 100. For STL-10, since it does not provide annotations for unlabeled data, we re-sample the labeled set from labeled data by β = N 1 /500 and set the imbalance ratio γ l = 10. As shown in Table 4, our L2AC consistently improves the baseline across different amounts of labeled data on both CIFAR-10 and STL-10. For example, STL-10 with β = 5% contains very scarce labeled data, where only 25 and 3 labeled samples belong to the most majority and minority classes, respectively. In such an extremely biased scenario, our L2AC significantly improves FixMatch by around 16% over bACC and 37% over GM. How does L2AC perform on the majority/minority classes? To explain the source of performance improvements, we further visualize the confusion matrices on the test set of CIFAR-10 with γ = 100. Noting that the diagonal vector of a confusion matrix represents per-class recall. As shown in Fig. 3, our L2AC provides a relatively balanced per-class recall compared with the baseline Fix-Match (Sohn et al., 2020) and other imbalanced SSL methods, e.g., DARP (Kim et al., 2020a) and ABC (Lee et al., 2021). It can also be observed that FixMatch easily tends to misclassify the samples of the minority classes into the majority classes, while our L2AC largely alleviates this bias. These results reveal that our L2AC provides an unbiased linear classifier for the test stage. Could L2AC improve the quality of pseudo-labels? Qualitative and quantitative experiment results have shown that the proposed L2AC can improve the performance of pseudo-labeling SSL methods under different settings. We attribute this to the fact that L2AC can generate unbiased pseudo-labels during training. To validate this, we show per-class recall of pseudo-labels for CIFAR-10 with γ l = γ u = 100 and γ l = 100, γ u = 100 (reversed) in Fig. 4. It is clear that L2AC significantly raises the final recall of the minority classes. Especially for the situation where the distribution of labeled and unlabeled data are severely mismatched, as shown in Fig. 4(b), our L2AC considerably improves the recall of the most minority class by around 60% upon FixMatch. Such a high-quality pseudo-label estimation probably benefits from a more unbiased classifier with a basically equal preference for each class. How about the learnt linear classifier and feature extractor? We revisit the toy experiment in Section 1 where the whole class-balanced training set is used to train an unbiased upper bound model. Instead, the baseline FixMatch and our L2AC are trained under the standard imbalanced SSL setups, and we compare the predicted class distributions on the test set with that of the upper bound model. As shown in Fig. 1(c . L2AC helps to discriminate tail classes from majority ones. Fig. 5, the features of the tail classes from FixMatch are scattered to the majority classes. However, L2AC can help the model to effectively discriminate the tail classes (e.g., Class 8, 9) from majority classes (e.g., Class 0, 1). Such a high-quality representation learning also benefits from an unbiased classifier during training. In Appendix E.1, we further present the unbiasedness of our L2AC by evaluating it on various imbalanced test sets. Ablation analysis. We conduct an ablation study to explore the contribution of each critical component in L2AC. We experiment with FixMatch on CIFAR-10 under γ l = γ u = 100 and γ l = 100, γ u = 100 (reversed). 1) We first verify whether the bias attractor is helpful. To this end, we apply the proposed bias attractor to FixMatch. It can be seen from Table 5 that the bias attractor helps improve the performance to a certain extent, indicating that the bias attractor is effective yet not significant. As analyzed in Section 3.2, there is no prior knowledge for the bias attractor to assimilate the training bias in this plain training manner. 2) Next, we study how important the role bi-level optimization plays in our method. Instead of using a bi-level learning framework, we disengage the hierarchy structure of our upper-level loss and lower-level loss, and reformulate a single level optimization problem as L + λL bal . The results in Table 5 show that such a degraded version of L2AC provides substantial performance gain over FixMatch w/ bias attractor while still inferior to our L2AC. This demonstrates the effect of the proposed bias adaptive classifier and bi-level learning framework on protecting the linear classifier from the training bias. As shown in CONCLUSION In this work, we propose a bias adaptive classifier to deal with the training bias problem in imbalanced SSL tasks. The bias adaptive classifier is consist of a linear classifier to predict unbiased labels and a bias attractor to assimilate the complicated training bias. It is learned with a bi-level optimization framework. With such a tailored classifier, the unlabeled data can be fully used by online pseudo-labeling to improve the performance of pseudo-labeling SSL methods. Extensive experiments show that our proposed method achieves consistent improvements over the baselines and current state-of-the-arts. We believe that our bias adaptive classifier can also be used for more complex data bias other than class imbalance. A ALGORITHM We give the training algorithm of the proposed L2AC method in Algorithm 1. Algorithm 1 learning to adapt classifier during training Input: labeled / unlabeled training data D l / D u , labeled / unlabeled batch size n / m, max iterations T Output: classification network parameters {θ, φ} 1: Initialize {θ 0 , φ 0 } ← {θ, φ} and ω 0 ← ω. 2: for t = 0 to T do 3:D l = {x i , y i } n i=1 ← SampleMiniBatch(D l , n). 4:D u = {x i } m i=1 ← SampleMiniBatch(D u , m). 5: B = {x i , y i } n i=1 ← SampleMiniBatch(D l , nφ t+1 = φ t − α 1 n n i=1 ∂Li(θ, φ, w) ∂φ | φ t .(8) In consequence, we have ω t+1 = ω t − η∇L bal (θ, φ t+1 )| ω t = ω t + ηα 1 n n i=1 ∂ 2 Li(θ, φ) ∂φ∂ω T | φ t ,ω t ∂L bal (θ, φ) ∂φ | φ t+1 = ω t + ηα 1 n n i=1 ∂∆fi ∂ω | ω t ∂ 2 Li(θ, φ) ∂φ∂∆fi T | φ t ∂L bal (θ, φ) ∂φ | φ t+1 ,(9) Denote by Ξ i = ∂Li(θ,φ) ∂∆fi , then Eq. (9) becomes ω t+1 = ω t + ηα 1 n n i=1 Gi ∂∆fi ∂ω | ω t ,(10) where G i = ∂Ξ i ∂φ | φ t , ∂L bal (θ, φ) ∂φ | φ t+1 . As the training loss is Li(θ, φ) = log d k=1 e z i,k +∆f i,k − zi,c i − ∆fi,c i ,(11) where c i is the class label of the i-th sample. Therefore Ξ i,k =          e z i,k +∆ i,k d s=1 e z i,s +∆ i,s , k = ci e z i,k +∆ i,k d s=1 e z i,s +∆ i,s − 1, k = ci = pi − yi(12) Meanwhile, the upper level loss is defined as L bal (θ, φ) = − 1 m m j=1 L bal i (θ, φ),(13) this finishes the proof. C COMPARISON METHODS To comprehensively evaluate the proposed method, we compare it with the Vanilla model that merely trained with labeled data and three other lines of methods: 1) re-balancing methods where only the class-imbalanced labeled data are used for training, including: Re-sampling (Japkowicz, 2000), LDAM-DRW (Cao et al., 2019) and cRT (Kang et al., 2020). 2) pseudo-labeling based SSL methods where both labeled and unlabeled data is used (without considering class-imbalance), including: Pseudo-labels (Lee et al., 2013), MixMatch (Berthelot et al., 2019b and FixMatch (Sohn et al., 2020). 3) imbalanced semi-supervised learning methods that consider class-imbalance and unlabeled data simultaneously, including: DARP (Kim et al., 2020a), CReST+ (Wei et al., 2021), ABC (Lee et al., 2021), SaR (Lai et al., 2022) and DASO (Oh et al., 2022). We herein give a brief introduction for all the comparison methods. • Vanilla, a plain classification network, e.g., Wide ResNet-28-2 (Oliver et al., 2018), trained with imbalanced labeled data by cross-entropy loss. • Re-sampling, a re-balancing method that uses re-sampling strategy to balance the distribution of training data. • LDAM-DRW, i.e., Label-distribution-aware margin, a re-weighting method where the classifier encourage to maintain large margin for tail classes. • cRT, i.e., Classifier re-training, a two-stage training method that first pretrains the entire network with all the imbalanced training data and re-train the classifier with a balanced objective. • MixMatch, a SSL method which combines pseudo-labeling and consistency regularization techniques via Mixup augmentation (Zhang et al., 2018). • FixMatch, a pseudo-labelling based SSL method of which the strongly augmented unlabeled samples (whose pseudo labels are generated from their weakly augmented versions) are used to train the network. • DARP, a recent state-of-the-art imbalanced SSL method that refines raw pseudo-labels via a convex optimization for alleviating distribution bias arisen by imbalanced and unlabeled training data. • CReST, a pseudo-labeling based imbalanced SSL method that combines re-balancing and distribution alignment techniques to alleviate the training bias. The method assumes that labeled and unlabeled data have roughly the same distribution. • ABC, which equips with two parallel linear classifiers with one fitting the imbalanced data and the other fitting the re-balanced data, and adds the consistency regularization to further improve the performance. • SaR, i.e., self-adaptive refinement, which proposes the concept of mitigating vector that refines the soft labels of unlabeled data before generating the one-hot pseudo labels to alleviate the confirmation bias brought about by unlabeled samples. • DASO, for an unlabeled sample, which combines its pseudo-label from the linear classifier with that from a similarity-based classifier to leverage their complementary properties in terms of bias. Moreover, a semantic alignment loss is proposed to balance the biased feature representation. For fair comparison, we use the same code base 3 as DARP (Kim et al., 2020a). As the training or evaluation protocols of CReST (Wei et al., 2021), ABC (Lee et al., 2021) and DASO (Oh et al., 2022) are different from that of DARP, we reproduce their results according to the official codes (i.e., CReST 4 , ABC 5 and DASO 6 ) released by the authors. Note that the results on CIFAR-100 in DARP (Kim et al., 2020a) are achieved under N 1 = 300, M 1 = 150, while this paper keeps N 1 = 150, M 1 = 300 for satisfying the common assumption that the amount of unlabeled data are larger than that of labeled data. Bias attractor: As aforementioned in Section 3.2, the bias attractor is a light-weight network, i.e., a multi-layer perceptron with one hidden layer in this paper. The input of the bias attractor is the prediction scores output by the linear classifier, so the input dimension is the same as the number of classes. We normalize the input through its L 2 norm or a softmax activation. Concretely, We use softmax operator on CIFAR-10 and STL-10, and L 2 norm on CIFAR-100 and SUN-397 due to its better and more stable performance than softmax operator. Note that the gradients of the input of the bias attractor are stopped in the training stage. The hidden layer dimension of the bias attractor is fixed as 256, which keeps stable and sound results through all our experiments. The parameters of bias attractor are updated by Eq. (6), where the learning rate η is set as 1 × 10 −4 . Baselines: We evaluate our L2AC based on two recent popular SSL methods, i.e., MixMatch (Berthelot et al., 2019b) and FixMatch (Sohn et al., 2020). Both the two baselines are the cornerstone of current state-of-the-art imbalanced SSL methods, such as DARP (Kim et al., 2020a), CReST (Wei et al., 2021) and DASO (Oh et al., 2022). Note that the two baselines can be uniformly formulated as Eq. (1). For MixMatch, the pseudo-label of one unlabeled example is produced by temperature sharpening to the the average prediction of its different augmented versions, and the objective function of unlabeled data is adopted as mean-squared loss (MSE) function. The threshold τ is kept as 0, and λ u is dynamically updated by a linear ramp-up strategy, i.e., λ u linearly increases from 0 to 75 during training. For FixMatch, unlabeled data are augmented by weak and strong augmentations via RandAugment (Cubuk et al., 2020). In particular, the weekly augmented data are used to generate pseudo-labels for the strongly augmented data, and these strongly augmented data are then used to compute the unlabeled loss in Eq. (1). We set τ as 0.95, and λ u as 1 without applying linear ramp-up strategy. D.2 IMPLEMENTATION DETAILS ON SUN397 SUN397 (Xiao et al., 2010) is an imbalanced real-world scene classification dataset, which originally consists of 108,754 RGB images labeled with 397 classes. Following the experimental setups in Kim et al. (2020a), we hold-out 50 samples per each class for testing because no official data split is provided. We then artificially construct the labeled and unlabeled dataset using the remaining dataset according to M k N k = 2. The comparison methods includes: Vanilla, classifier retraining (cRT) (Kang et al., 2020), FixMatch (Sohn et al., 2020), DARP (Kim et al., 2020a) and ABC (Lee et al., 2021). Training details. For pre-processing, we randomly crop and rescale to 224 × 224 size all labeled and unlabeled training images before applying augmentation. We use standard ResNet-34 (He et al., 2016) as our backbone network. During training, the model is trained with Adam optimizer (Kingma & Ba, 2015) with a batch-size of 128 labeled samples and 256 unlabeled samples, and a initial learning rate of 0.002 for 300 training epochs. For fair comparison, all the experiments are based on FixMatch (Sohn et al., 2020). We set unlabeled loss weight λ u as 1.0 and confidence threshold τ as 0.6, and utilize exponential moving average technique with decay rate 0.99. Following Kim et al. (2020a), we adopt RandAugment with random magnitude (Cubuk et al., 2020) for strong augmentation and random horizontal flip for weak augmentation. E ADDITIONAL EXPERIMENTS E.1 EVALUATION ON IMBALANCED TEST SETS Real-world test data is not always following an uniform distribution, and we herein simulate several imbalanced test sets to further study the generalization of the proposed method. As shown in Fig. 6, we construct three imbalanced test sets: (a) Test-1, which follows a long-tailed class distribution with an imbalance ratio of 10; (b) Test-2, which follows a reversed long-tailed class distribution; (c) Test-3, which follows a random class distribution. We evaluate the proposed L2AC upon FixMatch (Sohn et al., 2020) and compare it with the following methods: DARP (Kim et al., 2020a), CReST+ (Wei et al., 2021), ABC (Lee et al., 2021). All these models are trained on CIFAR-10 with imbalanced ratio γ = γ l = γ u = 100. As the evaluation metrics bACC and GM are insensitive to class imbalance of test sets, we add ACC to measure the recognition accuracy for all samples. The results are summarized in Tab. 6. It can be observed that the network learned with our L2AC achieves the best or the second best performance across all the test settings, which further indicates that the proposed bi-level learning framework provides a relatively unbiased classifier compared to other comparison methods. In the main text, we have evaluated the proposed L2AC under long-tailed imbalanced settings with various imbalance ratios. Here, we further validate its generalization in the case of step imbalance on CIFAR-10 under two typical setups, namely γ l = γ u = 100 and γ = 100, γ u = 1. As shown in Fig. 7, step imbalance assumes a more severely imbalanced class distribution than the long-tailed imbalance setups, as there are very scare data for half of the classes. The experimental results are summarized in Tab. 7. We can see that the proposed L2AC achieved the best performance compared with all the comparison methods for all settings. Especially in the case of distribution mismatch between labeled and unlabeled data, L2AC brings significant performance gain. Published as a conference paper at ICLR 2023 E.3 DETAILED ANALYSIS ON THE QUALITY OF PSEUDO-LABELS. Case of γ l = γ u . Fig. 8 visualizes the confusion matrices of pseudo-labels of Fixmatch (Sohn et al., 2020) and our L2AC under the imbalance ratio γ l = γ u = 100. Note that this requires to use true labels that are hidden in the training stage. We can observe that the original pseudolabels are highly biased toward majority classes of the labeled dataset. In contrast, our L2AC tends to a relatively equal per-class recall, especially on minority classes the performance is significantly improved compared to FixMatch (Sohn et al., 2020). This suggests that the proposed method readily improves the quality of pseudo-labels. Case of γ l = γ u . Fig. 9 visualizes the confusion matrices of pseudo-labels for Fixmatch (Sohn et al., 2020) and our L2AC under the imbalance ratio γ l = 100 while γ u = 100 (Reversed). Under this setting, labeled set is unchanged and the total number of the training data remains the same compared to γ l = γ u = 100. Unlabeled data follow a reversed class distribution, which means that more samples are added to the training set for the minority classes. According to the confusion matrix in Fig. 9, FixMatch (Sohn et al., 2020) remains to achieve high recall on the majority classes and low recall on the minority classes. Most samples of the last two tail classes are mislabeled as the two most majority classes. These findings reveal that the pseudo-labeling process becomes more biased as the mismatch of the distributions of labeled and unlabeled data become severe. The confusion matrix of L2AC clearly shows that it can exactly assimilate the training bias through the proposed bias attractor such that the linear classifier is able to predict correct label. E.4 TRAINING CONVERGENCE VERIFICATION To verify the convergence of our proposed L2AC approach, Fig. 10 visualizes the training curves of the lower-level loss Eq. (3) and the upper-level loss Eq. (5) with training iteration increasing from 0 to 2.5 × 10 5 . We can see that both lower-level loss and upper-level loss convergence fast within the first 100 epochs (5 × 10 4 iterations), and the test accuracy curve increases fast at the same time. When the test accuracy reached the peak value, our L2AC roughly remains the same test accuracy until termination, which verifies the robustness of the proposed method. Since the balanced set B is dynamically sampled from the labeled set D l , a nature question is whether the upper-level loss (over B) has the same convergence rate with the lower-level loss (over D l ) during training. To investigate this, we further visualize the curves of these two losses in Fig. 11, and we can observe that the two losses decrease differently and converge to values of different magnitudes at different iterations. E.5 RUNNING COST ANALYSIS In Section 3.2, we conduct a complexity analysis of the training algorithm of our L2AC, which shows that our method is very efficient in theory. To verify this, we herein measure floating point operations per second (FLOPS) using NVIDIA GeForce RTX 3090 to quantify the training cost. We compare our proposed algorithm with the baseline model FixMatch (Sohn et al., 2020) and two stateof-the-art methods (Kim et al., 2020a) and (Lee et al., 2021), as L2AC uses the same code base with these two methods. Besides, we also provide the training cost of L2AC (traditional), the algorithm that unrolls the gradient of the whole classification network to compute the second-order gradient of the bias attractor, just like most gradient-based bi-level optimization algorithm (Finn et al., 2017;Ren et al., 2018). It can be seen that: (1) our L2AC is much faster than L2AC (traditional); (2) The training cost of our L2AC is comparable to the current state-of-the-art method ABC (Lee et al., 2021). It can be observed that the computation cost increment of our proposed L2AC is nearly negligible compared with the baseline models especially considering the significant improvement performance of L2AC. And it is worth noting that our proposed L2AC is more efficient than traditional secondorder optimization, we think it is owing to two reasons: (1) the bias attractor only adds a very small number of parameters (about 0.68% of the total number of parameters); (2) To calculate the secondorder gradient of these parameters, we only need to unroll the gradient of the linear classifier as shown in Eq. (4). Note that in the test stage our L2AC requires no extra overhead compared with the baseline model FixMatch. E.6 FEATURE VISUALIZATION UNDER OTHER SETUPS In Fig. 5, we visualize the representations of training data through t-SNE (Van der Maaten & Hinton, 2008) on CIFAR-10 with γ l = 100, γ u = 1. Under such a setting, each class roughly has the same number of samples, which ensures a good visual visualization for each class. To verify that the proposed L2AC can generally obtain a high-quality representation, we further visualize the t-SNE of training data under other experimental setups, including γ l = γ u = 100 and γ l = 100, γ u = 100 (reversed). As shown in Fig. 12 and Fig. 13, compared with FixMatch (Sohn et al., 2020), our L2AC certainly improves the separability of the tail classes from the head classes. This verifies that the result in Fig. 5 is not owing to the specific choice of the experimental setups. F CONVERGENCE ANALYSIS OF ALGORITHM 1 We prove that our algorithm to minimize L bal converges at rate ofÕ( 1 √ T ), which meets the convergence results of similar work such as Ren et al. (2018). Theorem F.1 Assume that L bal in Eq. (5) is L-smooth with ρ-bounded gradients on the samples of balanced set B, and the bias attractor function ∆f w (·) is differentiable with a δ-bounded gradient and twice differentiable with bounded Hessian. Let α t in Eq. (4) satisfies α t = c1 t ≤ 2 L , and η t in Eq. (6) is set as η t = c2 σ √ t for c > 0 such that c σ √ t < 1 L . Then the loss function L bal converges to critical point at the rateÕ( 1 √ T ) as min 0≤t≤T E[ ∇L bal (φ t (ω t )) 2 2 ] ≤Õ( 1 √ T ).(14) Proof F.1 The parameter ω is updated by stochastic gradient descent as ω t+1 = ω t − ηt∇L bal (φ t (ω t ))|B t ,(15) where B t is a mini-batch of class-balanced data. We can write Eq. (15) as ω t+1 = ω t − ηt[∇L bal (φ t (ω t )) + ξ t ],(16) where ξ t is the random gradient noise with zero mean and finite variance σ. Observe that L bal (φ t+1 (ω t+1 )) − L bal (φ t (ω t )) ={L bal (φ t+1 (ω t+1 )) − L bal (φ t (ω t+1 ))} + {L bal (φ t (ω t+1 )) − L bal (φ t (ω t ))}.(17) According to L-smoothness of L bal with respect to φ, we have L bal (φ t+1 (ω t+1 )) − L bal (φ t (ω t+1 )) ≤ ∇ φ L bal (φ t (ω t+1 )), φ t+1 (ω t+1 ) − φ t (ω t+1 ) + L 2 φ t+1 (ω t+1 ) − φ t (ω t+1 ) 2 2 ,(18)meanwhile φ t+1 (ω t+1 ) − φ t (ω t+1 ) = −αt∇ φ L(φ(ω t+1 ), θ)| φ t (ω t+1 ) ,(19) we have L bal (φ t+1 (ω t+1 )) − L bal (φ t (ω t+1 )) ≤ ∇ φ L bal (φ t (ω t+1 )) − αt∇ φ L(φ(ω t+1 ), θ)| φ t (ω t+1 ) + L 2 − αt∇ φ L(φ(ω t+1 ), θ)| φ t (ω t+1 ) 2 2 , ≤αtρ 2 + L 2 α 2 t ρ 2 ,(20) the second inequality holds due to the assumption ∇ φ L bal (φ t (ω t+1 )) ≤ ρ and − α t ∇ φ L(φ(ω t+1 ), θ)| φ t (ω t+1 ) ≤ ρ. According to the L-smoothness of L bal with respect to ω, we have L bal (φ t (ω t+1 )) − L bal (φ t (ω t )) ≤ ∇ωL bal (φ t (ω t )), ω t+1 − ω t + L 2 ω t+1 − ω t 2 2 = ∇ωL bal (φ t (ω t )), −ηt[∇ωL bal (φ t (ω t )) + ξ t ] + Lη 2 t 2 ∇ωL bal (φ t (ω t )) + ξ t 2 2 =( Lη 2 t 2 − ηt) ∇ωL bal (φ t (ω t )) 2 2 + Lη 2 t 2 ξ t 2 2 + (Lη 2 t − ηt) ∇ωL bal (φ t (ω t )), ξ t ] . Then Eq. (17) becomes L bal (φ t+1 (ω t+1 )) − L bal (φ t (ω t )) ≤αtρ 2 + L 2 α 2 t ρ 2 + ( Lη 2 t 2 − ηt) ∇ωL bal (φ t (ω t )) 2 2 + Lη 2 t 2 ξ t 2 2 + (Lη 2 t − ηt) ∇ωL bal (φ t (ω t )), ξ t ] . Figure 1 : 1Experiments on CIFAR-10-LT. (a) Labeled set is class-imbalanced with imbalance ratio γ = 100, while the whole training data remains balanced. Analysis on (b) per-class recall and (c) predicted class distribution for Upper bound (trained with the whole training data with ground truth labels), Lower bound (trained with the labeled data only), FixMatch and Ours on the balanced test set. Note that predicted class distribution is averaged by the predicted scores for all samples. Figure 2 : 2Learning to adapt classifier for imbalanced SSL. Left: The schematic illustration of the bi-level learning framework of our method, which includes four main steps: i) Forward process to compute the lower level loss in Eq. (3); ii) Backward process to update the classification network parameters φ(ω), where ω are variables used for parameterizing φ; (iii) Forward process to compute the upper-level loss in Eq. (5); iv) Backward-on-backward to update ω. Right: The proposed bias adaptive classifier, which consists of the original linear classifier f cls φ and a bias attractor ∆f ω . (2020a), and further ablate the proposed L2AC under various labeled ratios in Section 4.4. The test set remains unchanged and class-balanced. Setups. The experimental setups are consistent with Kim et al. (2020a). Concretely, we employ Wide ResNet-28-2 (Oliver et al., 2018) as our backbone network and adopt Adam optimizer (Kingma & Ba, 2015) for 500 training epochs, each of which has 500 iterations. To evaluate the model, we use its exponential moving average (EMA) version, and report the average test accuracy of the last 20 epochs following Berthelot et al. (2019b). See Appendix D.1 for more details. (Japkowicz, 2000), LDAM-DRW (Cao et al., 2019) and cRT (Kang et al., 2020); 3) Recent imbalanced SSL methods, including: DARP (Kim et al., 2020a), CReST+ (Wei et al., 2021), ABC (Lee et al., 2021), SaR (Lai et al., 2022) and DASO Figure 3 :Figure 4 :Figure 5 345), the classifier learned by L2AC approximates the predicted class distribution of the upper bound model and much better than FixMatch, indicating that L2AC results in a relatively unbiased classifier. For the feature extractor, we further visualize the representations of training data through t-SNE (Van der Maaten & Hinton, 2008) on CIFAR-10 with γ l = 100, γ u = Confusion matrices of FixMatch (Sohn et al., 2020), DARP (Kim et al., 2020a), ABC (Lee et al., 2021), and ours on CIFAR-10 under the imbalance ratio γ = 100. Pseudo-label per-class recall of Fix-Match and ours on CIFAR-10 with γ l = 100 and (a) γ u = 100; (b) γ u = 100 (reversed). : t-SNE visualization of training data for (a) FixMatch and (b) L2AC experiments are implemented with the Pytorch platform(Paszke et al., 2019) and follows the experimental settings inKim et al. (2020a). We use Wide ResNet-28-2(Oliver et al., 2018) as our backbone network. During training, the model is trained with Adam optimizer(Kingma & Ba, 2015) under the default parameter setting, i.e., β 1 = 0.9, β 2 = 0.999, and = 10 −8 . The learning rate is set as 2 × 10 −3 and the batch size is set as 64. The total number of training iterations are 2.5 × 10 5 as inKim et al. (2020a). To evaluate the model, we follow the setting inBerthelot et al. (2019b) and use an exponential moving average (EMA) of its parameters with a decay rate of 0.999 at each iteration. We also follow the standard evaluation protocols in Berthelot et al. (2019b) that evaluates the performance at every 500 iterations and reports the average test accuracy of the last 20 evaluations. Figure 6 : 6Class distributions of three typical imbalanced test sets. Figure 7 : 7Class distributions of CIFAR-10 under step imbalance setups with (a) γ l = γ u = 100; (b) γ l = 100, γ u = 1. Figure 8 : 8Pseudo-label confusion matrices of (a) FixMatch and (b) Ours on CIFAR-10 under γ l = γ u = 100. Figure 9 : 9Pseudo-label confusion matrices of (a) FixMatch and (b) Ours on CIFAR-10 with γ l = 100, γ u = 100 (reversed). Figure 10 :Figure 11 : 1011Curves of Left: lower-level and upper-level losses and Right: test accuracy durning training of our approach on long-tailed CIFAR-10 under γ l = γ u = 100. Curves of upper-level loss (over B) and lower-level loss over labeled set D l . Figure 12 :Figure 13 : 1213t-SNE visualization of unlabeled data for (a) FixMatch and (b) L2AC on CIFAR-10 with γ l = γ u = 100. t-SNE visualization of unlabeled data for (a) FixMatch and (b) L2AC on CIFAR-10 with γ l = 100, γ u = 100 (reversed). ±0.57 / 81.5 ±0.64 77.6 ±0.53 / 75.8 ±0.71 89.5 ±0.18 / 89.2 ±0.19 82.2 ±1.23 / 81.7 ±1.364 EXPERIMENTSWe evaluate our approach on four benchmark datasets: CIFAR-10, CIFAR-100 (Krizhevsky et al., 2009), STL-10 (Coates et al., 2011) and SUN397(Xiao et al., 2010), which are broadly used in imbalanced learning and SSL tasks. We adopt balanced accuracy (bACC)(Huang et al., 2016;Wang et al., 2017) and geometric mean scores (GM) (Kubat et al.58.8 ±0.13 / 51.0 ±0.11 55.6 ±0.43 / 44.0 ±0.98 58.8 ±0.13 / 51.0 ±0.11 58.8 ±0.13 / 51.0 ±0.11 w/ Re-sampling 55.8 ±0.47 / 45.1 ±0.30 52.2 ±0.05 / 38.2 ±1.49 55.8 ±0.47 / 45.1 ±0.30 55.8 ±0.47 / 45.1 ±0.30 w/ LDAM-DRW 62.8 ±0.17 / 58.9 ±0.60 57.9 ±0.20 / 50.4 ±0.30 62.8 ±0.17 / 58.9 ±0.60 62.8 ±0.17 / 58.9 ±0.60 w/ cRT 63.2 ±0.45 / 59.9 ±0.40 59.3 ±0.10 / 54.6 ±0.72 63.2 ±0.45 / 59.9 ±0.40 63.2 ±0.45 / 59.9 ±0.40 MixMatch 64.8 ±0.28 / 49.0 ±2.05 62.5 ±0.31 / 42.5 ±1.68 41.5 ±0.76 / 12.0 ±1.34 47.9 ±0.09 / 20.5 ±0.85 w/ DARP 67.9 ±0.14 / 61.2 ±0.15 65.8 ±0.52 / 56.5 ±2.08 86.7 ±0.80 / 86.2 ±0.82 72.9 ±0.24 / 71.0 ±0.32 w/ SaR 66.8 ±0.92 / 59.9 ±1.32 64.4 ±2.21 / 57.3 ±1.95 68.4 ±3.20 / 62.0 ±2.17 65.5 ±1.01 / 64.2 ±0.95 w/ DASO 69.8 ±1.10 / 69.3 ±1.07 66.5 ±1.99 / 65.4 ±2.25 75.5 ±0.48 / 74.6 ±0.67 65.7 ±1.01 / 62.0 ±1.23 w/ ABC 75.7 ±0.76 / 74.7 ±0.47 68.5 ±0.40 / 56.4 ±1.50 72.1 ±0.53 / 41.2 ±4.40 62.9 ±0.36 / 59.9 ±0.60 w/ L2AC (ours) 76.6 ±0.73 / 75.7 ±1.08 72.1 ±0.62 / 70.3 ±0.93 87.2 ±0.09 / 86.7 ±0.08 74.0 ±0.82 / 72.9 ±1.01 FixMatch 71.5 ±0.72 / 66.8 ±1.51 68.4 ±0.15 / 59.9 ±0.43 68.9 ±1.95 / 42.8 ±8.11 65.5 ±0.05 / 26.0 ±0.44 w/ DARP 75.5 ±0.05 / 73.0 ±0.09 70.4 ±0.25 / 64.9 ±0.17 85.4 ±0.55 / 85.0 ±0.65 74.9 ±0.51 / 72.3 ±1.13 w/ CReST+ 77.5 ±0.15 / 76.1 ±0.15 72.1 ±0.74 / 68.9 ±1.29 N/A N/A w/ SaR 77.6 ±0.42 / 75.9 ±0.76 71.5 ±0.23 / 66.9 ±0.25 85.9 ±0.68 / 85.3 ±0.53 78.3 ±0.34 / 76.1 ±0.21 w/ DASO 78.3 ±0.55 / 76.5 ±0.57 74.6 ±0.74 / 71.7 ±0.52 87.9 ±0.41 / 87.7 ±0.43 79.5 ±0.91 / 78.9 ±0.96 w/ ABC 80.2 ±0.42 / 78.9 ±1.29 74.7 ±1.04 / 72.2 ±1.45 81.3 ±0.34 / 80.2 ±0.36 70.3 ±0.50 / 67.9 ±0.70 w/ L2AC (ours) 82.1 ±0.81 / 74.4 ±1.00 w/ L2AC (ours) 57.8 ±0.19 / 52.1 ±0.31 52.6 ±0.13 / 43.0 ±0.45 79.9 ±0.52 / 79.1 ±0.49 77.0 ±0.65 / 75.8 ±0.68FixMatch 55.1 ±0.09 / 46.7 ±0.53 49.5 ±0.38 / 34.2 ±1.01 69.6 ±0.60 / 62.6 ±1.11 65.5 ±0.05 / 26.0 ±0.44 w/ DARP 56.3 ±0.25 / 48.2 ±0.73 50.2 ±0.18 / 36.0 ±0.60 72.9 ±0.24 / 69.5 ±0.18 74.9 ±0.51 / 72.3 ±1.13 w/ ABC 58.2 ±0.08 / 51.8 ±0.25 53.1 ±0.19 / 42.2 ±0.82 78.2 ±0.35 / 77.3 ±0.30 72.7 ±0.08 / 70.6 ±0.22 w/ DASO 58.3 ±0.39 / 51.4 ±0.80 53.0 ±0.27 / 39.5 ±1.45 78.2 ±0.63 / 77.4 ±0.53 75.4 Table 3 : 3Comparison results on large-scale SUN397. The performance (bACC / GM) is reported in the form of mean±std across three random runs.Methods bACC/GM Methods bACC/GM Vanilla 38.3 ±0.05 / 29.9 ±0.08 DARP (Kim et al., 2020a) 45.5 ±0.32 / 37.5 ±0.04 cRT (Kang et al., 2020)] 39.3 ±0.21 / 33.7 ±0.37 ABC (Lee et al., 2021) 47.0 ±0.26 / 39.2 ±0.34 FixMatch Sohn et al. (2020) 44.9 ±0.11 / 35.7 ±0.66 L2AC (ours) 48.8 ±0.19 / 40.6 ±0.17 Table 4 : 4Performance (bACC / GM) on CIFAR10-LT and STL-10 under various label ratio β.CIFAR-10 (γ l = γu = 100) β = 1 β = 5 β = 10 β=20 β = 30 FixMatch 54.9 / 16.5 65.1 / 35.5 69.0 / 53.9 72.0 / 62.2 76.5 / 74.3 w/ L2AC (ours) 62.8 / 55.8 75.9 / 74.1 79.3 / 78.4 80.8 / 79.9 83.6 / 83.2 STL-10 (γ l = 10, γu = N/A) β = 5 β = 10 β = 20 β=40 β = 60 FixMatch 46.5 / 19.9 48.8 / 27.0 58.2 / 39.8 67.2 / 60.7 69.2 / 67.6 w/ L2AC (ours) 62.8 / 57.1 66.5 / 62.9 72.6 / 70.6 77.0 / 75.7 78.8 / 77.9 Kim et al. (2020a), we hold-out 50 samples per each class for testing because no official data split is provided. We then construct the labeled and unlabeled dataset according to M k /N k = 2. The comparison methods includes: Vanilla, cRT (Kang et al., 2020), FixMatch (Sohn et al., 2020), DARP (Kim et al., 2020a) and ABC (Lee et al., 2021). More training details are presented in Appendix D.2. Table 5 : 5Ablation study.Methods CIFAR-10 (γ l = 100) γ u = 100 1/100 (reversed) FixMatch 71.5 / 68.8 65.5 / 26.0 FixMatch w/ bias attractor 73.9 / 70.7 66.6 / 44.8 L2AC w/o bi-level training 78.4 / 76.6 79.3 / 78.0 L2AC (ours) 82.1 / 81.5 82.2 / 81.7 ).Proof B.1The update of φ is as:6: Estimate pseudo-labelŷ i for x i ∈D u . 7: Compute lower-level loss L by Eq. (3). 8: Update network parameters {θ t+1 , φ t+1 } by Eq. (4). 9: Compute upper-level loss L bal by Eq. (5). 10: Update bias attractor parameters ω t+1 by Eq. (6). 11: end for B PROOF OF PROPOSITION 3.1 Table 6 : 6Imbalanced test set results. ACC: accuracy for all samples.Methods Test-1 Test-2 Test-3 bACC GM ACC bACC GM ACC bACC GM ACC FixMatch (Sohn et al., 2020) 72.4 66.3 86.1 72.7 66.9 56.6 72.3 65.7 75.6 w/ DARP (Kim et al., 2020a) 74.8 72.5 86.3 75.5 73.2 63.9 75.6 73.3 77.4 w/ CReST (Wei et al., 2021) 77.8 76.5 86.3 77.2 74.8 68.9 77.5 76.3 80.3 w/ ABC (Lee et al., 2021) 80.2 79.2 88.1 80.2 79.0 71.7 80.1 79.0 82.8 w/ L2AC (ours) 82.6 82.0 87.2 82.4 81.8 78.6 82.7 82.1 83.9 E.2 EVALUATION ON STEP IMBALANCE SETUPS Table 7 : 7Performance (bACC / GM) on CIFAR-10 under step imbalance setups. Methods γ l = γu = 100 γ l = 100, γu = 1 FixMatch (Sohn et al., 2020) 55.0±0.84 / 24.4±2.97 60.1±1.97 / 18.32±3.16 w/ DARP (Kim et al., 2020a) 58.6±0.44 / 30.0±1.27 - w/ ABC (Lee et al., 2021) 75.1±0.78 / 67.5±0.97 75.8±0.98 / 69.2±1.35 w/ L2AC (ours) 76.7±0.41 / 74.5±0.61 81.8±0.87 / 80.5±1.01 Table 8 : 8Training cost analysis on CIFAR-10 and CIFAR-100.Methods Params FLOPS CIFAR-10 CIAFR-100 FixMatch (Sohn et al., 2020) 1.47 M 19.6 iter/sec 19.6 iter/sec w/ DARP (Kim et al., 2020a) 1.47 M 18.2 iter/sec 7.5 iter/sec w/ ABC (Lee et al., 2021) 1.47 M 15.1 iter/sec 14.9 iter/sec w/ L2AC (traditional) 1.48 M 9.9 iter/sec 9.7 iter/sec w/ L2AC (ours) 1.48 M 14.2 iter/sec 13.9 iter/sec In theory, an MLP can approximate almost any continuous function(Hornik et al., 1989). bACC and GM are defined as the arithmetic and geometric mean over class-wise sensitivity, respectively. https://github.com/bbuing9/DARP 4 https://github.com/google-research/crest 5 https://github.com/LeeHyuck/ABC 6 https://github.com/ytaek-oh/daso ACKNOWLEDGMENTSWe thank the anonymous reviewers for their constructive suggestions on improving this paper. This research was supported by National Key R&D Program of China (2020YFA0713900), the Macao Science and Technology Development Fund under Grant 0612020A2, The Major Key Project of PCL (PCL2021A12), the China NSFC projects under contract 61721002 and 61906144.Rearranging Eq. (22), we haveSumming up the above inequalities, we obtainTaking expectations with respect to ξ t on Eq.(24), we can obtainThe first inequality holds due to E[ ξ t 2 2 ] = σ 2 and the second equality holds due to E ξ t [ξ t ] = 0. Further more, we haveThe second inequality hods due to T t=1 (2η t − Lη 2 t ) ≥ T t=1 η t . The third inequality holds due to α t ≤ 2 L . As α t = c1 t and η t = c2 √ t , we have T t=1 η t = ( √ T ), T t=1 η 2 t = log(T ) and T t=1 α t = log(T ) thus the last equality holds, this finish our proof. Virtual adversarial training: a regularization method for supervised and semi-supervised learning. 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{'abstract': 'Pseudo-labeling has proven to be a promising semi-supervised learning (SSL) paradigm. Existing pseudo-labeling methods commonly assume that the class distributions of training data are balanced. However, such an assumption is far from realistic scenarios and thus severely limits the performance of current pseudolabeling methods under the context of class-imbalance. To alleviate this problem, we design a bias adaptive classifier that targets the imbalanced SSL setups. The core idea is to automatically assimilate the training bias caused by class imbalance via the bias adaptive classifier, which is composed of a novel bias attractor and the original linear classifier. The bias attractor is designed as a light-weight residual network and optimized through a bi-level learning framework. Such a learning strategy enables the bias adaptive classifier to fit imbalanced training data, while the linear classifier can provide unbiased label prediction for each class. We conduct extensive experiments under various imbalanced semi-supervised setups, and the results demonstrate that our method can be applied to different pseudo-labeling models and is superior to current state-of-the-art methods. Raffel. Mixmatch: A holistic approach to semi-supervised learning. NeurIPS, 32, 2019b. Paula Branco, Luís Torgo, and Rita P Ribeiro. A survey of predictive modeling on imbalanced domains. ACM Computing Surveys (CSUR), 49(2):1-50, 2016. Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance problem in convolutional neural networks. -supervised learning (chapelle, o. et al., eds.; 2006)[book reviews]. IEEE Transactions on Neural Networks, 20(3):542-542, 2009. . Smote: synthetic minority over-sampling technique. . Model-agnostic meta-learning for fast adaptation of deep networks.', 'arxivid': '2207.13856', 'author': ["Renzhen Wang [email protected] \nXi'an Jiaotong University\n\n", 'Xixi Jia \nXidian University\n\n', "Quanziang Wang \nXi'an Jiaotong University\n\n", 'Yichen Wu \nCity University of Hong\nKong\n', "Deyu Meng [email protected] \nXi'an Jiaotong University\n\n\nMacau University of Science and Technology\n\n\nPeng Cheng Laboratory\n\n"], 'authoraffiliation': ["Xi'an Jiaotong University\n", 'Xidian University\n', "Xi'an Jiaotong University\n", 'City University of Hong\nKong', "Xi'an Jiaotong University\n", 'Macau University of Science and Technology\n', 'Peng Cheng Laboratory\n'], 'corpusid': 257279756, 'doi': None, 'github_urls': ['https://github.com/renzhenwang/bias-adaptive-classifier.', 'https://github.com/bbuing9/DARP', 'https://github.com/google-research/crest', 'https://github.com/LeeHyuck/ABC', 'https://github.com/ytaek-oh/daso'], 'n_tokens_mistral': 24158, 'n_tokens_neox': 20340, 'n_words': 11541, 'pdfsha': 'cad82941268526b364b8b16d374aacf4573bc4eb', 'pdfurls': ['https://export.arxiv.org/pdf/2207.13856v2.pdf'], 'title': ['IMBALANCED SEMI-SUPERVISED LEARNING WITH BIAS ADAPTIVE CLASSIFIER', 'IMBALANCED SEMI-SUPERVISED LEARNING WITH BIAS ADAPTIVE CLASSIFIER'], 'venue': []}

