The article contains a brief essay on the scientific work of Naum Il’ich Feldman.

The article contains a brief essay on the scientific work of Askold Ivanovich Vinogradov.

The article is devoted to the life and scientific and pedagogical activity of the famous mathematician, doctor of physical and mathematical sciences, professor Alexander Vasilevich Malyshev (1928–1993) in connection with the 90th anniversary of his birth. It first provides brief biographical information from his life. The main part of our work is devoted to the achievements of A. V. Malyshev in number theory and scientific-pedagogical and editorialpublishing
activities.

This brief review contents the description of most important concept of geometry or numbers and its main application. It is not included the geometry of quadratic forms — interesting but the special part of a number theory (and a geometry of numbers) standing on joining point of the geometry of numbers and the quadratic forms theory.

The article contains personal reminiscences of the author about Boris Fadeevich Skubenko.

This work is devoted to the problem of stability of a small periodic solution of a normal Autonomous system of ordinary differential equations. It is natural to analyze the local dynamics of intersections of perturbed trajectories with orthogonal sections of the corresponding cycle when studying the stability of the periodic solution of an Autonomous system. The problem of orbital stability of the periodic solution is reduced to the problem of Lyapunov stability of the zero solution of an auxiliary system with a periodic t right-hand side by introducing a special coordinate system in which one of the axes is directed tangentially to the trajectory of the periodic solution. For an auxiliary system whose dimension is one less than the dimension of the original system, in a linear approximation, the question of the stability of the zero solution is reduced to an estimate of the multipliers of the monodromy matrix. Thus, according to the Andronov — Witt theorem, the classical approach to the study of the orbital stability of the periodic solution is realized. There is a non-critical case of orbital stability. This approach is traditionally used in Hopf-type bifurcation for systems with a parameter. In this paper, for an autonomous system with a parameter, the bifurcation conditions of a small solution whose period is close to the solution period of the corresponding linear homogeneous system are obtained. The determination of the orbital stability property by the parameter is formulated. According to this condition, the perturbed right half-vectors are arbitrarily close to the studied cycle not only due to the smallness of the initial values perturbations, but also due to the smallness of the parameter. In this case, the idea of weakening the requirements for determining the stability of the Lyapunov type, proposed by M. M. Khapaev, is used. The property of orbital stability with respect to the parameter can also take place in the presence of orbital instability of the studied cycle in the classical sense. A nonlinear approximation of the above-mentioned auxiliary system of perturbed motions is used to study the orbital stability of a small periodic solution with respect to the parameter.

Currently, there are a number of ways to determine trends and extremes in stochastic time series, which is not surprising, since time series trends are a fundamental characteristic of the dynamics of the process behind it.
Real stochastic trends are not at all like ideal mathematical ones, because they contain violations. This does not bother the researcher, who initially has an adaptive perception of the fundamental properties of extremeness, continuity, connectedness, trend, etc. He will understand when the violation is insignificant and the trend continues, and when the violation interrupts the trend.
In this paper, we propose a new approach to the recognition of stochastic trends, based on the mathematical construction of regression derivatives for a finite time series. Trends are sought using the derivative from the scenario of classical mathematical analysis.

Currently, there are a number of ways to determine trends and extremes in stochastic time series, which is not surprising, since time series trends are a fundamental characteristic of the dynamics of the process behind it.
Real stochastic trends are not at all like ideal mathematical ones, because they contain violations. This does not bother the researcher, who initially has an adaptive perception of the fundamental properties of extremeness, continuity, connectedness, trend, etc. He will understand when the violation is insignificant and the trend continues, and when the violation interrupts the trend.
In this paper, we propose a new approach to the recognition of stochastic trends, based on the mathematical construction of regression derivatives for a finite time series. Trends are sought using the derivative from the scenario of classical mathematical analysis.

In 1978 R. McEliece developed the first assymetric cryptosystem based on the use of Goppa's error-correctring codes and no effective key attacks has been described yet. Now there are many code-based cryptosystems known. One way to build them is to modify the McEliece cryptosystem by replacing Goppa's codes with other codes. But many variants of this modification were proven to be less secure.
In connection with the development of quantum computing code cryptosystems along with lattice-based cryptosystems are considered as an alternative to number-theoretical ones. Therefore, it is relevant to find promising classes of codes that are applicable in cryptography. It seems that for this non-commutative group codes, i.e. left ideals in finite non-commutative group algebras, could be used.
The Wedderburn theorem is useful to study non-commutative group codes. It implies the existence of an isomorphism of a semisimple group algebra onto a direct sum of matrix algebras. However, the specific form of the summands and the isomorphism construction are not explicitly defined by this theorem. Hence for each semisimple group algebra there is a task to explicitly construct its Wedderburn decomposition. This decomposition allows us to easily describe all left ideals of group algebra, i.e. group codes.
In this paper we consider one semidirect product $$Q_{m,n} = (\mathbb{Z}_m \times \mathbb{Z}_n) \leftthreetimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$$ of abelian groups and the group algebra $$\mathbb{F}_q Q_{m,n}$$. In the case when $$n \mid q -1$$ and $$\gcd(2mn, q) = 1,$$ the Wedderburn decomposition of this algebra is constructed. In the case when field is of characteristic $$2,$$ i.e. when this group algebra is not semisimple, a similar structure theorem is also obtained. Further in the paper, the primitive central idempotents of this group algebra are described. The obtained results are used to algebraically describe the group codes over $$Q_{m,n}.$$

