The paper is devoted to studying the frequencies at which first digits occur in series formed by powers of integer numbers. A number of generalizations of this problem are considered, and the relation between the distribution of first digits and Diophantine properties of logarithms is discussed. In conclusion of the article, several interesting problems in modern theory of Diophantine approximation are proposed.

We find a connection between the Schatten norm of the linear mapping A :V → W
of Euclidean spaces of dimensions n and m, respectively, and the power means generated by
the lengths of the images of vectors of the orthonormal basis V.

We consider the natural action of Galois groups of unramified Galois extensions of number fields on finite Galois stable subgroups of $GL_n$.

For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $G$ minus the generalized Fitting height of $M$ is not greater than 2 and the non-$p$-soluble length of $G$ minus the non-$p$-soluble length of $M$ is not greater than 1. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_\sigma(G, \mathfrak{F})-n_\sigma(M, \mathfrak{F})\mid G$ is finite $\sigma$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_\sigma(G, \mathfrak{F})$ denotes the $\sigma$-nilpotent length of the $\mathfrak{F}$-residual of  $G$.

This construction shows the results about the generalized lengths of maximal subgroups published in Math. Nachr. (1994) and Mathematics (2020) are not correct.

The paper proves the equivalence of two special matrix norms. Both norms arise in models formulated in terms of interactions between binary variables. One norm is associated with the interaction of these variables within a single group, while the other is related to the interaction of variables from different groups. The statement allows for an easy transfer of meaningful results from the second (simpler) case to the first.

The work is the fifth and final one in a series of articles, where for a set $\pi$ consisting of odd primes, finite $\pi$-solvable  irreducible complex linear groups of degree $2|H|+1$ are investigated, for which Hall $\pi$-subgroups are $TI$-subgroups and are not normal in groups. The purpose of the series is to prove solvability and to determine the conditions for factorization of such groups.

A group $L$ is called graded if it is represented as the union of a decreasing sequence of subgroups $L_m$. A general scheme for introducing the so-called sharp metric on such groups is proposed, with respect to which the algebraic operations are continuous and which is non-archimedean. It is shown that such a group is densely embedded in a complete group whose elements are series of a special type composed of elements of $L$. Similar constructions are considered for graded rings and graded vector spaces. As examples, it is shown that in concrete special cases, the application of the described construction leads to the construction of $p$-adic numbers and to the construction of Taylor and Laurent series.

In this paper, sufficient conditions for the existence of trigonometric Hermite–Jacobi approximations of a system of functions that are sums of convergent Fourier series are found. Based on these results, sufficient conditions are established under which nonlinear Hermite–Chebyshev approximations of systems of functions representable by Fourier series in Chebyshev polynomials of the first and second kind exist. When the found conditions are met, explicit formulas are obtained for the numerators and denominators of trigonometric Hermite–Jacobi approximations and nonlinear Hermite–Chebyshev approximations of the first and second kind of the specified systems of functions.

The paper considers the case of a stochastic differential equation in the sense of Ito with drift. For the equation under consideration, a formula for the approximate calculation of mathematical expectations from its solution is constructed. An estimate of the error of the constructed formula is obtained. A numerical experiment is performed.

The paper proposes a stochastic model, which describes the dynamics of biallelic polymorphism in a dairy herd. The proposed model assumes that the herd is formed under controlled mating conditions. The system of equations describing the model is based on the use of random processes containing jumps. A discrete analog of the stochastic system is constructed, for which a Monte Carlo simulation is performed.

A multi-server retrial queueing system with heterogeneous servers is analysed. The service times have a phase-type distribution with different irreducible representations. Customer arrival to the system is defined by a Markovian arrival process. When all servers are busy at an arrival moment, the customer moves to the virtual place called orbit to retry to reach the servers in exponentially distributed periods of time. The total retrial rate from the orbit infinitely increases when the number of customers residing in orbit grows. Upon arrival or retrial from the orbit, a customer occupies the server having the minimal number among all idle servers, if any. The dynamics of the system states is described by a multidimensional Markov chain having the special block structure of the infinitesimal generator. The explicit expression for this is presented. Ergodicity condition is derived. The expressions for computation of the key performance characteristics of the system are given. Numerical results, which highlight dependencies of these measures on the mean arrival rate for the system and its particular cases, when the arrivals are described by the stationary Poisson process or (and) service times follow the exponential distribution, are presented.