# Mathematics

## Wiener's theorem for periodic at infinity functions

In this article banach algebra of periodic at infinity functions is defined. For this class of functions notions of Fourier series and absolutely convergent Fourier series are introduced. As a result Wiener's theorem analog devoted to absolutely convergent Fourier series for periodic at infinity functions was proved.

## Mathematical model of dynamic chaos

The problem of analytical designing on the set mathematical model of dynamic system in space of states of mathematical model accompanying it in phase space is put and solved. It is shown, that the representing point of any decision of dynamic system of a general view in space of states conditions belongs to hypersphere with the displaced centre in phase space (or to central hypersphere of variable radius equivalent to it).

## Algorithm of integrating stiff problems using the explicit and implicit methods

An Lstable method order 3 and an explicit three-stage Runge–Kutta scheme order 1 are constructed. An integration algorithm of variable order and step is constructed that is based on of the two schemes The most effective numerical scheme is chosen for each step by means of stability control. The results are given that confirm the effectiveness of the algorithm.

## The conditions of invertibility of a class nonselfadjoint operators

In this work we obtain the conditions of invertibility of a class of nonselfadjoint operator that are difference between the nonbounded antisymmetric and the normal operator.

## A direct method of stochastic optimization

A direct method is proposed for stochastic programming. On each step the method uses solving of the linear program which is the linear approximation of input stochastic programs.

## On verification of Brauer's theorem concerning Artin's L-functions of number fields

This paper investigates problem of analytic continuation of Artin's L-functions. One refinement of Brauer's theorem was obtained. It states that in the case of non-main character all possible poles of Artin's L-functions should lay on the critical line.

## On spectrum of some classes of matrix operators

The paper is devoted to investigation of the spectrum of some classes of matrix operators. The relations between the parts of the spectrum of the matrix operators with corresponding parts of its elements are established.

## An Analogue of the Jordan–Dirichlet Theorem for the Integral Operator with Kernel Having Jumps on Broken Lines

In this paper the sufficient conditions (conditions such as Jordan–Dirichlet) expansion function f(x) in a uniformly convergent series of eigenfunctions and associated functions of the integral operator whose kernel is suffering jumps on the sides of the square, inscribed in the unit square. As is known, this expansion requires to f(x) is continuous and belong to the closure of the integral values operator. It turns out that if f(x) also is a function of bounded variation, these conditions are also sufficient.

## Integral operator with kernel having jumps on broken lines

In this paper we study equiconvergence expansions in trigonometric Fourier series, and in eigenfunctions and associated functions of an integral operator whose kernel suffers jumps at the sides of the square inscribed in the unit square.

## Frames and periodic groups of operators

In this paper some properties of periodic groups of operators which connected with frames theory are considered. We proof that there are no strongly continuous and uniformly bounded periodic one-parameter group of operators in Banach space which eigenvectors are cross-frame.