This article is devoted to p-extensions of complete discrete valuation fields of mixed characteristic where p is the characteristic of the residue field. It is known that any totally ramified Galois extension with a non-maximal ramification jump can be determined by an Artin-Schreier equation, and the upper bound for the ramification jump corresponds to the lower bound of the valuation in the right-hand side of the equation. The problem of construction of extensions with arbitrary Galois groups is not solved. Inaba considered p-extensions of fields of characteristic p-corresponding to a matrix equation $$X^{(p)}=AX$$ herein referred to as Inaba equation. Here $$X^{(p)}$$ is the result of raising each element of a square matrix X to power p, and A is a unipotent matrix over a given field. Such an equation determines a sequence of Artin-Schreier extensions. It was proved that any Inaba equation determines a Galois extension, and vice versa any finite Galois p-extension can be determined by an equation of this sort. In this article for mixed characteristic fields we prove that an extension given by an Inaba extension is a Galois extension provided that the valuations of the elements of the matrix A satisfy certain lower bounds, i.e., the ramification jumps of intermediate extensions of degree p are sufficiently small. This construction can be used in studying the field embedding problem in Galois theory. It is proved that any non-cyclic Galois extension of degree $$p^2$$ with sufficiently small ramification jumps can be embedded into an extension with the Galois group isomorphic to the group of unipotent $$3\times 3$$ matrices over $$\mathbb F_p$$. The final part of the article contains a number of open questions that can be possibly approached by means of this construction.

A general integral form of representation of five postulates in continuum mechanics, possible noncontradictory axiomatics of phenomenological construction of the fourth and fifth of them (namely, the laws of change of the internal energy and entropy) as well as the role of the Fourier law or its hyperbolic generalization in definition of temperature, are discussed. It is shown that in contrast to the statistical and molecular approaches, in this case, the internal energy and entropy of an individual (liquid) volume can be completely defined by specifying its source, flow through the surface, and production. Thus two thermodynamic postulates serve as definitions. The energy conjugate pairs of quantities of different physical nature and the possibility of expanding the table of postulates are discussed.

Let $$0<p\le \infty,$$ $$\mathcal{C}(n;p;r)=\sup_{T}\frac{\|T^{(r)}\|_{L^{\infty}[0,2\pi)}}{\|T\|_{L^{p}[0,2\pi)}}$$ and $$\mathcal{L}(p;r)=\sup_{F}\frac{\|F^{(r)}\|_{L^{\infty}(\mathbb{R})}}{\|F\|_{L^{p}(\mathbb{R})}}$$ be the sharp Nikolskii-Bernstein constants for r-th derivatives of trigonometric polynomials of degree n and entire functions of exponential type 1 respectively. Recently E.Levin and D.Lubinsky have proved that for the Nikolskii constants $$\mathcal{C}(n;p;0)=n^{1/p}\mathcal{L}(p;0)(1+o(1)),\quad n\to \infty.$$ M.Ganzburg and S.Tikhonov generalized this result to the case of Nikolskii-Bernstein constants: $$\mathcal{C}(n;p;r)=n^{r+1/p}\mathcal{L}(p;r)(1+o(1)),\quad n\to \infty.$$ They also showed the existence of the extremal polynomial $$\tilde{T}_{n,r}$$ and the function $$\tilde{F}_{r}$$ in this problem, respectively. Earlier, we gave more precise boundaries in the Levin-Lubinsky-type result, proving that for all p and n $$n^{1/p}\mathcal{L}(p;0)\le \mathcal{C}(n;p;0)\le (n+\lceil 1/p\rceil)^{1/p}\mathcal{L}(p;0).$$ Here we establish close facts for the case of Nikolskii-Bernstein constants, which also imply the asymptotic Ganzburg-Tikhonov equality. The results are stated in terms of extremal functions $$\tilde{T}_{n,r},$$ $$\tilde{F}_{r}$$ and the Taylor coefficients of a kernel of type Jackson-Fejer $$(\frac{\sin \pi x}{\pi x})^{2s}$$. We implicitly use Levitan-type polynomials arising from the Poisson summation formula. We formulate one hypothesis about the signs of the Taylor coefficients of the extremal functions.

