Recent Papers

On the definition of the capture regions in the case of a system of linear differential equations with periodic coefficients

INVARIANT COVERINGS OF SUBGROUPS

This paper studies finite nonnilpotent groups in which every proper subgroup lacking a specified special property is contained in a proper normal subgroup, generalizing earlier work on invariantly coverable groups. For special properties inherited by subgroups and homomorphic images, the paper prove…

DUALITY PRINCIPLES FOR BOUNDARY-VALUE PROBLEMS

This paper develops duality principles for nonlinear boundary-value problems for systems of ordinary differential equations with general continuous, possibly nonlinear, boundary functionals. The authors associate the problem both with a finite-dimensional vector field on the space of initial data, d…

ON GROUPS OF PRINCIPAL UNITS OF \(p\)-EXTENSIONS OF A LOCAL FIELD

The paper studies the structure, up to similarity, of the module of principal units of a normal p-extension of a finite p-adic field over the completed group ring of its Galois p-group. It relates this module to a Z-splitting extension of a known module by the augmentation ideal, identifies the rele…

ON THE EXISTENCE OF A COUNTABLE SET OF PERIODIC MOTIONS IN A NEIGHBORHOOD OF A HOMOCLINIC CURVE

This paper studies autonomous systems of differential equations possessing a saddle type periodic motion and a trajectory homoclinic to it. Under a rough, transverse dimensional condition on the intersection of the stable and unstable integral manifolds along the homoclinic curve, the author constru…

ON THE STABILITY OF THE ROOT NUMBER OF AN ANALYTIC OPERATOR-FUNCTION AND ON PERTURBATIONS OF ITS CHARACTERISTIC NUMBERS AND EIGENVECTORS

This note extends stability and analytic perturbation results for linear operators to analytic operator functions with values in bounded operators on a Banach space. It defines characteristic numbers, eigenvectors, associated chains, geometric and algebraic multiplicities, and the root number inside…

Partial Differential Equation with Retarded Argument

A solution to the equation \begin{equation} \frac{\partial^2u(t,x)}{\partial t^2}=a^2\frac{\partial^2u(t,x)}{\partial x^2}-b^2\frac{\partial^2u(t-\tau,x)}{\partial t^2}\tag{1} \label{1} \end{equation} for $t\ge\tau$ is sought, satisfying zero boundary conditions \begin{equation} u(t,0)\equiv0,\quad …

ON THE QUESTION OF CONFORMAL MAPPING OF CIRCULAR DOMAINS

The paper studies conformal mappings of the upper half-plane onto slit domains whose boundaries are composed of finitely many circular arcs and line segments, with particular attention to the Schwarz derivative and its accessory parameters. It derives evolution equations for the logarithmic derivati…

On Representations of a Continuous Group by Unbounded Operators

The paper develops elements of a theory of representations of continuous Lie groups by unbounded operators, motivated by infinite-dimensional and generally nonunitary representations arising in problems of complex angular momentum and spin. It defines representations, topology, irreducibility, and e…

ON THE THEORY OF DISTAL MINIMAL SETS AND DISTAL FUNCTIONS

This note studies distal dynamical systems on tori and related classes of distal functions. It proves that every finite or infinite dimensional torus admits a distal flow for which the torus is a minimal set, and gives a construction on the three dimensional torus that is distal and minimal but not …

Nonlinear differential equations in a Banach space, close to linear ones

The article investigates the exponential growth rates of solutions to the equations \begin{gather}\frac{dx}{dt}=Ax,\tag{1}\\\frac{dx}{dt}=Ax+h(x,t),\tag{2}\end{gather} where $A$ is a linear unbounded operator in a complex Banach space $X$, and $h(x,t)$ is a nonlinearity. It is assumed that the opera…

COMPARISON THEOREMS FOR FRACTIONAL POWERS OF OPERATORS

This paper develops comparison and perturbation estimates for fractional powers of weakly positive operators in Banach spaces. Using bounds on bilinear forms representing the difference between two operators and their adjoints, it derives inequalities for expressions involving \(A_1^\alpha - A_2^\al…

