Recent Papers

Corrections to the paper On the representation of large numbers by positive ternary quadratic forms with odd relatively prime invariants (DAN, vol. 118, no. 6, 1958)

A Further Strengthening of Jackson's Theorem on Approximation of Continuous Functions by Ordinary Polynomials

This note strengthens Jackson-type theorems on approximation of continuous functions by algebraic polynomials, refining earlier results of Nikol’skii and Timan. It introduces estimates in terms of the second modulus of smoothness of the r-th derivative and proves that, for a function with continuous…

On the uniqueness of determining the shape of an attracting body from the values of its external potential

This note studies uniqueness in the inverse problem of potential theory for determining the shape of attracting bodies from their exterior potentials. Extending methods of Novikov and Sretenskii, it proves that two bounded bodies of the same constant density with piecewise smooth boundaries must coi…

On the Approximate Solution of the Polyharmonic Problem

This note presents an approximate method for the principal boundary value problem for the polyharmonic equation in a simply connected plane domain with piecewise smooth boundary. The boundary conditions are reformulated in terms of iterated Laplacians and normal derivatives, and approximate solution…

Systems of complete singular integral equations of convolution type

This note studies systems of complete singular integral equations of convolution type with matrix kernels and completely continuous perturbations, extending earlier work on scalar equations and characteristic systems. Using Fourier transforms, matrix methods, and boundary value problems for analytic…

ON THE LEVITZKI PROBLEM

The paper addresses Levitzki’s problem for the class of associative rings with a strictly supporting monomial, a class introduced by Drazin that includes rings satisfying polynomial identities and rings with the minimal condition for right ideals. It proves a technical lemma on powers of ideals gene…

Boundary Properties of Differentiable and Harmonic Functions in Domains with Corner Points

The paper studies boundary behavior of harmonic and differentiable functions in spatial domains with angular points, especially cones and rectangular boxes. Using spherical logarithmic coordinates, Fourier series and Fourier integral representations with conical functions are derived for harmonic fu…

Polynomial Operations in Some Spaces

This paper studies linear trigonometric polynomial operations between normed spaces of periodic summable functions satisfying translation invariance and density of trigonometric polynomials. It establishes a fundamental integral formula relating any such operation of order n to its action on the nth…

On the Cauchy problem for quasilinear equations

The paper studies uniqueness for generalized, possibly discontinuous, solutions of the Cauchy problem for conservative quasilinear equations. It introduces potentials defined by integral conservation relations and shows that the Cauchy problem for generalized solutions can be reduced to a Cauchy pro…

On Perfect Compact Kernels of Sets Effectively Distinct from All $\Phi$-Sets

This paper studies a restricted form of the perfect compact kernel problem in descriptive set theory using Novikov’s notion of effective distinction. For broad classes of delta s-operations, including operations producing Borel, analytic, and projective sets of specified levels, it proves that sets …

On the Existence of a Solution to a Problem of F. I. Frankl

The paper studies the existence of a solution to F. I. Frankl’s boundary value problem for a mixed elliptic hyperbolic equation with degeneracy on the line separating the two types. Under specified smoothness and geometric assumptions on the boundary curve, the problem is reduced by symmetry to an a…

Use of Fractional Powers of Operators in the Study of Fourier Series in Eigenfunctions of Differential Operators

This paper develops an operator-theoretic approach to Fourier series in eigenfunctions of positive self-adjoint differential operators, using negative fractional powers to study convergence, rates of convergence, and termwise differentiation. The authors prove abstract criteria showing that if suita…

On the Rate of Convergence of the Iterative Process for the Difference Solution of the Dirichlet Problem for the Laplace Equation

This paper analyzes the convergence rate and computational cost of iterative finite-difference methods for the Dirichlet problem for the Laplace equation. It studies a family of modified averaging operators related to the standard Richardson and Liebmann iterations, with particular attention to the …