Many problems of Number Theory are connected with investigation of   Dirichlet series $$f(s)=\sum_{n=1}^{\infty} a_nn^{-s}$$ and the  adding functions $$\Phi(x)=\sum_{n\leq x} a_n$$ of their coefficients. The most famous Dirichlet series is the   Riemann zeta function $$\zeta(s),$$  defined for any $$s=\sigma+it$$ with $$\Re s=\sigma> 1$$ as $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.$$ The square of zeta function $$\zeta^{2}(s)=\sum_{n=1}^{\infty}\frac{\tau(n)}{n^s}, \,\, \Re s > 1,$$ is connected with  the  divisor function $$\tau (n)=\sum_ { d | n } 1,$$  giving the number of a  positive integer divisors of positive integer number n. The adding  function of the Dirichlet series  $$\zeta^2(s)$$ is the function $$D (x)=\sum_ { n\leq x}\tau(n)$$;   the  questions of the asymptotic behavior of this function  are known as  Dirichlet divisor problem. Generally, $$ \zeta^{k}(s)=\sum_{n=1}^{\infty}\frac{\tau_k(n)}{n^s}, \,\, \Re s > 1, $$ where function $$\tau_k (n)=\sum_{n=n_1\cdot...\cdot n_k}  1$$  gives the number of representations  of a  positive integer number  n as a product  of k  positive integer factors. The adding function of the Dirichlet series  $$ \zeta^k (s)$$ is the function  $$D_k (x)=\sum_ { n\leq x}\tau_k(n)$$;  its research   is known as the   multidimensional Dirichlet divisor problem.  The logarithmic derivative  $$\frac{\zeta^{'}(s)}{\zeta(s)}$$ of zeta function can be represented as $$\frac{\zeta^{'}(s)}{\zeta(s)}=-\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s},$$ $$\Re s >1.$$ Here  $$\Lambda(n)$$ is the  Mangoldt function, defined as $$\Lambda(n)=\log p,$$ if $$n=p^{k}$$ for a prime number p and a positive integer number k, and as  $$\Lambda(n)=0,$$ otherwise. So, the  Chebyshev function $$\psi(x)=\sum_{n\leq x}\Lambda(n)$$ is the adding function of the coefficients of the Dirichlet series $$\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s},$$ corresponding to logarithmic derivative  $$\frac{\zeta^{'}(s)}{\zeta(s)}$$ of zeta function. It is well-known in analytic Number Theory and  is closely connected with many important number-theoretical problems, for example, with  asymptotic law of distribution of prime numbers. In particular, the following representation of  $$\psi(x)$$  is very useful in many applications: $$\psi(x)=x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho}}{\rho}+O\left(\frac{x\ln^{2}x}{T}\right), $$ where x=n+0,5, $$n \in\mathbb{N},$$ $$2\leq T \leq x,$$  and $$\rho=\beta+i\gamma$$ are  non-trivial zeros of zeta function,  i.e., the zeros of $$\zeta(s),$$ belonging to the critical strip 0 < Res < 1.  We  obtain similar representations over non-trivial zeros of zeta function for an arithmetic function, relative to the Chebyshev function: $$\psi_{1}(x)=\sum_{n\leq x}(x-n)\Lambda(n).$$ In fact, we prove the following theorem: $$\psi_1(x)=\frac{x^2}{2}-\left(\frac{\zeta^{'}(0)}{\zeta(0)}\right)x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho+1}}{\rho(\rho+1)}+O\left(\frac{x^{2}}{T^2}\ln^2 x\right)+O\left(\sqrt{x}\ln^2x\right), $$ where x > 2, $$T \geq 2,$$ and  $$\rho=\beta+i\gamma$$ are  non-trivial zeros of zeta function, i.e., the zeros of $$\zeta(s),$$ belonging to the critical strip 0 < Res < 1.

The paper is devoted to the study of problems on the behavior of finite automata in mazes. For any n, a maze is constructed that can be bypassed with 2n stones but you can’t get around with n stones. The range of tasks is extensive and touches upon key aspects of theoretical Computer Science. Of course, the solution of such problems does not mean the automatic solution of complex problems of complexity theory, however, the consideration of these issues can have a positive impact on the understanding of the essence of theoretical Computer Science. It is hoped that the behavior of automata in mazes is a good model for non-trivial information theoretic problems, and the development of methods and approaches to the study of robot behavior will give more serious results in the future. Problems related to automaton analysis of geometric media have a rather rich history of study. The first work that gave rise to this kind of problems, it is necessary to recognize the work of Shannon [24]. It deals with a model of a mouse in the form of an automaton, which must find a specific target in the maze. Another early work, one way or another affecting our problems, is the work of Fisher [9] on computing systems with external memory in the form of a discrete plane. A serious impetus to the study of the behavior of automata in mazes was the work of Depp [7, 8], in which the following model is proposed: there is a certain configuration of cells from mathbbZ^2 (chess maze), in which finite automata, surveying some neighborhood of the cell in which they are, can move to an adjacent cell in one of four directions. The main question posed in such a model is whether there is an automaton that bypasses all such mazes. In [20], Muller constructed a flat trap for a given automaton (a maze that does not completely bypass) in the form of a 3-graph. Budach [5] constructed a chess trap for any given finite automaton. Note that Budach’s solution was quite complex (the first versions contained 175 pages). More visual solutions to this question are presented here [29, 31, 33, 34]. Antelman [2] estimated the complexity of such a trap by the number of cells, and in [1] Antelman, Budach, and Rollick made a finite trap for any finite automaton system. In the formulation with a chess maze and one automaton, there are a number of results related to the problems of traversability of labyrinths with different numbers of holes, with bundles of labyrinths by the number of States of the automaton, and other issues. An overview of such problems can be found for example here [35]. The impossibility of traversing all flat chess labyrinths with one automaton raised the question of studying the possible amplifications of the automaton model, which will solve the problem of traversal. The main way of strengthening can be the consideration of a collective of automata, instead of one automaton, interacting with each other. A special and widely used case is the consideration of a system of one full-fledged automaton and a certain number of automata of stones, which have no internal state and can move only together with the main automaton. Interaction between machines is a key feature of this gain, it is allowed to have a collective (or one machine with stones) external memory, thereby significantly diversifies its behavior. If you get rid of the interaction of automata, the resulting  independent system will be little better than a single machine. Next, we discuss the known results associated with the collective automata.

