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

# Mathematics

## 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\,--\,Pad\'e approximants (simultaneous Pad\'e 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\,--\,Pad\'e 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.

## Quasi-polynomials of Capelli. III

In this paper polynomials of Capelli type (double and quasi-polynomials of Capelli) belonging to a free associative algebra $F\{X\cup Y\}$ considering over an arbitrary field $F$ and generated by two disjoint  countable  sets $X, Y$  are investigated.  It  is shown  that  double Capelli's  polynomials $C_{4k,\{1\}}$, $C_{4k,\{2\}}$ are consequences of the standard polynomial $S^-_{2k}$.

## On maximal subformations of n-multiple Ω-foliated formations of finite groups

Only finite groups are considered in the article. Among the classes of groups the central place is occupied by classes closed regarding homomorphic images and subdirect products which are called formations. We study $\Omega$-foliateded formations constructed by V. A. Vedernikov in 1999 where $\Omega$ is a nonempty subclass of the class $\frak I$ of all simple groups.

## The research of some classes of almost periodic at infinity functions

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions.

## Subsystems and Automorphisms of Some Finite Magmas of Order k + k2

This work is devoted to the study of subsystems of some finite magmas S = (V, ∗) with a generating set of k elements and order k + k2. For k > 1, the magmas S are not semigroups and quasigroups. An element-by-element description of all magmas S subsystems is given. It was found that all the magmas S have subsystems that are semigroups. For k > 1, subsystems that are idempotent nonunit semigroups are explicitly indicated. Previously, a description of an automorphism group was obtained for magmas S.

## Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem.

## On Some Diagram Assertions in Preabelian and P-Semi-Abelian Categories

As is well known, many important additive categories in functional analysis and algebra are not abelian. Many classical diagram assertions valid in abelian categories fail in more general additive categories without additional assumptions concerning the properties of the morphisms of the diagrams under consideration. This in particular applies to the so-called Snake Lemma, or the KerCoker-sequence. We obtain a theorem about a diagram generalizing the classical situation of the Snake Lemma in the context of categories semi-abelian in the sense of Palamodov.

## Ωζ-foliated Fitting Classes

All groups under consideration are assumed to be finite. For a nonempty subclass of Ω of the class of all simple groups I and the partition ζ = {ζi | i ∈ I}, where ζi is a nonempty subclass of the class I, I = ∪iI ζi and ζi ∩ ζj = ø for all i ≠ j, ΩζR-function f and ΩζFR-function φ are introduced.