On linear dimensionality of topological vector spaces

The paper introduces a notion of linear dimension for topological vector spaces of type F, defined through continuous linear images of closed linear subspaces, and compares it with Banach’s dimension. It proves general structural properties, including dual invariance for reflexive Banach spaces, and…

On a Necessary and Sufficient Criterion for a Function to Belong to the Sobolev Class $W_p^{(1)}$

This note extends Riesz’s one-dimensional characterization of Sobolev functions by bounded p-variation to functions of several variables. For domains where Sobolev embedding theorems apply, it defines a multidimensional p-variation through sums of coordinate difference quotients over grid partitions…

Zonal spherical functions and Laplace operators on some symmetric spaces

This paper computes radial parts of invariant differential, or Laplace, operators and the corresponding zonal spherical functions for several symmetric spaces related to complex Grassmann manifolds. For the compact complex Grassmann space and its Cartan dual, the authors express radial operators thr…

A Proof of the Theorem on the Homeomorphism of Polyhedra and Point Sets

This paper proves a homeomorphism criterion, proposed by P. S. Aleksandrov, for point sets in Euclidean spaces and for triangulated polyhedra in terms of their geometric or combinatorial spectra. It reformulates the needed notion of cofinality in the ordinary sense and shows that two sets are homeom…

Adjoint Operators of a Generalized Shift

Levitan studies adjoint operators associated with generalized translations on a differentiable manifold, emphasizing how their deeper properties depend on the choice of measure. For translations defined through Cauchy problems for second order differential operators, the paper derives the correspond…

Asymptotic representation of solutions of second order linear differential equations

The paper develops a general scheme for deriving large-parameter asymptotic representations of solutions to second-order linear differential equations, including cases with singular points. The method reduces the original equation by changes of variables to an equation without a first derivative, co…

Application of Semigroup Theory to the Study of Differential Equations in Banach Spaces

The paper studies initial value problems for linear differential equations in Banach spaces of the form du/dt + Au = f(t), where A is a closed unbounded operator. Using sectorial resolvent assumptions, it constructs a holomorphic semigroup generated by negative A, derives estimates for integer and f…

On Mixed Problems for Partial Differential Equations with Two Independent Variables

This paper studies mixed initial and boundary value problems for constant coefficient partial differential equations in two independent variables satisfying Petrovskii’s condition for uniform well posedness of the Cauchy problem. Using Laplace transforms in time and space, the problem is reduced to …

Corrections to the Article On a Certain Generalization of the Variational Method (DAN, Vol. 117, No. 3, 1957)

Study of a New Type of Two-Dimensional Integral Equations

The paper studies a two-dimensional singular integral equation with kernel involving both a Cauchy-type singularity and a pole at the origin, for complex-valued functions on a bounded plane domain. By differentiating in the Sobolev sense and integrating the resulting differential equation, the probl…

Hypersurface in a Space of Projective Connection

This paper constructs a differential geometry of hypersurfaces in multidimensional spaces of projective connection with curvature and torsion, using a group-theoretic invariant method. Starting from the Pfaffian structure equations and the adapted moving frame of a hypersurface, it derives a sequenc…

On the Estimation of Rational Trigonometric Sums

Korobov studies nontrivial bounds for rational trigonometric sums with polynomial phase modulo q, focusing on the range P = q^{1/r} where earlier estimates of Vinogradov type applied but included large exponential factors and weaker savings. The paper proves estimates replacing that coefficient by a…

On a method for counting integer points in $n$ -dimensional polytopes

The paper develops a weighted method for counting lattice points in n-dimensional polyhedra by assigning fractional interior weights to boundary lattice points according to the directions and higher-dimensional extensions entering the polyhedron. It proves that the resulting polyhedron weight is add…

A New Method for Proving Incompleteness Theorems for Systems with Carnap’s Rule and Its Application to the Question of the Relationship between Classical and Constructive Analysis

