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In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function σ α (n) : = ∑ d | n d α. We obtain an exact identity relating the Dirichlet series ζ (s) ζ (s − α) and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip. [ABSTRACT FROM AUTHOR]

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL 2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL 2 ≅ SO 3 associated with the prehomogenous vector space of symmetric matrices of degree 2. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a more general class of equations called functional Volterra integral equations involving measures, which encompass many types of equations such as functional Volterra equations, functional Volterra equations with impulses, functional Volterra delta integral equations on time scales, functional fractional differential equations with and without impulses, among others. Also we present some important results such as: local existence, uniqueness and prolongation of solutions, which play an important role for further investigations. We provide many examples and applications. [ABSTRACT FROM AUTHOR]

This work is devoted to Poisson stable (including stationary, periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent) solutions for stochastic functional evolution equations (SFEEs) with infinite delay. First, we prove the existence of bounded mild solutions for SFEEs with infinite delay. Then, according to the relationship between the solution and (drift and diffusion) coefficients, we obtain such Poisson stable solutions. Because the solutions of the delay equations are not Markov, we employ the solution maps in some phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability. [ABSTRACT FROM AUTHOR]

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to C r and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K , but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K = F q (T) , and describe the denominators and support of the numerators. [ABSTRACT FROM AUTHOR]

Let M n , m be a rectangular finite dimensional Cartan factor, i.e. the space L (C n , C m) with 1 ≤ n ≤ m , and let δ : M n , m → M n , m be a quadratic Jordan derivation of M n , m , i.e., a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ { A B A } = { δ (A) B A } + { A δ (B) A } + { A B δ (A) } , (A , B ∈ M n , m) , where (A , B , C) ↦ { A B , C } : = 1 2 (A B ⁎ C + C B ⁎ A) stands for the Jordan triple product in M n , m. We prove that then δ automatically is a continuous complex linear map on M n , m. More precisely we show that δ admits a representation of the form δ (A) = U A + A V , (A ∈ M n , m) , for a suitable pair U , V of square skew symmetric matrices with complex entries U ∈ M n , n and V ∈ M m , m. [ABSTRACT FROM AUTHOR]

We introduce an algorithm to compute the functions belonging to a suitable set F defined as follows: f ∈ F means that f (s , x) , s ∈ A ⊂ R being fixed and x > 0 , has a power series expansion centred at x 0 = 1 with convergence radius greater or equal than 1; moreover, it satisfies a functional equation of step 1 and the Euler-Maclaurin summation formula can be applied to f. Denoting the Euler gamma-function as Γ, we will show that, for x > 0 , log ⁡ Γ (x) , the digamma function ψ (x) , the polygamma functions ψ (w) (x) , w ∈ N , w ≥ 1 , and, for s > 1 being fixed, the Hurwitz ζ (s , x) -function and its first partial derivative ∂ ζ ∂ s (s , x) are in F. In all these cases the coefficients of the involved power series will depend on the values of ζ (u) , u > 1 , where ζ is the Riemann zeta-function. As a by-product, we will also show how to compute the Dirichlet L -functions L (s , χ) and L ′ (s , χ) , s > 1 , χ being a primitive Dirichlet character, by inserting the reflection formulae of ζ (s , x) and ∂ ζ ∂ s (s , x) into the first step of the Fast Fourier Transform algorithm. Moreover, we will obtain some new formulae and algorithms for the Dirichlet β -function and for the Catalan constant G. Finally, we will study the case of the Bateman G -function and of the alternating Hurwitz zeta-function, also known as the η -function; we will show that, even if they are not in F , our approach can be adapted to handle them too. In the last section we will also describe some tests that show a performance gain with respect to a standard multiprecision implementation of ζ (s , x) and ∂ ζ ∂ s (s , x) , s > 1 , x > 0. [ABSTRACT FROM AUTHOR]

