Recent Papers

ON A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH ALGEBRAIC MOVING SINGULAR POINTS

GENERALIZED APPELL FUNCTIONS WITH FUNCTIONAL COEFFICIENTS

The paper studies generalized Appell functions on a fundamental polygon of a closed Riemann surface, replacing constant transition coefficients by functional boundary data for meromorphic solutions of a Vekua-type generalized analytic equation. Using boundary value problem methods, Cauchy-type integ…

ON THE BEST QUADRATURE FORMULAS

The paper studies optimal quadrature formulas for classes of functions on the unit interval whose rth derivative is bounded in an Lq norm, using formulas involving function values and derivatives up to order r minus 2 at fixed numbers of nodes. For even r, it reduces the minimization of the sharp qu…

ON THE THEORY OF BOUNDARY VALUE PROBLEMS FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH TYPE DEGENERATION ON THE BOUNDARY OF THE DOMAIN

ON A CLASS OF DUAL INTEGRAL EQUATIONS

On the First Transcendental Integrals of the Equations of Motion of a Heavy Rigid Body About a Fixed Point

The paper studies time-independent transcendental first integrals of the equations of motion of a heavy rigid body about a fixed point, a problem for which explicit integrals are generally unavailable outside classical special cases. Using a small-parameter transformation and earlier results on peri…

ON THE EMBEDDING OF DISCRETE FIELDS IN CONNECTED ONES

The paper proves that every discrete field can be embedded in a connected topological field. Starting from an arbitrary field, it constructs a rational function field in variables indexed by the unit interval and defines a norm using finite equivalence relations on the interval and associated length…

EXPANSIONS OF MATRIX FUNCTIONS IN SOLUTIONS OF A SPECTRAL PROBLEM

Zum 60. Geburtstag von Boris Pavlovic Demidovic

GENERAL ADDITIONAL CONDITIONS FOR A BOUNDARY-VALUE PROBLEM WITH OBLIQUE DERIVATIVE

The paper characterizes additional conditions needed to make the heat equation in a plane domain with an oblique derivative boundary condition well posed when the oblique field has exceptional boundary points where its normal component changes sign. Using the language of positive contraction semigro…

ON THE GEOMETRIC LOCATION AND ESTIMATION OF THE GROWTH OF SOLUTIONS OF PERTURBED SYSTEMS

ON SOME CONDITIONS FOR MUTUAL CONVERGENCE OF FUNCTIONS

This paper studies mutual convergence of two principal functions within finite systems of auxiliary functions subject to uniform increment bounds and sublinear error terms. It defines comparison sets and extremal auxiliary quantities that measure how closely auxiliary functions approximate the princ…

On generalized-function solutions of ordinary differential equations with polynomial coefficients

This note studies solutions, in the space of generalized functions \((S_0^\beta)'\) with \(\beta>1\), of a class of ordinary differential equations with polynomial coefficients of the form \(\sum_{p=0}^{n}\sum_{k=0}^{m_p} a_{kp}x^{k+p}y^{(k)}=0\). It first characterizes solutions concentrated at the…

ON THE APPROXIMATION OF A PROBLEM OF OPTIMAL CONTROL IN A SYSTEM WITH AFTEREFFECT

This paper studies finite-dimensional approximation of an infinite-horizon optimal control problem for linear systems with aftereffect, where the control must stabilize the system while minimizing a quadratic integral cost. Krasovskii formulates approximating systems and cost functionals, specifies …

INTEGRABILITY CONDITIONS IN FINITE FORM FOR CERTAIN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

ON A PROBLEM OF BIOPHYSICS

The paper studies a mathematical problem arising in biophysics, namely diffusion and transformation of substances inside and outside a living cell with membrane permeability conditions. It formulates a more general system of parabolic equations in two adjoining domains with special discontinuities i…

RESONANCES IN MULTIFREQUENCY OSCILLATIONS

The paper studies resonances in multifrequency oscillatory systems with slow variables and fast phase variables, defining resonance surfaces, resonance phases, and a complexity index for states satisfying independent integer resonance relations. It shows that, by an integer change of phase variables…

ON THE QUESTION OF CYCLING IN LINEAR PROGRAMMING

The paper addresses cycling in the simplex method for linear programming, noting that existing anti-cycling procedures increase the number of divisions relative to the nondegenerate case. It introduces a lexicographic polynomial framework for simplex tableaux and proves a main theorem showing how, u…

APPROXIMATE SOLUTION OF THE WIENER–HOPF EQUATION IN AN EXCEPTIONAL CASE

INVESTIGATION OF A CLASS OF NONLINEAR INTEGRAL EQUATIONS

ON THE SOLVABILITY OF THE BASIC BOUNDARY-VALUE PROBLEM FOR AN EQUATION OF PARABOLIC TYPE WITH DEGENERATING PRINCIPAL PART

The paper studies the main homogeneous boundary-value problem for a parabolic equation whose principal part degenerates on part of the boundary, in a cylindrical domain with a planar cross-section satisfying geometric and elliptic solvability conditions. Weighted Sobolev spaces adapted to the degene…

