ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

# Mathematics

## On a concrete characterization problem of universal graphic semiautomata

Automata theory is one of the branches of mathematical cybernetics, that studies information transducers that arise in many applied problems. The major objective of automata theory is to develop methods by which one can describe and analyze the dynamic behavior of discrete systems. In this paper, we consider automata without output signals (called semiautomata). Depending on study tasks, semiautomata are considered, for which the set of states is equipped with additional mathematical structure preserved by the transition function of semiautomata.

## Kolyada inequality for partial moduli of smoothness of functions with lacunary Fourier coefficients

The problem of estimating the moduli of smoothness of functions from $L_q$  in terms of moduli of smoothness from $L_p$  is well known. The initial stage in estimating the moduli of smoothness was the study of properties of functions from Lipschitz classes and obtaining the corresponding embeddings in the works of Titchmarsh, Hardy, Littlewood, and Nikolsky. P. L. Ulyanov for the moduli of continuity of functions of one variable proved an inequality later named after him — "Ulyanov's inequality".

## Representation of Green’s functions of the wave equation on a segment in finite terms

Solutions of initial-boundary value problems on the excitation of oscillations of a finite segment by an instantaneous point sourse are investigated. Solutions to these problems, called Green's functions of the equation of oscillations on a segment, are known in the form of infinite Fourier series or series in terms of Heaviside functions. A. N.

## Representation of functions on a line by a series of exponential monomials

In this work, we consider the weight spaces of integrable functions $L_p^\omega$ ($p\geq 1$) and continuous functions $C^\omega$ on the real line. Let $\Lambda=\{\lambda_k,n_k\}$ be an unbounded increasing sequence of positive numbers $\lambda_k$ and their multiplicities $n_k$, $\mathcal{E}(\Lambda)=\{t^n e^{\lambda_k t}\}$ be a system of exponential monomials constructed from the sequence $\Lambda$.

## Divergent series and generalized mixed problem for a wave equation of the simplest type

With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series  which corresponds to the simplest mixed problem with a summable initial function.

## Application of queueing network models in insurance

The purpose of this paper is to study the issues of the functioning of insurance companies using the methods of the queueing networks theory. The introduction provides a brief overview of scientific publications in this area. In particular, research based on the use of Markov stochastic processes and queueing systems are considered. In the first section of the article, a closed exponential queueing network is proposed as a model for the process of processing insurance claims. A detailed description of the corresponding network model is given.

## On a solution of a nondegenerate boundary value problem of Carleman type for quasiharmonic functions in circular domains

This paper considers a Carleman type boundary value problem for quasiharmonic functions. The boundary value problem is an informal model of a Carleman type differential problem for analytic functions of a complex variable.This paper presented a complex-analytical method for solving the problem under consideration in circular domains, which makes it possible to establish the instability of its solutions concerning small contour changes.

## Elementary definability of the class of universal hypergraphic automata in the class of semigroups

Hypergraphic automata are automata, state sets and output symbol sets of which are hypergraphs, being invariant under actions of transition and output functions. Universally attracting objects in the category of hypergraphic automata are called universal hypergraphic automata. The  semigroups of input symbols of such automata are derivative algebras of  mappings for such automata. So their properties are interconnected with  the properties of the algebraic structures of the automata.

## On the continuity of some classes and subclasses of maps with an s-averaged characteristic

According to the well-known theorem of S. L. Sobolev, if $G$ is a bounded domain of Euclidean space and a function is a function having the first generalized derivatives summable with degree  $p$, then it is continuous in $G$. If $1<p\le n$  this property, generally speaking, may not be. In this paper, we find the necessary conditions under which some classes and subclasses of maps with an $s$-averaged characteristic will be continuous. Examples of subclasses of such mappings with the above properties are given in our papers.

## Forcing total outer connected monophonic number of a graph

For a connected graph $G = (V,E)$ of order at least two, a subset $T$ of a minimum total outer connected monophonic set $S$ of $G$ is a forcing total outer connected monophonic subset for $S$ if $S$ is the unique minimum total outer connected monophonic set containing $T$. A forcing total outer connected monophonic subset for $S$ of minimum cardinality is a minimum forcing total outer connected monophonic subset of $S$.