# Mathematics

## The M.A. Lavrentiev Inverse Problem on Mapping of Half-Plane Onto Polygon with Infinite Set of Vertices

The authors consider a generalization of the M.A.Lavrentiev inverse problem on a conformal mapping of half-plane onto interiority of a polygon for the case where the set of vertices of this polygon is infinite. We assume that the inner angles at unknown vertices and the image of the vertices under the conformal mapping on the real line are given. Under certain restrictions on values of the angles and on the sequence of points of the real line that are preimages of the vertices the formula for such a mapping is obtained.

## The Approached Calculation of Eigenvalues of the Discrete Operator by Means of Spectral Traces of Resolvent Degree

Letadiscreteself-adjointoperatorT actsinaseparableHilbertspace and have the kernel resolvent, and eigenvalues and eigenfunctions of the operator T be known. In the paper the method of calculation of eigenvalues of the perturbed operator T + P is considered. Resolvent of this operator is presented as convergent Neumann series on eigenfunctions of the operator T. The point of the method is that at ﬁrst is found a set of numbers which approximate traces of the resolvent degrees of the operator T + P.

## About the Congruences of Two-Generated Monoid

The congruences of two-generated monoid which generated by pair of words of length 2 are considered over two-letter alphabet. It is shown that number of equivalence classes for words of length n is equal to n + 1. The number of words in each class is found.

## The Problem of Convergence in Point Trigonometric Interpolation Process of Lagrange

An analogue of the characteristic of R. Salem is obtained for a trigonometric Lagrange interpolation process on the matrix of equally spaced nodes.

## On Inverse Problem for Sturm – Liouville Operator with Discontinuous Coefﬁcients

In the paper uniqueness of reconstruction of the Sturm – Liouville operator with discontinuous coefﬁcients by spectral data is proved and algorithm of construction of the potential is provided.

## The Full Class of Smooth Axially Symmetric Longitudinal-Vortex Unit Vector Fields

In the paper, two vector ﬁelds are constructed by means of transformation method. The ﬁrst describes the axially symmetric unit solutions (ASUS) of the Gromeka problem to ﬁnd out vector ﬁelds which ﬂow lines coincide in R3 with vortex lines. The second describes the smooth ASUS of the extended in this paper Gromeka problem of ﬁnding a vector ﬁelds with different vortex properties in adjacent parts of R3.

## About Solution of Discrete Linear-Quadratic Optimal Control Problem

This paper is focuseson development of the method forexact solution of the optimal control problem for discrete linear system with quadratic criteria, with boundary conditions and constraints on control. This method give sasolution of ﬁnite number of systems of linear algebraic equations.

## The Error of Approximation of Differentiable Functions of Several Variables by Means of Interpolatory Shape-Preserving Operators

The article deals with the estimation of the error of uniform approximation of differentiable functions of several variables with limited second derivations by means of linearinterpolation operators, which preserve the properties of positivity and convexity of approximated functions.

## Repair Technology Basis of Turbine Disks by Using StressStrain State Parameters

Stress-strain state of power steam turbine disks under operation conditions including both contouran dtighten loadings is considered. Full-size elastic-plastic stress-strain state analysis of turbine disk for different variants of considering key geometries is represented. As a result of numerical calculations three critical zones of turbine disk are deﬁned. Proposed design modiﬁcations and repair technology to existing in-service power steam turbine disks by removing of damaged material volume are analyzedand substantiate donastress state parameters basis.

## About Completeness of Products of Functions, Initiated by Singular Differential Equations

In this article we introduced the completeness theorem for special vector-functions, initiated by products of Weil solutions of forth order differential equation and its derivatives on the halfline. We prove that such nonlinear combinations of Weil solutions and its derivatives form the linear subspace of solutions, which decrease to infinity, of linear singular Kamke-type differential system.