We study properties of metric segments in the class of all metric spaces considered up to an isometry, endowed with Gromov–Hausdorff distance. On the isometry classes of all compact
metric spaces, the Gromov-Hausdorff distance is a metric. A metric segment is a class that consists of points lying between two given ones. By von Neumann–Bernays–G¨odel (NBG) axiomatic set theory, a proper class is a “monster collection”, e.g., the collection of all sets.
We prove that any metric segment in the proper class of isometry classes of all metric spaces with the Gromov-Hausdorff distance is a proper class if the segment contains at least one metric
space at positive distances from the segment endpoints. If the distance between the segment endpoints is zero, then the metric segment is a set. In addition, we show that the restriction of
a non-degenerated metric segment to compact metric spaces is a non-compact set.

A quantitative estimate for the measure of the set of 𝑝-adic numbers for which the inequality |𝑃(𝑥)|𝑝 < 𝑄−𝑤 for 𝑤 > 3𝑛/2 + 2 has a solution in integral polynomials P of degree n and of height 𝐻(𝑃) at most 𝑄 ∈ N, is established.

The article describes a method for finding a solution to a linearized ellipsoidal-statistical kinetic equation (ES) with a homogeneous boundary condition based on the Chebyshev polynomial approximation in the framework of the problem of modeling the axial flow of a
rarefied gas in a long channel. The channel is formed from two cylinders having a common central axis. Diffuse Maxwell reflection is used as a model for the reflection of gas molecules from cylinders. The gas flow is due to a small absolute value of the pressure gradient directed along the axis of the cylinders. The calculation of the mass flow of gas in the channel is carried out
depending on the rarefaction parameter and the ratio of the radii of the cylinders. The unknown function approximating the solution of the linearized ES equation is represented as a partial sum of the expansion in Chebyshev polynomials of the first kind. By choosing interpolation nodes and applying the properties of finite sums of Chebyshev polynomials, the problem is reduced
to a system of linear algebraic equations with respect to the values of the desired function at these nodes. The expressions for the gas mass velocity in the channel and the gas mass flow are obtained in terms of the partial sums of the series of Chebyshev polynomials.

In this paper theorems on the expression of real numbers on multiplicative number system, Fibonacci sequence and integral valued sequences satisfiing recurrent correlations and connected
with Pisot–Vidgajraghavan, are proven. It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations. We note that unifiing of an expression of
a real number over inverse values of a multiplicaticative system permits to get the estimation of the form

$$𝑒 −Σ︁𝑛𝑘=0 1/𝑘!=𝑥𝑛/𝑛!, 1/𝑛 + 1≤ 𝑥𝑛 <1/𝑛.$$

Expressions of numbers over the sequence of inverse of Fibonacci numbers essentially uses these representation throw powers of “the gold section” 𝜙 = (1+√5)/2 .
Systems numbers connected with Pisot–Vidgajraghavana were considered less than in details, as demands to make a properties of examinated numbers more concrete.

The problem of topological classification of real algebraic curves is a classical problem in fundamental mathematics that actually arose at the origins of mathematics. The problem gained particular fame and modern formulation after D. Hilbert included it in his famous list of mathematical problems at number 16 in 1900. This was the problem of classifying curves of
the sixth degree, solved in 1969 by D.A. Gudkov [1]. In the same place, Gudkov posed the problem of the topological classification of real algebraic curves of degree 6 decomposing into a product of two non-singular curves under certain natural conditions of maximality and general position of quotient curves. Gudkov’s problem was solved in 1977 by G.M. Polotovsky [2], [3].
At present, after a large series of works by several authors (exact references can be found in [4]), the solution of a similar problem on curves of degree 7 is almost complete. In addition, in [5] a topological classification of curves of degree 6 decomposing into a product of any possible number of irreducible factors in general position, and in [6] a classification of mutual arrangements of M-quintics, a couple of lines were found.
The present paper is devoted to the case when the irreducible factors of the curve of degree 7 have degrees 3, 2, and 2, and is a continuation of the study begun in [7].

