# Mathematics

## Stability of three-layer differential-difference schemes with weights in the space of summable functions with supports in a network-like domain

## On the question of the residual of strong exponents of oscillation on the set of solutions of third-order differential equations

In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of non-strict signs, zeros, and roots of non-zero solutions of linear homogeneous differential equations of the third order with continuous and bounded coefficients on the positive semi-axis. A nonzero solution of a linear homogeneous equation cannot be zeroed due to the existence and uniqueness theorem. Therefore, the spectra of all the listed exponents of oscillation (i.e. their sets of values on nonzero solutions) consist of one zero value.

## On functions of van der Waerden type

Let $\omega(t)$ be an arbitrary modulus of continuity type function, such that $\omega(t)/t\to+\infty$, as $t\to+0$.

## Unitary extension principle on zero-dimensional locally compact groups

In this article, we obtain methods for constructing step tight frames on an arbitrary locally compact zero-dimensional group. To do this, we use the principle of unitary extension. First, we indicate a method for constructing a step scaling function on an arbitrary zero-dimensional group. To construct the scaling function, we use an oriented tree and specify the conditions on the tree under which the tree generates the mask $m_0$ of a scaling function.

## Classic and generalized solutions of the mixed problem for wave equation with a summable potential. Part I. Classic solution of the mixed problem

The resolvent approach and the using of the idea of A. N. Krylov on the acceleration of convergence of Fourier series, the properties of a formal solution of a mixed problem for a homogeneous wave equation with a summable potential and a zero initial function are studied. This method makes it possible to obtain deep results on the convergence of a formal series with arbitrary boundary conditions and without overestimating the requirements for the smoothness of the initial data.

## On the iterative method for solution of direct and inverse problems for parabolic equations

The paper is devoted to approximate methods for solution of direct and inverse problems for parabolic equations. An approximate method for the solution of the initial problem for multidimensional nonlinear parabolic equation is proposed. The method is based on the reduction of the initial problem to a nonlinear multidimensional intergral Fredholm equation of the second kind which is approximated by a system of nonlinear algebraic equations with the help of the method of mechanical quadratures.

## Risky investments and survival probability in the insurance model with two-sided jumps: Problems for integrodifferential equations and ordinary differential equation and their equivalence

We consider a model of an insurance portfolio that includes both non-life and life annuity insurance while assuming that the surplus (or some of its fraction) is invested in risky assets with the price dynamics given by a geometric Brownian motion. The portfolio surplus (in the absence of investments) is described by a stochastic process involving two-sided jumps and a continuous drift.

## The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution

An initial boundary value problem for an inhomogeneous second-order hyperbolic equation on a finite segment with constant coefficients and a mixed derivative is investigated. The case of fixed ends is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. The classical solution of the initial boundary value problem is determined.

## On the approximation of bounded functions by trigonometric polynomials in Hausdorff metric

The article discusses a method for constructing a spline function to obtain estimates that are exact in order to approximate bounded functions by trigonometric polynomials in the Hausdorff metric. The introduction provides a brief history of approximation of continuous and bounded functions in the uniform metric and the Hausdorff metric. Section 1 contains the main definitions, necessary facts, and formulates the main result. An estimate for the indicated approximations is obtained from Jackson's inequality for uniform approximations.

## Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions

We consider the problems of multiple interpolation of analytic functions $f(z)=f_0+f_1z+\dots$ in the unit disk with node $z=0$ by means of simple partial fractions (logarithmic derivatives of algebraic polynomials) with free poles and with all poles on the circle $|z|=1$. We obtain estimates of the interpolation errors under a condition of the form $|f_{m-1}|<C/\sqrt{m}$, $m=1,2,\dots$.