arxiv

A SPECTRAL THEORY OF POLYNOMIALLY BOUNDED SEQUENCES AND APPLICATIONS TO THE ASYMPTOTIC BEHAVIOR OF DISCRETE SYSTEMS 11 Mar 2020 Nguyen Van Minh NGUYENHideaki Matsunaga Duc Huy Vu Trong Luong A SPECTRAL THEORY OF POLYNOMIALLY BOUNDED SEQUENCES AND APPLICATIONS TO THE ASYMPTOTIC BEHAVIOR OF DISCRETE SYSTEMS 11 Mar 2020 In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by n ν , where ν is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form ∆ α x(n) = T x(n) + y(n), n ∈ N, where 0 < α ≤ 1. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the αresolvent operator Sα satisfies sup n∈N Sα(n) /n ν < ∞ and the set of z 0 ∈ C such that (z −k α (z)T ) −1 exists, and together withk α (z), is holomorphic in a neighborhood of z 0 consists of at most 1, wherek α (z) is the Z-transform of k α (n) := Γ(α + n)/(Γ(α)Γ(n + 1)), then lim n→∞ Introduction Let us consider difference equations of the form x(n + 1) = T x(n) + y(n), n ∈ N, (1.1) where T is a bounded operator in a Banach space X, {x(n)} ∞ n=1 and {y(n)} ∞ n=1 are sequences in X. The asymptotic behavior of solutions of the above mentioned equations is a central topic in Analysis and Dynamical Systems. There are numerous methods for this study of this topic. The reader is referred to [8] and its references for information on classical methods of Dynamical Systems in the finite dimensional case. On the other hand, in the infinite dimensional case, by Harmonic Analysis and Operator Theory, many results on the asymptotic behavior of solutions of Eq. (1.1) have been obtained, see e.g. [1,3,4,5,6,7,9,11,14,15,16]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [9]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. [1,3,5,7,9,11,14,15,16] and the references therein. As shown in [15] the above mentioned Katznelson-Tzafriri Theorem is equivalent to its weaker version for individual orbits. Namely, the following statement: Let T be a bounded operator in a Banach space X such that (1.2) holds and σ(T ) ⊂ {1}. Then, for each x ∈ X lim n→∞ (T − I)T n x = 0. (1.4) In [11] a simple proof of this weaker version is given, based on a transform associated with the translation operator of sequences. The main concern of this paper is to extend the above mentioned Katznelson-Tzafriri Theorem to fractional difference equations of the form ∆ α x(n) = T x(n) + y(n), n ∈ N, (1.5) where 0 < α ≤ 1, the operator ∆ α (the fractional difference operator in the sense of Riemann-Liouville) and other operators are defined as follows (see [10] and its references for more details): for each n ∈ N, (∆ α )f (n) = ∆ 1 • ∆ −(1−α) f (n), (∆ 1 f )(n) = f (n + 1) − f (n), (∆ −α )f (n) = n k=0 k α (n − k)f (k), k α (j) = Γ(α + j) Γ(α)Γ(j + 1) , where Γ(·) is the Gamma function defined below. Our method relies on a spectral theory of polynomially bounded sequences that will be presented in the next sections, and that would be of independent interest. The obtained results will be illustrated in simple cases of ordinary difference equations and then stated for fractional difference equations. Our main result is Theorem 4.16. To our best knowledge, it is a new extension of the Katznelson-Tzafriri Theorem to fractional difference equations. Preliminaries and Notations 2.1. Notations. Throughout this paper we will denote by N, Z, R, C the set of natural numbers, integers, real numbers and the complex plane, respectively. For z ∈ C, ℜz stands for its real part. The gamma function Γ(z) is defined to be Γ(z) = ∞ 0 x z−1 e −x dx, ℜz > 0. For a Banach space X, L(X) denotes the space of all bounded linear operators from X to itself. We also use the following standard notations: ρ(T ) denotes the resolvent set of a given operator T , that is, ρ(T ) := {λ ∈ C : (λ − T ) −1 exists}, and σ(T ) := C\ρ(T ). For each λ ∈ ρ(T ) we denote R(λ, T ) := (λ − T ) −1 . Moreover, we will denote by Γ the unit circle in the complex plane. For a given nonnegative integer ν, we denote by l ν ∞ (X) the space of all sequences in a Banach space X such that sup n∈N x(n) n ν < ∞. It is easy to see that l ν ∞ (X) is a Banach space with norm x ν := sup n∈N x(n) n ν for each x = {x(n)} n∈N . We will denote by c ν 0 (X) the subspace of l ν ∞ (X) consisting of all sequences {x(n)} n∈N such that lim n→∞ x(n) n ν = 0. We can check that c ν 0 (X) is a complete subspace of l ν ∞ (X), so the quotient space Y := l ν ∞ (X)/c ν 0 (X) is well defined as a Banach space. If f ∈ l ν ∞ (X) we will denote its equivalence class byf . In the space l ν ∞ (X) let us consider the translation operator S defined as [Sx](n) = x(n + 1), n ∈ N, x ∈ l ν ∞ (X). This is a bounded operator. Moreover, this operator leaves c ν 0 (X) invariant. Hence, it induces an operatorS on Y. 2.2. Vector-valued holomorphic functions. In this paper we say that a function f (z) defined for all z ∈ Ω ⊂ C with values in a complex Banach space X is holomorphic (or analytic) for z ∈ Ω if for each z 0 ∈ Ω f ′ (z 0 ) := lim h→0,h =0 f (z 0 + h) − f (z 0 ) h exists. A family of continuous functionals W ⊂ X * is said to be separating if x ∈ X and x, φ = 0 for all φ ∈ W , then x = 0. We will need the following whose proof can be found in [2, We will need an auxiliary result that is a special kind of maximum principle for holomorphic functions (for the proof see e.g. [3, Lemma 4.6.6]): Lemma 2.2. Let U be an open neighborhood of iη such that U contains the closed diskB(iη, 2r) = {z ∈ C : |z − iη| ≤ 2r}. Let h : U → X be holomorphic and c ≥ 0, k ∈ N such that h(z) ≤ c |ℜz| k , if |z − iη| = 2r, ℜz = 0. Then h(z) ≤ 4 3 k c r k , for all z ∈B(iη, r). Spectrum of a polynomially bounded sequence The following lemma is the key for us to set up a spectral theory for polynomially bounded sequences. Moreover, for each |λ| = 1 with |λ| < 2 and f ∈ l ν ∞ (X), the following estimate is valid: R(λ,S)f ≤ C ||λ| − 1| ν+1 f ν , (3.1) where C is a certain positive number, independent of f . Proof. We will prove that if |λ| = 1, then λ ∈ ρ(S). In other words, σ(S) ⊂ Γ. And after that, we will give estimates of the resolvent R(λ,S)f of a given sequence f ∈ l ν ∞ (X). To study the invertibility of the operator (λ −S), we consider the non-homogeneous linear difference equation x(n + 1) − λx(n) = f (n), n ∈ N. (3.2) To prove that λ ∈ ρ(S) for each |λ| = 1 we will show that this equation (3.2) has a unique solution x ∈ l ν ∞ (X) modulo c ν 0 (X) given f ∈ l ν ∞ (X). We first consider the case |λ| < 1. In this case, we will use the Variation of Constants Formula x(n) = λ n−1 x(1) + n−1 k=1 λ n−1−k f (k), n ∈ N. Since the sequence f grows polynomially, the series ∞ k=1 λ n−1−k f (k) is absolutely convergent. Also, by |λ| < 1 the sequence {λ n−1 x(1)} n∈N is in c ν 0 (X). Therefore, Eq. (3.2) has a unique solution x f (n) := n−1 k=1 λ n−1−k f (k) n∈N modulo c ν 0 (X). Now suppose that g is any element in the classf . We will show thatx g =x f . Or equivalently, we have to show that whenever h ∈ c ν 0 (X), the sequence {x h (n)} n∈N = n−1 k=1 λ n−1−k h(k) n∈N ∈ c ν 0 (X). In fact, as h ∈ c ν 0 (X), given ε > 0 there exists a natural number M such that for all k ≥ M , |h(k)| k ν < 1 − |λ| 2 ε. Therefore, for all n ≥ M + 1, x h (n) n ν ≤ M−1 k=1 |λ| n−1−k n ν h(k) + n−1 k=M |λ| n−1−k n ν h(k) ≤ |λ| n n ν M−1 k=1 |λ| −1−k h(k) + n−1 k=M |λ| n−1−k h(k) k ν ≤ |λ| n n ν M−1 k=1 |λ| −1−k h(k) + n−1 k=M |λ| n−1−k 1 − |λ| 2 ε ≤ |λ| n n ν M−1 k=1 |λ| −1−k h(k) + ε 2 . As M is a fixed natural number and |λ| < 1 there exists a natural number K ≥ M +1 such that for all n ≥ K, |λ| n n ν M−1 k=1 |λ| −1−k h(k) ≤ ε 2 . Consequently, given any ε > 0 there exists a number K such that for all n ≥ K, x h (n) n ν ≤ ε. This means lim n→∞ x h (n) n ν = 0. By this we have proved thatx f =x g wheneverf =ḡ. Namely, we have showed that if |λ| < 1, thenx f = (λ −S) −1 f . In other words, λ ∈ ρ(S). Moreover, for any representative g of the classf R(λ,S)f ν = x f ν = inf g∈f x g ν ≤ x g ν = sup n∈N n−1 k=1 λ n−1−k g(k) n ν ≤ sup n∈N n−1 k=1 |λ| n−1−k g(k) k ν ≤ sup n∈N n−1 k=1 |λ| n−1−k g ν ≤ g ν 1 − |λ| . Finally, as g is any representative of the classf , we have R(λ,S)f ≤ inf g∈f g ν 1 − |λ| = f ν 1 − |λ| . Next, we consider the case |λ| > 1. We can verify that the formula x(n) = λ n−1 x(1) − ∞ k=n λ n−k−1 f (k), n ∈ N, (3.3) gives the general solution to Eq. (3.2). In fact, since |λ| > 1 and f grows polynomially the series ∞ k=n λ n−k−1 f (k) is absolutely convergent for each n ∈ N. Moreover, by (3.3), for each n ∈ N, x(n + 1) = λ n x(1) − ∞ k=n+1 λ n−k f (k) = λ n x(1) − ∞ k=n λ n−k f (k) + f (n) = λx(n) + f (n). Given f ∈ l ν ∞ (X), the only solution of Eq. (3.2) in l ν ∞ (X) is x f := − ∞ k=n λ n−k−1 f (k) n∈N . Indeed, x f ν ≤ sup n∈N ∞ k=n |λ| n−k−1 f (k) n ν = sup n∈N ∞ k=n |λ| n−k−1 k ν n ν f (k) k ν ≤ sup n∈N ∞ k=n |λ| n−k−1 k ν n ν f ν = sup n∈N ∞ j=1 |λ| −j 1 + j − 1 n ν f ν ≤ ∞ j=1 |λ| −j j ν f ν . (3.4) We are interested in the behavior of ∞ j=1 |λ| −j j ν as |λ| gets closer and closer to 1 (and ∞, respectively). To this end, we note that for each j ∈ N, |λ| −j−1 j ν ≤ j+1 j |λ| −t t ν dt. Therefore, ∞ j=1 |λ| −j j ν ≤ |λ| ∞ 0 |λ| −t t ν dt = |λ| ∞ 0 e −t·ln(|λ|) t ν dt = |λ|ν! | ln(|λ|)| ν+1 . Consequently, since ln(|λ|) is equivalent to |λ| − 1 as |λ| is close to 1, there exists a number C independent of f such that for 1 < |λ| < 2 x f ν ≤ |λ|ν! | ln(|λ|)| ν+1 f ν ≤ C ||λ| − 1| ν+1 f ν . (3.5) Similarly as in the previous case where |λ| < 1, we will prove thatx f =x g wheneverf =ḡ. Namely, ifh = 0, thenx h = 0. In fact, for a given ε > 0, there exists a natural number N such that for all k ≥ N , h(k) /k ν < ε. Therefore, for all n ≥ N , x h (n) n ν ≤ ∞ k=n |λ| n−k−1 n ν h(k) ≤ ∞ k=n |λ| n−k−1 n ν εk ν ≤ ε|λ| −1 ∞ j=0 |λ| −j 1 + j n ν . Since |λ| > 1 is fixed, the series ∞ j=0 |λ| −j (1 + j/n) ν is convergent, so this shows that lim n→∞ x h (n) n ν = 0. That is,x h = 0. This yields that λ ∈ ρ(S) andx f = (λ −S) −1 f . Finally, with (3.5) the proof of the lemma is complete. Definition 3.2. Let f ∈ l ν ∞ (X) be a given sequence in X. Then its spectrum is defined to be the set of all complex ξ 0 ∈ Γ such that the complex function R(λ,S)f has no analytic extension to any neighborhood of ξ 0 . The spectrum of a sequence f ∈ l ν ∞ (X) will be denoted by σ ν (f ). Before we proceed we introduce some notations: D |z|>1 := {z ∈ C : |z| > 1}, B(ξ 0 , δ) := {z ∈ C : |z − ξ 0 | < δ}. Lemma 3.3. Let f ∈ l ν ∞ (X). Then, ξ 0 ∈ Γ is in σ ν (f ) if and only if the function g : D |z|>1 ∋ λ → R(λ,S)f ∈ l ν ∞ (X) cannot be extended to an analytic function in any neighborhood of ξ 0 . Proof. It suffices to show that if g can be extended to an analytic function in a neighborhood of ξ 0 , then ξ 0 / ∈ σ ν (f ). Suppose that g(λ) = h(λ) for all λ ∈ D |z|>1 ∩ B(ξ 0 , δ) where h is an analytic function in a small disk B(ξ 0 , δ). Then, the function (λ −S)h(λ) is analytic in B(ξ 0 , δ). We observe that, for λ ∈ D |z|>1 ∩ B(ξ 0 , δ) (λ −S)h(λ) = (λ −S)g(λ) = (λ −S)R(λ,S)f =f . That is, the function (λ −S)h(λ) is a constant in an open and connected subset D |z|>1 ∩ B(ξ 0 , δ) of the disk B(ξ 0 , δ). Hence, (λ −S)h(λ) =f for all λ in B(ξ 0 , δ). In particular, when |λ| < 1 and λ ∈ B(ξ 0 , δ), h = R(λ,S)f . That means, h(λ) is an analytic extension of the function R(λ,S)f as a complex function on {z ∈ C : |z| = 1} to a neighborhood of ξ 0 . Proposition 3.4. Let f ∈ l ν ∞ (X) be a given sequence in X. Then the following assertions are valid: i) σ ν (f ) is a closed subset of Γ; ii) The sequence f is in c ν 0 (X) if and only if σ ν (f ) = ∅; iii) If ξ 0 is an isolated element of σ ν (f ), then the point ξ 0 is a pole of the complex function R(λ,S)f of order up to ν + 1. Proof. Part (i) is obvious from the definition of the spectrum of x. Part (ii): Clearly, if f ∈ c ν 0 (X), then σ ν (f ) = ∅. Conversely, if σ ν (f ) = ∅, then the complex functionf (λ) := R(λ,S)f is an entire function. Moreover, it is bounded. In fact, from (3.4) for large |λ| > 2, f (λ) ν ≤ x f ≤ ∞ j=1 |λ| −j j ν f ν = |λ| −1 ∞ k=0 |λ| −k (k + 1) ν f ν ≤ |λ| −1 ∞ k=0 (k + 1) ν 2 k f ν . Since the series ∞ k=0 (k + 1) ν /2 k is convergent, we have lim |λ|→∞ f (λ) ν = 0. By the Liouville Theorem, this complex functionf (λ) := R(λ,S)f is the zero function, sof = 0 since R(λ,S) is injective for each large |λ|. That means f ∈ c ν 0 (X). Part (iii): Without loss of generality we may assume that ξ 0 = 1. Consider λ in a small neighborhood of 1 in the complex plane. We will express λ = e z with |z| < δ 0 . Choose a small δ 0 > 0 such that if |z| < δ 0 , then 1 |1 − |λ|| ≤ 2 |ℜz| . It follows from Lemma 3.1 that for 0 < |ℜz| < δ 0 , R(λ,S)x ≤ C |1 − |λ|| ν+1 x ≤ C2 ν+1 |ℜz| ν+1 x . Set f (z) = R(e z ,S)x with |z| < δ 0 . Since 1 is a singular point of R(λ,S)x , 0 is a singular point of f (z) in {|z| < δ 0 } . For each n ∈ Z and 0 < r < δ 0 , we have 1 2πi |z|=r 1 + z 2 r 2 ν+1 f (z)dz ≤ 1 2π |z|=r 1 + z 2 r 2 ν+1 f (z) |dz|. If z = re iϕ , where ϕ is real, one has 1 + z 2 r 2 ν+1 = |1 + e 2iϕ | ν+1 = |e −iϕ + e iϕ | ν+1 = (2| cos ϕ|) ν+1 = 2 ν+1 r −ν−1 |ℜz| ν+1 . Therefore, 1 2πi |z|=r 1 + z 2 r 2 ν+1 f (z) z n+1 dz ≤ 1 2π |z|=r 2 ν+1 r −n−ν−2 |ℜz| ν+1 C2 ν+1 |ℜz| ν+1 x |dz| = C4 ν+1 r −n−ν−2 2π |z|=r |dz| x = C4 ν+1 r −n−ν−1 x . (3.6) Consider the Laurent series of f (z) at z = 0, f (z) = ∞ n=−∞ a n z n , where a n = 1 2πi |z|=r f (z) z n+1 dz, n ∈ Z. It follows that for each n ∈ Z, 1 2πi |z|=r 1 + z 2 r 2 ν+1 f (z) z n+1 dz = 1 2πi |z|=r ν+1 k=0 (ν + 1)! k!(ν + 1 − k)! r −2k f (z) z n+1−2k dz = ν+1 k=0 (ν + 1)! k!(ν + 1 − k)! r −2k 1 2πi |z|=r f (z) z n+1−2k dz = ν+1 k=0 (ν + 1)! k!(ν + 1 − k)! r −2k a n−2k This, together with (3.6), shows ν+1 k=0 (ν + 1)! k!(ν + 1 − k)! r −2k a n−2k ≤ C4 ν+1 r −n−ν−1 x . Multiplying both sides by r 2ν gives ν+1 k=0 (ν + 1)! k!(ν + 1 − k)! r 2ν−2k a n−2k ≤ C4 ν+1 r ν−n−1 x . Observe that in the left side is a polynomial in terms of r whose zero power term is a n−2ν . Therefore, when ν − n − 1 ≥ 1 if we let r to get closer and closer to zero, then a n−2ν must be zero. That is for all ν ≥ n + 2, the coefficients a n−2ν = 0. This yields that for all j ≤ −ν − 2, a j = 0. In other words, z = 0, or λ = 1 is a pole of the complex functionf (λ) := R(λ,S)f with order up to ν + 1. Before proceeding we introduce a notation: Let 0 = z ∈ C such that z = re iϕ with reals r, ϕ, and F (z) be any complex function. Then we define R(λ,S)f = ∞ j=−ν−1 a j (λ − ξ 0 ) j+1 . If (3.8) is satisfied, then for any k ≥ 1 the following is also valid: lim λ↓z (λ − ξ 0 ) k R(λ,S)f = 0. If we let k take on the values 1, 2, . . . , ν + 1, then we see that a j = 0 for all j = −ν−1, −ν, . . . . That is, the function R(λ,S)f is zero (so analytic) in a neighborhood of ξ 0 . From the properties of analytic functions this function must be zero in the connected open subset of its domain as well. Applications 4.1. Asymptotic behavior of polynomially bounded solutions of difference equations. In this subsection we will apply the results obtained in the previous section to study the polynomially bounded solutions of difference equations of the form x(n + 1) = T x(n) + F (n), n ∈ N, (4.1) where T is a bounded linear operator in X and F ∈ c ν 0 (X). Definition 4.1. A bounded operator T from a Banach space X to itself is said to be ν-polynomially power bounded, if sup n∈N T n k ν < ∞, where ν is a nonnegative integer. Lemma 4.2. Let x ∈ l ν ∞ (X) be a solution of (4.1). Then σ ν (x) ⊂ σ(T ) ∩ Γ. (4.2) Moreover, for λ ∈ ρ(S) ∩ ρ(T ), R(λ,S)x = R(λ,T )x. (4.3) Proof. Consider the operator of multiplication by T in the spaces l ν ∞ (X). It is easy to see that the operator is bounded and preserves c ν 0 (X), so it induces an operator T in the quotient space l ν ∞ (X)/c ν 0 (X). Moreover, σ(T ) ⊂ σ(T ). Since x is a solution of (4. If ξ 0 ∈ Γ and ξ 0 ∈ σ(T ), then there exists a neighborhood of ξ 0 (in C) such that for any λ ∈ U and |λ| = 1, R(λ,T )x = R(λ,S)x. (4.4) As the left hand side function is an analytic extension in a neighborhood U of ξ 0 , by (4.4), the complex function R(λ,S)x has an analytic extension to the neighborhood U of ξ 0 , that is ξ 0 ∈ σ ν (x). In other words, σ ν (x) ⊂ σ(T ) ∩ Γ. Moreover, (4.3) is proved. We will prove the following that extends the famous Katznelson-Tzafriri to the case of ν-polynomially bounded operator. Proof. We consider the sequence x := {x(n) := T n } ∞ n=1 in L(X). Obviously, {x(n)} ∞ n=1 ∈ l ν ∞ (L(X)). Let us denote byT the operator of multiplication by T in Y := L(X). Then, we have an equation in Y: x(n + 1) =T x(n), n ∈ N. Note that σ(T ) ⊂ σ(T ). Therefore, by Lemma 4.2, we have σ ν (x) ⊂ {1}. For each λ ∈ ρ(S), we have the identity R(λ,S)Sx = λR(λ,S)x −x. By a simple induction we can show that for each j ∈ N, R(λ,S)S jx = λ j R(λ,S)x − P (λ,x,Sx), where P (λ,x,Sx) is a polynomial of λ,x,Sx. Hence, is extendable analytically to a neighborhood of 1. Therefore, for the sequence y := (S − I) ν+1 x we have σ ν (y) = ∅. By Proposition 3.4, y = (S − I) ν+1 x ∈ c ν 0 (L(X)), that is, (4.5) is valid. There are many extensions of this theorem (see e.g. [15] and its references). An elementary proof of this theorem is given in [11]. In Theorem 4.3, when ν = 0, we obtain the above mentioned Katznelson-Tzafriri Theorem. R(λ,S)(S − I) ν+1x = (λ − 1) ν+1 R(λ,S)x + Q(λ,x,Sx), where Q(λ,x,Sx) is a polynomial of λ,x,Sx. Note that σ ν ((S − I) ν x) ⊂ σ ν (x) ⊂ {1}. By Below is an individual version of Katznelson-Tzafriri Theorem for possibly non-ν polynomially bounded operator T . where T ∈ L(X). Each solution {x(n)} ∞ n=1 of this equation (4.7) is of the form x(n) = T n−1 x 0 , n ∈ N, for some x 0 ∈ X. Theorem 4.6. Let T ∈ L(X), and let x be a ν polynomially bounded solution of Eq. (4.1). Assume further that the following conditions are satisfied: i) σ(T ) ∩ Γ is countable ii) For each ξ 0 = e iφ0 ∈ σ(T ) ∩ Γ {z = re iφ0 , r > 1} ⊂ ρ(T ); (4.8) lim λ↓z (λ − ξ 0 )R(λ,T )x = 0. (4.9) Then lim n→∞ x(n) n ν = 0. (4.10) Proof. Since σ ν (f ) ⊂ σ(T ) ∩ Γ, if σ(T ) ∩ Γ is empty, then the claim of the theorem is clear. Next, if it is not, then from the countability of σ ν (f ) as a closed subset of Γ there must be an isolated point, say ξ 0 of σ ν (f ). However, by condition (4.9) and Corollary 3.5 the set of non-removable singular points of the complex function R(λ,S)x = R(λ,T )x cannot have an isolated point. That means, σ ν (x) must be empty set, so by Proposition 3.4, the sequence x = {x(n)} ∞ n=1 must be in c ν 0 (X), that is (4.10). The following result gives a sufficient condition for the stability of polynomially bounded solutions that is well known as Arendt-Batty-Ljubich-Vu Theorem (see [3]): Proof. It is clear that x(n) = T n x is a solution of Eq. (4.7). By the Spectral Radius Theorem the spectral radius r σ (T ) of T must satisfy r σ (T ) ≤ 1 because of the polynomial boundedness of T , so (4.8) is satisfied. By Theorem 4.6, we only need to check condition (4.9). We have 0 ≤ lim λ↓z (λ − ξ 0 )R(λ,T )x ν ≤ lim λ↓z sup n∈N (λ − ξ 0 )R(λ, T )T n x n ν ≤ lim λ↓z sup n∈N T n n ν (λ − ξ 0 )R(λ, T )x = sup n∈N T n n ν lim λ↓z (λ − ξ 0 )R(λ, T )x . Since T is ν-polynomially power bounded sup n∈N ( T n /n ν ) is finite, so (4.11) yields that condition (4.9) is satisfied. Asymptotic behavior of solutions of fractional difference equations. Consider fractional difference equations of the form ∆ α x(n) = T x(n) + y(n), n ∈ N, (4.13) where 0 < α ≤ 1, T ∈ L(X) and y ∈ c ν 0 (X). Definition 4.8. ([10, Definition 3.1]) Let T be a bounded operator defined on a Banach space X and α > 0. We call T the generator of an α-resolvent sequence if there exists a sequence of bounded and linear operator {S α (n)} n∈N ⊂ L(X) that satisfies the following properties i) S α (0) = I; ii) S α (n + 1) = k α (n + 1)I + T n j=0 k α (n − j)S α (j), for all n ∈ N. As shown in [10, Theorem 3.4] and a note before it, S α is determined by one of the following formulas: S α (n) = n j=0 Γ(n − j + (j + 1)α) Γ(n − j + 1)Γ(jα + α) T j ; iii) S α (n) = 1 2πi C z n ((z − 1) α z 1−α − T ) −1 dz, where C is a circle, centered at the origin of the complex plane, that encloses all spectral values of (z − 1) α z 1−α − T. Recall that the Z-transform of a sequence x := {x(n)} ∞ n=0 is defined as x(z) := ∞ j=0 x(j)z −j . (4.14) Let us denote D |z|>1 := {z ∈ C : |z| > 1}, and D |z|<1 := {z ∈ C : |z| < 1}. For each {x(n)} n∈N ∈ l ν ∞ (X) we will set x(0) = 0, so some properties of the Z-transform of sequences can be stated in the following: Proposition 4.11. Let {x(n)} n∈N and {y(n)} n∈N be in l ν ∞ (X). Then i)x(z) is a complex function in z ∈ D |z|>1 ; ii) Sx(z) = zx(z) − zx(0); iii) x * y(z) =x(z) ·ỹ(z). Proof. For the proof see e.g. [8,Chapter 6]. To study fractional difference equations (4.13) we will need the following analog of [12,Lemma 3.3]: Lemma 4.12. Let {x(n)} n∈N ∈ l ν ∞ (X) . If the Z-transformx(z) of the sequence x has a holomorphic extension to a neighborhood of z 0 ∈ Γ, then z 0 ∈ σ ν (x). Proof. Assume thatx(z) (with |z| > 1) can be extended to a holomorphic function g 0 (z) in B(z 0 , δ) with a sufficiently small positive δ. We will show that R(z, S)x (with |z| > 1) has a holomorphic extension in a neighborhood of z 0 . By setting x(0) = 0 we define a sequence {g k (z)} ∞ k=1 as follows: g k (z) := z k−1x (z) − k−1 j=0 z k−1−j x(j), k ∈ N. (4.15) We are going to prove that this defines a bounded function g(z) with z in a small disk B(z 0 , δ) := {z ∈ C : |z −z 0 | < δ}, and then, applying a necessary and sufficient condition for a locally bounded function to be holomorphic to prove that R(z, S)x is holomorphic. To prove the boundedness of g(z) in a small disk B(z 0 , δ) we will use a special maximum principle as in [12]. We have R(z, S)x := (z − S) −1 x = z −1 (I − z −1 S) −1 x = z −1 ∞ n=0 z −n S(n)x = ∞ n=0 z −n−1 S(n)x. Therefore, for z ∈ B(z 0 , δ) ∩ D |z|>1 and for each k ∈ N, [R(z, S)x](k) = ∞ n=0 z −n−1 x(n + k) = z −1 z kx (z) − k−1 j=0 z k−j x(j) = g k (z). By (3.4) and (3.5), for z ∈ B(z 0 , δ) ∩ D |z|>1 , there is a certain number C such that sup k∈N g k (z) k ν = g(z) ν = ∞ n=0 z −n−1 x(n + k) ∞ k=1 ν ≤ C (|z| − 1) ν+1 z ν . (4.16) On the other hand, for z ∈ B(z 0 , δ) ∩ D |z|<1 we have for all k ∈ N, g k (z) ≤ |z| k−1 g 0 (z) + k−1 j=0 |z| k−1−j x(j) ≤ |z| k−1 g 0 (z) + k−1 j=0 |z| k−1−j j ν x ν ≤ sup z∈B(z0,δ) g 0 (z) + x ν k−1 j=0 |z| k−1−j j ν = M k−1 j=0 |z| k−1−j j ν where M := sup z∈B(z0,δ) g 0 (z) + x ν . Hence, for all k ∈ N, g k (z) k ν ≤ M k−1 j=0 |z| k−1−j j k ν ≤ M k−1 j=0 |z| k−1−j ≤ M 1 − |z| . (4.17) By (4.16) and (4.17) we have proved that there is a positive number K such that for z ∈ B(z 0 , δ) and and for each k ∈ N, this estimate is valid: g k (z) k ν ≤ K ||z| − 1| ν+1 . (4.18) Applying the maximum principle Lemma 2.2 as in [12] to the function g k (z)/k ν gives the boundedness of g k (z)/k ν in B(z 0 , δ/2). In fact, it is clear that for each k ∈ N the function g k (z)/k ν is holomorphic in z ∈ B(z 0 , δ). Therefore, g k (z)/k ν is bounded by a number independent of k, so g(z) is bounded in B(z 0 , δ/2). We are now ready to apply Theorem 2.1, a criterion for a locally bounded function to be holomorphic. In fact, since the family W := {x * • p k , x * ∈ X * , p k : {x n } → x k , k ∈ N} is separating and x * • p k (g(·)) = x * (g k (·)) is holomorphic, the complex function g(z) is holomorphic for z ∈ B(z 0 , δ/2). At this point we have shown that g(z) is holomorphic for z ∈ B(z 0 , δ/2), and as g(z) = R(z, S)x for |z| > 1. This yields that R(z, S)x has a holomorphic extension g(z) to a neighborhood of z 0 . This completes the proof of the lemma. Definition 4.13. We denote by σ Z,ν (x) the set of all points ξ 0 on Γ such that the Z-transform of a sequence x := {x(n)} n∈N ∈ l ∞ ν (X) cannot be extended holomorphically to any neighborhood of ξ 0 , and call this set the Z-spectrum of the sequence x. In the simplest case where ν = 0, σ ν (x) may be different from σ Z,ν (x). In fact, the following numerical sequence x := {x(n)} n∈N ∈ l ∞ 0 (R), where x(n) := 0, n = 0, 1/n, n ∈ N is in c 0 (R). Obviously,x = 0, so σ(x) = ∅. However, 1 ∈ σ Z,ν (x) becausex(z) = ∞ j=1 z −j /j cannot be extended holomorphically to a neighborhood of 1. In general, we only have the following inclusion. Corollary 4.14. For each x := {x(n)} n∈N ∈ l ∞ ν (X), σ ν (x) ⊂ σ Z,ν (x). Proof. The corollary is an immediate consequence of Lemma 4.12 and the definitions of the spectra mentioned in the statement. Before we proceed, we introduce a notation Then σ ν (S α ) ⊂ Σ. Proof. It suffices to show that if z 0 ∈ Σ 0 , then z 0 ∈ σ ν (S α ). Taking the Z-transform of S α from the equation in Definition 4.8 gives zS α (z) − zS α (0) = SS α (z) = zk α (z)I − zk α (0)I +k α (z) · TS α (z). Therefore, for z ∈ D |z|>1 , (z −k α (z)T )S α (z) = zS α (0) + zk α (z)I − zk α (0)I. Let z 0 ∈ Σ 0 . Then (z −k α (z)T ) −1 exists. Hence, S α (z) = (z −k α (z)T ) −1 (zS α (0) + zk α (z)I − zk α (0)I). And, it is clear thatS α (z) has a holomorphic extension to a neighborhood of z 0 because bothk α (z) and (z −k α (z)T ) −1 are holomorphic in a neighborhood of z 0 , so z 0 ∈ σ Z,ν (S α ). By Corollary 4.14 this yields z 0 ∈ σ ν (S α ). This completes the proof of the lemma. Lemma 3. 1 . 1Assume thatS is the operator induced by the translation S in the quotient space l ν ∞ (X)/c ν 0 (X). Then σ(S) ⊂ Γ. . Let f ∈ l ν ∞ (X), and ξ 0 ∈ Γ be an isolated point in σ ν (f ).Then the singular point ξ 0 of R(λ,S)f is removable and the complex function R(λ,S)f is zero in the connected open subset of its domain that contains ξ 0 . Proof. By Proposition 3.4, ξ 0 is a pole of order up to ν + 1. Consider the Laurent series of R(λ,S)f in a neighborhood of ξ 0 we have 1), for each |λ| = 1 we have R(λ,S)Sx = R(λ,S)Tx + R(λ,S)F =T R(λ,S)x, This, together with the identity λR(λ,S)x −x = R(λ,S)Sx, shows λR(λ,S)x −x =T R(λ,S)x.Therefore,x = λR(λ,S)x −T R(λ,S)x = (λ −T )R(λ,S)x. Theorem 4 . 3 . 43Let T ∈ L(X) be ν-polynomially bounded such that σ(T ) ∩ Γ ⊂ {1}, where ν is a nonnegative integer. Then lim n→∞ 1 n ν (T − I) ν+1 T n = 0. (4.5) Proposition 3.4, 1 is a pole of ν + 1 order of the complex function g(λ) := R(λ,S)x, so the complex function λ → R(λ,S)(S − I) ν+1x Remark 4.4. A famous Katznelson-Tzafriri Theorem (see[9]) is stated as follows: Let T ∈ L(X) satisfy sup n∈N T n < ∞ and (σ(T ) ∩ Γ) ⊂ {1}. Then lim n→∞ (T n+1 − T n ) = 0. Theorem 4 . 5 . 45Let T ∈ L(X) satisfy σ(T ) ∩ Γ ⊂ {1}. Then, for each x 0 ∈ X, lim n→∞ 1 n ν (T − I) ν+1 T n x 0 Proof. The proof is similar to that of Theorem 4.3. In analogy to the sequence x in the proof of Theorem 4.3 we can use the sequence {T n x 0 } ∞ n=1 . Let us consider homogeneous linear difference equations of the form x(n + 1) = T x(n), n ∈ N, (4.7) Corollary 4 . 7 . 47Let T ∈ L(X) be ν-polynomially power bounded. Assume further that i) σ(T ) ∩ Γ is countable, ii) For each ξ 0 of σ(T ) ∩ Γ, and each x ∈ X, lim λ↓z (λ − ξ 0 )R(λ, T )x Theorem 4 . 9 . 49Let α > 0 and T be a bounded operator defined on a Banach space X. The following properties are equivalent: i) T is the generator of an α-resolvent sequence {S α (n)} n∈N ; ii) Theorem 4 . 410. ([10, Theorem 3.7]) Let 0 < α < 1 and {y(n)} n∈N is given. The unique solution of Eq. (4.13) with initial condition u(0) = x can be represented by u(n) = S α (n)u(0) + (S α * y)(n − 1), for all n ∈ N. Σ 0 0:={z 0 ∈ Γ ⊂ C : (z −k α (z)T ) −1 exists and (z −k α (z)T ) −1 andk α (z)are holomorphic in a neighborhood of z 0 } and Σ = Γ\Σ 0 . Lemma 4 . 15 . 415Let α > 0 and S α := {S α (n)} n∈N ⊂ L(X) be the resolvent of Eq. (4.13) that satisfies sup n∈N S α (n) n ν < ∞. Theorem 4 . 16 . 416Let 0 < α ≤ 1 and Σ ⊂ {1}. Assume further that the α-resolvent S α of Eq. (4.ν+1+k S α (n + k) = 0. (4.19)Proof. As in the proof of Theorem 4.3, we can show thatλ → R(λ,S)(S − I) ν+1Sα has a holomorphic extension to a neighborhood of 1 in the complex plane. Moreover, since σ ν (S α ) ⊂ Σ this function has a holomorphic extension to a neighborhood of all points of Γ. Namely, σ ν ((S − I) ν+1S α ) = ∅. Therefore, (S − I) ν+1 S α ∈ c ν 0 (X). In other words,lim n→∞ 1 n ν (S − I) ν+1 S α (n) − I) ν+1 = (−1) ν+1 (I − S) ν+1 = (k S k .This, together with (4.20), yields(4.19). The theorem is proved. Remark 4. 17 . 17When α = 1 Eq. (4.13) becomes x(n + 1) = (I + T )x(n) + y(n), n ∈ N. As shown in [10] S α (n) = (I + T ) n , n ∈ N. With this formula, (4.19) becomes lim n→∞ 1 n ν T ν+1 (I + T ) n . 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Anal., 68 (1986), 313-328. lp-maximal regularity for fractional difference equations on UMD spaces. C Lizama, Math. Nachr. 288C. Lizama, lp-maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288 (2015), 2079-2092. Asymptotic behavior of individual orbits of discrete systems. Minh Nguyen Van, Proc. Amer. Math. Soc. 1379Nguyen Van Minh, Asymptotic behavior of individual orbits of discrete systems, Proc. Amer. Math. Soc., 137 (2009), no. 9, 3025-3035. On the asymptotic behaviour of Volterra difference equations. Minh Nguyen Van, J. Difference Equ. Appl. 19Nguyen Van Minh, On the asymptotic behaviour of Volterra difference equations, J. Difference Equ. Appl., 19 (2013), 1317-1330. Boundedness and almost periodicity in dynamical systems. T Naito, Nguyen Van Minh, R Miyazaki, Y Hamaya, J. Difference Equ. Appl. 74T. Naito, Nguyen Van Minh, R. Miyazaki, Y. Hamaya, Boundedness and almost periodicity in dynamical systems, J. Difference Equ. Appl., 7 (2001), no. 4, 507-527. Some improvements of the Katznelson-Tzafriri theorem on Hilbert space. D Seifert, Proc. Amer. Math. Soc. 1439D. Seifert, Some improvements of the Katznelson-Tzafriri theorem on Hilbert space, Proc. Amer. Math. Soc., 143 (2015), no. 9, 3827-3838. A short proof of Y. Katznelson's and L. Tzafriri's theorem. Phong Quoc, Vu, Proc. Amer. Math. Soc. 1154Quoc Phong Vu, A short proof of Y. Katznelson's and L. Tzafriri's theorem, Proc. Amer. Math. Soc., 115 (1992), no. 4, 1023-1024. Theorems of Katznelson-Tzafriri type for semigroups of operators. Phong Quoc, Vu, J. Funct. Anal. 103Quoc Phong Vu, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Funct. Anal., 103 (1992), 74-84.

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{'abstract': 'In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by n ν , where ν is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form ∆ α x(n) = T x(n) + y(n), n ∈ N, where 0 < α ≤ 1. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the αresolvent operator Sα satisfies sup n∈N Sα(n) /n ν < ∞ and the set of z 0 ∈ C such that (z −k α (z)T ) −1 exists, and together withk α (z), is holomorphic in a neighborhood of z 0 consists of at most 1, wherek α (z) is the Z-transform of k α (n) := Γ(α + n)/(Γ(α)Γ(n + 1)), then lim n→∞', 'arxivid': '2003.05314', 'author': ['Nguyen Van Minh ', 'NGUYENHideaki Matsunaga ', 'Duc Huy ', 'Vu Trong Luong '], 'authoraffiliation': [], 'corpusid': 212658031, 'doi': '10.1619/fesi.65.261', 'github_urls': [], 'n_tokens_mistral': 14420, 'n_tokens_neox': 12783, 'n_words': 6901, 'pdfsha': '34a5c61e58afd65b6ff3da81d68d450435b9936d', 'pdfurls': ['https://arxiv.org/pdf/2003.05314v1.pdf'], 'title': ['A SPECTRAL THEORY OF POLYNOMIALLY BOUNDED SEQUENCES AND APPLICATIONS TO THE ASYMPTOTIC BEHAVIOR OF DISCRETE SYSTEMS', 'A SPECTRAL THEORY OF POLYNOMIALLY BOUNDED SEQUENCES AND APPLICATIONS TO THE ASYMPTOTIC BEHAVIOR OF DISCRETE SYSTEMS'], 'venue': []}

