Only finite groups are considered. A subgroup $H$ of a group $G$ is called an $n\Phi$-subgroup if there exists a normal subgroup $K$ such that $G=HK$ and $H\cap K$ is contained in the Frattini subgroup of $H$. The structure of a finite group is obtained in the following cases: all normal subgroups are $n\Phi$-subgroups; the Frattini subgroup of the group is trivial and every $n\Phi$-subgroup is normal; every $2$-maximal subgroup is an $n\Phi$-subgroup; every $3$-maximal subgroup is an $n\Phi$-subgroup; for all primes $p$, every subgroup of order $p^2$ is an $n\Phi$-subgroup. For an arbitrary formation $\mathfrak F$, it is established that in the $\mathfrak F$-residual of the a group, every non-trivial $\mathfrak F$-subgroup is not an $n\Phi$-subgroup.

We study irreducible representations of simple algebraic groups over the field $\mathbb{C}$ of~complex numbers. For groups of type $A_2$, $A_3$ and $C_2$ a generalization of the famous Dynkin theorem on the spindle property of the weight system is obtained. This has made it possible to describe the Jordan normal form of unipotent elements in irreducible representations of such groups, i.e., to determine the dimensions of all Jordan blocks of the images of unipotent elements without finding the number of these blocks.

For an odd prime number $r$, we found conditions under which a Sylow $r$-subgroup $G_{r}$ is Abelian and normal in an irreducible complex linear group $G$.

In this paper, we consider Banach spaces of operators from $\ell^p$ to $\ell^q$ that can be realized as infinite matrices. We show that for $p>1$ and $q<\infty$, for almost all subspaces formed by randomly chosen matrix units, the canonical projectors onto these subspaces will be unbounded. Moreover, these projectors will be unbounded even on the class of matrices with elements $a_{ij}\in\{-1,0,1\}$.

Generalized Riemann boundary problems are studied on a closed curve located on the complex plane. The boundary condition of the problems, along with the limit values of the desired functions, includes the limit values of their derivatives. The boundary conditions is written using determinants close to Vronsky's determinants. The solution of the problems is reduced to solving the classical Riemann problem and solving linear differential equations in areas of the complex plane with some restrictions on the solutions. The conditions for the solvability of the initial problems are indicated explicitly, and when they are fulfilled, explicit formulas for solutions are indicated. Examples are given.

The object of the paper's research is the stochastic differential equation of Ito with drift. The paper proposes a method for approximate calculation of mathematical expectations of functions from the solution of such equation. The method is based on the use of an auxiliary random process of a special kind that depends only on the Wiener process. This approach allows us to use the already known approximate calculation formulas for the case when the functional depends only on the Wiener process to obtain an approximate value of the desired mathematical expectation. The paper presents the results of a numerical experiment.

A linear periodic system with a constant non-degenerate matrix under control is considered. The program control is periodic, and its period is incommensurable with the period of the coefficient matrix. The admissible set of such periodic controls is called irregular. The problem is to choose such a control from this admissible set so that the now quasi-periodic system has a partially irregular periodic solution with a given frequency spectrum whose period coincides with the period of the control. Such problem is called the control problem of the asynchronous spectrum with an irregular admissible set. To solve it, the original system is reduced to some linear nonhomogeneous system of lower dimension. The non-resonant case is studied, when the corresponding homogeneous system has no irregular periodic solutions. A necessary condition for the solvability of the control problem of the asynchronous spectrum with an irregular admissible set is obtained.

For a one-dimensional mildly quasilinear wave equation given in the first quadrant, we consider a mixed problem in which Cauchy conditions are specified on the spatial semi-axis and a Dirichlet condition is specified on the temporal semi-axis. The nonlinearity contains independent variables, the unknown function, and its derivatives. We construct the solution in implicit analytical form as the solution of some integro-differential equations. We prove the solvability of these integro-differential equations using a generalization of the Banach fixed-point theorem. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists

The article focuses on the development of a computational procedure for determining the effective Young’s modulus of composite powder materials based on the analysis of their microstructure. The proposed approach includes constructing a two-dimensional geometric model of the composite microstructure as a periodicity cell, formulating a mathematical model that simulates a physical tension–compression test of a representative volume element, and performing numerical implementation using the control volume method on structured quadrilateral grids. The properties of the matrix and inclusions, their volume fractions, and morphology are taken into account. The effective elastic modulus is calculated based on an energy balance: the work done by external forces is equated to the sum of the strain energies of all grid cells. The results of computational experiments are presented for copper-matrix composites with tungsten carbide and Teflon inclusions, as well as for NiCr-based gradient coatings with TiC additives. It is shown that the procedure allows predicting the variation of the elastic modulus as a function of the filler volume fraction and can be used in the design of layered wear-resistant coatings.

Detailed proof of ergodicity for a multi-server retrial queueing system with heterogeneous servers, service times having a phase-type distribution with different irreducible representations and customer arrival defined by a Markovian arrival process is given. The proof consists of the use of the asymptotically quasi-Toeplitz Markov chains and Markov renewal processes theory

A metrizable topologycal space $X$ and the set $\Omega_X$ of all metrics generating the topology of $X$ are considered. It is well known that, in general, equivalent metrics $\rho$ and $\sigma$ from $\Omega_X$ can determine different Hausdorff metric topologies $\tau_{\widehat{\rho}}$ and $\tau_{\widehat{\sigma}}$, different proximal topologies $\tau_{\delta (\rho)}$ and $\tau_{\delta (\sigma)}$ and different Wijsman topologies $\tau_{W(\rho)}$ and $\tau_{W(\sigma)}$ on $\exp X$. Thus the families of topologies $\mathcal{T}_H = \{\tau_{\widehat{\rho}}\: | \: \rho\in\Omega_X\}$, $\mathcal{T}_{\delta} = \{\tau_{\delta (\rho)}\: | \: \rho\in\Omega_X\}$ and $\mathcal{T}_W = \{\tau_{W({\rho})}\: | \: \rho\in\Omega_X\}$ appear on the $\exp X$. We've described the cases when the infimums of this families, i.e. the topologies $\tau_{H(\inf)} = \inf\mathcal{T}_H$, $\tau_{\delta (\inf)} = \inf\mathcal{T}_{\delta}$ and $\tau_{W(\inf)} = \inf\mathcal{T}_W$ coincide: $\tau_{H(\inf)} =\tau_{\delta (\inf)}$ if and only if the space $X$ is second countable, $\tau_{H(\inf)} =\tau_{W(\inf)}$ and $\tau_{\delta (\inf)} = \tau_{W(\inf)}$ if and only if the space $X$ is compact. Besides it is found that the topologies $\tau_{\delta (\inf)}$ and $\tau_{W(\inf)}$ are sequential if and only if the space $X$ is second countable.