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Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

In this work, we examine a third-order nonlinear neutral integro-differential equation with constant delay. By building a Lyapunov functional, we obtain some sufficient criteria that ensure the asymptotic stability and boundedness of solutions for an analyzed equation. We present two examples to demonstrate the applicability of our conclusions, which extend and improve several well-known results in the literature. [ABSTRACT FROM AUTHOR]

We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left minima) on separable and irreducible separable permutations. The equations are used to find a third degree equation for joint distribution of ascents and descents on separable permutations that generalizes the respective known result for the descent distribution. Moreover, our general functional equations allow us to derive explicitly (joint) distribution of any subset of maxima and minima statistics on irreducible, reducible and all separable permutations. In particular, there are two equivalence classes of distributions of a pair of maxima or minima statistics. Finally, we present three unimodality conjectures about distributions of statistics on separable permutations. [ABSTRACT FROM AUTHOR]

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and cumbersomely prove uniqueness in stability studies. [ABSTRACT FROM AUTHOR]

We verify the recently developed method by Kisil (Kisil 2024 SIAM J. Appl. Math. 84, 464-476 (doi: 10.1137/23M1562445)) through its application in the study of the anti-plane dynamic response of an elastic square-cell lattice quadrant having a free lateral boundary that is subjected to a sinusoidal point load. The vertical boundary of the quadrant is assumed to be either fixed or free. The approach adopted utilizes the discrete Fourier transform in both principal lattice directions to reduce the governing equations to a functional equation linking both the bulk and boundary response of the system. For both problems, this functional equation is shown to reduce to scalar Wiener-Hopf equation whose solution can be exploited to determine the full response of the lattice system. The analytical solutions based on the Wiener-Hopf method are also implemented in numerical investigations that demonstrate the lattice response to various positions and frequencies of the load. In particular, this study includes dynamic regimes induced by the load that possess strong anisotropy and shows how the associated waveforms interact with the boundaries of the quadrant. [ABSTRACT FROM AUTHOR]

A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction. [ABSTRACT FROM AUTHOR]

Iteration is one of the most important topics in computer science and attentions are paid to functional equations involving iterates, one of which is the linear combination of alternative iterates. Recently existence, uniqueness and dependence for increasing Lipschitzian solutions were obtained for linear combination of alternative iterates on $ [0, 1] $ [ 0 , 1 ] and $ \mathbb {R} $ R in the case that all given functions are strictly increasing. In this paper we work in various cases of monotonicity (increasing and decreasing) among given functions and the unknown function. [ABSTRACT FROM AUTHOR]

It is known that many physical systems undergoing (nuclear, atomic, etc.) decay do not obey the standard decreasing exponential formula which corresponds to the solution of a first-order, linear ODE having constant coefficients. We propose and solve a new functional equation mathematical model whose solutions are consistent with current experimental data. The basis of our functional representation is centred on the critical role played by the concept of the decay half-life. [ABSTRACT FROM AUTHOR]

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay. [ABSTRACT FROM AUTHOR]

The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. Special attention is given to modelling such functions by systems of functional equations. [ABSTRACT FROM AUTHOR]

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions f , g , h ∈ C (G) of the Levi-Civita equation f (x y) = f (x) h (y) + g (x) f (y) , x , y ∈ G , which is an extension of the sine addition law. Representations of G on C 2 play an important role. As a corollary we get the solutions f , g ∈ C (G) of the sine subtraction law f (x y ∗) = f (x) g (y) - g (x) f (y) , x , y ∈ G , in which x ↦ x ∗ is a continuous involution, meaning that (x y) ∗ = y ∗ x ∗ and x ∗ ∗ = x for all x , y ∈ G . [ABSTRACT FROM AUTHOR]

