Recent Papers

On Linear Partial Differential Equations of Infinite Order with Constant Coefficients

The paper studies Cauchy problems for linear differential systems of infinite order with constant coefficients and applies these results to linear partial differential equations of infinite order. It reduces an infinite system of order m to a first order matrix equation, represents the formal soluti…

On the behavior of solutions to the Cauchy problem for certain quasilinear equations as time increases without bound

This paper studies the large-time behavior of solutions to the Cauchy problem for a scalar quasilinear parabolic conservation law with viscosity and for the corresponding first-order conservation law, under bounded measurable initial data with prescribed limits at spatial infinity. Using comparison …

On Chebyshev Approximation in a Hilbert Ring

This paper studies Chebyshev approximation of continuous functions on a compact set with values in a Hilbert ring by functions of the form given by left multiplication by an arbitrary ring element and a fixed function. Using a more general framework of approximation by operator functions in Hilbert …

Hellinger Operator Integral and Some of Its Applications

The paper develops an operator version of the Hellinger integral for systems formed from block operator matrices on an orthogonal decomposition of a Hilbert space. It gives criteria for positivity of block operators and operator measures in terms of a positive type system and the integral \(\int dF_…

Differential equation with an abstract elliptic operator in a Hilbert space

This paper studies the Cauchy problem for a first order differential equation in a Hilbert space whose operator is an abstract elliptic operator, defined as a positive self-adjoint operator plus a perturbation of fractional order. Using semigroup methods, Mellin representations of negative powers, a…

Ill-Posedness of the Dirichlet Problem for Mixed-Type Equations in Mixed Domains

The paper studies the Dirichlet problem for Lavrent’ev’s model equation of mixed elliptic-hyperbolic type in simply connected mixed domains. It argues that this Dirichlet problem is ill-posed independently of the size and shape of the hyperbolic part by reducing the analysis to an auxiliary mixed bo…

TENSOR PRODUCTS OF SYSTEMS OF GROUPS AND UNIVERSAL COEFFICIENT THEOREMS FOR HOMOLOGY AND COHOMOLOGY

This paper revisits the universal coefficient problem for Čech spectral cohomology, noting that earlier arguments established only an exact sequence in general and a direct formula only under additional hypotheses. It introduces a tensor product for conjugate systems of Abelian groups and uses the m…

ON COMPUTATIONS CONNECTED WITH THE VERIFICATION OF THE RIEMANN HYPOTHESIS

This note reports computations performed on the Strela computer to test the Riemann hypothesis for zeros of the zeta function numbered 15,000 through 35,337, corresponding to the range of the parameter tau from about 2234 to 4735. The verification divides the critical line into intervals, computes t…

OSCILLATION THEOREMS FOR DIFFERENTIAL EQUATIONS OF HIGHER ORDERS AND THE SPECTRUM OF THE CORRESPONDING DIFFERENTIAL OPERATORS

This note studies the relation between oscillation of higher order differential equations on a half-line and the spectra of the associated self-adjoint differential operators. Using Glazman’s decomposition method, it derives spectral criteria for oscillation and nonoscillation without relying on asy…

Fractional powers of self-adjoint extensions of operators and some boundary value problems

This paper studies parabolic equations in Hilbert space generated by self-adjoint differential operators whose boundary conditions depend on time, a setting in which the operator domains may vary. It derives estimates for fractional powers of time-dependent self-adjoint extensions, first for regular…

On the Relationship Between Proximity Spaces and Bicompact Extensions of Completely Regular Spaces

This note derives Smirnov’s correspondence between proximities compatible with a completely regular topology and bicompact extensions of the space from the theory of commutative normed rings. It defines continuous functions on a general proximity space, proves that they form a uniformly closed conju…

On the Representation of Linear Functionals in a Class of Discontinuous Functions

This paper extends the Riesz representation problem from continuous functions to the class of bounded piecewise continuous functions on an interval, where extensions of functionals from continuous functions are generally nonunique. It introduces characteristic functions associated with interval step…

