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For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach to calculating the coefficients in the procedure of quantum averaging is proposed. The procedure of quantum averaging is transferred to the space of the graded algebra of symbols by using twisted product introduced in the paper. As a result, the averaged Hamiltonian is represented as a function of generators of the quantum symmetry algebra of the harmonic part of the Hamiltonian. The proposed method is applied to the spectral problem for the Hamiltonian of the cylindrical Penning trap. [ABSTRACT FROM AUTHOR]

The paper deals with the solvability analysis of boundary value problems for degenerate higher-order quasi-hyperbolic equations. The problems in question have the specific feature that the manifolds on which the equations characteristically degenerate are not freed from carrying boundary data. The aim of this paper is to prove the existence and uniqueness of regular solutions of the problems under study, that is, solutions all of whose generalized derivatives occurring in the corresponding equations exist as generalized derivatives in the sense of Sobolev. [ABSTRACT FROM AUTHOR]

We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in [4]. In addition, we give an answer to a question posed by the authors. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence and multiplicity of positive solutions of the boundary value problem for the fourth-order semipositone nonlinear Euler–Bernoulli beam equation where and are constants, is a parameter, and is a function satisfying for some positive constant ; here . The paper is concentrated on applications of the Green's function of the above problem to the derivation of the existence and multiplicity results for the positive solutions. One example is also given to demonstrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the subelliptic equation with weight where , , and is a subelliptic operator of the form Here is the homogeneous dimension on associated with the operator , and the , , satisfy some general hypotheses. Our purpose is to establish the nonexistence of nontrivial stable solutions for [ABSTRACT FROM AUTHOR]

The paper deals with a general problem of convex stochastic optimization in a space of small dimension (for example, 100 variables). It is known that for deterministic problems of convex optimization in small dimensions, the methods of centers of gravity type (for example, Vaidya's method) provide the best convergence. For stochastic optimization problems, the question of the possibility of applying Vaidya's method can be reduced to the question of how it accumulates inaccuracies in the subgradient. A recent result of the authors stating that there is no accumulation of inaccuracies at the iterations of Vaidya's method allows the authors to propose its analog for solving stochastic optimization problems. The main technique is to replace the subgradient in Vaidya's method by its batched analogue (the arithmetic mean of stochastic subgradients). In the present paper, this plan is implemented, which results in an efficient method (under conditions of the possibility of parallel calculations with batching) for solving problems of convex stochastic optimization in spaces of small dimensions. The work of the algorithm is illustrated by a numerical experiment. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the uniqueness and certain qualitative properties of the solution of a class of integral equations with sum-difference kernel on the positive half-line and with a convex nonlinearity. This class of equations arises in a particular case in the dynamical theory of -adic closed-open strings for the scalar field of tachyons. Such equations also play a very important role in the study of the existence and uniqueness of solutions of nonlinear integral equations in the mathematical theory of the geographical distribution of an epidemic within the framework of the Diekmann–Kaper model. We prove the uniqueness theorem for the solution of the equation under consideration for a class of nonnegative (nonzero) and bounded functions on , thereby obtaining a definitive solution of Vladimirov's open problem on the uniqueness of rolling solutions of nonlinear -adic equations. Under an additional constraint on the kernel of the equation, we also prove that the solution is a concave function on whose derivative belongs to the space . At the end of the paper, we give specific model equations from the above-mentioned applications, to which our results are applied. [ABSTRACT FROM AUTHOR]

We present an explicit formula for calculating the volume of an arbitrary hyperbolic 4-simplex in terms of the coordinates of its vertices; by this formula, the volume can be expressed in terms of one-dimensional integrals of real-valued integrands over closed intervals of the real line. In addition, it is proved in the paper that the volume of a hyperbolic 5-simplex cannot be expressed as the double integral of an elementary function of the coordinates of its vertices (of edge lengths). [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of locally dominated orthogonally additive operators on lattice-normed spaces (LNS). In the first part of the paper, sufficient conditions for the existence of a local exact majorant of a locally dominated operator and formulas for its calculation are given. The second part shows that the -compactness of a dominated orthogonally additive operator acting from a decomposable lattice-normed space to a Banach space with mixed norm implies the -compactness of its exact majorant. [ABSTRACT FROM AUTHOR]