We present a linear stability analysis of the Richtmyer-Meshkov instability that develops when a shock wave reaches a sinusoidally perturbed premixed flame from behind. In the hydrodynamic regime, when acoustic contributions dominate the flame growth rate, the problem is analytically addressed by the direct integration of the sound wave equations at both sides of the flame, which are bounded by the reflected and transmitted shock waves and the flame front that acts as a contact surface in the hydrodynamic limit. The resolution involves: a hyperbolic change of variables to modify the triangular spatio-temporal domain, a transformation in the Laplace variable, the resolution of the functional equations in the frequency domain, and the final inverse Laplace transform. The latter involves a novel resolution method that is proven beneficial for long-time dynamics. Asymptotic analysis is also carried out to describe the early time and late time hydrodynamic response. The nonuniform flow field resulting from distorted oscillating shocks is characterized by acoustic, rotational, and entropic disturbances, each of which exerts a substantial influence on flame dynamics. These disturbances contribute to the intricate interplay of factors shaping the behavior of the flame in response to the nonuniform flow. In particular, the sensitivity of local flame propagation to temperature disturbances is investigated. This work contributes to a deeper understanding of Richtmyer-Meshkov instability dynamics and offers insights into instability reduction through the modulation of temperature disturbances generated by the transmitted shock. • Introduction of a self-consistent analytical model for Richtmyer-Meshkov instability within shock-flame interactions. • Development of an enhanced resolution method for the functional equation in the Laplace variable. • Introduction of a new resolution method for the transient evolution via the inverse Laplace transform. • Provision of a comprehensive mechanism regarding Richtmyer-Meshkov instability reduction. [ABSTRACT FROM AUTHOR]

This study investigates weakly singular nonlinear functional Volterra integral equations (WSNFVIEs) of Urysohn type involving Riemann–Liouville operator. By imposing specific smoothness conditions on the involved functions, we establish both the existence and uniqueness of the solution using a fixed point approach. Subsequently, we employ discretization methods such as the trapezoidal and Euler methods to approximate the solution, resulting in a system of nonlinear algebraic equations. To ascertain the convergence order of the Euler method (first order) and the trapezoidal method (second order), we utilize the Grönwall inequality and its discrete counterpart. Additionally, we introduce a novel Grönwall inequality to establish the convergence of the trapezoidal method. Thoroughly we examine the Hyers–Ulam–Rassias and Hyers–Ulam stability of the integral equations within the specified domain. Finally, the efficacy of the proposed methods is validated through numerical examples accompanied by comparative analyses. [ABSTRACT FROM AUTHOR]

A general theory for solving electromagnetic diffraction problems with impenetrable/penetrable wedges immersed in/made of an arbitrary linear (bianistropic) medium is presented. This novel and general spectral theory handles complex scattering problems by using transverse equations for layered planar and angular structures, the characteristic Green function procedure, the Wiener–Hopf technique, and a new methodology for solving GWHEs. The technique has been proven effective for analyzing problems involving wedges immersed in isotropic media; in this study, we extend the theory to more general cases while providing all necessary mathematical tools and corresponding validations. We obtain generalized Wiener–Hopf equations (GWHEs) from spectral functional equations in angular regions filled by arbitrary linear media. The equations can be interpreted with a network formalism for a systematic view. We recall that spectral methods (such as the Sommerfeld–Malyuzhinets (SM) method, the Kontorovich–Lebedev (KL) transform method, and the Wiener–Hopf (WH) method) are well-consolidated, fundamental, and effective tools for the correct and precise analysis of electromagnetic diffraction problems constituted by abrupt discontinuities immersed in media with one propagation constant, although they are not immediately applicable to multiple-propagation-constant problems. To the best of our knowledge, the proposed mathematical technique is the first extension of spectral analysis to electromagnetic problems in the presence of angular regions filled by complex arbitrary linear media, thereby providing novel mathematical tools. Validation through fundamental examples is proposed. [ABSTRACT FROM AUTHOR]

We study the Kneser-type oscillation criterion for a class of second-order half-linear functional dynamic equations on an arbitrary time scale utilizing the integral averaging approach and the Riccati transformation method. The results show an improvement in Kneser-type when compared to some known results. We provide some illustrative examples to demonstrate the significance of our main results. [ABSTRACT FROM AUTHOR]

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi- β -Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases. [ABSTRACT FROM AUTHOR]