The main purpose of this note is to provide an algorithm for approximating the value of the balanced p -adic L -function, as constructed in [Hsi21] , at the point (2 , 1 , 1) , which is lying outside of the interpolation region. The algorithmic procedure is obtained building on the work of [FM14] and considering finite-length geodesics on the Bruhat–Tits tree for G L 2 (Q p). We are interested in the case where at least one of the Hida families is associated with an elliptic curve over the rationals and we further restrict ourselves to the case where only one finite local sign of the functional equation is −1. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field with class number 1, in this paper we obtain the functional equation of the p -adic L -function of small slope p -stabilised Bianchi modular forms. Then, using p -adic families of Bianchi modular forms, we extend our result to Σ-smooth base-change Bianchi modular forms. [ABSTRACT FROM AUTHOR]

We fix an elliptic curve E / F q (t) and consider the family { E ⊗ χ D } of E twisted by quadratic Dirichlet characters. The one-level density of their L -functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (− 1 , 1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ ≠ 2 coprime to q. We obtain a restriction of (− 1 / 2 , 1 / 2) on the support with a unitary symmetry. [ABSTRACT FROM AUTHOR]

We consider functional differential equations (FDEs) which are perturbations of smooth ordinary differential equations (ODEs). The FDE can involve multiple state-dependent delays, distributed delays, or implicitly defined delays (forward or backward). We show that, under some mild assumptions on the perturbation, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, when the perturbation depends on some parameters, we get smooth dependence of the periodic orbit and its frequency on the parameters with high regularity. The method can also be applied to treat equations with small delays appearing in electrodynamics and FDEs which are perturbations of some evolutionary partial differential equations (PDEs). The proof consists in solving functional equations satisfied by the parameterization of the periodic orbit and the frequency using a fixed-point approach. We do not need to consider the smoothness of the evolution or even the phase space of the FDEs. [ABSTRACT FROM AUTHOR]

An abstract formulation of a duality principle established by Krasnoselskii is presented. Under appropriate conditions, it shall be shown that if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators share some topological properties. An explicit construction of such dual viewpoints is presented for a class of nonlinear functional equations which includes in the same framework some cases previously treated in the literature as well as some novel ones. [ABSTRACT FROM AUTHOR]

We study the cancellation of zeros between the Riemann zeta function and certain Artin L -functions. To do so, we develop a converse theorem for Maass forms of Laplace eigenvalue 1/4 in which the twisted L -functions are not assumed to be entire. We do not require the conjectural automorphy of Artin L -functions, only their established meromorphic continuation and functional equation. [ABSTRACT FROM AUTHOR]

In this paper, new conditions are found for the representability of a continuous multivariate function as a sum of ridge functions. Using these conditions, we give a new proof for the earlier theorem solving the problem, posed by A.Pinkus in his monograph "Ridge Functions", up to a multivariate polynomial. That is, we show that if a continuous multivariate function has a representation as a sum of arbitrarily behaved ridge functions, then it can be represented as a sum of continuous ridge functions and some multivariate polynomial. [ABSTRACT FROM AUTHOR]

This work is concerned with solving high-dimensional Fokker-Planck equations with the novel perspective that solving the PDE can be reduced to independent instances of density estimation tasks based on the trajectories sampled from its associated particle dynamics. With this approach, one sidesteps error accumulation occurring from integrating the PDE dynamics on a parameterized function class. This approach significantly simplifies deployment, as one is free of the challenges of implementing loss terms based on the differential equation. In particular, we introduce a novel class of high-dimensional functions called the functional hierarchical tensor (FHT). The FHT ansatz leverages a hierarchical low-rank structure, offering the advantage of linearly scalable runtime and memory complexity relative to the dimension count. We introduce a sketching-based technique that performs density estimation over particles simulated from the particle dynamics associated with the equation, thereby obtaining a representation of the Fokker-Planck solution in terms of our ansatz. We apply the proposed approach successfully to three challenging time-dependent Ginzburg-Landau models with hundreds of variables. • Proposes a novel perspective that solving the PDE can be reduced to density estimation tasks based on particle methods. • Provides an end-to-end procedure that efficiently performs density estimation with functional hierarchical tensor. • Provides detailed numerical techniques to solve the challenging time-dependent Ginzburg-Landau models. [ABSTRACT FROM AUTHOR]

