Recent Papers

METRIZABILITY, NORMABILITY, AND MULTINORMABILITY OF CONSTRUCTIVE LOCALLY CONVEX SPACES

This note studies metrizability, normability, and multinormability for constructive uniform and locally convex spaces, using constructive interpretations of equality, entourages, metrics, seminorms, and locally convex topologies. It formulates criteria and counterexamples showing that several classi…

Sufficient Conditions for Isochronism of Canonical Systems of Two Differential Equations

Information about the Izhevsk Seminar

This chronicle note surveys the development of the Izhevsk mathematical seminar, founded in 1954 at the Izhevsk Mechanical Institute under N. V. Azbelev. It traces the seminar’s shift from integral and differential inequalities, especially Urysohn and Chaplygin type theorems, to applications in exis…

BRIEF COMMUNICATIONS

Conditions for a Center in the Case of Homogeneous Nonlinearities of Third Degree

SOME ASYMPTOTIC PROPERTIES OF ORTHOGONAL POLYNOMIALS IN THE COMPLEX DOMAIN

This note studies asymptotic properties of polynomials orthogonal on a closed contour in the complex plane under smoothness assumptions on the contour and the weight. The method replaces the usual orthogonalization of powers by orthogonalization of generalized Faber polynomials, introducing normal m…

Short-Wave Asymptotics of Eigenfunctions of the Helmholtz Equation

This note derives short-wave asymptotic formulas for selected eigenvalues and eigenfunctions of the Helmholtz equation in a smooth convex plane domain with Dirichlet or Neumann boundary conditions. Using the parabolic equation method, it identifies two classes of stable ray systems, those concentrat…

Estimates of $H$-condition numbers of matrices of the Petrov–Galerkin method

The paper studies conditioning in the Petrov-Galerkin approximation of linear operator equations between Hilbert spaces. It establishes a necessary and sufficient convergence criterion in terms of uniform boundedness of the inverses of matrices formed after orthonormalizing the trial and test system…

On Finding Boundary Solutions of a System of Linear Inequalities and on the Method of Equalizing Descents

The paper develops a computational procedure for finding boundary solutions of finite systems of linear inequalities of closed type. Using iterative Jordan transformations, and when necessary a one phase simplex auxiliary problem with a single artificial variable, the method determines the rank of t…

On the Application of Spectrally Equivalent Operators to the Solution of Difference Analogs of Strongly Elliptic Systems

The paper develops an iterative framework for solving finite-difference analogues of strongly elliptic systems by replacing the original discrete operator with a spectrally equivalent positive operator. It defines spectral equivalence estimates, analyzes Richardson-type iterations and Chebyshev para…

ON PARABOLIC SYSTEMS WITH COEFFICIENTS SATISFYING THE DINI CONDITION

The paper studies linear Petrovskii parabolic systems whose coefficients are continuous in the spatial variables with a modulus satisfying the Dini condition. Using Hopf’s method and, under a stronger iterated Dini condition, the Levi parametrix method, it establishes correct solvability of the Cauc…

REPRESENTATION OF THE SOLUTION OF A CERTAIN CAUCHY PROBLEM IN THE FORM OF AN INTEGRAL RESIDUE

On the Asymptotic Behavior as $t\to\infty$ of Solutions to Some Nonstationary Boundary Value Problems

The paper studies the long-time behavior of solutions to a nonstationary exterior boundary-value problem for the equation involving the polyharmonic operator with time-harmonic forcing. It establishes existence, uniqueness, smoothness, boundedness, and spatial decay estimates for generalized solutio…

ON INTEGRAL INEQUALITIES FOR A SYSTEM OF VOLTERRA EQUATIONS WHOSE RIGHT-HAND SIDE IS A DISCONTINUOUS OPERATOR

ON THE CARDINALITY OF HAUSDORFF SPACES

This note studies how the cardinality of a Hausdorff space constrains the sizes of its discrete, separated, and scattered subspaces. Using transfinite decreasing sequences of closed sets, caliber arguments, and cardinal estimates including Pospisil’s theorem, it proves that for a Hausdorff space X t…

On the Equidecomposability of Two Equicomplementary Polyhedra

This paper proves that in any homogeneous n-dimensional Riemannian space, two polyhedra that are equicomplementable are also equicomposable. The argument reduces the comparison to two congruent completed polyhedra and then, by successive finite rearrangements of pieces corresponding to the complemen…

