Throughout this paper, all groups are finite. A group class closed under taking homomorphic images and finite subdirect products is called a formation. The symbol σ denotes some partition of the set of all primes. V. G. Safonov, I. N. Safonova, A. N. Skiba (Commun. Algebra. 2020. Vol. 48, № 9. P. 4002–4012) defined a generalized formation σ-function. Any function f of the form f : σ È {Ø} → {formations of groups}, where f(Ø) ≠ ∅, is called a generalized formation σ-function. Generally local formations or so-called Baer-σ-local formations are defined by means of generalized formation σ-functions. The set of all such formations partially ordered by set inclusion is a lattice. In this paper it is proved that the lattice of all Baerσ-local formations is algebraic and modular.

This article is devoted to the development of a numerical-analytical method for constructing extremes in the Chebyshev norm polynomials, given on the square of the complex plane. The studied polynomials are a generalization of the classical Chebyshev polynomials of the first kind. In the complex case there are no classical Chebyshev alternance conditions, and the Kolmogorov criterion along with the Ivanov – Remez criterion are difficult to prove for establishing the extremality property of specific polynomials. On the basis of the subdifferential construction developed by the authors of the article the extremal polinomials on the squares of the complex plane are calculated in an explicit way. The basic research methods are the methods of functional and complex mathematical analysis, as well as the Maple 2021 computer mathematics system. Methods of function theory and some general results of optimization theory are also used.

In this article, we study a mixed problem in a quarter-plane for one system of differential equations, which describes vibrations in the string from viscoelastic material, which corresponds to the Maxwell model. At the bottom of the boundary, we pose the Cauchy conditions, and one of them has a discontinuity of the first kind at one point. We set a smooth boundary condition on the lateral boundary. We derive the Klein – Gordon – Fock equation for one function of the studied system. We use the method of characteristics to build the classical solution as a solution of some integral equation. We prove the uniqueness and establish conditions under which a piecewise smooth solution exists. The Cauchy problem is considered the system’s second function. We determine the conditions under which the solution of the system has sufficient smoothness

The subject of this paper is the mixing time of random walks on minimal Cayley graphs of complex reflection groups G(m,1,n). The key role in estimating it is played by the coupling of distributions, which has been used before for the same task on symmetric groups. The difficulty with its adaptation for the current case is that there are now two components in a walk, which are to be coupled, and they influence each other’s behaviour. To solve this problem, random walks are split into several blocks for each of which the time needed for their states to match is estimated separately. The result is upper and lower bounds on mixing times of random walks on complex reflection groups, analogous to those obtained by Aldous for a symmetric group.

In recent years, the eigenvalues of the distance matrix of a graph have attracted a lot of attention of mathematicians, since there is a close connection between its spectrum and the structural properties of the graph. Thus, quite recently an interesting result was obtained, relating the Hamiltonicity of a graph to the distance spectral radius of the graph, on the basis of which a more general conjecture about the Hamiltonicity of a graph was formulated. We confirm this conjecture put forward for a k-connected graph, when k Î{2;3}, and also establish similar sufficient conditions for the traceability of a k-connected graph, when k Î{1; 2}.

In this paper, we consider the problem of the classical and quantum movement of a charged particle in a two-dimensional Lobachevsky space in the presence of analogues of uniform magnetic and electric fields. Based on this consideration, equations for the conductivity for the classical and quantum Hall effect are obtained. It is shown that in Lobachevsky space the presence of a small electrical field leads to a shift of the stair structure of the quantum Hall conductivity.

AgIn₇S₁₁ single crystals are herein grown by the vertical Bridgman method. The composition of the obtained single crystals is determined by X-ray microprobe analysis as well as the crystal structure – by X-ray diffraction analysis. It is shown that the obtained single crystals are crystallized in the cubic spinel structure. Using transmission spectra in the tem- perature range 10–320 K we determined the band gap of these single crystals and plotted its temperature dependence. This dependence is similar to that of the majority of semiconductor materials, namely, Eg increases with decreasing the tempera- ture. We showed the agreement of the calculated and experimental values.