The paper establishes a relationship between the values of two integer polynomials without common roots on disjoint intervals of fixed length with the main characteristics of the polynomials – degree and height. The proved theorem can be considered as a two-dimensional generalization of Gelfond’s lemma from the theory of transcendental numbers. The theorem can be used to estimate from above the Hausdorff dimension of a set of vectors that are approximated by conjugate algebraic numbers in a given order.

Let $G$ be a finite group and ${\cal L}_{sn}(G)$ be the lattice of all subnormal subgroups of $G$. Let $A$ and $N$ be subgroups of $G$ and $G\in {\cal L}$ be a sublattice of ${\cal L}_{sn}(G)$, that is, $A\cap B$, $\langle A, B \rangle \in {\cal L}$ for all $A, B \in {\cal L} \subseteq {\cal L}_{sn}(G)$. Then: $A^{{\cal L}}$ is the $\cal L$-closure of $A$ in $G$, that is, the intersection of all subgroups in $ {\cal L}$ containing $A$ and $A_{{\cal L}}$ is the $\cal L$-core of $A$ in $G$, that is, the subgroup of $A$ generated by all subgroups of $A$ belonging $\cal L$. We say that $A$ is an $N$-${\cal L}$-subgroup of $G$ if either $A\in {\cal L}$ or $A_{{\cal L}} < A < A^{\cal L}$ and $N$ avoids every composition factor $H/K$ of $G$ between $A_{{\cal L}}$ and $ A^{\cal L}$, that is, $N\cap H=N\cap K$. Using this concept, we give new characterizations of soluble and supersoluble finite groups. Some know results are extended.

   The canonical locals definitions of the classes of groups defined by the systems of generalized subnormal subgroups in the case when these classes are local are constructed in the paper. Conditions are found under which a class of groups defined by a system of generalized subnormal subgroups is a Fitting formation.

All groups under consideration are finite. Let $\sigma =\{\sigma_{i} \mid i\in I \}$ be some partition of the set of all primes, $G$ be a group, $\sigma (G)=\{\sigma_i\mid \sigma_i\bigcap \pi (G)\ne \varnothing\} $, $\mathfrak F$ be a class of groups, and $\sigma (\mathfrak{F})=\bigcup_{G\in \mathfrak{F}}\sigma (G).$ A function $f$ of the form $f:\sigma \to\{\text{formations of groups}\}$ is called a formation σ‑function. For any formation σ‑function $f$ the class $LF_{\sigma}(f)$ is defined as follows: $ LF_{\sigma}(f)=(G \mid G=1 \ \text{or }\ G\ne 1\ \text{and }\ G/O_{\sigma_i', \sigma_i}(G) \in f(\sigma_{i}) \ \text{ for all } \sigma_i \in \sigma(G)). $ If for some formation σ‑function $f$ we have $\mathfrak{F}=LF_{\sigma}(f),$ then the class $\mathfrak{F}$ is called $\sigma $-local and $f$ is called a σ‑local definition of $ \mathfrak{F}.$ Every formation is called 0-multiply $\sigma $-local. For $n \geqslant 1,$ a formation $\mathfrak{F}$ is called $n$-multiply $\sigma $-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f),$ where $f(\sigma_i)$ is $(n-1)$-multiply σ‑local for all $\sigma_i\in \sigma (\mathfrak{F}).$ Let $\tau(G)$ be a set of subgroups of $G$ such that $G\in \tau(G).$ Then $\tau$ is called a {subgroup functor} if for every epimorphism $\varphi$ : $A \to~B$ and any groups $H \in \tau (A)$ and $T\in \tau (B)$ we have $H^{\varphi}\in\tau(B)$ and $T^{{\varphi}^{-1}}\in\tau(A).$ A class of groups $\mathfrak{F}$ is called $\tau$-closed if $\tau(G)\subseteq\mathfrak{F}$ for all $G\in\mathfrak F.$ In this paper, necessary and sufficient conditions for $n$-multiply σ‑locality $(n\geqslant 1)$ of a non-empty $\tau$-closed formation are obtained.

Approximations of Riemann–Liouville integral on a segment by rational integral operators Fourier–Chebyshev are investigated. An integral representation of the approximations is found. Rational approximations Riemann–Liouville integral with density $\varphi_\gamma(x) = (1-x)^\gamma,$ $\gamma \in (0,+\infty)\backslash\mathbb{N},$ are studied, estimates of pointwise and uniform approximations are established. In the case of one pole in an open complex plane, an asymptotic expression is obtained for the approximating function majorants of uniform approximations and the optimal value of the parameter at which the majorant has the asymptotically highest rate of decrease. As a consequence, estimates of approximations of Riemann–Liouville integral with density belonging to some classes of continuous functions on the segment by partial sums of the polynomial Fourier–Chebyshev series are obtained.

   Distance-regular graph Γ with strongly regular graphs Γ₂ and Γ₃ has intersection array {r(c₂ +1)+a₃, rc₂, a₃ +1; 1, c₂, r(c₂ +1)} (M. S. Nirova). For distance-regular graph with diameter 3 and degree 44 there are 7 fisiable intersection arrays. For each of them the graph Γ₃ is strongly regular. For intersection array {44,30,5;1,3,40} we have a₃ = 4, c₂ = 3 and r = 10, Γ₂ has parameters (540, 440, 358, 360) and Γ₃ has parameters (540, 55, 10, 5). This graph does not exist (Koolen-Park). For intersection array {44, 35, 3; 1, 5, 42} the graph Γ₃ has parameters (375, 22, 5, 1). Graph Γ₃ does nor exist (local subgraph is the union of isolated 6-cliques). In this paper it is proved that distance-regular graphs with intersection arrays {44, 36, 5; 1, 9, 40}, {44, 36, 12; 1, 3, 33} and {44, 42, 5; 1, 7, 40} do not exist.

   We study a linear homogeneous algebraic system with the quasiperiodic matrix of coefficients for the existence strongly irregular quasiperiodic solution. If there is such a solution, then there is a linear relationship between its components. An algorithm for finding this dependence is given.

   The paper provides an existence criterion and a complete description of continuos solutions f : R → R of the linear second-order functional equation f(f(x))+a f(x)+bx = 0 with constant coefficients.

   A comprehensive method has been developed for finding the effective thermal conductivity coefficients of dispersed-filled composite materials, taking into account their structure and depending on the thermophysical properties of temperature. Computational experiments were carried out.

   A numerical method for solving the initial boundary value problem of shielding of pulsed electromagnetic fields by a flat magnetized permalloy screen is developed for the case when the energy flow of the pulsed field is orthogonal to the screen. Numerically investigated the dynamics of pulse transformation during their passage through the screen. The shielding efficiency coefficient is calculated.

This paper is dedicated to construction and study of three-layer of compact difference schemes for linear and quasi-linear parabolic equations of order $O(h^4+\tau^2)$. In the linear case, a priori stability estimates from the initial data on the right side are obtained. The basic scheme for constructing difference schemes of a given quality is the asymptotic stability of the second order of accuracy $O(h^4+\tau^2)$ by A. A. Samarsky. The results are generalized to the case of boundary conditions of the third kind, variable coefficients. A three-layer scheme of approximation order $O(h^6+\tau^3)$ is also constructed on a three-point stencil in space, which allows to use an economical sweep method to solve the corresponding system of algebraic equations. Numerical experiments are presented to illustrate the correctness of our theoretical conclusions. Simulation of nonlinear processes with traveling waves showed that these algorithms can also be used for differential problems that have features in solution.