arxiv

Gate tunable lateral 2D pn junctions: an analytical study of its electrostatics Ferney A Chaves Anibal Pacheco-Sanchez David Jiménez Gate tunable lateral 2D pn junctions: an analytical study of its electrostatics 1Index Terms 2D pn junctionelectrostatic potentialdepletion width The electrostatics of two-dimensional (2D) lateral pn homojunctions considering the impact of electrostatic doping by means of two split bottom-gates are studied here. Analytical expressions are obtained from the solution of the 2D Poisson equation considering a depletion approximation. Straightforward analytical models for the electrostatic potential and the depletion width within both the dielectric and the 2D semiconductor are obtained for both the symmetrical and asymmetrical cases. In contrast to the case of devices with chemical doping, the obtained depletion width model of devices with electrostatic doping do not depend on the dielectric constant but only on the electrostatic potential and oxide thickness. The models describe the electrostatics of gate-tunable 2D pn junctions at arbitrary bias. A benchmark against numerical device simulations of MoS 2 -based pn junctions validate the analytical models. I. INTRODUCTION Two-dimensional (2D) based devices have been in the spotlight of the continuous downscaling trend of electronics due to their semiconductor thin-body, high-performance and integration feasibility into standard production lines [1]. 2D pn junctions are among the most versatile emerging devices for both electronics and optoelectronics applications [2]. Homojunctions and heterojunctions have been demonstrated with 2D materials with various levels of reproducibility depending on the fabrication approach [2], [3]. In contrast to chemical doping, electrostatic doping by a pair of bottom-gate contacts separated by a gap has been demonstrated to be a more straightforward solution in controlling the different carrier concentration regions in 2D lateral (L) PN junctions. Proof-of-concept devices based on these gate tunable (GT) 2D lateral pn junctions have been experimentally proven to be suitable for optoelectronics [4]- [10], electronics [11] and neuromorphic applications [11], [12]. Modeling approaches have been presented elsewhere [13]- [16] for chemically doped 2D pn junctions, however, descriptions of the internal physical phenomena in GT 2D junctions are scarce due to the few theoretical studies on this structure [17]. In contrast to three-dimensional PN junctions where only a one-dimensional (1D) Poisson equation and the completedepletion approximation are considered , electrostatics analysis in 2D junctions involves solving a 2D Poisson equation and the consideration of partial depletion in the transition between the depletion and quasi-neutral regions due to a weaker screening of the electric field [14], [16]. Numerical device simulations and semi-analytical solutions have been proposed for GT junctions previously for the specific case of symmetrical gate voltages [17]. In this work, a general analysis of the electrostatics of GT lateral 2D pn homojunctions, embracing both symmetric and asymmetric gate voltage tunning, yields a compact analytical model for the electrostatic potential at any location of the device cross-section. This work is organized as follows. The analytical models for the electrostatic potential and depletion width of the device under study are presented in section II. In section III, the models are applied to MoS 2 lateral pn junctions under different bias and with different device geometry and the results are discussed benchmarked with the output of an in-house numerical device simulation (NDS) tool [17]. A conclusion is provided in section IV followed by three useful appendix with details on the procedures to solve and evaluate equations in the main text. II. DEVICE ELECTROSTATICS The electrostatics of a GT 2D junction depicted in Fig. 1(a) can be made analytically treatable under the depletion approximation. Specifically, the electrostatic potential in the depletion region of the 2D semiconductor without chemical doping, as the one considered here, comes from the solution of the 2D Poisson equation under the assumption that there is no mobile charge inside. Fig. 1(b) shows a sketch of such a region with the boundary conditions assumed in this work. Outside of the depletion region, the electrostatic potential at each side of the pn junction, far away from the depletion region, is considered a constant and equal to the lowest and highest of the potential inside of the depletion region, those extreme values reached at the pand n-boundary of the quasi neutral region, respectively. For this study, the physical gap between bottom-gates, l gap , has been considered to be much shorter than the depletion width W d and hence, it has been neglected. The latter consideration has no impact on the results here as discussed below (cf. section III) and shown elsewhere [17]. The device width along the y-axis is large enough to consider that the junction is uniform in that direction. Thus, models presented here do not depend on the device width. Furthermore, an infinitesimally thin semiconductor considered here reduces the 2D Poisson equation to the Laplace's equation within the total computing region constrained by the following general boundary conditions: homogeneous Neumann boundary conditions at x = 0 and x = W d forcing a zero electric field outside the depletion region; Dirichlet boundary conditions φ = φ g (x) at the bottom-gate contacts (z = 0) whereas a non-homogeneous Neumman condition φ z = σ (φ )/ε ox has been set at the semiconductor plane (z = t ox ), where σ = q(p − n) is the charge density of the 2D semiconductor with p(n) as the hole( (electron) carrier density, and ε ox the dielectric permittivity. Notice that σ within the depletion region is zero for the 2D semiconductor without chemical doping as the one considered here. The electrostatic potential profile φ (x, z), obtained from the solution of the 2D Poisson's equation within the computing region (cf. Fig.1(b)), can be expressed as (see Appendix A for details on the solving process) t ox (a) V g1 V g2 l gap V 1 V 2 W d E C E V E F (0,0) (W d ,0) (0,t ox ) (w 1 ,0) f x =0 f x =0 f=f g (x) f z =s(f)/e ox z x (b)φ (x, z) = ∞ ∑ k=1 {A k cos (λ k x) cosh [λ k (t ox − z)]} ,(1) where A k = 2 λ k W d sin (λ k w 1 ) (φ 1 − φ 2 ) cosh (λ k t ox ) ,(2) with k = 1, 2, 3, . . ., λ k = (kπ) /W d , t ox the oxide thickness, w 1 the splitting point between the two gates where the electrostatic potential changes from φ 1 = V g1 in the gate 1 to φ 2 = V g2 in the gate 2. It can be inferred that for a symmetric electrostatic doping (V g1 = −V g2 ), w 1 = W d /2. Eqs. (1) and (2) provide a general solution of the electrostatic potential of GT 2D junctions for both symmetrical (φ 1 = −φ 2 ) and asymmetrical (φ 1 = −φ 2 ) electrostatic doping, provided that the depletion region parameters, namely, W d and w 1 , could be previously obtained. The latter parameters are obtained by considering the following: (i) the electrostatic potential is known at a quasi-neutral region along the 2D semiconductor, φ o1 = φ (0,t ox ), calculated at either the por n-type region (e.g., p-type), and (ii) that φ 1 (0, 0) = V g1 . For (i), an analysis of the 1D electrostatics in the z-direction is required. φ o1(o2) can be estimated from an analytical solution of a 1D metal-oxide-semiconductor (MOS) model obtained by considering the band profile shown in Fig. 2 and analyzed in detail in Appendix B. The 1D MOS model [17] results in a non-linear equation for φ o (cf. Eq. (B.2)), whose solution can be expressed by analytical piecewise functions. By considering an overdrive gate voltage V g embracing flat-band conditions (see definition in Appendix B), for a V g V th , φ o< = V g ,(3) and φ o> = E g 2q + kT q log exp (V g −V th )C ox qn 0 − 1 ,(4) for V g > V th , with the 2D semiconductor bandgap E g , the Boltzmann constant k, the absolute temperature T , the electric charge q, the oxide capacitance C ox , n 0 = g 2D kT with the band-edge density of states g 2D (see definition in Appendix B) and the threshold voltage V th defined as the 60% of E g /(2q) [17]. In order to obtain a smooth and continous analytical function for any gate voltage, an educated solution combining Eqs. (3) and (4) yields where φ o = φ m − φ 2 m − φ o> φ o< ,(5)V g f (x) dielectric x f o B A x z B A W m qVg>0 qV ox <0 0 Ei EF V f o E g (a) (b)φ m = 1 2 (φ o> + µφ o< ) ,(6) with µ as a fitting parameter. As shown in Fig By applying the conditions (i) and (ii) into Eq. (1), it is found that φ o1 = ∞ ∑ k=1   2 πk sin πk w 1 W d (φ 1 − φ 2 ) cosh πk t ox W d   ,(7) and φ 1 = ∞ ∑ k=1 2 πk sin πk w 1 W d (φ 1 − φ 2 ) ,(8) which are non-linear equations for the variables W d and w 1 . From Eq. (8) and by considering the convergence of the series ∑ ∞ k=1 [(1/(πk)) sin(πkx)] → (−1/2)(x − 1) ∀ x < 1, the following ratio is obtained w 1 W d = 1 − φ 1 φ 1 − φ 2 ,(9) from which it can be seen that for the symmetric case, i.e., φ 1 = −φ 2 , w 1 = W d /2 holds. By replacing Eq. (9) in Eq. (7), the following equation results (10) can be solved numerically to obtain W d and, consequently, w 1 from Eq. (9). Alternatively, a general analytical solution to obtain W d is proposed in this work by considering the first term (k = 1) of the sum in Eq. (10), which leads to − 1 2 φ o1 φ 2 = ∞ ∑ k=1   1 πkr sin (πkr) cosh πk t ox W d   ,(10)with r = φ 2 /(φ 2 − φ 1 ). Eq.W d = πt ox sech −1 − 1 2sinc(r) φ o1 φ 2 ,(11) being sinc(r) the normalized sinc function. Eq. (11) is valid for all values of r between 0 and 1 as long as t ox /W d ≥ 0.3 (cf. Appendix C). For the case of a GT 2D junction with symmetrical applied gate voltages, the depletion width is symmetrical. Hence, after some algebra, an expression for the depletion width for this case (r = 1/2) obtained from Eq. (10) reads W d = πt ox sech −1 − π 4 φ o1 φ 2 φ 1 =−φ 2 ,(12) where in contrast to a model for chemically doped 2D pn-junctions suggested elsewhere [13], [15], [16] there is no dependence on the oxide dielectric constant but only on its thickness. A phenomenological expression previously presented in [17] for W d of 2D junctions in this same scenario, is a particular case of the general analytical solution presented here (cf. Eq. (12)). The physics-based and straightforward W d expressions obtained here can be used also to calculate transport-related parameters of 2D junctions [18], [19], however, this is out of the scope of the present study. III. RESULTS AND DISCUSSION The model presented here has been evaluated considering a 2 µm-long MoS 2 lateral pn-junction. Unless stated otherwise, a 300 nm-thick SiO 2 oxide separates the back-gates and the 2D semiconductor. l gap is considered 0 as in the analytical model. Furthermore, previous studies have shown that l gap defines mostly W d by a linear relation for long enough values [17]. The device has been studied in two different scenarios: with symmetric (V g1 = −V g2 ) and asymmetric (V g1 = −V g2 ) electrostatic doping. Hereinafter, V gx indicates the overdrive gate voltage unless stated otherwise. Numerical device simulations (NDS) have been performed to benchmark the analytical approach presented here (cfs. Figs. [4][5]. W d in the NDS tool corresponds to the semiconductor V g -induced charge Q sc (x) is equal to 43.6% of the induced charge at each side of induced charge regions of the junction. Further details on the experimentally-calibrated in-house physics-based simulation tool can be found elsewhere [17]. In Fig. 4 and Fig. 5 both the 2D electrostatic potential inside the dielectric and the 1D electrostatic potential in the semiconductor for the symmetric and asymmetric cases are shown, respectively. Fig. 6 shows the depletion width dependence on the gate voltages for some symmetric and asymmetric cases with different values of t ox . For the symmetric case, gate voltages of ±1.5 V and ±10 V have been used, yielding r = 1/2 for each electrostatic doping. The symmetric electrostatic potential (cf. Eq. (1)) has been obtained with the model within the device depletion region for both inside the oxide and at the 2D semiconductor, provided W d (≈ 0.6 µm) has been calculated with Eq. The equipotential line at x = w 1 ≈ 0.3 µm in Fig. 4(a) corresponds to φ = 0. The symmetric distribution of the total electrostatic potential within the oxide can be observed along the x-direction, i.e., φ (0 ≤ x ≤ w 1 , z) = −φ (w 1 ≤ x ≤ W d , z). Interestingly, the electric field lines 1 in gate tunable lateral 2D pn-junctions here studied are different to those generated in chemically doped lateral 2D pn-junctions, as studied in refs. [14]- [16] where there is a non-zero component of the electric field perpendicular to the semiconductor plane inside the dielectric next to the semiconductor in the depletion region. However, for GT devices, such electric field component is vanished, and hence, there is a very weak dependence of W d (Eq. (11)) on ε ox in contrast with the chemical doped case. The analytical model successfully describes the symmetric electrostatic potential profile along the 2D semiconductor (at z = t ox ) as shown in Fig. 4(b) by comparing it with numerical simulation results of the same device at different bias. These curves exhibit depletion widths of ∼ 0.6 µm and 0.27 µm for V g1 = −V g2 = −1.5 V and V g1 = −V g2 = −10 V, respectively, which correspond to the calculated values by means of Eq. (12) with φ o1 = −0.75 V and −0.82 V from Eq. (5). The left (x < 0 in the computational region) and right (x > W D ) 2D semiconductor quasi-neutral regions are described by the minimum and maximum values of φ (obtained with the analytical approach used here, cf. Eq. (5)), respectively, in the corresponding depletion region. For the asymmetric case, the same MoS 2 -based pn junction studied above (t ox = 300 nm) has been biased at three differenct configurations. Fig. 5(a) shows the potential contour plot with equipotential lines from the analytical model for the case with V g1 = −20 V and V g2 = 10 V (r = 0.33) exhibiting W d ≈ 0.23 µm and w 1 ≈ 0.09 µm, as given by Eq. (9). The different electrostatic doping induced by each gate breaks the symmetry of φ within the computational region as observed by the analytical modeling (cf. Eq.(1)). A zero perpendicular electric field component next to the semiconductor can be observed, similar to the symmetric case, explaining the weak dependence of W d on ε ox even in the asymmetric case. x ( m) analyt. num. z=t ox V g1 =-40 V V g2 =20 V V g2 =45 V (b)w 1 from W d /2. Hence, φ (x − w 1 = 0,t ox ) = 0 V in contrast to the symmetric scenario. The analytical depletion width model (cf. Eqs. (11) and (12)) is able to describe NDS results of both symmetric and asymmetric GT MoS 2 junctions with different t ox . Fig. 6(a) shows the analytical (cf. Eq. (12)) and NDS results of W d for symmetric pn-junctions as a function of φ o1 /φ 2 . As example cases, the ratio takes values of 0.5 and 0.082 for the two symmetrical devices described in Fig. 4(b). The depletion widths obtained with the analytical model (cf. Eq. (11)) for asymmetric GT MoS 2 junctions with different t ox are compared to NDS results in Fig. 6(b). The electrostatic doping is enabled by applying V g1 = −40 V and sweeping V g2 from 20 V to 60 V, yielding a minimum and maximum values of r equal to 0.3 and 0.6, respectively. The V g2 -dependence of W d at different t ox is qualitatively captured by the model. The maximum relative error for the analytical results in comparison to the NDSs is of ∼ 21% for the devices under study. The origin of this deviation is discussed next. W d values from NDS could, in fact, be lower than the ones reported here in the asymmetric case due to the not unique definition of the computational space related to depletion region, specially its beginning and ending, i.e., the x-limits. In this work, we have assumed the same criterium of the 43.6% of the induced charge density in the quasi-neutral regions like in the symmetric case [16], [17]. However, due to the large transition region separating the fully depletion region and the quasineutral depletions regions exhibited by 2D lateral pn-junctions in comparison to 3D pn-junctions, as discussed elsewhere [16], this criterium could be relaxed for devices with asymmetric electrostatic doping and pristine 2D semiconductors towards an improved description of the NDS by the analytical model. Additionally, the 1D analytical model of φ o1(o2) might be another source of error since it only considers the impact of one gate voltage in contrast to the 2D numerical solution depending on both applied bias. By considering these discussions and by showing that all conditions are fulfilled, i.e., 0 < r < 1 and t ox /W d > 0.3, the analytical model of W d in the asymmetric scenario, can be used to set reliable minimum limit of values. IV. CONCLUSION The 2D Poisson equation has been solved analytically yielding models for the electrostatic potential and depletion width of GT 2D lateral pn-junctions with symmetric and asymmetric electrostatic doping applied by two separated bottom-gates. The analyitical φ model enables to elucidate the electrostatic potential within both the oxide and the 2D semiconductor, regardless the symmetric bias conditions. The analytical φ and W d models have been benchmarked against numerical device simulations of GT MoS 2 -based junctions. For the symmetric electrostatic doping scenario (V g1 = −V g2 ), the physics-based analytical model is able to describe the NDS results within the 2D semiconductor at all bias and for different oxide thickness; whereas for the asymmetric scenario (V g1 = −V g2 ), the model captures the bias dependence of practical cases for different oxide thicknesses while yielding reliable W d minimum values within a confidence range of 80%. The straightforward analytical models proposed here for the electrostatics of GT 2D junctions intend to be of aid for technology improvements by unveiling some aspects of the internal physical mechanisms as well as for the modeling community by considering the expressions provided here in electrostatics-dependent transport models. APPENDIX A. SOLUTION OF THE 2D POISSON EQUATION The electrostatic potential within the depletion region can be expressed as the product of an x-dependent function X(x) and a z-dependent function Z(z), i.e., φ (x, z) = XZ. Hence, by separation of variables, the Laplace's equation (∇ 2 φ = 0) reads Z ∂ 2 X ∂ x 2 + X ∂ 2 Z ∂ z 2 = 0, (A.1) from which a differential partial equation system can be obtained by dividing Eq. (A.1) by XZ such as ∂ 2 X ∂ x 2 + λ 2 X = 0, (A.2a) ∂ 2 Z ∂ z 2 − λ 2 Z = 0, (A.2b) with λ 2 = − 1 X ∂ 2 X ∂ x 2 = 1 Z ∂ 2 Z ∂ z 2 . (A.3) The solutions of Eq. (A.2) can be expressed in a general form as X(x) = a k sin (λ k x) + b k cos (λ k x) , (A.4a) Z(z) = c k exp (−λ k z) + d k exp (λ k z) , (A.4b) for λ 2 > 0 and X(x) = a k exp (−λ k x) + b k exp (λ k x) , (A.5a) Z(z) = c k sin (λ k z) + d k cos (λ k z) , (A.5b) for λ 2 < 0. In this work, the former case (λ 2 > 0) has been considered without loss of generality. By using Eq. (A.4a) and the Neumann boundary conditions φ x (x = 0) = 0 and φ x (x = W d ) = 0 (cf. Fig. 1(b)) lead to find a k = 0 from the former boundary condition and consequently, sin(λ k W d ) = 0 for the second one, from which the λ = kπ/W d is obtained where k = 1, 2, 3, . . .. Therefore, X(x) = b k cos kπ W d x . (A.6) Similarly, by considering Eq. (A.4b), the evaluation of φ z = 0 at z = t ox yields c k = d k exp(2λ k t ox ), leading to Z(z) = 2d k exp(λ k t ox ) cosh [λ k (t ox − z)] . (A.7) By using Eqs. (A.6) and (A.7) in the proposed definition of the electrostatic potential, Eq. (1) is found, whereas the definition of A k is obtained as follows. The value of the electrostatic potential at z = 0 along the x-direction φ (x, 0) = φ g (x) reads as φ g (x) = ∞ ∑ k=1 [A k cos(λ k x) cosh(λ k t ox )] . (A.8) Solving Eq. (A.8) for A k leads to A k = W d 0 φ g (x) cos(λ k x)dx cosh(λ k t ox ) W d 0 cos 2 (λ k x)dx . (A.9) The integral in the denominator yields W d /2. Since φ g (x) = φ 1 for x < w 1 and φ g (x) = φ 2 for x > w 1 , the integral in the numerator can be split in a sum of integrals such as W d 0 = w 1 0 + W d w 1 . Hence, w 1 0 φ 1 cos(λ k x)dx = φ 1 λ k sin(λ k w 1 ), (A.10a) W d w 1 φ 2 cos(λ k x)dx = − φ 2 λ k sin(λ k w 1 ). (A.10b) Eq. (2) is hence determined by substituing the solutions of the integrals in Eq. (A.9). APPENDIX B. 1D MOS MODEL The band profile of a transversal 1D section of the GT 2D pn-junction ( Fig. 2(a)) is shown in Fig. 2(b). The section is chosen in a way that the Fermi level energy E F considered is far away from the depletion region. From this band profile and by considering the charge conservation law, i.e., Q m + Q sc = 0, and the voltage Kirchoff's law, i.e., W m − qV ox = χ + E g /2 − qφ o + qV g , the following relation can be obtained χ + E g /2 − qφ o + qV g −W m C ox q + Q sc (φ o −V ) = 0 (B.1) with the semiconductor electron affinity χ, the band gap of the 2D semiconductor E g , the metal work function W m , the gate charge density Q m and the V g -induced carrier charge density Q sc (= σ ) in the semiconductor which is a function of the local electrostatic potential φ o and of an electrochemical potential V (arbitrarily referred). In order to calculate φ o , for instance in the n-region and assuming thermal equilibrium with V = 0, Q sc ≈ −qn = −qn 0 log 1 + exp (qφ o − E g /2)/(kT ) , where n 0 = g 2D kT = g v g s m/(2πh 2 ) kT defined by the bandedge effective mass, spin and valley degeneracy factors m, g v , g s of the 2D semiconductor, respectively. Therefore, Eq. (B.1) can be written as −φ o +V g C ox − qn 0 log 1 + exp qφ o − E g /2 /(kT ) = 0, (B.2) which is a transcendental equation for φ o with qV g = qV g −W m + χ + E g /2, where the term W g − χ − E g/2 describes the flat-band voltage. A piecewise analytical solution for Eq. (B.2) is given by Eqs. (3)-(5) in the main text. The model works regardless the device geometry and dielectric properties since in the hypothetical worst case where C ox = −qn 0 /(V g −V th ), i.e., φ o> = E g /(2q) (cf. Eq. (4)), a solution still exists for Eq. (5). For the analytical results presented in this work, the following parameter values have been considered: T = 300 K, g v = g s = 2, E g = 1.8 eV and m = 0.57m 0 where m 0 is the free electron mass. The same values have been used in the numerical device simulations in [17]. APPENDIX C: VALIDITY OF CONDITIONS FOR THE ANALYTICAL W d MODEL Fig. 7 shows results of Eq. (10) as a function of ξ = t ox /W d with k-terms. For the symmetric case ( Fig. 7(a)), φ o1 depends on V g1 = V g2 whereas, for the asymmetric case ( Fig. 7(b)), φ o1 is constant since V g1 is fixed and V g2 varies. f (ξ ) and g(ξ ) correspond to the summatory on the right side of Eq. (10) calculated with one (k=1) and several terms (k from 1 to 1200), respectively. It is observed that g(ξ ) ≈ f (ξ ) for ξ > 0.3 for the symmetric (Fig. 7(a)) and asymmetric cases ( Fig. 7(b)). The left side of Eq. (10) (cf. insets of Fig. 7) has values below 0.4 and 0.03 in the symmetric and asymmetric scenarios for the gate voltages used in each case. The latter indicates also that g(ξ ) ≈ f (ξ ) holds for all the devices and cases under study. These results validate Eq. (11) as equivalent to Eq. (10). Fig. 1 . 1(a) Bottom: schematic cross section of a gate tunable 2D pn-junction (not drawn to scale). Top: sketch of the device band diagram showing the depletion width. (b) Sketch of a symmetric depletion region showing critical coordinates and the boundary conditions required for solving the Poisson's equation within it, where φ x,z = ∂ φ /∂ x, z. For an asymmetric depletion region, w 1 is not the middle point in the x-direction. Fig. 2 . 2(a) Bottom: a cross-section of the GT 2D pn-junction showing only one gate. Top: sketch of the electrostatic potential profile over the x-direction along the semiconductor. (b) Band profile of the metal-oxide-semiconductor structure considering the layers across the cut AB shown in the bottom part of (a). Fig. 3 . 3. 3, the proposed analytical solution matches with the numerical device simulations of the 1D MOS model considering MoS 2 as the 2D semiconductor. φ o versus V g : numerical simulations (symbols) and analytical results with Eq. (5) (solid line). The model parameters are E g = 1.8 eV, t ox = 300 nm, ε ox = 3.9ε 0 and µ = 1.003. Fig. 4 . 4(12), as shown in Figs. 4(a) for V g1 = −V g2 = −1.5 V, and (b) for both gate voltages, respectively. Electrostatic potential of a 2D MoS 2 junction with symmetrical applied voltages. (a) Analytical results within the depletion region inside the oxide obtained with Eq. (1) for V g1 = −V g2 = −1.5 V. Dashed lines are the equipotential lines. (b) Analytical (lines) and numerical simulation results (symbols) along the 2 µm 2D semiconductor for V g1 = −V g2 = −1.5 V and -10 V. Fig. 5 . 5Electrostatic potential of a 2D MoS 2 junction with asymmetrical applied voltages (V g1 = −V g2 ). (a) Analytical results within the computational region obtained with Eq. (1) at V g1 = −20 V and V g2 = 10 V. (b) Analytical (lines) and numerical simulation results (symbols) along the 2D semiconductor at different gate voltages. Electrostatic potentials for the case with smallest asymmetry have been shifted to the right by an amount of 0.5 µm for visualization purposes. Fig. 5 ( 5b) shows analytical and numerical results of the electrostatic potential along the 2D semiconductor of GT MoS 2 junction at two different asymmetric bias sets: V g1 = −40 V; V g2 = 20 V (strong asymmetry r = 0.33) and for V g1 = −40 V; V g2 = 45 V (small asymmetry r = 0.53) exhibiting predicted W d equal to ∼ 0.23 µm and ∼ 0.20 µm, respectively. The more asymmetric the electrostatic doping is the more distant is Fig. 6 . 6Variation of the depletion width of (a) symmetric GT 2D junction with the ratio φ o1 /φ 2 for t ox equal to 60 nm, 100 nm, 200 nm and 300 nm, and of (b) asymmetric GT 2D junction with V g2 for 3 different t ox . Symbols are numerical device simulation results and lines are results from the analytical model proposed here. Fig. 7 . 7Right side of Eq. (10) calculated with one (k=1) and several terms (k from 1 to 1200) terms for (a) symmetric and (b) asymmetric electrostatic doping. Insets: results of the left side of Eq. (10). Dotted line indicates ξ = 0.3 and is added as a guide for the eye. Values considered for this study are t ox = 300 nm for all cases whereas φ o1 is equal to 0.89 for the asymmetric case. Electric field lines are perpendicular to the equipotential lines. Graphene and two-dimensional materials for silicon technology. 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{'abstract': 'The electrostatics of two-dimensional (2D) lateral pn homojunctions considering the impact of electrostatic doping by means of two split bottom-gates are studied here. Analytical expressions are obtained from the solution of the 2D Poisson equation considering a depletion approximation. Straightforward analytical models for the electrostatic potential and the depletion width within both the dielectric and the 2D semiconductor are obtained for both the symmetrical and asymmetrical cases. In contrast to the case of devices with chemical doping, the obtained depletion width model of devices with electrostatic doping do not depend on the dielectric constant but only on the electrostatic potential and oxide thickness. The models describe the electrostatics of gate-tunable 2D pn junctions at arbitrary bias. A benchmark against numerical device simulations of MoS 2 -based pn junctions validate the analytical models.', 'arxivid': '2302.02794', 'author': ['Ferney A Chaves ', 'Anibal Pacheco-Sanchez ', 'David Jiménez ', 'Ferney A Chaves ', 'Anibal Pacheco-Sanchez ', 'David Jiménez '], 'authoraffiliation': [], 'corpusid': 256213807, 'doi': '10.1109/tnano.2023.3238815', 'github_urls': [], 'n_tokens_mistral': 12269, 'n_tokens_neox': 10191, 'n_words': 6018, 'pdfsha': '9cad2e38d3a6f91b766be1c4302baf1786aafd3a', 'pdfurls': ['https://export.arxiv.org/pdf/2302.02794v1.pdf'], 'title': ['Gate tunable lateral 2D pn junctions: an analytical study of its electrostatics', 'Gate tunable lateral 2D pn junctions: an analytical study of its electrostatics', 'Gate tunable lateral 2D pn junctions: an analytical study of its electrostatics', 'Gate tunable lateral 2D pn junctions: an analytical study of its electrostatics'], 'venue': []}

arxiv

EXACT: How to Train Your Accuracy Ivan Karpukhin Stanislav Dereka Sergey Kolesnikov EXACT: How to Train Your Accuracy Classification tasks are usually evaluated in terms of accuracy. However, accuracy is discontinuous and cannot be directly optimized using gradient ascent. Popular methods minimize cross-entropy, hinge loss, or other surrogate losses, which can lead to suboptimal results.In this paper, we propose a new optimization framework by introducing stochasticity to a model's output and optimizing expected accuracy, i.e. accuracy of the stochastic model. Extensive experiments on linear models and deep image classification show that the proposed optimization method is a powerful alternative to widely used classification losses. Introduction Accuracy is one of the most used evaluation metrics in computer vision (LeCun et al., 1998;Krizhevsky et al., 2009;, natural language processing (Maas et al., 2011;Zhang et al., 2015), and tabular data classification (Arık & Pfister, 2021). While accuracy naturally appears in classification tasks, it is discontinuous and difficult to optimize directly. To tackle this problem, multiple surrogate losses were proposed, including cross-entropy and hinge loss (Wang et al., 2022). However, as shown below, in our toy example from Figure 1, decreasing cross-entropy or hinge loss can lead to a drop in accuracy. In other words, there is no direct connection between surrogate losses and accuracy optimization. In this work, we take a different approach to accuracy optimization. Our idea is to introduce stochasticity to the model's output and then optimize the expected accuracy, i.e. accuracy of the stochastic model, via gradient methods. We call the proposed method EXpected ACcuracy opTimization (EXACT). It directly optimizes the accuracy of the stochastic model in contrast to surrogate losses. Contributions of this work can be summarized as follows: 1 Tinkoff. Correspondence to: Ivan Karpukhin <[email protected]>. Hinge Cross-entropy EXACT Max accuracy Class 1 Class -1 Figure 1. The toy example, which demonstrates importance of accuracy optimization. The model consists of a single bias parameter (decision threshold), while scaling weight is assumed to be 1. EX-ACT achieves 100% accuracy, while cross-entropy and hinge loss misclassify one element. 1. We propose a new optimization framework for classification tasks. To the best of our knowledge, it is the first work, where the classification model's accuracy is directly optimized via gradient methods. 2. We provide an efficient method for evaluating the proposed loss function and its gradient. We do this by presenting a new algorithm for gradient propagation through the orthant integral of the multivariate normal PDF 2 . 3. We compare the quality of the proposed EXACT method with cross-entropy and hinge losses. According to our results, EXACT improves accuracy in multiple tabular and image classification tasks, including SVHN, CIFAR10, and CIFAR100. Model Logits distribution Accuracy loss Input Figure 2. EXACT training pipeline. The model predicts the mean and variance of the logit vector. EXACT's training objective estimates accuracy, which is differentiable for the stochastic model. Related Work Classification Losses One of the most used classification loss functions is crossentropy (CE), also known as negative log-likelihood (Wang et al., 2022). Minimization of cross-entropy reduces the difference between predicted class probabilities and true posteriors. If a model predicts true posteriors, then selecting the class with maximum probability will lead to maximum accuracy classification (Mitchell & Mitchell, 1997b). In practice, cross-entropy can lead to suboptimal results due to several reasons. First, we usually don't have to predict true posteriors in order to achieve maximum accuracy. Any model with logits of the true class exceeding other logits leads to the optimal performance. On the other hand, overfitting and local optima prevent cross-entropy training from true posterior prediction. The limitations of cross-entropy gave a rise to other classification losses. One of the widely used classification losses is hinge loss (Gentile & Warmuth, 1998). Unlike cross-entropy, hinge loss stops training when scores of ground truth classes exceed alternative scores with the required margin. In some problems, hinge loss shows results on par or better than cross-entropy (Epalle et al., 2021;Jin et al., 2014;Ozyildirim & Kiran, 2021;Peng & Liu, 2018). Loss functions, such as cross-entropy and hinge loss, correlate with accuracy but do not optimize it directly (Grabocka et al., 2019). For this reason, these methods are referred to as proxy or surrogate losses. In this work, we propose a different approach that directly optimizes the accuracy of a specially designed stochastic model. Surrogate Losses Beyond Accuracy Many machine learning methods, especially deep learning approaches, rely on gradient descent optimization (Mitchell & Mitchell, 1997a), which is applicable only to differentiable loss functions. Surrogate losses provide differentiable approximations for non-differentiable target metrics. However, optimization of surrogate losses does not necessary leads to a target metric optimization. Previous works pro-pose differentiable surrogate losses for metrics beyond accuracy, including ROC AUC (Calders & Jaroszewicz, 2007;Yuan et al., 2021) and F1 scores (Bénédict et al., 2021). For example, AUC ROC can be rewritten in terms of the Heaviside step function, which is approximated via a logistic function or polynomials. Unlike the above-mentioned approaches, we focus on a direct target metric optimization. Surrogate loss functions were proposed in domains such as metric learning and ranking (Kaya & Bilge, 2019;Calauzenes et al., 2012). Many of them use margin-based losses, similar to hinge loss (Weinberger & Saul, 2009). Some surrogate losses formulate target metrics in terms of the Heaviside step function, which is then approximated by smooth functions (Patel et al., 2021). The proposed EXACT method is designed for direct accuracy optimization and does not require any surrogate loss function. Furthermore, the idea of using stochastic prediction can potentially be extended beyond accuracy optimization and applied in other machine learning domains. Stochastic Prediction There are multiple ways to introduce stochasticity to the model's output. One way is to use probabilistic embeddings (Shi & Jain, 2019), where the model predicts a distribution of outputs rather than a single vector. For example, the model can predict a mean and covariance matrix of the multivariate normal distribution. This concept was used in many deep learning approaches, such as regression via mixture density networks (Bishop, 1994), variational autoencoders (Kingma & Welling, 2014), and metric learning (Chang et al., 2020). Another way to introduce stochasticity is to use Bayesian neural networks (BNNs) (Goan & Fookes, 2020). In BNNs, the model weights are treated as random variables, and the output depends on particular weights' values. One part of the proposed EXACT method is stochastic prediction, which is similar to probabilistic embeddings. However, unlike previous approaches, we use stochastic prediction for accuracy optimization. Motivation Let's consider a toy example where cross-entropy and hinge loss minimization produces suboptimal results in terms of accuracy. Suppose we solve a binary classification problem via the simple threshold model f (x) = x − b. The model predicts classỹ based on the sign of f (x): y = −1, f (x) < 0 1, f (x) ≥ 0 .(1) The goal of training is to fit the threshold parameter b. Suppose the training set consists of three points: -0.25, 0, and 0.25. The first two points have the label y = −1, and the last point has the label 1, as shown in Figure 1. Cross-entropy loss for binary classification is usually applied to the output of the logistic function g, which converts f (x) to the probability of the positive class: L CE (x, y) = − log g(yf (x)),(2)g(x) = 1 1 + e −x .(3) Binary hinge loss is computed as follows: L Hinge (x, y) = max(0, 1 − yf (x)).(4) Minimization of cross-entropy and hinge losses fails to find the optimal value of b, which must be in the interval (0, 0.25). The global minimum of the cross-entropy loss is b = 0.7, and 0.75 ≤ b ≤ 1 for hinge loss. Gradient ascent optimization of the proposed EXACT method, which we describe below, achieves perfect classification in this example, leading to b = 0.125. EXACT The accuracy metric is not differentiable and cannot be directly optimized via gradient ascent. The core idea behind EXACT is to introduce stochasticity to the model's output and then optimize the expected accuracy, i.e. accuracy of the stochastic model. Definitions The goal of classification training is to minimize the empirical risk of the mapping x −→ỹ, where x ∈ R d is an input feature vector andỹ ∈ 1, C is an output class label. In machine learning with gradient methods, including deep learning, the problem is usually solved by a parametric differentiable function f θ : R d −→ R C with parameters θ, which predicts score vectors (or logits) of output classes. Prediction is obtained by taking the class with the maximum score:ỹ In some equations below we will omit θ subscript for simplicity. θ (x) = arg max i∈1,C f θ (x) i .(5 In this work, we consider accuracy maximization, i.e. maximization of the correct classification probability of elements (x, y) from some dataset: A(θ) = E x,y 1(ỹ θ (x) = y). (6) θ = arg max θ A(θ),(7) whereθ is an optimal set of parameters, which we estimate using gradient ascent. Stochastic Model's Accuracy Let's consider a modification of the function f : x → (µ, σ) with σ > 0, which predicts the multivariate normal distribution of a score vector s rather than a single vector: s ∼ N (µ(x), σ 2 (x)I),(8)y θ (s) = arg max i∈1,C s i .(9) In this case, the predicted classỹ θ (s) is a discrete random variable, which depends on the value of the score vector s. We call such a model stochastic since it produces random output. Accuracy of the stochastic model is an expectation of accuracy w.r.t. values of the score vector: A(θ) = E x,y E s 1(ỹ θ (s) = y).(10) Since the indicator function is a Bernoulli random variable, we can replace the indicator's expectation with probability: A(θ) = E x,y P(ỹ θ (s) = y).(11) By taking into account Equation 9, we can rewrite accuracy as A(θ) = E x,y P(s y > max i =y s i ).(12) Here we assume that classification is incorrect if scores of two or more classes are equal to each other. Examples of the expected accuracy for different values of σ are presented in Figure 3. As we will show below, the accuracy of the stochastic model is differentiable w.r.t. mean vector µ. This property makes optimization via gradient ascent possible. Optimization In this section, we propose an effective approach to expected accuracy computation and differentiation. We base this method on reducing the original problem to an orthant integration of the multivariate normal probability density function (PDF). The latter can be effectively computed using algorithms proposed in previous works. Definition 4.1. Delta matrix D y of the order C for the ground truth label y is a matrix of size C − 1 × C with D y i,j =          1, j = y −1, j < y, i = j −1, j > y, i = j − 1 0, otherwise .(13) In other words, the delta matrix is used to compute differences between scores of different classes and has the following form: D y =             −1 0 . . . 0 1 0 . . . 0 0 −1 . . . 0 1 0 . . . 0 . . . . . . . . . 0 0 . . . −1 1 0 . . . 0 0 0 . . . 0 1 −1 . . . 0 . . . . . . . . . 0 0 . . . 0 1 0 . . . −1             .(14) Now we can state a key result, necessary for expected accuracy evaluation. Theorem 4.2. Suppose the scores vector s is distributed according to multivariate normal distribution N (µ, σ 2 I) in R C . In this case, the probability of the y-th score exceeding other scores can be represented as P(s y > max i =y s i ) = Ω+ N (t; D y µ σ , D y D T y )dt,(15) where N (t; µ, Σ) denotes multivariate normal PDF, D y is a delta matrix of the order C for the label y and Ω + : {t ∈ R C−1 , t i ≥ 0, i = 1, C − 1} is an orthant in R C−1 . The orthant integral in equation 15 can be effectively computed using Genz algorithm (Genz, 1992). The Equation 15 represents correct classification probability for a single element of the dataset. Total accuracy is computed according to equation 12 as an average of correct classification probabilities for all elements. Gradient ascent requires gradient computation w.r.t. parameters µ and σ. Both parameters are included only in the mean part of the normal distribution in Equation 15. Let's define m = D y µ σ ,(16)Σ = D y D T y .(17) Note, that the matrix Σ is constant for each element of the dataset and doesn't require gradients. Note also that Σ is a covariance matrix by construction and thus can be used as a parameter of a multivariate normal distribution. Gradients w.r.t. mean vector m can be computed using the following theorem. Theorem 4.3. Suppose the scores vector s is distributed according to multivariate normal distribution N (m, Σ) in R C−1 andŝ i is a random vector of all elements in s ex- cept i-th, conditioned on s i = −m i . Thenŝ i is normally- distributed with some parametersm i andΣ i and    Ω+ N (t; m, Σ)dt    mi = N (0; m i , Σ i,i ) Ω + N (t;m i ,Σ i )dt,(18) where N (t; µ, Σ) denotes multivariate normal PDF and Ω + : {t ∈ R C−2 , t i ≥ 0, i = 1, C − 2} is an orthant in R C−2 . Same as before, the orthant integral is computed using Genz algorithm (Genz, 1992). According to Theorem 4.3 and Equation 16, accuracy of the stochastic model is differentiable w.r.t. both µ(x) and σ(x). The gradient descent method can be applied to minimize the negative value of accuracy. Inference During inference, EXACT predicts a score vector with maximum density, i.e. µ vector. Variance prediction can be excluded from the model in inference time, leading to zero computational overhead. Improvements In this section, we analyze corner cases and training with EXACT loss and propose improvements aimed at training stabilization and better final optima. When σ is close to zero, the stochastic model's accuracy reduces to a single-point prediction. In this case EXACT loss resembles the accuracy of the traditional model and suffers from local accuracy optima. To achieve better optimum during training, EXACT has to start with large σ. However, if σ is very large, then the surface of EXACT loss function will interpolate prior class probabilities, as shown in Figure 4. In the latter case, EXACT completely smooths optima. To make the method more robust to large initial σ values, we incorporate margin to the loss function. EXACT margin is similar to that of hinge loss and affects the computation of the mean vector from Equation 16: m r = min (D y µ, r) σ ,(19) where min is applied element-wise and r is a margin value. The obtained m r value is then used instead of m in Equation 16 for loss and gradient computation. As shown in Figure 4, margin affects training with large σ values, while has almost no effect as σ decreases. RATIO AMBIGUITY According to Equation 15, µ and σ are used in EXACT loss only as a fraction µ σ . This leads to an ambiguity in the model's prediction because the multiplication of both values by the same number will not affect stochastic accuracy. To decouple the effects of µ and σ, we apply batch normalization (Ioffe & Szegedy, 2015) to the µ branch of the model. Batch normalization is a part of the loss function and is removed during inference. VARIANCE SCHEDULER While σ prediction subnetwork can be trained simultaneously with other parameters, we found it more beneficial to change σ according to the predefined schedule. One reason behind this is that gradient descent rapidly reduces predicted variance, converging to the closest (usually poor) local optimum. Slow reduction of the variance forces the algorithm to make more steps with large σ and results in higher final accuracy. GRADIENT NORMALIZER In practice, EXACT gradient norm largely depends on the output variance. As shown in Figure 5, cross-entropy and hinge losses achieve plateau during training, while EXACT gradient norm exponentially growth. To better control the update step, we divide EXACT's gradient by running mean statistics. The running mean is computed using exponential smoothing with a factor 0.9. Experimental Setup In this section, we describe data preparation and training details of the proposed method and the baselines. Model Architectures For tabular data classification, we use linear models. Scores (or logits) of the first class are fixed to zero. Scores of the remaining classes are predicted via linear transform Ax + b of input features x. This setup is equivalent to a binary classification model when the number of classes is equal to 2. We also evaluate logistic regression implementation from Sklearn (Pedregosa et al., 2011). For image classification datasets we use different neural network models to better match data complexity. For MNIST we use 10-layer plain M3 CNN architecture from the recent work , which achieved top-performing results on this dataset. For SVHN and CIFAR we use Wide ResNet (WRN) architecture (Zagoruyko & Komodakis, 2016), which achieved better performance than ResNet (He et al., 2016). Particularly, we use WRN-16-8 for SVHN and WRN-28-10 for CIFAR-10 / CIFAR-100. Tabular Data Preparation We evaluate linear models using datasets from UCI machine learning repository (Dua & Graff, 2017 Table 2. Test set accuracy (%) of linear models trained with different loss functions on 10 tabular datasets. Mean and STD of 5 runs with different seeds are reported. Sklearn training doesn't depend on the random seed and thus STD is always zero. numeric data with mean values and missing categorical data with most frequent elements. We also apply one-hot encoding to categorical data. All features are normalized using per-feature mean and STD statistics. For datasets without original test part, we split 20% randomly selected items for testing. Image Data Preparation We conduct experiments on MNIST (LeCun et al., 1998), SVHN (Netzer et al., 2011), CIFAR-10, and CIFAR-100 (Krizhevsky et al., 2009) datasets. We prefer small and medium-size datasets because hyperparameter tuning on large datasets is practically unattainable. For MNIST we use simple data transform consisting of random translation of the image by at most 20% of the original size and random rotation by at most 20 degrees . For SVHN we use auto augmentation (Cubuk et al., 2019) from PyTorch (Paszke et al., 2019a) with parameters designed for CIFAR-10. For CIFAR-10 and CIFAR-100 we use CIFAR-10 auto augmentations along with random horizontal flipping. All input images are scaled to the fixed size depending on the neural architecture to achieve spatial tensor size 8 × 8 before the final pooling layer. For M3, used for MNIST, images are scaled to 28 pixels on each side. For Wide ResNet architectures, used for SVHN, CIFAR-10, and CIFAR-100, images are scaled to 32 pixels on each side. Hyperparameters and Training The list of hyperparameters includes initial learning rate, gradient clipping threshold, and margin (for hinge loss and EXACT). Hyperparameters are tuned via 50 runs of random search with 20% of the training set reserved for validation. For linear models we also tune L2 regularization coefficient. All models, except Sklearn, are trained with batch size equal to 256. Linear models are trained for 8000 steps to balance training for different dataset sizes. The logistic regression from Sklearn is trained using standard L-BFGS solver (Byrd et al., 1995). Neural networks are trained for 150 epochs on MNIST and SVHN and for 500 epochs on CIFAR10 and CIFAR100. Models are trained via stochastic gradient descent with momentum equal to 0.9 and weight decay equal to 10 −4 . Learning rate exponentially decays from initial value to 10 −4 at last epoch. We also train neural networks with Adam (Kingma & Ba, 2015) and ASAM (Kwon et al., 2021) optimizers. EXACT variance scheduler reduces σ from 10 in the first epoch to the final value, which is 0.01 for linear models and 1 for neural networks. We use the PyTorch deep learning framework (Paszke et al., 2019b) for all models except logistic regression from Sklearn. Each experiment is performed using a single NVIDIA V100 GPU. EXACT implementation depends on the sample size used in the Genz integration algorithm. We use sample size 16 in all comparisons. Experiments In this work, we aim to answer the following questions: (1) how does EXACT performance correspond to that of crossentropy and hinge losses, (2) how to choose EXACT sample size, and (3) how big is the computational overhead of the proposed EXACT method? The answers to these questions are summarized in the subsections below. Classification Quality LINEAR MODELS We compared linear models trained with Sklearn, crossentropy loss, hinge loss, and the proposed EXACT loss. The comparison on 10 tabular datasets is presented in Tables 1 and 2. On 9 out of 10 datasets EXACT achieves the highest accuracy on the training part of the dataset. It demonstrates the effectiveness of the proposed loss function in accuracy maximization. During testing EXACT achieves the highest accuracy in 6 out of 10 cases. On 3 benchmarks EXACT strictly outperforms all other methods, including Sklearn's logistic regression trained with L-BFGS solver. DEEP IMAGE CLASSIFICATION We compared EXACT with surrogate losses on multiple datasets with different optimizers. The results are presented in Table 3. Hinge loss achieves the highest accuracy on MNIST dataset. On SVHN and CIFAR-10, results are mixed, but EXACT outperforms other methods in most cases. On CIFAR-100 EXACT strictly outperforms all baselines. Generally, EXACT outperforms other methods in 7 out of 12 comparisons. Sample Size EXACT depends on the sample size used in the Genz integration algorithm (Genz, 1992). Training results for different sample sizes are presented in Figure 6. On MNIST, SVHN, and CIFAR-10, the sample size has minor effects on accuracy. Even training with sample size 1 produces results on par with larger samples. On CIFAR-100, a larger sample size generally leads to higher accuracy. Computational Complexity EXACT memory consumption and computational efficiency depend on the number of classes C and sample size N . The most expensive part of the algorithm is gradient computation, which requires C calls to Genz algorithm with complexity O(N C). Thus the total complexity of EXACT loss is O(N C 2 ) in terms of both operations and memory. Empirical memory usage per data element and computational efficiency (milliseconds per data element) are presented in Figure 7. Evaluations for different numbers of classes were made with a sample size equal to 16. Different sample sizes were evaluated with the number of classes equal to 32. The relative EXACT overhead largely depends on the num- Discussion While previous works on deep classification minimized surrogate losses, EXACT directly optimizes the accuracy of the model with stochastic prediction. The benefits of accuracy optimization were illustrated in our toy example, where both cross-entropy and hinge losses failed to solve the problem. The proposed EXACT method treats non-differentiable accuracy metric in a novel way which can potentially be applied to domains beyond classification, such as metric learning, ranking, etc. According to our experiments, EXACT leads to competitive results in tabular and image classification tasks. In many cases, EXACT achieves higher classification accuracy with a computational overhead of about 0−6% for neural networks. While computational efficiency is dependent on sample size, we show that EXACT can be trained even with single-point estimation. Extra computational resources can be used to further increase EXACT accuracy. Conclusion We presented EXACT, a novel approach for optimizing stochastic model accuracy via gradient ascent. Our results show that EXACT achieves higher accuracy than crossentropy and hinge losses in several tabular and image classification tasks, including SVHN, CIFAR10, and CIFAR100. EXACT can be effectively implemented in popular deep learning frameworks, including PyTorch, and leads to the computational overhead of about 3% depending on the number of classes and neural model complexity. A. Proofs of Theorems Theorem 4.2. Suppose the scores vector s is distributed according to multivariate normal distribution N (µ, σ 2 I) in R C . In this case, the probability of the y-th score exceeding other scores can be represented as P(s y > max i =y s i ) = Ω+ N (t; D y µ σ , D y D T y )dt,(20) where N (t; µ, Σ) denotes multivariate normal PDF, D y is a delta matrix of the order C for the label y and Ω + : {t ∈ R C−1 , t i ≥ 0, i = 1, C − 1} is an orthant in R C−1 . Proof. Let's rewrite Equation 20: P(s y > max i =y s i ) = P(s y − max i =y s i > 0) (21) = P(min i =y (s y − s i ) > 0).(22) Due to the definition of the delta matrix D y , vector d = D y s is a (C − 1)-dimensional vector with elements equal to differences between the score of the ground truth class y and other scores: d i = s y − s i , i < y s y − s i+1 , i >= y ,(23) then P(s y > max i =y s i ) = P(min i d i > 0) (24) = P(min i d i σ > 0) (25) = P(min i w i > 0),(26) where w = Dys σ . Due to the properties of the multivariate normal distribution, random vector w is also normally distributed: w ∼ N ( D y µ σ , D y D T y ).(27) Finally: P(s y > max i =y s i ) = P(w 1 > 0, . . . , w C−1 > 0) (28) = Ω+ N (t; D y µ σ , D y D T y )dt.(29) Theorem 4.3. Suppose the scores vector s is distributed according to multivariate normal distribution N (m, Σ) in R C−1 andŝ i is a random vector of all elements in s except i-th, conditioned on s i = −m i . Thenŝ i is normallydistributed with some parametersm i andΣ i and    Ω+ N (t; m, Σ)dt    mi = N (0; m i , Σ i,i ) Ω + N (t;m i ,Σ i )dt,(30) where N (t; µ, Σ) denotes multivariate normal PDF and Ω + : {t ∈ R C−2 , t i ≥ 0, i = 1, C − 2} is an orthant in R C−2 . Proof. Ω+ N (t; m, Σ)dt (31) = ∞ 0 · · · ∞ 0 N (t; m, Σ)dt C−1 . . . dt 1 (32) = ∞ 0 · · · ∞ 0 N (t − m; 0, Σ)dt C−1 . . . dt 1 (33) = ∞ −m1 · · · ∞ −m C−1 N (t; 0, Σ)dt C−1 . . . dt 1 (34) Assume, without loss of generality, that i = 1. Then we can find the derivative using Leibniz integral rule:    Ω+ N (t; m, Σ)dt    m1 (35) = ∞ −m2 · · · ∞ −m C−1 N (t; 0, Σ)dt C−1 . . . dt 2 t1=−m1 (36) = ∞ 0 · · · ∞ 0 N (t; m, Σ)dt C−1 . . . dt 2 t1=0(37) Integration region in the last integral is a positive orthant Ω + , which is a lower dimension subset of Ω + :    Ω+ N (t; m, Σ)dt    m1 = Ω + N (t; m, Σ)dt C−1 . . . dt 2 t1=0 .(38) We can decompose inner density using properties of the multivariate normal distribution: P s (t) = P s (t 1 , . . . , t C−1 ) (39) = P s (t 1 , t 2:C−1 ) (40) = P s1 (t 1 )P s 2:C−1 (t 2:C−1 |s 1 = t 1 ), : m 1 = m 2:C−1 + Σ 2:C−1,1 Σ −1 1,1 (t 1 − m 1 ),(43) B. Implementation Notes EXACT loss computation algorithm with all improvements is presented in Listing 1. Listing 1: Example PyTorch Code for EXACT computation 1 mu = empty(batch_size, dim) # Input. 2 sigma = empty(batch_size) # Input. 3 mu_bn = (mu -mu.mean()) / mu.std() 4 deltas = mu_bn @ Dy.T 5 deltas_margin = deltas.clip(max=r) 6 ratio = deltas_margin / sigma [:, None] 7 loss = 1 -genz_integral(mean=ratio, cov=eye(dim) + 1) A considerable boost in EXACT performance can be obtained from analysis of the matrix Σ = D y D T y in Equation 20. Matrix D y is a delta matrix for the label y: D y =             −1 0 . . .            .(45) It can be seen, that D y D T y is independent of y: Σ = D y D T y =      2 1 . . . 1 1 2 . . . 1 . . . 1 1 . . . 2      .(46) Matrix Σ and it's Cholesky decomposition, used in Genz algorithm, can be computed before training. On the other hand, the special form of Σ largely simplifies equations for computingm i andΣ i . Cholesky decomposition of the matrix Σ has the following form: L =        α 1 0 . . . 0 β 1 α 2 . . . 0 . . . β 1 β 2 . . . 0 β 1 β 2 . . . α C−1        .(47) Note, that elements in each column below the diagonal are the same. Genz algorithm can be optimized to make use of this property of the matrix L. C. Backpropagation with Log-derivative Trick EXACT loss evaluates expectation of the form: L(x, y, θ) = E s∼N (µ,Σ) 1(ỹ(s) = y), where µ = µ(x, θ) and Σ = Σ(x, θ). While EXACT exploits properties of the multivariate normal distribution, there is a more general approach to compute the expectation and its gradients, called REINFORCE (Williams, 1992). The expectation itself can be computed using Monte-Carlo integration: L(x, y, θ) ≈ 1 N N i=1 1(ỹ(s i ) = y),(49) where s i are drawn from N (µ, Σ). Gradient computation can be performed using the log-derivative trick: ∇ θ L(x, y, θ) = E s∼N (µ,Σ) 1(ỹ(s) = y)∇ log N (s; µ, Σ),(50) where N (s; µ, Σ) is a multivariate normal PDF function. The latter expectation can also be evaluated using Monte-Carlo integration. We compared the performance of Genz integration algorithm (Genz, 1992) and EXACT extensions for gradient computation, with Monte-Carlo integration and logderivative trick. For this purpose we estimated the integral from Equation 20 and its gradient from Equation (30) µ was set to (1, 2, 0.5, 10, 6, −3, −4, 5, 1, 0). Each method was applied 1000 times with different random seeds. After that, we computed root-mean-square error (RMSE) of the integral values and ground truth value, computed with the sample size equal to 1000000. The results are presented in Figure 8 and Figure 9. It can be seen that EXACT algorithm produces an order of magnitude smaller error than Monte-Carlo with log-derivative trick. For example, Monte-Carlo requires a sample size 256 to achieve the gradient error of EXACT with sample size 1. This study highlights the importance of the proposed EXACT computation algorithm for stable training. D. Toy Example Derivation Cross-entropy loss for the toy example has the following form: L CE (b) = − log 1 1 + exp(−(0.25 + b)) − log 1 1 + exp(−b) − log 1 1 + exp(−(0.25 − b) ) . (51) By taking derivative of the loss function w.r.t. b, we get d db L CE (b) = −3.933e b + 1.284e 3b − 2.568 (e b + 1) (1.284e b + 1) (1.284 + e b ) . (52) The derivative equals zero at single point b ≈ 0.7. As L CE (b) is convex, it has single global minimum at b ≈ 0.7. The same procedure can be applied to minimize Hinge loss: L Hinge (b) = max(0, 1 + (−0.25 − b)) + max(0, 1 − b) + max(0, 1 − (0.25 − b)) (53) d db L Hinge (b) =                −2 b < − 3 4 −1 − 3 4 < b < 3 4 0 3 4 < b < 1 1 b > 1 N/A otherwise( Hinge Cross-entropy EXACT Max accuracy Figure 10. The toy example, which demonstrates importance of accuracy optimization. The model is a linear binary classifier with weight and bias parameters. EXACT achieves 80% accuracy, while cross-entropy and hinge loss minimization leads to 60% accuracy (global optimum). L Hinge (b) reaches minimum at 0.75 ≤ b ≤ 1. E. Toy Example with Learned Weight In the toy example from the main paper, we used the model with a single bias parameter, while weight was fixed to 1. Here we demonstrate another example, presented in Figure 10, with a trainable weight parameter. The dataset consists from 5 points in positions -6, -5, -4, 0, and 2. The labels are -1, 1, 1, -1, and 1. We minimized cross-entropy and hinge losses using grid search. Cross-entropy minimum is achieved for the threshold equal to -7.75, hinge loss is minimized at -11. Both cross-entropy and hinge loss lead to an accuracy 60%. Gradient descent optimization of EXACT leads to the threshold -5.35 and accuracy 80%. Figure 3 . 3Dependency of the expected accuracy on the model parameter in our toy example for different values of σ. Figure 4 . 4EXACT loss dependency on the model parameter with and w/o margin. Margin affects training with large σ, creating a better optimization landscape in early epochs. 4.5.1. MARGIN Figure 5 . 5Gradient norm during training on CIFAR-100 for different loss functions. Figure 7 . 7EXACT loss memory consumption (MB per element) and computation speed (ms per element) for different numbers of classes and sample sizes. Performance is compared to the Wide ResNet 16-8 neural network. soN (t; m, Σ) = N (t 1 ; m 1 , Σ 1,1 )N (t 2:C−1 ;m 1 ,Σ 1 ),(42)wherem 1 andΣ 1 are parameters of the conditional distribution from Equation 41 1 . (44) Substitution of the Equation 42 to Equation 38 finally proofs the theorem. . . . 0 1 −1 . . . 0 . . . . . . . . . 0 0 . . . 0 1 0 . . . −1 Figure 8 . 8. The covariance matrix Σ was computed using Equation 46 and Integral value error for Genz and Monte-Carlo algorithms with different sample sizes. Figure 9 . 9Gradient error for EXACT and Log-derivative trick with different sample sizes. Table 6 . 6Comparison of hinge loss with and without gradient normalizer on tabular datasets.Table 7. Hyperparameters for vision classification datasets.EXACT: How to Train Your AccuracyMethod Dataset SGD/ASAM Learning rate Adam Learning rate Gradient clipping Margin Range 0.001 -1 0.001 -1 0.01 -10 0 -10 Cross-entropy MNIST 0.5 0.001 10.0 N/A SVHN 1.0 0.001 0.1 N/A CIFAR-10 0.1 0.001 10.0 N/A CIFAR-100 0.5 0.001 1.0 N/A Hinge MNIST 0.05 0.001 1.0 5.0 SVHN 0.5 0.001 0.1 10.0 CIFAR-10 0.1 0.001 10.0 1.0 CIFAR-100 1.0 0.001 10.0 0.1 EXACT MNIST 0.5 0.001 N/A 0.5 SVHN 1.0 0.001 N/A 10.0 CIFAR-10 1.0 0.001 N/A 0.5 CIFAR-100 1.0 0.001 N/A 5.0 Table 8 . 8Hyperparameters for tabular classification datasets. https://github.com/tinkoff-ai/exact arXiv:2205.09615v3 [cs.LG] 21 Sep 2022 F. Ablation StudiesIn this section we compare methods with and without EXACT improvements on UCI datasets. First, we compare EXACT variants without margin, without variance scheduler, and without gradient normalizer with original EXACT method. Results are presented inTable 4. 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{'abstract': "Classification tasks are usually evaluated in terms of accuracy. However, accuracy is discontinuous and cannot be directly optimized using gradient ascent. Popular methods minimize cross-entropy, hinge loss, or other surrogate losses, which can lead to suboptimal results.In this paper, we propose a new optimization framework by introducing stochasticity to a model's output and optimizing expected accuracy, i.e. accuracy of the stochastic model. Extensive experiments on linear models and deep image classification show that the proposed optimization method is a powerful alternative to widely used classification losses.", 'arxivid': '2205.09615', 'author': ['Ivan Karpukhin ', 'Stanislav Dereka ', 'Sergey Kolesnikov '], 'authoraffiliation': [], 'corpusid': 248887344, 'doi': '10.48550/arxiv.2205.09615', 'github_urls': ['https://github.com/tinkoff-ai/exact'], 'n_tokens_mistral': 16020, 'n_tokens_neox': 13596, 'n_words': 7881, 'pdfsha': '4dc0980891488c9899d442b3370d9cf29f6fb56c', 'pdfurls': ['https://export.arxiv.org/pdf/2205.09615v3.pdf'], 'title': ['EXACT: How to Train Your Accuracy', 'EXACT: How to Train Your Accuracy'], 'venue': []}