The asymptotic behavior of solutions of certain nonlinear differential equations

The differential equation \begin{equation} u^{(n)}+a(x)u=f(x,u,u',\dots,u^{(n)}) \tag{1} \label{1} \end{equation} is considered. A series of sufficient conditions is established under which equation \eqref{1} has solutions with the same asymptotic form as the solutions of the equation $u^{(n)}+a(x)u…

SOLVABILITY CONDITIONS FOR SOME BOUNDARY-VALUE PROBLEMS FOR AN ORDINARY LINEAR DIFFERENTIAL EQUATION OF SECOND ORDER

The paper derives solvability and sign preservation conditions for Green’s functions of second order linear ordinary differential boundary value problems with summable real coefficients. Using the fork principle, it formulates integral inequalities involving the coefficients that ensure existence of…

A sufficient test for the stability of the trivial solution of a third-order differential equation with periodic coefficients

This article considers a third-order differential equation: $$x'''+P_1(t)x''+P_2(t)x'+P_3(t)x=0,$$ where $P_1(t)$ and $P_3(t)$ are $\omega$-periodic odd coefficients that are non-negative on the interval $[0,\omega/2]$, and $P_2(t)$ is an $\omega$-periodic even positive coefficient satisfying the co…

On a property of the Blaschke function

This note studies when a Blaschke product with zeros in the unit disk belongs to the class of analytic functions defined by bounded Riemann, Liouville fractional mean integrals for a parameter between minus one and zero. It proves that membership in this class is equivalent to the finiteness of the …

The structure of a solution of the equation $u'=U_0(\nu)u+\sum_{n=1}^\infty(\nu)u^{1+\alpha_n}\equiv U(u,\nu)$ in a small neighborhood of the origin

The article constructs a solution to the equation \begin{equation} u'=U_0(\nu)u+\sum_{n=1}^\infty(\nu)u^{1+\alpha_n}\equiv U(u,\nu)\tag{1} \label{1} \end{equation} where $u$ and $\nu$ are polar coordinates. At the same time, there is one case where the solution to equation \eqref{1} cannot be obtain…

ON THE BRANCHING OF PERIODIC SOLUTIONS OF AUTONOMOUS SYSTEMS AND DIFFERENTIAL EQUATIONS IN BANACH SPACES

The paper studies branching of periodic solutions for analytic autonomous systems with a small parameter and for nonautonomous differential equations in Banach spaces. By applying Poincaré’s reduction to initial data and period corrections, the existence problem is transformed into finite-dimensiona…

ON A CLASS OF INTERPOLATION PROBLEMS

The paper studies interpolation problems for analytic functions in which derivatives of orders divisible by a fixed integer are prescribed at rotating systems of points. It reduces the problem for entire functions of exponential type to lacunary auxiliary functions satisfying infinite order linear d…

THE RIEMANN—HILBERT BOUNDARY-VALUE PROBLEM WITH DISCONTINUOUS BOUNDARY CONDITIONS FOR QUASILINEAR ELLIPTIC SYSTEMS OF EQUATIONS

The paper studies the Riemann-Hilbert boundary value problem for uniformly elliptic first order quasilinear systems in complex form, focusing on the case of discontinuous boundary coefficients and initially with vanishing lower order term. By reducing the boundary data to piecewise constant form, th…

A priori estimate of solutions of boundary value problems for second order ordinary nonlinear differential equations

Problems $$N[y]=y''f(t,y,y')=0,\\\alpha_0 y(a)+\alpha_1 y'(a)=A,\quad\beta_0 y(b)+\beta_1 y'(b)=0$$ are considered under the assumption that $f(t,y,y')$ satisfies the Carathéodory condition, the Lipschitz condition with respect to $y$, and there exists a continuous $\partial f(t,y,y')/\partial y'$. …

The asymptotic equivalence of the solutions of certain linear systems of differential equations

The question of the equivalence of systems of two differential equations \cite{4} \begin{gather}\frac{dx}{dt}=X[R(t)+Q(t)],\label{1}\\\frac{dx}{dt}=YR(t)\label{2}\end{gather} in the sense of $x=YA(t)$, where $A(t)\to A=\operatorname{const}$ as $t\to\infty$, is considered. Here $R+Q=P_0+P_1t^{-1}+P_2…