The paper deals with a significant extension of the recently proposed class of relatively strongly convex optimization problems in spaces of large dimension. In the present paper, we introduce an analog of the concept of relative strong convexity for variational inequalities (relative strong monotonicity) and study estimates for the rate of convergence of some numerical first-order methods for problems of this type. The paper discusses two classes of variational inequalities depending on the conditions related to the smoothness of the operator. The first of these classes of problems contains relatively bounded operators, and the second, operators with an analog of the Lipschitz condition (known as relative smoothness). For variational inequalities with relatively bounded and relatively strongly monotone operators, a version of the subgradient method is studied and an optimal estimate for the rate of convergence is justified. For problems with relatively smooth and relatively strongly monotone operators, we prove the linear rate of convergence of an algorithm with a special organization of the restart procedure of a mirror prox method for variational inequalities with monotone operators. [ABSTRACT FROM AUTHOR]

This paper deals with the following singular problem: where () are a bounded smooth domain, is positively homogeneous of degree , , for almost every , , , , , and . Under appropriate conditions on the function , we establish the existence of multiple solutions by using the Nehari manifold method. [ABSTRACT FROM AUTHOR]

The vertices of the commuting graph of a semigroup S are the noncentral elements of this semigroup, and its edges join all pairs of elements g, h that satisfy the relation gh = hg. The paper presents a proof of the fact that the diameter of the commuting graph of the semigroup of real matrices of order n ≥ 3 is equal to 4. A survey of results in that subject matter is presented, and several open problems are formulated. [ABSTRACT FROM AUTHOR]

In this paper, we study bounds for the total variation distance between distributions of second order polynomials in normal random variables provided that they essentially depend on at least three variables. Moreover, we partially extend some recent bounds for the Kolmogorov distance between the distributions of norms of Gaussian random vectors to the case of the total variation distance. [ABSTRACT FROM AUTHOR]

This paper presents some generalizations of a Thébault conjecture, provides an analog of the Thébault conjecture for the -simplex, and also solves a conjecture in a 2022 paper by the authors by using linear algebra. [ABSTRACT FROM AUTHOR]

This paper answers the question of V. M. Buchstaber on the growth function in case of certain -valued group. This question is in close relation to specific discrete integrable systems. In the present paper, we find a specific formula for the growth function in the case of prime . We also prove a polynomial asymptotic estimate of the growth function in the general case. At the end, we pose new conjectures and questions regarding growth functions. [ABSTRACT FROM AUTHOR]

The paper investigates an inverse problem for finding the order of the fractional derivative in the sense of Gerasimov–Caputo in the wave equation with an arbitrary positive self-adjoint operator having a discrete spectrum. By means of the classical Fourier method, it is proved that the value of the projection of the solution onto some eigenfunction at a fixed time uniquely restores the order of the derivative. Several examples of the operator are discussed, including a linear system of fractional differential equations, fractional Sturm–Liouville operators, and many others. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of a solution of a problem involving Leray–Lions type operators in Sobolev spaces with variable exponent. The proofs of our main results combine variational methods with energy estimates and Ekeland's variational principle. This paper improves and generalizes previous ones in literature, more precisely those of K. Kefi and V. Rãdulescu (Z. Angew. Math. Phys. 68, 80 (2017)) and M. M Boureanu (Discrete Contin. Dyn. Syst. Ser S, 12 (2) : 231-243 (2019)). [ABSTRACT FROM AUTHOR]

A multi-agent system is a system of multiple interacting entities, known as intelligent agents, who possibly have different information and/or diverging interests. The agents could be robots, humans, or human teams. Opinion dynamics is a process of individual opinions in which a group of interacting agents continuously fuse their opinions on the same issue based on established rules to reach a consensus at the final stage. Historically, the idea of reaching consensus through repeated averaging was introduced by DeGroot for a structured time-invariant and synchronous environment. Since then, consensus, which is the most ubiquitous phenomenon of multi-agent systems, has become popular in various scientific fields such as biology, physics, control engineering, and social science. To some extent, a Krause mean process is a general model of opinion sharing dynamics in which the opinions are represented by vectors. In this paper, we represent opinion sharing dynamics by means of Krause mean processes generated by diagonally primitive cubic doubly stochastic matrices, and then we establish a consensus in the multi-agent system. [ABSTRACT FROM AUTHOR]

The paper deals with maximal operators associated with a family of singular hypersurfaces in the space . The boundedness of these operators in the space of integrable functions is proved for the case in which the singular hypersurfaces are given by parametric equations. The boundedness exponent of maximal operators for spaces of integrable functions is found. [ABSTRACT FROM AUTHOR]