arxiv

BEYOND AMLS: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING * 26 Nov 2017 Vassilis Kalantzis Yuanzhe Xi ANDYousef Saad BEYOND AMLS: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING * 26 Nov 2017Domain decompositionSchur complementsymmetric generalized eigenvalue prob- lemrational filteringparallel computing AMS subject classifications 65F1515A1865F50 This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem associated with each subdomain into two disjoint subproblems. The first subproblem is associated with the interface variables and accounts for the interaction among neighboring subdomains. To compute the solution of the original eigenvalue problem at the interface variables we leverage ideas from contour integral eigenvalue solvers. The second subproblem is associated with the interior variables in each subdomain and can be solved in parallel among the different subdomains using real arithmetic only. Compared to rational filtering projection methods applied to the original matrix pencil, the proposed technique integrates only a part of the matrix resolvent while it applies any orthogonalization necessary to vectors whose length is equal to the number of interface variables. In addition, no estimation of the number of eigenvalues lying inside the interval of interest is needed. Numerical experiments performed in distributed memory architectures illustrate the competitiveness of the proposed technique against rational filtering Krylov approaches. 1. Introduction. The typical approach to solve large and sparse symmetric eigenvalue problems of the form Ax = λM x is via a Rayleigh-Ritz (projection) process on a low-dimensional subspace that spans an invariant subspace associated with the nev ≥ 1 eigenvalues of interest, e.g., those located inside the real interval [α, β]. One of the main bottlenecks of Krylov projection methods in large-scale eigenvalue computations is the cost to maintain the orthonormality of the basis of the Krylov subspace; especially when nev runs in the order of hundreds or thousands. To reduce the orthonormalization and memory costs, it is typical to enhance the convergence rate of the Krylov projection method of choice by a filtering acceleration technique so that eigenvalues located outside the interval of interest are damped to (approximately) zero. For generalized eigenvalue problems, a standard choice is to exploit rational filtering techniques, i.e., to transform the original matrix pencil into a complex, rational matrix-valued function [8,29,21,34,22,30,37,7]. While rational filtering approaches reduce orthonormalization costs, their main bottleneck is the application the transformed pencil, i.e., the solution of the associated linear systems. An alternative to reduce the computational costs (especially that of orthogonalization) in large-scale eigenvalue computations is to consider domain decompositiontype approaches (we refer to [32,35] for an in-depth discussion of domain decomposition). Domain decomposition decouples the original eigenvalue problem into two separate subproblems; one defined locally in the interior of each subdomain, and one defined on the interface region connecting neighboring subdomains. Once the original eigenvalue problem is solved for the interface region, the solution associated with the interior of each subdomain is computed independently of the other subdomains [28,10,9,26,25,19,4]. When the number of variables associated with the interface region is much smaller than the number of global variables, domain decomposition approaches can provide approximations to thousands of eigenpairs while avoiding excessive orthogonalization costs. One prominent such example is the Automated Multi-Level Substructuring (AMLS) method [10,9,15], originally developed by the structural engineering community for the frequency response analysis of Finite Element automobile bodies. AMLS has been shown to be considerably faster than the NASTRAN industrial package [23] in applications where nev ≫ 1. However, the accuracy provided by AMLS is good typically only for eigenvalues that are located close to a user-given real-valued shift [9]. In this paper we describe the Rational Filtering Domain Decomposition Eigenvalue Solver (RF-DDES), an approach which combines domain decomposition with rational filtering. Below, we list the main characteristics of RF-DDES: 1) Reduced complex arithmetic and orthgonalization costs. Standard rational filtering techniques apply the rational filter to the entire matrix pencil, i.e., they require the solution of linear systems with complex coefficient matrices of the form A − ζM for different values of ζ. In contrast, RF-DDES applies the rational filter only to that part of A − ζM that is associated with the interface variables. As we show later, this approach has several advantages: a) if a Krylov projection method is applied, orthonormalization needs to be applied to vectors whose length is equal to the number of interface variables only, b) while RF-DDES also requires the solution of complex linear systems, the associated computational cost is lower than that of standard rational filtering approaches, c) focusing on the interface variables only makes it possible to achieve convergence of the Krylov projection method in even fewer than nev iterations. In contrast, any Krylov projection method applied to a rational transformation of the original matrix pencil must perform at least nev iterations. 2) Controllable approximation accuracy. Domain decomposition approaches like AMLS might fail to provide high accuracy for all eigenvalues located inside [α, β]. This is because AMLS solves only an approximation of the original eigenvalue problem associated with the interface variables of the domain. In contrast, RF-DDES can compute the part of the solution associated with the interface variables highly accurately. As a result, if not satisfied with the accuracy obtained by RF-DDES, one can simply refine the part of the solution that is associated with the interior variables. 3) Multilevel parallelism. The solution of the original eigenvalue problem associated with the interior variables of each subdomain can be applied independently in each subdomain, and requires only real arithmetic. Moreover, being a combination of domain decomposition and rational filtering techniques, RF-DDES can take advantage of different levels of parallelism, making itself appealing for execution in high-end computers. We report results of experiments performed in distributed memory environments and verify the effectiveness of RF-DDES. Throughout this paper we are interested in computing the nev ≥ 1 eigenpairs (λ i , x (i) ) of Ax (i) = λ i M x (i) , i = 1, . . . , n, for which λ i ∈ [α, β], α ∈ R, β ∈ R. The n × n matrices A and M are assumed large, sparse and symmetric while M is also positive-definite (SPD). For brevity, we will refer to the linear SPD matrix pencil A − λM simply as (A, M ). The structure of this paper is as follows: Section 2 describes the general working of rational filtering and domain decomposition eigenvalue solvers. Section 3 describes computational approaches for the solution of the eigenvalue problem associated with the interface variables. Section 4 describes the solution of the original eigenvalue problem associated with the interior variables in each subdomain. Section 5 combines all previous discussion into the form of an algorithm. Section 6 presents experiments performed on model and general matrix pencils. Finally, Section 7 contains our concluding remarks. 2. Rational filtering and domain decomposition eigensolvers. In this section we review the concept of rational filtering for the solution of real symmetric generalized eigenvalue problems. In addition, we present a prototype Krylov-based rational filtering approach to serve as a baseline algorithm, while also discuss the solution of symmetric generalized eigenvalue problems from a domain decomposition viewpoint. Throughout the rest of this paper we will assume that the eigenvalues of (A, M ) are ordered so that eigenvalues λ 1 , . . . , λ nev are located within [α, β] while eigenvalues λ nev+1 , . . . , λ n are located outside [α, β]. 2.1. Construction of the rational filter. The classic approach to construct a rational filter function ρ(ζ) is to exploit the Cauchy integral representation of the indicator function I [α,β] , where I [α,β] (ζ) = 1, iff ζ ∈ [α, β], and I [α,β] (ζ) = 0, iff ζ / ∈ [α, β]; see the related discussion in [8,29,21,34,22,30,37,7] (see also [17,37,36,5] for other filter functions not based on Cauchy's formula). Let Γ [α,β] be a smooth, closed contour that encloses only those nev eigenvalues of (A, M ) which are located inside [α, β], e.g., a circle centered at (α + β)/2 with radius (β − α)/2. We then have (2.1) I [α,β] (ζ) = −1 2πi Γ [α,β] 1 ζ − ν dν, where the integration is performed counter-clockwise. The filter function ρ(ζ) can be obtained by applying a quadrature rule to discretize the right-hand side in (2.1): (2.2) ρ(ζ) = 2Nc ℓ=1 ω ℓ ζ − ζ ℓ , where {ζ ℓ , ω ℓ } 1≤ℓ≤2Nc are the poles and weights of the quadrature rule. If the 2N c poles in (2.2) come in conjugate pairs, and the first N c poles lie on the upper half plane, (2.2) can be simplified into 2], where {ζ ℓ , ω ℓ } 1≤ℓ≤2Nc are obtained by numerically approximating I [−1,1] (ζ) by the Gauss-Legendre rule (left) and Midpoint rule (right). Notice that as N c increases, ρ(ζ) becomes a more accurate approximation of I [−1,1] (ζ) [37]. Throughout the rest of this paper, we will only consider the Midpoint rule [2]. The eigenvectors of ρ(M −1 A) are identical to those of (A, M ), while the corresponding eigenvalues are transformed to {ρ(λ j )} j=1,...,n . Since ρ(λ 1 ), . . . , ρ(λ nev ) are all larger than ρ(λ nev+1 ), . . . , ρ(λ n ), the eigenvalues of (A, M ) located inside [α, β] become the dominant ones in ρ(M −1 A). Applying a projection method to ρ(M −1 A) can then lead to fast convergence towards an invariant subspace associated with the eigenvalues of (A, M ) located inside [α, β]. (2.3) ρ(ζ) = 2ℜe Nc ℓ=1 ω ℓ ζ − ζ ℓ , when ζ ∈ R.(ζ) (scaled such that ρ(α) ≡ ρ(β) ≡ 1/2) in the interval ζ ∈ [−2, One popular choice as the projection method in rational filtering approaches is that of subspace iteration, e.g., as in the FEAST package [29,21,22]. One issue with this choice is that an estimation of nev needs be provided in order to determine the dimension of the starting subspace. In this paper, we exploit Krylov subspace methods to avoid the requirement of providing an estimation of nev. Compute w = 2ℜe Nc ℓ=1 ω ℓ (A − ζ ℓ M ) −1 M q (µ) 3. For κ = 1, . . . , µ 4. h κ,µ = w T q (κ) 5. w = w − h κ,µ q (κ) 6. End 7. h µ+1,µ = w 2 8. If h µ+1,µ = 0 9. generate a unit-norm q (µ+1) orthogonal to q (1) , . . . , q (µ) 10. putation of all eigenvalues of (A, M ) located inside [α, β] and associated eigenvectors. Line 2 computes the "filtered" vector w by applying the matrix function ρ(M −1 A) to q (µ) , which in turn requires the solution of the N c linear systems associated with matrices A − ζ ℓ M, ℓ = 1, . . . , N c . Lines 3-12 orthonormalize w against the previous Arnoldi vectors q (1) , . . . , q (µ) to produce the next Arnoldi vector q (µ+1) . Line 13 checks the sum of those eigenvalues of the upper-Hessenberg matrix H µ which are no less than 1/2. If this sum remains constant up to a certain tolerance, the outer loop stops. Finally, line 16 computes the Rayleigh quotients associated with the approximate eigenvectors of ρ(M −1 A) (the Ritz vectors obtained in line 15). Throughout the rest of this paper, Algorithm 2.1 will be abbreviated as RF-KRYLOV. Else 11. q (µ+1) = w/h µ+1, 2.3. Domain decomposition framework. Domain decomposition eigenvalue solvers [18,19,10,9] compute spectral information of (A, M ) by decoupling the original eigenvalue problem into two separate subproblems: one defined locally in the interior of each subdomain, and one restricted to the interface region connecting neighboring subdomains. Algebraic domain decomposition eigensolvers start by calling a graph partitioner [27,20] to decompose the adjacency graph of |A| + |M | into p non-overlapping subdomains. If we then order the interior variables in each subdomain before the interface variables across all subdomains, matrices A and M then take the following block structures: (2.5) A =        B 1 E 1 B 2 E 2 . . . . . . B p E p E T 1 E T 2 . . . E T p C        . . . M (p)T E M C         . If we denote the number of interior and interface variables lying in the jth subdomain by d j and s j , respectively, and set s = Under the permutation (2.5), A and M can be also written in a compact form as: (2.6) A = B E E T C , M = M B M E M T E M C . The block-diagonal matrices B and M B are of size d × d, where d = p i=1 d i , while E and M E are of size d × s. 2.3.1. Invariant subspaces from a Schur complement viewpoint. Domain decomposition eigenvalue solvers decompose the construction of the Rayleigh-Ritz projection subspace Z is formed by two separate parts. More specifically, Z can be written as (2.7) Z = U ⊕ Y, where U and Y are subspaces that are orthogonal to each other and approximate the part of the solution associated with the interior and interface variables, respectively. Let the ith eigenvector of (A, M ) be partitioned as (2.8) x (i) = u (i) y (i) , i = 1, . . . , n, where u (i) ∈ R d and y (i) ∈ R s correspond to the eigenvector part associated with the interior and interface variables, respectively. We can then rewrite Ax (i) = λ i M x (i) in the following block form (2.9) B − λ i M B E − λ i M E E T − λ i M T E C − λ i M C u (i) y (i) = 0. Eliminating u (i) from the second equation in (2.9) leads to the following nonlinear eigenvalue problem of size s × s: (2.10) [C − λ i M C − (E − λ i M E ) T (B − λ i M B ) −1 (E − λ i M E )]y (i) = 0. Once y (i) is computed in the above equation, u (i) can be recovered by the following linear system solution (2.11) (B − λ i M B )u (i) = −(E − λ i M E )y (i) . In practice, since matrices B and M B in (2.5) are block-diagonal, the p sub-vectors u (i) j ∈ R dj of u (i) = [(u (i) 1 ) T , . . . , (u (i) p ) T ] T can be computed in a decoupled fashion among the p subdomains as (2.12) (B j − λ i M (j) B )u (i) j = −(Ê j − λ iM (j) E )y (i) j , j = 1, . . . , p, where y (i) j ∈ R sj is the subvector of y (i) = [(y (i) 1 ) T , . . . , (y (i) p ) T ] T that corresponds to the jth subdomain. By (2.10) and (2.11) we see that the subspaces U and Y in (2.7) should ideally be chosen as Y = span y (1) , . . . , y (nev) , (2.13) U = span (B − λ1MB) −1 (E − λ1ME)y (1) , . . . , (B − λnevMB) −1 (E − λnevME)y (nev) . (2.14) The following two sections propose efficient numerical schemes to approximate these two subspaces. (1) , . . . , y (nev) }. In this section we propose a numerical scheme to approximate span{y (1) , . . . , y (nev) }. Approximation of span{y 3.1. Rational filtering restricted to the interface region. Let us define the following matrices: B ζ ℓ = B − ζ ℓ M B , E ζ ℓ = E − ζ ℓ M E , C ζ ℓ = C − ζ ℓ M C . Then, each matrix (A − ζ ℓ M ) −1 in (2.4) can be expressed as (3.1) (A − ζ ℓ M ) −1 = B −1 ζ ℓ + B −1 ζ ℓ E ζ ℓ S −1 ζ ℓ E H ζ ℓ B −1 ζ ℓ −B −1 ζ ℓ E ζ ℓ S −1 ζ ℓ −S −1 ζ ℓ E H ζ ℓ B −1 ζ ℓ S −1 ζ ℓ , where (3.2) S ζ ℓ = C ζ ℓ − E H ζ ℓ B −1 ζ ℓ E ζ ℓ , denotes the corresponding Schur complement matrix. Substituting (3.1) into (2.4) leads to ρ(M −1 A) =2ℜe    Nc ℓ=1 ω ℓ   B −1 ζ ℓ + B −1 ζ ℓ E ζ ℓ S −1 ζ ℓ E H ζ ℓ B −1 ζ ℓ −B −1 ζ ℓ E ζ ℓ S −1 ζ ℓ −S −1 ζ ℓ E H ζ ℓ B −1 ζ ℓ S −1 ζ ℓ      M. (3.3) On the other hand, we have for any ζ / ∈ Λ(A, M ): (3.4) (A − ζM ) −1 = n i=1 x (i) (x (i) ) T λ i − ζ . The above equality yields another expression for ρ(M −1 A): ρ(M −1 A) = n i=1 ρ(λ i )x (i) (x (i) ) T M (3.5) = n i=1 ρ(λ i ) u (i) (u (i) ) T u (i) (y (i) ) T y (i) (u (i) ) T y (i) (y (i) ) T M. (3.6) Equating the (2,2) blocks of the right-hand sides in (3.3) and (3.6), yields (3.7) 2ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ = n i=1 ρ(λ i )y (i) (y (i) ) T . Equation (3.7) provides a way to approximate span{y (1) , . . . , y (nev) } through the information in S −1 ζ ℓ . The coefficient ρ(λ i ) can be interpreted as the contribution of the direction y (i) in 2ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ . In the ideal case where ρ(ζ) ≡ ±I [α,β] (ζ), we have n i=1 ρ(λ i )y (i) (y (i) ) T = ± nev i=1 y (i) (y (i) ) T . In practice, ρ(ζ) will only be an approximation to ±I [α,β] (ζ), and since ρ(λ 1 ), . . . , ρ(λ nev ) are all nonzero, the following relation holds: (3.8) span{y (1) , . . . , y (nev) } ⊆ range ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ . The above relation suggests to compute an approximation to span{y (1) , . . . , y (nev) } by capturing the range space of ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ . A Krylov-based approach. To capture range ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ we consider the numerical scheme outlined in Algorithm 3.1. In contrast with RF-KRYLOV, Algorithm 3.1 is based on the Lanczos process [31]. Variable T µ denotes a µ × µ symmetric tridiagonal matrix with α 1 , . . . , α µ as its diagonal entries, and β 1 , . . . , β µ−1 as its off-diagonal entries, respectively. Line 2 computes the "filtered" vector w by applying ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ to q (µ) by solving the N c linear systems associated with matrices S ζ ℓ , ℓ = 1, . . . , N c . Lines 4-12 orthonormalize w against vectors q (1) , . . . , q (µ) in order to generate the next vector q (µ+1) . Algorithm 3.1 terminates when the trace of the tridiagonal matrices T µ and T µ−1 remains the same up to a certain tolerance. Algorithm 3.1. Krylov restricted to the interface variables 0. Start with q (1) ∈ R s , s.t. q (1) 2 = 1, q 0 := 0, b 1 = 0, tol ∈ R 1. For µ = 1, 2, . . . 2. Compute w = ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ q (µ) − b µ q (µ−1) 3. a µ = w T q (µ) 4. For κ = 1, . . . , µ 5. w = w − q (κ) (w T q (κ) ) 6. End 7. b µ+1 := w 2 8. If b µ+1 = 0 9. generate a unit-norm q (µ+1) orthogonal to q (1) , . . . , q (µ) 10. Else 11. q (µ+1) = w/b µ+1 12 EndIf 13. If the sum of eigenvalue of T µ remains unchanged (up to tol) during the last few iterations; BREAK; EndIf 14. End 15. Return Q µ = [q (1) , . . . , q (µ) ] Algorithm 3.1 and RF-KRYLOV share a few key differences. First, Algorithm 3.1 restricts orthonormalization to vectors of length s instead of n. In addition, Algorithm 3.1 only requires linear system solutions with S ζ instead of A−ζM . As can be verified by (3.1), a computation of the form (A − ζM ) −1 v = w, ζ ∈ C requires -in addition to a linear system solution with matrix S ζ -two linear system solutions with B ζ as well as two Matrix-Vector multiplications with E ζ . Finally, in contrast to RF-KRYLOV which requires at least nev iterations to compute any nev eigenpairs of the pencil (A, M ), Algorithm 3.1 might terminate in fewer than nev iterations. This possible "early termination" of Algorithm 3.1 is explained in more detail by Proposition 3.1. Proposition 3.1. The rank of the matrix ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ , (3.9) r(S) = rank ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ , satisfies the inequality (3.10) rank y (1) , . . . , y (nev) ≤ r(S) ≤ s. Proof. We first prove the upper bound of r(S). Since ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ is of size s× s, r(S) can not exceed s. To get the lower bound, let ρ(λ i ) = 0, i = nev + κ, . . . , n, where 0 ≤ κ ≤ n − nev. We then have ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ = nev+κ i=1 ρ(λ i )y (i) (y (i) ) T , and rank ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ = rank y (1) , . . . , y (nev+κ) . Since ρ(λ i ) = 0, i = 1, . . . , nev, we have r(S) = rank y (1) , . . . , y (nev+κ) ≥ rank y (1) , . . . , y (nev) . By Proposition 3.1, Algorithm 3.1 will perform at most r(S) iterations, and r(S) can be as small as rank y (1) , . . . , y (nev) . We quantify this with a short example for a 2D Laplacian matrix generated by a Finite Difference discretization with Dirichlet boundary conditions (for more details on this matrix see entry "FDmesh1" in Table 6.1) where we set [α, β] = [λ 1 , λ 100 ] (thus nev = 100). After computing vectors y (1) , . . . , y (nev) explicitly, we found that rank y (1) , . . . , y (nev) = 48. N c increases, the trailing s − rank y (1) , . . . , y (nev) singular values approach zero. Moreover, even for those singular values which are not zero, their magnitude might be small, in which case Algorithm 3.1 might still converge in fewer than r(S) iterations. Indeed, when N c = 16, Algorithm 3.1 terminates after exactly 36 iterations which is lower than r(S) and only one third of the minimum number of iterations required by RF-KRYLOV for any value of N c . As a sidenote, when N c = 2, Algorithm 3.1 terminates after 70 iterations. (1) , . . . , u (nev) }. Recall the partitioning of eigenvector x (i) in (2.8) and assume that its interface part y (i) is already computed. A straightforward approach to recover u (i) is then to solve the linear system in (2.11). However, this entails two drawbacks. First, solving the linear systems with each different B − λ i M B for all λ i , i = 1, . . . , nev might become prohibitively expensive when nev ≫ 1. More importantly, Algorithm 3.1 only returns an approximation to span{y (1) , . . . , y (nev) }, rather than the individual vectors y (1) , . . . , y (nev) , or the eigenvalues λ 1 , . . . , λ nev . Approximation of span{u In this section we alternatives for the approximation of span u (1) , . . . , u (nev) . Since the following discussion applies to all n eigenpairs of (A, M ), we will drop the superscripts in u (i) and y (i) . The basic approximation. To avoid solving the nev different linear systems in (2.11) we consider the same real scalar σ for all nev sought eigenpairs. The part of each sought eigenvector x corresponding to the interior variables, u, can then be approximated by (4.1)û = −B −1 σ E σ y. In the following proposition, we analyze the difference between u and its approximationû obtained by (4.1). Lemma 4.1. Suppose u andû are computed as in (2.11) and (4.1), respectively. Then: (4.2) u −û = −[B −1 λ − B −1 σ ]E σ y + (λ − σ)B −1 λ M E y. Proof. We can write u as (4.3) u = −B −1 λ E λ y = −B −1 λ (E σ − (λ − σ)M E )y = −B −1 λ E σ y + (λ − σ)B −1 λ M E y. The result in (4.2) follows by combining (4.1) and (4.3). We are now ready to compute an upper bound of u−û measured in the M B -norm. 1 Theorem 4.2. Let the eigendecomposition of (B, M B ) be written as (4.4) BV = M B V D, where D = diag(δ 1 , . . . , δ d ) and V = [v (1) , . . . , v (d) ] . Ifû is defined as in (4.1) and (δ ℓ , v (ℓ) ), ℓ = 1, . . . , d denote the eigenpairs of (B, M B ) with (v (ℓ) ) T M B v (ℓ) = 1, then (4.5) u −û MB ≤ max ℓ |λ − σ| |(λ − δ ℓ )(σ − δ ℓ )| ||E σ y|| M −1 B + max ℓ |λ − σ| |λ − δ ℓ | ||M E y|| M −1 B , Proof. Since M B is SPD, vectors E σ y and M E y in (4.3) can be expanded in the basis M B v (ℓ) as: (4.6) E σ y = M B ℓ=d ℓ=1 ǫ ℓ v (ℓ) , M E y = M B ℓ=d ℓ=1 γ ℓ v (ℓ) , where ǫ ℓ , γ ℓ ∈ R are the expansion coefficients. Based on (4.4) and noting that V T M B V = I, shows that (4.7) B −1 σ = V (D − σI) −1 V T , B −1 λ = V (D − λI) −1 V T . Substituting (4.6) and (4.7) into the right-hand side of (4.2) gives u −û = − V (D − λI) −1 − (D − σI) −1 V T M B ℓ=d ℓ=1 ǫ ℓ v (ℓ) + (λ − σ)V (D − λI) −1 V T M B ℓ=d ℓ=1 γ ℓ v (ℓ) = − ℓ=d ℓ=1 ǫ ℓ (λ − σ) (δ ℓ − λ)(δ ℓ − σ) v (ℓ) + ℓ=d ℓ=1 γ ℓ (λ − σ) δ ℓ − λ v (ℓ) . Now, taking the M B -norm of the above equation, we finally obtain ||u −û|| MB ≤ ℓ=d ℓ=1 ǫ ℓ (λ − σ) (δ ℓ − λ)(δ ℓ − σ) v (ℓ) MB + ℓ=d ℓ=1 γ ℓ (λ − σ) δ ℓ − λ v (ℓ) MB = ℓ=d ℓ=1 (λ − σ) (δ ℓ − λ)(δ ℓ − σ) ǫ ℓ v (ℓ) MB + ℓ=d ℓ=1 (λ − σ) δ ℓ − λ γ ℓ v (ℓ) MB ≤ max ℓ |λ − σ| |(λ − δ ℓ )(σ − δ ℓ )| ℓ=d ℓ=1 ǫ ℓ v (ℓ) MB + max ℓ |λ − σ| |λ − δ ℓ | ℓ=d ℓ=1 γ ℓ v (ℓ) MB = max ℓ |λ − σ| |(λ − δ ℓ )(σ − δ ℓ )| M −1 B E σ y MB + max ℓ |λ − σ| |λ − δ ℓ | M −1 B M E y MB = max ℓ |λ − σ| |(λ − δ ℓ )(σ − δ ℓ )| E σ y M −1 B + max ℓ |λ − σ| |λ − δ ℓ | M E y M −1 B . Theorem 4.2 indicates that the upper bound of ||u−û|| MB depends on the distance between σ and λ, as well as the distance of these values from the eigenvalues of (B, M B ). This upper bound becomes relatively large when λ is located far from σ, while, on the other hand, becomes small when λ and σ lie close to each other, and far from the eigenvalues of (B, M B ). Enhancing accuracy by resolvent expansions. Consider the resolvent expansion of B −1 λ around σ: (4.8) B −1 λ = B −1 σ ∞ θ=0 (λ − σ)M B B −1 σ θ . By (4.2), the error u −û consists of two components: i) (B −1 λ − B −1 σ )E σ y; and ii) (λ − σ)B −1 λ M E y. An immediate improvement is then to approximate B −1 λ by also considering higher-order terms in (4.8) instead of B −1 σ only. Furthermore, the same idea can be repeated for the second error component. Thus, we can extractû by a projection step from the following subspace (4.9) u ∈ {B −1 σ E σ y, . . . , B −1 σ M B B −1 σ ψ−1 E σ y, B −1 σ M E y, . . . , B −1 σ M B B −1 σ ψ−1 M E y}. The following theorem refines the upper bound of u −û MB whenû is approximated by the subspace in (4.9) and ψ ≥ 1 resolvent expansion terms are retained in (4.8). Theorem 4.3. Let U = span {U 1 , U 2 } where U 1 = B −1 σ E σ y, . . . , B −1 σ M B B −1 σ ψ−1 E σ y , (4.10) U 2 = B −1 σ M E y, . . . , B −1 σ M B B −1 σ ψ−1 M E y . (4.11) Ifû := arg min g∈U u − g MB , and (δ ℓ , v (ℓ) ), ℓ = 1, . . . , d denote the eigenpairs of (B, M B ), then: (4.12) u −û MB ≤ max ℓ |λ − σ| ψ ||E σ y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | + max ℓ |λ − σ| ψ+1 ||M E y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | . Proof. Define a vector g : = U 1 c 1 + U 2 c 2 where (4.13) c 1 = − 1, λ − σ, . . . , (λ − σ) ψ−1 T , c 2 = λ − σ, . . . , (λ − σ) ψ T . If we equate terms, the difference between u and g satisfies u − g = − B −1 λ − B −1 σ ψ−1 θ=0 (λ − σ)M B B −1 σ θ E σ y (4.14) + (λ − σ) B −1 λ − B −1 σ ψ−1 θ=0 (λ − σ)M B B −1 σ θ M E y. Expanding B −1 σ and B −1 λ in the eigenbasis of (B, M B ) gives B −1 λ − B −1 σ ψ−1 θ=0 (λ − σ)M B B −1 σ θ = (λ − σ) ψ V (D − λI) −1 (D − σI) −ψ V T ,(4.15) and thus (4.14) can be simplified as u − g = − (λ − σ) ψ V (D − λI) −1 (D − σI) −ψ V T E σ y + (λ − σ) ψ+1 V (D − λI) −1 (D − σI) −ψ V T M E y. Plugging in the expansion of E σ y and M E y defined in (4.6) finally leads to (4.16) u − g = ℓ=d ℓ=1 −ǫ ℓ (λ − σ) ψ (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) + ℓ=d ℓ=1 γ ℓ (λ − σ) ψ+1 (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) . Considering the M B -norm gives u − g MB ≤ ℓ=d ℓ=1 −ǫ ℓ (λ − σ) ψ (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) MB + ℓ=d ℓ=1 γ ℓ (λ − σ) ψ+1 (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) MB ≤ max ℓ |λ − σ| ψ |(λ − δ ℓ )(σ − δ ℓ ) ψ | ℓ=d ℓ=1 ǫ ℓ v (ℓ) MB + max ℓ |λ − σ| ψ+1 |(λ − δ ℓ )(σ − δ ℓ ) ψ | ℓ=d ℓ=1 γ ℓ v (ℓ) MB = max ℓ |λ − σ| ψ ||E σ y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | + max ℓ |λ − σ| ψ+1 ||M E y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | . Sinceû is the solution of min g∈U u − g MB , it follows that u −û MB ≤ u − g MB . A comparison of the bound in Theorem 4.3 with the bound in Theorem 4.2 indicates that one may expect an improved approximation when σ is close to λ. Numerical examples in Section 6 will verify that this approach enhances accuracy even when |σ − λ| is not very small. Enhancing accuracy by deflation. Both Theorem 4.2 and Theorem 4.3 imply that the approximation error u −û might have its largest components along those eigenvector directions associated with the eigenvalues of (B, M B ) located the closest to σ. We can remove these directions by augmenting the projection subspace with the corresponding eigenvectors of (B, M B ). Theorem 4.4. Let δ 1 , δ 2 , . . . , δ κ be the κ eigenvalues of (B, M B ) that lie the closest to σ, and let v (1) , v (2) , . . . , v (κ) denote the corresponding eigenvectors. Moreover, let U = span {U 1 , U 2 , U 3 } where Ifû := arg min g∈U u − g MB and (δ ℓ , v (ℓ) ), ℓ = 1, . . . , d denote the eigenpairs of (B, M B ), then: U 1 = B −1 σ E σ y, . . . , B −1 σ M B B −1 σ ψ−1 E σ y , (4.17) U 2 = B −1 σ M E y, . . . , B −1 σ M B B −1 σ ψ−1 M E y , (4.18) U 3 = v (1) , v (2) , . . . , v (κ) .(4.20) u −û MB ≤ max ℓ>κ |λ − σ| ψ ||E σ y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | + max ℓ>κ |λ − σ| ψ+1 ||M E y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | . Proof. Let us define the vector g := U 1 c 1 + U 2 c 2 + U 3 c 3 where c 1 = − 1, λ − σ, . . . , (λ − σ) ψ−1 T , c 2 = λ − σ, . . . , (λ − σ) ψ T , c 3 = γ 1 (λ − σ) ψ+1 − ǫ 1 (λ − σ) ψ (δ 1 − λ)(δ 1 − σ) ψ , . . . , γ κ (λ − σ) ψ+1 − ǫ κ (λ − σ) ψ (δ κ − λ)(δ κ − σ) ψ T . Since c 1 and c 2 are identical to those defined in (4.13), we can proceed as in (4.16) and subtract U 3 c 3 . Then, u − g = ℓ=d ℓ=1 −ǫ ℓ (λ − σ) ψ (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) + ℓ=d ℓ=1 γ ℓ (λ − σ) ψ+1 (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) − ℓ=κ ℓ=1 −ǫ ℓ (λ − σ) ψ (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) − ℓ=κ ℓ=1 γ ℓ (λ − σ) ψ+1 (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) = ℓ=d ℓ=κ+1 −ǫ ℓ (λ − σ) ψ (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) + ℓ=d ℓ=κ+1 γ ℓ (λ − σ) ψ+1 (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) Considering the M B -norm of u − g gives u −ĝ MB ≤ ℓ=d ℓ=κ+1 −ǫ ℓ (λ − σ) ψ (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) MB + ℓ=d ℓ=κ+1 γ ℓ (λ − σ) ψ+1 (δ ℓ − λ)(δ ℓ − σ) ψ v (ℓ) MB ≤ max ℓ>κ |λ − σ| ψ ||E σ y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | + max ℓ>κ |λ − σ| ψ+1 ||M E y|| M −1 B |(λ − δ ℓ )(σ − δ ℓ ) ψ | , where, as previously, we made use of the expression of E σ y and M E y in (4.6). Sincê u is the solution of min g∈U u − g MB , it follows that u −û MB ≤ u −ĝ MB . The RF-DDES algorithm. In this section we describe RF-DDES in terms of a formal algorithm. B ), and stores these eigenvectors in V j ∈ R dj×nev (j) B , j = 1, . . . , p. As our current implementation stands, these eigenvectors are computed by Lanczos combined with shift-and-invert; see [16]. Moreover, while in this paper we do not consider any special mechanisms to set the value of nev (j) B , it is possible to adapt the work in [38]. The next step of RF-DDES is to call Algorithm 3.1 and approximate span{y (1) , . . . , y (nev) } by range{Q}, where Q denotes the orthonormal matrix returned by Algorithm 3.1. RF-DDES then builds an approximation subspace as described in Section 5.1 and performs a Rayleigh-Ritz (RR) projection to extract approximate eigenpairs of (A, M ). The complete procedure is shown in Algorithm 5.1. Compute the eigenvectors associated with the nev The Rayleigh-Ritz eigenvalue problem at step 7) of RF-DDES can be solved either by a shift-and-invert procedure or by the appropriate routine in LAPACK [11]. 5.1. The projection matrix Z. Let matrix Q returned by Algorithm 3.1 be written in its distributed form among the p subdomains, (5.1) Q =        Q 1 Q 2 . . . Q p        , where Q j ∈ R si×µ , j = 1, . . . , p is local to the jth subdmonain and µ ∈ N * denotes the total number of iterations performed by Algorithm 3.1. By defining (5.2) B (j) σ = B j − σM (j) B , Φ (j) σ = E j − σM (j) E Q j , Ψ (j) = M (j) E Q j , the Rayleigh-Ritz projection matrix Z in RF-DDES can be written as: (5.3) Z =           V 1 −Σ (ψ) 1 Γ (ψ) 1 V 2 −Σ (ψ) 2 Γ (ψ) 2 . . . . . . . . . V p −Σ (ψ) p Γ (ψ) p [Q, 0 s,(ψ−1)µ ]           , where 0 χ,ψ denotes a zero matrix of size χ × ψ, and (5.4) Σ (ψ) j = (B (j) σ ) −1 Φ (j) σ , (B (j) σ ) −1 M (j) B (B (j) σ ) −1 Φ (j) σ , . . . , (B (j) σ ) −1 M (j) B (B (j) σ ) −1 ψ−1 Φ (j) σ , Γ (ψ) j = (B (j) σ ) −1 Ψ (j) , (B (j) σ ) −1 M (j) B (B (j) σ ) −1 Ψ (j) , . . . , (B (j) σ ) −1 M (j) B (B (j) σ ) −1 ψ−1 Ψ (j) . When M E is a nonzero matrix, the size of matrix Z is n × (κ + 2ψµ). However, when M E ≡ 0 d,s , as is the case for example when M is the identity matrix, the size of Z reduces to n × (κ + ψµ) since Γ (ψ) j ≡ 0 dj,ψµ . The total memory overhead associated with the jth subdomain in RF-DDES is at most that of storing d j (nev (j) B + 2ψµ) + s i µ floating-point numbers. Main differences with AMLS. Both RF-DDES and AMLS exploit the domain decomposition framework discussed in Section 2.3. However, the two methods have a few important differences. In contrast to RF-DDES which exploits Algorithm 3.1, AMLS approximates the part of the solution associated with the interface variables of (A, M ) by solving a generalized eigenvalue problem stemming by a first-order approximation of the nonlinear eigenvalue problem in (2.10). More specifically, AMLS approximates span y (1) , . . . , y (nev) by the span of the eigenvectors associated with a few of the eigenvalues of smallest magnitude of the SPD pencil (S(σ), −S ′ (σ)), where σ is some real shift and S ′ (σ) denotes the derivative of S(.) at σ. In the standard AMLS method the shift σ is zero. While AMLS avoids the use of complex arithmetic, a large number of eigenvectors of (S(σ), −S ′ (σ)) might need be computed. Moreover, only the span of those vectors y (i) for which λ i lies sufficiently close to σ can be captured very accurately. In contrast, RF-DDES can capture all of span y (1) , . . . , y (nev) to high accuracy regardless of where λ i is located inside the interval of interest. Another difference between RF-DDES and AMLS concerns the way in which the two schemes approximate span u (1) , . . . , u (nev) . As can be easily verified, AMLS is similar to RF-DDES with the choice ψ = 1 [9]. While it is possible to combine AMLS with higher values of ψ, this might not always lead to a significant increase in the accuracy of the approximate eigenpairs of (A, M ) due to the inaccuracies in the approximation of span y (1) , . . . , y (nev) . In contrast, because RF-DDES can compute a good approximation to the entire space span y (1) , . . . , y (nev) , the accuracy of the approximate eigenpairs of (A, M ) can be improved by simply increasing ψ and/or nev 6. Experiments. In this section we present numerical experiments performed in serial and distributed memory computing environments. The RF-KRYLOV and RF-DDES schemes were written in C/C++ and built on top of the PETSc [14, 13,6] and Intel Math Kernel (MKL) scientific libraries. The source files were compiled with the Intel MPI compiler mpiicpc, using the -O3 optimization level. For RF-DDES, the computational domain was partitioned to p non-overlapping subdomains by the METIS graph partitioner [20], and each subdomain was then assigned to a distinct processor group. Communication among different processor groups was achieved by means of the Message Passing Interface standard (MPI) [33]. The linear system solutions with matrices A− ζ 1 M, . . . , A− ζ Nc M and S(ζ 1 ), . . . , S(ζ Nc ) were performed by the Multifrontal Massively Parallel Sparse Direct Solver (MUMPS) [3], while those with the block-diagonal matrices B ζ1 , . . . , B ζN c , and B σ by MKL PARDISO [1]. The quadrature node-weight pairs {ω ℓ , ζ ℓ }, ℓ = 1, . . . , N c were computed by the Midpoint quadrature rule of order 2N c , retaining only the N c quadrature nodes (and associated weights) with positive imaginary part. Unless stated otherwise, the default values used throughout the experiments are p = 2, N c = 2, and σ = 0, while nev (1) B = . . . = nev (p) B = nev B . The stopping criterion in Algorithm 3.1, was set to tol = 1e-6. All computations were carried out in 64-bit (double) precision, and all wall-clock times reported throughout the rest of this section will be listed in seconds. Table 6.2 Maximum relative errors of the approximation of the lowest nev = 100 eigenvalues returned by RF-DDES for the matrix pencils in Table 6.1. nev B = 50 nev B = 100 nev B = 200 ψ = 1 ψ = 2 ψ = 3 ψ = 1 ψ = 2 ψ = 3 ψ = 1 ψ = 2 ψ = 3 bcsst24 2.2e-2 1.8e-3 3.7e-5 9.2e-3 1.5e-5 1.4e-7 7.2e-4 2.1e-8 4.1e-11 Kuu/Muu 2.4e-2 5.8e-3 7.5e-4 5.5e-3 6.6e-5 1.5e-6 1.7e-3 2.0e-6 2.3e-8 FDmesh1 1.8e-2 5.8e-3 5.2e-3 6.8e-3 2.2e-4 5.5e-6 2.3e-3 1.3e-5 6.6e-8 bcsst39 2.5e-2 1.1e-2 8.6e-3 1.2e-2 7.8e-5 2.3e-6 4.7e-3 4.4e-6 5.9e-7 qa8fk/qa8fm 1.6e-1 9.0e-2 2.0e-2 7.7e-2 5.6e-3 1.4e-4 5.9e-2 4.4e-4 3.4e-6 6.1. Computational system. The experiments were performed on the Mesabi Linux cluster at Minnesota Supercomputing Institute. Mesabi consists of 741 nodes of various configurations with a total of 17,784 compute cores that are part of Intel Haswell E5-2680v3 processors. Each node features two sockets, each socket with twelve physical cores at 2.5 GHz. Moreover, each node is equipped with 64 GB of system memory. Numerical illustration of RF-DDES. We tested RF-DDES on the matrix pencils listed in Table 6.1. For each pencil, the interval of interest [α, β] was chosen so that nev = 100. Matrix pencils 1), 2), 4), and 5) can be found in the SuiteSparse matrix collection (https://sparse.tamu.edu/) [12]. Matrix pencil 3) was obtained by a discretization of a differential eigenvalue problem associated with a membrane on the unit square with Dirichlet boundary conditions on all four edges using Finite Differences, and is of the standard form, i.e., M = I, where I denotes the identity matrix of appropriate size. Table 6.2 lists the maximum (worst-case) relative error among all nev approximate eigenvalues returned by RF-DDES. In agreement with the discussion in Section 4, exploiting higher values of ψ and/or nev B leads to enhanced accuracy. Figure 6.1 plots the relative errors among all nev approximate eigenvalues (not just the worst-case errors) for the largest matrix pencil reported in Table 6.1. Note that "qa8fk/qa8fm" is a positive definite pencil, i.e., all of its eigenvalues are positive. Since σ = 0, we expect the algebraically smallest eigenvalues of (A, M ) to be approximated more accurately. Then, increasing the value of ψ and/or nev B mainly improves the accuracy of the approximation of those eigenvalues λ located farther away from σ. A similar pattern was also observed for the rest of the matrix pencils listed in Table 6.1. Table 6.3 lists the number of iterations performed by Algorithm 3.1 as the value of N c increases. Observe that for matrix pencils 2), 3), 4) and 5) this number can be less than nev (recall the "early termination" property discussed in Proposition Table 6.1. N c = 2 N c = 4 N c = 8 N c = 12 N c = 16 A comparison of RF-DDES and RF-KRYLOV in distributed computing environments. In this section we compare the performance of RF-KRYLOV and RF-DDES on distributed computing environments for the matrices listed in Table 6.4. All eigenvalue problems in this section are of the form (A, I), i.e., standard eigenvalue problems. Matrices "boneS01" and "shipsec8" can be found in the SuiteSparse matrix collection. Similarly to "FDmesh1", matrices "FDmesh2" and "FDmesh3" were generated by a Finite Differences discretization of the Laplacian operator on the unit plane using Dirichlet boundary conditions and two different mesh sizes so that n = 250, 000 ("FDmesh2") and n = 1, 000, 000 ("FDmesh3"). Throughout the rest of this section we will keep N c = 2 fixed, since this option was found the best both for RF-KRYLOV and RF-DDES. 6.3.1. Wall-clock time comparisons. We now consider the wall-clock times achieved by RF-KRYLOV and RF-DDES when executing both schemes on τ = 2, 4, 8, 16 and τ = 32 compute cores. For RF-KRYLOV, the value of τ will denote the number of single-threaded MPI processes. For RF-DDES, the number of MPI processes will be equal to the number of subdomains, p, and each MPI process will utilize τ /p compute threads. Unless mentioned otherwise, we will assume that RF-DDES is executed with ψ = 3 and nev B = 100. Table 6.7 lists the wall-clock time required by RF-KRYLOV and RF-DDES to approximate the nev = 100, 200 and nev = 300 algebraically smallest eigenvalues of the matrices listed in Table 6.4. For RF-DDES we considered two different values of p; p = 2 and p = 4. Overall, RF-DDES was found to be faster than RF-KRYLOV, with an increasing performance gap for higher values of nev. Table 6.5 lists the number of iterations performed by RF-KRYLOV and Algorithm 3.1 in RF-DDES. For all Table 6.5 Number of iterations performed by RF-KRYLOV (denoted as RFK) and Algorithm 3.1 in RF-DDES (denoted by RFD(2) and RFD (4), with the number inside the parentheses denoting the value of p) for the matrix pencils listed in Table 6.4. The convergence criterion in both RF-KRYLOV and Algorithm 3.1 was tested every ten iterations. nev = 100 nev = 200 nev = 300 RFK RFD(2) RFD(4) RFK RFD(2) RFD(4) RFK RFD (2) 6.2e-5 8.5e-6 4.3e-6 6.3e-4 1.1e-4 3.1e-5 9.1e-4 5.3e-4 5.3e-5 Table 6.8 Time elapsed to apply to apply the rational filter in RF-KRYLOV and RF-DDES using τ = 2, 4, 8, 16 and τ = 32 compute cores. RFD (2) and RFD(4) denote RF-DDES with p = 2 and p = 4 subdomains, respectively. For RF-KRYLOV the times listed also include the amount of time spent on factorizing matrices A − ζ ℓ M, ℓ = 1, . . . , Nc. For RF-DDES, the times listed also include the amount of time spent in forming and factorizing matrices S ζ ℓ , ℓ = 1, . . . , Nc. nev = 100 nev = 200 nev = 300 Matrix RFK RFD(2) RFD(4) RFK RFD(2) RFD(4) RFK RFD (2) matrices but "boneS01", Algorithm 3.1 required fewer iterations than RF-KRYLOV. Table 6.6 lists the maximum relative error of the approximate eigenvalues returned by RF-DDES when p = 4. The decrease in the accuracy of RF-DDES as nev increases is due the fact that nev B remains bounded. Typically, an increase in the value of nev should be also accompanied by an increase in the value of nev B , if the same level of maximum relative error need be retained. On the other hand, RF-KRYLOV always computed all nev eigenpairs up to the maximum attainable accuracy. Table 6.8 lists the amount of time spent on the triangular substitutions required to apply the rational filter in RF-KRYLOV, as well as the amount of time spent on forming and factorizing the Schur complement matrices and applying the rational filter in RF-DDES. For the values of nev tested in this section, these procedures were found to be the computationally most expensive ones. Figure 6.5 plots the total amount of time spent on orthonormalization by RF-KRYLOV and RF-DDES when applied to matrices "FDmesh2" and "FDmesh3". For RF-KRYLOV, we report results for all different values of nev and number of MPI processes. For RF-DDES we only report the highest times across all different values of nev, τ and p. RF-DDES was found to spend a considerably smaller amount of time on orthonormalization than what RF-KRYLOV did, mainly because s was much smaller than n (the values of s for p = 2 and p = 4 can be found in Table 6.4). Indeed, if both RF-KRYLOV and Algorithm 3.1 in RF-DDES perform a similar number of iterations, we expect the former to spend roughly n/s more time on orthonormalization compared to RF-DDES. Figure 6.6 lists the wall-clock times achieved by an MPI-only implementation of RF-DDES, i.e., p still denotes the number of subdomains but each subdomain is handled by a separate (single-threaded) MPI process, for matrices "shipsec8" and "FDmesh2". In all cases, the MPI-only implementation of RF-DDES led to higher wall-clock times than those achieved by the hybrid implementations discussed in Tables 6.7 and 6.8. More specifically, while the MPI-only implementation reduced the cost to construct and factorize the distributed S ζ ℓ matrices, the application of the rational filter in Algorithm 3.1 became more expensive due to: a) each linear system solution with S ζ ℓ required more time, b) a larger number of iterations had to be performed as p increased (Algorithm 3.1 required 190, 290, 300, 340 and 370 iterations for "shipsec8", and 160, 270, 320, 350, and 410 iterations for "FDmesh2" as p = 2, 4, 8, 16 and p = 32, respectively). One more observation is that for the MPIonly version of RF-DDES, its scalability for increasing values of p is limited by the scalability of the linear system solver which is typically not high. This suggests that reducing p and applying RF-DDES recursively to the local pencils (B 7. Conclusion. In this paper we proposed a rational filtering domain decomposition approach (termed as RF-DDES) for the computation of all eigenpairs of real symmetric pencils inside a given interval [α, β]. In contrast with rational filtering Krylov approaches, RF-DDES applies the rational filter only to the interface variables. This has several advantages. First, orthogonalization is performed on vectors whose length is equal to the number of interface variables only. Second, the Krylov projection method may converge in fewer than nev iterations. Third, it is possible to solve the original eigenvalue problem associated with the interior variables in real arithmetic and with trivial parallelism with respect to each subdomain. RF-DDES can be considerably faster than rational filtering Krylov approaches, especially when a large number of eigenvalues is located inside [α, β]. In future work, we aim to extend RF-DDES by taking advantage of additional levels of parallelism. In addition to the ability to divide the initial interval [α, β] into non-overlapping subintervals and process them in parallel, e.g. see [24,21], we can also assign linear system solutions associated with different quadrature nodes to different groups of processors. Another interesting direction is to consider the use of iterative solvers to solve the linear systems associated with S(ζ 1 ), . . . , S(ζ Nc ). This could be helpful when RF-DDES is applied to the solution of symmetric eigenvalue problems arising from 3D domains. On the algorithmic side, it would be of interest to develop more efficient criteria to set the value of nev (j) B , j = 1, . . . , p in each subdomain, perhaps by adapting the work in [38]. In a similar context, it would be interesting to also explore recursive implementations of RF-DDES. For example, RF-DDES could be applied individually to each matrix pencil (B Figure 2 . 21 plots the modulus of the rational function ρ 2. 2 .Fig. 2 . 1 . 221Rational filtered Arnoldi procedure. Now, consider the rational matrix function ρ(M −1 A) with ρ(.) defined as in (2.2): (2.4) ρ(M −1 A) = 2ℜe Nc ℓ=1 ω ℓ (A − ζ ℓ M ) −1 M . The modulus of the rational filter function ρ(ζ) when ζ ∈ [−2, 2]. Left: Gauss-Legendre rule. Right: Midpoint rule. Algorithm 2 . 1 . 21RF-KRYLOV 0. Start with q (1) ∈ R n s.t. q (1) 2 = 1 1. For µ = 1, 2, . . . p j=1 s j , then B j and M (j) B are square matrices of size d j ×d j , E j and M (j) E are rectangular matrices of size d j ×s j , and C and M C are square matrices of size s × s. Matrices E i , M (j) E have a special nonzero pattern of the form E j = [0 dj,ℓj ,Ê j , 0 dj,νj ], and M (j) E = [0 dj,ℓj ,M (j) E , 0 dj,νj ], where ℓ j = k<j k=1 s k , ν j = k=p k>j s k , and 0 χ,ψ denotes the zero matrix of size χ × ψ. Fig. 3 . 1 . 31The leading singular values of ℜe Nc ℓ=1 ω ℓ S −1 ζ ℓ for different values of Nc. Figure 3 3 RF-DDES starts by calling a graph partitioner to partition the graph of |A| + |M | into p subdomains and reorders the matrix pencil (A, M ) as in (2.5). RF-DDES then proceeds to the computation of those eigenvectors associated with the nev(j) B smallest (in magnitude) eigenvalues of each matrix pencil (B j − σM B ) and store them in V j 4. End 5. Compute Q by Algorithm 3.1 6. Form Z as in(5.3) 7. Solve the Rayleigh-Ritz eigenvalue problem: Z T AZG = Z T M ZGΛ 8. If eigenvectors were also sought, permute the entries of each approximate eigenvector back to their original ordering B and repeating the Rayleigh-Ritz projection. Fig. 6 . 1 . 61Relative errors of the approximation of the lowest nev = 100 eigenvalues for the "qa8fk/qafm" matrix pencil. Left: nev B = 50. Center: nev B = 100. Right: nev B = 200. , even for values of N c as low as N c = 2. Moreover, Figure 6.2 plots the 150 leading 2 singular values of matrix ℜe Nc ℓ=1 ω ℓ S(ζ ℓ ) −1 for matrix pencils "bcsst24" and "Kuu/Muu" as N c = 4, 8, 12 and N c = 16. In agreement with the discussion in Section 3.2, the magnitude of the trailing s − rank y (1) , . . . , y (nev) singular values approaches zero as the value of N c increases. Except the value of N c , the number of subdomains p might also affect the number of iterations performed by Algorithm 3.1. Figure 6.3 shows the total number of iterations performed by Algorithm 3.1 when applied to matrix "FDmesh1" for p = 2, 4, 8 and p = 16 subdomains. For each different value of p we considered N c = 2, 4, 8, 12, and N c = 16 quadrature nodes. The interval [α, β] was set so that it included only eigenvalues λ 1 , . . . , λ 200 (nev = 200). Observe that higher values of p might lead to an increase in the number of iterations performed by Algorithm 3.1. For example, when the number of subdomains is set to p = 2 or p = 4, setting N c = 2 Fig. 6 . 3 . 63Total number of iterations performed by Algorithm 3.1 when applied to matrix "FDmesh1" (where [α, β] = [λ 1 , λ 200 ]). Results reported are for all different combinations of p = 2, 4, 8 and p = 16, and Nc = 1, 2, 4, 8 and Nc = 16.is sufficient for Algorithm 3.1 to terminate in less than nev iterations On the other hand, when p ≥ 8, we need at least N c ≥ 4 if a similar number of iterations is to be performed.This potential increase in the number of iterations performed by Algorithm 3.1 for larger values of p is a consequence of the fact that the columns of matrix Y = y (1) , . . . , y (nev) now lie in a higher-dimensional subspace. This might not only increase the rank of Y , but also affect the decay of the singular values of ℜe Nc ℓ=1 ω ℓ S(ζ ℓ ) −1 . This can be seen more clearly inFigure 6.4 where we plot the leading 250 singular values of ℜe Nc ℓ=1 ω ℓ S(ζ ℓ ) −1 of the problem in Figure 6.3 for two different values of p, p = 2 and p = 8. Note that the leading singular values decay more slowly for the case p = 8. Similar results were observed for different values of p and for all matrix pencils listed in Fig. 6 . 4 . 64The leading 250 singular values of ℜe Nc ℓ=1 ω ℓ S(ζ ℓ ) −1 for the same problem as in Figure 6.3. Left: p = 2. Right: p = 8. For both values of p we set Nc = 1, 2, 4, and Nc = 8. Fig. 6 . 5 . 65Time spent on orthonormalization in RF-KRYLOV and RF-DDES when computing the nev = 100, 200 and nev = 300 algebraically smallest eigenvalues and associated eigenvectors of matrices "FDmesh2" and "FDmesh3". Fig. 6. 6 . 6Amount of time required to apply the rational filter ("Interface"), form the subspace associated with the interior variables ("Interior"), and total wall-clock time ("Total") obtained by an MPI-only execution of RF-DDES for the case where nev = 300. Left: "shipsec8". Right: "FDmesh2". B ), j = 1, . . . , p might be the best combination when only distributed memory parallelism is considered. B ), j = 1, . . . , p to compute the nev (j) B eigenvectors of interest. This could be particularly helpful when either d j , the number of interior variables of the jth subdomain, or nev (j) B , are large. 8. Acknowledgments. Vassilis Kalantzis was partially supported by a Gerondelis Foundation Fellowship. The authors acknowledge the Minnesota Supercomputing Institute (MSI; http://www.msi.umn.edu) at the University of Minnesota for providing resources that contributed to the research results reported within this paper. Algorithm 5.1. RF-DDES 0. Input: A, M, α, β, σ, p, {ω ℓ , ζ ℓ } ℓ=1,...,Nc , {nev(j) B } j=1,...,p , ψ 1. Reorder A and M as in (2.5) 2. For j = 1, . . . , p: 3. Table 6 . 61 n: size of A and M , nnz(X): number of non-zero entries in matrix X.# Mat. pencil n nnz(A)/n nnz(M )/n [α, β] nev 1. bcsst24 3,562 44.89 1.00 [0, 352.55] 100 2. Kuu/Muu 7,102 47.90 23.95 [0, 934.30] 100 3. FDmesh1 24,000 4.97 1.00 [0, 0.0568] 100 4. bcsst39 46,772 44.05 1.00 [-11.76, 3915.7] 100 5. qa8fk/qa8fm 66,127 25.11 25.11 [0, 15.530] 100 Table 6 .3 6Number of iterations performed by Algorithm 3.1 for the matrix pencils listed inTable 6.1. 's' denotes the number of interface variables.Mat. pencil s s/n N c = 2 N c = 4 N c = 8 N c = 12 N c = 16 bcsst24 449 0.12 164 133 111 106 104 Kuu/Muu 720 0.10 116 74 66 66 66 FDmesh1 300 0.01 58 40 36 35 34 bcsst39 475 0.01 139 93 75 73 72 qa8fk/qa8fm 1272 0.01 221 132 89 86 86 2 4 6 8 10 12 14 16 100 200 300 # of subdomains (p) # of iterations Table 6 . 64 n: size of A, nnz(A): number of non-zero entries in matrix A. s 2 and s 4 denote the number of interface variables when p = 2 and p = 4, respectively.# Matrix n nnz(A)/n s2 s4 [λ1, λ101, λ201, λ300] 1. shipsec8 114,919 28.74 4,534 9,001 [3.2e-2, 1.14e-1, 1.57e-2, 0.20] 2. boneS01 172,224 32.03 10,018 20,451 [2.8e-3, 24.60, 45.42, 64.43] 3. FDmesh2 250,000 4.99 1,098 2,218 [7.8e-5, 5.7e-3, 1.08e-2, 1.6e-2] 4. 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{'abstract': 'This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem associated with each subdomain into two disjoint subproblems. The first subproblem is associated with the interface variables and accounts for the interaction among neighboring subdomains. To compute the solution of the original eigenvalue problem at the interface variables we leverage ideas from contour integral eigenvalue solvers. The second subproblem is associated with the interior variables in each subdomain and can be solved in parallel among the different subdomains using real arithmetic only. Compared to rational filtering projection methods applied to the original matrix pencil, the proposed technique integrates only a part of the matrix resolvent while it applies any orthogonalization necessary to vectors whose length is equal to the number of interface variables. In addition, no estimation of the number of eigenvalues lying inside the interval of interest is needed. Numerical experiments performed in distributed memory architectures illustrate the competitiveness of the proposed technique against rational filtering Krylov approaches.', 'arxivid': '1711.09487', 'author': ['Vassilis Kalantzis ', 'Yuanzhe Xi ', 'ANDYousef Saad '], 'authoraffiliation': [], 'corpusid': 13724383, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 24098, 'n_tokens_neox': 21455, 'n_words': 11811, 'pdfsha': '4e3ffd2f91eb28d672fc037cfd208932ad18a12f', 'pdfurls': ['https://export.arxiv.org/pdf/1711.09487v1.pdf'], 'title': ['BEYOND AMLS: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING *', 'BEYOND AMLS: DOMAIN DECOMPOSITION WITH RATIONAL FILTERING *'], 'venue': []}