The work refers to the analytical theory of numbers and it deals with the application of number theory to problems of approximate analysis. The concept of the hyperbolic parameter of grids with weights and the analogue of Bakhvalov’s theorem for the hyperbolic parameter of grids with weights and the hyperbolic Zeta function of grids are considered. In this paper the following results are obtained:
1. a strengthened generalized Bakhvalov–Korobov theorem for the hyperbolic Zeta function of three-dimensional grids is proved;
2. the number of nodes of the resin grid is calculated taking into account their multiplicity; the number of nodes taking into account their weights.
3. the number of nodes of the resin grid is calculated without taking into account their multiplicity;
4. the number of nodes of the resin grid is calculated taking into account their weights;
5. the form of a quadrature formula with a resin grid without multiple nodes is found and explicit formulas for the weights of this quadrature formula are found. It is shown that the number of nodes of such a quadrature formula is 7 times less than in the case of a formula with multiple nodes.

A review of works on solving the inverse problems of scattering of sound waves by elastic bodies is presented. The theoretical foundations for solving inverse problems of sound diffraction are based on fundamental studies of the problem of inverse problems for partial differential equations performed by Russian scientists. In the most general classification, inverse acoustic problems are divided into inverse radiation problems (IRP) and inverse scattering problems (ISP). When solving problems of the first class, the parameters of the sound field determine some parameters of the source. When solving the problems of the second class, the parameters of the scattered sound field are used to identify the properties of the scattering object. Most applications of acoustic methods are based on solving inverse diffraction problems when the parameters of an object or medium are judged by the parameters of the emitted or reflected sound field. Analysis of sound fields forms the basis of methods in hydro- and aeroacoustics; researches in biology and medicine; non-destructive testing and diagnostics of objects; ultrasonic flaw detection; inspection and testing of materials, structures and structures. The solutions of all inverse problems are based on the solution of direct diffraction problems. The paper presents the most significant results in solving direct problems of scattering of sound waves by elastic objects. The works devoted to the problems of inverse problems of sound scattering by inhomogeneous elastic bodies are singled out. This direction is the subject of interest in the research of the authors.

The Arnoux-Ito theory of geometric substitutions allows to construct sequences of generalized exchanged tilings of the d-dimensional torus. These tilings consist of parallelepipeds of d + 1 type, and the action of a certain toric shift on the tiling reduces to exchanging of the d + 1 central parallelepipeds. Moreover, the set of vertices of all parallelepipeds of the tiling is a fragment of the orbit of zero point under considered toric shift. The considered tilings are actively used in various problems of number theory, combinatorics, and the theory of dynamical systems. In this paper, we study the local structure of toric tilings obtained using geometric substitutions. The n-corona of the parallelepiped is a set of all parallelepipeds located at a distance of not greater than n from a given parallelepiped in the natural metric of the tiling. The problem is to describe all possible types of n-coronas. With each parallelepiped in the tiling we can naturally assigned a number — its number in the orbit of the corresponding central parallelepiped with respect to the toric shift. It is proved that the set of all parallelepipeds numbers splits into a finite number of half-intervals defining possible types of n-coronas. Moreover, it is proved that the boundaries of the corresponding half-open intervals are determined by the numbers of the parallelepipeds in the n-corona of the set of d + 1 central parallelepiped. It is shown that this result can be considered as some multi-dimensional generalization of the famous three lengths theorem. Earlier, a similar description was obtained for 1-coronas of the toric tilings obtained using one specific geometric substitution: the Rauzy substitution. In addition, similar results were previously obtained for some quasiperiodic plane tilings. In conclusion, some directions for further research are formulated.

An n-tuple semigroup  is a nonempty set G equipped with n binary operations $$\fbox{1}\,, \fbox{2}\,, ..., \fbox{n}\,,$$ satisfying the axioms $$(x\fbox{r} \, y) \fbox{s}\, z=x\fbox{r}\,(y\fbox{s}\,z)$$ for all $$x,y,z \in G$$ and $$r,s\in \{1,2,...,n\}.$$ This notion was considered by Koreshkov in the context of the theory of  n-tuple algebras of associative type. Doppelsemigroups are  2-tuple semigroups. The n-tuple semigroups are related to interassociative semigroups, dimonoids, trioids, doppelalgebras, duplexes, G-dimonoids, and restrictive bisemigroups. If operations of an n-tuple semigroup coincide, the  n-tuple semigroup becomes a semigroup. So, n-tuple semigroups are a generalization of semigroups. The class of all n-tuple semigroups forms a variety. Recently, the constructions of the free n-tuple semigroup, of the free commutative n-tuple semigroup, of the free k-nilpotent n-tuple semigroup and of the free product of arbitrary n-tuple semigroups were given. The class of all rectangular n-tuple semigroups, that is,   n-tuple semigroups  with n  rectangular semigroups, forms a subvariety of the variety of  n-tuple semigroups. In this paper, we construct the free rectangular n-tuple semigroup and characterize the least rectangular congruence on the free n-tuple semigroup.