PROOF OF THE NONISOMORPHISM OF THE SPACES OF SMOOTH FUNCTIONS ON AN INTERVAL AND ON A SQUARE

This note proves that, for any integers p greater than or equal to 0, s greater than or equal to 1, and n greater than or equal to 2, the Banach space of s times continuously differentiable functions on an n-dimensional cube is not isomorphic to the corresponding space of p times differentiable func…

Conditions for the existence of bounded solutions of a difference equation in a Banach space

We consider the difference equation \begin{equation} x(m+1)=A(m)x(m)+u(m),\tag{1} \label{1} \end{equation} where $x(m)$, $u(m)$ are functions of the argument $m$ ($m=0,1,2,\dots$), the values of which belong to some Banach space $E$; $A(m)$ is a linear operator-function mapping (for each $m$) the sp…

On the existence of an upper and a lower solution of a boundary value problem for a system of ordinary differential equations

Under certain conditions imposed on the right-hand sides of the system of equations, the existence of upper and lower solutions to the boundary value problem for finite and countable systems of ordinary differential equations on an interval of a specific length is proven. Bibliography 2.

ON THE INDEX AND NORMAL SOLVABILITY OF A GENERAL LINEAR BOUNDARY-VALUE PROBLEM FOR A SYSTEM OF EQUATIONS OF COMPOSITE TYPE

This note studies a general linear boundary value problem for a first order system of composite type in a simply connected planar domain, with one real and one complex unknown and boundary operators involving derivatives up to orders m and n. Under nonvanishing and endpoint compatibility conditions …

On a One-Parameter Apparatus for Function Approximation

The paper studies a one-parameter family of positive approximation operators for continuous functions on the interval [0,1], extending a previously considered case corresponding to alpha equal to 2. It proves uniform convergence and gives estimates of the approximation error in terms of the modulus …

A COMBINATORIAL PROBLEM OF N. V. SMIRNOV

The paper addresses a combinatorial problem concerning arrangements of objects of several prescribed types in a chain such that no two adjacent objects have the same type, motivated by multidimensional order statistics and run distributions. Using a formula for chains with fixed transition numbers b…

A LIMIT THEOREM FOR THE NUMBER OF LEVEL CROSSINGS BY A STATIONARY GAUSSIAN PROCESS

This paper proves a central limit theorem for the number of upcrossings of a fixed level by a stationary Gaussian process. The argument studies the variance and higher moments of the crossing count, first showing that under decay of the correlation function the variance diverges, and then that, unde…

CLASSIFICATION OF INDECOMPOSABLE INFINITESIMAL REPRESENTATIONS OF THE LORENTZ GROUP

This paper gives an explicit classification framework for indecomposable Harish-Chandra modules over the Lie algebra of the proper Lorentz group. Decomposing a module into rotation-subgroup components, the authors express the Lorentz generators through auxiliary operators and derive the relations th…

ON THE COMPLETE REGULARITY OF STATIONARY PROCESSES

This note studies complete regularity of stationary random processes through conditions on their spectral densities. For one-dimensional continuous-time processes, it gives sufficient and, in special factored cases, necessary criteria involving entire functions of finite exponential type, continuity…

ON REGULARIZATION OF THE CAUCHY PROBLEM

This note develops a modification of Maslov’s regularizer construction for Cauchy problems involving pseudodifferential, pseudohyperbolic systems. It formulates a canonical operator on Lagrangian manifolds with values in Hilbert space scales associated with an unbounded self-adjoint operator, includ…

Duality of Functors in the Category of Locally Convex Spaces

This note studies duality of functors in the category of separated locally convex spaces with morphisms continuous on bicompact sets, using internal Hom and tensor product constructions. It describes the dual functor through conjugate spaces, extremal topologies, and a special topology on values of …

SOME QUESTIONS ON APPROXIMATION OF FUNCTIONS BY ALGEBRAIC POLYNOMIALS IN THE INTEGRAL METRIC

The paper studies constructive characterizations of nonperiodic function classes in integral metrics by approximation with algebraic polynomials on the interval [-1,1]. It compares the classes defined by ordinary moduli of smoothness and those defined by endpoint-adapted difference conditions, estab…

ON THE SOLUTION OF A DIFFUSION PROBLEM WITH A NONLINEAR BOUNDARY CONDITION

The paper studies a diffusion equation on a semi-infinite domain with a nonlinear, piecewise boundary condition arising in models of self-oscillations in electrochemical systems with a falling characteristic. It reduces the periodic boundary-value problem to a nonlinear integral equation by represen…

