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

# Mathematics

## Binary basic splines in MRA

$B$-splines were introduced by Carry and Schoenberg. Constructed on a uniform mesh and defined in terms of convolutions, such splines generate a Riesz MRA. We constructed splines $varphi_n$, where $n$ is the order of integration of the Walsh function with the number $2^n - 1$. We called these splines binary basic splines. We know that binary basic splines form a basis in the space of functions that are continuous on the segment $[0, 1]$ and $0$ outside of it.

## Harmonic analysis of functions almost periodic at infinity in Banach modules

The article is devoted to homogeneous spaces of functions defined on a locally compact Abelian group and with their values in a complex Banach space. These spaces include a~umber of well-known spaces such as the spaces of Lebegue-measurable summable functions, substantially limited functions, bounded continuous functions, continuous vanishing at infinity functions, Stepanov and Holder spaces. It is important that they can be endowed with structure of Banach modules, defined by the convolution of functions.

## The structure of groups with cyclic commutator subgroups indecomposable to a subdirect product of groups

The article studies finite groups indecomposable to subdirect product of groups (subdirectly irreducible groups), commutator subgroups of which are cyclic subgroups. The article proves that extensions of a primary cyclic group by any subgroup of its automorphisms completely describe the structure of non-primary finite subdirectly irreducible groups with a cyclic commutator subgroup.

## Distance between strongly and weakly convex sets

The problem of finding the distance between non-intersecting strongly convex and weakly convex (as defined by J.-F. Viall) sets of finite-dimensional space is considered. Three alternative formalizations in the form of extremal problems are used in presenting the results. We obtained the necessary conditions for the solution of the problem taking into account the constants of strong and weak convexity of the sets and their other characteristics.

## Two-dimensional limit series in ultraspherical Jacobi polynomials and their approximative properties

Let $C[-1,1]$ be the space of functions continuous on the segment $[-1,1]$, $C[-1,1]^2$ be the space of functions continuous on the square $[-1,1]^2$. We denote by $P_n^\alpha(x)$ the ultraspherical Jacobi polynomials. Earlier, for function $f$ from the space $C[-1,1]$ limit series were constructed by the system of polynomials $P_n^\alpha(x)$ and the approximative properties of their partial sums were investigated. In particular, an upper bound for the corresponding Lebesgue function was obtained.

## 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.