arxiv

Programming Language Approaches to Concurrency and communication-cEntric Software 2010 (PLACES'10) EPTCS 69 2011 Thomas T Hildebrandt Logic and Semantics Group Rued Langgaards Vej 7 IT University of Copenhagen Programming DK-2300Copenhagen SDenmark Raghava Rao Mukkamala Logic and Semantics Group Rued Langgaards Vej 7 IT University of Copenhagen Programming DK-2300Copenhagen SDenmark Programming Language Approaches to Concurrency and communication-cEntric Software 2010 (PLACES'10) EPTCS 69 201110.4204/EPTCS.69.5 We present Dynamic Condition Response Graphs (DCR Graphs) as a declarative, event-based process model inspired by the workflow language employed by our industrial partner and conservatively generalizing prime event structures. A dynamic condition response graph is a directed graph with nodes representing the events that can happen and arrows representing four relations between events: condition, response, include, and exclude. Distributed DCR Graphs is then obtained by assigning roles to events and principals. We give a graphical notation inspired by related work by van der Aalst et al. We exemplify the use of distributed DCR Graphs on a simple workflow taken from a field study at a Danish hospital, pointing out their flexibility compared to imperative workflow models. Finally we provide a mapping from DCR Graphs to Büchi-automata. Introduction A key difference between declarative and imperative process languages is that the control flow for the first kind is defined implicitly as a set of constraints or rules, and for the latter is defined explicitly, e.g. as a flow diagram or a sequence of state changing commands. There is a long tradition for using declarative logic based languages to schedule transactions in the database community, see e.g. [15]. Several researchers have noted [12,23,24,7,3,4,22] that it could be an advantage to use a declarative approach to achieve more flexible process descriptions in other areas, in particular for the specification of case management workflow and ad hoc business processes. The increased flexibility is obtained in two ways: Firstly, since it is often complex to explicitly model all possible ways of fulfilling the requirements of a workflow, imperative descriptions easily lead to over-constrained control flows. In the declarative approach any execution fulfilling the constraints of the workflow is allowed, thereby leaving maximal flexibility in the execution. Secondly, adding a new constraint to an imperative process description often requires that the process code is completely rewritten, while the declarative approach just requires the extra constraint to be added. In other words, declarative models provide flexibility for the execution at run time and with respect to changes to the process. As a simple motivating example, consider a hospital workflow extracted from a real-life study of paper-based oncology workflow at danish hospitals [19,21]. As a start, we assume two events, prescribe and sign, representing a doctor adding a prescription of medicine to the patient record and signing it respectively. We assume the constraints stating that the doctor must sign after having added a prescription of medicine to the patient record and not to sign an empty record. A naive imperative process description may simply put the two actions in sequence, prescribe;sign, which allows the doctor first to prescribe medicine and then sign the record. In this way the possibilities of adding several prescriptions before or after signing and signing multiple times are lost, even if they are perfectly legal according to the constraints. The most general imperative description should start with the prescribe event, followed by a loop allowing either sign or prescribe events and only allow termination after a sign event. If the execution continues forever, it must be enforced that every prescription is eventually followed by a sign event. With respect to the second type of flexibility, consider adding a new event give, representing a nurse giving the medicine to the patient, and the rule that a nurse must give medicine to the patient if it is prescribed by the doctor, but not before it has been signed. For the most general imperative description we should add the ability to execute the give event within the loop after the first sign event and not allow to terminate the flow if we have had a prescribe event without a subsequent give event. So, we have to change the code of the loop as well as the condition for exiting it. In [4,22], van der Aalst and Pesic propose to use Linear-time Temporal Logic (LTL) as a declarative language for describing the constraints of the workflow. LTL allows for describing a rich set of constraints on the execution flow. In particular, the first example workflow above is expressed as (FPrescribe =⇒ ¬Sign U Prescribe) ∧ (G(Prescribe =⇒ FSign)), in words: "(Future Prescribe implies (not Sign Until Prescribe)) and (Globally (Prescribe implies Future Sign))". The expression becomes slightly more readable if the past modality is used: (G(Sign =⇒ PPrescribe) ∧ (G(Prescribe =⇒ FSign)), in words: "(Globally (Sign implies Past Prescribe)) and (Globally (Prescribe implies Future Sign))". Since the notation of LTL is likely to be too difficult to use directly by the end user it is suggested to use a graphical notation for common patterns of temporal constraints which are then compiled to LTL. The example is then a combination of a preceedence pattern, (G(Sign =⇒ PPrescribe) and a response pattern (G(Prescribe =⇒ FSign)) represented graphically in [4,22] as shown in Fig. 1. However, this approach suffers from the fact that the subsequent tools for execution and analysis will refer to the LTL expression (or further compilations to e.g. Büchi automata) and not the graphical notation. Also, the full generality of LTL may lead to a poor execution time. [4,22]. This motivates researching the problem of finding an expressive declarative process language where both the constraints as well as the run time state can be easily visualized and understood by the end user and also allows an effective execution. We believe that the declarative process model language of dynamic condition response graphs and its graphical representation proposed in this paper is a promising candidate. The model language is inspired by and a conservative generalization of the declarative process matrix model language [19,21] used by our industrial partner and prime event structures [26]. It is similar to [4,22] in that it is based on a graph of constraints between events. The crucial difference is that only a fixed set of four primitive constraints is allowed and that the process semantics can be expressed directly as transitions between markings of the graph instead of via a translation to LTL. We present distributed dynamic condition response graphs as a sequence of three generalizations of prime event structures. A prime event structure can be regarded as a minimal, declarative model for concurrent processes. It consists of a (possibly infinite) set of events (that can happen at most once), a (partial order) causality relation between events corresponding to the precedence LTL pattern above and a conflict relation stating which events can not happen in the same execution. The first generalization, named condition response event structures, is obtained by adding a response relation between events and a set Re of initially required response events. The initially required response events can be regarded as goals that must be fulfilled (or falsified) in order for an execution to be accepting. That is, for any event e ∈ Re, either e must eventually happen or it must become in conflict with an event that has happened in the past. The response relation in some sense corresponds to the response LTL pattern above as a dual relation to the usual causality relation: If an event b is a response to an event a then b must happen at some point after event a happens or become in conflict. However, note that the response pattern does not allow for conflicts. Operationally, as we will see in the following section, one can think of the event b as being added to the set Re of required responses when a happens. Next we generalize condition response event structures by allowing each event to happen many times and replacing the symmetric conflict relation by an asymmetric relation which dynamically determines which events are included in or excluded from the structure. To allow the graphs to represent intermediate run time state (e.g. like the marking of a Petri Net) we also add sets I and E of respectively included and executed events and refer to the triple of sets of pending responses, included and executed events as the marking of the graph. This results in the model of Dynamic Condition Response Graphs, short DCR Graphs. Finally, we reach the model of Distributed Dynamic Condition Response Graphs allowing for role based distribution by adding a set of principals and a set of roles assigned to both principals and events, and define that an event can only be executed by a principal assigned one of the roles assigned to the event. Being based on only four relations between events (condition, response, include, exclude) and the role assignment, the distributed dynamic condition response graphs can be simply visualized as a directed graph with a box for each event as nodes and four different kinds of arrows. We base our graphical notation for the condition and response relations on the notation suggested in [3] for precendence and response LTL patterns, since they coincide when no events are excluded. The inclusion and exclusion relations are denoted by arrows with a + and % sign at the head respectively. We label each node with the activity of the event and add a small box to the top containing the roles that can execute the event. We annotate the graph by the marking, showing if an event is required as a response by adding a small exclamation mark, if it has happened in the past by a small check sign, and if it is excluded by making the box dashed. In addition we found it useful to show (by a small no-entry sign) if an event is blocked by an unfulfilled condition event, even though this information can be inferred from the condition relations and the currently included and executed events. We formalize the execution of dynamic condition response graph as a labelled transition system, which is finite state if the graph is finite. Indeed, the states of transition system will be markings consisting of triples of sets of executed, included, and required response events. We define a (finite or infinite) run of the labelled transition system to be accepting if no response event is forever continuously included and pending without being executed. We end by characterizing the execution semantics by providing a mapping of dynamic condition response graph to Büchi-automata. The rest of the paper is structured as follows. In Sec. 2 below we recall the definition of prime event structures and introduce condition response event structures as the first generalization. We show how the response relation allows to represent the notion of weak fairness. In Sec. 3 we introduce the model of dynamic condition response graph (DCR Graphs) and distributed DCR Graphs. In Sec. 4 we provide a mapping from DCR Graphs to Büchi-automata with τ-events. In Sec. 5 we briefly address related work. Finally, we conclude and discuss current and future work in Sec. 6. This paper replaces and extends the work presented in the two previous short papers [16] and [20]. The paper [16] introduced condition response event structures and dynamic condition response structures, which are essentially dynamic condition response graph without markings.The paper [20] provided a mapping from dynamic condition response structures to Büchi automata, but only capturing acceptance for the infinite runs. The mapping from dynamic condition response graphs to Büchi automata provided in the present paper characterizes also the acceptance of finite runs by introducing silent (τ) transitions in the Büchi automata. Condition Response Event Structures As an intermediate step towards dynamic condition response graphs, we generalize prime event structures to allow for a notion of progress based on a response relation. This model is interesting in itself as an extensional event-based model with progress, abstracting away from the intentional representation of repeated behavior. In particular we show that it allows for an elegant characterization of weakly fair runs of event structures. First let us recall the definition of a prime event structure and configurations of such [26]. Definition 1 A labeled prime event structure (ES) is a 5-tuple E = (E, Act, ≤, #, l) where (i) E is a (possibly infinite) set of events (ii) Act is the set of actions (iii) ≤ ⊆ E × E is the causality relation between events which is a partial order (iv) # ⊆ E × E is a binary conflict relation between events which is irreflexive and symmetric (v) l : E → Act is the labeling function mapping events to actions The causality and conflict relations must satisfy the conditions that 1. ∀e, e , e ∈ E.e#e ≤ e =⇒ e#e , 2. ∀e ∈ E.e ↓= {e | e < e} is finite. A configuration of E is a set c ⊆ E of events satisfying the conditions 1. conflict-free: ∀e, e ∈ c.¬e#e , 2. downwards-closed: ∀e ∈ c, e ∈ E.e ≤ e =⇒ e ∈ c. A run ρ of E is a (possibly infinite) sequence of labelled events (e 0 , l(e 0 )), (e 1 , l(e 1 )), . . . such that for all i ≥ 0. ∪ 0≤ j≤i {e j } is a configuration. A run (e 0 , l(e 0 )), (e 1 , l(e 1 )), . . . is maximal if any enabled event eventually happen or become in conflict, formally ∀e ∈ E, i ≥ 0.e ↓⊆ (e i ↓ ∪{e i }) =⇒ ∃ j ≥ 0.(e#e j ∨ e = e j ). Action names a ∈ Act represent the actions the system might perform, an event e ∈ E labelled with a represents occurrence of action a during the run of the system. The causality relation e ≤ e means that event e is a prerequisite for the event e and the conflict relation e#e implies that events e and e both can not happen in the same run, more precisely one excludes the occurrence of the other. The definition of maximal runs follows the definition of weak fairness for concurrency models in [8] and is equivalent to stating that the configuration defined by the events in the run is maximal with respect to inclusion of configurations. We now generalize prime event structures to condition response event structures, by adding a dual response relation •→, such that {e | e •→ e } is the set of events that must happen (or be in conflict) after the event e has happened for a run to be accepting. The resulting structures, named condition response event structures, in this way add the possibility to state progress conditions. We also introduce a subset of the events Re of initial responses, which are events that are initially required eventually to happen (or become in conflict). In this way the structures can represent the state after an event has been executed. As we will see below, it also allows us to capture the notion of maximal runs. Definition 2 A labeled condition response event structure (CRES) over an alphabet Act is a tuple (E, Re, Act, →•, •→, #, l) where (i) (E, →•, #, l) is a labelled prime event structure, referred to as the underlying event structure (ii) •→ ⊆ E × E is the response relation between events, satisfying that →• ∪ •→ is acyclic. (iii) Re ⊆ E is the set of initial responses. We define a configuration c and run ρ of a CRES to be a respectively a configuration and run of the underlying event structure. We define a run (e 0 , l(e 0 )), (e 1 , l(e 1 )), . . . to be accepting if ∀e ∈ E, i ≥ 0.e i •→ e =⇒ ∃ j ≥ 0.(e#e j ∨ (i < j ∧ e = e j ) and ∀e ∈ R.∃ j ≥ 0.(e#e j ∨ e = e j ) . In words, any pending response event must eventually happen or be in conflict. A prime event structure can trivially be regarded as a condition response event structure with empty response relation. This provides an embedding of prime event structures into condition response event structures which preserves configurations and runs. Proposition 1 The labelled prime event structure (E, Act, ≤, #, l) has the same runs as the CRES (E, / 0, Act, ≤ , / 0, #, l) for which all runs are accepting. We can also embed event structures into CRES by considering every condition to be also a response and all events with no conditions to be initial responses. This characterizes the interpretation in [8] where only maximal runs are accepting. In other words, the embedding captures the notion of weakly fair execution of event structures. Proposition 2 The labelled prime event structure (E, Act, ≤, #, l) has the same runs and maximal runs as respectively the runs and the accepting runs of the CRES (E, {e | e ↓= / 0}, Act, ≤, ≤, #, l). Distributed Dynamic Condition Response Graphs We now go on to generalize condition response event structures to dynamic condition response graphs (DCR Graphs). As opposed to event structures, a dynamic condition response graph allows events to be executed multiple times and there are no constraints on the condition and response relations. This allows for finite representations of infinite behavior, but also for introducing deadlocks. Moreover, the conflict relation is generalized to two relations for dynamic exclusion and inclusion of events, which is more appropriate in a model where events can be re-executed and has shown useful in practice as a primitive for skipping events and constraints. (vii) l : E → Act is a labelling function mapping every event to an action. Definition 3 A dynamic condition response graph is a tuple G = (E, M, Act, →•, •→, ±, l) where (i) E is the set of events (ii) M ∈ M (G) = P(E) × P(E) × P(E) is the marking and M (G) is the set of all markings (iii) Act is the set of actions (iv) →•⊆ E× E is the condition relation (v) •→⊆ E× E is the response relation (vi) ± : E × E {+, We let DCR Graphs refer to the model of dynamic condition response graphs. The condition and response relations in DCR Graphs are similar to the corresponding relations in CRES, except that they are not constrained in any way. In particular, we may have cyclic relations. The marking M = (Ex, Re, In) ∈ M (G) consists of three sets of events, capturing respectively which events have previously been executed (Ex), which events are pending responses required to be executed or excluded (Re), and finally which events are currently included (In). The set of pending responses Re of DCR Graphs thus plays the same role as the set of initial responses in the CRES. The dynamic inclusion/exclusion relations →+ and →%, represented by the (partial map) ± : E × E {+, %} , allow events to be included and excluded dynamically in the graph. The intuition is that only the currently included events are considered in evaluating the constraints. This means that if event a has event b as condition, but event b is excluded from the graph then it is not required that b has happened for a to happen. Also, if event a has event b as response and event b is excluded then it is not required that b happens for the flow to be acceptable. Formally, the relation e →+ e expresses that, whenever event e happens, it will include e in the graph. On the other hand, e →% e expresses that when e happens it will exclude e from the graph. We define the execution semantics of DCR Graphs by a labelled transition system with markings as states and define the set of accepting runs by requiring that no event must be continuously included and pending. We define a run to be accepting if ∀i ≥ 0, e ∈ Re i .∃ j ≥ i.(e = e j ∨ e ∈ In j ). In words, a run is accepting if no response event is continuously included and pending without it happens. The first two items in the above definition are markings before and after the transition. The third item expresses that only events e that are currently included can be executed. The requirement saying that all currently included condition events for e should have been executed previously is expressed in (iv). The next two items are the updates to the sets of included events and pending responses respectively. Note that an event e can not be both included and excluded by the same event e, but an event may exclude itself. Also an event may trigger itself as a response and/or has itself as condition. If one only want to consider finite runs, which is common for workflows, the acceptance condition degenerates to requiring that no pending response is included at the end of the run. This corresponds to defining all states where Re ∩ In = / 0 to be accepting states and define the accepting runs to be those ending in an accepting state. If infinite runs are also of interest (as e.g. for reactive systems and the LTL logic) the acceptance condition can be captured by a mapping to a Büchi-automaton with τ-events which we give in Sec. 4 below. A CRES can be represented as a dynamic condition response graph by making every event exclude itself and encode the conflict relation by defining any two conflicting events to mutually exclude each other as shown in figure 2(b). events in DCR Graphs We now define distributed dynamic condition response graphs by adding roles and principals. Definition 5 A distributed dynamic condition response graph is a tuple (G, Roles, P, as) where 1. G = (E, M, Act, →•, •→, ±, l) is a dynamic condition response graph, Roles is a set of roles, 3. P is a set of principals (e.g. persons or processors) and 4. as ⊆ (P ∪ Act) × Roles is the role assignment relation to principals and actions. For a distributed DCR Graphs, the role assignment relation indicates the roles (access rights) assigned to principals and which roles gives right to execute which actions. As an example, assume Peter ∈ P and Doctor ∈ Roles, then if Peter as Doctor and Sign as Doctor then Peter as a doctor can sign as a doctor. This is formalized by defining the labelled transition semantics for a distributed dynamic condition response graph D = (G, Roles, P, as) to have the same states as the underlying dynamic condition response graph G, and the transitions →⊆ M (G) × E × (P × Act × Roles) × M (G) defined by We define a run to be (finite or infinite) sequence of labels (e 0 , (p 0 , a 0 , r 0 ))(e 1 , (p 1 , a 1 , r 1 )) . . . of a sequence of transitions M i (e i ,(p i ,a i ,r i )) − −−−−−− → M i+1 starting from the initial marking. We define a run to be accepting if the underlying run of the DCR Graphs is accepting. We are now ready to give the small example workflow from the introduction graphically as a distributed dynamic condition response graph shown in Fig. 3(a). It contains three events: prescribe medicine (the doctor calculates and writes the dose for the medicine), sign (the doctor certifies the correctness of the calculations) and give medicine (the nurse administers medicine to patient). The events are also labelled by the assigned roles (D for Doctor and N for Nurse). The arrow •→• between prescribe medicine and sign indicates that the two events are related by both the condition relation and the response relation. The condition relation means that the prescribe medicine event must happen at least once before the sign event. The response relation enforces that, if the prescribe medicine event happen, subsequently at some point the sign event must happen for the run to be accepted. Similarly, the response relation between prescribe medicine and give medicine enforces that, if the prescribe medicine event happen, subsequently at some point the give medicine event must happen for the flow to be accepted. Finally, the condition relation between sign and give medicine enforces that the signature event must have happened before the medicine can be given. Note the nurse can give medicine many times, and that the doctor can at any point choose to prescribe new medicine and sign again. (This will not block the nurse from continue to give medicine. The interpretation is that the nurse may have to keep giving medicine according to the previous prescription). The transition system for the prescribe medicine example is shown in Fig. 4. For simplicity we only show the actions as labels. The green states are the states with no included pending responses. The dynamic inclusion and exclusion of events is illustrated by an extension to the scenario (also taken from the real case study): If the nurse distrusts the prescription by the doctor, it should be possible to indicate it, and this action should force either a new prescription followed by a new signature or just a new signature. As long the new signature has not been added, medicine must not be given to the patient. This scenario can be modeled as shown in Fig. 3(b), where one more event labelled don't trust is added. Now, the nurse have a choice to indicate distrust of prescription and thereby exclude give medicine until the doctor execute sign again. Executing don't trust action will make sign a pending response. So the only way to reach an accepting run is to re-execute sign which will include give medicine. The doctor may choose to re-do prescribe medicine followed by sign (if the reason for distrusting the prescription was indeed valid) or simply re-do sign. In Fig. 5 below we propose a graphical notation that illustrates the run-time information during two different runs of the extended scenario in Fig. 3(b). We use three different small icons (Ø, √ ,!) above the boxes to show if the event is not enabled (i.e. it is blocked by an included condition event that has not been executed), if it has been executed (i.e. included in the set E in the marking), and if it is required as a response (i.e. included in the set R in the marking). We indicate that an event is excluded (i.e. not included in the set I in the marking) by making the box around the event dashed. (a) Prescribe Medicine Example (b) Prescribe Medicine Example Don't trust Figure 5: DCR Graphs Runtime state graphical notation Fig 5(a) shows four states of a run in the workflow process in Fig. 3(b), starting in the initial state where all events except prescribe medicine is blocked. The second state is the result of executing prescribe medicine, now showing that sign and give medicine are required as responses and that sign is no longer blocked. The third state is the result of executing the sign event, which enables give medicine and don't trust. Finally, the fourth state is the result of executing the give medicine event, excluding the don't trust event. Similarly, Fig. 5(b) shows the six states of a run where the nurse executes don't trust in the third step, leading to a different fourth state where give medicine is excluded (but still required as response if it gets included again) and sign is required as response. The fifth state shows the result of the doctor executing sign, which re-includes give medicine, which is then executed, leading to the final state where all events have been executed, and don't trust is excluded. From DCR Graphs to Büchi-automata In this section, we show how to characterize the acceptance condition for DCR Graphs by a mapping to Büchi-automata with τ-event. Recall that a Büchi-automaton is a finite state automaton accepting only infinite runs, and only the runs that pass through an accepting state infinitely often. Acceptance of finite runs can be represented in the standard way by introducing a special silent event, e.g. a τ-event, which may be viewed as a delay. If an infinite accepting run contains infinitely many delays it then represent an accepting run containing only a finite number of (real) events. We define a Büchi-automaton with τ-event as follows. Definition 6 A Büchi-automaton with τ-event is a tuple (S, s, Ev τ , →⊆ S × Ev τ × S, F) where S is the set of states, s ∈ S is the initial state, Ev τ is the set of events containing the special event τ, →⊆ S × Ev τ × S is the transition relation, and F is the set of accepting states. A (finite or infinite) run is a sequence of labels not containing the τ event that can be obtained by removing all τ events from a sequence of labels of transitions starting from the initial state. The run is accepting if the sequence of transitions passes through an accepting state infinitely often. The mapping from DCR Graphs to Büchi-automata is not entirely trivial, since we at any given time may have several pending responses and thus must make sure that all of them are eventually executed or excluded. To make sure we progress, we assume any fixed order of the finite set of events E of the given dynamic condition response graph and enforce the execution (or exclusion) of response events in that order. For an event e ∈ E we write rank(e) for its rank in that order and for a subset of events E ⊆ E we write min(E ) for the event in E with the minimal rank. for M r = {e ∈ In ∩ Re | rank(e) > i}. In the marking M, the set Ex records the events that have been executed, where as In and Re records the events that are currently included and pending responses respectively. The index i is used to make sure that no event stays forever included and in the pending response set without being executed. Finally, the flag j indicates if the state is accepting or not. The conditions (a) and (iii) define when a state is accepting. Either there are no included pending responses in the resulting state (iiia) or the included pending response with the minimal rank above the index i was either excluded or executed (iiib). Alternatively, if the set of included pending responses with rank above the index i is empty and the included pending response with the minimal rank is excluded or executed (iiic), then also the resulting state will be accepting. Condition (iv) records the new rank if the resulting state is accepting according to condition (iiib) and similarly when the state is accepting according to condition (iiic), the condition (v) records the new rank. Figure 6: The Büchi-automaton for DCR Graph from Fig. 3 (a) annotated with state information To give a simple example of the mapping, let us consider the dynamic condition response graph in Fig. 3(a) and the corresponding Büchi-automaton in Fig. 6. The key point to note is that the automaton enters an accepting state if there is no pending responses, or if the pending response which is the minimal ranked event according to the index i is executed or excluded. State S7 and S11 illustrate the use of the rank: Both states have the two events s (having rank 1) and gm as pending responses. In state S7 only executing event s leads to an accepting state (S10). The result of executing event gm is to move to state S9 which is not accepting. Dually, in state S11 only executing event gm leads to an accepting state (S16). The result of executing event s is to move to state S12 which is not accepting. Fig. 7 shows a stratified view of the automaton, dividing the state sets according to the rank i in order to emphasize the role of the rank in guaranteeing progress. We end by stating the main theorem that the mapping from dynamic condition response graph to Büchi-automata characterizes the execution semantics. Theorem 1 For a finite distributed dynamic condition response graph D the Büchi-automaton with τevent B(D) has the same runs and accepting runs as D. There exists many different approaches to formally specify and enact business processes and workflows. As it is not possible to provide a complete overview of all related work, we give here just a brief overview of some of the formalisms which are related to our work and compare them to DCR Graphs. Related Work As already described in the introduction, the authors in [4,3] have proposed ConDec, a declarative language for modeling and enacting the dynamic business processes based on Linear Temporal Logic (LTL). In [5], the authors have proposed Declarative Service Flow language (Dec-SerFlow) to specify, enact and monitor service flows, which is a sister language for ConDec. Both the languages share the same concepts and are supported in the Declare [3] tool. They specifies what should be done, instead of specifying how it should be done, there by leaving more flexibility to users. The enactment in both the languages is defined by translating the constraints specified in LTL, into a Buchi automaton and executing the workflow/service by executing the referring Buchi automaton. LTL being a very expressive language, the Declare tool suffers from efficiency problems in executing models with large specification [3]. Even though our approach is related to the work in [4,5], DCR Graphs has a fixed set of constraints that makes it simpler to learn and possible to describe the execution semantics directly as transitions between markings of the graph. The Event Calculus [10,9,18] is another logic-based methodology for specification and execution of workflows. It is a logic programming formalism for representing events and their effects in the context of database applications. The authors have expressed the basic control flow primitives as a set of logical formulas and used axioms of Event Calculus to specify activity dependency execution and agent assignments rules. Their workflow model also supports enactment and iteration of activities, but does not support verification of global and temporal constraints on workflow activities. Also, their approach is limited to imperative/procedural workflow modeling languages. Concurrent Transaction Logic (CTR) is used in [12] as a language for specifying, analysis, scheduling and verification of workflows. The authors have used CTR formulas for expressing the local and global properties of workflows. Reasoning about the workflows has been done with the help of proof theory and semantics of logic. In [23], the authors have used Concurrent Constraint Transaction Logic (CCTR) which is a flavor of CTR integrated with Constraint Logic Programming for scheduling workflows. Like the other logic programming systems, the authors in [12,23] have used the proof theory of CTR as run-time environment for enactment of workflows. The CTR approach mainly aims at developing an algorithm for consistency checking and verification of properties of workflows, but only limited to imperative modeling languages. Petri nets have been studied and used extensively in the domain of workflows and business processes, see e.g. [2,13,1]. The correspondence between event structures and Petri nets are well studied, see e.g. [27]. It is however not possible to express the notion of pending responses directly in standard Petri Nets. Our formal model DCR Graphs also relates to the declarative approaches used by Guard-Stage-Milestone (GSM) model [17] by Hull et al, presented as an invited talk at the WS-FM 2010 workshop. The GSM meta-model uses declarative approach for specification of life cycles, which is part of research on data driven artifact-centric business processes [6,14,11], carried out by the IBM Research. The operational semantics of GSM are based on Event-Condition-Action (ECA) rules, and provide a basis for formal verification and reasoning. Conclusion and Future Work We have presented dynamic condition response graphs, short DCR Graphs, as a new declarative, eventbased workflow process model inspired by the workflow language employed by our industrial partner [21]. We have demonstrated the use and flexibility of the model on a small example taken from a field study on danish hospitals [19] and proposed a graphical notation for presenting both the processes and their run-time state. The model was presented as a sequence of generalizations of the classical model for concurrency of prime event structures [26]. The first generalization introduced a notion of progress to event structures by replacing the usual causal order by two dual relations, a condition relation →• expressing for each event which events it has as preconditions and a response relation •→ expressing for each event which events that must happen (or become in conflict) after it has happened. We demonstrated that the resulting model, named condition response event structures, can express the standard notion of weak concurrency fairness. The next generalization is to allow for finite representations of infinite behaviours by allowing multiple execution, and dynamic inclusion and exclusion of events, resulting in the model of dynamic condition response graphs. Finally, we extended the model to allow distribution of events via roles and presented a graphical notation inspired by related work by van der Aalst et al. [4,3], but extended to include information about the run-time state (e.g. markings). We prove that all generalizations conservatively contain the previous model. Moreover, we provide a mapping from dynamic condition response graphs to Büchi-automata characterising the acceptance condition for finite and infinite runs. One key advantage of the dynamic condition response graphs compared to the related work explored in [4,3,12,9] is that the latter logics are more complex to visualize and understand by people not trained in logic. Another advantage, illustrated in the given mapping to Büchi-automata and our graphical visualization of the run time state, is that the execution of dynamic condition response graphs can be based on a relatively simple information about the run-time state, which can also be visualized directly as annotations (marking) on the graph. We have implemented a prototype engine and mapping to the input format for the SPIN model checker, and are currently working on implementing a simulator for DCRS able to visualize the state graphically in this way. Current and future work include studying extensions of the DCR Graphs model with time, exceptions, nesting (sub processes) and data (as publish/subscribe events to changes of data) and the relationship between DCR Graphs and Petri Net with time and infinite behavior [25]. Also we are investigating the expressiveness of the model compared to LTL and the work in [4]. Along the line of work in [27] we investigate the definition of interfaces between concurrently interacting dynamic condition response graphs, a (categorical) theory of simulations and defining unfolding (forgetful) mappings from DCR Graphs to CRES and from CRES to event structures. We expect the theory to be useful in achieving compositional design and verification of workflow processes, as well as studying the impact of adapting or adding new interacting workflow processes to a pool of processes. Finally we intend to explore the relation to the recent work on event-based business processes. Figure 1 : 1Graphical notation proposed in %} defines the dynamic inclusion/exclusion relations by e →+ e if ±(e, e ) = + and e →% e if ±(e, e ) = %. Definition 4 4For a dynamic condition response graph G = (E, M, Act, →•, •→, ±, l) we define the corresponding labelled transition systems T (G) to be the tuple (M (G), M, →⊆ M(G) × Act × M (G)) where M (G) is the set of markings of G, M ∈ M (G) is the initial marking, →⊆ M (G) × (E × Act) × (G) isthe transition relation given by M (e,a) −−→ M where (i) M = (Ex , Re , In ) is the marking before transition (ii) M = (Ex ∪ {e}, Re , In ) is the marking after transition (iii) e ∈ In and l(e) = a (iv) {e ∈ In | e →• e} ⊆ Ex (v) In = (In ∪ {e | e →+ e }) \ {e | e →% e } (vi) Re = (Re \ {e}) ∪ {e | e •→ e } We define a run (e 0 , a 0 ), (e 1 , a 1 ), . . . of the transition system to be a sequence of labels of a sequence of transitions M i (e i ,a i ) − −− → M i+1 , where M i = (Ex i , Re i , In i ) and M 0 = M. Proposition 3 3The CRES (E, Re, Act, →•, •→, #, l) has the same runs and accepting runs as the dynamic condition response graph (E, M, Act, →•, •→, ±, l) where M = ( / 0, Re, E), ±(e, e ) = % if e = e or e#e and undefined otherwise. Figure 2 : 2Encoding conflicting events in CRES as mutual excluding → M if p as r and a as r and M (e,a) −−→ M in the underlying dynamic condition response graph. Figure 3 : 3DCR Graphs example in graphical notation Figure 4 : 4Transition system for DCR graph fromFig 3(a) Definition 7 7For G = (E, M, Act, →•, •→, ±, l, Roles, P, as) a finite distributed dynamic condition response graph where E = {e 1 , . . . , e n } and rank(e i ) = i, we define the corresponding Büchi-automaton with τ-event to be the tuple B(G) = (S, s, →⊆ S × Ev τ × S, F) where• S = M (G) × {1, . . . , n} × {0, 1} is the set of states, • Ev τ = (E × (P × Act × Roles)) ∪ {τ} is the set of events, • s = (M, 1, 1) if I ∩ Re = / 0, and s = (M, 1, 0) otherwise • F = M (G) × {1, . . . , n} × {1} isthe set of accepting states and • →⊆ S × Ev τ × S is the transition relation given by (M , i, j) τ − −− → (M , i, j ) where (a) j = 1 if In ∩ Re = / 0 otherwise j = 0. and (M , i, j) (e,(p,a,r)) − −−−−−−−− → (M , i , j ) where (i) M = (Ex , Re , In ) and M = (Ex ∪ {e}, Re , In ) (ii) M (e,(p,a,r)) − −−−−−−−− → M is a transition of T (D) (iii) j = 1 if (a) In ∩ Re = / 0 or (b) min(M r ) ∈ (In ∩ Re \(In ∩ Re )) ∪ {e} or (c) M r = / 0 and min(In ∩ Re ) ∈ (In ∩ Re \(In ∩ Re )) ∪ {e} otherwise j = 0. 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An Event Algebra for Specifying and Scheduling Workflows. P Munindar, Greg Singh, Christine Meredith, &amp; Tomlinson, C Paul, Attie, Proceedings of DASFAA. DASFAAWorld Scientific PressMunindar P. Singh, Greg Meredith, Christine Tomlinson & Paul C. Attie (1995): An Event Algebra for Specifying and Scheduling Workflows. In: Proceedings of DASFAA, World Scientific Press, pp. 53-60. Infinite Behaviour and Fairness in Petri Nets. Rüdiger Valk, &amp; Heino Carstensen, Advances in Petri Nets. G. RozenbergSpringer-Verlag88Rüdiger Valk & Heino Carstensen (1985): Infinite Behaviour and Fairness in Petri Nets. In G. Rozenberg, editor: Advances in Petri Nets 1984, Lecture Notes in Computer Science 88, Springer-Verlag, pp. 83-100. Event Structures. Glynn Winskel, Wilfried Brauer, Wolfgang Reisig & Grzegorz Rozenberg. Springer255Advances in Petri NetsGlynn Winskel (1986): Event Structures. In Wilfried Brauer, Wolfgang Reisig & Grzegorz Rozenberg, editors: Advances in Petri Nets, Lecture Notes in Computer Science 255, Springer, pp. 325-392. Models for concurrency. Glynn Winskel, &amp; Mogens Nielsen, Handbook of logic in computer science. Oxford, UKOxford University Press4semantic modellingGlynn Winskel & Mogens Nielsen (1995): Models for concurrency. In: Handbook of logic in computer science (vol. 4): semantic modelling, Oxford University Press, Oxford, UK, pp. 1-148.

