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

# Mathematics

## Differential operators on graphs with a cycle

An inverse problem of spectral analysis is studied for Sturm – Liouville differential operators on a graph with a cycle. We pay the main attention to the most important nonlinear  inverse problem of recovering coefficients of differential equations provided that the structure of the graph is known a priori. We use the standard matching conditions in the interior vertices  and Robin boundary conditions in the boundary vertices.

## On the convergence of the order-preserving weak greedy algorithm for subspaces generated by the Szego kernel in the Hardy space

In this article we consider representing properties of subspaces generated by the Szego kernel. We examine under which conditions on the sequence of points of the unit disk the order-preserving weak greedy algorithm for appropriate subspaces generated by the Szego kernel converges. Previously, we constructed a representing system based on discretized Szego kernels.

## The explicit solution of the Neumann boundary value problem for Bauer differential equation in circular domains

The article is devoted to the boundary value problem of Neumann problem's type for solutions of one second-order elliptic differential equation. Based on the general representation of the solutions of the differential equation as two analytical functions of a complex variable, and also taking into account the properties of the Schwarz equations for circles, it is established that in the case of circular domains, the boundary value problem is solved explicitly, i.e., its general solution can be found using only the F. D.

## Solutions of the Loewner equation with combined driving functions

The paper is devoted to the multiple chordal Loewner differential equation with different driving functions on two time intervals. We obtain exact implicit or explicit solutions to the Loewner equations with piecewise constant driving functions and with combined constant and square root driving functions. In both cases, there is an analytical and geometrical description of generated traces.

## Non-reductive spaces with equiaffine connections of nonzero curvature

The introduction of this article states the object of our investigation which is structures on homogeneous spaces. The problem of establishing links between the curvature and the structure of a manifold is one of the important problems of geometry. In general, the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem in a narrower class of non-reductive homogeneous spaces. If a homogeneous space is reductive, then the space admits an invariant connection.

## Analytic embedding of pseudo-Helmholtz geometry

For modern geometry, the study of maximal mobility geometries is of great importance. Some of these geometries are well studied (Euclidean, pseudo-Euclidean, symplectic, spherical, Lobachevsky, etc.), and others are poorly understood (Helmholtz, pseudo-Helmholtz, etc.). There is no complete classification of geometries for maximum mobility. In this paper part of this large classification problem is solved.

## Reconstruction formula for differential systems with a singularity

Our studies concern some aspects of scattering theory of the singular differential systems $y'-x^{-1}Ay-q(x)y=\rho By$, $x>0$ with $n\times n$ matrices $A,B, q(x), x\in(0,\infty)$, where $A,B$ are constant and $\rho$ is a spectral parameter. We concentrate on the important special case when $q(\cdot)$ is smooth and $q(0)=0$ and\linebreak  derive a formula that express such $q(\cdot)$ in the form of some special contour integral, where the kernel can be written in terms of the Weyl-type solutions of the considered differential system.

## On periodic solutions of Rayleigh equation

New sufficient conditions for the existence and uniqueness of a periodic solution of a system of differential equations equivalent to the Rayleigh equation are obtained. In contrast to the known results, the existence proof of at least one limit cycle of the system is based on applying curves of the topographic Poincare system. The uniqueness of the limit cycle surrounding a complex unstable focus is proved by the Otrokov method.

## About the convergence rate Hermite – Pade approximants of exponential functions

This paper studies uniform convergence rate of Hermite – Pade approximants (simultaneous Pade approximants) $\{\pi^j_{n,\overrightarrow{m}}(z)\}_{j=1}^k$ for a system of exponential functions $\{e^{\lambda_jz}\}_{j=1}^k$, where $\{\lambda_j\}_{j=1}^k$ are different nonzero complex numbers. In the general case a research of the asymptotic properties of Hermite – Pade approximants is a rather complicated problem.

## Numerical solution of linear differential equations with discontinuous coefficients and Henstock integral

We consider the  problem of approximate solution of linear differential equations with discontinuous coefficients. We assume that  these coefficients have $f$-primitive. It means that  these coefficients are Henstock integrable only. Instead of the original Cauchy problem,  we consider a different problem with piecewise-constant coefficients. The sharp solution of this new problem is the approximate solution of the original Cauchy problem. We found the degree of approximation in terms of $f$-primitive for Henstock integrable coefficients.