The theory of recurrence relations is an important component of modern mathematical science. Many numerical sequences have a recurrent nature. Often they are naturally related to Number Theory (Fibonacci numbers, figurate numbers, Mersenne and Fermat numbers, amicable numbers, etc.) or have combinatorial “roots”(elements of the Pascal triangle, Stirling
numbers, Bell numbers, Catalan numbers, etc.). The generating functions used for the study of recurrent sequences are considered in detail in Mathematical Analysis, providing a wide range of practical-oriented examples of the use of classical analytical constructions. Recursive functions play an important role in the Theory of Algorithms.
Applications of the theory of recurrence relations are extremely in demand in Cryptography (generation of pseudo-random sequences over finite fields), digital signal processing (feedback modeling in a system where the output simultaneously becomes input for future time), Economy (models of various sectors of the economy - financial, commodity, etc., in which the current values of key variables (interest rate, real GDP, etc.) are analyzed in terms of past and current values
of other variables), Biology (for example, models of growth dynamics of a particular population;
recall Fibonacci numbers), etc.
We consider several aspects of this topic, including:
- history of the issue, place of recurrent numerical sequences in the development of mathematical science and mathematical education;
- examples of using a recurrent approach when constructing various classes (and subclasses)
of special numbers (figurate numbers, amicable numbers, etc.);
- theoretical aspects of using of sequences of large periods over finite fields in radar-location and methods for generating pseudo-random sequences to provide cryptographic protection of information transmitted over long distances.

In particular, the paper presents a recurrent scheme for constructing so-called centered 𝑘-pyramidal numbers 𝐶𝑆3𝑘(𝑛), 𝑛 = 1, 2, 3, . . ., which present configurations of points that form the 𝑘-gonal pyramid, at the base of which lies the centered 𝑘-gonal number 𝐶𝑆𝑘(𝑛).
Based on the definition, we get for the sequence 𝐶𝑆3
𝑘(𝑛), 𝑛 = 1, 2, 3, . . ., recurrence formula 𝐶𝑆3𝑘(𝑛 + 1) =𝐶𝑆3𝑘(𝑛) + 𝐶𝑆𝑘(𝑛 + 1), 𝐶𝑆3𝑘(1) = 1. Noting that 𝐶𝑆𝑘(𝑛 + 1) = )𝑘𝑛2+𝑘𝑛+2)/2 , and using standard approaches, we prove that the generating function 𝑓(𝑥) of the sequence 𝐶𝑆3𝑘(𝑛), 𝑛 = 1, 2, 3, . . ., has the form 𝑓(𝑥)=(𝑥(1+𝑘−2)𝑥+𝑥2)/(1−𝑥)2 , |𝑥| < 1, while the closed formula for 𝐶𝑆3 𝑘(𝑛) has the form 𝐶𝑆3𝑘(𝑛) = (𝑘𝑛3+𝑛(6−𝑘))/6.

The paper studies algebraic structures arising with respect to the multiplication operationof two sets of natural numbers. The main objects of study are the monoid MN of monoids of natural numbers and the monoid SN of products of arbitrary subsets of a natural series. Also, the monoid will be SN* = SN ∖ {∅}.
An important property of these monoids is the fact that the set of all idempotents in the monoid SN except for the zero element coincides with the set of idempotents of the monoid SN* forms the monoid MN.
The presence of such a fact allowed us to consider the order. With respect to the order of 𝐴 6 𝐵 and binary operations inf, sup the monoid MN is an irregular, complete A-lattice.
The paper distinguishes the concepts of A-lattice as an object of general algebra and Tlattice as an object of number theory and geometry of numbers.The paper defines the structure of a complete metric space with a non-Archimedean metric on the monoid SN. This made it possible to prove a theorem on the convergence of a sequence of Dirichlet series over convergent sequences of natural numbers.
If we consider the product of two zeta functions of monoids of natural numbers, then it will be a zeta function of a monoid of natural numbers only when these monoids are mutually simple.
In general, their product will be a Dirichlet series with natural coefficients over a monoid equal to the product of the monoids of the cofactors. This monoid generated by the zeta functions of
the monoids of natural numbers is denoted by MD. It is shown that the monoids MN and MD are non-isomorphic.
The paper defines two small categories ℳ𝒩 and 𝒮𝒩 and studies some of their properties.