EXISTENCE OF A SOLUTION OF A BOUNDARY-VALUE PROBLEM FOR AN ORDINARY NONLINEAR DIFFERENTIAL EQUATION OF ORDER \(n\)

The paper studies existence of solutions for a two-point boundary-value problem for a nonlinear ordinary differential equation of order n, with initial conditions imposed at one endpoint up to order n minus 2 and one derivative condition at the other endpoint. It proves an existence theorem under a …

ON LIMITS OF MOLECULAR MEASURES IN SOME SPACES

The paper studies when molecular measures, finite linear combinations of point masses, approximate broader classes of measures under locally convex topologies defined by families of test functions. It formulates a general duality between a free linear space on a set and a function space, then gives …

LETTER TO THE EDITOR

This note responds to a critical remark in K. S. Sibirskii’s article on the number of limit cycles near a singular point, specifically concerning Bautin’s earlier work on focus and center type equilibria. Bautin argues that the remark is incorrect because the relevant divisibility condition for the …

ON THE COMPLETENESS OF A SYSTEM OF ANALYTIC FUNCTIONS \(\{z^n F^{(n)}(\lambda_n z)\}\)

This paper studies completeness of systems of analytic functions of the form z^n F^(n)(lambda_n z), where F is an entire function and the moduli of the complex parameters lambda_n increase to infinity. Using the Abel-Goncharov interpolation formula, the authors derive conditions under which the inte…

FOURIER SERIES IN EIGENFUNCTIONS OF MULTIDIMENSIONAL DOMAINS, DIVERGING ALMOST EVERYWHERE

The paper studies almost everywhere divergence of Fourier series in eigenfunctions of the Laplace operator for functions on multidimensional domains, focusing on the ball in dimensions at least two with homogeneous Dirichlet or Neumann boundary conditions. Using spherically symmetric eigenfunctions …

ON THE EXPRESSION OF A FUNCTION THROUGH ITS INTEGRALS OVER ELLIPSOIDS

The paper studies the reconstruction of a smooth compactly supported function in odd-dimensional Euclidean space from its integrals over ellipsoids determined by a positive definite quadratic form. Using generalized functions, convolution identities, and Fourier transforms involving Bessel functions…

ANISOTROPIC BESSEL POTENTIALS.

The paper studies anisotropic Sobolev spaces with fractional Bessel derivatives in different coordinate directions and introduces corresponding anisotropic Bessel potentials defined through Fourier multipliers. It proves that these potentials of functions in \(L_p\) coincide with the spaces \(L_p^{(…

ON THE REPRESENTATION OF CUSP FORMS BY THETA SERIES

This note studies representations of cusp forms for the group of prime level by quaternary theta series, refining Eichler’s results on Hecke’s conjecture. For cusp forms of type with trivial character that are simultaneous eigenfunctions of the Hecke operators, the argument compares the Hecke repres…

STUDY OF A NEIGHBORHOOD OF A SINGULAR POINT OF A MULTIDIMENSIONAL DIFFERENTIAL EQUATION IN THE ANALYTIC CASE

This paper studies the local behavior near a singular point of a completely solvable multidimensional first order differential equation with analytic nonlinear terms. It extends classical one-dimensional linearization results of Poincaré, Picard, and Dulac by deriving conditions under which a formal…

ON THE SIGNATURES OF TOEPLITZ FORMS

This note extends Jacobi’s rule for determining the numbers of positive and negative squares of a Hermitian form to arbitrary finite Hermitian Toeplitz forms, including cases where leading principal minors vanish. It analyzes the structure of zero blocks in the sequence of Toeplitz determinants, pro…

ON THE APPROXIMATION AND CONVERGENCE OF THE GRID METHOD IN ELLIPTIC PROBLEMS

This paper studies approximation by piecewise linear extensions of grid functions on simplicial complexes and applies the results to grid methods for elliptic boundary value problems in n-dimensional domains. It constructs families of complexes adapted to smooth boundaries and proves estimates for a…

ON PSEUDODIFFERENTIAL OPERATORS OF PRINCIPAL TYPE

This note studies pseudodifferential operators on smooth manifolds for which Sobolev estimates of subelliptic type hold with loss parameter between 0 and 1. It characterizes such operators, termed operators of principal type, by conditions depending only on the principal symbol, reducing the verific…

ON THE INTERPRETATION OF LOGICAL FORMULAS BY MEANS OF FINITARY PROBLEMS

The paper studies the interpretation of logical formulas by finitary problems and gives algebraic criteria for finite universal validity. It represents reduced systems of finitary problems by closed families in a Boolean algebra with closure, obtains corresponding Brouwer algebras, and proves that a…

PERIODIC NILPOTENT LINEAR GROUPS OVER THE FIELD OF RATIONAL NUMBERS

This note studies periodic nilpotent subgroups of the full linear group over the field of rational numbers. It first classifies irreducible such subgroups, showing that they can occur only in degrees equal to Euler’s function of an integer and describing maximal irreducible cases by Kronecker produc…

