# Mathematics

## 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.

## Tolerant Bundle of Ways and Gurevich’s Theorem for Tolerant Spaces

In the article the tolerant bundle of ways with tolerant collapsible bundle space is constructed and the tolerant analogue of the Gurevich’s the oremisproved by mean sofexact homotopic sequence and homology spectral sequence of tolerant bundles .

## Solution of Inverse Problem for the Diffusion Operator in a Symmetric Case

In the paper uniqueness of reconstruction of the diffusion operator by aspectrum is proved and sufﬁcient solvability conditions are provided.

## On Riesz Basises of Eigenfunctions of Integral Operators with Kernels Discontinuous on Broken Lines

For the integral operator, which kernel has jump discontinuities on the sides and diagonals of the four equal subsquares of the unit square 0 ≤ x, t ≤ 1, Riesz basisness of its eigen and associated functions is proved.

## On Spectral Property of Matrix Operators in Banach Space

The paper is covers to the investigation of spectral property of matrix operators in Banach space. One matrix operator obtained on linearization of a polynomial operator bundle is being searched resolution of identity for its spectral properties.

## On the Same Theorem on a Equiconvergence at the Whole Segment for the Functional Differential Operators

The equiconvergence of expansions in eigen- and adjoint functions of functional-differential operator with involution, containing the potentials, and simplest functional-differential operator at the whole segment of Fourier series is established.

## Absolute Convergence of Single and Double Fourier Series on Multiplicative Systems

Two-dimensional analogs of famous Zygmund and Szasz tests for absolute convergence of Fourier – Vilenkin series are established. Also it is proved that two-dimensional Szasz test is the best possible in the certain sense.