On Integral Perturbations in Differential Equations with Rapidly Oscillating Solutions

One-Dimensional Singular Equations in Generalized Functions

This paper studies one-dimensional singular integral equations on a closed smooth contour when solutions are sought not only as functions in a basic countably normed space but also as generalized functions in its dual. It formulates solvability criteria for the nondegenerate case and for cases in wh…

On the Existence of an Interval of Stability of Motion According to the First Approximation

On uniqueness classes for solutions of the Cauchy problem and representations of positive definite kernels

The paper establishes a general uniqueness theorem for the Cauchy problem for systems of differential equations with constant coefficients, extending earlier results to systems of reduced order greater than one under exponential growth bounds involving slowly increasing functions satisfying a diverg…

POSITIVE DEFINITE FUNCTIONS ON THE GROUP OF MATRICES WITH ELEMENTS FROM A DISCRETE FIELD

Homogeneous Difference Schemes on Irregular Meshes for a Differential Equation of the Fourth Order with Discontinuous Coefficients

Local Minima of the Density of Lattice Coverings of Four-Dimensional Euclidean Space by Equal Balls

This note determines the local minima of the density of lattice coverings of four-dimensional Euclidean space by equal spheres, expressed as a function of the scalar products in a reduced Selling frame. The argument uses the simplicial L-decomposition of a lattice and proves a necessary extremality …

ON SOLVING THE DIRICHLET PROBLEM BY THE METHOD OF REFINEMENTS USING DIFFERENCES OF HIGHER ORDERS

The paper studies finite difference refinement methods using higher order differences for the Dirichlet problem for the Poisson and Laplace equations. For rectangular domains and for domains with sufficiently smooth boundaries, it derives improved representations of the refined grid solutions and sh…

Some Properties of Unconditional Bases

This note studies when the topology of a Banach space generated by a fixed sequence is determined by the numerical coefficient sequences for which the corresponding series converges. It gives a criterion, using positive inclination between spans of complementary subsequences, for a sequence to be an…

A priori $L^p$ estimates for solutions of parabolic equations

The paper develops a class of function spaces \(V_p\), defined through dyadic Fourier decompositions, for use in boundary value problems for parabolic equations. It establishes embeddings and comparison estimates with Sobolev spaces, proves trace and extension theorems, and gives multiplier and mult…

A new $D$-theorem in the theory of finite groups

This note proves a new D-theorem for finite groups concerning Hall subgroups relative to a set of primes. The main result states that if a normal subgroup has a Hall subgroup whose Sylow subgroups are cyclic, and the corresponding quotient satisfies a D-condition with an Ore-dispersive Hall subgroup…

On tensor products of $C^*$-algebras

This paper studies tensor products of \(C^*\)-algebras by introducing a norm defined through representations of the algebraic tensor product arising from commuting representations on a common Hilbert space. It shows that this norm is the largest \(C^*\)-cross norm with the standard tensor product pr…

On the Number of Limit Cycles in the Neighborhood of a Singular Point

ON THE CONSTRUCTION OF A ZERO APPROXIMATION IN SOLVING THE CAUCHY PROBLEM FOR A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS BY S. A. CHAPLYGIN’S METHOD

On the Method of Characteristic Coefficients

The paper develops a general method of characteristic coefficients for comparing distribution functions relative to a fixed continuous strictly increasing distribution function. By transforming the real line to a finite interval and using Fourier and Chebyshev expansions, it represents the differenc…

On the Algebraic Structure of the Field of Siegel Modular Functions

This paper studies the algebraic structure of fields of Siegel modular functions, with emphasis on determining when such fields fail to be rational function fields. The authors use plurigenera as birational invariants, asymptotic dimension formulas for scalar and vector-valued automorphic forms, and…

On the Problem of Minimizing a Smooth Functional under Convex Constraints

This paper studies minimization of a smooth functional on a closed bounded convex subset of a Banach space, deriving first and second order conditions for local minima under convex constraints. It defines stationary points through a variational inequality involving the gradient, proves this conditio…

ON AN ARITHMETIC FUNCTION HAVING AN APPLICATION IN CODING THEORY

The paper studies the arithmetic function K_{n,q}(z), which counts bounded integer solutions of a linear congruence and is connected with the size of certain asymmetric error-correcting codes. Using generating functions, recurrence relations for coefficient sums, and auxiliary inequalities for cycli…