The main difficulty one has to deal with while investigating arithmetic nature of the values of the generalized hypergeometric functions with irrational parameters consists in the fact that the least common denominator of several first coefficients of the corresponding power series increases too fast with the growth of their number. The last circumstance makes it impossible to apply known in the theory of transcendental numbers Siegel’s method for carrying out the above mentioned investigation. The application of this method implies usage of pigeon-hole principle for the construction of a functional linear approximating form. This construction is the first step in a long and complicated reasoning that leads ultimately to the required arithmetic result. The attempts to apply pigeon-hole principle in case of functions with irrational parameters encounters insurmountable obstacles because of the aforementioned fast growth of the least common denominator of the coefficients of the corresponding Taylor series. Owing to this difficulty one usually applies effective construction of the linear approximating form (or a system of such forms in case of simultaneous approximations) for the functions with irrational parameters. The effectively constructed form contains polynomials with algebraic coefficients and it is necessary for further reasoning to obtain a satisfactory upper estimate of the modulus of the least common denominator of these coefficients. The known estimates of this type should be in some cases improved. This improvement is carried out by means of the theory of divisibility in quadratic fields. Some facts concerning the distribution of the prime numbers in arithmetic progression are also made use of. In the present paper we consider one of the versions of effective construction of the simultaneous approximations for the hypergeometric function of the general type and its derivatives. The least common denominator of the coefficients of the polynomials included in these approximations is estimated subsequently by means of the improved variant of the corresponding lemma. All this makes it possible to obtain a new result concerning the arithmetic values of the aforesaid function at a nonzero point of small modulus from some imaginary quadratic field.

In the paper a new version of the Hadamard’s method in the theory of Dirichlet’s L-functions
is given. We prove of this method of the absence of the L-functions zeroes on the unit line. We
show that the Hadamard’s method allow to get results, which on the accuracy correspond to the
Vallee Poussin results in the asymptotical law of the distribution of primes. Of this we extend
possibilities of the Hadamard’s method. New estimations of the zeta-sum twisted together with
the Dirichlet’s character by modulo, equals to the degree of an odd prime number are obtained
that permits to get the modern limit of zeroes for the corresponding Dirichlet’s L-function.

There is a wide range of problems devoted to the possibility of traversing the maze by finite automatons. They can differ as the type of maze (it can be any graph, even infinite), and the automata themselves or their number. In particular, a finite state machine can have a memory (store) or a random bit generator. In the future, we will assume that the robot — is a finite automaton with a random bit generator, unless otherwise stated. In addition, in this system, there can be stones-an object that the finite state machine can carry over the graph, and flags-an object whose presence the finite state machine can only "observe". This topic is of interest due to the fact that some of these problems are closely related to problems from probability theory and computational complexity.

This paper continues to address some of the open questions posed in Ajans’s thesis: traversal by a robot with a random bit generator of integer spaces in the presence of a stone and a subspace of [4] flags. Such problems help to develop the mathematical apparatus in this area, in addition, in this work we investigate the almost unexplored behavior of a robot with a random number generator. It is extremely important to transfer combinatorial methods developed by A. M. Raigorodsky in the problems of this topic.

This work is devoted to the maze traversal by a finite automaton with a random bit generator. This problem is part of the actively developing theme of traversing the maze by various finite automata or their teams, which is closely related to problems from the theory of complexity of calculations and probability theory. In this work it is shown what dimensions a robot with a generator of random bits, and you can get around stone integer space with flag subspace. In this paper, we will study the behavior of a finite automaton with a random bit generator on integer spaces. In particular, it is proved that the robot bypasses Z² and cannot bypass Z³; the c ++ robot bypasses Z⁴ and cannot bypass Z⁵; a robot with a stone and a flag bypasses Z⁶ and cannot bypass Z⁷; a robot with a stone and a flag plane bypasses Z⁸ and cannot bypass Z⁹.

We evolve an approach to construction and classification of semifield projective planes with the use of the linear space and spread set. This approach is applied to the problem of existance for a projective plane with the fixed restrictions on collineation group.

A projective plane is said to be semifield plane if its coordinatizing set is a semifield, or division ring. It is an algebraic structure with two binary operation which satisfies all the axioms for a skewfield except (possibly) associativity of multiplication. A collineation of a projective plane of order p²ⁿ (p > 2 be prime) is called Baer collineation if it fixes a subplane of order pⁿ pointwise. If the order of a Baer collineation divides pⁿ − 1 but does not divide pⁱ − 1 for i < n then such a collineation is called p-primitive. A semifield plane that admit such collineation is a p-primitive plane.