Corrections to the article On a method for counting integer points in $n$-dimensional polyhedra (DAN, vol. 118, no. 2, 1958)

On the Extraction of Singularities of Analytic Functions

This note studies when an analytic function can be decomposed into a sum of analytic functions whose singular sets are prescribed parts of the original singular set. Using a lemma on the decomposition of continuous linear functionals in locally convex topological vector spaces, it proves a general s…

On the convergence of Fourier series with respect to systems of the type $\{\varphi(nx)\}$ close to the trigonometric system

The paper studies convergence of Fourier series in the odd space L2 with respect to systems of dilates of a function, {phi(nx)}, that are close to the trigonometric sine system. It defines such T-systems through absolute convergence and nonvanishing of an associated multiplicative-function series, t…

On the Asymptotic Behavior of Analytic Functions

This paper studies how the asymptotic behavior on the positive real axis of a power series with alternating coefficients depends on analytic properties of the coefficient function treated as a function of a complex variable. It introduces property A, meaning analytic continuation to the nonnegative …

Some General Theorems on the Behavior of Spectral Functions of Second-Order Differential Systems

This paper studies the high energy asymptotic behavior of spectral functions for second order differential systems with measurable coefficients and separated initial conditions at the left endpoint. It formulates comparison theorems showing that, under local asymptotic equivalence of the coefficient…

Asymptotic expression for the confluent hypergeometric function

The paper develops a method for deriving uniform asymptotic solutions of second order differential equations by replacing the original equation with a nearby equation whose solutions are expressed through Bessel functions. The method is applied to Whittaker’s confluent hypergeometric function for la…

On the convergence of some finite-difference processes for the equations $y'=f(x,y)$ and $y'(x)=f(x,y(x),y(x-\tau(x)))$

This paper studies convergence of general linear finite-difference processes for the Cauchy problem for ordinary differential equations and for equations with retarded argument. It formulates consistency conditions for a multistep difference equation, introduces a notion of uniform convergence based…

EXTRAPOLATION OF RANDOM VECTOR PROCESSES AND THE AMOUNT OF INFORMATION CONTAINED IN ONE VECTOR STATIONARY RANDOM PROCESS RELATIVE TO ANOTHER STATIONARILY CONNECTED WITH IT

This paper studies stationary vector random processes and the information rate between two stationarily connected vector processes using spectral methods. It establishes a determinant formula for one-step linear extrapolation errors in terms of the spectral density matrix, yielding criteria for full…

On the Convergence of Galerkin-Type Methods

This paper presents a general Hilbert space framework for studying convergence of Galerkin type approximation methods for equations of the form Au plus lambda Ku equals v, where A is invertible and K is a linear perturbation. Using Kantorovich’s abstract scheme, it formulates determinant and compact…

On the Expansion in Eigenfunctions of Self-Adjoint Operators

This note extends constructions of generalized eigenfunction expansions for self-adjoint operators from function spaces with differential basic operators to an abstract separable Hilbert space setting. The author defines basic elements as the domain of a closed densely defined operator with bounded …

Estimating the domain of convergence of periodic series solutions of differential equations with a small parameter

The paper develops a Lyapunov-based method for estimating the convergence domain of small-parameter series representing periodic solutions of nonlinear differential systems near a generating periodic solution, including resonant cases where characteristic roots are integral imaginary values. It deri…

On the Application of the Variational Principle to Certain Nonlinear Differential Equations

The paper develops a variational framework for solving nonlinear operator equations in a Hilbert space under symmetry and positivity assumptions on the Gâteaux derivative. It proves uniqueness of classical solutions, characterizes them as minimizers of an associated functional, and introduces genera…

A Well-Posed Boundary Value Problem for Some Nonclassical Operators

This paper constructs a well-posed boundary-value problem for a nonclassical differential operator of the form \(-D_0^3+b\) in a bounded cylindrical domain, where \(b\) is an elliptic operator with constant coefficients in a generalized sense. Using Sobolev-type Hilbert spaces, the argument defines …