Analytic Capacity and the Cauchy Integral

This paper establishes estimates for the Cauchy integral over the boundary of a Lyapunov domain in terms of the analytic capacity, or analytic C-capacity, of the set of singularities of the integrand. It proves bounds for continuous functions analytic off a closed exceptional set and for bounded fun…

ON THE STABILITY OF A BOUNDARY-VALUE PROBLEM IN THE THEORY OF ANALYTIC FUNCTIONS

The paper studies the stability of a boundary value problem for analytic functions with conjugation on the unit circle, under general assumptions on the coefficient functions. It reduces the problem, via analytic continuation, to a Riemann boundary value problem for two pairs of unknown functions an…

Solvability Conditions for a Class of Pseudodifferential Equations

This paper studies local solvability for a class of pseudodifferential equations extending differential operators of constant strength and allowing certain nonconstant-strength operators with characteristics of arbitrary multiplicity. It defines operators with principal and subprincipal homogeneous …

RATIONAL \(G\)-SURFACES

This note studies rational surfaces equipped either with a Galois action over a perfect field or with an action of a finite group, with emphasis on birational invariants and applications to the Cremona group. Using the method of adjunction, it introduces standard models, including surfaces with rati…

Isothermal Coordinates on Surfaces of Bounded Integral Mean Curvature

This note studies analytic representations of disk type surfaces in Euclidean space with bounded integral mean curvature, using isothermic parametrizations whose generalized Laplacian is a vector valued measure of finite total variation. For polyhedral surfaces, the Laplacian of an isothermic parame…

ON THE DIMENSION OF THE PRODUCT OF ORDERED CONTINUA

This note studies the covering, small inductive, and large inductive dimensions of finite products of ordered continua, viewed as bicompact topological spaces. It proves that for a product of n ordered continua, all three dimensions are equal to n. The lower bound is obtained by constructing separat…

ON THE EQUATION \(\Delta U[x(t)] + P[x(t)]U[x(t)] = 0\) IN FUNCTION SPACE

The paper studies a boundary value problem for a generalized Laplace equation in Wiener function space, involving the Lévy Laplacian of a functional and a potential term. For domains defined by a smooth functional with constant positive Lévy Laplacian, it constructs solution functionals using Fourie…

Application of the Contour Integral Method to the Solution of Mixed Problems for a Parabolic System

This note applies the contour integral method to a mixed boundary value problem for a parabolic system describing heat and mass transfer in a three-dimensional domain. The argument reduces the problem to an associated spectral problem, constructs a fundamental matrix and Green matrix, and represents…

The theory of characteristic vectors and its application to the study of the asymptotic behavior of solutions of differential systems. I

ERGODIC THEOREMS FOR GENERAL DYNAMICAL SYSTEMS

This paper extends classical ergodic theorems to dynamical systems whose time parameter is a measurable semigroup acting by measure-preserving transformations. It formulates conditions on averaging sets and measures, including Følner-type invariance and size bounds, and proves maximal inequalities t…

Two-point and multi-point Taylor formulas

A formula is considered by which the value of the function $f(x)\in C^n$ on a given interval is determined through its values and the values of its derivatives at $m$ points ($m\ge2$) of the given interval. The considered examples demonstrate the possibility of effectively using the obtained general…

ON PERTURBATIONS OF A DIFFERENTIAL EQUATION

ON CLASSES OF REGULAR FUNCTIONS OF SEVERAL COMPLEX VARIABLES

This note extends previously defined classes of regular functions of several complex variables by introducing classes associated with bounded rotation and contracted distortion, formulated through restrictions to analytic plane sections of a bounded complete bicircular domain. It gives necessary and…

SIMPLE EXAMPLES OF UNDECIDABLE ASSOCIATIVE CALCULI

The paper studies the equivalence problem for associative calculi, focusing on how few defining relations are needed to obtain undecidability for equivalence to a fixed word. Starting from Tseitin’s calculi with seven relations, it constructs encodings by normal algorithms and proves reductions betw…

ON A QUESTION OF UNIQUENESS OF SOLUTIONS OF LINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS