# Vol 67, No 2 (2021): Dedicated to the memory of Professor N. D. Kopachevsky

**Year:**2021**Articles:**11**URL:**https://journals.rudn.ru/CMFD/issue/view/1481**DOI:**https://doi.org/10.22363/2413-3639-2021-67-2

## Full Issue

## Articles

### In Memory of Nikolay Dmitrievich Kopachevsky, a Mathematician and a Human

#### Abstract

This paper is dedicated to the scientific and pedagogical activities of Nikolay Dmitrievich Kopachevsky (1940-2020) - a well-known mathematician, head of the Department of Mathematical Analysis of the V. I. Vernadsky Crimean Federal University, organizer and head of the Crimean Autumn Mathematical School-Symposium (KROMSH).

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):193-207

### Right-Sided Invertibility of Binomial Functional Operators and Graded Dichotomy

#### Abstract

In this paper, we consider the right-sided invertibility problem for binomial functional operators. It is known that such operators are invertible iff there exists dichotomy of solutions of the homogeneous equation. New property of solutions of the homogeneous equation named graded dichotomy is introduced and it is proved that right-sided invertibility of binomial functional operators is equivalent to existence of graded dichotomy.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):208-236

### Deficiency Indices of Block Jacobi Matrices: Survey

#### Abstract

The paper is a survey and concerns with infinite symmetric block Jacobi matrices **J** with *m*×*m*-matrix entries. We discuss several results on general block Jacobi matrices to be either self-adjoint or have maximal as well as intermediate deficiency indices. We also discuss several conditions for **J** to have discrete spectrum.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):237-254

### Investigation of Integrodifferential Equations by Methods of Spectral Theory

#### Abstract

This paper provides a survey of results devoted to the study of integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are operator models of integrodifferential partial differential equations arising in numerous applications: in the theory of viscoelasticity, in the theory of heat propagation in media with memory (Gurtin-Pipkin equations), and averaging theory. The most interesting and profound results of the survey are devoted to the spectral analysis of operator functions that are symbols of the integrodifferential equations under study.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):255-284

### Stochastic Lagrange Approach to Viscous Hydrodynamics

#### Abstract

The work is a survey of the author’s results with modifications and preliminary information on the use of stochastic analysis on Sobolev groups of diffeomorphisms of a flat n-dimensional torus to describe the motion of viscous fluids (nonrandom ones). The main idea is to replace the covariant derivatives on the groups of diffeomorphisms in the equations introduced by D. Ebin and J. Marsden to describe ideal fluids by the so-called mean derivatives of random processes.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):285-294

### On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations

#### Abstract

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator *P _{λ}(t)* depending on

*t*and the nonlinear operator

*Q*. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differentialdifference operators.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):316-323

### Equations Related to Stochastic Processes: Semigroup Approach and Fourier Transform

#### Abstract

The work is devoted to integro-differential equations related to stochastic processes. We study the relationship between differential equations with random perturbations - stochastic differential equations (SDEs) - and deterministic equations for the probabilistic characteristics of processes determined by random perturbations. The resulting deterministic pseudodifferential equations are investigated by semigroup methods and Fourier transform methods.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):324-348

### Asymptotics of the Spectrum of Variational Problems Arising in the Theory of Fluid Oscillations

#### Abstract

This work is a survey of results on the asymptotics of the spectrum of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. The problems were posed by N. D. Kopachevsky in the late 1970s and cover various fluid models. The formulations of problems are given both in the form of boundary-value problems for eigenvalues in the domain *Ω⊂R ^{3}*, which is occupied by the fluid in the equilibrium state, and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an “elliptic” constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface Γ. The spectrum in the considered problems is discrete; the spectrum distribution functions have power-law asymptotics.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):363-407

### Direct and Inverse Problems of Spectral Analysis for Arbitrary-Order Differential Operators with Nonintegrable Regular Singularities

#### Abstract

A short review is presented of results on the spectral theory of arbitrary order ordinary differential operators with non-integrable regular singularities. We establish properties of spectral characteristics, prove theorems on completeness of root functions in the corresponding spaces, prove expansion and equiconvergence theorems, and provide a solution of the inverse spectral problem for this class of operators.

**Contemporary Mathematics. Fundamental Directions**. 2021;67(2):408-421