DETERMINATION OF A DIRAC SYSTEM FROM THE SCATTERING PHASE

The paper addresses the inverse scattering problem for a canonical half-line Dirac system with real decaying coefficients and boundary condition at the origin. It introduces scattering data consisting of the scattering phase, discrete eigenvalues in the spectral gap, and corresponding norming consta…

AN ANALOGUE OF STONE’S THEOREM IN A SPACE WITH AN INDEFINITE METRIC

The paper formulates an analogue of Stone’s theorem for spaces with an indefinite metric, denoted by \(\Pi_\varkappa\), and for strongly continuous one-parameter groups of \(J\)-unitary operators. It proves that the generator of such a group has the form \(iH\), where \(H\) is \(J\)-self-adjoint and…

A BOUNDARY-VALUE PROBLEM FOR AN EQUATION OF MIXED TYPE WITH TWO LINES OF DEGENERATION

The paper studies a boundary value problem for the mixed type equation y(y minus 1)u_xx plus u_yy equals zero in a specially constructed finite domain containing intervals of both degeneration lines y equals 0 and y equals 1. It formulates admissible singular behavior and boundary data on elliptic a…

THE SAMPLE MEAN AS AN ESTIMATE OF A SHIFT PARAMETER

This paper studies when the sample mean is an admissible or optimal unbiased estimator of a shift parameter in families of one-dimensional distributions with finite moments. It introduces nested classes of unbiased estimators obtained by adding to the sample mean polynomials of degree at most k in t…

SUBGROUPS OF PERIODIC GROUPS

The paper studies infinite periodic groups in which any two elements generate a finite subgroup, termed F2-groups, a class containing all locally finite groups. Using normalizer criteria, conjugacy class arguments, and finite group results including the Feit-Thompson theorem, it proves that every F2…

EQUIAFFINE GEOMETRY OF A DUAL FIELD AND ITS APPLICATION TO THE THEORY OF STEADY FLUID FLOW

This paper develops an equiaffine geometric description of a dual field in three-dimensional space, consisting of a vector and a covector with unit contraction, using Cartan’s exterior forms. It derives invariant vectors, covectors, and tensors associated with such fields and relates them to line co…

THE TRICOMI PROBLEM FOR AN ELLIPTIC-HYPERBOLIC EULER—POISSON—DARBOUX EQUATION

The paper studies a Tricomi-type boundary value problem for an elliptic-hyperbolic Euler-Poisson-Darboux equation and for a related equation reducible to it by a power substitution. Under stated restrictions on the parameters, the problem is transformed by a one-to-one change of variables into the s…

ON LOGIC OF HIGHER LEVELS

This note strengthens earlier results on logics of higher levels, focusing on definability, spectra of formulas, and the expressive limits of finite and transfinite levels of higher-order languages. It proves that for each natural level there are formulas not equivalent to any set of lower-level for…

THE METHOD OF DIFFERENTIAL DESCENT FOR SOLVING MULTIDIMENSIONAL VARIATIONAL PROBLEMS

This paper studies differential descent trajectories for minimizing functionals and solving nonlinear operator equations in Hilbert spaces, with extensions indicated for Hilbert and Banach manifolds. It establishes local convergence criteria based on inequalities relating a functional or residual no…

COMPARISON THEOREMS FOR VOLTERRA EQUATIONS WITH RETARDED ARGUMENT

ON THE APPROXIMATION OF FUNCTIONS BY FOURIER–JACOBI SUMS

This note studies uniform approximation of functions by partial sums of Fourier series in orthonormal Jacobi polynomials on the interval with Jacobi weight. For functions whose r-th derivative satisfies a Lipschitz condition of order gamma, it derives upper estimates for the supremum norm of the Fou…

A CRITERION FOR THE DISCRETENESS OF THE SPECTRUM OF A SELF-ADJOINT SYSTEM OF FIRST-ORDER DIFFERENTIAL EQUATIONS WITH SLOWLY VARYING COEFFICIENTS

This paper studies the spectrum of a self-adjoint first-order differential system on the real line with a constant diagonal principal part and continuous Hermitian, slowly varying coefficients, including a form related to the Dirac operator considered by Titchmarsh. Using splitting principles and lo…

AN ITERATIVE METHOD FOR SOLVING A CONVEX PROGRAMMING PROBLEM

The paper develops an iterative method for minimizing a linear functional over a closed convex set, including sets defined by systems of smooth convex inequalities. It introduces Fejér and convergent Fejér operators, gives constructions of such operators from defining convex functions and gradients,…

On the initial direction of the crystallization process when immersing a plate in a melt

This paper analyzes an idealized Stefan problem for the initial phase changes that occur when an infinite plate below its melting temperature is immersed in a stirred melt above its crystallization temperature. Three possible initial regimes are formulated mathematically: simultaneous crystallizatio…

ON CONDITIONS FOR THE SOLVABILITY OF THE CHAPLYGIN PROBLEM

The paper studies solvability conditions for the Chaplygin comparison problem for differential equations in a partially ordered functional setting. It gives a necessary condition, and in the linear case a sufficient one, in terms of preservation of negativity for solutions of an adjoint variational …