Recent Papers

ON SINGULAR PERIODIC SOLUTIONS OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

THE METHOD OF STRAIGHT LINES

Three Cylinders Theorem and Its Applications

The paper proves a three-cylinder theorem for solutions of a second-order uniformly parabolic equation with bounded sufficiently smooth coefficients, giving a logarithmic convexity type estimate analogous to Hadamard’s three-circle theorem. The proof uses a coordinate transformation, weighted integr…

Approximately Convex Sets in Banach Spaces

The paper generalizes the notion of a Chebyshev set in a normed linear space by studying approximately convex sets, defined through convex-valued metric projections. It develops basic properties of the associated majorization relation, uses fixed point results for multivalued mappings, and proves th…

Relaxation on Surfaces with Saddle Points

This note constructs descent algorithms for minimizing differentiable functionals in Euclidean space whose surfaces may contain saddle points, under qualitative assumptions including coercivity and finitely many stationary points. The principal algorithm combines gradient relaxation with controlled …

COUNTABLE SYSTEMS OF DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS[^1]

THE FOURIER METHOD FOR NONSTATIONARY EQUATIONS WITH GENERAL BOUNDARY CONDITIONS

Hartogs' Theorem in Some Normed Fields

This note extends Hartogs’ theorem to power series over certain complete non-Archimedean normed fields of rank one, especially non-locally compact fields whose value group is dense or whose residue field is infinite. After defining holomorphy by convergent power series on disks and polydisks, the pa…

Solution of the Dirichlet Problem for a Finite Wedge Using a Special Integral Transform in Cylindrical Functions

The paper develops an exact solution of the Dirichlet problem for a finite cylindrical wedge bounded by two radial planes, two axial planes, and a cylindrical surface. By separating variables, the radial problem is reduced to a singular spectral problem whose admissible eigenfunctions are expressed …

ON IRREDUCIBLE EXCEPTIONAL SUBMANIFOLDS OF THE FIRST KIND ON THREE-DIMENSIONAL COMPLEX-ANALYTIC MANIFOLDS

The paper studies irreducible exceptional submanifolds of the first kind on three-dimensional complex-analytic manifolds, extending the surface-theoretic notion of an exceptional curve contractible to a nonsingular point. It proves that such a contraction is inverse to a sigma-process, and gives a c…

On compactness criteria in spaces of abstract functions and the complete continuity of the embedding operator

The paper studies compactness criteria for spaces of abstract, Banach-valued set functions that generalize Sobolev and Nikolsky type function spaces. It establishes analogues of Arzela’s theorem and derives necessary and sufficient compactness conditions in the space of functions with translation-co…

ON BOUNDED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS

Uniqueness and Approximation Theorems for the Local Laplace Transform

The paper studies local Laplace transforms with values in an arbitrary Banach space, especially when the transform is entire or meromorphic, and relates their analytic growth to uniqueness and approximation properties of the original function. After extending the definition of the local Laplace tran…

On Estimates of the Growth of Solutions to Hyperbolic Partial Differential-Difference Equations

The paper establishes a necessary and sufficient criterion for exponential growth estimates of solutions to a class of hyperbolic partial differential-difference equations in a complex Banach space, with variable compact operator coefficients and bounded delays. The criterion is expressed through th…

Certain Theorems on the Existence of a Periodic Solution and on Determining the Boundary of the Region of Asymptotic Stability of the Trivial Solution of a System of Ordinary Differential Equations

On Limit Relations and Asymptotic Formulas for Polynomials Orthogonal on the Unit Circle

This note studies boundary limit relations for polynomials orthonormal on the unit circle with respect to an absolutely continuous or locally absolutely continuous measure. It gives new sufficient conditions, expressed through integral and ordinary moduli of continuity of the weight and its reciproc…

ON THE PROBLEM OF THE OBLIQUE DERIVATIVE

This note studies an oblique derivative boundary value problem for harmonic functions in the unit ball, with Hölder continuous coefficients on the sphere depending on \(x^2+y^2\) and \(z\). By expanding the solution and boundary data in angular Fourier series, the problem is reduced to a family of t…

ON A THEORY OF LINEAR DIFFERENTIAL EQUATIONS WITH AFTER-EFFECT

ON THE NONUNIQUENESS OF STATIONARY SOLUTIONS FOR A SYSTEM OF EQUATIONS OF THE THEORY OF COMBUSTION WITH PIECEWISE CONSTANT RATE CONSTANT AND COEFFICIENTS OF THERMAL CONDUCTIVITY AND DIFFUSION

