# Mathematics

## Approximating Properties of Solutions of the Differential Equation with Integral Boundary Condition

With the use of the solution of the first-order differential equation the approximations to the continuous functions with integral boundary conditions are constructed.

## Hilbert Generalizations b-Bessel Systems

The notion of b-Bessel systems that generalizes the known classic notion of Bessel systems is introduced, the criteria of Bessel property of the systems are established. Some properties of the space of coefficients corresponding to the b-basis generalizing the classic notion of Schauder basis are studied.

## On One Special Mapping

The paper deals with the study of a specialmapping in connection with the structure of extremal functions in exact Kolmogorov inequalities on the half-line in the uniform metrics.

## On Weighted Analogs of Wiener’s and Levy’s Theorems for Fourier – Vilenkin Series

In this paper we find the general form of complex homomorphism for some subalgebras of absolutely convergent Fourier – Vilenkin series algebra. As a corollary, we obtain weighted analogs of Wiener’s and Levy’s theorems for Fourier – Vilenkin series.

## Variable Order and Step Algorithm Based on a Stages of Runge – Kutta Method of Third Order of Accuracy

An inequality for the stability control of 3-stage Runge – Kutta method of 3th order of accuracy is obtained. Method of first order with expanded stability domain is constructed. Algorithm of variable order is formulated. The results of stiff system computations are provided, which confirm an increase in efficiency for the variable order method as compared to a calculation with fixed scheme.

## Analysis of Heterogeneous Queueing Networks with Batch Movements of Customers

Closed exponential queueing network with different classes of customers and batch movements is considered. To model evolution of given network Markov chains are used. Two approaches to stationary distribution calculation for given type queueing networks are presented. Formulas for basic stationary characteristics are given.

## About Asymptotic Polynomials, Orthogonal on Any Grids

Asymptotic properties of polynomials orthogonal ln(x), with weight e −xj ∆tj on any infinite set points from semi-axis [0, ∞) are investigated. Namely an asymptotic formula is proved in which asymptotic behaviour of these polynomials as n tends to infinity together with N is closely related to asymptotic behaviour of the polynomials by Lagerra.

## Nonorthogonal Multiresolution Analysis on Zero-Dimensional Locally Compact Groups

We given necessary and sufficient condition under which the solution of refinement equation with compactly supported Fourier transform generate the multiresolution analysis.

## On Characterization Determining Entire Functions and Consistent with Riman’s Type Equation Dirichlet’s Series with Finetly-Valued Coefficients

In the investigation were founded specifications for Dirichlet’s series coefficients, wherein this series determine entire function and measure up functional Riman’s type equation. Were shown that exist infinit multitude of such series that are different from Dirichlet’s L-functions.

## Approximation of Functions by Borel Means of Fourier Series with Respect to Multiplicative Systems

In the present paper we consider Borel means of Fourier series with respect to Vilenkin systems with bounded generating sequence and obtain some estimates of approximation by this means in L p , uniform and Holder type norm in classes of functions with given majorant of best approximation or modulus of continuity. In the trigonometric case similar results were established by P.Chandra, L.Rempulska and K.Tomczak.