M. Cordero in 1997 construct 4 examples of 3-primitive semifield planes of order 81 with the nucleus of order 9, using a spread set formed by 2 × 2-matrices. In the paper we consider the general case of 3-primitive semifield plane of order 81 with the nucleus of order ≤ 9 and a spread set in the ring of 4 × 4-matrices. We use the earlier theoretical results obtained independently to construct the matrix representation of the spread set and autotopism group. We determine 8 isomorphism classes of 3-primitive semifield planes of order 81 including M. Cordero examples. 

We obtain the algorithm to optimize the identification of pair-isomorphic semifield planes, and computer realization of this algorithm. It is proved that full collineation group of any semifield plane of order 81 is solvable, the orders of all autotopisms are calculated. 

We describe the structure of 8 non-isotopic semifields of order 81 that coordinatize 8 nonisomorphic 3-primitive semifield planes of order 81. The spectra of its multiplicative loops of non-zero elements are calculated, the left-, right-ordered spectra, the maximal subfields and automorphisms are found. The results obtained illustrate G. Wene hypothesis on left or right primitivity for any finite semifield and demonstrate some anomalous properties.

The methods and algorithsm demonstrated can be used for construction and investigation of semifield planes of odd order pⁿ for p ≥ 3 and n ≥ 4.

One of the varieties of radio engineering systems is considered in the work, namely, a frequency-phase locked loop system FPLL. The mathematical model of such a system is described by a system of differential equations with a cylindrical phase space. For the FPLL system, the conditions for the formation of latent synchronization modes are defined. Despite the numerous works devoted to FPLL systems, the questions of finding hidden synchronization, determining the mechanisms of its occurrence, finding the conditions of bifurcation of cycles and studying their scenarios, the occurrence of complex modulated oscillations remain open. The conditions for the formation of hidden synchronization are the presence in the phase-locked loop system of the frequency of the beating modes, vibrational-rotational cycles, and the presence of multistability. By multistability we understand the coexistence of several attractors in the phase space, in particular, limit cycles can be attractors. One of the cases of multistability is phase multistability, when the attractors differ from each other by the values of the phase difference between the oscillations of the system. The phase space in systems with phase multistability is more complicated than in systems with a single stable limit cycle. In the formation of multistability, the decisive role is played by unstable limit sets corresponding to oscillations not observed in the experiment. In this regard, the development of methods for determining multistability and determining the mechanisms of its appearance is relevant.In connection with the above, the urgent task is to develop numerical algorithms that allow one to find complex modulated oscillations in radio engineering systems and determine the mechanisms of their occurrence.Analytical methods for determining the latent synchronization of the PLL system are proposed, which allow developing effective computational methods for studying mathematical models of radio engineering systems using computer technologies.

In [11], the authors considered the realization T of SO(2, 2)-representation in a space of homogeneous functions on 2×4-matrices. In this sequel, we aim to compute matrix elements of the identical operator T(e) and representation operator T(g) for an appropriate g with respect to the mixed basis related to two different bases in the SO(2, 2)-carrier space and evaluate some improper integrals involving a product of Bessel-Clifford and Whittaker functions. The obtained result can be rewritten in terms of Hankel-Clifford integral transforms and their analogue. The first and the second Hankel-Clifford transforms introduced by Hayek and Pérez–Robayna, respectively, play an important role in the theory of fractional order differential operators (see, e.g., [6, 8]). The similar result have been derived recently by the authors for the regular Coulomb function in [12].

In this paper are investigated a polynomial formal group analogues of the operators introduced by Coleman for the Lubin Tate formal group and the multiplicative formal group. Explicit constructions of the norm and trace operators for Laurent series are given, their main propirtes are checked. The eigenvalues and root values of these operators are also studied, and a homomorphism is constructed that connects the additive structure and the structure of the formal module on the set of formal power series.

Rauzy introduced a fractal set associted with the toric shift by the vector (β⁻¹, β⁻²), where β is the real root of the equation β³ = β² + β + 1. He show that this fractal can be partitioned into three fractal sets that are bounded remaider sets with respect to the considered toric shift. Later, the introduced set was named as the Rauzy fractal. Further, many generalizations of Rauzy fractal are discovered. There are many applications of the generalized Rauzy fractals to problems in number theory, dynamical systems and combinatorics.

Zhuravlev propose an infinite sequence of tilings of the original Rauzy fractal and show that these tilings also consist of bounded remainder sets. In this paper we consider the problem of constructing similar tilings for the generalized Rauzy fractals associated with algebraic Pisot units.

We introduce an infinite sequence of tilings of the d−1-dimensional Rauzy fractals associated with the algebraic Pisot units of the degree d into fractal sets of d types. Each subsequent tiling is a subdivision of the previous one. Some results describing the self-similarity properties of the introduced tilings are proved.

Also, it is proved that the introduced tilings are so called generalized exchanding tilings with respect to some toric shift. In particular, the action of this shift on the tiling is reduced to exchanging of d central tiles. As a corollary, we obtain that the Rauzy tiling of an arbitrary order consist of bounded remainder sets with respect to the considered toric shift.