This paper studies stationary one-dimensional combustion waves governed by a nonlinear temperature and concentration system with cut-off reaction rate and piecewise constant thermal conductivity and diffusion coefficients. Building on earlier uniqueness results for constant coefficients when the con…

On a class of preimages of metric spaces

The paper studies spaces that admit, for each open cover, weak mappings into metric spaces, where point preimages are contained in elements of the cover, and relates these spaces to inverse spectra of metric spaces. It gives necessary and sufficient conditions for the existence of such mappings in t…

Information about the Samarkand Symposium

This chronicle reports on the inter-university symposium on the qualitative theory of differential equations and its applications, held in Samarkand in October 1964. It summarizes the organization of the meeting, its three thematic sections, and the principal plenary reports on singular points in mu…

Nonlinear parabolic equations which have nondivergent form

This paper studies nonlinear parabolic equations in nondivergence form on a space-time rectangle, with boundary and initial conditions formulated in Sobolev-type spaces. Using Galerkin approximations, a priori estimates, monotonicity arguments, and compactness results, it proves homeomorphism theore…

On a Class of Interpolation Polynomials with Unfixed Nodes

This paper generalizes a theorem of Mycielski and Paszkowski on interpolation polynomials with unfixed nodes, replacing algebraic polynomials by generalized polynomials generated by systems of functions satisfying Chebyshev-type zero-counting conditions. For prescribed positive interpolation values …

THE HYPERPLANE METHOD IN NON-STATIONARY PROBLEMS WITH A SELF-ADJOINT OPERATOR

Applicability Conditions and Error Estimate for the Averaging Method for Systems Passing Through Resonances During Evolution

This note studies the validity of the averaging method for analytic slow-fast systems with two angular variables when the evolving frequencies pass through resonances. Arnold shows by examples that naive averaging can give wrong results and that, even under suitable hypotheses, the discrepancy betwe…

EXTREMAL PROBLEMS ON CLASSES OF ANALYTIC FUNCTIONS HAVING A STRUCTURAL FORMULA

This paper formulates a general extremal problem for compact classes of analytic functions and weakly differentiable systems of functionals, with the aim of describing the boundary of the corresponding value set. For classes whose elements admit a representation as sums of Stieltjes integrals, the a…

Generalization of the du Bois-Reymond Theorem to Bounded Orthonormal Systems of Functions

This note studies Fourier expansions with respect to orthonormal systems of uniformly bounded functions, addressing whether such systems can provide everywhere or uniformly convergent expansions for all continuous functions. Extending the classical Du Bois-Reymond theorem, it proves that for any uni…

ON THE REPRESENTATION OF ARBITRARY ENTIRE FUNCTIONS BY DIRICHLET SERIES

The paper studies representation of entire functions by Dirichlet series with complex exponents determined by the zeros of an auxiliary entire function of finite order. It establishes absolute and locally uniform expansions for entire functions of order less than \(\rho/(\rho-1)\), gives remainder e…

Density theorem for finite functions in weighted classes

The paper proves a density theorem for compactly supported smooth functions in weighted Sobolev-type classes on bounded and unbounded domains with simple boundary structure in the sense of Sobolev. It formulates boundary and growth conditions on the weights, including behavior near lower-dimensional…

Asymptotics of Eigenfunctions of the Many-Particle Schrödinger Equation

This paper studies symmetric and antisymmetric eigenfunctions of the many-particle one-dimensional Schrödinger equation with pair interactions whose potentials admit an exponential integral representation. Using two-particle scattering data, local solutions in ordered polyhedral sectors, and resolve…

On Ponomarev's Theory of Absolutes of Topological Spaces

This note reformulates Ponomarev’s spectral theory of absolutes for paracompact spaces by giving a direct definition of “absolute” zero-dimensional projection spectra. It proves that every Ponomarev absolute is the space of a unique absolute spectrum, namely the maximal spectrum of that absolute, an…

On eigenfunctions of an elliptic operator in some unbounded domains

The paper studies eigenfunctions and eigenvalues of a real elliptic operator in an unbounded domain that coincides outside a bounded part with a semi-infinite cylinder, with Dirichlet boundary conditions. It formulates solvability criteria for the inhomogeneous problem in exponentially weighted Sobo…

On the Existence and Uniqueness of Solutions to Certain Boundary Value Problems for Degenerate Second-Order Elliptic Equations

This paper studies existence and uniqueness for boundary-value problems for second-order elliptic equations that degenerate on part of the boundary, with boundary conditions imposed only on the nondegenerate portion or supplemented by suitable nonhomogeneous data. Extending a method of Schechter, th…

