The article is devoted to the scientific and pedagogical activities of Pafnuty Lvovich Chebyshev (1821 – 1894) – one of the greatest mathematicians of the 19th century, the founder of the theory of synthesis of mechanisms, the author of fundamental results in thefield of probability theory, who transformed this science into one of the most dynamically developing branches of mathematics and mathematical natural science, the author of remarkable achievements in number theory, function theory and mathematical analysis and in other branches of mathematics, the creator of the brilliant Petersburg mathematical school, who,
together with Ch. Hermite, K. Weierstrass, J. Sylvester, G. Mittag-Leffler, acted as one of the leaders of the newly emerging world mathematical community.

The aim of this work is to find optimal estimates for the Bergman kernels for the classical domains ℜ𝐼 (𝑚, 𝑘) ,ℜ𝐼𝐼 (𝑚) ,ℜ𝐼𝐼𝐼 (𝑚) and ℜ𝐼𝑉 (𝑛) through the Bergman kernels of balls in the spaces C𝑚𝑘,C𝑚(𝑚+1) 2 ,C𝑚(𝑚−1) 2 and C𝑛, respectively. For this, we use the statements of the Summer-Mehring theorem on the extension of the Bergman kernel and some properties of the Bergman kernel.

In the theory of bi-Hamiltonian systems, the generalized Mischenko–Fomenko conjecture is known. The conjecture states that there exists a complete set of polynomial functions in
involution with respect to a pair of naturally defined Poisson structures on a dual space of a Lie algebra. This conjecture is closely related to the argument shift method proposed by
A. S. Mishchenko and A. T. Fomenko in [10]. In research works devoted to this conjecture, a connection was found between the existence of a complete set in bi-involution and the algebraic
type of the pencil of compatible Poisson brackets, defined by a linear and constant bracket. The numbers that describe the algebraic type of the generic pencil of brackets on the dual space
to a Lie algebra are called the Jordan–Kronecker invariants of a Lie algebra. The notion of Jordan–Kronecker invariants was introduced by A. V. Bolsinov and P. Zhang in [2]. For some
classes of Lie algebras (for example, semisimple Lie algebras and Lie algebras of low dimension), the Jordan–Kronecker invariants have been computed, but in the general case the problem of
computation of the Jordan–Kronecker invariants for an arbitrary Lie algebra remains open. The problem of computation of the Jordan–Kronecker invariants is frequently mentioned among the
most interesting unsolved problems in the theory of integrable systems [4, 5, 6, 11]. In this paper, we compute the Jordan–Kronecker invariants for the series 𝐵𝑠𝑝(2𝑛) and construct complete sets of polynomials in bi-involution for each algebra of the series. Also, we calculate the Jordan–Kronecker invariants for the Borel subalgebras 𝐵𝑠𝑜(𝑛) for any 𝑛. Thus, together with the results obtained in [2] for 𝐵𝑠𝑙(𝑛), this paper presents a solution to the problem of computation of Jordan–Kronecker invariants for Borel subalgebras of classical Lie algebras.

A contour integration method, used to study the asymptotic of the sums of coefficients of Dirichlet series, is based on the Inversion formula. It allows you to express the sum of the
coefficients in terms of the sum of the series. This approach gives effective estimates if the abscissa of absolute convergence 𝜎𝑎 > 1. In some cases, when studying arithmetical functions in
generating Dirichlet series, this value is less than 1. As a rule, in this case, the Tauberian Delange theorem, which gives only the main term of asymptotic, is applied. However, generating Dirichlet series have better analytical properties than we need for the Delange theorem application. The contour integration method allows to count on precise results, but it need the inversion formula which is effective for series with 𝜎𝑎 < 1.
In this paper the such inversion formula is presented and is proved to be an effective tool on examining the distribution of d(n) function values in the residue classes coprim with a module.
W. Narkievicz used Delange theorem to obtain the main term of the asymptotic for frequency of hits of the values of function d(n) in residue classes. Application of the inversion formula allowed us to obtain more precise results.