For 0 < a ≤ 1 / 2 , we define the quadrilateral zeta function Q (s , a) using the Hurwitz and periodic zeta functions and show that Q (s , a) satisfies Riemann's functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove that for any 0 < a ≤ 1 / 2 , there exist positive constants A (a) and T 0 (a) such that the number of zeros of the quadrilateral zeta function Q (s , a) on the line segment from 1/2 to 1 / 2 + i T is greater than A (a) T whenever T ≥ T 0 (a). [ABSTRACT FROM AUTHOR]

Let U be any of the two exceptional Cartan factors V and VI and let δ : U → U be a Jordan *-triple derivation of U , that is, a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ { a b ⁎ a } = { δ (a) b ⁎ a } + { a δ (b) ⁎ a } + { a b ⁎ δ (a) } , (a , b ∈ U) , where (a , b) ↦ { a b ⁎ a } stands for the Jordan triple product in U. We give an explicit representation of δ as certain multipliers on U and prove that δ automatically is a continuous real linear map on U. This gives a new description of the real Banach Lie algebra of Jordan triple derivations of U. [ABSTRACT FROM AUTHOR]

For a compact Riemann surface M of genus g ≥ 2 , we study the functional equations of the Selberg zeta functions attached with the Tate motives f. We prove that certain functional equations hold if and only if f has the absolute automorphy. [ABSTRACT FROM AUTHOR]

Generalized ordinary differential equations (we write generalized ODEs), introduced by J. Kurzweil in 1957, are known to encompass several other types of equations as measure functional differential equations, for instance. In this paper, we obtain converse Lyapunov theorems for generalized ODEs and, in particular, for measure functional differential equations which, in turn, encompass impulsive functionals differential equations as well as functional dynamic equations on time scales. We also relate uniform stability to boundedness of solutions. As an application, we establish necessary and sufficient conditions for a system of non–homogeneous nonlinear generalized ODEs defined in a Banach space and for a system of non–homogeneous measure functional differential equations to be asymptotically controllable. We include an example which illustrates the main results. [ABSTRACT FROM AUTHOR]

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of admissible representations. We summarize the basic properties of these integrals such as the convergence, meromorphic continuation and an abstract functional equation. In the second part, we prove a generalization that accommodates certain non-spherical spaces with spherical quotients. As an application, the resulting theory applies to the prehomogeneous vector space underlying Bhargava's cubes, which is also considered by F. Sato and Suzuki–Wakatsuki in their study of toric periods. [ABSTRACT FROM AUTHOR]

Let η (s) = ∑ n = 1 ∞ (− 1) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m (τ) and consider finite Dirichlet series υ M (τ , s) = ∑ m = 1 M b M , m (τ) m − s and η N (τ , s) = ∑ M = 1 N υ M (τ , s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N (τ , s) approximates η (s) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η (s) = ∑ M = 1 ∞ υ M (τ , s). Moreover, it looks that lim M → ∞ ⁡ m υ M (τ , 1 − σ + i t) ‾ m υ M (τ , σ + i t) = η (σ + i t) m η (1 − σ + i t) ‾ m ; in other words, the individual summands in expected expansion (⁎) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M (τ , σ + i t) = υ M (τ , σ + i t) / η (σ + i t). When M , τ , and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M (τ , σ + i t) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function. [ABSTRACT FROM AUTHOR]

For x irrational, we study the convergence of series of the form ∑ n − s f (n x) where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal who studied the examples f (t) = cot ⁡ (π t) and f (t) = sin − 2 ⁡ (π t). [ABSTRACT FROM AUTHOR]

This work, Part II, together with its companion-Part I develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the problems, it is natural to divide our contributions into two parts to answer a long-standing question in biology and ecology. What are the minimal conditions for long-term persistence and extinction of a population? Part I of our work provides characterization of persistence, whereas in this part, extinction is the main focus. The techniques used in this paper are combination of the newly developed functional Itô formula and a dynamic system approach. Compared to the study of stochastic Kolmogorov systems without delays, the main difficulty is that infinite dimensional systems have to be treated. The extinction is characterized after investigating random occupation measures and examining behavior of functional systems around boundaries. Our characterizations of long-term behavior of the systems reduce to that of Kolmogorov systems without delay when there is no past dependence. A number of applications are also examined. [ABSTRACT FROM AUTHOR]