In the article, the author continues to consider issues related to the problem of freedom in Artin groups with a woody structure, and published jointly with V. N. Bezverkhnim in the Chebyshev Collection in 2014. In particular, the following subgroup theorem is proved for Artin groups with a tree structure: if 𝐻 is a finitely generated subgroup of the Artin group with a tree structure, and the intersection of 𝐻 with any subgroup conjugate to a cyclic subgroup.
generated by the generating element of the group, there is a unit subgroup, then there is an algorithm describing the process of constructing free subgroups in 𝐻.
The study of free subgroups in various classes of groups was carried out by many outstanding mathematicians, the fundamental results are presented in a number of textbooks on group theory, monographs and articles.
Artin’s groups have been actively studied since the beginning of the last century. If the Artin group corresponds to a finite tree graph such that its vertices correspond to generating groups, and every edge connecting the vertices corresponds to a defining relation connecting
the corresponding generators, then we have an Artin group with a tree structure.
An Artin group with a woody structure can be represented as a tree product of twogenerators Artin groups united by infinite cyclic subgroups.
In the process of proving the main result, the following methods were used: the reduction of the set of generators to a special set introduced by V. N. Bezverkhnim as a generalization of the Nielsen set to amalgamated products of groups, as well as the representation of a subgroup as a free product of groups and the assignment of a group using a graph.

In various sections of modern mathematics and theoretical physics find their wide application of geometry of constant curvature. These geometries include spherical geometry, Lobachevsky geometry, de Sitter geometry. 𝑛-dimensional geometries of constant curvature are defined by metric functions that are invariants of motion groups of dimension 𝑛(𝑛+1)/2, therefore they are geometries of local maximum mobility. In this article, by the example of geometries of constant
curvature, the embedding problem is solved, the essence of which is to find (𝑛+1) -dimensional geometries of local maximum mobility from 𝑛-dimensional geometries of constant curvature. We search for all functions of a pair of points of the form 𝑓(𝐴,𝐵) = 𝜒(𝑔(𝐴,𝐵),𝑤𝐴,𝑤𝐵) that define (𝑛 + 1)-dimensional geometries with motion groups of dimension (𝑛 + 1)(𝑛 + 2)/2 by the wellknown metric functions of 𝑔(𝐴,𝐵) 𝑛-dimensional geometries of constant curvature. This problem
reduces to solving functional equations of a special form in the class of analytic functions. The solution is sought in the form of Taylor series. To simplify the analysis of coefficients, the Maple 17 mathematical program package is used. The results of this embedding of 𝑛-dimensional geometries of constant curvature are (𝑛 + 1)-dimensional extensions of Euclidean and pseudo-Euclidean 𝑛-dimensional spaces. In addition to the main theorem, auxiliary statements of independent significance are proved.

This article is concerned with the study of connections between the Jordan–Kronecker invariants and free generatedness of the ring of Ad*-invariant polynomials of Lie algebras of dimension less than or equal to seven. At the dual space of the Lie algebra it is possible to define the Poisson bracket with the constant coefficients and the Lie-Poisson bracket. Thus, any pair of points from this dual space defines an one-parameter family of skew-symmetric bilinear forms, called a pencil. For any two bilinear forms from the pencil there exists a basis, in which
their matrices can be simultaneously reduced to the block-diagonal form with the blocks of two types. This form is called the Jordan-Kronecker decomposition. At the same time, the number and sizes of blocks will be the same for any pair of bilinear forms from the pencil. The algebraic type of a pencil is the number and sizes of blocks in the Jordan-Kronecker decomposition of any pairs of bilinear forms from the pencil. Almost all pencils of the same Lie algebra have the same algebraic type, which is the Jordan-Kronecker invariant of a given Lie algebra. There is a theorem that states that for a nilpotent Lie algebra, the existence of two Kronecker pencils of the same rank but of different algebraic types means that the ring of Ad*-invariant polynomials
must be non-freely generated. In this paper, we considered all Kronecker Lie algebras (from the certain list of 7-dimensional nilpotent Lie algebras) for which there was a possibility of the existence of a Kronecker pencils of the same rank as the rank of the algebra. As a result of the research, a negative answer was obtained to the question of whether the converse statement to the previous theorem is true.