In this article we consider questions of graph enumeration for some graphs of a special form.
In fact, a number of new results have been proved on the number of spanning trees and spanning forests of graphs that play an important role in the applied problems of Information Theory.
On the one hand, the properties of the spanning converging forests of oriented graphs involved in the construction of the mean first passage time quasi-metric, a generalized metric structure closely related to ergodic homogeneous Markov chains, are considered. On the other hand, the characteristics of spanning rooted forests and spanning converging forests of non-oriented and oriented graphs needed for the construction of a matrix of relative connectivity via forests, one of the measures of proximity of the vertices of graph structures, which plays an important role in solving of applied problems, have been studied. The consideration is based on several simple graph models, including a simple cycle, a simple path and their oriented analogues.
The first section (introduction) presents the history of the problem and provides an overview of the main ideas and results presented in the article. The role of graph models in the
presentation and study of ergodic homogeneous Markov chains is considered. In fact, a Markov chain is a mathematical model of some random process describing a sequence of possible events
in which the probability of each event depends only on the state attained in the previous event.
The Markov chain can be depicted as an oriented weighted graph of transitions whose vertices correspond to the states of the chain and the arcs correspond to the transitions between them.
On the other hand, any connected graph can be used as a basis for building a model of the simplest Markov chain: if a vertex 𝑖 has degree 𝑘, all incident edges are converted into arcs
with the weights 1/𝑘 . The matrix of relative connectivity via forests for non-oriented and oriented graphs is defined; its role for solving important applied problems of Information Theory is
disclosed.The second section contains the basic definitions of Graph Theory necessary to formulate and prove the main results of the article. The definitions of a graph and an oriented graph, a
spanning subgraph, a spanning rooted forest (for non-orentied graphs) and a spanning converging forest (for oriented graphs) are given. Some examples are represented.
In the third section, the definition of Fibonacci numbers is given, a number of properties of Fibonacci numbers necessary to obtain the main results of the article for undirected paths and
cycles are formulated and proved.
In the fourth section, two theorems on the enumeration of graphs related to the construction of the mean first passage time matrix for a homogeneous ergodic Markov chain are proved. In
fact, the number of spanning converging trees for the oriented path and the oriented cycle and the number of spanning rooted trees for the non-oriented path and the non-oriented cycle are
given; the spanning forests consisting of two trees for the same graph structures are counted.
Results for the oriented case are formulated in terms of values 2𝑘, 𝑘 ≥ 0; results for the nonoriented case are formulated in terms of Fibonacci numbers 𝑢𝑘, 𝑘 ≥ 1. The proofs are based on
elementary methods of enumerating Combinatorics.
The fifth section presents the results related to enumeration of spanning forests needed for construction of the matrix of relative connectivity via forests for the non-oriented paths and
cycles and their oriented analogues. Total number of spanning converging forests (for oriented paths and cycles) and total number of spanning rooted forests (for non-oriented paths and
cycles) are found; enumeration of the spanning converging forests, in which a vertex 𝑖 belongs to a tree converging to a vertex 𝑗 (for the oriented paths and cycles), and enumeration of the spanning rooted forests, in which a vertex 𝑖 belongs to a tree with a root 𝑗 (for the non-oriented paths and cycles) are represented. As before, results for the oriented case are formulated in terms of values 2𝑘, 𝑘 ≥ 0; results for the non-oriented case are formulated in terms of Fibonacci
numbers 𝑢𝑘, 𝑘 ≥ 1. The sixth section (conclusion) presents the main conclusions of the article, outlines the ideas of further studies.

This is the second article in a series dedicated to Smolyak grids. The paper relates to analytical number theory and it deals with the application of number theory to problems of approximate analysis.
In this paper, it was shown that for an arbitrary Smolyak grid, the trigonometric sum of the Smolyak grid is 𝑆𝑞(⃗0) = 1. It follows that the norm of the linear functional of approximate integration on the class 𝐸𝛼 𝑠 is equal to the value of the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the resin grid. It is shown that the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the Smolyak grid
is a Dirichlet series. This raises the question of the analytic continuation of the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the Smolyak grid as a function of an arbitrary complex 𝛼 = 𝜎 + 𝑖𝑡.
Since the Smolyak grid belongs to the number of rational grids, it turns out that it has an analytical continuation of the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the Smolyak grid on the entire complex plane except for the point 𝛼 = 1, in which it has a pole of order 𝑠.
It follows from the work that the following questions remain open:
1. is the linear operator 𝐴𝑞 of weighted grid averages over the Smolyak grid at dimension 𝑠 > 3 normal?
2. what are the true values of the trigonometric sums 𝑆𝑞(𝑚1, . . . ,𝑚𝑠) Smolyak grids with dimension 𝑠 > 3?

A beautiful and meaningful harmonic analysis has been constructed on the Euclidean space R𝑑 with Dunkl weight. The classical Fourier analysis on R𝑑 corresponds to the weightless case.
The Dunkl–Riesz potential and the Dunkl–Riesz transforms play an important role in the Dunkl harmonic analysis. In particular, they allow one to prove the Sobolev type inequalities for
the Dunkl gradient. Earlier we proved (𝐿𝑞,𝐿𝑝)-inequalities for the Dunkl–Riesz potential with two radial piecewise power weights. For the Dunkl–Riesz transforms, we proved 𝐿𝑝-inequality
with one radial power weight and, as a consequence, we obtained (𝐿𝑞,𝐿𝑝)-inequalities for the Dunkl gradient with two radial power weights. In this paper, these results for the Dunkl–Riesz transforms and the Dunkl gradient for radial power weights are generalized to the case of radial piecewise power weights.

During last 20 years, an essential part of the theory of Hausdorff operators is concentrated on their boundedness on the real Hardy space 𝐻1(R𝑑). The spaces introduced by Sweezy are,
in many respects, natural extensions of this space. They are nested in full between 𝐻1(R𝑑) and 𝐿10 (R𝑑). Contrary to 𝐻1(R𝑑), they are subject only to atomic characterization. For the estimates of Hausdorff operators on 𝐻1(R𝑑), other characterizations have always been applied.
Since this option is excluded in the case of Sweezy spaces, in this paper an approach to the estimates of Hausdorff operators is elaborated, where only atomic decompositions are used.
While on 𝐻1(R𝑑) this approach is applicable to the atoms of the same type, on the Sweezy spaces the same approach is not less effective for the sums of atoms of various types. For a
single Hausdorff operator, the boundedness condition does not depend on the space but only on the parameters of the operator itself. The space on which this operator acts is characterized
by the choice of atoms. An example is given (two-dimensional, for simplicity), where a matrix dilates the argument only in one variable.

For a positive integer 𝑄 > 0, let 𝐼 ⊂ R denote an interval of length 𝜇1𝐼 = 𝑄−𝛾1 (where 𝜇1 is the Lebesgue measure) and 𝜇2𝐾 = 𝑄−𝛾2 , 𝛾2 > 0 (where 𝜇2 is the Haar measure of a measurable cylinder 𝐾 ⊂ Q𝑝). Let us denote the set of polynomials of degree ≤ 𝑛 and height 𝐻 (𝑃) ≤ 𝑄 as 𝒫𝑛 (𝑄) = {𝑃 ∈ Z[𝑥] : deg 𝑃 ≥ 𝑛, 𝐻 (𝑃) ≤ 𝑄} .
Let 𝒜(𝑛,𝑄) denote the set of real and 𝑝-adic roots of such polynomials 𝑃 (𝑥) lying in the space 𝑉 = 𝐼 ×𝐾. In this paper it is proved that the following inequality holds for a suitable constant
𝑐1 = 𝑐1 (𝑛) and 0 ≤ 𝑣1, 𝑣2 6 1 2 : #𝒜(𝑛,𝑄) > 𝑐1𝑄𝑛+1−𝛾1−𝛾2 .
The proof relies on methods of metric theory of Diophantine approximation developed by V.G. Sprindzuk to prove Mahler’s conjecture and by V.I. Bernik to prove A. Baker’s conjecture.