We consider a function-analytic approach to study synchronizing automata, primitive and ergodic matrix families. This gives a new way to establish some criteria for primitivity and for ergodicity of families of nonnegative matrices. We introduce a concept of canonical partition and use it to construct a polynomial-time algorithm for finding a positive matrix product and an ergodic matrix product whenever they exist. This also provides a generalization of some results of the Perron-Frobenius theory from one nonnegative matrix to families of matrices. Then we define the h -synchronizing automata and prove that the existence of a reset tuple is polynomially decidable. The question whether the functional-analytic approach can be extended to the h -primitivity is addressed and several open problems are formulated. [ABSTRACT FROM AUTHOR]

Recently, Dixit et al. [24] established a very elegant generalization of Hardy's theorem concerning the infinitude of zeros that the Riemann zeta function possesses at its critical line. By introducing a general transformation formula for the theta function involving the Bessel and modified Bessel functions of the first kind, we extend their result to a class of Dirichlet series satisfying Hecke's functional equation. In the process, we also find new generalizations of classical identities in Analytic number theory. [ABSTRACT FROM AUTHOR]

For any d ∈ { 1 , ... , 6 } , we prove that the web of conics on a del Pezzo surface of degree d carries a functional identity whose components are antisymmetric hyperlogarithms of weight 7 − d. Our approach is uniform with respect to d and relies on classical results about the action of the Weyl group on the set of lines on the del Pezzo surface. These hyperlogarithmic functional identities are natural generalizations of the classical 3-term and (Abel's) 5-term identities satisfied by the logarithm and the dilogarithm, which correspond to the cases when d = 6 and d = 5 respectively. [ABSTRACT FROM AUTHOR]

We codify some aspects of Deuring's zero-spacing phenomenon, namely that in the presence of an exceptional Landau/Siegel zero, various product L -functions have their low-height zeros lie in a very regular pattern, nearly an arithmetic progression on the central line. This piece of "analytic number theory folklore" seems to have had its actual proof escape the literature in generality, yet we still claim to provide a "different" proof in that we use Lavrik's general method for approximate functional equations, rather than (say, in the imaginary quadratic case) a decomposition into Epstein ζ -functions and analysis with K -Bessel functions and Kloosterman sums. [ABSTRACT FROM AUTHOR]

We consider the hybrid fourth shifted moment of Dirichlet L -functions over rational function fields, where the moment average is taken over all odd primitive characters of modulus Q ∈ F q [ t ] and over the critical circle, which is the symmetry line of the corresponding functional equation. We obtain an asymptotic formula for this moment with the full main term for arbitrary modulus Q , as deg ⁡ Q → ∞ and q is fixed. Moreover, in case of an irreducible modulus Q we get an exact formula for this hybrid moment. [ABSTRACT FROM AUTHOR]

We prove various converse theorems for automorphic forms on Γ 0 (N) , each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N ∈ { 18 , 20 , 24 } – these integers are the smallest multiples of 4 or 9 not covered by earlier work of Conrey–Farmer. This development is a consequence of finding generating sets for Γ 0 (N) such that each generator can be written as a product of special matrices. As for real-analytic Maass forms of even (resp. odd) weight we prove the analogous statement for 1 ≤ N ≤ 12 and N ∈ { 16 , 18 } (resp. 1 ≤ N ≤ 12 , 14 ≤ N ≤ 18 and N ∈ { 20 , 23 , 24 }). [ABSTRACT FROM AUTHOR]

The aim of the paper is to define an infinitesimal analog of the Bloch regulator, which attaches to a pair of meromorphic functions on a Riemann surface, a line bundle with connection on the punctured surface. In the infinitesimal context, we consider a pair (X , X _) of schemes over a field of characteristic 0, such that the regular scheme X _ is defined in X by a square-zero sheaf of ideals which is locally free on X _. We propose a definition of the weight two motivic cohomology of X based on the Bloch group, which is defined in terms of the functional equation of the dilogarithm. The analog of the Bloch regulator is a map from a subspace of the infinitesimal part of H M 2 (X , Q (2)) to the first cohomology group of the Zariski sheaf associated to an André-Quillen homology group. Using Goodwillie's theorem, we deduce that this map is an isomorphism, which is an infinitesimal analog of the injectivity conjecture for the Bloch regulator. [ABSTRACT FROM AUTHOR]