In this paper, we study the functional that arises in numerous economic applications, in particular, in the monopolist problem. A special feature of these problems is that the domains of such functionals are nonclassical (in our case, increasing convex functions). We use an appropriate minimax theorem to prove the duality relation for . In particular, an important corollary is obtained stating that the dual functional (defined on a space of measures and known as the "Beckmann functional) attains its minimum. The present approach also provides simpler proofs of some previously known results. [ABSTRACT FROM AUTHOR]

The paper deals with the study of the model of point vortices proposed by the German scientist Hermann Helmholtz. Necessary and sufficient conditions for the existence of infinitely many nonequivalent stationary configurations are found for a system consisting of two point vortices of intensity and an arbitrary number of point vortices of intensity . A classification of such configurations is carried out. For the first time, a connection is discovered between the negative Diophantine Pell equation and stationary configurations of point vortices on the plane. [ABSTRACT FROM AUTHOR]

Even Poincaré had constructed an example which implies the existence of an irrational rotation of the circle and a function continuous on it with zero mean for which the Birkhoff sums at some points tend to infinity as the number of iterations grows. The strict ergodicity in this case is a natural constraint on the growth rate of Birkhoff sums: the sequence of Birkhoff means uniformly tends to zero on the circle. The paper shows that any prescribed admissible rate of growth of Birkhoff sums within the ergodic theorem can be realized, and the set of points at which the sums grow with a given speed is massive: it has the Hausdorff dimension one. [ABSTRACT FROM AUTHOR]

Let be an Abelian group. A class is said to be a -class if, for any groups , a group isomorphism of and implies an isomorphism of the groups and . In the paper, conditions on a completely decomposable Abelian group are investigated under which a class of certain completely decomposable torsion-free Abelian groups is a -class. [ABSTRACT FROM AUTHOR]

Michael Somos discovered several theta-function identities of various levels by computer and offered no proof for them. These identities highly resemble some of Ramanujan's identities. The main focus of this paper is to prove some of these theta-function identities, in particular those of level 6 that have been discovered using computational searches. Some of the the Somos identities that we are discussing in this paper cannot be expressed in the form of P −−Q type. Furthermore, we establish certain colored partition identities for them. [ABSTRACT FROM AUTHOR]

On a uniform grid on the real axis , we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space of two-way real sequences with the least value of the norm of a linear formally s This problem is considered for the class of sequences for which the generalized finite differences of order corresponding to the operator are bounded in the space . In this paper, the least value of the norm is calculated exactly if the grid step and the averaging step of the function to be interpolated in the mean are related by the inequalities . The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the -times differentiation operator . [ABSTRACT FROM AUTHOR]

We study the solvability of initial-boundary value problems for second-order hyperbolic and parabolic equations with a boundary condition that integrally connects the values of the solution on the lateral boundary with the values of the solution inside the domain. To study such problems, it was previously established that their solvability is ensured by the bijectivity of a certain Fredholm operator constructed from an integral condition. In this paper, we show that the condition of predecessors is not required for the existence and uniqueness of regular solutions (solutions with all derivatives generalized in the sense of Sobolev that are contained in the equation) of integral analogues of the first initial-boundary value problem for second-order hyperbolic and parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we extend the -like iteration method for a new generalized absolute value equation and obtain its convergence properties. What is more, the optimal parameter of the -like iteration is obtained. The result of numerical experiments shows that the proposed method is reliable and feasible. [ABSTRACT FROM AUTHOR]

We present new closed-form formulas for the matrix logarithm. Our method is direct and elementary, and it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Using these results, an elegant explicit formula for the principal matrix logarithm can be obtained. Several special cases and examples are stated to illustrate the method presented in this paper. [ABSTRACT FROM AUTHOR]

The paper introduces a function defined by a power series in the probabilities of multivariate records in a sequence of independent identically distributed random vectors with continuous components. In the univariate case, this function is always the same, but in the multivariate case there exists a broad variety of such functions determined by distribution copulas. The probabilistic meaning of this function and its derivatives is presented. A calculation method using the Kendall distribution function is given. The concepts of favorableness of copulas for records, record time distribution, and mean record time (without taking into account the order) are introduced. Examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of the Hardy averaging operator between weighted variable grand Lebesgue spaces for nonincreasing functions. [ABSTRACT FROM AUTHOR]