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{'abstract': 'We present Dynamic Condition Response Graphs (DCR Graphs) as a declarative, event-based process model inspired by the workflow language employed by our industrial partner and conservatively generalizing prime event structures. A dynamic condition response graph is a directed graph with nodes representing the events that can happen and arrows representing four relations between events: condition, response, include, and exclude. Distributed DCR Graphs is then obtained by assigning roles to events and principals. We give a graphical notation inspired by related work by van der Aalst et al. We exemplify the use of distributed DCR Graphs on a simple workflow taken from a field study at a Danish hospital, pointing out their flexibility compared to imperative workflow models. Finally we provide a mapping from DCR Graphs to Büchi-automata.', 'arxivid': '1110.4161', 'author': ['Thomas T Hildebrandt \nLogic and Semantics Group Rued Langgaards Vej 7\nIT University of Copenhagen Programming\nDK-2300Copenhagen SDenmark\n', 'Raghava Rao Mukkamala \nLogic and Semantics Group Rued Langgaards Vej 7\nIT University of Copenhagen Programming\nDK-2300Copenhagen SDenmark\n'], 'authoraffiliation': ['Logic and Semantics Group Rued Langgaards Vej 7\nIT University of Copenhagen Programming\nDK-2300Copenhagen SDenmark', 'Logic and Semantics Group Rued Langgaards Vej 7\nIT University of Copenhagen Programming\nDK-2300Copenhagen SDenmark'], 'corpusid': 14353309, 'doi': '10.4204/eptcs.69.5', 'github_urls': [], 'n_tokens_mistral': 14484, 'n_tokens_neox': 12939, 'n_words': 8471, 'pdfsha': '9091a3b5fb1f2ee994eee2665258c0ac63645eb8', 'pdfurls': ['https://arxiv.org/pdf/1110.4161v1.pdf'], 'title': ["Programming Language Approaches to Concurrency and communication-cEntric Software 2010 (PLACES'10) EPTCS 69", "Programming Language Approaches to Concurrency and communication-cEntric Software 2010 (PLACES'10) EPTCS 69"], 'venue': []}

arxiv

Video Surveillance System Incorporating Expert Decision-making Process: A Case Study on Detecting Calving Signs in Cattle Ryosuke Hyodo Waseda University TokyoJapan Susumu Saito Waseda University TokyoJapan Intelligent Framework Lab TokyoJapan Teppei Nakano Waseda University TokyoJapan Intelligent Framework Lab TokyoJapan Makoto Akabane Waseda University TokyoJapan Ryoichi Kasuga Farmers Support KagoshimaJapan Tetsuji Ogawa Waseda University TokyoJapan Video Surveillance System Incorporating Expert Decision-making Process: A Case Study on Detecting Calving Signs in Cattle 1Index Terms-XAImachine learninguser studyprecision livestock farming Through a user study in the field of livestock farming, we verify the effectiveness of an XAI framework for video surveillance systems. The systems can be made interpretable by incorporating experts' decision-making processes. AI systems are becoming increasingly common in real-world applications, especially in fields related to human decisionmaking, and its interpretability is necessary. However, there are still relatively few standard methods for assessing and addressing the interpretability of machine learning-based systems in realworld applications. In this study, we examine the framework of a video surveillance AI system that presents the reasoning behind predictions by incorporating experts' decision-making processes with rich domain knowledge of the notification target. While general black-box AI systems can only present final probability values, the proposed framework can present information relevant to experts' decisions, which is expected to be more helpful for their decision-making. In our case study, we designed a system for detecting signs of calving in cattle based on the proposed framework and evaluated the system through a user study (N=6) with people involved in livestock farming. A comparison with the black-box AI system revealed that many participants referred to the presented reasons for the prediction results, and five out of six participants selected the proposed system as the system they would like to use in the future. It became clear that we need to design a user interface that considers the reasons for the prediction results. I. INTRODUCTION In recent years, artificial intelligence (AI) and machine learning (ML) have been used in areas related to human decision-making, including the medical field [1]. When developing AI systems, an end-to-end approach is generally considered to be an indispensable technique due to its simplicity. However, it has been pointed out that this approach is blackbox in its internal behavior and is not a sufficient system for supporting human decision-making [2]. Meanwhile, there has been growing interest in the interpretability of AI systems as explainable AI (XAI) [3], [4]. Representative technologies of XAI include visualization techniques such as CAM [5] and Grad-CAM [6], and post-hoc explanations such as local interpretable model agnostic explanations (LIME) [7] and Shapley additive explanations (SHAP) [8]. However, many developments in AI and ML tend to suffer from a lack of usability and practical interpretability for real decisionmakers [3], [4], [9]. In recent years, the human-computer interaction (HCI) community has recognized the importance of human-centered evaluation, which incorporates user evaluation into the interpretability of AI systems. There has been increasing research on the user evaluation of these explanatory techniques using experimental data sets [10], [11], [12]. However, when implementing AI systems for highly specialized tasks in industry, the aforementioned generic explanatory techniques and the findings on general user evaluations using experimental datasets do not lead to interpretability for experts. Thus, system design and empirical experiments in highly specialized tasks are essential; however, there is still a lack of research on XAI frameworks for industrial use and their validation through user studies [13]. This study examines the framework of a video surveillance system that can provide relevant explanations for expert judgments by incorporating their decision-making processes into the neural networks. The experts watch a video and determine whether there are any abnormalities after considering various attributes based on their domain knowledge. We design a neural network to incorporate this process into anomaly detection. Specifically, we extract different features related to an anomaly based on domain knowledge, and each stream with extracted features as input determine whether the features are anomalous. Here, explicitly extracting the features that experts use to make judgments enables them to be presented as interpretable information for the experts The above framework is applied to video-based detection for calving signs in cattle thorough a user study with livestock farmers (Fig. 1). Since calf deaths during calving can be very costly for farmers [14], accurately detecting the signs of calving is important for livestock management. Although contact-type sensors, which are attached directly to cattle, are widely used, a detection system using a camera, which is a non-contact-type sensor, is valuable in terms of management efficiency and animal welfare. First, to apply the above framework, we reviewed the literature on animal studies and conducted farmer interviews to collect domain knowledge about calving signs. Using our findings, we designed a system that explicitly extracts statistical information on the posture, rotation, and movement of cattle relevant to calving and then identifies the signs from these features. We designed the user interface of the notification screen using the information provided by the feature extractor. In the user interface, the fre-quency of posture, amount of rotation, amount of movement, and the pattern of movement (bird's eye view) of cattle are presented to farmers. Compared with the user interface of the end-to-end system, which only presents the probability values of calving signs, the presentation of the internal state of the system is expected to be more interpretable for farmers. The user interface of the end-to-end system and the proposed system were evaluated by people who were involved in livestock farming (N=6). We anticipate that our findings will be useful in developing human-centered explainable AI-based systems that effectively incorporate experts' knowledge. The rest of the paper is organized as follows. In Section II, we briefly explain the framework which enables us to understand the reasoning behind predictions. Section III describes the case study on a calving detection system in livestock farming based on our framework. In Section IV, we discuss the findings from the case study and future work of the proposed framework. We conclude in Section V with a brief summary. II. FRAMEWORK The proposed framework incorporates the users' decisionmaking process into a prediction network and allows for interpretation on the basis of their domain knowledge. A typical machine leaning-based system uses an end-to-end approach with experts' annotations to model the notification target. In contrast, the proposed framework uses a humancentric approach and models the notification target through user interviews. This section describes the following four phases of the design procedure for the proposed framework: • Interview with an expert • Designing streams to extract the information that characterizes what is being monitored • Designing detectors by integrating information from multiple streams • Designing the notification interface In the first phase, we interview experts on the notification target. In interviews, we ask them to verbalize their decisionmaking processes when detecting the target, and determine the features they focus on in daily operations. Then we divide those features into more detailed features that can be judged even by non-experts. In the next phase, we extract more detailed features from the ones that experts focus on when they detect the notification target, and then we develop a stream network based on the extracted features. Dividing the features into more detailed features that can be judged even by non-experts, enables the addition of crowd-sourced annotations. This is important in terms of making the surveillance system sustainable, as it allows us to increase the amount of training data without having to rely on experts in the domain. In the third phase, we develop a network that integrates multiple stream networks on the basis of different attribute features. The users are notified based on the posterior probability of the network. Here, even if the detection of each stream network is not necessarily effective, if the predictions of each stream are complementary, the detection can be improved. Finally, we design the user interface that is presented to the users. In addition to the final posterior probability of the notification target, multiple features extracted from each stream network can be presented as the reasons for the prediction results. These features are designed with consideration for the responses in the user interviews, so they provide information necessary for users' decision-making processes. III. CASE STUDY In this section, we verify the effectiveness of the proposed framework for detecting signs of calving in cattle through a user study with livestock farmers. First, we describe the system design based on our framework introduced in Section II, and then we describe the procedure for the user experiments and the results and findings. A. System Design In this section, we describe the design of our calving detection system based on the proposed framework. We describe the domain knowledge of calving signs and the network structure that explicitly extracts features relevant to calving. Finally, we explain the user interface that presents the farmers with the reasons for the prediction results. 1) Calving Signs Observable from Video: This section presents an overview of the domain knowledge about calving signs. We interviewed farmers and reviewed the animal science literature to identify cattle behaviors related to calving. Changes in postures and behaviors related to calving have been extensively investigated in animal science. This part corresponds to the implementation of the first phase of the proposed framework. The following are typical posture-based calving signs that can be observed from images: • Switching between standing and lying postures: About two to six hours before calving, the number of posture changes (e.g., switching between standing and lying) become more frequent [15], [16], [17] and the time spent lying increases two hours before calving [17]. • Tail raising: About four to six hours before calving, tail raising becomes more frequent [17] and the position of the tail before calving is elevated [18], [19]. The following action-based calving signs were observed in the video: • Increase in the number of rotations and turns: Characteristic walking patterns (e.g., rotations and turns) can be observed four hours before calving and become more frequent three hours before calving [20]. • Increase in aimless walking time: The duration of walking on the calving day increases [16], [17]. Aimless walking time apparently increases about 140 minutes before calving [19]. 2) System Architecture: We designed a multi-stream network [21] that extracts statistical information on posture, rotation, and movement based on calving signs identified in prior studies and integrates the three streams depending on the situation. This part corresponds to the implementation of the second and third phases of the proposed framework. Specifically, each stream identifies calving signs for each 30minute input video based on features as follows: • Posture-based feature: The appearance of a cattle standing, lying, and raising its tail are captured for each video frame using ResNet-50 [22] and then accumulated into the relevant frequencies using temporal pooling techniques. • Rotation-based feature: Information on body direction is extracted from each video frame using ResNet-50 and accumulated into a statistic on the cattle's rotation by measuring the changes in the body direction using the M-measure [23], [24]. • Movement-based feature: The region of the cattle's body is detected in each video frame using YOLOv3 [25] and differences in locations across frames are accumulated into a statistic on the cattle's movement. The calving-relevant features are designed to be extracted from information that can be judged by non-experts, such as posture, neck and tail positions, and positional coordinates. This makes it possible to collect data using crowdsourcing, and re-training the feature extraction mechanism becomes easier. 3) User Interface of Notification Screen: In this section, we describe the design of the user interface of the proposed system. This part corresponds to the implementation of the fourth phase of the proposed framework. In addition to the final posterior probability of calving signs, the proposed framework provides the following information related to calving signs for each frame: • Posterior probability of cattle's posture, 1) standing cattle with tail raised, 2) standing cattle without tail raised, 3) lying cattle, and 4) can't tell • Heatmaps of cattle's body direction • Position coordinates of cattle Displaying these data in an easy-to-understand representation will help farmers to estimate when cattle start calving. Fig. 2 shows the user interface designed using the information described above. In addition to the monitoring video and the posterior probability values of the system in this scene, the interface also displays information on the frequency of posture, amount of rotation, amount of movement, and trajectory of the cattle as seen from directly above. The upper right bar graph shows the posterior probability of calving signs which the system output using the 30-minute video frames. Four graphs are presented, each with posterior probabilities and their mean values based on the statistics on posture, rotation, and movement. Users can check which information the system considers to be a calving sign. The statistics on posture, rotation, and movement are presented at the bottom of the screen. The circle graph on the left shows the frequency of these 30-minute postures calculated from the posterior probability of the posture classification obtained by the feature extractor. This graph can be interpreted as the reasons for the prediction results of the posture-based stream. The comparison with the normal state in the amount of rotation (shown in the center) is calculated from the estimated heatmaps of the cattle's body direction obtained from the feature extractor. It is possible to quantify how much the cattle turn during these 30 minutes compared with their default state, which can then be interpreted as reasons for the prediction results. In the prototype user interface, we visualized the time-series changes in the angle of body direction. After interviewing a farmer, a bar graph was used to simplify the representation. The comparison with the normal state in the amount of movement (shown on the right side) is calculated from the amount of change in the positional coordinates of the cattle. Compared with the normal state, the amount of movement of cattle during this 30-minute can be quantified and interpreted as the reasons for the prediction results of the movement-based stream. Finally, a bird's eye view of the cattle's position is displayed on the right side of the monitoring video, which shows the trajectory of the cattle's position as seen from directly above the room. This allows us to understand the general movement pattern without continuously watching the video. B. User Study To evaluate the proposed framework, we experimentally compared it with the end-to-end system were conducted. The experiment took about 60-90 minutes in total and consisted of instruction, practice, experiment part 1, part 2, and a postexperiment survey (Fig. 3). First, the participants watched a Characteristic behaviors related to posture and rotation were observed before calving. 5-minute instructional video on the experiment. Afterwards, we obtained written informed consent signed by the participants before the experiments, and the participants were granted a gift card of ¥ 5,000 JPY (roughly $50 USD). As an exercise, the participants responded to two notifications from the proposed system and the end-to-end system. In this section, the participants were instructed to think aloud [26] and practiced answering questions by actively commenting on what they saw and thought during the experiment. The experiment consisted of two parts with a break in between, and the participants responded to 36 notifications in total. The participants were divided into two experimental groups, and each group was shown the user interface in a pseudo-random order to account for the sequential effects of the interface order. After the experiment, a qualitative post-experiment survey was conducted using Google Forms. In the experiment, the two user interfaces were presented to the participants in a pseudo-random order on the basis of the experimental group. The user interface of the end-toend system for comparison is shown in Fig. 4. Compared with the user interface of the proposed system, the endto-end system presents only the posterior probability graph of the prediction result. For the same 30-minute sequence, the value of the posterior probability presented in the user interface of the end-to-end system was the same value as the average posterior probability of each stream in the user interface of the proposed system. To avoid bias from the system names, we designated the end-to-end system as system A and the proposed system as system B when presenting the user interfaces to the participants. The unit of the notification was a 30-minute video sequence, and a total of 18 sequences were prepared. A positive case was defined as the time from three to zero hours before calving and a negative case as the time from 24 to 27 hours before calving. The prepared sequences consisted of six sequences each of true positives and false positives, and three sequences each of false negatives and true negatives. In other words, we assume that the notification threshold is significantly lowered and contains a certain number of false positives. The participants are asked to answer the following two questions (maximum of four) at the bottom of the interface: Q1-1 Did you recognize calving signs in this 30-minutes scene? (-3: Absolutely Not, ... 3: Absolutely Yes) Q1-2 What are the reasons for your answer? (Verbal answer) The participants responded to the following prompts in Q1-2: 1) What did you see on the screen?, 2) What did you think?, and 3) How did you reach your decision? The intention of this format was to reveal as much of their thought process as possible. After answering the above questions, if the answer to Q1-1 was "neither" or "predictive" (0-3), the participant was asked to answer the following questions as well: Q2-1 What action would you take after seeing this user interface? (1: Begin assisting immediately, 2: Start making arrangements, 3: Do nothing) Q2-2 What are the reasons for your answer? (Verbal answer) The candidate answers to Q2-1 are terms referring to common decision-making behaviors for livestock farmers. The terms are used in the contact-type sensor, Gyuonkei 1 , a calving notification sensor widely used in Japan. Here, "Begin assisting immediately" refers to the decision to immediately begin assisting with calving, and "Start making arrangements" refers to the preparations made about 24 hours before calving. These questions are designed to encourage participants to use the interface for their decision-making. Each participant is asked Table I. The participants belonged to two groups: farmers (P1-4) and people from an agricultural college (P5-6). Here, P5 is a professor, and P6 is a student of animal science. The experiment was administered on a computer, and the participants accessed the URL to the experiment page provided in advance. We used a web conference application (Zoom) to record the participants' responses. To record the participant's speech during their think-aloud, we recorded online on Zoom and locally as a backup. C. Results and Findings Participants judged whether a behavior was a calving sign from the monitoring video, but the behavior of their responses varied depending on the type of the user interface presented. When the interface of the end-to-end system was presented, most of the verbal comments were related to the video. However, when the interface of the proposed system was presented, some of the comments were related to the statistics. One participant shared their judgment, referring to a graph comparing the amount of movement with normal conditions. P5 stated "I think the cattle is near to calving because the data shows a lot of movement and an increase in rotation.". Given the trend of the responses, we believe that presenting the internal state of the proposed system was effective for judging whether a behavior was a calving sign. The results of the responses to the post-experimental survey are shown in Fig 5. Five out of six participants selected the proposed system as the system they would like to use in the future (Fig. 5, Q'1). They found that the internal state of the system was helpful in identifying calving signs. P1 said "It is easier to understand when the amount of movement, which is usually judged by the senses, is visualized in a graph." P2 said "It's helpful to see detailed information about the cattle." P3 said "The presented graphs were helpful in determining whether a behavior was a calving sign." , and P5 said "Because the data is presented in detail, I think less experienced farmers can make decisions with more certainty." In addition, some of the comments fit the hypothesis that the user interface of the end-to-end system, which is a blackbox, is insufficient for decision-making. P1 stated, "The [endto-end system's] predicted probability alone does not tell us what we should do. I don't know what to do with it.". In contrast, P4, who was the only one to choose the end-to-end system in this question, said, "I did not find the indicators in the proposed system to be judgmental because there were many situations where the information presented differed from my perceptions." P4 was a farmer who was asked to assist in the interviews to collect domain knowledge about calving signs, and he stated that the information presented was less accurate than he expected. One shortcoming is that a user evaluation should have been conducted in the second phase of the framework design. Because we did not receive sufficient feedback from experts in this phase, the extracted features were not always sufficiently useful indicators. In addition, it was suggested that actively presenting the internal state of the system, including the errors, may cause noise in judgment and cause participants to distrust the system. Next, we discuss which information on the user interface was useful (Fig. 5, Q'3, Q'4). We found that the trend in the probability of calving signs varied between participants. Only P4 responded that the probability of the end-to-end system was more useful, while the other participants found the proposed system to be more useful or about the same. We now turn our attention to the results of the user interface of each statistic on the posture, rotation, and movement of the proposed system (Fig. 5, Q'4). Two participants from the agricultural college (P5, P6) responded that the graph of the posture frequency was useful, while the remaining four farmers responded that it was generally not useful. This may be because the farmers could view the raw data to obtain more accurate information on posture frequency, so there was less of a need to refer to the information presented by the system. On the other hand, participants from an agricultural college with an academic background found it useful to be able to quantitatively visualize the posture frequency. A higher percentage of participants found the graph of the amount of movement to be more useful than the posture frequency graph because the meta-information, which cannot be captured from the raw data, is useful for judgment. In fact, many of the participants said that the comparison with normal conditions was helpful during their responses. P1 said "Before, I judged the amount of rotation and movement intuitively, so it's easier to see when graphed." and P6 said "The graph [of the amount of movement] makes it easier to notice the degree of change compared with the normal conditions." The rotation graph tended to be less useful than the movement graph because the participants paid attention to the intensity of the cattle's movement as one of the criteria for judging whether it was a calving sign, and the amount of rotation was not directly related to their judgment. Thus, it is important to present features in a way that users can easily understand, and a more abstract expression such as "intensity of movement" may be more suitable. In their responses to the last question (Q'7), participants pointed out issues related to the accuracy of the presented information and that the user interface made the video smaller when information is presented. For the former issue, P4 said "I felt that the graph was not directly linked to the calving signs." and P6 said "I was worried when the AI identified an indeterminable posture in many cases. I would prefer to judge it myself in such situations." For the latter issue, P5 stated, "Readability. I don't want the video to be too small." IV. DISCUSSION Black-box AI systems are considered inadequate for supporting human decision-making. One farmer pointed out that the end-to-end system lacked instructions on what they should do after seeing the value for the probability of calving signs. Several farmers pointed out the advantages of our proposed system, stating that the detailed information would help novice farmers make correct decisions confidently and the reason visualization was helpful. In particular, statistical information including meta-information that cannot be obtained from the raw data tended to be particularly useful. In contrast, we found that the reliability of the system is adversely affected when the presented information is incorrect. This has been reported in several studies as algorithm aversion [27], [28], which is a phenomenon in which users stop trusting algorithms after seeing its mistakes. To present more information to users, the Q'1. Which AI system would you like to use in the future? Q'3. How useful was the information presented in end-to-end system? Q'5. Which AI system provided more accurate predictions? Q'4. How useful was the information presented in proposed system? proposed framework needs to take into account the accuracy of the information. In addition, we should have conducted a user evaluation in the second phase of the framework design because the extracted features may not have been sufficiently useful indicators for users. In the participants' feedback, they noted that the user interface became more complicated as more information was presented. When considering development for actual use, it is necessary to consider elderly users as well as smartphones which may make the screen more difficult to view. A more sophisticated user interface design is needed, such as the separation of the summary and detailed analysis screens. We anticipate that our evaluation of an XAI framework through user studies will contribute to the integration of XAI in various industrial domains. V. CONCLUSION In this study, we proposed a framework for a video surveillance system that incorporates experts' decision-making processes into the architecture such that the reasons for the prediction results can be interpreted. We evaluated the calving sign detection systems based on our proposed framework through a user study with people involved in livestock farming (N=6). The proposed framework was compared with an endto-end system, and five out of six participants selected the proposed framework as the system they would like to use in the future. In addition, the proposed framework is used to present the internal state of the system, which can be used to help users make decisions and identify system errors. However, we found that presenting an inaccurate internal state of the system could interfere with the user's judgment and cause them to distrust the system. In future work, we intend to study the accuracy of the information presented to users. Fig. 1 . 1User study of livestock farmers. Fig. 2 . 2User interface of proposed system (System B). User interface presented to participants was in Japanese. Fig. 3 . 3Schematic diagram of experimental procedure. Order of tasks differed for each experimental group. to respond to the above questions for a total of 36 notifications (2 UIs x 18 sequences). The questions in the post-experiment survey are as follows: Q'1 Which AI system would you like to use in the future? (1. System A, 2. System B) Q'2 Why did you choose that system? (Orally, if you prefer.) Q'3 How useful was the information presented in System A? (-2. Useless, ..., 2. Useful) a Graph of the probability of calving signs. Q'4 How useful was the information presented in System B? (-2. Useless, ..., 2. Useful) a Graph of the probability of calving signs. b Graph of the posture frequency. c Graph of the amount of rotation. d Graph of the amount of movement. e Bird's eye view (position of the cattle as seen from directly above). Q'5 Which system provided more accurate predictions?(1. System A, 2. System B, 3. Neither) Q'6 What are the advantages of the system you chose in Q'1 over the other? (Orally, if you prefer.) Q'7 What improvements (if any) would you make to System B? (Orally, if you prefer.) Six people who involved in livestock farming participated in the user experiment. Age, sex, and years of experience in the livestock industry of the participants are shown in Fig. 4 . 4User interface of end-to-end system (System A). User interface presented to participants was in Japanese. Fig. 5 . 5Responses to select questions in post-experiment survey. TABLE I DETAILS IOF EXPERIMENT PARTICIPANTS. FOUR ARE FARMERS (P1-P4), AND OTHER TWO STUDIES ARE IN AGRICULTURAL COLLEGE (P5-P6).ID Group Age Sex Years of experience P1 II 30's M 10-19 P2 I 50's M 20-29 P3 I 20's F 5-9 P4 II 40's M 20-29 P5 I 40's M 20-29 P6 II 10's F 0-1 Gyuonkei, http://www.gyuonkei.jp ACKNOWLEDGMENTWe would like to thank the farmers who participated in the experiment and Kagoshima Prefecture Agricultural College. We also thank Kagoshima Brain Center for fruitful discussions. A human-centered evaluation of a deep learning system deployed in clinics for the detection of diabetic retinopathy. E Beede, E Baylor, F Hersch, A Iurchenko, L Wilcox, P Ruamviboonsuk, L M Vardoulakis, Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems (CHI '20). the 2020 CHI Conference on Human Factors in Computing Systems (CHI '20)E. Beede, E. 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Wort- man Vaughan, "Interpreting interpretability: Understanding data scien- tists' use of interpretability tools for machine learning," in Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems (CHI '20), 2020, p. 1-14. Cogam: Measuring and moderating cognitive load in machine learning model explanations. A Abdul, C Der Weth, M Kankanhalli, B Y Lim, Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems (CHI '20). the 2020 CHI Conference on Human Factors in Computing Systems (CHI '20)A. Abdul, C. von der Weth, M. Kankanhalli, and B. Y. Lim, "Cogam: Measuring and moderating cognitive load in machine learning model explanations," in Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems (CHI '20), 2020, pp. 1-14. Understanding the effect of accuracy on trust in machine learning models. M Yin, J Wortman Vaughan, H Wallach, Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19). the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19)M. Yin, J. Wortman Vaughan, and H. Wallach, "Understanding the effect of accuracy on trust in machine learning models," in Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19), 2019, pp. 1-12. Designing theory-driven user-centric explainable ai. D Wang, Q Yang, A Abdul, B Y Lim, Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19). the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19)D. Wang, Q. Yang, A. Abdul, and B. Y. Lim, "Designing theory-driven user-centric explainable ai," in Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems (CHI '19), 2019, pp. 1-15. Economic cost of difficult calvings in the united kingdom dairy herd. 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Jensen, "Behaviour around the time of calving in dairy cows," Applied Animal Behaviour Science, vol. 139, no. 3, pp. 195-202, 2012. Detection of parturition in cow and mare by a useful warning system. L P Bueno, D J Tainturier, Y Ruckebusch, Theriogenology. 166L. P. Bueno, D. J. Tainturier, and Y. Ruckebusch, "Detection of partu- rition in cow and mare by a useful warning system," Theriogenology, vol. 16, no. 6, pp. 599-605, 1981. Parturient behaviour and calf survival in a herd selected for twinning. J L Owens, T N Edey, B M Bindon, L R Piper, Applied Animal Behaviour Science. 134J. L. Owens, T. N. Edey, B. M. Bindon, and L. R. Piper, "Parturient behaviour and calf survival in a herd selected for twinning," Applied Animal Behaviour Science, vol. 13, no. 4, pp. 321-333, 1985. Calving prediction from video: Exploiting behavioural information relevant to calving signs in japanese black beef cows. 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Her- mansky, "A new efficient measure for accuracy prediction and its appli- cation to multistream-based unsupervised adaptation," in International Conference on Pattern Recognition (ICPR 2017), 2017, pp. 2222-2227. Yolov3: An incremental improvement. J Redmon, A Farhadi, abs/1804.02767CoRR. J. Redmon and A. Farhadi, "Yolov3: An incremental improvement," CoRR, vol. abs/1804.02767, 2018. [Online]. Available: http://arxiv.org/ abs/1804.02767 The think aloud method: A guide to user interface design. M Jaspers, T Steen, C Bos, M Geenen, International journal of medical informatics. 73M. Jaspers, T. Steen, C. Bos, and M. Geenen, "The think aloud method: A guide to user interface design," International journal of medical informatics, vol. 73, pp. 781-95, 12 2004. Algorithm aversion: People erroneously avoid algorithms after seeing them err. B Dietvorst, J Simmons, C Massey, Journal of experimental psychology. General. 1442014B. Dietvorst, J. Simmons, and C. Massey, "Algorithm aversion: People erroneously avoid algorithms after seeing them err," Journal of experi- mental psychology. General, vol. 144, 11 2014. The perceived utility of human and automated aids in a visual detection task. M Dzindolet, L Pierce, H Beck, L Dawe, Human factors. 44M. Dzindolet, L. Pierce, H. Beck, and L. Dawe, "The perceived utility of human and automated aids in a visual detection task," Human factors, vol. 44, pp. 79-94, 2 2002.

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{'abstract': "Through a user study in the field of livestock farming, we verify the effectiveness of an XAI framework for video surveillance systems. The systems can be made interpretable by incorporating experts' decision-making processes. AI systems are becoming increasingly common in real-world applications, especially in fields related to human decisionmaking, and its interpretability is necessary. However, there are still relatively few standard methods for assessing and addressing the interpretability of machine learning-based systems in realworld applications. In this study, we examine the framework of a video surveillance AI system that presents the reasoning behind predictions by incorporating experts' decision-making processes with rich domain knowledge of the notification target. While general black-box AI systems can only present final probability values, the proposed framework can present information relevant to experts' decisions, which is expected to be more helpful for their decision-making. In our case study, we designed a system for detecting signs of calving in cattle based on the proposed framework and evaluated the system through a user study (N=6) with people involved in livestock farming. A comparison with the black-box AI system revealed that many participants referred to the presented reasons for the prediction results, and five out of six participants selected the proposed system as the system they would like to use in the future. It became clear that we need to design a user interface that considers the reasons for the prediction results.", 'arxivid': '2301.03926', 'author': ['Ryosuke Hyodo \nWaseda University\nTokyoJapan\n', 'Susumu Saito \nWaseda University\nTokyoJapan\n\nIntelligent Framework Lab\nTokyoJapan\n', 'Teppei Nakano \nWaseda University\nTokyoJapan\n\nIntelligent Framework Lab\nTokyoJapan\n', 'Makoto Akabane \nWaseda University\nTokyoJapan\n', 'Ryoichi Kasuga \nFarmers Support\nKagoshimaJapan\n', 'Tetsuji Ogawa \nWaseda University\nTokyoJapan\n'], 'authoraffiliation': ['Waseda University\nTokyoJapan', 'Waseda University\nTokyoJapan', 'Intelligent Framework Lab\nTokyoJapan', 'Waseda University\nTokyoJapan', 'Intelligent Framework Lab\nTokyoJapan', 'Waseda University\nTokyoJapan', 'Farmers Support\nKagoshimaJapan', 'Waseda University\nTokyoJapan'], 'corpusid': 255569825, 'doi': '10.48550/arxiv.2301.03926', 'github_urls': [], 'n_tokens_mistral': 10580, 'n_tokens_neox': 9453, 'n_words': 6496, 'pdfsha': '22d0e1d4b60ca17886df87ca5141f6976d9e2e9e', 'pdfurls': ['https://export.arxiv.org/pdf/2301.03926v1.pdf'], 'title': ['Video Surveillance System Incorporating Expert Decision-making Process: A Case Study on Detecting Calving Signs in Cattle', 'Video Surveillance System Incorporating Expert Decision-making Process: A Case Study on Detecting Calving Signs in Cattle'], 'venue': []}