Let p , q be distinct primes, with p > 2. We classify the Hopf-Galois structures on Galois extensions of degree p 2 q , such that the Sylow p -subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups (G , ⋅) of order p 2 q , in the case when the Sylow p -subgroups of G are cyclic. This is equivalent to classifying the skew braces (G , ⋅ , ∘). Furthermore, we prove that if G and Γ are groups of order p 2 q with non-isomorphic Sylow p -subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G. Our method relies on the alternate brace operation ∘ on G , which we use mainly indirectly, that is, in terms of the functions γ : G → Aut (G) defined by g ↦ (x ↦ (x ∘ g) ⋅ g − 1). These functions are in one-to-one correspondence with the regular subgroups of the holomorph of G , and are characterised by the functional equation γ (g γ (h) ⋅ h) = γ (g) γ (h) , for g , h ∈ G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual. [ABSTRACT FROM AUTHOR]

We deepen the study of the relations previously established by Mayer, Lewis and Zagier, and the authors, among the eigenfunctions of the transfer operators of the Gauss and the Farey maps, the solutions of the Lewis-Zagier three-term functional equation and the Maass forms on the modular surface P S L (2 , Z) \ H. In particular we introduce an "inverse" of the integral transform studied by Lewis and Zagier, and use it to obtain new series expansions for the Maass cusp forms and the non-holomorphic Eisenstein series restricted to the imaginary axis. As corollaries we obtain further information on the Fourier coefficients of the forms, including a new series expansion for the divisor function. [ABSTRACT FROM AUTHOR]

We introduce a new type of summation formula for central values of GL (4) × GL (2) L -functions, when varied over Maaß forms. By rewriting such a sum in terms of GL (4) × GL (1) L -functions and applying a new "balanced" Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The "balanced" Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL (4) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL (4 , Z) , there exist infinitely many Maaß forms π for SL (2 , Z) such that L (1 / 2 , Π × π) ≠ 0. [ABSTRACT FROM AUTHOR]

We prove the absolute convergence, functional equations and meromorphic continuation of local intertwining periods on parabolically induced representations of finite length for certain symmetric spaces over local fields of characteristic zero, including Galois pairs as well as pairs of Prasad and Takloo-Bighash type. Furthermore, for a general symmetric space we prove a sufficient condition for distinction of an induced representation in terms of distinction of its inducing data. Both results generalize previous results of the first two named authors. In particular, for both we remove a boundedness assumption on the inducing data and for the second we further remove any assumption on the symmetric space. Moreover, we extend the field of coefficients from p -adic to any local field of characteristic zero. In the case of p -adic symmetric spaces, combined with the necessary conditions for distinction that follow from the geometric lemma, this provides a necessary and sufficient condition for distinction of representations induced from cuspidal. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following superlinear elliptic problem (P) { − Δ u = λ | u | p − 2 u + f (x , u) , in Ω , u = 0 , on ∂ Ω , where λ > 0 and 2 < p < 2 + δ for some δ > 0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u = 0 is a local minimizer of the associated energy functional of equation (P). Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ = 1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ. [ABSTRACT FROM AUTHOR]

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F (x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study F k , N (x) , an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by F k , N (x). Radchenko and Zagier gave a beautiful relation between the integral ∫ 0 1 log ⁡ (1 + t x) 1 + t d t and F (x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between F k , N (x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of F k , N (x) and some generalized Lambert series are also obtained along with other supplementary results. [ABSTRACT FROM AUTHOR]