On the Representation of an Arbitrary Analytic Function by Special Series

The paper studies representation of arbitrary analytic functions in rho-convex domains by series and integral expansions associated with entire functions of finite order, extending earlier work on Dirichlet series with complex exponents. It introduces an associated function defined through coefficie…

On Nonlinear Equations of the First Kind

The paper develops regularization methods for nonlinear equations of the first kind with a continuous one-to-one operator between metric spaces, using a smoothing functional that combines data misfit with a quadratic norm penalty. Under compact embedding and Hilbert space convexity assumptions, it p…

ON THE CAUCHY PROBLEM FOR SINGULAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

\(C^{1,\alpha}\)-Isometric Immersions of Riemannian Spaces

This paper studies local isometric immersions of analytic Riemannian spaces into codimension one Euclidean space with prescribed Hölder regularity, focusing on whether classes \(C^{1,\alpha}\) can be universal beyond the analytic case. It formulates universality results and obstructions: Nash and Ku…

A THEOREM ON A DIFFERENTIAL INEQUALITY FOR A PERIODIC BOUNDARY VALUE PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION OF THE SECOND ORDER

Remarks on the Realization of Integral Curves of Equations of the Form \(\dot x=f(t,x,x(t-\tau))\) on Certain Two-Dimensional Manifolds

A PRIORI ESTIMATES AND SOLVABILITY OF GENERAL BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC SYSTEMS WITH DISCONTINUOUS COEFFICIENTS

This paper studies general boundary value and conjugation problems for Douglis, Nirenberg elliptic systems with discontinuous coefficients on a bounded domain divided by an internal surface. It formulates the relevant Sobolev, Hölder, and Slobodetskii spaces, defines a joint covering condition combi…

Parametric Method for Solving Fractional Programming Problems on Analog Computers

The paper presents a parametric method for solving a class of fractional programming problems with a convex numerator, positive denominator, convex inequality constraints, linear equalities, and bounded variables. The authors introduce an auxiliary convex function depending on a parameter, whose zer…

On the Existence of Unbounded Analytic Constructive Functions

The paper addresses whether there exist constructive functions that are analytic in the unit disk, continuous on the closed disk, and nevertheless unbounded in the disk. It formulates complex constructive functions and analyticity, then adapts Zaslavskii’s unbounded continuous function using a speci…

Embedding of Nonorientable Three-Dimensional Manifolds in Five-Dimensional Euclidean Space

This note proves that every nonorientable closed three-dimensional smooth manifold can be embedded in five-dimensional Euclidean space, completing the corresponding result for all three-dimensional manifolds in combination with Hirsch’s theorem for the orientable case. The argument reduces the close…

Difference method and uniqueness of the generalized solution for the Tricomi problem

This note studies the Tricomi boundary value problem for a second order equation of mixed type with variable coefficients in a domain bounded partly by characteristics. It constructs a finite difference scheme adapted to the elliptic and hyperbolic parts of the domain, formulates coefficient conditi…

On Stable Solvable Extensions of Partial Differential Operators

This paper studies stability of solvable extensions of partial differential operators under small perturbations of differential coefficients and boundary conditions. For constant-coefficient operators in bounded domains, it gives a necessary and sufficient criterion for the existence of a stable sol…

On Some Criteria for Non-Oscillation and Boundedness of Solutions of Linear Differential Equations

The paper studies criteria for nonoscillation and boundedness of solutions of linear differential equations of the form x^(n) + p(t)x = 0, with emphasis on estimating intervals on which a distinguished solution has no zeros. Using integral inequalities, comparison functions, and reduction to Volterr…

Qualitative Investigation in the Large of Integral Curves of Isochronous Systems of Two Differential Equations

THE GIBBS PHENOMENON

This note studies the occurrence of the Gibbs phenomenon for triangular summation methods of trigonometric Fourier series whose multipliers are approximated by a function of k/n with small perturbations. It proves a lemma giving sufficient conditions in terms of the limiting integral associated with…

On the Theory of Finite Matrices

This paper develops comparison estimates for finite complex matrices using elementwise majorization, induced matrix norms, and logarithmic norms. It proves relations between spectral abscissa and the limiting behavior of the spectral radius, extends Perron type comparison arguments to real majorants…

On the Structure of Analytic Solutions of One Class of Systems of Linear Partial Differential Equations with Constant Coefficients

This paper studies local analytic solutions of systems of linear partial differential equations with constant coefficients, using the ideal generated by the symbolic forms of the differential operators. Assuming a primary decomposition of the associated polynomial ideal, the author relates membershi…