arxiv

Symmetric Rank-k Methods Chengchang Liu Luo Luo ‡ Cheng Chen Luo Luo ‡ Symmetric Rank-k Methods This paper proposes a novel class of block quasi-Newton methods for convex optimization which we call symmetric rank-k (SR-k) methods. Each iteration of SR-k incorporates the curvature information with k Hessian-vector products achieved from the greedy or random strategy. We prove SR-k methods have the local superlinear convergence rate of O (1 − k/d) t(t−1)/2 for minimizing smooth and strongly self-concordant function, where d is the problem dimension and t is the iteration counter. This is the first explicit superlinear convergence rate for block quasi-Newton methods and it successfully explains why block quasi-Newton methods converge faster than standard quasi-Newton methods in practice. Introduction We study the quasi-Newton methods for solving the minimization problem min x∈R d f (x),(1) where f : R d → R is smooth and strongly self-concordant. Quasi-Newton methods [2,3,4,6,8,33,36] are widely recognized for their fast convergence rates and efficient updates, which attracts growing attention in many fields such as statistics [1,16,37], economics [20,24] and machine learning [12,15,19,22,23]. Unlike standard Newton methods which need to compute the Hessian and its inverse, quasi-Newton methods go along the descent direction by the following scheme x t+1 = x t − G −1 t ∇f (x t ), where G t ∈ R d×d is an estimator of the Hessian ∇ 2 f (x t ). The most popular ways to construct the Hessian estimator are the Broyden family updates, including the Davidon-Fletcher-Powell (DFP) method [8,10], the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [3,4,33], and the symmetric rank 1 (SR1) method [2,8]. The classical quasi-Newton methods with Broyden family update [3,4] find the Hessian estimator G t+1 for the next round by the secant equation G t+1 (x t+1 − x t ) = ∇f (x t+1 ) − ∇f (x t ).(2) These methods have been proven to exhibit local superlinear convergence in 1970s [5,9,28], and their nonasymptotic superlinear rates were established in recent years [17,30,31,35]. For example, Rodomanov and Nesterov [31] showed classical BFGS method enjoys the local superlinear rates of O (dκ/t) t /2 which has been later improved to O (exp(d ln(κ)/t) − 1) t/2 [30], and Ye et al. [35] showed classical SR1 method converges with local rate of O (d ln(κ)/t) t/2 , where κ is the condition number of the objective. Some recent works [13,29] proposed new types of quasi-Newton methods, which construct the Hessian estimator by the following equation G t+1 u t = ∇ 2 f (x t+1 )u t ,(3) where u t ∈ R d is chosen by greedy or random strategies. Rodomanov and Nesterov [29] established the local superlinear rate of O (1 − 1/(κd)) t(t−1)/2 for greedy quasi-Newton methods with Broyden family updates. Later, Lin et al. [21] provided the condition-number free superlinear rate of O (1 − 1/d) t(t−1)/2 for greedy and random quasi-Newton methods with specific BFGS and SR1 updates. Block quasi-Newton methods construct the Hessian estimator along multiple directions at per iteration. The study of these methods dates back to 1980s. Schnabel [32] proposed first block BFGS method by extending equation (2) to multiple secant equations G t+1 (x t+1 − x t+1−j ) = ∇f (x t+1 ) − ∇f (x t+1−j ) for j = 1, · · · , k. Although block quasi-Newton methods usually have better empirical performance than classical ones [11,13,14,18,27,32], their theoretical guarantees are mystery until Gao and Goldfarb [11] proved block BFGS method has asymptotic local superlinear convergence. On the other hand, Gower and Richtárik [13], Gower et al. [14], Kovalev et al. [18] introduced the randomized block BFGS by generalizing condition (3) to G t+1 U t = ∇ 2 f (x t+1 )U t , where U t ∈ R d×k is some random matrix. The empirical studies show randomized block BFGS performs well on real-world applications. Kovalev et al. [18] showed randomized block BFGS method also has asymptotic local superlinear convergence, but its advantage over vanilla BFGS methods is still unclear in theory. The known results cannot explain why block quasi-Newton methods enjoy faster convergence behavior than vanilla quasi-Newton methods in practice. This naturally leads to the following question: Can we design a block quasi-Newton method with explicit superior convergence rate? In this paper, we give an affirmative answer to above question by proposing symmetric rank-k (SR-k) methods. The construction of Hessian estimators in SR-k methods are based on generalizing the idea of symmetric rank 1 (SR1) [2,8,35] methods and the equation of the form (3). We provide the random and greedy strategies to determine U t for SR-k. Both of these strategies lead to the explicit local superlinear convergence rate of O (1 − k/d) t(t−1)/2 , where k is the number of directions used to approximate Hessian at per iteration. For k = 1, our convergence rate reduces to the one of greedy and random SR1 methods [21]. For k ≥ 2, it is clear that the convergence rate of SR-k methods is better than existing greedy and random quasi-Newton methods [21,29]. We also follow the design of SR-k to propose a variant of randomized block BFGS method [13,14,18], resulting an explicit superlinear convergence of O (1 − k/(κd)) t(t−1) . We compare proposed methods with existing quasi-Newton methods for minimizing strongly convex function in Table 1. The remainder of this paper is organized as follows. In Section 2, we introduce the notation and the preliminaries throughout this paper. In Section 3, we introduce the SR-k update in the view of matrix approximation. In Section 4, we propose the quasi-Newton methods with SR-k updates for solving the strongly self-concordant function and provide their the superior local superlinear convergence rates. In Section 5, we propose a variant of randomized block BFGS method with explicit local superlinear convergence rate. In Section 6, we conduct numerical experiments to show the outperformance of proposed methods. Finally, we conclude our work in Section 7. Preliminaries We use {e 1 , · · · , e d } to present the the standard basis in space R d and let I d ∈ R d×d be the identity matrix. We denote the trace of a square matrix by tr (·). We use · to present the spectral norm and Euclidean norm of matrix and vector respectively. Given a positive definite matrix A ∈ R d×d , we denote the corresponding weighted norm as x A (x Ax) 1/2 for some x ∈ R d . We use the notation x z to present x ∇ 2 f (z) for positive definite Hessian ∇ 2 f (z), if there is no ambiguity for the reference function f (·). We also define E k (A) [e i1 ; · · · ; e i k ] ∈ R d×k ,(4) where i 1 , . . . , i k are the indices for the largest k entries in the diagonal of A. Throughout this paper, we suppose the objective in problem (1) satisfies the following assumptions. O (1 − 1/d) t Greedy/Randomized SR1 [21] 1 or 2 O (1 − 1/d) t Multi-Secant Block-BFGS [11, 32] k ∈ [d] implicit Randomized Block-BFGS (v1) [13,18] k ∈ [d] implicit Randomized Block-BFGS (v2) Algorithm 3 k ∈ [d] O (1 − k/(κd)) t SR-k Algorithm 1 k ∈ [d] O (1 − k/d) t , k ∈ [d − 1] O λ t , k = d Assumption 2.1. We assume the objective function f : R d → R is L-smooth, i.e., there exists some constant L ≥ 0 such that ∇f (x) − ∇f (y) 2 ≤ L x − y 2 for any x, y ∈ R d . Assumption 2.2. We assume the objective function f : R d → R is µ-strongly-convex, i.e., there exists some constant µ > 0 such that f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) − λ(1 − λ)µ 2 x − y 2 2 for any x, y ∈ R d and λ ∈ [0, 1]. We define the condition number as κ L/µ. The following proposition shows the objective function has bounded Hessian under Assumption 2.1 and 2.2. µI d ∇ 2 f (x) LI d(5) for any x ∈ R d . We also impose the assumption of strongly self-concordance [21,29] as follows. Assumption 2.4. We assume the objective function f : R d → R is M -strongly self-concordant, i.e., there exists some constant M > 0 such that ∇ 2 f (y) − ∇ 2 f (x) M y − x z ∇ 2 f (w),(6) for any x, y, w, z ∈ R d . The strongly-convex function with Lipschitz-continuous Hessian is strongly self-concordant. f (x) − ∇ 2 f (y) ≤ L 2 x − y , for all x, y ∈ R d , then f satisfies is M -strongly self-concordant with M = L 2 /µ 3/2 . Symmetric Rank-k Updates We propose the symmetric rank-k (SR-k) update as follows. Definition 3.1 (SR-k Update). Let A ∈ R d×d and G ∈ R d×d be two positive-definite matrices with A G. For any full rank matrix U ∈ R d×k with k ≤ d, we define SR-k(G, A, U) G if GU = AU. Otherwise, we define SR-k(G, A, U) G − (G − A)U(U (G − A)U) −1 U (G − A).(7) We provide two strategies to select U ∈ R d×k for SR-k update: 1. For randomized strategy, we sample each entry of U according to N (0, 1) independently. For greedy strategy, we construct U = E k (G − A) , where E k (·) follows the notation of (4). For k = 1, SR-k updates with above two strategies reduce to randomized or greedy SR1 updates [21,31]. The remain of this section shows the multiple directions in U ∈ R d×k provably make SR-k update has the advantage over SR1 update in the view of estimating the target matrix A. First, we provide the following lemma to show the output G of SR-k update does not increase the deviation from A, which is similar to ordinary Broyden family updates [29]. Lemma 3.2. For any positive-definite matrices A ∈ R d×d and G ∈ R d×d with A G ηA for some η ≥ 1, we let G + = SR-k(G, A, U) for some full rank matrix U ∈ R d×k . Then it holds that A G + ηA.(8) Then we introduce the quantity [21,35] τ A (G) tr (G − A) (9) to characterize the difference between A and G. We can prove SR-k updates with randomized or greedy strategies enjoy explicit faster convergence rate than SR1 updates for estimating A. Theorem 3.3. Let G + = SR-k(G, A, U)(10) with G A ∈ R d×d and select U ∈ R d×k by one of the following strategies: 1. Sample each entry of U according to N (0, 1) independently. Construct U = E k (G − A). Then, we have E [τ A (G + )] ≤ 1 − k d τ A (G).(11) The term (1 − k/d) in inequality (11) reveals the advantage of block-type update in SR-k methods, since the larger k leads to the faster decay of τ A (G + ). As a comparison, the results of randomized or greedy SR1 updates [21] match the special case of Theorem 3.3 with k = 1. Algorithm 1 Symmetric Rank-k Method 1: Input: G 0 , M and k. 2: for t = 0, 1 . . . 3: x t+1 = x t − G −1 t ∇f (x t ) 4: r t = x t+1 − x t xt 5:G t = (1 + M r t )G t 6: Construct U t ∈ R d×k by (a) randomized strategy: [U t ] ij i.i.d ∼ N (0, 1) (b) greedy strategy: U t = E k (G t − ∇ 2 f (x t+1 )) 7: G t+1 = SR-k(G t , ∇ 2 f (x t+1 ), U t ) 8: end for Minimization of Strongly Self-Concordant Function We propose SR-k methods for minimizing strongly self-concordant function in Algorithm 1, where M > 0 follows the notation in Assumption 2.4. Then we provide the convergence analysis for SR-k methods and show its superiority to existing quasi-Newton methods. The convergence of SR-k methods (Algorithm 1) is measured by the local gradient norm [29] λ (x) ∇f (x) (∇ 2 f (x)) −1 ∇f (x).(12) The theoretical analysis starts from the following result for quasi-Newton iterations. [29]). Suppose that the twice differentiable function f : R d → R is strongly self-concordant with constant M > 0 and the positive definite matrix G t ∈ R d×d satisfies ∇ 2 f (x t ) G t η t ∇ 2 f (x t )(13) for some η t ≥ 1. Then the update formula x t+1 = x t − G −1 t ∇f (x t )(14) holds that x t+1 − x t xt ≤ λ(x t ) and λ(x t+1 ) ≤ 1 − 1 η t λ(x t ) + M 2 (λ(x t )) 2 + M 2 4η t (λ(x t )) 3 .(15) Applying Lemma 4.1 with fixed η t = 3η 0 /2 and Lemma 3.2, we can establish the linear convergence rate of SR-k methods. λ(x 0 ) ≤ ln(3/2) 4η 0 M and ∇ 2 f (x 0 ) G 0 η 0 ∇ 2 f (x 0 ) for some η 0 ≥ 1. Then it holds that ∇ 2 f (x t ) G t 3η 0 2 ∇ 2 f (x t ) and λ(x t ) ≤ 1 − 1 2η 0 t λ(x 0 ).(16) Note that the choice of η t in inequality (15) is very important for guarantee the convergence rate of the quasi-Newton method. Specifically, we can obtain the superlinear rate for iteration (14) if there exists some η t ≥ 1 that converges to 1. For example, the randomized and greedy SR1 methods [21] corresponds to some η t such that E[η t − 1] ≤ O 1 − 1 d t . As the results shown in Theorem 3.3, the proposed SR-k updates have the superiority in matrix approximation. So it is natural to construct some η t ≥ 1 for SR-k methods (Algorithm 1) such that E[η t − 1] ≤ O 1 − k d t . Based on above intuition, we derive the local superlinear convergence rate for SR-k methods, which is explicitly sharper than existing randomized and greedy quasi-Newton methods [21,29]. λ(x 0 ) ≤ ln 2 2 · (d − k) M η 0 d 2 κ and ∇ 2 f (x 0 ) G 0 η 0 ∇ 2 f (x 0 )(17) for some η 0 ≥ 1. Then we have E λ(x t+1 ) λ(x t ) ≤ 2dκη 0 1 − k d t ,(18) which naturally indicates the following two stage convergence: • For SR-k method with randomized update, we have λ(x t0+t ) ≤ 1 − k d + k t(t−1)/2 · 1 2 t · 1 − 1 2η 0 t0 λ(x 0 ), with probability at least 1 − δ for some δ ∈ (0, 1), where t 0 = O(d ln(η 0 κd/δ)/k). • For SR-k method with greedy update, we have λ(x t0+t ) ≤ 1 − k d t(t−1)/2 · 1 2 t · 1 − 1 2η 0 t0 λ(x 0 ), where t 0 = O(d ln(η 0 κd)/k). Additionally, SR-k methods with k = d have the local quadratic convergence rate. λ(x 0 ) ≤ ln(3/2) 4M η 0 and ∇ 2 f (x 0 ) G 0 η 0 ∇ 2 f (x 0 ).(19) Then we have E [λ(x t+1 )] ≤ M (λ(x t )) 2(20) for any t ≥ 1. Algorithm 2 Randomized Block BFGS Method (v1) 1: Input: G 0 and k. 2: for t = 0, 1 . . . 3: x + = x t − G −1 t ∇f (x t ) 4: x t+1 = arg min x∈{xt,x+} f (x) 5: Construct U t by [U t ] ij i.i.d ∼ N (0, 1) 6: G t+1 = BlockBFGS(G t , ∇ 2 f (x t ), U t ) 7: end for Algorithm 3 Randomized Block BFGS Method (v2) 1: Input: G 0 , M and k. 2: for t = 0, 1 . . . 3: x t+1 = x t − G −1 t ∇f (x t ) 4: r t = x t+1 − x t xt 5:G t = (1 + M r t )G t 6: Construct U t by [U t ] ij i.i.d ∼ N (0, 1) 7: G t+1 = BlockBFGS(G t , ∇ 2 f (x t+1 ), U t ) 8: end for Improved Results for Block BFGS In this section, we present the non-asymptotic superlinear convergence rate of randomized block BFGS method [13,14] by following the idea of SR-k. The block BFGS update [13,14,32] is defined as follows. Definition 5.1. Let A ∈ R d×d and G ∈ R d×d be two positive-definite symmetric matrices with A G. For any full rank matrix U ∈ R d×k with k ≤ d, we define BlockBFGS(G, A, U) G if GU = AU. Otherwise, we define BlockBFGS(G, A, U) G − GU U GU −1 U G + AU U AU −1 U A.(21) Gower et al. [14], Kovalev et al. [18] proposed randomized block BFGS method (Algorithm 2) by constructing the Hessian estimator with formula (21) and showed it has asymptotic local superlinear convergence rate. For achieving the explicit superlinear convergence rate, we require providing some properties of randomized BFGS update, which is similar to the counterpart of SR-k updates. First, we observe that randomized block BFGS update also has non-increasing deviation from the target matrix. Lemma 5.2. For any positive-definite matrices A ∈ R d×d and G ∈ R d×d with A G ηA for some η ≥ 1, we let G + = BlockBFGS(G, A, U) for some full rank matrix U ∈ R d×k . Then, it holds that A G + ηA.(22) Then we introduce the quantity [29] σ A (G) tr A −1 (G − A) ,(23) to measure the difference of two positive definite matrices. We show that randomized block BFGS update converges to the target matrix with a faster rate than the ordinary randomized BFGS update [21,29]. Theorem 5.3. Consider the block BFGS update G + = BlockBFGS(G, A, U),(24) where G A ∈ R d×d . If µI d A LI d and U ∈ R d×k is selected by sample each entry of U according to N (0, 1) independently. Then, we have E [σ A (G + )] ≤ 1 − k dκ σ A (G).(25) We proposed a variant randomized block BFGS method in Algorithm 3. Based on the observation in Theorem 5.3, we establish its explicit superlinear convergence rate as follows. Figure 1: We demonstrate "#iteration vs. ∇f (x) 2 " and "running time (s) vs. ∇f (x) 2 " on datasets "a9a", "w8a" and "madelon", where we take k = 5 for all of the block quasi-Newton methods. 10 4 GrSR1 RaSR1 BlockBFGSv1 BlockBFGSv2 GrSR-k RaSR-k (a) a9a (iteration) (b) w8a (iteration) (c)10 4 GrSR1 RaSR1 BlockBFGSv1 BlockBFGSv2 GrSR-k RaSR-k (d) a9a (time) (e) w8a (time) (f) madelon (time) Theorem 5.4. Under Assumption 2.1, 2.2 and 2.4, we run Algorithm 3 and set the initial x 0 and G 0 such that λ(x 0 ) ≤ ln 2 4 · 1 M η 0 d and ∇ 2 f (x 0 ) G 0 η 0 ∇ 2 f (x 0 ),(26) for some η 0 ≥ 1. Then we have E λ(x t+1 ) λ(x t ) ≤ 2dη 0 1 − k dκ t . Remark 5.5. For k = 1, Theorem 5.3 and 5.4 match the results of ordinary randomized BFGS methods [21]. Numerical Experiments We conduct the experiments on the model of regularized logistic regression, which can be formulated as min x∈R d f (x) 1 n n i=1 ln(1 + exp(−b i a i x)) + γ 2 x 2 ,(27)where {a i , b i } n i=1 are the training set with a i ∈ R d , b i ∈ {−1, +1} and γ > 0 is the regularization hyperparameter. We refer to SR-k methods (Algorithm 1) with randomized and greedy strategies as RaSR-k and GrSR-k respectively. The corresponding SR1 methods with randomized and greedy strategies are referred as RaSR1 and GrSR1 [21,Algorithm 4] respectively. We also refer to randomized block BFGS (Algorithm 2 [13,14]) and its variant (Algorithm 3) as BlockBFGSv1 and BlockBFGSv2. We compare the proposed RaSR-k, GrSR-k and BlockBFGSv2 with baseline methods on problem (27). For all methods, We set the parameters G 0 and M from {I d , 10 · I d , 10 2 · I d , 10 3 · I d , 10 4 · I d } and {2, 20, 200, 2000} respectively. We evaluate the performance for all of methods on four real-world datasets "a9a", "w8a" and "madelon". We conduct our experiments on a PC with Apple M1 and implement all algorithms in Python 3.8.12. Figure 2: We demonstrate "#iteration vs. ∇f (x) 2 " and "running time (s) vs. ∇f (x) 2 " on datasets "a9a", "w8a" and "madelon", where we take k = 10 for all of the block quasi-Newton methods. We present the results of "iteration numbers vs. gradient norm" and "running time (second) vs. gradient norm" in Figure 1 and Figure 2, which corresponds to the settings of k = 5 and 10 for block quasi-Newton methods RaSR-k, GrSR-k, BlockBFGSv1 and BlockBFGSv2. We observe that the proposed SR-k methods (RaSR-k and GrSR-k) always significantly outperform baselines. Conclusion In this paper, we have proposed symmetric rank-k (SR-k) methods for convex optimization. We have proved SR-k methods enjoy the explicit local superlinear convergence rate of O (1 − k/d) t(t−1)/2 . Our result successfully reveals the advantage of block-type updates in quasi-Newton methods, building a bridge between the theories of ordinary quasi-Newton methods and standard Newton method. As a byproduct, we also provide the convergence rate of O (1 − k/(κd)) t(t−1)/2 for randomized block BFGS method. In future work, it would be interesting to establish the global convergence of SR-k methods and study the convergence for limited memory block quasi-Newton methods. A The Proofs in Section 3 We provide the proofs for the properties of SR-k updates shown in Section 3. We focus on the case of GU = AU, since the results are obvious for GU = AU. A.1 The Proof of Lemma 3.2 Proof. Define R = G − A 0. According to the update rule, we have G + − A =R − RU(U RU) −1 U R = I d − RU(U RU) −1 U R I d − U (URU) −1 U R = I d − RU(U RU) −1 U R I d − U(U RU) −1 U R 0, which means G + A. The condition G ηA means G + ηA − RU(U RU) −1 U R 0 ηA, which finish the proof. A.2 The Proof of Theorem 3.3 We first provide several lemmas for random matrix and the trace of positive definite matrix. Lemma A.1. Let U ∈ R d×k be a random matrix and each of its entry is independent and identically distributed according to N (0, 1), then it holds that E U(U U) −1 U = k d I d .(28) Proof. We use V d,k to present the Stiefel manifold which is the set of all d × k column orthogonal matrices. We denote P k,d−k as the set of all m × m orthogonal projection matrices idempotent of rank k. According to Theorem 2.2.1 (iii) of Chikuse [7], the random matrix Z = U(U U) −1/2 is uniformly distributed on the Stiefel manifold V d,k . Applying Theorem 2.2.2 (iii) of Chikuse [7], the random matrix P = ZZ = U(U U) −1 U is uniformly distributed on P k,d−k . Combining above results with Theorem 2.2.2 (i) of Chikuse [7] on P achieves E[P] = k d I d . Remark A.2. The above proof requires the knowledge for statistics on manifold. For the readers who are not familiar with this, we also present a elementary proof of Lemma A.1 by induction in Appendix A.3. Lemma A.3. For positive semi-definite matrix S ∈ R d×d and the column orthonormal matrix Q ∈ R d×k , we have tr Q SQ ≤ tr (S) .(29) Proof. Since matrix Q is column orthonormal, we have QQ = Q(Q Q) −1 Q I d . According to Lemma C.1, we have tr Q SQ = tr SQQ ≤ tr (S) . Lemma A.4. For positive semi-definite matrix B ∈ R d×d and full rank matrix U ∈ R d×k with d ≥ k, it holds that tr BU U BU −1 U B ≥ tr U U U −1 U B .(30) Proof. We denote SVD of U as U = QΣV , where Q ∈ R d×k , V ∈ R k×k are (column) orthogonal and Σ ∈ R k×k is diagonal. We have tr U U U −1 U B = tr QΣV VΣ 2 V −1 VΣQ B =E tr (U BU) −1 (U B 2 U) (30) ≥ E tr U U U −1 U B = tr E U U U −1 U B = k d tr (B) , for any positive semi-definite B ∈ R d×d . The update rule of SR-k update leads to G + − A = G − A − (G − A)U(U (G − A)U) −1 U (G − A) = B − BU(U BU) −1 U B, where we set B = G − A 0. Then, we have E [tr (G + − A)] = E tr B − BU(U BU) −1 U B = tr (B) − E tr BU(U BU) −1 U B = tr (B) − E tr (U BU) −1 (U B 2 U) ≤ tr (B) − k d tr (B) = 1 − k d tr (B) . Greedy SR-k Update: Given positive semi-definite matrix B ∈ R d×d , we use {b i } k i=1 to denote its diagonal entries such that b 1 ≥ b 2 ≥ · · · ≥ b d , which implies d i=1 b i = tr (B) and k i=1 b i ≥ k d tr (B) .(31) Then we have tr (U U) −1 U BU = tr I k U BU = tr U BU = k p=1 u p Bu p = b 1 + b 2 + · · · + b k (31) ≥ k d d i=1 b i = k d tr (B) ,(32) where u p is the p-th column of U. Setting B = G − A and applying Lemma A.4, we have tr (G + − A) = tr (B) − tr (U BU) −1 (UB 2 U) (30) ≤ tr (B) − tr (U U) −1 U BU (32) ≤ 1 − k d tr (B) ≤ tr (G − A) . A.3 An Elementary Proof of Lemma A.1 We first provide the following lemma for multivariate normal distribution. Lemma A.5. Assume P ∈ R d×k is column orthonormal (k ≤ d) and p ∼ N (0, PP ) is a d-dimensional multivariate normal distributed vector. Then we have E pp p p = 1 k PP . Proof. The distribution p ∼ N (0, PP ) implies there exists a k-dimensional multivariate normal distributed vector p 1 ∼ N (0, I k ) such that p = Pp 1 . Thus we have E pp p p = E (Pp 1 )(Pp 1 ) (Pp 1 ) (Pp 1 ) = E Pp 1 p 1 P p 1 p 1 = PE p 1 p 1 p 1 p 1 P = 1 k PP . Then we provide an elementary proof of Lemma A.1. Proof. We prove inequality (28) by induction on k. The induction base k = 1 is easily verified. Now we assume E U(U U) −1 U = k d I d holds for any U ∈ R d×k that each of its entries are independently distributed according to N (0, 1). We define the random matrixŪ = U q ∈ R d×(k+1) , where q ∼ N (0, I d ) is independent distributed to U. Then we havē U(Ū Ū )Ū = U q U q U q −1 U q = A + (I d − A)qq (I d − A) q (I d − A)q , where A = U(U U) −1 U . Since the rank of projection matrix I d − A is d − k, we have I d − A = QQ for some column orthonormal matrix Q ∈ R d×(d−k) . Thus, we achieve E[Ū(Ū Ū )Ū ] = k d I d + E U E q (I d − A)qq (I d − A) q (I d − A)q U = k d I d + E U E q (QQ q)(q QQ ) (q QQ )(QQ q) U = k d I d + 1 d − k E U [QQ ] = k d I d + 1 d − k E U [I d − A] = k d I d + 1 d − k d − k d I d = k + 1 d I d , which completes the induction. In above derivation, the second equality is due to Lemma A.5 and the fact QQ q ∼ N (0, QQ ) for given Q; the third equality comes from the inductive hypothesis. B The Proof of Section 4 We provide the proofs for the results of SR-k methods shown in Section 4. B.1 Auxiliary Lemmas We first provide some auxiliary lemmas which will be used in our later proof. Lemma B.1. Let {λ t } and {δ t } be two non-negative random sequences that satisfy λ t+1 ≤ (1 + mλ t ) 2 (δ t + bλ t )λ t , λ t ≤ 1 − 1 β t λ 0 , δ 0 + aλ 0 ≤ s(33) and E t [δ t+1 ] ≤ 1 − 1 α (1 + mλ t ) 2 (δ t + cλ t ),(34) for some b, c, m, s, β ≥ 0 and α > 1, where a = max{b, c} > 1 and E t [ · ] E[ · |δ 0 , · · · , δ t , λ 0 , · · · , λ t ]. If λ 0 is sufficient small such that λ 0 ≤ ln 2 β(2m + a(α/(α − 1))) (35) then it holds that E λ t+1 λ t ≤ 1 − 1 α t 2s. Proof. We denote θ t δ t + aλ t .(36) Since the index t + 1 ≥ 1, we have E t [δ t+1 ] (34) ≤ 1 − 1 α (1 + mλ t ) 2 (δ t + aλ t ) ≤ 1 − 1 α e 2mλt θ t and λ t+1 (33) ≤ e 2mλt θ t λ t .(37) Then it holds that E t [θ t+1 ] (36) ≤ 1 − 1 α 1 + αa α − 1 λ t e 2mλt θ t ≤ 1 − 1 α e (2m+aα/(α−1))λt θ t = 1 − 1 α e m λt θ t (33) ≤ 1 − 1 α e m (1−1/β) t λ0 θ t ,(38) where m = 2m + aα/(α − 1). Taking expectation on both sides of (38), we have E[θ t+1 ] ≤ 1 − 1 α e m (1−1/β) t λ0 E[θ t ],(39) where we use the fact E[E t [δ t+1 ]] = E[δ t+1 ]. Therefore, we have E λ t+1 λ t (37) ≤ E[e 2mλt θ t ]≤ 1 − 1 α e m (1−1/β) t λ0 E [θ t ] (39) ≤ 1 − 1 α e (m (1−1/β) t +m (1−1/β) t−1 )λ0 E [θ t ] (39) ≤ 1 − 1 α t e m t p=0 (1−1/β) p λ0 E [θ 0 ] (35) ≤ 1 − 1 α t e m βλ0 E [θ 0 ] (33) ≤ 1 − 1 α t 2s.λ t+1 ≤ 1 − 1 η t λ t + m 1 2 λ 2 t + m 2 1 4η t λ 3 t andη t+1 ≤ (1 + m 2 λ t ) 2η t ,(40) for some m 1 and m 2 > 0. If mλ 0 ≤ ln(3/2) 4η 0 ,(41) where m max{m 1 , m 2 }, then it holds that η t ≤ e 2m t−1 i=0 λtη 0 ≤ 3η 0 2(42) and λ t ≤ 1 − 1 2η 0 t λ 0 .(43) Proof. We prove results of (42) and (43) by induction. In the case of t = 0, inequalities (42) and (43) are satisfied naturally. Now we suppose inequalities (42) and (43) holds for t = 0, . . . , t , then we have m t i=0 λ i (43) ≤ mλ 0 t i=0 1 − 1 2η 0 i ≤ 2η 0 mλ 0 (41) ≤ 1.(44) In the case of t = t + 1 we have 1 − m 1 λ t /2 η t ≥ e −m1λ t η t (42) ≥ e −2m t i=0 λĩ η 0 ≥ 2 3η 0 and mλ t ≤ mλ 0 (41) ≤ 1 8η 0 .(45) According to condition (40), we have λ t +1 (40) ≤ 1 + m 1 λ t 2 1 − 1 − m 1 λ t /2 η t λ t (45) ≤ 1 − 1 2η 0 λ t ≤ 1 − 1 2η 0 t +1 λ 0 , where the last step is based on induction. We also havẽ Lemma 25]). If the twice differentiable function f : R d → R is M -strongly self-concordant and µ-strongly convex and the positive definite matrix G ∈ R d×d and x ∈ R d satisfy ∇ 2 f (x) G η∇ 2 f (x) for some η > 1, then we have η t +1 (40) ≤ (1 + m 2 λ t ) 2η t ≤ e 2mλ t η t (42) ≤ e 2m t i=0 λ t η 0 (41) ≤ 3η 0 2 . Lemma B.3 ([21,∇ 2 f (x + ) G η(1 + M r) 2 ∇ 2 f (x + ),(46)σ ∇ 2 f (x+) (G) ≤ (1 + M r) 2 (σ ∇ 2 f (x) (G) + 2dM r),(47) and τ ∇ 2 f (x+) (G) ≤ (1 + M r) 2 τ ∇ 2 f (x) (G) tr (∇ 2 f (x)) + 2M r tr ∇ 2 f (x + )(48) for any x + ∈ R d , whereG = (1 + M r)G, r = x − x + x and the notations of τ H (G) and σ H (G) follow the expressions of (9) and (23) respectively. whereκ is the condition number of H and the notations of τ H (G) and σ H (G) follow the expressions of (9) and (23) respectively. Lemma B.5. For any positive definite symmetric matrices G, H ∈ R d×d such that H G ηH for some η ≥ 1 , it holds that σ H (G) ≤ d(η − 1) and τ H (G) tr (H) ≤ η − 1,(50) where the notations of τ H (G) and σ H (G) follow the expressions of (9) and (23) respectively. Proof. We obtain the statements directly from the definition, that is σ H (G) = tr H −1 (G − H) = tr H −1/2 (G − H)H −1/2 ≤ tr (η − 1)H −1/2 HH −1/2 = (η − 1)d, and τ H (G) tr (H) = tr ((G − H)) tr (H) ≤ tr ((η − 1)H) tr (H) = η − 1. Lemma B.6 ([21, Lemma 26]). Suppose the nonnegative random sequences {X t } satisfies E[X t ] ≤ a (1 − 1/α) t for all t ≥ 0 and some constants a ≥ 0 and α > 1. Then for any δ ∈ (0, 1), we have X t ≤ aα 2 δ 1 − 1 1 + α t for all t with probability at least 1 − δ. B.2 The Proof of Theorem 4.2 Proof. We denote λ t λ(x t ) andη t min ∇ 2 f (xt) ηGt η, which means ∇ 2 f (x t ) G t η t ∇ 2 f (x t ). According to Lemma 4.1, we have λ t+1 ≤ 1 − 1 η t λ t + M 2 λ 2 t + M 2 4η t λ 3 t . According to Lemma B.3, we have ∇ 2 f (x t+1 ) G t (1 + M r t ) 2η t ∇ 2 f (x t ). According to Lemma 3.2, we have ∇ 2 f (x t+1 ) (8) G t+1 (8) (1 + M r t ) 2η t ∇ 2 f (x t+1 )(15)(1 + M λ t ) 2η t ∇ 2 f (x t+1 ), which meansη t+1 ≤ (1 + M λ t ) 2η t . Hence, the sequences {η t } and {λ t } satisfy the conditions of Lemma B.2 with m 1 = m 2 = M , then we obtain ∇ 2 f (x t ) G t 3η 0 2 ∇ 2 f (x t ) 3η 0 2 ∇ 2 f (x t ), and λ(x t ) ≤ 1 − 1 2η 0 t λ(x 0 ) ≤ 1 − 1 2η 0 t λ(x 0 ). B.3 The Proof of Theorem 4.3 Proof. Denote g t = tr G t − ∇ 2 f (x t ) /tr ∇ 2 f (x t ) , δ t = dκg t , λ t = λ(x t ) and E t [ · ] E[ · | U 0 , · · · , U t−1 ]. From Theorem 3.3, we have E t tr G t+1 − ∇ 2 f (x t+1 ) ≤ 1 − k d tr G t − ∇ 2 f (x t+1 ) ,(51)From Lemma B.3, we have E t [τ ∇ 2 f (xt+1) (G t )](48)≤ (1 + M r t ) 2 (g t + 2M r t )tr ∇ 2 f (x t+1 ) , which means E t [δ t+1 ] (15), (51) ≤ 1 − k d (1 + M λ t ) 2 (δ t + 2κdM λ t ).(52) The initial condition (17) means the results of Theorem 4.2 hold, that is λ t ≤ 1 − 1 2η 0 t λ 0 and ∇ 2 f (x t ) G t .(53) According to Lemma B.4 and the definition of δ t , we have ∇ 2 f (x t )(53) G t (1 + δ t )∇ 2 f (x t ). According to Lemma 4.1, we have λ t+1 (15) ≤ 1 − 1 1 + δ t λ t + M η t λ 2 t + M 2 4η t λ 3 t ≤ 1 + M λ t 2 δ t + M λ t /2 1 + δ t λ t ≤ (1 + M λ t ) 2 δ t + M 2 λ t λ t . According to Lemma B.5 and the initial condition (17), we have δ 0 = dκg 0 (50) ≤ (η 0 − 1)dκ and θ 0 = δ 0 + 2dκM λ 0 (17) ≤ η 0 dκ.(54) Hence, the random sequences of {λ t } and {δ t } satisfies the conditions of Lemma B.1 with m = M, b = M 2 , c = 2κdM, α = d k , β = 1 2η 0 and s = η 0 dκ, which means we can obtain inequality (18). Now, we prove the two-stage convergence of SR-k methods. For SR-k method with randomized strategy [U t ] ij i.i.d ∼ N (0, 1), we apply Lemma B.6 with α = d/k and a = 2dκη 0 to obtain λ t+1 λ t ≤ 2d 2 κη 0 kδ 1 − k d + k t (55) holds for all t with probability at least 1 − δ. Take t 0 = O(d ln(η 0 κd)/k), which satisfies that 2d 2 κη 0 kδ 1 − k d + k t0 ≤ 1 2 ,(56) together with the linear rate (16), we have λ t+t0 (55) ≤ 1 − k d + k t+t0 2dκη 0 dλ t+t0−1 (56) ≤ 1 − k d + k t 1 2 λ t+t0−1 ≤ · · · (55),(56) ≤ 1 − k d + k t(t−1)/2 1 2 t λ t0 (16) ≤ 1 − k d + k t(t−1)/2 1 2 t 1 − 1 2η 0 t0 λ 0 . with probability at least 1 − δ. For SR-k method with greedy strategy U t = E k (G t − ∇ 2 f (x t+1 )), we choose t 0 = O (d ln(η 0 κd)/k) such that 1 − k d t0 2dκη 0 ≤ 1 2 ,(57) together with the linear rate (16), we have λ t+t0 (18) ≤ 1 − k d t+t0 2dκη 0 dλ t+t0−1 (57) ≤ 1 − k d t 1 2 λ t+t0−1 ≤ · · · (18),(57) ≤ 1 − k d t(t−1)/2 1 2 t λ t0(16)≤ 1 − k d t(t−1)/2 1 2 t 1 − 1 2η 0 t0 λ 0 . B.4 The Proof of Corollary 4.4 Proof. According to Theorem 4.3, we have E[τ ∇ 2 f (xt+1) (G t+1 )](11) = 0. According to Theorem 4.2 and Lemma B.4, we have ∇ 2 f (x t+1 ) (16) G t+1 1 + dκτ ∇ 2 f (xt+1) (G t+1 ) tr (∇ 2 f (x t+1 )) at+1 ∇ 2 f (x t+1 ) and λ t+1 ≤ λ 0(58) and E[a t+1 ] = 1. According to Lemma 4.1, we have λ t+2 ≤ 1 − 1 a t+1 bt+1 λ t+1 + M 2 λ 2 t + M 2 4(a t+1 ) λ 3 t+1 .(59) Then it holds that 0 ≤ E[b t+1 ] = 1 − E[1/(a t+1 )]≤1 − 1/E[a t+1 ] = 0, where the first inequality comes from the fact that a t+1 ≥ 1 and the second inequality comes from the fact E[1/X] ≥ 1/E[X] for positive random variable X > 0 by using Jensen's inequality. Hence, we have proved E[b t+1 ] = 0, which means E[λ t+2 ] (59) ≤ M 2 λ 2 t+1 + M 2 4 λ 3 t+1 (58) ≤ M 2 λ 2 t+1 + M 2 λ 2 t+1 (M λ 0 /2) (19) ≤ M λ 2 t+1 , for all t ≥ 0. C The Proof of Section 5 We provide the proofs for the results of randomized block BFGS method shown in Section 5. C.1 The Proof of Lemma 5.2 Proof. According to Woodbury formula [34], we have G −1 + = U(U AU) −1 U + I d − U(U AU) −1 U A G −1 I d − AU(U AU) −1 U . The condition (22) means we have 1 η A −1 G A −1 .(60) Thus, we have G −1 + (60) U(U AU) −1 U + I d − U(U AU) −1 U A A −1 I d − AU(U AU) −1 U A −1 , and G −1 + (60) U(U AU) −1 U + 1 η I d − U(U AU) −1 U A A −1 I d − AU(U AU) −1 U = 1 η A −1 + 1 − 1 η U(U AU) −1 U 1 η A. Thus, we can obtain 1 η A −1 G −1 + A −1 , which is equivalent to the desired result (22). C.2 The Proof of Theorem 5.3 We first provide a lemma for the traces of positive definite matrices. Lemma C.1. For any positive semi-definite matrices B, P 1 , P 2 ∈ R d×d such that P 1 P 2 , we have tr (P 1 B) ≥ tr (P 2 B) .(61) Proof. We denote R = P 1 − P 2 0, then R 1/2 BR 1/2 0, which means tr (P 1 − P 2 ) 1/2 B(P 1 − P 2 ) 1/2 ≥ 0. So we have tr (P 1 B) − tr (P 2 B) =tr ((P 1 − P 2 )B) =tr (P 1 − P 2 ) 1/2 B(P 1 − P 2 ) 1/2 ≥0. Then we provide the proof of Theorem 5.3 Proof. We rewrite the term of σ A (G + ) by substituting G + = BlockBFGS(G, A, U), that is σ A (G + ) =tr (G + − A)A −1 =σ A (G) − tr GU U GU −1 U GA −1 + tr AU U AU −1 U .(62)Let P = A 1/2 U, then U G A −1 − U U AU −1 U GU =U GA −1/2 I d − A 1/2 U U AU −1 U A 1/2 A −1/2 GU =U GA −1/2 I d − P P P −1 P 0 A −1/2 GU 0.(63) So we have tr U GU −1 U GA −1 GU − tr U AU −1 U GU =tr U GU −1 U GA −1 GU − (U GU) U AU −1 U GU =tr U GU −1/2 U G A −1 − U U AU −1 U GU U GU −1/2 (63) ≥ 0.(64) Recall we assume that µI d A LI d .(65) Lemma C.1 leads to tr U AU −1 U GU − tr AU U AU −1 U =tr U AU −1 U (G − A)U (65) ≥ 1 L tr U U −1 U (G − A)U = 1 L tr (G − A)U U U −1 U (65) ≥ µ L tr A −1 (G − A)U U U −1 U .(66) Combining with Lemma A.1, we have E tr GU U GU −1 U GA −1 − tr AU U AU −1 U =E tr GU U GU −1 U GA −1 − tr U AU −1 U GU + E tr U AU −1 U GU − tr AU U AU −1 U ≥ µ L E tr A −1 (G − A)U U U −1 U = µ L tr E U U U −1 U A −1 (G − A) = k dκ σ A (G),(67) where the inequality is based on the results of (64) and (66). Finally, we have E[σ A (G + )] (62) = σ A (G) − E U tr GU U GU −1 U GA −1 − tr AU U AU −1 U (67) ≤ 1 − k dκ σ A (G). C.3 The Proof of Theorem 5.4 We first provide a lemma which shows the linear convergence for the proposed block BFGS method (Algorithm 3). λ(x t ) ≤ 1 − 1 2η 0 t λ(x 0 ) and ∇ 2 f (x t ) G t 3η 0 2 ∇ 2 f (x t )(68) for all t ≥ 0. Proof. We can obtain this result by following the proof of Theorem 4.2 by replacing the use of Lemma 3.2 with Lemma 5.2. Now we prove Theorem 5.4. Proof. We denote δ t tr (G t − ∇ 2 f (x t ))(∇ 2 f (x t )) −1 and λ t λ(x t ). The initial condition means we have the results of Lemma C.2. According to Lemma B.4, we have ∇ 2 f (x t ) (68) G t (1 + δ t )∇ 2 f (x t ). Using Theorem 5.3, we have E t [δ t+1 ] = E t [σ ∇ 2 f (xt+1) (G t+1 )] (25) ≤ 1 − k dκ σ ∇ 2 f (x k+1 ) (G t ).(69) Using Lemma B.3, we have σ ∇ 2 f (x k+1 ) (G t )(47) ≤ (1 + M r t ) 2 (δ t + 2dM r t ). Thus, we can obtain following result E t [δ t+1 ](15)≤ 1 − k dκ (1 + M λ t ) 2 (δ t + 2dM λ t ). According to Lemma 4.1, we have λ t+1 (15) ≤ 1 + M λ t 2 δ t + M λ t /2 1 + δ t λ t ≤ (1 + M λ t ) 2 (δ t + 2dM λ t )λ t . According to Lemma C.2, we have λ t ≤ 1 − 1 2η 0 t λ 0 . According to Lemma B.5 and the initial condition (26), we have δ 0 = tr (G 0 − ∇ 2 f (x 0 ))(∇ 2 f (x 0 )) −1(50) ≤ d(η 0 − 1) and θ 0 = δ 0 + 2dM λ 0 H t+1 = SR-k(H t , J(z t+1 ) J(z t+1 ), U t ) 8: end for D Extension for Solving Nonlinear Equations In this section, we apply SR-k methods to solve the nonlinear equations F(z) = 0,(71) where F : R d → R d is a differentiable vector-valued function. We use J(z) to represent the Jacobian of F(·) at z ∈ R d and impose the following assumptions. Assumption D.1. We assume the vector-valued function F : R d → R d is differentiable and its Jacobian is L 2 -Lipschitz continuous, i.e., there exists someL 2 ≥ 0 such that J(z) − J(z ) ≤L 2 z − z .(72) for any z, z ∈ R d . Assumption D.2. We assume there exists equation (71) has a solution z * such that J(z * ) is non-degenerate. According to Assumption D.2, we denotẽ µ σ min (J(z * )) √ 2 ,L 2σ max (J(z * )) andκ L µ , where σ min (·) and σ max (·) are the smallest and the largest singular values of given matrix respectively. We present SR-k methods for solving nonlinear equations in Algorithm 4. The design of this algorithm is inspired by the recent work of Liu and Luo [22], which applies the quasi-Newton methods to estimate the information of non-degenerate indefinite matrix by its square. We use the Euclidean normλ(z) F(z) 2 to measure the convergence of our algorithm. The advantage of block updates in SR-k updates results a faster superlinear convergence than Liu and Luo [22]'s methods. Following the analysis of SR-k methods for convex optimization, we obtain the results for solving nonlinear equations as follows. Proposition 2. 3 . 3Suppose the objective function f : R d → R satisfies Assumptions 2.1 and 2.2, then it holds Lemma 4. 1 ( 1Lemma 4.3 of Rodomanov and Nesterov Theorem 4. 2 . 2Under Assumption 2.1, 2.2 and 2.4, we run Algorithm 1 with initial x 0 and G 0 such that Theorem 4 . 3 . 43Under Assumption 2.1, 2.2 and 2.4, if we run Algorithm 1 with k < d and set the initial x 0 and G 0 such that Corollary 4. 4 . 4Under Assumption 2.1, 2.2 and 2.4, we run Algorithm 1 with k = d and set the initial x 0 and G 0 such that Lemma B. 2 ( 2Following Rodomanov and Nesterov [29, Theorem 4.7] and Lin et al. [21, Theorem 23]). Let {λ t } and {η t } be two positive sequences whereη t ≥ 1 that satisfy Lemma B. 4 4([21, Lemma 25]). For any positive definite symmetric matrices G, H ∈ R d×d such that H G , it holds that G (1 + σ H (G))H and G 1 +κ dτ H (G) tr (H) H, ≤ dη 0 . 0Hence, the random sequences of {λ t } and {δ t } satisfy the conditions of Lemma B.1 with m = M, b = 2dM, c = 2dM, α = dκ k , β = 2η 0 and s = η 0 d, Theorem D. 3 . 3Under Assumption D.1 and D.2, we run Algorithm 4 with k < d,M = 2κ 2L 2 /L and set the initial z 0 and H 0 such thatλ(z 0 ) ≤ ln 2 8 · (d − k)μ M η 0 d 2κ2 and J(z 0 ) J(z 0 ) H 0 η 0 J(z 0 ) J(z 0 )for some η 0 ≥ 1. Then we haveE λ (z t+1 ) λ(z t ) ≤ 2dκ 2 η 0 1 − k d t . Table 1 : 1We summarize the properties of quasi-Newton methods for convex optimizationMethod Rank E [λ t+1 /λ t ] Newton [25, 26] d O(λ t ) Classical Quasi-Newton [17, 30, 31, 35] 1 or 2 O (1/t) Greedy/Randomized Broyden Family [21, 29] 1 or 2 O (1 − 1/(κd)) t Greedy/Randomized BFGS [21] 1 or 2 tr QQ B ,and tr BU U BU −1 U B = tr BQΣV VΣQ BQΣV −1 VΣQ B = tr BQ Q BQ −1 Q B (29) ≥ tr Q BQ Q BQ −1 Q BQ = tr Q BQ , which leads to inequality (30). Now, we present the proof of Theorem 3.3. Proof. Randomized SR-k Update: Combining Lemma A.1 with Lemma A.4, we have Lemma C.2. Under the setting of Theorem 5.4, Algorithm 3 holds that Parameter estimation in stochastic differential equations. P N Jaya, Bishwal, SpringerJaya P.N. Bishwal. Parameter estimation in stochastic differential equations. Springer, 2007. Quasi-Newton methods and their application to function minimisation. Charles G Broyden, 21Mathematics of ComputationCharles G. Broyden. 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On the local and superlinear convergence of quasi-Newton methods. IMA Journal of Applied Mathematics, 12(3):223-245, 1973. Global convergence of a cass of quasi-Newton methods on convex problems. H Richard, Jorge Byrd, Ya-Xiang Nocedal, Yuan, SIAM Journal on Numerical Analysis. 245Richard H. Byrd, Jorge Nocedal, and Ya-Xiang Yuan. Global convergence of a cass of quasi-Newton methods on convex problems. SIAM Journal on Numerical Analysis, 24(5):1171-1190, 1987. Statistics on special manifolds. Yasuko Chikuse, Springer1Yasuko Chikuse. Statistics on special manifolds, volume 1. Springer, 2003. Variable metric method for minimization. C William, Davidon, SIAM Journal on Optimization. 11William C. Davidon. Variable metric method for minimization. SIAM Journal on Optimization, 1(1): 1-17, 1991. A characterization of superlinear convergence and its application to quasi-Newton methods. J E Dennis, Jr , Jorge J Moré, Mathematics of Computation. 28126J. E. Dennis, Jr., and Jorge J. Moré. A characterization of superlinear convergence and its application to quasi-Newton methods. Mathematics of Computation, 28(126):549-560, 1974. A rapidly convergent descent method for minimization. Roger Fletcher, Micheal J D Powell, The Computer Journal. 6Roger Fletcher and Micheal J.D. Powell. A rapidly convergent descent method for minimization. The Computer Journal, 6:163-168, 1963. Block bfgs methods. Wenbo Gao, Donald Goldfarb, SIAM Journal on Optimization. 282Wenbo Gao and Donald Goldfarb. Block bfgs methods. SIAM Journal on Optimization, 28(2):1205- 1231, 2018. Practical quasi-Newton methods for training deep neural networks. Donald Goldfarb, Yi Ren, Achraf Bahamou, NeurIPS. Donald Goldfarb, Yi Ren, and Achraf Bahamou. Practical quasi-Newton methods for training deep neural networks. NeurIPS, 2020. Randomized quasi-Newton updates are linearly convergent matrix inversion algorithms. M Robert, Peter Gower, Richtárik, SIAM Journal on Matrix Analysis and Applications. 384Robert M. Gower and Peter Richtárik. Randomized quasi-Newton updates are linearly convergent matrix inversion algorithms. SIAM Journal on Matrix Analysis and Applications, 38(4):1380-1409, 2017. Stochastic block BFGS: Squeezing more curvature out of data. Robert M Gower, Donald Goldfarb, Peter Richtárik, ICML. Robert M. Gower, Donald Goldfarb, and Peter Richtárik. Stochastic block BFGS: Squeezing more curvature out of data. In ICML, 2016. Quasi-Newton methods: A new direction. Philipp Hennig, Martin Kiefel, Journal of Machine Learning Research. 141Philipp Hennig and Martin Kiefel. Quasi-Newton methods: A new direction. Journal of Machine Learning Research, 14(1):843-865, 2013. Acceleration of the EM algorithm by using quasi-Newton methods. Mortaza Jamshidian, Robert I Jennrich, Journal of the Royal Statistical Society: Series B (Statistical Methodology). 593Mortaza Jamshidian and Robert I. Jennrich. Acceleration of the EM algorithm by using quasi-Newton methods. 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Greedy quasi-Newton methods with explicit superlinear con- vergence. SIAM Journal on Optimization, 31(1):785-811, 2021. New results on superlinear convergence of classical quasi-Newton methods. Anton Rodomanov, Yurii Nesterov, Journal of Optimization Theory and Applications. 1883Anton Rodomanov and Yurii Nesterov. New results on superlinear convergence of classical quasi-Newton methods. Journal of Optimization Theory and Applications, 188(3):744-769, 2021. Rates of superlinear convergence for classical quasi-Newton methods. Anton Rodomanov, Yurii Nesterov, Mathematical Programming. 194Anton Rodomanov and Yurii Nesterov. Rates of superlinear convergence for classical quasi-Newton methods. Mathematical Programming, 194:159-190, 2021. Quasi-Newton methods using multiple secant equations. Robert B Schnabel, Colorado University at BoulderTechnical reportRobert B. Schnabel. Quasi-Newton methods using multiple secant equations. Technical report, Colorado University at Boulder, 1983. 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{'abstract': 'This paper proposes a novel class of block quasi-Newton methods for convex optimization which we call symmetric rank-k (SR-k) methods. Each iteration of SR-k incorporates the curvature information with k Hessian-vector products achieved from the greedy or random strategy. We prove SR-k methods have the local superlinear convergence rate of O (1 − k/d) t(t−1)/2 for minimizing smooth and strongly self-concordant function, where d is the problem dimension and t is the iteration counter. This is the first explicit superlinear convergence rate for block quasi-Newton methods and it successfully explains why block quasi-Newton methods converge faster than standard quasi-Newton methods in practice.', 'arxivid': '2303.16188', 'author': ['Chengchang Liu \nLuo Luo ‡\n\n', 'Cheng Chen \nLuo Luo ‡\n\n'], 'authoraffiliation': ['Luo Luo ‡\n', 'Luo Luo ‡\n'], 'corpusid': 257771276, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19580, 'n_tokens_neox': 16672, 'n_words': 9807, 'pdfsha': '601447f059a01dd7f595a20ef9a454729a8c9d7f', 'pdfurls': ['https://export.arxiv.org/pdf/2303.16188v3.pdf'], 'title': ['Symmetric Rank-k Methods', 'Symmetric Rank-k Methods'], 'venue': []}