On the theory of fractional differentiation and integration of periodic functions belonging to the class $L_p$, $p>1$

This paper studies Weyl fractional integration and differentiation of periodic functions in \(L_p\), \(p>1\), through the behavior of best trigonometric approximation in the \(L_p\) metric. It proves that fractional integration increases, and fractional differentiation decreases, the order of best a…

ON THE STABILITY OF QUASILINEAR SYSTEMS WITH AFTEREFFECT

This paper extends Lyapunov stability criteria for ordinary quasilinear systems to functional differential systems with aftereffect in a Banach space. The linear delayed equation is represented as an abstract ordinary equation on a space of continuous history segments, generated by an unbounded oper…

On the Theory of Newton's and Chaplygin's Methods

This paper develops operator-theoretic generalizations of Newton and Chaplygin methods for solving nonlinear equations in ordered Banach-type and K-spaces. It replaces derivative inverses in iterative schemes by more general auxiliary operators, not necessarily invertible, and gives convergence cond…

Genus of a Fibered Space

This note introduces the genus of a locally trivial fiber space as the least cardinality of an open cover of the base over which local cross-sections exist, generalizing both earlier equivariant genus notions and the Lusternik-Schnirelmann category. It establishes basic functorial properties, compar…

THE MEMBERSHIP PROBLEM FOR DIRECT PRODUCTS OF GROUPS

The paper studies strong and weak membership problems for finitely presented groups under direct products, distinguishing them from the identity problem. It proves that the weak membership problem is unsolvable for the direct product of two free groups on two generators, by reducing Novikov’s unsolv…

On the Connection of the Method of Nearby Systems in Special Linear Topological Spaces with Certain Questions in the Perturbation Theory of Linear Operators in Banach Spaces

The paper studies how results from the perturbation theory of linear operators in Banach spaces extend to linear topological spaces formed from nested families of Banach spaces. It proves that Fredholm type properties of Phi operators, including normal solvability, finite defect characteristics, and…

Toward a General Theory of Equations with Deviating Argument

This paper develops a general framework for differential equations with deviating argument, reducing them locally to equations with delays and classifying them as retarded, neutral, or advanced according to the relative orders of present and delayed derivatives. It extends the method of successive i…

Forms of Dependencies with Additional Possibilities for Transforming a Nomogram with an Oriented Transparent Overlay

The paper identifies additional functional forms whose nomograms with an oriented transparent overlay allow expanded transformations beyond those previously described. It derives several classes of dependencies by eliminating auxiliary variables from systems of equations, including forms involving b…

On repeated differentiation with respect to the parameter of solutions of systems of ordinary differential equations with a small parameter multiplying the derivative

This note studies higher order differentiation with respect to a small parameter in singularly perturbed systems of ordinary differential equations where the parameter multiplies the derivative of the fast variable. Building on earlier convergence and first derivative results, it formulates the dege…

Some Functional Inequalities of Embedding Theorem Type

This note establishes functional inequalities analogous to Sobolev embedding theorems for functions on domains satisfying a uniform interior sector condition, under additional local integral bounds on highest derivatives that depend on powers of diameters of sections or neighborhoods of sections. Fo…

On spaces that are nuclear in various senses

This note compares two notions of nuclearity for locally convex spaces, those of Grothendieck and Gel'fand, building on Raikov's result that Grothendieck nuclearity implies Gel'fand nuclearity for barrelled spaces. It proves that the two notions are equivalent in the classes of Frechet spaces and co…

Description of a Certain Class of Representations of the Lorentz Group

This note addresses the problem of describing nonirreducible bounded representations of the Lorentz group in reflexive Banach spaces, focusing on a broad class denoted K0. It constructs a canonical model from a representation of a one-parameter subgroup, an integer parameter, and a Banach space, and…