The paper develops a method for proving Gödel and Rosser style incompleteness results for formal systems equipped with restricted Carnap rules of transfinite rank. It defines a hierarchy of systems, constructs primitive recursive well-orderings and provability predicates by transfinite induction, an…

On the Dependence of the Index of an Operator of a Boundary Value Problem for an Elliptic System of Second-Order Linear Differential Equations on the Leading Coefficients

This paper studies the index of the operator associated with a boundary value problem for elliptic systems of second order linear differential equations. Building on earlier notation and constructions, it proves that the index is invariant under arbitrary changes of the leading coefficients of the d…

Mean Value Theorems and Multiplication of Summable Series

The paper studies when the Cauchy product of two series is summable by a triangular matrix method, given that the factors are summable or absolutely summable by other triangular methods. Using mean value type estimates for summability matrices, it formulates necessary and sufficient conditions under…

On the generalization of a Moore's theorem

This note generalizes Moore’s theorem on convergence multipliers for summable series from scalar Voronoi-Nörlund methods to series with terms in a Banach space and operator-valued multipliers between Banach spaces. For a normal numerical matrix satisfying a summability condition on its inverse coeff…

On the Completeness of the System $\{z^{\lambda_k}\}$ on Curves in the Complex Plane

The paper studies completeness of systems of powers \(z^{\lambda_k}\), with nonnegative integer exponents, on unbounded curves in the complex plane under weighted \(L_2\) approximation. Extending earlier polynomial completeness results, it gives sufficient conditions in terms of the density of the o…

General solutions of differential-difference equations with constant coefficients

The paper studies generalized solutions of linear differential-difference equations with constant coefficients on bounded domains, formulated in spaces of compactly supported smooth functions and their duals. Using the Fourier transform, the equation is reduced to multiplication by an exponential po…

On the Oscillation of Solutions of Third- and Fourth-Order Linear Differential Equations

The paper studies oscillation and nonoscillation of solutions to the linear equations \(y''' + p(x)y = 0\) and \(y^{\mathrm{IV}} + q(x)y = 0\), where the coefficients are continuous on a half-line. It establishes Sturm-type separation bounds for zeros under sign restrictions on the coefficients, pro…

On a Stable System of Partial Indices of the Hilbert Problem for Several Unknown Functions

This note studies the stability of partial indices for the Hilbert boundary value problem with several unknown functions, where the coefficient is a nonsingular Hölder continuous matrix on a smooth contour. The authors characterize stable systems of partial indices: stability holds exactly when the …

On the Stability of Solutions of Integral Equations under Discontinuous Variation of the Kernel

This note studies the stability and limiting behavior of Fredholm integral equations when a kernel is discontinuously perturbed, in the case where the unperturbed equation has an eigenvalue at one and hence a family of solutions. It analyzes perturbations obtained by removing a small interval from t…

Some Properties of Singular Integrals of Cauchy Type and Their Applications

This note examines limiting cases of the Plemelj-Privalov theorem for singular Cauchy type integrals on a Lyapunov curve, focusing on the endpoints of the usual Hölder scale where the classical equivalence fails. It introduces function classes defined by second-difference conditions and by best appr…

On the Boundedness of Solutions to Boundary Value Problems for First-Order Linear Elliptic Systems in the Plane

This paper studies a boundary value problem for first order linear elliptic systems in a simply connected plane domain, reduced to a complex equation in the unit disk with measurable principal coefficients satisfying a uniform ellipticity bound. The analysis treats separately nonnegative and negativ…

Milne Problem with an Arbitrary Indicatrix

This paper studies the Milne problem for radiative transfer in a plane parallel half space with absorption and an arbitrary scattering indicatrix, a case not reducible in general to the classical scalar integral equation treated by Hopf-Wiener methods. Working directly with the integral equation, it…

Asymptotic Behavior of Solutions of Differential Equations with a Small Parameter Whose Systems of “Fast Motions” Are Close to Hamiltonian Ones