An important problem in discrete geometry and computational mathematics is the estimation of the minimum number of nodes 𝑁(𝑠) of a quadrature formula (weighted 𝑠- design) of the form 1
|S2| ∫︀ S2 𝑓(𝑥) 𝑑𝑥 = Σ︀𝑁 𝜈=1 𝜆𝜈𝑓(𝑥𝜈) with positive weights, exact for all spherical polynomials of degree at most 𝑠. P. Delsarte, J.M Goethals, and J.J. Seidel (1977) to estimate 𝑁(𝑠) from below formulated an extremal problem 𝐴𝑠 for expansions in terms
of orthogonal Gegenbauer (Legendre for S2) polynomials with restrictions on the sign of the Fourier–Gegenbauer coefficients. Using a version of this problem 𝐴𝑠,𝑛 on polynomials of degree
𝑛 = 𝑠, they proved the classical estimate for tight designs. This estimate is sharp and gives a solution to 𝐴𝑠 only in exceptional cases (𝑠 = 0, 1, 2, 3, 5 for S2). For general dimensions, there
are cases when 𝐴𝑠,𝑛 > 𝐴𝑠,𝑠 for 𝑛 > 𝑠, which leads to better estimates for 𝑁(𝑠). In particular, N.N. Andreev (2000) proved in this way the minimality of an 11-design on the sphere S3. Related
Delsarte problems are also formulated for estimating the cardinality of spherical codes. In this direction, V.V. Arestov and A.G. Babenko (1997), based on the methods of infinite-dimensional linear programming, solved an analog of the 𝐴𝑠 problem for the case of spherical 0.5-codes on the sphere S3 (the kissing number problem). Then this method was developed in the works of D.V. Shtrom, N.A. Kuklin. A.V. Bondarenko and D.V. Gorbachev (2012) showed that 𝑁(4) = 10. This fact follows from the estimate 𝐴4,7 > 9, previously obtained by P. Boyvalenkov and S. Nikova (1998), and the existence of weighted 4-designs of 10 nodes. Nevertheless, it is of interest to solve the problem 𝐴4 exactly, aiming to transfer the method of calculating 𝐴𝑠 to the general dimensions and orders of designs. In this paper, it is proved that 𝐴4 = 𝐴4,22 = 9.31033 . . .
For this, the Arestov–Babenko–Kuklin method is adapted and the problem is reduced to the construction of a special quadrature formula for [−1, 1], consistent with the form of the assumed extremal function (polynomial). The proposed method is based on the use of nonlinear programming, in particular, semidefinite programming, and the solution of a polynomial system of equations arising from a quadrature formula. To prove the existence of an analytical solution of such a system in the neighborhood of the numerical solution, interval Krawczyk’s method from HomotopyContinuation.jl is used.

The paper continues the authors ’ research on the evaluation of trigonometric sums of an algebraic grid with weights. The case of an arbitrary weight function of infinite order is considered.
For the parameter ⃗𝑚 of the trigonometric sum 𝑆𝑀(𝑡),⃗𝜌∞(⃗𝑚), three cases are highlighted. If ⃗𝑚 belongs to the algebraic lattice Λ(𝑡·𝑇(⃗𝑎)), then for any natural 𝑟 the asymptotic formula
is valid 𝑆𝑀(𝑡),⃗𝜌∞(𝑡(𝑚, . . . ,𝑚)) = 1 + 𝑂 (︂ ln𝑠−1 det Λ(𝑡) (detΛ(𝑡))𝑟+1)︂.
If ⃗𝑚 does not belong to the algebraic lattice Λ(𝑡·𝑇(⃗𝑎)), then two vectors are defined ⃗𝑛Λ(⃗𝑚) =(𝑛1, . . . , 𝑛𝑠) and ⃗𝑘Λ(⃗𝑚) from the conditions ⃗𝑘Λ(⃗𝑚) ∈ Λ, ⃗𝑚 = ⃗𝑛Λ( ⃗𝑀)+ ⃗𝐾𝜆(⃗𝑚) and the product𝑞(⃗𝑛𝜆(⃗𝑚)) = 𝑛1 · . . . · 𝑛𝑠 is minimal. Asymptotic estimation is proved|𝑆𝑀(𝑡),⃗𝜌∞(⃗𝑚)| 6 𝐵(𝑟,∞)(︃1 − 𝛿(⃗𝑘Λ(⃗𝑚))(𝑞(⃗𝑛Λ(⃗𝑚)))𝑟+1 + 𝑂(︂𝑞(⃗𝑛Λ(⃗𝑚))𝑟+1 ln𝑠−1 det Λ(𝑡)(det Λ(𝑡))𝑟+1)︂)︃.

In this paper, we find an asymptotic formula with power-saving remainder term for the number of solutions of one non-linear ternary problem with primes. The proof is based on the "precise formula"for Chebyshev’s function involving the zeros of Riemann zeta function. In fact, a ternary problem "at each zero"is solved. I. M. Vinogradov’s solution of the ternary Goldbach problem (1937, see [1], [2]) opened the way of solving a wide class of problems of the above type. In 1938, he found a power-saving estimate (with respect to the length of the summation interval) for the mean value of the modulus of the exponential sum with primes (see [2], theorem 3, p.82; theorems 6 and 7, p.86). Starting at 1996, G.I.Arkhipov, K.Buriev and the author have obtained several results concerning the exceptional sets in some binary problems of Goldbach’s type. These results used both the tools of the theory of Diophantine approximations and the "precise formulas"from Riemann’s zeta function theory. At the same time, the method of estimating of linear sums with primes based on the measure theory was derived in the papers of G. L. Watson, D. Bruedern, R. D. Cook and A. Perelli.