arxiv

s talks at the Lisbon School on Superstrings II July 13-17, 2001. July 15-27, 2001 Christopher P Herzog Department of Physics Princeton University Princeton 08544NJUSA Igor R Klebanov Department of Physics Princeton University Princeton 08544NJUSA Peter Ouyang Department of Physics Princeton University Princeton 08544NJUSA s talks at the Lisbon School on Superstrings II and at the Benasque Workshop "Physics in the Pyrenees: Strings, Branes and Field Theory July 13-17, 2001. July 15-27, 2001 We assemble a few remarks on the supergravity solution of hep-th/0007191, whose UV asymptotic form was previously found in hep-th/0002159. First, by normalizing the R-R fluxes, we compare the logarithmic flow of couplings in supergravity with that in field theory, and find exact agreement. We also write the 3-form field strength G 3 = F 3 − τ H 3 present in the solution in a manifestly SO(4) invariant (2, 1) form. In addition, we discuss various issues related to the chiral symmetry breaking and wrapped branes. Introduction The warped deformed conifold [1] is a solution of type IIB supergravity that is dual to a certain N = 1 supersymmetric SU(N + M) × SU(N) gauge theory in the limit of strong 't Hooft coupling. This solution encodes various interesting gauge theory phenomena in a dual geometrical language, such as the duality cascade in the UV and chiral symmetry breaking and confinement in the IR. In these notes we assemble a few remarks on the solution of [1], whose UV asymptotic form was previously found in [2]. Our intention is to compromise between presenting a self-contained review of supergravity conifold solutions and their field theory duals, and presenting three or four new results that we believe to be of general interest. A summary of the new results follows. First, by normalizing the R-R fluxes, we compare the logarithmic flow of couplings in supergravity with that in field theory, and find exact agreement. 1 Next, we show that the 3-form field strength G 3 = F 3 − τ H 3 present in the solution is an SO(4) invariant (2, 1) form on the deformed conifold. (It was shown previously that G 3 must be (2,1) in order to preserve SUSY [6], [7], [8].) Although G 3 has appeared explicitly in the literature before [9], the basis in which we write G 3 and some other differential forms important to the supergravity solutions of [2] and [1] is a particularly simple one in which many of the properties of these forms become completely obvious. We also discuss various issues related to the chiral symmetry breaking and wrapped branes. For example, we develop the gauge field/string dictionary for this system by showing the correspondence between a certain wrapped D5-brane in supergravity and domain walls in the field theory that interpolate between inequivalent vacua. Finally, in the style of [10], we discuss various UV/IR relations for the conifold. We now review some basic facts about the AdS/CFT correspondence first because we would like these notes to be self-contained and second because it is important to understand the normalizations here in view of the more complicated solutions ahead. The duality between N = 4 supersymmetric SU(N) gauge theory and the AdS 5 × S 5 background of type IIB string theory [11,12,13] is usually motivated by considering a stack of a large number N of D3-branes. The SYM theory is the low-energy limit of the gauge theory on the stack of D3-branes. On the other hand, the curved background produced by the stack is ds 2 = h −1/2 −dt 2 + dx 2 1 + dx 2 2 + dx 2 3 + h 1/2 dr 2 + r 2 dΩ 2 5 ,(1) where dΩ 2 5 is the metric of a unit 5-sphere and h(r) = 1 + L 4 r 4 . This 10-dimensional metric may be thought of as a "warped product" of the R 3,1 along the branes and the transverse space R 6 . Note that the dilaton, Φ = 0, is constant, and the selfdual 5-form field strength is given by F 5 = F 5 + ⋆F 5 , F 5 = 16π(α ′ ) 2 Nvol(S 5 ) .(3) The normalization above is dictated by the quantization of Dp-brane tension which implies S 8−p ⋆F p+2 = 2κ 2 τ p N g s ,(4) where τ p = √ π κ (4π 2 α ′ ) (3−p)/2(5) and κ = 8π 7/2 g s α ′2 is the 10-dimensional gravitational constant. In particular, for p = 3 we have S 5 F 5 = (4π 2 α ′ ) 2 N ,(6) which is consistent with (3) since the volume of a unit 5-sphere is Vol(S 5 ) = π 3 . Note that the 5-form field strength may also be written as g s F 5 = d 4 x ∧ dh −1 − r 5 dh dr vol(S 5 ) .(7) Then it is not hard to see that the Einstein equation R M N = g 2 s F M P QRS F N P QRS /96 is satisfied. Since −r 5 dh dr = 4L 4 , we find by comparing with (3) that L 4 = 4πg s Nα ′2 .(8) A related way to determine the scale factor L is to equate the ADM tension of the supergravity solution with N times the tension of a single D3-brane [14]: 2 κ 2 L 4 Vol(S 5 ) = √ π κ N ,(9) This way we find L 4 = κN 2π 5/2 = 4πg s Nα ′2 (10) in agreement with the preceding paragraph. The radial coordinate r is related to the scale in the dual gauge theory. The low-energy limit corresponds to r → 0. In this limit the metric becomes ds 2 = L 2 z 2 −dt 2 + d x 2 + dz 2 + L 2 dΩ 2 5 ,(11) where z = L 2 r . This describes the direct product of 5-dimensional Anti-de Sitter space, AdS 5 , and the 5-dimensional sphere, S 5 , with equal radii of curvature L. An interesting generalization of the basic AdS/CFT correspondence [11,12,13] is found by studying branes at conical singularities [15,16,17,18]. Consider a stack of D3-branes placed at the apex of a Ricci-flat 6-d cone Y 6 whose base is a 5-d Einstein manifold X 5 . Comparing the metric with the D-brane description leads one to conjecture that type IIB string theory on AdS 5 × X 5 is dual to the low-energy limit of the world volume theory on the D3-branes at the singularity. The equality of tensions now requires [19] L 4 = √ πκN 2Vol(X 5 ) = 4πg s Nα ′2 π 3 Vol(X 5 ) ,(12) an important normalization formula which we will use in the following subsection. The simplest examples of X 5 are the orbifolds S 5 /Γ where Γ is a discrete subgroup of SO(6) [15]. In these cases X 5 has the local geometry of a 5-sphere. The dual gauge theory is the IR limit of the world volume theory on a stack of N D3-branes placed at the orbifold singularity of R 6 /Γ. Such theories typically involve product gauge groups SU(N) k coupled to matter in bifundamental representations [20]. Constructions of the dual gauge theories for Einstein manifolds X 5 which are not locally equivalent to S 5 are also possible. The simplest example is the Romans compactification on X 5 = T 1,1 = (SU(2) × SU(2))/U(1) [21,17]. The dual gauge theory is the conformal limit of the world volume theory on a stack of N D3-branes placed at the singularity of a Calabi-Yau manifold known as the conifold [17], which is a cone over T 1,1 . Let us explain this connection in more detail. D3-branes on the Conifold The conifold may be described by the following equation in four complex variables, 4 a=1 z 2 a = 0 .(13) Since this equation is invariant under an overall real rescaling of the coordinates, this space is a cone. Remarkably, the base of this cone is precisely the space T 1,1 [22,17]. In fact, the metric on the conifold may be cast in the form [22] ds 2 6 = dr 2 + r 2 ds 2 T 1,1 ,(14) where ds 2 T 1,1 = 1 9 dψ + 2 i=1 cos θ i dφ i 2 + 1 6 2 i=1 dθ 2 i + sin 2 θ i dφ 2 i .(15) is the metric on T 1,1 . Here ψ is an angular coordinate which ranges from 0 to 4π, while (θ 1 , φ 1 ) and (θ 2 , φ 2 ) parametrize two S 2 's in a standard way. Therefore, this form of the metric shows that T 1,1 is an S 1 bundle over S 2 × S 2 . Now placing N D3-branes at the apex of the cone we find the metric ds 2 = 1 + L 4 r 4 −1/2 −dt 2 + dx 2 1 + dx 2 2 + dx 2 3 + 1 + L 4 r 4 1/2 dr 2 + r 2 ds 2 T 1,1(16) whose near-horizon limit is AdS 5 × T 1,1 . Using the metric (15) it is not hard to find that the volume of T 1,1 is 16π 3 27 [23]. From (12) it then follows that L 4 = 4πg s N(α ′ ) 2 27 16 = 27κN 32π 5/2 .(17) The same logic that leads us to the N = 4 version of the AdS/CFT correspondence now shows that the type IIB string theory on this space should be dual to the infrared limit of the field theory on N D3-branes placed at the singularity of the conifold. Since Calabi-Yau spaces preserve 1/4 of the original supersymmetries this should be an N = 1 superconformal field theory. This field theory was constructed in [17]: it is SU(N) × SU(N) gauge theory coupled to two chiral superfields, A i , in the (N, N) representation and two chiral superfields, B j , in the (N, N) representation. In order to match the two gauge couplings to the moduli of the type IIB theory on AdS 5 × T 1,1 , one notes that the integrals over the S 2 of T 1,1 of the NS-NS and R-R 2-form potentials, B 2 and C 2 , are moduli. In particular, the two gauge couplings are determined as follows [17,18]: 2 4π 2 g 2 1 + 4π 2 g 2 2 = π g s e Φ ,(18)4π 2 g 2 1 − 4π 2 g 2 2 g s e Φ = 1 2πα ′ S 2 B 2 − π (mod 2π) .(19) These equations are crucial for relating the SUGRA background to the field theory β-functions when the theory is generalized to SU(N + M) × SU(N) [3,2]. From the quantization condition on H 3 , 1 2πα ′ ( S 2 B 2 ) must be a periodic variable with period 2π. This periodicity is crucial for the cascade phenomenon that we discuss in the next section. The RG cascade The addition of M fractional 3-branes (wrapped D5-branes) at the singular point changes the gauge group to SU(N + M) × SU(N). Let us consider the effect on the dual supergravity background of adding M wrapped D5-branes. The D5-branes serve as sources of the magnetic RR 3-form flux through the S 3 of T 1,1 . Therefore, the supergravity dual of this field theory involves M units of the 3-form flux, in addition to N units of the 5-form flux: 1 4π 2 α ′ S 3 F 3 = M , 1 (4π 2 α ′ ) 2 T 1,1 F 5 = N .(20) The coefficients above follow from the quantization rule (4). The warped conifold solution with such fluxes was constructed in [2]. It will be useful to employ the following basis of 1-forms on the compact space [24]: g 1 = e 1 − e 3 √ 2 , g 2 = e 2 − e 4 √ 2 , g 3 = e 1 + e 3 √ 2 , g 4 = e 2 + e 4 √ 2 , g 5 = e 5 ,(21) where e 1 ≡ − sin θ 1 dφ 1 , e 2 ≡ dθ 1 , e 3 ≡ cos ψ sin θ 2 dφ 2 − sin ψdθ 2 , e 4 ≡ sin ψ sin θ 2 dφ 2 + cos ψdθ 2 , e 5 ≡ dψ + cos θ 1 dφ 1 + cos θ 2 dφ 2 .(22) In terms of this basis, the Einstein metric on T 1,1 assumes the form ds 2 T 1,1 = 1 9 (g 5 ) 2 + 1 6 4 i=1 (g i ) 2 .(23) Keeping track of the normalization factors, in order to be consistent with the quantization conditions (20), F 3 = Mα ′ 2 ω 3 , B 2 = 3g s Mα ′ 2 ω 2 ln(r/r 0 ) ,(24)H 3 = dB 2 = 3g s Mα ′ 2r dr ∧ ω 2 ,(25) where ω 2 = 1 2 (g 1 ∧ g 2 + g 3 ∧ g 4 ) = 1 2 (sin θ 1 dθ 1 ∧ dφ 1 − sin θ 2 dθ 2 ∧ dφ 2 ) ,(26)ω 3 = 1 2 g 5 ∧ (g 1 ∧ g 2 + g 3 ∧ g 4 ) .(27) In Appendix A we show that S 2 ω 2 = 4π , S 3 ω 3 = 8π 2(28) where the S 2 is parametrized by ψ = 0, θ 1 = θ 2 and φ 1 = −φ 2 , and the S 3 by θ 2 = φ 2 = 0. As a result, the quantization condition for RR 3-form flux is obeyed. Both ω 2 and ω 3 are closed. Note also that g s ⋆ 6 F 3 = H 3 , g s F 3 = − ⋆ 6 H 3 ,(29) where ⋆ 6 is the Hodge dual with respect to the metric ds 2 6 . Thus, the complex 3-form G 3 satisfies the self-duality condition ⋆ 6 G 3 = iG 3 , G 3 = F 3 − i g s H 3 .(30) Note that the self-duality fixes the relative factor of 3 in (24) (see (14), (15)). We will see that this geometrical factor is crucial for reproducing the well-known factor of 3 in the N = 1 beta functions. It follows from (29) that g 2 s F 2 3 = H 2 3 ,(31) which implies that the dilaton is constant, Φ = 0. Since F 3µνλ H µνλ 3 = 0, the RR scalar vanishes as well. The 10-d metric found in [2] has the structure of a "warped product" of R 3,1 and the conifold: ds 2 10 = h −1/2 (r)dx n dx n + h 1/2 (r)(dr 2 + r 2 ds 2 T 1,1 ) . The solution for the warp factor h may be determined from the trace of the Einstein equation: R = 1 24 (H 2 3 + g 2 s F 2 3 ) = 1 12 H 2 3 .(33) This implies − h −3/2 1 r 5 d dr (r 5 h ′ ) = 1 6 H 2 3 .(34) Integrating this differential equation, we find that h(r) = 27π(α ′ ) 2 [g s N + a(g s M) 2 ln(r/r 0 ) + a(g s M) 2 /4] 4r 4(35) with a = 3/(2π). An important feature of this background is thatF 5 acquires a radial dependence [2]. This is becauseF 5 = F 5 + B 2 ∧ F 3 , F 5 = dC 4 ,(36) and ω 2 ∧ ω 3 = 54vol(T 1,1 ). Thus, we may writẽ F 5 = F 5 + ⋆F 5 , F 5 = 27πα ′2 N ef f (r)vol(T 1,1 ) ,(37) and N ef f (r) = N + 3 2π g s M 2 ln(r/r 0 ) .(38) The novel phenomenon in this solution is that the 5-form flux present at the UV scale r = r 0 may completely disappear by the time we reach a scale where N ef f = 0. The non-conservation of the flux is due to the type IIB SUGRA equation dF 5 = H 3 ∧ F 3 .(39) A related fact is that S 2 B 2 is no longer a periodic variable in the SUGRA solution once the M fractional branes are introduced: as the B 2 flux goes through a period, Due to the non-vanishing RHS of (39), N ef f (r) → N ef f (r) − M1 (4π 2 α ′ ) 2 T 1,1F5 is not quantized. We may identify this quantity with N ef f defining the gauge group SU(N ef f + M) × SU(N ef f ) only at special radii r k = r 0 exp(−2πk/3g s M) where k is an integer. Thus, N ef f = N −kM. Furthermore, we believe that the continuous logarithmic variation of N ef f (r) is related to continuous reduction in the number of degrees of freedom as the theory flows to the IR. Some support for this claim comes from studying the high-temperature phase of this theory using black holes embedded into asymptotic KT grometry [26]. The effective number of degrees of freedom computed from the Bekenstein-Hawking entropy grows logarithmically with the temperature, in agreement with (38). The metric (32) has a naked singularity at r = r s where h(r s ) = 0. Writing h(r) = L 4 r 4 ln(r/r s ) , L 2 = 9g s Mα ′ 2 √ 2 ,(40) we find a purely logarithmic RG cascade: ds 2 = r 2 L 2 ln(r/r s ) dx n dx n + L 2 ln(r/r s ) r 2 dr 2 + L 2 ln(r/r s )ds 2 T 1,1 .(41) Since T 1,1 expands slowly toward large r, the curvatures decrease there so that corrections to the SUGRA become negligible. Therefore, even if g s M is very small, this SUGRA solution is reliable for sufficiently large radii where g s N ef f (r) ≫ 1. In this regime the separation between the cascade steps is very large, so that the SUGRA calculation of the β-functions may be compared with SU(N ef f + M) × SU(N ef f ) gauge theory. We will work near r = r 0 where N ef f may be replaced by N. Matching of the β-functions In gauge/gravity duality the 5-dimensional radial coordinate defines the RG scale of the dual gauge theory [11,12,13,27,10]. There are different ways of establishiing the precise relation. The simplest one is to identify the field theory energy scale Λ with the energy of a stretched string ending on a probe brane positioned at radius r. For all metrics of the form (32) this gives Λ ∼ r .(42) In this section we adopt this UV/IR relation, which typically corresponds to the Wilsonian renormalization group. We will discuss UV/IR relations in more detail in Section 5. Now we are ready to interpret the solution of [2] in terms of RG flow in the dual SU(N + M) × SU(N) gauge theory. The constancy of the dilaton translates into the vanishing of the β-function for 8π 2 g 2 1 + 8π 2 g 2 2 . Substituting the solution for B 2 into (19) we find 8π 2 g 2 1 − 8π 2 g 2 2 = 6M ln(r/r s ) + const .(43) Since ln(r/r s ) = ln(Λ/µ), (43) implies a logarithmic running of 1 g 2 1 − 1 g 2 2 in the SU(N + M) × SU(N) gauge theory. As we mentioned earlier, this SUGRA result is reliable for any value of g s M provided that g s N ≫ 1. We may consider g s M ≪ 1 so that the cascade jumps are well-separated. Let us compare with the Shifman-Vainshtein β-functions [28]: 3 d dlog(Λ/µ) 8π 2 g 2 1 = 3(N + M) − 2N(1 − γ) ,(44)d dlog(Λ/µ) 8π 2 g 2 2 = 3N − 2(N + M)(1 − γ) ,(45) where γ is the anomalous dimension of operators TrA i B j . The conformal invariance of the field theory for M = 0, and symmetry under M → −M, require that γ = − 1 2 + O[(M/N) 2n ] where n is a positive integer [1]. Taking the difference of the two equations in (44) we then find 8π 2 g 2 1 − 8π 2 g 2 2 = M ln(Λ/µ)[3 + 2(1 − γ)] = 6M ln(Λ/µ)(1 + O[(M/N) 2n ]) .(46) Remarkably, the coefficient 6M is in exact agreement with the result (43) found on the SUGRA side. This consitutes a geometrical explanation of a field theory β-function, including its normalization. We may also trace the jumps in the rank of the gauge group to a well-known phenomenon in the dual N = 1 field theory, namely, Seiberg duality [31]. The essential observation is that 1/g 2 1 and 1/g 2 2 flow in opposite directions and, according to (44) theory which resembles closely the theory we started with [1]. As the theory flows to the IR, the cascade must stop, however, because negative N is physically nonsensical. Thus, we should not be able to continue the solution (41) to the region where N ef f is negative. To summarize, the fact that the solution of [2] is singular tells us that it has to be modified in the IR. Deformation of the Conifold It was shown in [1] that, to remove the naked singularity found in [2] the conifold (13) should be replaced by the deformed conifold 4 i=1 z 2 i = ε 2 ,(47) in which the singularity of the conifold is removed through the blowing-up of the S 3 of T 1,1 . We now review the deformed conifold in order to be able to normalize properly the field strengths and to prepare for a discussion of a new and simple SO(4) invariant way of writing the field strengths. The 10-d metric of [1] takes the following form: ds 2 10 = h −1/2 (τ )dx n dx n + h 1/2 (τ )ds 2 6 ,(48) where ds 2 6 is the metric of the deformed conifold (49). This is the same type of "Dbrane" ansatz as (32), but with the conifold replaced by the deformed conifold as the transverse space. The metric of the deformed conifold was discussed in some detail in [22,24,32]. It is diagonal in the basis (21): ds 2 6 = 1 2 ε 4/3 K(τ ) 1 3K 3 (τ ) (dτ 2 + (g 5 ) 2 ) + cosh 2 τ 2 [(g 3 ) 2 + (g 4 ) 2 ] + sinh 2 τ 2 [(g 1 ) 2 + (g 2 ) 2 ] ,(49) where K(τ ) = (sinh(2τ ) − 2τ ) 1/3 2 1/3 sinh τ .(50) For large τ we may introduce another radial coordinate r via r 2 = 3 2 5/3 ε 4/3 e 2τ /3 ,(51) and in terms of this radial coordinate ds 2 6 → dr 2 + r 2 ds 2 T 1,1 . At τ = 0 the angular metric degenerates into dΩ 2 3 = 1 2 ε 4/3 (2/3) 1/3 [ 1 2 (g 5 ) 2 + (g 3 ) 2 + (g 4 ) 2 ] ,(52) which is the metric of a round S 3 [22,24]. The additional two directions, corresponding to the S 2 fibered over the S 3 , shrink as 1 8 ε 4/3 (2/3) 1/3 τ 2 [(g 1 ) 2 + (g 2 ) 2 ] .(53) The simplest ansatz for the 2-form fields is F 3 = Mα ′ 2 g 5 ∧ g 3 ∧ g 4 + d[F (τ )(g 1 ∧ g 3 + g 2 ∧ g 4 )] = Mα ′ 2 g 5 ∧ g 3 ∧ g 4 (1 − F ) + g 5 ∧ g 1 ∧ g 2 F + F ′ dτ ∧ (g 1 ∧ g 3 + g 2 ∧ g 4 ) ,(54) with F (0) = 0 and F (∞) = 1/2, and B 2 = g s Mα ′ 2 [f (τ )g 1 ∧ g 2 + k(τ )g 3 ∧ g 4 ] ,(55)H 3 = dB 2 = g s Mα ′ 2 [dτ ∧(f ′ g 1 ∧g 2 + k ′ g 3 ∧g 4 ) + 1 2 (k −f )g 5 ∧(g 1 ∧g 3 + g 2 ∧g 4 )] . (56) As before, the self-dual 5-form field strength may be decomposed asF 5 = F 5 +⋆F 5 . We have F 5 = B 2 ∧ F 3 = g s M 2 (α ′ ) 2 4 ℓ(τ )g 1 ∧ g 2 ∧ g 3 ∧ g 4 ∧ g 5 ,(57) where ℓ = f (1 − F ) + kF ,(58) and ⋆ F 5 = 4g s M 2 (α ′ ) 2 ε −8/3 dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ∧ dτ ℓ(τ ) K 2 h 2 sinh 2 (τ ) .(59) The First-Order Equations and Their Solution In searching for BPS saturated supergravity backgrounds, the second order equations should be replaced by a system of first-order ones. Luckily, this is possible for our ansatz [1]: f ′ = (1 − F ) tanh 2 (τ /2) , k ′ = F coth 2 (τ /2) , F ′ = 1 2 (k − f ) ,(60) and h ′ = −α f (1 − F ) + kF K 2 (τ ) sinh 2 τ ,(61) where α = 4(g s Mα ′ ) 2 ε −8/3 .(62) These equations follow from a superpotential for the effective radial problem [33]. Once we have the solutions to these differential equations, we can check that the large τ limit of the properly normalized B 2 , F 3 and F 5 field strengths agree with their simpler counterparts of section 2. Also, we can understand precisely the large and small τ behavior of the warp factor h(τ ). Note that the first three of these equations, (60), form a closed system and need to be solved first. In fact, these equations imply the self-duality of the complex 3-form with respect to the metric of the deformed conifold: ⋆ 6 G 3 = iG 3 . The solution is F (τ ) = sinh τ − τ 2 sinh τ , f (τ ) = τ coth τ − 1 2 sinh τ (cosh τ − 1) , k(τ ) = τ coth τ − 1 2 sinh τ (cosh τ + 1) .(63) Now that we have solved for the 3-forms on the deformed conifold, the warp factor may be determined by integrating (61). First we note that ℓ(τ ) = f (1 − F ) + kF = τ coth τ − 1 4 sinh 2 τ (sinh 2τ − 2τ ) .(64) This behaves as τ 3 for small τ . For large τ we impose, as usual, the boundary condition that h vanishes. The resulting integral expression for h is h(τ ) = α 2 2/3 4 I(τ ) = (g s Mα ′ ) 2 2 2/3 ε −8/3 I(τ ) ,(65) where I(τ ) ≡ ∞ τ dx x coth x − 1 sinh 2 x (sinh(2x) − 2x) 1/3 .(66) We have not succeeded in evaluating this integral in terms of elementary or well-known special functions, but it is not hard to see that I(τ → 0) → a 0 + O(τ 2 ) ; I(τ → ∞) → 3 · 2 −1/3 τ − 1 4 e −4τ /3 ,(67) where a 0 ≈ 0.71805. This I(τ ) is nonsingular at the tip of the deformed conifold and, from (51), matches the form of the large-τ solution (40). The small τ behavior follows from the convergence of the integral (65), while at large τ the integrand becomes ∼ xe −4x/3 . Thus, for small τ the ten-dimensional geometry is approximately R 3,1 times the deformed conifold: ds 2 10 → ε 4/3 2 1/3 a 1/2 0 gsM α ′ dx n dx n + a 1/2 0 6 −1/3 (g s Mα ′ ) 1 2 dτ 2 + 1 2 (g 5 ) 2 + (g 3 ) 2 + (g 4 ) 2 + 1 4 τ 2 [(g 1 ) 2 + (g 2 ) 2 ] .(68) Note that we have suppressed the O(τ 2 ) corrections for all but the (g 1 ) 2 and (g 2 ) 2 components of the metric. This metric will be useful in section 4 where we investigate various infrared phenomenon of the gauge theory. Very importantly, for large g s M the curvatures found in our solution are small everywhere. This is true even far in the IR, since the radius-squared of the S 3 at τ = 0 is of order g s M in string units. This is the 't Hooft coupling of the gauge theory found far in the IR. As long as this is large, the curvatures are small and the SUGRA approximation is reliable. SO(4) invariant expressions for the 3-forms In [7,8] it was shown that the warped background of the previous section preserves N = 1 SUSY if and only if G 3 is a (2, 1) form on the CY space. Perhaps the easiest way to see the supersymmetry of the deformed conifold solution is through a Tduality. Performing a T-duality along one of the longitudinal directions, and lifting the result to M-theory maps our background to a Becker-Becker solution supported by a G 4 which is a (2, 2) form on T 2 × CY. G-flux of this type indeed produces a supersymmetric background [34]. While writing G 3 in terms of the angular 1-forms g i is convenient for some purposes, the (2, 1) nature of the form is not manifest. That G 3 is indeed (2, 1) was demonstrated in [9] with the help of a holomorphic basis. Below we write the G 3 found in [1] in terms of the obvious 1-forms on the deformed conifold: dz i and dz i , i = 1, 2, 3, 4: G 3 = Mα ′ 2ε 6 sinh 4 τ sinh(2τ ) − 2τ sinh τ (ǫ ijkl z izj dz k ∧ dz l ) ∧ (z m dz m ) +2(1 − τ coth τ )(ǫ ijkl z izj dz k ∧ dz l ) ∧ (z m dz m ) .(69) We also note that the NS-NS 2-form potential is an SO(4) invariant (1, 1) form: B 2 = ig s Mα ′ 2ε 4 τ coth τ − 1 sinh 2 τ ǫ ijkl z izj dz k ∧ dz l .(70) The derivation of these formulae is given in Appendix B. Our expressions for the gauge fields are manifestly SO(4) invariant, and so is the metric. This completes the proof of SO(4) invariance of the KS solution. Infrared Physics We have now seen that the deformation of the conifold allows the solution to be non-singular. In the following sections we point out some interesting features of the SUGRA background we have found and show how they realize the expected phenomena in the dual field theory. In particular, we will now demonstrate that there is confinement; that the theory has glueballs and baryons whose mass scale emerges through a dimensional transmutation; that there is a gluino condensate that breaks the Z 2M chiral symmetry down to Z 2 , and correspondingly there are domain walls separating inequivalent vacua. Other stringy approaches to infrared phenomena in N = 1 SYM theory have recently appeared in [35,36]. Dimensional Transmutation and Confinement The resolution of the naked singularity via the deformation of the conifold is a supergravity realization of the dimensional transmutation. While the singular conifold has no dimensionful parameter, we saw that turning on the R-R 3-form flux produces the logarithmic warping of the KT solution. The scale necessary to define the logarithm transmutes into the the parameter ε that determines the deformation of the conifold. From (51) we see that ε 2/3 has dimensions of length and that τ = 3 ln(r/ε 2/3 ) + const . Thus, the scale r s entering the UV solution (40) should be identified with ε 2/3 . On the other hand, the form of the IR metric (68) makes it clear that the dynamically generated 4-d mass scale, which sets the tension of the confining flux tubes, is ε 2/3 α ′ √ g s M .(72) The reason the theory is confining is that in the metric for small τ (68) the function multiplying dx n dx n approaches a constant. This should be contrasted with the AdS 5 metric where this function vanishes at the horizon, or with the singular metric of [2] where it blows up. Consider a Wilson contour positioned at fixed τ , and calculate the expectation value of the Wilson loop using the prescription [37,38]. The minimal area surface bounded by the contour bends towards smaller τ . If the contour has a very large area A, then most of the minimal surface will drift down into the region near τ = 0. From the fact that the coefficient of dx n dx n is finite at τ = 0, we find that a fundamental string with this surface will have a finite tension, and so the resulting Wilson loop satisfies the area law. A simple estimate shows that the string tension scales as T s ∼ ε 4/3 (α ′ ) 2 g s M .(73) The masses of glueball and Kaluza-Klein (KK) states scale as m glueball ∼ m KK ∼ ε 2/3 g s Mα ′ .(74) Comparing with the string tension, we see that T s ∼ g s M(m glueball ) 2 .(75) Due to the deformation, the full SUGRA background has a finite 3-cycle. We may interpret various branes wrapped over this 3-cycle in terms of the gauge theory. Note that the 3-cycle has the minimal volume near τ = 0, hence all the wrapped branes will be localized there. A wrapped D3-brane plays the role of a baryon vertex which ties together M fundamental strings. Note that for M = 0 the D3-brane wrapped on the S 3 gave a dibaryon [23]; the connection between these two objects becomes clearer when one notes that for M > 0 the dibaryon has M uncontracted indices, and therefore joins M external charges. Studying a probe D3-brane in the background of our solution show that the mass of the baryon scales as M b ∼ M ε 2/3 α ′ .(76) Chiral Symmetry Breaking and Gluino Condensation Our SU(N + M) × SU(N) field theory has an anomaly-free Z 2M R-symmetry at all scales. The UV (large τ ) limit of our metric, which coincides with the solution found in [2], has a U(1) R-symmetry associated with the rotations of the angular coordinate ψ. However, the background value of the R-R 2-form C 2 does not have this continuous symmetry. Although F 3 = dC 2 given in (24) is U(1) symmetric, there is no smooth global expression for C 2 . Locally, we may write for large τ , C 2 → Mα ′ 2 ψω 2 .(77) Under ψ → ψ + ǫ, C 2 → C 2 + Mα ′ 2 ǫω 2 .(78) This modification of the asymptotic value of C 2 is dual to the appearance of opposite θ-angles for the two gauge groups, which is a manifestation of the anomaly in the U(1) R-symmetry [39,35]. Let us show that only the discrete shifts ψ → ψ + 2πk M , k = 1, 2, . . . , M(79) are symmetries of the UV theory. To this end we may consider domain walls made of k D5-branes wrapped over the finite-sized S 3 at τ = 0, with remaining directions parallel to R 3,1 . Such a domain wall is obviously a stable object in the KS background and crossing it takes us from one ground state of the theory to another. Indeed, the wrapped D5-brane produces a discontinuity in B F 3 , where B is the cycle dual to the S 3 . If to the left of the domain wall B F 3 = 0, as in the basic solution derived in the preceding sections, then to the right of the domain wall B F 3 = 4π 2 α ′ k ,(80) as follows from the quantization of the D5-brane charge. The B-cycle is bounded by a 2-sphere at τ = ∞, hence B F 3 = S 2 ∆C 2 . Therefore from (28) it is clear that to the right of the wall ∆C 2 → πα ′ kω 2 (81) for large τ . This change in C 2 is produced by the Z 2M transformation (79) on the original field configuration (77). Recalling that ψ ranges from 0 to 4π, we see that the full solution, which depends on ψ through cos ψ and sin ψ, has the Z 2 symmetry generated by ψ → ψ + 2π. Therefore, a domain wall made of M D5-branes returns the solution to itself. There are exactly M different discrete orientations of the solution, corresponding to breaking of the Z 2M UV symmetry through the IR effects. The domain walls constructed out of the wrapped D5-branes separate these inequivalent vacua. As we expect, flux tubes can end on these domain walls [40], and baryons can dissolve in them. By studying a probe D5-brane in the metric, we find that the domain wall tension is T wall ∼ 1 g s ε 2 (α ′ ) 3 .(82) In supersymmetric gluodynamics the breaking of chiral symmetry is associated with the gluino condensate λλ . A holographic calculation of the condensate was carried out by Loewy and Sonnenschein in [41] (see also [42] for previous work on gluino condensation in conifold theories.) They looked for the deviation of the complex 2-form field C 2 − i gs B 2 from its asymptotic large τ form that enters the KT solution: δ C 2 − i g s B 2 ∼ M α ′ 4 τ e −τ [g 1 ∧ g 3 + g 2 ∧ g 4 − i(g 1 ∧ g 2 − g 3 ∧ g 4 )] ∼ M α ′ ε 2 r 3 ln(r/ε 2/3 )e iψ (dθ 1 − i sin θ 1 dφ 1 ) ∧ (dθ 2 − i sin θ 2 dφ 2 ) . (83) In a space-time that approaches AdS 5 a perturbation that scales as r −3 corresponds to the expectation value of a dimension 3 operator. The presence of an extra ln(r/ε 2/3 ) factor is presumably due to the fact that the asymptotic KT metric differs from AdS 5 by such logarithmic factors. From the angular dependence of the perturbation we see that the dual operator is SU(2) × SU(2) invariant and carries R-charge 1. These are precisely the properties of λλ. Thus, the holographic calculation tells us that λλ ∼ M ε 2 (α ′ ) 3 .(84) Thus, the parameter ε 2 which enters the deformed conifold equation has a dual interpretation as the gluino condensate. 4 UV/IR Relations and the RG Flow In this section we investigate some of the consequences of compactifying the conifold. If the cascade is embedded inside a compact manifold, as in [25], then the radius τ is effectively cut off at some large τ c . The radial cutoff is a scale in the theory, which becomes an ultraviolet cutoff in the boundary gauge theory. The precise relation of these distance and energy scales depends in general on the physics one is investigating. We are aware of three schemes for relating the two scales: first, one could consider the energy of a string stretched from the tip of the conifold to the regularized boundary as a function of τ c [11]. Second, one can try to think about the warp factor h(τ ) as a redshift factor which relates the energy of a probe in the bulk of AdS space to its apparent energy as seen by an observer on the boundary. Third, one can consider the equations of motion for supergravity probes; this is sometimes called the holographic scheme [27]. In conformal backgrounds, the various distance/energy relations differ only by their normalization; for AdS 5 ×S 5 , E ∝ r in all three schemes. However, in non-conformal backgrounds the distance/energy relations can have different functional forms [10], and we will see that this is the case for the KT and KS solutions. One prescription for relating distance and energy scales comes from considering the energy of a string stretched from some fixed τ to the cutoff radius τ c , where it is stabilized by an external force -by a probe D-brane at τ c , for example. The energy of such a string is proportional to its worldvolume per unit time: E ∼ τc √ g tt g τ τ dτ ∼ e τc/3 + const ∼ r c .(85) The energy of this string corresponds to the linearly divergent self-energy of a quark in the boundary gauge theory, and the radial cutoff of the geometry regulates the divergence. Decreasing the radial cutoff removes high energy string modes and thus corresponds to integrating out high energy gauge theory modes in the Wilsonian sense. An appealing feature of this prescription is that ln Λ µ ∼ ln(r c )+ const, so that the difference of the couplings as predicted by supergravity agrees exactly with the gauge theory expectation, with no additional ln ln terms. An alternative prescription is to interpret the warp factor h(τ ) as a redshift factor. An object with energy E τ at radial position τ will appear to an observer at τ ′ to have energy E τ ′ , where the energies are related by E τ h 1/4 (τ ) = E τ ′ h 1/4 (τ ′ ). In terms of the renormalization scale, the distance/energy relation becomes Λ ∼ µ I(τ c ) −1/4 ∼ µ [τ −1/4 c e τc/3 . . .] .(86) This redshift relation introduces corrections to (46) of the form ln ln(Λ/µ). They have the same form as corrections to the flow due to two-loop β-functions. We can derive a third distance/energy relation by considering a massless supergravity probe in the KT background. For a massless scalar field the equation of motion is ∇ 2 φ = 1 L 2 ln(r/r s ) L 4 ln(r/r s ) r 2 ∂ i ∂ i + 1 r 3 ∂ ∂r r 5 ∂ ∂r φ = 0 .(87) The second term is invariant under a rescaling of the radius. Thus a solution of (87) which is wavelike on a radial slice of AdS depends on the radius through the quantity L 4 ln(r/rs)k 2 r 2 . For this prescription Λ ∼ r c L 2 ln(r c /r s ) .(88) We can obtain the same result by another physical argument which seems quite different on its surface. Let us consider instead this theory at finite temperature, as was studied in [26]. At sufficiently high temperature, the system develops a horizon, and the Hawking temperature is related to the horizon radius r H by T H ∼ r H L 2 ln(r H /r s ) ,(89) in the limit of high temperature. In the theory with a large radius cutoff, the maximum temperature is simply given by setting r H = r c . Then if we identify the UV cutoff Λ as the maximum Hawking temperature, we recover the result (88). The agreement between these two methods for relating the RG scale to the cutoff radius is a sign that holography is at work. Let us note that the relations we find between µ and Λ are exactly of the form one finds in an asymptotically free gauge theory. To make contact with the standard dimensional transmutation formula, we have to identify τ c /3 = 8π 2 /(β 0 g 2 0 ). If the beta function is dg d log(Λ/µ) = −β 0 g 3 16π 2 − β 1 g 5 128π 4 − . . . , then we find µ ∼ Λe −8π 2 /(β 0 g 2 0 ) (8π 2 /β 0 g 2 0 ) β 1 /β 2 0 .(90) To make contact with the SUGRA result (86) we have to identify τ c /3 = 8π 2 /(β 0 g 2 0 ). We can reexpress τ c in terms of the NS-NS flux τ <τc S 2 H 3 = 4π 2 α ′ K . One quickly finds τ c ≈ 2πK/(g s M). In order to achieve the continuum limit, we have to take τ c → ∞, Λ → ∞ while keeping the physical scale µ fixed. The exponential factor e −τc/3 = e −2πK/(3gsM ) , which may give rise to a large hierarchy of scales in compactified F-theory, was derived in [25]. It was observed that the type IIB supergravity picture of gluino condensation is reminiscent of the gluino condensation in the hidden sector of the heterotic string [43]. Here we note that a more precise SUGRA analysis may produce a power-law prefactor, which is analogous to the prefactor due to the 2-loop β-function in an asymptotically free gauge theory. With the stretchedstring relation, we find β 1 /β 2 0 = 0; the redshift relation gives β 1 /β 2 0 = 1/4; and the holographic relation gives β 1 /β 2 0 = 1/2. For pure N = 1 supersymmetric gauge theory with gauge group some simple Lie group G, β 0 = 3C 2 (G) and β 1 = 3C 2 (G) 2 (see for example [28]). The quantity C 2 (G) is the quadratic Casimir. Unfortunately, none of our distance/energy relations give the required value of 1/3. Perhaps adding the right kind of matter will fix the ratio to the correct value. It is also possible that a different identification between the SUGRA and field theory couplings may fix the prefactor. In any case, it would be very interesting to find out if the analogy with the gluino condensation in the hidden sector of the heterotic string [25] is in fact an exact duality. A Volume of the Two and Three Cycles The manifold T 1,1 can be identified as the intersection of the conifold 4 i=1 z 2 i = 0 and the sphere 4 i=1 |z i | 2 = 1 , where z i ∈ C. Dividing up z i = x i + iy i into real and imaginary parts, we see that T 1,1 can be thought of as the set of points satisfying x 2 i = 1/2 and y 2 i = 1/2 along with the constraint x · y = 0. If we use this constraint to eliminate one of the x, we can see, at least in a heuristic way, that the manifold T 1,1 can be thought of as a S 2 defined by the remaining x i fibered over an S 3 base defined by the y i . We now use this observation to parametrize the two cycle C 2 . An explicit parametrization of the whole T 1,1 is known in terms of the angles 0 ≤ ψ < 4π, 0 ≤ θ i ≤ π, and 0 ≤ φ i < 2π where i = 1, 2. Indeed z 1 = e iψ/2 √ 2 cos θ 1 + θ 2 2 cos φ 1 + φ 2 2 + i cos θ 1 − θ 2 2 sin φ 1 + φ 2 2 z 2 = e iψ/2 √ 2 − cos θ 1 + θ 2 2 sin φ 1 + φ 2 2 + i cos θ 1 − θ 2 2 cos φ 1 + φ 2 2 z 3 = e iψ/2 √ 2 − sin θ 1 + θ 2 2 cos φ 1 − φ 2 2 + i sin θ 1 − θ 2 2 sin φ 1 − φ 2 2 z 4 = e iψ/2 √ 2 − sin θ 1 + θ 2 2 sin φ 1 − φ 2 2 − i sin θ 1 − θ 2 2 cos φ 1 − φ 2 2 To describe the fiber, we would like to stay on one point on the base S 3 . Thus, we want to keep the imaginary part of the z i constant while still keeping two degrees of freedom available to trace out the S 2 fiber. For convenience, we begin by choosing ψ = 0. From the parametrization, we can trace out the S 2 by setting θ 1 = θ 2 and φ 1 = −φ 2 . Integrating using these coordinates, C 2 ω 2 = 4π. Next we consider the three cycle C 3 . First recall that g 5 ∧ g 3 ∧ g 4 = ω 3 − 1 2 d(g 1 ∧ g 3 + g 2 ∧ g 4 ) .(91) Moreover, C 3 has no boundary so C 3 ω 3 = C 3 g 5 ∧ g 3 ∧ g 4 .(92) We recall from [24] that ds 2 = 1 2 (g 5 ) 2 + (g 3 ) 2 + (g 4 ) 2(93) is the standard metric on a S 3 with radius √ 2. Moreover Vol(S 3 ) = 2π 2 r 3 . It follows then that C 3 ω 3 = 8π 2 . B Complex notation for the forms Our strategy is to guess differential forms, written in terms of the z i , with the appropriate symmetries and properties. To refine further and check the guess, we use computer assisted algebra to rewrite the differential forms in terms of the angular coordinates on the conifold. We can then compare the guess with the differential forms given in previous sections in terms of the g i . First, we need to review the construction of the angular coordinates on the deformed conifold. We define W = z 3 + iz 4 z 1 − iz 2 z 1 + iz 2 −z 3 + iz 4 .(94) The defining relation of the deformed conifold (47) becomes det W = −ε 2 . We introduce the angular coordinates with two SU(2) j = 1, 2 matrices L j = cos θ j 2 e i(ψ j +φ j )/2 − sin θ j 2 e −i(ψ j −φ j )/2 sin θ j 2 e i(ψ j −φ j )/2 cos θ j 2 e −i(ψ j +φ j )/2 .(95) The idea is then to take some representative point p ∈ C corresponding to W 0 = 0 εe τ /2 εe −τ /2 0 .(96) We can generate all of C by acting on W 0 with L 1 and L 2 W = L 1 · W 0 · L † 2 .(97) As the coordinates ψ 1 and ψ 2 only appear in W as ψ 1 + ψ 2 , we may define a new variable ψ = ψ 1 + ψ 2 . It is natural to define a radial coordinate ρ 2 ≡ 4 i=1 z izi = 1 2 Tr(W · W † ) .(98) With this definition, one straightforwardly obtains ρ 2 = ε 2 cosh τ . The singular case, ε = 0 is recovered by taking the large τ limit. Equivalently, we may start with a slightly different W sing 0 = 0 √ 2ρ 0 0 .(99) To summarize, then, the angular coordinates on the conifold are 0 ≤ ψ < 4π, 0 ≤ θ j < π, 0 ≤ φ j < 2π and a radius ρ. In the case where ε = 0, we can substitute τ for ρ. In the case ε = 0, ρ is typically expressed as r 3 ∼ ρ 2 in order to make the conical nature of the metric evident (14). In principle, we have an explicit coordinate transformation between the angular variables and the complex coordinates z i . The goal of this appendix, to rewrite the important supergravity quantities in terms of the z i , should be a straightforward task. Given any quantity written in terms of the angles, we should be able to write down the same quantity in terms of the z i . In practice, this variable change is difficult for two reasons. First, and especially in the case ε = 0, the variable change is quite complicated and nearly impossible to do without some computer assisted algebra. Second, there are more z i than one needs. By choosing three out of the four z i , one explicitly breaks the SO(4) symmetry. The formulae involving only three z i are typically messy and uninformative. Moreover, it is usually not completely obvious how to reintroduce the fourth z i in a way that symmetrizes the quantity of interest. B.1 Forms on the Singular Conifold We begin with the easier case, the singular conifold (13). We would like to express the forms important to the KT solution [2] and discussed in section 2 in terms of the z i . It is a fact that g 5 , ω 2 , and ω 3 all transform as singlets under the SO(4) action. Another important symmetry that holds when ε = 0 is the scaling z i → λz i where λ ∈ C * . The real part of this scaling, i.e. λ ∈ R + , corresponds to scaling the radius ρ, while the complex U(1) part, i.e. λ = e iα , corresponds to shifting the angle ψ. Cursory inspection of the vielbeins and the defining relations for ω 2 and ω 3 (21,26,27) show that the forms g 5 , ω 2 , and ω 3 are invariant under the full scaling. Begin with g 5 . Using the z i , there are two ways to write down singlet one forms which obey the scaling symmetries: z i dz i /ρ 2 orz i dz i /ρ 2 where summation on the indices is implied. 5 Any singlet one form must be some linear combination of these two, and all that need be done is fix the constants. Indeed dρ ρ + i 2 g 5 = 1 ρ 2z i dz i(100) and dρ ρ − i 2 g 5 = 1 ρ 2 z i dz i .(101) Next, we consider the two form ω 2 . There are several ways of writing SO(4) invariant two forms. Indeed η 1 = ǫ ijkl z izj dz k ∧ dz l , η 2 = ǫ ijkl z izj dz k ∧ dz l , η 3 = ǫ ijkl z izj dz k ∧ dz l , η 4 = (z i dz i ) ∧ (z j dz j ) , η 5 = (dz i ∧ dz i ) .(102) We can eliminate η 2 and η 3 immediately because they explicitly break the U(1) symmetry z i → e iα z i . The situation is even simpler. The form ω 2 transforms with a minus sign under the spatial inversion z 1 → −z 1 , keeping all the other z i fixed, while the forms η 4 and η 5 are invariant under the full O(4) symmetry. Our constraints leave only η 1 as a candidate for ω 2 : ω 2 = i ρ 4 η 1 .(103) As mentioned in [44], this form is closed, as we may check explicitly: ∂ω 2 = − i ρ 6 2χ 1 + ρ 2 χ 4 ,(104) where χ 1 = (ǫ ijkl z izj dz k ∧ dz l ) ∧ (z m dz m ) , χ 4 = ǫ ijklzi dz j ∧ dz k ∧ dz l ,(105) are new (2, 1) SO(4) invariant forms, labeled to conform with the notation used in the next section. With computer assisted algebra, using (13), it is easy to see that the expression on the right side of (104) vanishes. We also provide a symmetry argument for the vanishing. Because χ 1 and χ 4 are SO(4) and scale invariant, we are free to check the vanishing for a specific point p on the singular conifold, z 1 = 1, z 2 = i, z 3 = 0, and z 4 = 0, and then invoke the SO(4) and scaling symmetry to prove the vanishing for all points. Finally, we turn to ω 3 , and in fact we already know the answer because ω 3 = g 5 ∧ω 2 . The most important form in the KT solution is not F 3 or H 3 (and hence ω 3 or ω 2 ) independently but their combination G 3 = F 3 − iH 3 /g s , which needs to be a (2, 1) form in order to preserve supersymmetry. We may check that G 3 = α ′ 2 M g 5 − 2idρ ρ ∧ ω 2 = α ′ ρ 6 M (ǫ ijkl z izj dz k ∧ dz l ) ∧ (z m dz m )(106) which is explicitly a (2, 1) form, as required. Reassuringly, changing the sign of dρ/ρ in the expression above produces instead a (1, 2) form. B.2 Forms on the Deformed Conifold The deformed conifold is more difficult, not only because the coordinate transformation is more complicated but also because the nonzero ε explicitly breaks the U(1) and scale invariance. Before tackling the (2,1) form G 3 = F 3 −iH 3 /g s , let us warm up with some simpler one and two forms. First, we check what happens to z i dz i andz i dz i when ε is turned on:z i dz i = ε 2 sinh τ 2 dτ + ig 5 .(107) Comfortingly, in the large τ limit, we recover the singular conifold result (100). The result for z i dz i is, not surprisingly, the complex conjugate. Next we consider what happens to ω 2 . Remember that we have broken the U(1) symmetry and as a result, the forms η 2 and η 3 (102) may contribute. Indeed, they do as ω 2 = i cosh τ ε 4 sinh 3 τ ǫ ijkl z izj dz k ∧ dz l − 1 2 cosh τ (dz k ∧ dz l + dz k ∧ dz l ) . In the large τ limit, the reader may check that we recover the result for the singular conifold (103). Although ω 2 becomes a combination of different U(1) breaking differential forms, the NS-NS potential B 2 is actually more closely related to the old ω 2 of the singular conifold. Indeed B 2 = ig s Mα ′ 2ε 4 τ coth τ − 1 sinh 2 τ η 1 .(109) Again, in the large τ limit, we recover B 2 on the singular conifold. Now we are ready to attack the harmonic (2, 1) form G 3 . We begin by writing down all of the SO(4) invariant (2, 1) forms, of which there are five, χ 1 = (ǫ ijkl z izj dz k ∧ dz l ) ∧ (z m dz m ) , χ 2 = (ǫ ijkl z izj dz k ∧ dz l ) ∧ (z m dz m ) , χ 3 = (ǫ ijkl z i dz j ∧ dz k ∧ dz l ) , χ 4 = (ǫ ijklzi dz j ∧ dz k ∧ dz l ) , χ 5 = (dz i ∧ dz i ) ∧ (z j dz j ) .(110) Fortunately we can eliminate χ 5 because ∂ (h(ρ 2 )η 4 ) = h(ρ 2 )χ 5 . Based on experience with the singular conifold, and in particular the demonstration that ∂ω 2 = 0, one may wonder if the remaining χ i are linearly independent on the deformed conifold. They are not. The equation χ ≡ αχ 1 + βχ 2 + γχ 3 + δχ 4 = 0 is easy to satisfy. We choose a convenient point p ∈ C, for example the point corresponding to the matrix W 0 . If χ vanishes at p, it vanishes on all of C by SO (4) invariance. The two conditions that must be met for χ to vanish are α cosh τ + 2β − 2δ/ε 2 = 0 , α + 2β cosh τ + 2γ/ε 2 = 0 . We choose the ansatz for the (2, 1) form G 3 = αχ 1 + βχ 2 + γχ 3 + δχ 4 .(112) An intensive computer assisted computation reveals α cosh τ + 2β − 2δ/ε 2 = Mα ′ 2ε 6 sinh 5 τ [sinh τ (cosh(2τ ) + 5) − 6τ cosh τ ] , α + 2β cosh τ + 2γ/ε 2 = − Mα ′ ε 6 sinh 5 τ [τ (cosh(2τ ) + 2) − 3 sinh τ cosh τ ] . Because of the linear dependence of the χ i , (111), we are free to choose any two of the four parameters α, β, γ, and δ freely. Said another way, we can express G 3 as the sum of any two χ i , i = 1, . . . , 4. Let us choose γ = 0 and δ = 0. In this case, α = Mα ′ 2ε 6 sinh(2τ ) − 2τ sinh 5 τ and β = Mα ′ 2ε 6 2(1 − τ coth τ ) sinh 4 τ . In the large τ limit, β becomes vanishingly small compared to α. If it did not, the U(1) symmetry on the singular conifold would not be preserved! Moreover, α → Mα ′ /ρ 6 , in agreement with (106). which has the effect of decreasing the 5-form flux by M units. Note from (38) that for a single cascade step N ef f (r) → N ef f (r) − M the radius changes by a factor r 2 /r 1 = exp(−2π/3g s M), agreeing with a result of[25]. , there is a scale where the SU(N + M) coupling, g 1 , diverges. To continue past this infinite coupling, we perform a N = 1 duality transformation on this gauge group factor. The SU(N + M) gauge factor has 2N flavors in the fundamental representation. Under a Seiberg duality transformation, this becomes an SU(2N − [N + M]) = SU(N − M) gauge group. Thus we obtain an SU(N) × SU(N − M) For N = 2 supersymmetric gauge theories realized on fractional branes at orbifold singularities, the agreement of supergravity with field theory β-functions was demonstrated in[3,4,5]. Exactly the same relations apply to the N = 2 supersymmetric Z 2 orbifold theory[15,5]. These expressions for the β-functions differ from the standard NSVZ form[29] by a factor of 1/(1 − g 2 N c /8π 2 ). The difference comes from the choice of normalization of the vector superfields. We choose the normalization so that the relevant kinetic term in the field theory action is 1 2g 2 d 4 xd 2 θT r(W α W α )+ h.c.; this choice is dictated by the form of the supergravity action and differs from the canonical normalization by a factor of 1/g 2 . With this convention the additional factor in the β-function does not appear. A nice review of the derivation of the exact β-functions is in[30]. It would be nice to understand the relative factor of g s M between T wall and λλ . Note that z i dz i and its complex conjugate vanish by the defining relation on the conifold (47). Acknowledgements I.R.K. is grateful to S. Gubser, N. Nekrasov, M. Strassler, A. Tseytlin and E. Witten for collaboration on parts of the material reviewed in these notes and for useful input. We also thank Sergey Frolov and John Pearson for useful discussions. This work was supported in part by the NSF grant PHY-9802484. 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{'abstract': 'We assemble a few remarks on the supergravity solution of hep-th/0007191, whose UV asymptotic form was previously found in hep-th/0002159. First, by normalizing the R-R fluxes, we compare the logarithmic flow of couplings in supergravity with that in field theory, and find exact agreement. We also write the 3-form field strength G 3 = F 3 − τ H 3 present in the solution in a manifestly SO(4) invariant (2, 1) form. In addition, we discuss various issues related to the chiral symmetry breaking and wrapped branes.', 'arxivid': 'hep-th/0108101', 'author': ['Christopher P Herzog \nDepartment of Physics\nPrinceton University Princeton\n08544NJUSA\n', 'Igor R Klebanov \nDepartment of Physics\nPrinceton University Princeton\n08544NJUSA\n', 'Peter Ouyang \nDepartment of Physics\nPrinceton University Princeton\n08544NJUSA\n'], 'authoraffiliation': ['Department of Physics\nPrinceton University Princeton\n08544NJUSA', 'Department of Physics\nPrinceton University Princeton\n08544NJUSA', 'Department of Physics\nPrinceton University Princeton\n08544NJUSA'], 'corpusid': 9890622, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 22725, 'n_tokens_neox': 18774, 'n_words': 11733, 'pdfsha': 'e4cd839be10b131222b9c5187f05bbb69f087e89', 'pdfurls': ['https://arxiv.org/pdf/hep-th/0108101v3.pdf'], 'title': ['s talks at the Lisbon School on Superstrings II', 's talks at the Lisbon School on Superstrings II'], 'venue': ['and at the Benasque Workshop "Physics in the Pyrenees: Strings, Branes and Field Theory']}