# Mathematics

## 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 $S$.

## On the solvability of a class of nonlinear Hammerstein integral equations on the semiaxis

The paper studies a class of nonlinear integral equations on the semiaxis with a non-compact Hammerstein operator. It is assumed that the kernel of the equation decreases exponentially on the positive part of the number axis. Equations of this kind arise in various fields of natural science. In particular, such equations are found in the theory of radiation transfer in spectral lines, in the mathematical theory of the space-time propagation of an epidemic, in the kinetic theory of gases.

## On generation of a limit cycle from a separatrix loop of a sewn saddle-node

The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node — an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system.

## Stochastic model of innovation diffusion that takes into account the changes in the total market volume

The article proposes a stochastic mathematical model of the diffusion of consumer innovations, which takes into account changes over time in the total number of potential buyers of an innovative product. A stochastic differential equation is constructed for a random value of the number of consumers of an innovative product. The interaction of random changes in the number of consumers with changes in the total market volume of the product under consideration is investigated.

## On the approximation of class C(0) components of physical quantities in curved coordinate systems

In numerical methods for calculating the strength of technospheric objects approximating expressions of the desired values in terms of their nodal values are widely used.