On W. Blaschke’s Problem on Quasi-Translation Surfaces

This paper studies Blaschke’s problem for quasi-translation surfaces in three-dimensional quasi-elliptic space, equivalently two-parameter motions of the Euclidean plane that decompose into products of one-parameter motions in more than one way. Using the notions of quasi-conjugate and Chebyshev net…

New books

This bibliographic notice surveys recently published books and journal articles relevant to differential equations, boundary value problems, mathematical physics, integral equations, numerical methods, and stability theory. It lists monographs, lecture texts, edited proceedings, and research papers …

SYSTEMS OF CONGRUENCES AND EQUATIONS OF WARING TYPE

This paper studies solution counts for systems of congruences and Waring type equations associated with equal sums of powers, with emphasis on estimates for smaller numbers of variables than those covered directly by Vinogradov’s mean value theorem. It proves that a previously known estimate for a s…

MINIMUM PRINCIPLE OF A QUADRATIC FUNCTIONAL FOR A HYPERBOLIC EQUATION

Differential survival games with many participants

This paper studies differential pursuit and survival games in a convex planar set with multiple pursuers and evaders moving at constant speeds and able to change direction continuously. It extends earlier two-player formulations to games with several pursuers against one evader, one pursuer against …

ON THE CONVERGENCE OF TRANSFORMATIONS OF DIFFERENTIAL EQUATIONS TO NORMAL FORM

This paper studies when a formal change of variables reducing an analytic system of differential equations near a singular point to normal form is convergent. Building on earlier normal form results, it formulates conditions on the eigenvalues and on the resonant terms of the normal form, including …

On the Asymptotic Properties of the Spectral Function of Hypoelliptic Operators

This note studies the high energy asymptotic behavior of the spectral function for semibounded self-adjoint extensions of constant-coefficient hypoelliptic differential operators on domains in Euclidean space. It introduces a broad class of operators characterized by a growth condition on the model …

ON A CERTAIN CASE OF A VOLTERRA INTEGRAL EQUATION

S. N. Bernstein's interpolation process in the complex domain

The paper studies S. N. Bernstein’s modified interpolation process in the complex domain, extending a method previously considered mainly for real variables. For functions analytic in the unit disk and continuous on its closure, it proves uniform convergence inside the disk for interpolation polynom…

On the Strong Mixing Condition for Stationary Gaussian Processes

The paper studies spectral conditions associated with strong mixing for stationary Gaussian processes in discrete and continuous time, using analytic functions built from the logarithm of the spectral density and Hardy space approximation methods. For discrete time, it relates mixing to properties o…

EXTREMAL SPACES

This paper introduces extremal spaces, algebraic structures equipped with maximum and minimum operations and scalar multiplication over an extremal field, as a framework for studying extremization independently of ordinary addition. It develops basic order, independence, decomposition, and metrizati…

On Graphs of a Special Type

The paper introduces a class of regular graphs represented by vertices placed on an infinite rectangular lattice, with edges allowed only between nearest horizontal, vertical, or diagonal neighbors, motivated by problems in automated design of computing units and systems. It derives basic structural…

A priori estimates for solutions of second-order linear elliptic and parabolic equations

This paper develops a priori estimates for generalized solutions of linear second-order elliptic and parabolic equations with measurable coefficients under integrability assumptions. It introduces classes of functions satisfying energy-type inequalities and derives integral, distributional, exponent…

On the Structure of the Complete Class of Unbiased Estimators for Families of Distributions of a Special Form

This paper studies unbiased estimation for finite mixture type families of probability measures whose coefficients are nonnegative linearly independent functions of an abstract parameter. For estimable parameter functions, the authors characterize admissibility under quadratic loss by reducing the p…

ON THE REDUCTION METHOD FOR SYSTEMS OF EQUATIONS OF WIENER–HOPF TYPE

This paper extends earlier results on the reduction method from scalar equations to systems of Wiener-Hopf type equations and related operator equations. It gives necessary and sufficient conditions for finite truncated systems to be uniquely solvable and for their solutions to converge in the corre…

CLASSIFICATION OF MULTIDIMENSIONAL SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS BY THE METHOD OF LYAPUNOV FUNCTIONS