The paper focuses on establishing sufficient conditions for the existence of the solutions in some functional q-integral equations, particularly in Banach spaces. In this method, the technique of measures of noncompactness and Petryshyn's fixed point theorem in Banach space is used. We provide some examples of equations, which confirm that our result is applicable to a wide class of integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new integral transform called the HK Transform and employ it to analytically solve the Volterra integral equations (VIE.) of the first kind. To achieve this First we derive the HK transform of basic algebraic and transcendental mathematical functions. The fundamental characteristics of the HK transform, which can be used to solve a variety of functional equations, including, we then discuss the fundamental properties, such as ordinary differential equations and integral equations, then we discover the exact solution for general first class V.I.E. To illustrate the applicability of the HK transform, numerical problems are examined and handled carefully step-by-step. The results show that the proposed HK Transform new integral transform produces accurate solutions for first-kind VIE. without requiring time-consuming computations. [ABSTRACT FROM AUTHOR]

Let Y (+) be a group, D ... ℤ² where ℤ (+, ≤) denotes the ordered group of all integers, and ℤ² : = ℤxℤ. We shall use the notations Dx: = ... . The main purpose of the article is to find sets D ... ℤ² that the general solution of the functional equation f(x+y) = g(x)+h(y) for all (x, y) ∈ D with unknown functions f: Dx+y → Y, g : Dx → Y, h : Dy → Y is in the form of f(u) = a(u) + C1 + C2 for all u ∈ Dx+y, g(v) = a(v) + C1 for all v ∈ Dx, h(z) = a(z)+C2 for all z ∈ Dy where a: ℤ → Y is an additive function, C1, C2 ∈ Y are constants. [ABSTRACT FROM AUTHOR]

The Z-function is the real function given by Z (t) = e i θ (t) ζ 1 2 + i t , where ζ (s) is the Riemann zeta function, and θ (t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z (t) and its zeros. The sections of Z (t) are given by Z N (t) : = ∑ k = 1 N cos (θ (t) − ln (k) t) k for any N ∈ N . Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z (t) ≈ 2 Z N ˜ (t) (t) for N ˜ (t) = t 2 π . While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z (t) ≈ Z N (t) (t) for N (t) = t 2 , which is Spira's approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira's conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal "Symmetry" focused on qualitative studies of fractional differential equations, fractional integro-differential equations, and related topics. The editorial provides an overview of the published articles in the special issue, summarizing their contents and contributions. The articles cover a range of topics, including numerical simulations of fractional differential equations, analysis of cancer models using stem cells and chemotherapy, analysis of image segmentation algorithms, and control of bioreactors using fractional order models. The editorial concludes by listing the authors and titles of the contributions. [Extracted from the article]

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials. [ABSTRACT FROM AUTHOR]

This paper investigates second-order functional dynamic equations with mixed nonlinearities on an arbitrary unbounded above-time scale, T. We will use a unified time scale approach and the well-known Riccati technique to derive oscillation criteria of the Nehari-type for second-order dynamic equations. The findings demonstrate a significant improvement in the literature on dynamic equations. The symmetry is beneficial and influential in defining the right style of study for the qualitative behavior of solutions to dynamic equations. We include an example to demonstrate the significance of our results. [ABSTRACT FROM AUTHOR]

We present some remarks on [Carpathian J. Math. 39 (2023), No 2, 541-551] in order to obtain a unique non trivial solution. [ABSTRACT FROM AUTHOR]

Given a rational number α ≠ 2, we establish a criterion for the existence of the general solution of an alternative quadratic functional equation of the form f (x y) + f (x y-1) = 2f (x) + 2f (y) or f (x y) + f (x y-1) = αf (x) + 2f (y), where f is a mapping from an abelian group (...) to a uniquely divisible abelian group (...). We also find the general solution in the cases when G is a 6-divisible abelian group and G is a cyclic group. [ABSTRACT FROM AUTHOR]

We define L-series of weakly holomorphic quasimodular forms and we derive functional equations of those L-series. We also prove a converse theorem for weakly holomorphic quasimodular forms. [ABSTRACT FROM AUTHOR]

In this work, we discuss the existence and uniqueness of solutions to the initial value problem for perturbed functional fractional q-difference equations involving q-derivative of the Caputo sense. By applying Banach contraction principle and Burton and Kirk's fixed point theorems. Further, we present the Ulam-Hyers and Ulam-Hyers-Rassias stabilities results by using direct analysis methods. Finally, we give two examples illustrating of the results. [ABSTRACT FROM AUTHOR]

