# Mathematics

## 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 + k^{2}. 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 = {ζ_{i} | i ∈ I}, where ζ_{i} is a nonempty subclass of the class I, I = ∪_{i}_{∈}_{I} ζ_{i} and ζ_{i} ∩ ζ_{j} = ø for all i ≠ j, ΩζR-function f and ΩζFR-function φ are introduced. The domain of these functions is the set Ωζ ∪ {Ω′}, where Ωζ = { Ω ∩ ζ_{i} | Ω ∩ ζ_{i} ≠ ø }, Ω′ = I \ Ω.

## On the Uniform Convergence of the Fourier Series by the System of Polynomials Generated by the System of Laguerre Polynomials

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and L^{p}_{w} be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by W^{r}_{Lpw} the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space L^{p}_{w}.

## On Determination of Functional-Differential Pencils on Closed Sets from the Weyl-Type Function

Second-order functional-differential pencils on closed sets are considered with nonlinear dependence on the spectral parameter. Properties of their spectral characteristics are obtained and the inverse problem is studied, which consists in recovering coefficients of the pencil from the given Weyl-type function. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points.

## Smooth Approximations in C[0, 1]

The first orthonormal basis in the space of continuous functions was constructed by Haar in 1909. In 1910, Faber integrated the Haar system and obtained the first basis of continuous functions in the space of continuous functions. Schauder rediscovered this system in 1927. All functions of Faber – Shauder are piecewise linear, and partial sums are inscribed polygons. There was many attempts to build smooth analogues of the Faber – Schauder basis. In 1965, K. M. Shaidukov succeeded. The functions he constructed were smooth, but consisted of parabolic arcs.

## On the Approximate Solution of a Class of Weakly Singular Integral Equations

The work is devoted to the study of the solution of one class of weakly singular surface integral equations of the second kind. First, a Lyapunov surface is partitioned into “regular” elementary parts, and then a cubature formula for one class of weakly singular surface integrals is constructed at the control points. Using the constructed cubature formula, the considered integral equation is replaced by a system of algebraic equations.

## New Method for Investigating the Hilbert Boundary Value Problem with an Infinite Logarithmic Order Index

We consider the problem of identification of the analytical in the complex upper half plane by boundary condition on the entire real axis, according to which, the real part of the product, by the given on the real axis complex function and the boundary values of the desired analytical function equal zero everywhere on the real axis.

## On Customary Spaces of Leibniz –Poisson Algebras

Let K be a base field of characteristic zero. It is well known that in this case all information about varieties of linear algebras V contains in its polylinear components P_{n}(V), n ∈ N, where P_{n}(V) is a linear span of polylinear words of n different letters in a free algebra K(X,V). D. Farkas defined customary polynomials and proved that every Poisson PI-algebra satisfies some customary identity. Poisson algebras are special case of Leibniz –Poisson algebras.