In addition, some self-similarity property of the orbit of considered toric shift is established.

The object of this note is to use PI-theory to simplify the results of Dicks and Lewin [4] on the automorphisms of the free algebra F{X}, namely that if the Jacobian is invertible, then every endomorphism is an epimorphism. We then show how the same proof applies to a somewhat wider class of rings.

We study the interrelation between the extremal Turán-type problems and Nikolskii – Bernstein problems for nonnegative functions on R^d with the Dunkl weight. The Turán problem is to find the supremum of a given moment of a positive definite (with respect to the Dunkl transform) function with a support in the Euclidean ball and a fixed value at zero. In the sharp L¹-Nikolskii–Bernstein inequality, the supremum norm of the Dankl Laplacian of an entire function of exponential spherical type with the unit L¹-norm is estimated. Extremal Feuér and Beaumann problems is also mentioned. The Dunkl transform covers the case of the classical Fourier transform in the case of unit weight.

Nikolskii–Bernstein inequalities are classical in approximation theory, and the Turán-type problems have applications in metric geometry. Nevertheless, we prove that they have the same answer, which is given explicitly. The easy proof is relied on our old results from the theory of solving extremal problems to the Dunkl transform.

This paper is devoted to the development of a new approach for estimating from below the constant of the best Diophantine approximations. The history of this problem dates back to P. G. Dirichlet. Over time, the approaches used to solve this problem have undergone major changes. From algebra (P. G. Dirichlet, A. Hurwitz, F. Furtwengler) this problem has moved into the field geometry of numbers (H. Davenport, J. W. S. Cassels). One cannot fail to note such an interesting component of this problem as the close relationship of diophantine approximations with geometry of numbers in general, and algebraic lattices in particular (J. W. S. Cassels, A. D. Bruno). This provided new opportunities, both for applying the already known results and for application of new approaches to the problem of the best Diophantine approximations (A. D. Bruno, N. G. Moshchevitin).

In the mid-twentieth century, H. Davenport found a fundamental relationship between the value of the constant of the best joint Diophantine approximations and critical determinant of a stellar body of a special kind. Later, J. W. S. Cassels went from directly calculating the critical determinant to estimating its value by calculating the largest value of V_n,s – the volume of the parallelepiped centered at the origin with certain properties. This approach allowed us to obtain estimates of the constant of the best joint Diophantine approximations for n = 2, 3, 4 (see the works of J. W. S. Kassels, T. Cusick, S. Krass).

In this paper, based on the approach described above, the estimates n = 5 and n = 6 are obtained. The idea of constructing estimates differs from the work of T. Cusick. Using numerical experiments, approximate and then exact values of the estimates V_n,s were obtained. The proof of these estimates is rather cumbersome and is primarily of technical complexity. Another difference between constant estimates is the ability to generalize them to any dimension.

As part of the proof of estimates of the constant of the best Diophantine approximations, we have solved a number of multidimensional optimization problems. In solving them, we used the mathematical package Wolfram Mathematica quite actively. These results are an intermediate step for analytical proofs of the estimates of V_n,s and the constant of the best Diophantine approximations C_n for n ≥ 3.

In the process of numerical experiments, interesting information was also obtained on the structure of the values of V_n,s. These results are in good agreement with the results obtained in the works of S. Krass. The question of the structure of the values of V_n,s for large dimensions has been little studied and can be of considerable interest both from the point of geometry of numbers and from the point of theory of diophantine approximations.

The paper presents a brief overview of the history of new branches of mathematics and their impact on theoretical studies of mechanics of composite materials. The contribution of Russian and Soviet mathematicians and mechanics is shown, which allowed to create a functional basis for the study of mechanical properties of composites – new materials that have been widely used in engineering and the national economy. Composite materials were created in the second half of the twentieth century. They are multicomponent structures composed of various homogeneous materials. The most common are two-component structures of matrix and filler. Technologically, these components can constitute deterministic or random structures. By changing the structure and properties of the components, it is possible to obtain materials with predetermined macroscopic properties (effective properties) necessary for a particular application. The emergence of composite materials has caused a rapid growth of research on mechanical properties, allowing the design of these materials. These studies were conducted in both theoretical and practical terms. Theoretical studies were mainly reduced to the construction of mathematical models of the mechanical behavior of composites as structurally inhomogeneous materials.

We discuss the evolution of the course of the mathematical analysis in the Russian universities during the first half of the XXth century and the role in this process of the professor of the Leningrad University G. M. Fikhtengol’ts (1888–1959), who was the author of the classical works — of the three-volume treatise «Differential and Integral Calculus» (1947–1949) and of the two-volumes text-book «The Fundamentals of Mathematical Analysis» (1955–1956).

The paper presents comparative schemes of classical production of complex products and their manufacturing with using additive technologies, including the main positive and negative aspects of using additive technologies. The article listes the main technologies of additive manufacturing of products, indicating the specifics of their application. The paper describes the methods of production and properties of powder materials. The spheroidization technology of powder materials and its post-processes is described. The article presents the concept of full cycle additive manufacturing. The basic software packages for modeling the processes of additive production of products from various metal systems are presented.