We obtain an asymptotic formula for the number of primes 𝑝 ≤ 𝑥1, 𝑝 ≤ 𝑥2 such that
𝑝1(𝑝2 + 𝑎) ≡ 𝑙 (mod 𝑞) with 𝑞 ≤ 𝑥æ0 , 𝑥1 ≥ 𝑥1−𝛼, 𝑥2 ≥ 𝑥𝛼, 

$$æ0 =1/(2, 5 + 𝜃 + 𝜀), 𝛼 ∈[︂(𝜃 + 𝜀)ln 𝑞/ln 𝑥, 1 − 2, 5(ln 𝑞/ln 𝑥)]︂,$$

where 𝜃 = 1/2, if 𝑞 is a cube free and 𝜃 = 5
6 otherwise. This is the refinement and generalization
of the well-known formula of A.A.Karatsuba.

The article is devoted to the question whether the orbit space of a compact linear group is a topological manifold and a homological manifold. In the paper, the case of a simple three-dimensional group is considered. An upper bound is obtained for the sum of the halfdimension integral parts of the irreducible components of a representation whose quotient space is a homological manifold, that enhances an earlier result giving the same bound if the quotient
space of a representation is a smooth manifold. The most of the representations satisfying this bound are also researched before. In the proofs, standard arguments from linear algebra, theory of Lie groups and algebras and their representations are used.

Currently, one of the main problems of using the heat-resistant nickel alloy ZhS6U is associated with the presence of expensive components in its composition, such as Ni, Ti, Mo, Co, etc. and the need to reuse it by grinding. One of the effective, but insufficiently studied metallurgical methods of grinding metal waste is electrodispersion. To date, in the modern scientific and technical literature there is no complete information about the composition, structure and properties of the particles of the ZhS6U alloy obtained in the conditions of
electroerosive metallurgy.
In order to predict the high physical and mechanical properties of products from the resulting charge, it was necessary to optimize the modes of electroerosive dispersion of waste of the ZhS6U
alloy by the method of experiment planning. For a charge with a spherical particle shape, one of the main technological parameters is the optimal granulometric composition, therefore, the optimization of the process of obtaining the charge from the waste of the ZhS6U alloy was carried out according to the average particle size. The electroerosive dispersion of the waste of the ZhS6U alloy was carried out on an experimental installation (RF Patent No. 2449859). As a
result of exposure to short-term electrical discharges, particles of various shapes and sizes were formed. Optimization of the process of electrodispersion of particles obtained by the EED of the waste of the ZhS6U alloy was carried out by experimental determination of a combination of levels of factors at which the required value of the average diameter of the particles of the electroerosion charge was achieved. To do this, the method of steep ascent of Box and Wilson
was used. Optimization of the process of electrodispersion of waste of the ZhS6U alloy in distilled water and lighting kerosene was carried out taking into account such factors as the voltage at the electrodes, the capacity of the discharge capacitors and the pulse repetition frequency.
According to the conducted series of experiments, the limiting values of the optimization parameter for the average size of electroerosive particles were determined, which were: for distilled water – 50.4 microns with a capacity of discharge capacitors of 65.5 UF, a voltage
at the electrodes of 200 V, a pulse repetition frequency of 200 Hz; for lighting kerosene - 58.4 microns with a capacity of discharge capacitors of 65.5 UF, a voltage at the electrodes of 200 V, a pulse repetition frequency of 200 Hz.
Carrying out the planned measures will solve the problem of recycling heat-resistant nickel alloy waste and their reuse in the manufacture of critical parts of mechanical engineering.

In this paper we consider the first special boundary value problem in the mechanics of inhomogeneous deformable solids, when the defining relations connecting the stress tensor with the strain tensor are a nonlinear operator from the strain tensor. The type of the defining operator in an inhomogeneous body depends on at which point the stresses are determined. At the boundary of the body, at each boundary point, the displacements are defined as a convolution
of an arbitrary constant symmetric tensor of rank 2 with the coordinates of this point. In our study, it is assumed that the deformations, arising in the body from such a boundary action are small. As a consequence, the average value of the strain tensor in the body coincides with the constant tensor defined at the boundary, independently of type of the defining relations. The
displacement of a point inside the body is represented as a sum of two terms. The first term is the convolution of the boundary tensor with the point coordinates, and the second term is an unknown vector function (structural function) that depends on the coordinates of the point and the boundary tensor. This function is zero at the boundary of the body. A nonlinear operator
differential equation is obtained for the structural function in the general case. To solve this equation, the method of successive approximations is applied and approximate expressions for the structural functions and, through them, the strains and stresses at each point of the body are found. Stresses are then averaged over the body volume and compared with average strains, i.e., the type of effective defining relations expressing average stresses through average strains
is determined. The case of an inhomogeneous in thickness, infinite in plan, plate is considered in detail.