This paper studies singularly perturbed differential systems whose fast variables form, in the limit of a small positive parameter, a Hamiltonian system with closed phase trajectories. By introducing action-like variables based on the Hamiltonian level curves and a phase variable along each periodic…

General Representation of Solutions of an Elliptic System of $2n$ Equations in the Plane

This paper develops a representation theory for uniformly elliptic first-order systems of 2n real equations in the plane, written in a complex canonical form involving a matrix coefficient Q. Extending Vekua’s theory for the scalar case, it introduces Q-holomorphic vectors, constructs a Cauchy kerne…

ON THE QUESTION OF THE INVERSE PROBLEM OF POTENTIAL THEORY

This note addresses an inverse problem in potential theory concerning how a mass-bearing region must change when its density is increased while its exterior potential is kept fixed. Using a harmonic test-function method attributed to V. P. Simonov, it proves that for two star-shaped plane regions wi…

On the Feasibility of Applying the Alternating Schwarz Method on Electronic Digital Computers

The paper evaluates the practicality of applying the alternating Schwarz method on electronic digital computers to the finite difference Dirichlet problem for the Laplace equation in a rectangular domain subject to internal storage limitations. It derives convergence estimates for a two-subdomain de…

Fundamental Solution of a Linear Partial Differential Equation with Constant Coefficients

This note studies the fundamental solution of a homogeneous linear partial differential equation with constant coefficients, under the assumption that the characteristic cone of the associated polynomial has no singular points away from the origin. Using a regularized spherical integral representati…

RELATION BETWEEN THE INVERSION FORMULAS FOR THE SOMMERFELD INTEGRAL AND THE KONTOROVICH–LEBEDEV FORMULAS

This paper examines the connection between inversion formulas for the Sommerfeld integral, used to represent solutions of the two-dimensional Helmholtz equation, and the Kontorovich-Lebedev transformation. Starting from the odd form of the Sommerfeld density and assuming suitable analyticity and dec…

Residue method for solving mixed problems and some related expansion formulas

This note develops a residue method for solving mixed problems for systems of linear differential equations with piecewise smooth coefficients on several intervals, under boundary conditions that may connect values and derivatives at multiple endpoints and include time derivatives. By reducing assoc…

A modification of the abstract analogue of Chaplygin's method

This paper proposes simplified variants of an abstract Chaplygin method for approximate solution of operator equations in semi-ordered spaces. It replaces exact residuals in the iterative construction by suitably bounded elements, using additive positively invertible majorants and, for uniqueness, m…

Representations of Solutions of the Euler–Poisson–Darboux Equation with a Negative Coefficient

This paper studies representations of solutions to the Euler, Poisson, Darboux equation with negative coefficient in domains adjoining the boundary line y = 0. Extending earlier results for positive coefficients, it proves integral representations in terms of analytic functions of one complex variab…

On the Square Root of a Linear Operator in Locally Convex Spaces

This paper studies square roots and factorizations of bounded linear operators between dual locally convex spaces embedded around a Hilbert space, extending earlier results that required complete continuity. Under compatibility assumptions on the triples of spaces, the authors prove that a positive …

$L_p$ Estimates for Solutions of Elliptic Systems

This note extends trace and a priori estimate results for Sobolev spaces \(W_p^{(l)}\), \(1<p<\infty\), on smooth domains. It proves that normal derivatives up to order \(l-1\) of a function in \(W_p^{(l)}(\Omega)\) belong to the corresponding fractional spaces on the boundary, and conversely constr…

On the behavior of cycles not homologous to zero under a mapping of an $n$-dimensional manifold into $n$-dimensional Euclidean space

This note studies continuous mappings of closed orientable n-dimensional manifolds into Euclidean n-space, or into n-dimensional orientable manifolds with degree zero, focusing on homology classes that contain cycles mapped to zero. It introduces numerical invariants measuring independent image cycl…