The brittle destruction of high-strength metals and alloys used in the chemical and oil refining industry, caused by the influence of aggressive hydrogen-containing media, is a serious scientific issue, the relevance of which has increased dramatically in recent decades due to the discovery of the anomalous hydrogen effects on the complex properties of metals and alloys (abnormal plastic auto-deformation of iron, structural-phase transformations, synergistic effects of microplasticity, effect of reversible shape loss in amorphous metal alloys, and many others). A significant number of hydrogen sources (corrosion in aqueous solutions, hydrogen absorption in the production of welding operations and application of technological protective coatings or cathodic protection of underground pipelines) causes significant difficulties in describing the processes of hydrogen degradation of metal materials. Degradation is manifested in various ways, such as: hydrogen cracking of high-strength steels; hydrogen participation in the process of stress corrosion cracking of stainless steels; cracking of nuclear reactor tubes made of zirconium alloys and embrittlement of titanium alloys by hydride formation, GaAs degradation of monolithic microwave integrated circuits on satellites, etc. The harmful effect of hydrogen on mechanical properties was first noted by Johnson in 1875. Since then, scientists have made many advances in the development of metals with optimal parameters of strength and plasticity. Despite many years of research, the problem of interaction of metal-hydrogen systems remains open due to the variety of approaches and techniques to the assessment of embrittlement effects of hydrogen and hydrogen-containing media. So far it has not been possible to establish a single mechanism of interaction of hydrogen with metal materials, which would explain the whole set of phenomena, related to hydrogen destruction. Therefore, to analyze the mechanisms of hydrogen cracking of metal systems and to develop methods of steel products protection from corrosion-mechanical destruction are relevant areas of scientific and practical activities.

The article is devoted to the appearance of the famous Warsaw mathematical School of V. Sierpinski and Lviv School of Functional Analysis of S. Banach. These schools have played an important role in the development of new areas of mathematics in the first half of the 20th century. Particular attention is paid to the interrelations between Polish and Moscow mathematicians in the period between the two world wars. Most of their outstanding results, especially in topology, Moscow scientists published in the newly created Polish mathematical journals. The leaders of the schools constantly maintained close friendly relations, as evidenced by the surviving letters of N. N. Luzin and V. Sierpinsky, P. S. Uryson and K. Kuratovsky, N. K. Bari and A. Rajchman, in which, among others, the organization of mathematical research was discussed.

The participation of Polish scientists in the work of several important mathematical forums held at that time in the USSR was covered: the First Congress of Mathematicians of the USSR in Kharkov (1930), the International Conference on Differential Geometry and Tensor Analysis (Moscow, 1934) and the International Topological Conference (Moscow, 1935). It is noted that in Moscow and in Polish universities in the first half of the 20th century scientific student seminars were starting to work, the themes of which also have indicated the constant interest of both Polish and Moscow mathematicians in their colleagues research.

This article analyzes the largest and most important works of N. E. Zhukovsky (1847-1921), related to the development of theoretical hydrodynamics: on the kinematics of a liquid body, on the motion of a solid body having a cavity filled with a homogeneous droplet liquid and on the modification of the Kirchhoff method.

The work is devoted to the 170th anniversary of the birth of the outstanding Russian scientist Nikolai Egorovich Zhukovsky.

Against the background of the formation and development of one of the promising methods of physical materials science – mechanical spectroscopy - the history of the discovery of Tula metal physicists N.N. Sergeev and V.S. Ageev previously unknown effect of inelastic behavior of steels. Their first attempt of theoretical description of the mechanism of its formation is described. Set out the further fate of the detected effect in the interplay with the lives of its researchers. The history of the renovation of the forgotten for almost 30 years of discovery is proposed. A multi-faceted program of large-scale study of the mechanism of undeservedly forgotten effect, implemented in decades by the followers of the pioneers, is presented in detail. Disputes with skeptics are described. The main alternative ideas as the reasons of scientific disputes around the nature of effect are stated. Answers to critical questions are given, which allowed the authors to convince skeptics of the reality of the discovered phenomenon and to create a new direction in the study of steels and alloys – a method of assessing their damage by the results of mechanical spectroscopy. The examples of industrial application of the created direction are described. The areas of application of the method of application of the complex of inelastic effects developed by the authors on the basis of the revived method for a multiscale description of structural changes in steels and alloys in the course of external destructive effects of different nature are listed. New ways of development and improvement of the method proposed by the authors in the study of products obtained both by standard ingot technology and in the conditions of additive 3D printing technologies are described.

The most important scientific problem solved under the guidance of Professor Krishtal M. A. was the problem of corrosion-mechanical destruction of high-strength reinforcing iron alloys. Much effort has been expended to solve this scientific problem of great applied importance. Complex regularities were established and the physical nature and mechanisms of hydrogen embrittlement and destruction of reinforcing high-strength steels used in composite reinforced concrete structures and structures in the form of fibrous steel reinforcing fillers were revealed. In the Tula region in solving this problem a significant contribution was made by a student of Mikhail Aronovich Krishtal – Professor Nikolay Nikolaevich Sergeev, who defended his PhD and doctoral dissertations under his leadership.