In the article the direct and inverse problems of scattering of a harmonic plane sound wave by a homogeneous isotropic cylinder with inhomogeneous anisotropic elastic coating in the presence of the underlying flat surface are considered. It is assumed that the coating material of cylinder is radially inhomogeneous and transverse-isotropic, the inhomogeneity laws of the coating material described by continuous radial coordinate functions, the body is placed in an ideal fluid, underlying surface is perfect (absolutely hard or acoustically soft).
An analytical solution of the direct diffraction problem is obtained. The scattered acoustic field and wave fields in the cylinder and its coating are defined.
Based on the solution of the direct problem a mathematical modeling of an inhomogeneous
anisotropic coating of a elastic cylinder providing the least sound reflection done.
The inhomogeneity laws of the coating material ensuring the minimum sound scattering in the given frequency range at a fixed angle of observation and also in the given angular sector of observation at a fixed frequency are obtained. The functionals expressing the average intensity of sound scattering are built. Minimization of the functionals are implemented with the help of the burnout simulation algorithm.
The results of numerical calculations of frequency dependencies of intensity of the scatter acoustic field at the optimal parabolic inhomogeneity laws are presented for different types of transverse-isotropic coatings.

Analysis of the deformation processes of both long-known and new polymer, composite and synthetic materials used in building structures, parts of apparatuses, machines, as well as power plants revealed their specific properties. It is established that many similar materials have orthotropy of the structure with simultaneous manifestation of deformation anisotropy
or heterogeneity. Induced deformation anisotropy or mechanical inhomogeneity is caused by the dependence of stiffness and strength characteristics on the type of stress state. In previous works of the authors, it has been shown that traditional models of deformation of such materials and their mathematical representations lead to gross errors that are clearly manifested in the
calculation of various structures. At the same time, the theories of deformation of composite materials with "complicated properties specially developed for them by other authors in the last 40 years, are very contradictory and have insurmountable disadvantages. The authors of
the presented work have previously developed nonlinear energy relations of strain and stress tensors, for determining the constants of which a wide range of experiments is recommended.
However, among the experimental tests, it is necessary to involve experiments on complex stress states, many of which are currently practically unrealizable. Therefore, in 2021, a quasi-linear deformation potential was postulated, represented in the main axes of orthotropy of materials.
For this option, it turned out to be sufficient to calculate constants according to the simplest experiments. Despite the undoubted advantages of this potential, nevertheless, real nonlinear diagrams were approximated by direct rays using the least squares method, and this, with qualitative adequacy, led to quantitative errors. In this regard, the presented article attempts to avoid the general rules for the formulation of a complete nonlinear potential relationship of
strain and stress tensors. In this direction, a nonlinear mathematical model of the connection of two second-rank tensors is postulated, combining the form of the generalized Hooke’s law for orthotropic material, the theory of small elastic-plastic deformations and the tensor space technique of normalized stresses. This approach allowed us to determine nonlinear material
functions, limiting ourselves to a set of traditional simplest experiments. A remark is made about the uniqueness of solutions to boundary value problems, which boils down to checking the stability of the equations of state in the small Drucker. Within the framework of the proposed mathematical model, widely known experimental diagrams for a carbon-graphite composite are processed, for which nonlinear material functions are obtained.

The subgroups 𝐴 and 𝐵 of a group 𝐺 are called mutually permutable if 𝐴 permutes with all subgroups of 𝐵 and 𝐵 permutes with all subgroups of 𝐴. The sufficient conditions of wsupersolubility of a group 𝐺 = 𝐴𝐵 that is factorized by two mutually permutable w-supersoluble subgroups 𝐴 and 𝐵 were obtained. Besides we found the construction of w-supersoluble residual of such group.