On the Theory of Equations with Concave Operators

This paper studies the convergence of successive approximations for fixed point equations with concave nonlinear operators in ordered Banach spaces. It formulates several convergence theorems under assumptions involving invariant cones, monotonicity, complete continuity, uniqueness of a nonzero solu…

UNIQUENESS OF THE SYMMETRIC SOLUTION OF THE PROBLEM OF LARGE DEFLECTIONS OF A SYMMETRICALLY LOADED CIRCULAR PLATE

The paper addresses uniqueness for the axisymmetric large deflection problem of a circular plate under symmetric loading, formulated through the reduced Kármán ordinary differential system with clamped boundary conditions and no membrane stress on the contour. It derives a priori estimates, rewrites…

On the Expansion of Regularly Monotone Functions into a Goncharov Series

This note studies expansions of regularly monotone functions on an interval in Goncharov series associated with a prescribed sign sequence. Using classical arguments based on compactness, Rolle’s theorem, and estimates for the corresponding Goncharov polynomials, it generalizes an earlier result obt…

Corrections to the article “Adsorption of Water on Quartz Crushed in Vacuum” (DAN, Vol. 115, No. 5, 1957)

On the Archimedean Principle in Semi-Ordered Quotient Lineals

This note studies when the quotient of an Archimedean partially ordered K-lineal by a normal sublineal remains Archimedean. It gives a necessary and sufficient condition in terms of bounded sequences from the normal sublineal and then derives simplified criteria for K-lineals with projections, K-pri…

An Algorithm for Constructing the Chebyshev Approximation of a Continuous Function by a Polynomial

This paper presents an algorithm for constructing a Chebyshev approximation of a continuous real function on a compact set by a linear combination of given continuous, linearly independent functions. Instead of first replacing the compact set by a finite net and solving a discrete inconsistent linea…

Estimates for Solutions of Elliptic and Parabolic Systems

The paper develops a priori estimates for solutions of elliptic and parabolic systems in generalized Sobolev type spaces allowing fractional orders and arbitrary integrability exponent in part of the results. It first defines anisotropic spaces with respect to groups of variables and proves an inter…

Lie Theorems for Generalized Shift Operators

This note extends the first and second direct Lie theorems from Lie groups to families of generalized translation operators on a differentiable manifold. It defines higher-order infinitesimal operators through derivatives of the translated function with respect to the translation and base variables,…

Asymptotic Expansion of Solutions of Second-Order Ordinary Differential Equations in Banach Spaces

The paper studies singularly perturbed second order ordinary differential equations in Banach spaces, with a small parameter multiplying the highest derivative. It defines asymptotic expansions in the Banach space setting and constructs fundamental solutions for a linear equation with a bounded oper…

On the rigidity of an open surface of nonnegative curvature under a nonorthogonal sleeve connection

This paper studies infinitesimal rigidity of open convex surfaces of nonnegative curvature when holes in the surface are filled by rigid sleeves imposing a nonorthogonal contact constraint on boundary displacements. Using Blaschke’s integral formula, boundary decompositions of the bending and rotati…

On the Arithmetic of Matrices

This paper develops arithmetic invariants for integral nonsingular matrices and integral skew-symmetric matrices, using canonical elementary-divisor forms under unimodular equivalence. It defines left and right matrix divisors, proves their equality in number, and gives an explicit product formula f…

S. L. Sobolev spaces of fractional order and their application to boundary value problems for partial differential equations

The paper introduces anisotropic Sobolev spaces of fractional order defined by square integrability of derivatives and fractional difference quotients, including versions on product domains and smooth product surfaces. It proves embedding and trace-extension theorems for these spaces, giving norm es…

THE NEMYTSKII OPERATOR AND ITS POTENTIAL IN ORLICZ SPACES

This paper studies the Nemytskii operator defined by hu equals g(u(x), x) and its potential functional in generalized Orlicz spaces, extending earlier results for Lp spaces relevant to nonlinear Hammerstein integral equations. It establishes necessary and sufficient conditions, expressed through You…