We consider the magnetic Ginzburg–Landau equations in R 4 { − ε 2 (∇ − i A) 2 u = 1 2 (1 − | u | 2) u , ε 2 d ⁎ d A = 〈 (∇ − i A) u , i u 〉 , formally corresponding to the Euler–Lagrange equations for the energy functional E (u , A) = 1 2 ∫ R 4 | (∇ − i A) u | 2 + ε 2 | d A | 2 + 1 4 ε 2 (1 − | u | 2) 2. Here u : R 4 → C , A : R 4 → R 4 and d denotes the exterior derivative acting on the one-form dual to A. Given a minimal surface M 2 in R 3 with finite total curvature and non-degenerate, we construct a solution (u ε , A ε) which has a zero set consisting of a smooth surface close to M × { 0 } ⊂ R 4. Away from the latter surface we have | u ε | → 1 and u ε (x) → z | z | , A ε (x) → 1 | z | 2 (− z 2 ν (y) + z 1 e 4) , x = y + z 1 ν (y) + z 2 e 4 for all sufficiently small z ≠ 0. Here y ∈ M and ν (y) is a unit normal vector field to M in R 3. [ABSTRACT FROM AUTHOR]

Given a semigroup S and a uniquely divisible group H , for every h ∈ S we define the right difference operator Δ h on functions f : S ⟶ H as follows: Δ h f (g) = f (g h) − f (g). Each of the following two conditions on a function f can be considered as a characterization of polynomial mappings on S : 1) Δ h 1 ⋯ Δ h m f = 0 for some m ∈ N and every h 1 , ... , h m ∈ S. 2) there is m ∈ N such that Δ h m f = 0 for every h ∈ S ; We prove that if S possesses the property g S = S g for every g ∈ S , then two classes of functions, defined by the above conditions respectively, coincide. [ABSTRACT FROM AUTHOR]

We investigate the structure of the generalized Weierstrass semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to associate them with combinatorial objects, the Poincaré series and the semigroup polynomial. We show that this Poincaré series determines completely the generalized Weierstrass semigroup and it is entirely determined by the semigroup polynomial. We finish the paper by describing the functional equations occurring to the Poincaré series under the hypothesis of a symmetric generalized Weierstrass semigroup. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a generalized action integral of both gravity and matter is defined on the sphere bundle over Finsler space–time manifold M with a Lorentz–Finsler metric. The Euler–Lagrange equation of this functional, a generalization of the Riemann–Einstein gravity equation is obtained by using some divergence theorems. Fibres of the sphere bundle are unbounded according to the pseudo-Finsler metric. Moreover, solutions of vacuum Finsler gravity equation under the weakly Landsberg condition are discussed and some concrete examples are provided. At last, we raise some questions for further study. [ABSTRACT FROM AUTHOR]

Abstract In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third author. On the other hand, we investigate geometric programming methods for constrained polynomial optimization problems, which were recently developed by Ghasemi and Marshall. We show that the combination of both results yields a new method to solve certain classes of constrained polynomial optimization problems. We test the new method experimentally and compare it to semidefinite programming in various examples. [ABSTRACT FROM AUTHOR]

In this article, we undertake the study of the function F (x ; u , v) , which we refer to as the Herglotz-Zagier-Novikov function. This function appears in Novikov's work on the Kronecker limit formula, which was motivated by Zagier's paper where he obtained the Kronecker limit formula in terms of the Herglotz function F (x). Two, three, and six-term functional equations satisfied by F (x ; u , v) are exhibited. These are cohomological relations coming from the action of an involution and SL 2 (Z) on C × D 1 2 (the unit circle D 1). We also provide the special values of F (x ; u , v) at rational arguments of x. Importantly, F (x ; u , v) serves as a unified generalization of three other interesting functions, namely F (x) , J (x) , and T (x) , which also appear in various Kronecker limit formulas and are previously studied by Cohen, Herglotz, Muzaffar and Williams, and Radchenko and Zagier. Consequently, our study not only reveals the numerous elegant properties of F (x ; u , v) but also helps us to further develop the theories of functions related to its special cases such as J (x) and T (x). [ABSTRACT FROM AUTHOR]

In this paper we study the Fréchet functional equation in the n -dimensional Euclidian space as well as in the context of distributions. We also generalize the Cauchy functional equation for distributions to any natural order. [ABSTRACT FROM AUTHOR]

