Vol 204 (2022)

In this paper, we discuss the construction of bi-invariant subspaces for a self-adjoint, linear, closed operator with discrete spectrum perturbed by a bounded operator. The main result is the theorem on the similarity of this operator to a block diagonal operator. This theorem implies results concerning biinvariant subspaces and formulas for projectors and weighted average eigenvalues. In addition, we construct the corresponding group of operators and propose a new modification of the method of similar operators.

In this paper, we introduce the concept of k-potent sets in monoids, k ∈ $\mathrm{\mathbb{N}}$, establish their simplest properties, and indicate a class of homogeneous monoids with a set of generating elements. We find simple necessary conditions of the k-potency of a fixed set in such a monoid. For commutative monoids, we establish an isormorphism between them and the monoid ${\mathrm{\mathbb{N}}}_{+}^{\mathfrak{I}}$ with the corresponding label set $\mathfrak{I}$. For commutative homogeneous monoids with sets of generators, we prove necessary and sufficient conditions for the k-potency of their subsets. Finally, we apply this result to the binary Goldbach problem in analytic number theory.

We consider elliptic boundary-value problems for differential-difference equations containing incommensurable shifts of arguments in leading terms. Using the reduction of the original problem to a certain nonlocal problem, we examine the solvability of boundary-value problems, the smoothness of solutions, and spectral properties.

In this paper, we discuss the possibility of using the method of integral representations (the Hankel method) for solving the nonstationary problem of heat and mass transfer in a semiconductor target. Some features of this approach to problems of heat and mass transfer in homogeneous and multilayer media are studied. We consider the example of two-dimensional diffusion of minority charge carriers generated by an electron probe. We show that a number of practical problems for multilayer targets with different layer parameters can be solved by the approach developed earlier for problems of heat and mass transfer in homogeneous semiconductor targets.

In this paper, we examine the problem of finding all locally doubly transitive extensions of the translation group of a two-dimensional space. This problem is reduced to the search for finding Lie algebras of locally doubly transitive extensions of the translation group. The basis operators of such Lie algebras are found from solutions of systems of second-order differential equations. We prove that the matrices of these systems commute with each other and can be simplified by reduction to the Jordan form. From the solutions of systems of differential equations, the Lie algebras of all locally doubly transitive extensions of the translation group of the plane are obtained. Using the exponential mapping, we calculate locally doubly transitive Lie transformation groups.

We consider a class of Volterra integro-algebraic equations with two integrable power singularities in the kernel and indicate fundamental difficulties in studying such equations. In terms of matrix pencils, we formulate sufficient conditions for the existence of a unique continuous solution. Also, we propose multi-step methods for solving such equations based on the method of integrating products and Adams quadrature formulas and present the results of numerical experiments.

In this paper, we consider a mixed problem for a second-order hyperbolic equation with constant coefficients and a mixed partial derivative. We assume that the boundary conditions are splitted (i.e., one condition is posed at the left endpoint of the main interval and the other at the right endpoint) and the roots of the characteristic equation are simple and lie on the positive half-line. The coefficients of the equation and the boundary conditions are constrained by conditions that guarantee the absence of the two-fold completeness of eigenfunctions of the corresponding spectral problem for the differential quadratic pencil. Using the Poincare-Cauchy contour integral method, we to obtain sufficient conditions for the solvability of this problem.

We consider the first initial-boundary-value problem for the heat equation in a bounded domain Q with curvilinear lateral boundaries. Using the method of boundary integral equations, we prove the existence of a solution to this problem in the class C₂₁ (Q).

The splitting transformation is a generalization of the well-known Chang transformation for linear, stationary, singularly perturbed system with many delays in slow-state variables; it reduces the original two-speed system to two independent subsystems of smaller dimensions with different rates of change of variables. The splitting transformation leads us to Riccati and Sylvester equations for functional matrices, which can be found in the form of asymptotic series in powers of the small parameter. In this work, we prove that asymptotic approximations of any order of accuracy based on these series can be represented as finite sums in powers of А. We compare exact solutions with approximations obtained by the method proposed.