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model’s parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g (x σ (y)) = g (x) g (y) + f (x) f (y) , x , y ∈ S , f (x σ (y)) = f (x) g (y) + f (y) g (x) , x , y ∈ S , and f (x σ (y)) = f (x) g (y) - f (y) g (x) , x , y ∈ S , where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications. [ABSTRACT FROM AUTHOR]

We calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximizes throughput. Our method works with many observables and can be used as a blueprint for further analysis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new integral transform, namely, Sumudu transform and we apply the transform to investigate the Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of second order linear differential equations. [ABSTRACT FROM AUTHOR]

The relationship is studied here between the 3D incompressible Brinkman–Forchheimer problem with delay and its generalized steady state. First, with some restrictive condition on the delay term, the global well-posedness of 3D Brinkman–Forchheimer problem and its steady state problem are obtained by compactness method and Brouwer fixed point method respectively. Then the global L p (2 ≤ p < ∞) decay estimates are established for weak solution of non-autonomous Brinkman–Forchheimer equations with delay by using a retarded integral inequality. The global decay estimates can be proved for strong solution as well. Finally, the exponential stability property is investigated for weak solution of the 3D non-autonomous Brinkman–Forchheimer problem by a direct approach and also for the autonomous system by using a retarded integral inequality. Furthermore, the Razumikhin approach is utilized to achieve the asymptotic stability for strong solution of autonomous system under a relaxed restriction. [ABSTRACT FROM AUTHOR]

This paper mainly studies new additive functional equations f(x + y/n0) = f(x) + f(y)/n0 f(λx +(λy/n0) = f(λx) + f(λy/n0 where n0 is a positive integer with n0 ≠ 1, and λ > 0 is a fixed constant. We prove the Hyers-Ulam stability for the above additive functional equations by using fixed point method in complex normed spaces. Further, we establish some new results about partial multipliers related to additive functional equations in complex Banach *-algebras. [ABSTRACT FROM AUTHOR]

We consider a transient Brownian motion reflected obliquely in a two-dimensional wedge. A precise asymptotic expansion of Green's functions is found in all directions. To this end, we first determine a kernel functional equation connecting the Laplace transforms of the Green's functions. We then extend the Laplace transforms analytically and study its singularities. We obtain the asymptotics applying the saddle point method to the inverse Laplace transform on the Riemann surface generated by the kernel. [ABSTRACT FROM AUTHOR]

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]

In this paper, we focus on studying the non-probability Gibbs measures for a Hard-Core (HC) model on a Cayley tree of order k ≥ 2, where the set of integers ℤ is the set of spin values. It is well known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function (with strictly positive coordinates) defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalizable boundary law corresponds to a Gibbs measure. However, a non-normalizable boundary law can define the gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC model on a Cayley tree of any order k ≥ 2. [ABSTRACT FROM AUTHOR]

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The spatial Three-Dimensional (3D) edge network is one of the typical rank-lossless networks. The current network adjustment usually uses Least Squares (LS) algorithm, which has the complexity of linearization derivation, computational volume and other problems. It is based on high-precision ranging values. This study aims to minimize the sum of the difference between the inverse distance of the control point coordinates and the observation distance, the composition of the non-linear system of equations to build a functional model. Considering the advantages of the intelligent optimization algorithm in the non-linear equation system solving method, such as no demand derivation and simple formula derivation, the Particle Swarm Optimization (PSO) algorithm is introduced and the improved PSO algorithm is constructed; at the same time, the improved Gauss-Newton (G-N) algorithm is studied for the calculation of the 3D control network adjustment function model to solve the problems of computational volume and poor convergence performance of the algorithm with large residuals of the unknown parameters. The results show that the improved PSO algorithm and the improved G-N algorithm can guarantee the accuracy of the solution results. Compared with the traditional PSO algorithm, the improved PSO algorithm has a faster optimization speed. When the residuals of the unknown parameters are too large, the improved G-N algorithm is more stable than the improved PSO algorithm, which not only provides a new way to solve the spatial 3D network, but also provides theoretical support for the establishment of the spatial 3D network. [ABSTRACT FROM AUTHOR]

Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of C 1 solutions of a class of iterative functional equations which involves iterates of the unknown functions and a nonlinear term. [ABSTRACT FROM AUTHOR]

The note is concerned with the functional equation λ 1 H 1 (f (x)) + λ 2 H 2 (f 2 (x)) + ⋯ + λ n H n (f n (x)) = F (x) , which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability. [ABSTRACT FROM AUTHOR]

How to free a road from vehicle traffic as efficiently as possible and in a given time, in order to allow for example the passage of emergency vehicles? We are interested in this question which we reformulate as an optimal control problem. We consider a macroscopic road traffic model on networks, semi-discretized in space and decide to give ourselves the possibility to control the flow at junctions. Our target is to smooth the traffic along a given path within a fixed time. A parsimony constraint is imposed on the controls, in order to ensure that the optimal strategies are feasible in practice. We perform an analysis of the resulting optimal control problem, proving the existence of an optimal control and deriving optimality conditions, which we rewrite as a single functional equation. We then use this formulation to derive a new mixed algorithm interpreting it as a mix between two methods: a descent method combined with a fixed point method allowing global perturbations. We verify with numerical experiments the efficiency of this method on examples of graphs, first simple, then more complex. We highlight the efficiency of our approach by comparing it to standard methods. We propose an open source code implementing this approach in the Julia language. [ABSTRACT FROM AUTHOR]

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin's Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner's 6j symbols, in details. The later are rather well known in the theory of 3 D indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state. [ABSTRACT FROM AUTHOR]

For any N ≥ 3 , we study invariant measures of the dynamics of N hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold M of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all N hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable N-body scattering maps which generate a billiard dynamics on M admitting a canonical weighted Hausdorff measure on M (that we term the Liouville measure on M ) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linearN-body scattering map yields a billiard dynamics which admits the Liouville measure on M as an invariant measure. [ABSTRACT FROM AUTHOR]

We examine the fundamental Kazakov–Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on a graph similar to that in the small coupling region. We estimate the phase structure of the FKM model in both the small and large coupling regions by comparing it with the Gross–Witten–Wadia model. We further validate the theoretical analysis through detailed numerical simulations. [ABSTRACT FROM AUTHOR]

Some properties of the constant in asymptotic expansion of iterates of a cubic function were investigated. This paper analyzed the monotonicity, differentiability of the constant with respect to the initial value and the functional equation that is satisfied. [ABSTRACT FROM AUTHOR]

In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable. [ABSTRACT FROM AUTHOR]

In the process of solving the problems of synthesis of antennas with given amplitude characteristics, it is often necessary to apply nonlinear spectral theory. The practical statement of the problems of synthesis is based on the use of the amplitudes of desired functions. A standard procedure of optimization is to deduce the Euler equation for the functional used as an optimization criterion. As a rule, the indicated equation is integral and nonlinear due to the specific features of the problem. This equation is characterized by the nonuniqueness of solutions and their branching or bifurcations. Finding branched solutions requires the investigation of the corresponding homogeneous equations and the corresponding eigenvalue problem. The analysis of the problem makes it possible to determine the set of points of the input parameters at which the corresponding eigenvalues are equal to one, which determines the points of branching of the solutions. These calculations demonstrate the ability of the proposed approach to numerically determine the solutions of nonlinear equations, their properties, and branching points of the solutions with relatively low computational costs. [ABSTRACT FROM AUTHOR]

For the inner product space, we have Appolonius' identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers' direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces. [ABSTRACT FROM AUTHOR]

The motion of the polymer center of mass (CM) is driven by two stochastic terms that are Gaussian white noise generated by standard thermal stirring and chain polymerization processes, respectively. It can be described by the Langevin equation and is Brownian non-Gaussian by calculating the kurtosis. We derive the forward Fokker–Planck equation governing the joint distribution of the motion of CM and the chain polymerization process. The backward Fokker–Planck equation governing only the probability density function (PDF) of CM position for a given number of monomers is also derived. We derive the forward and backward Feynman–Kac equations for the functional distribution of the motion of the CM, respectively, and present some of their applications, which are validated by a deep learning method based on backward stochastic differential equations (BSDEs), i.e. the deep BSDE method. [ABSTRACT FROM AUTHOR]