Abstract Let D be a division ring with characteristic different from 2. In this paper we will describe an additive map f satisfying the identity f (x) y + x f (y) = l for every x , y ∈ D such that x y = a , where l , a ∈ D and a is nonzero. Additionally, for nonzero m , k ∈ D , we will describe the additive bijection f satisfying the identity f (x) f (y) = m for every x , y ∈ D such that x y = k ; this description is the solution to a problem posed by Chebotar, Ke, Lee, and Shiao in 2005. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the general solution of the following general radical quintic functional equation f ( a x 5 + b y 5 5 ) = r f ( x ) + s f ( y ) for f a mapping from the field of real numbers into a vector space, where a , b , r , s are fixed nonzero reals. Also, we prove the generalized hyperstability results for the general radical quintic functional equation by using the fixed point theorem (cf. Dung and Hang (2018) [15] , Theorem 2.1) in quasi- β -Banach spaces. Namely, we show, under some weak natural assumptions, functions satisfying the above equation approximately (in some sense) must be actually solutions to it. [ABSTRACT FROM AUTHOR]

We develop a stochastic model for the sum X and the maximum Y of dependent, heavy-tail Pareto components. Our results include explicit forms of the probability density and cumulative distribution functions, marginal and conditional distributions, moments and related parameters, parameter estimation, and stochastic representations. We also derive mixed conditional tail expectations, E ( X | Y > y ) and E ( Y | X > x ) , which provide measures of risk frequently used in finance and insurance. An extension incorporating a random number N of components in the sum and the maximum, along with its basic properties, is included as well. Two data examples from finance illustrate modeling potential of these new multivariate distributions. [ABSTRACT FROM AUTHOR]

The unit interval vertex deletion problem asks for a set of at most k vertices whose deletion from a graph makes it a unit interval graph. We develop an O ( k 4 ) -vertex kernel for the problem, significantly improving the O ( k 53 ) -vertex kernel of Fomin et al. (2013) [11] . We start from a constant-approximation solution and study its interaction with other vertices, which induce a unit interval graph. We partition vertices of this unit interval graph into cliques in a naive way, and pick a small number of representatives from each of them. Our constructive proof for the correctness of our algorithm, using interval models, greatly simplifies the “destructive” proofs, based on destroying forbidden structures, for similar problems in literature. Our algorithm can be implemented in O ( m n + n 2 ) time, where n and m denote respectively the numbers of vertices and edges of the input graph. [ABSTRACT FROM AUTHOR]

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. Extra connectivity and conditional diagnosability are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processors. However, the extra connectivity and conditional diagnosability of many well-known networks have been independently investigated. Fault diagnosis of general regular graph is more meaningful than that of special graph. In this paper, the relationship between extra connectivity and conditional diagnosability of regular graphs is explored. First, we determine that the conditional diagnosability under the PMC model equals 3-extra connectivity plus 1 or 3-extra connectivity plus 2. Finally, we give empirical analysis on extra connectivity and conditional diagnosability of some graphs by our proposed relationship. [ABSTRACT FROM AUTHOR]

We investigate a generalization of stacks that we call C - machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that C -machines generate, and how these systems of functional equations can be iterated and sometimes solved. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by C -machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their counting sequences, seem to not have D-finite generating functions. [ABSTRACT FROM AUTHOR]

Let f : R n → R be a function. Assume that for a measurable set Ω and almost every x ∈ Ω there exists a vector ξ x ∈ R n such that lim inf h → 0 f ( x + h ) − f ( x ) − 〈 ξ x , h 〉 | h | 2 > − ∞ . Then we show that f satisfies a Lusin-type property of order 2 in Ω , that is to say, for every ε > 0 there exists a function g ∈ C 2 ( R n ) such that L n ( { x ∈ Ω : f ( x ) ≠ g ( x ) } ) ≤ ε . In particular every function which has a nonempty proximal subdifferential almost everywhere also has the Lusin property of class C 2 . We also obtain a similar result (replacing C 2 with C 1 ) for the Fréchet subdifferential. Finally we provide some examples showing that these kinds of results are no longer true for Taylor subexpansions of higher order. [ABSTRACT FROM AUTHOR]