The Fejes T´oth problem about the maximum 𝐸* of the mean value of the sum of angles between lines in R3 with a common center is considered. L. Fejes T´oth suggested that 𝐸* = 𝜋/3 = 1.047 . . .. This conjecture has not yet been proven. D. Bilyk and R.W. Matzke proved that 𝐸*<= 1.110 . . .. We refine this estimate using an extremal problem of the Delsarte type: 𝐸* <= 𝐴* <= 1.08326. Using the dual problem 𝐵* we show that the solution of the 𝐴* problem does not allow us to prove the Fejes T´oth conjecture, since 1.05210 < 𝐴*.

This is the third article in a series dedicated to Smolyak grids. The work relates to analytical number theory and it deals with the application of number theory to problems of approximate analysis.
The paper shows that:
1. the linear operator 𝐴𝑞 of weighted grid averages over the Smolyak grid at dimension 𝑠 > 3
is not normal;
2. found the values of some trigonometric sums 𝑆𝑞(𝑚1, . . . ,𝑚𝑠) of the resin grid at the dimension 𝑠 >= 3.

The probability distribution density of an aggregated random variable is constructed, which is used to estimate the parameters of an aggregated production function determined by a quadratic convolution of production functions characterizing the particular results of the functioning of elements of a complex system. The relations in quadratures for the threedimensional
case are obtained.

In order to investigate arithmetic properties of the values of generalized hypergeometric functions with rational parameters one often makes use of Siegel’s method. By means of this method have been achieved the most general results concerning this problem. The
main deficiency of Siegel’s method consists in the impossibility of its application in case of hypergeometric functions with irrational parameters. In this situation the investigation is usually based on the effective construction of the functional approximating form (in Siegel’s method the existence of such a form is proved by means of pigeon-hole principle). The construction and investigation of an approximating form is the first step to the achievement of arithmetic result.
Applying effective method we encounter at least two problems which make considerably narrow the area of its employment. First, the more or less general effective construction of the approximating form for the products of hypergeometric functions is unknown. While using
Siegel’s method one doesn’t deal with such a problem. Hence the investigator is compelled to consider only questions of linear independence of the values of hypergeometric functions over some algebraic field. Choosing this field is the second problem. The great majority of published results concerning corresponding questions deals with imaginary quadratic field (or the field of rational numbers). Only in exceptional situations it is possible to investigate the case of some other algebraic field. We consider here the case of a field of the fourth degree. By means of a special technique we
establish linear independence over such a field of the values of some hypergeometric function with irrational parameter from that field.

Established in November 1922, the Research Institute of Mathematics and Mechanics of Moscow University can be placed among the largest mathematical institutes in the first third of the 20th century. By its very position – a metropolitan mathematical institute with a powerful
scientific potential – it became the leading mathematical institute of the Soviet Union, which determined the social life of the Soviet mathematical community. With the relocation in 1934 from Leningrad to Moscow of the Presidium of the Academy of Sciences of the USSR and
the V. A. Steklov Mathematical Institute, the general situation has undergone fundamental changes – «Steklovka» became the head scientific institute in the USSR, which included leading scientists from the university research institute of mathematics and mechanics, whose work was reduced mainly to organizing postgraduate studies. At the same time, of course, we should not forget that, having transferred its leading scientists to «Steklovka», the Research Institute of Mathematics and Mechanics of Moscow University actually became one of its co-founders. And when we talk today about the history of the Steklov Institute, we must consider the history of the Research Institute of Mathematics and Mechanics of Moscow University as an integral part of it. So we are not talking about the “death” of the Institute, but about its new life, about the
synthesis of the ideas of the two main Russian schools that gave life to the Soviet Mathematical School, one of the leading mathematical schools of the second half of the 20𝑡ℎ century.

The article gives a brief outline of the history of the department of Number Theory of MSPU from its creation to the present. In connection with the 150-th anniversary of Moscow State Pedagogical University, a short overvew about the main milestones in history, leading specialists, scientific and educational activities of one of the oldest departments of the Institute of Mathematics and Informatics (until 2018 — the Faculty of Mathematics) of Moscow State Pedagogical University (until 1990 — Moscow State Pedagogical Institute named after V. I. Lenin) is presented.