In the previous work, the authors laid the foundations of the theory of smooth varieties of number-theoretic lattices. The simplest case of one-dimensional lattices is considered.
This article considers the case of one-dimensional shifted lattices. First of all, we consider the construction of a metric space of shifted lattices by mapping one-dimensional shifted lattices
to the space of two-dimensional lattices.
In this paper, we define a homeomorphic mapping of the space of one-dimensional shifted lattices to an infinite two-dimensional cylinder. Thus, it is established that the space of onedimensional
shifted lattices 𝐶𝑃𝑅2 is locally a Euclidean space of dimension 2.
Since the metric on these spaces is not Euclidean, but is "logarithmic" , unexpected results are obtained in the one-dimensional case about derivatives of basic functions, such as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
The paper considers the relationship of these functions with the issues of studying the error of approximate integration over parallelepipedal grids as the determinant of the lattice, the
hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
Note that the geometry of metric spaces of multidimensional lattices and shifted multidimensional lattices is much more complex than the geometry of an ordinary Euclidean space.
This can be seen from the paradox of nonadditivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox, it follows that there is
an open problem of describing geodesic lines in the spaces of multidimensional lattices and multidimensional shifted lattices, as well as in finding a formula for the length of the arcs of
lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts.
A further direction of research may be the study of the analytical continuation of the hyperbolic zeta function on the spaces of lattices and multidimensional lattices. As is known, an
analytical continuation of the hyperbolic zeta function of lattices is constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytic continuations in the left
half-plane on the lattice space has not been studied. All these, in our opinion, are relevant areas for further research.

Only finite groups are considered. The work is devoted to the study of formations which are classes of groups that are closed with respect to homomorphic images and subdirect products.
For a non-empty set 𝜔 of primes V.A. Vedernikov, using two types of functions, defined 𝜔-fibered formations of finite groups. Developing this functional approach, in the paper for an arbitrary
partition ¯𝜔 of the set 𝜔 we constructed ¯𝜔-fibered formations. The construction uses the 𝜎- concept of A.N. Skiba for the study of finite groups and their classes, where 𝜎 is an arbitrary partition of the set P of all primes. We gave examples of ¯𝜔-fibered formations, established their properties (existence of ¯𝜔-satellites of different types; sufficient conditions for a group 𝐺 to belong to an ¯𝜔-fibered formation; relationship with 𝜔-fibered and P𝜎-fibered formations).

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The author considered the series
𝑓0(𝜆) =∞Σ︁𝑛=0(𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) =∞Σ︁𝑛=0(𝜆 + 1)𝑛𝜆𝑛,
where 𝜆 is a certain polyadic Liouville number. The series considered converge in any field Q𝑝 . Here (𝛾)𝑛 denotes Pochhammer symbol, i.e. (𝛾)0 = 1 , and for 𝑛 ≥ 1 we have(𝛾)𝑛 =
𝛾(𝛾 + 1)...(𝛾 + 𝑛 − 1). The values of these series were also calculated at polyadic Liouville number. The canonic expansion of a polyadic number 𝜆 is of the form 𝜆 =∞Σ︁𝑛=0𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z, 0 ≤ 𝑎𝑛 ≤ 𝑛.
This series converges in any field of 𝑝-adic numbers Q𝑝.
We call a polyadic number 𝜆 a polyadic Liouville number, if for any 𝑛 and 𝑃 there exists a positive integer 𝐴 such that for all primes 𝑝 ,satisfying 𝑝 ≤ 𝑃 the inequality |𝜆 − 𝐴|𝑝 < 𝐴−𝑛
holds.
The paper gives a simple proof that the Liouville polyadic number is transcendental in any field Q𝑝. In other words,the Liouville polyadic number is globally transcendental. We prove
here a theorem on approximations of a set of 𝑝−adic numbers and it’s corollary — a sufficient condition of the algebraic independence of a set of 𝑝−adic numbers. We also present a theorem on global algebraic independence of polyadic numbers.

In this paper, a new method is constructed for solving partial differential equations using a sequence of nested generalized parallelepiped grids.
This method is a generalization and development of the V. S. Ryaben’kii and N. M. Korobov method for the approximate solution of partial differential equations for the case of using
arbitrary generalized parallelepiped grids for integer lattices. The error of this method was also found. In the case of using an infinite sequence of nested generalized parallelepiped grids,
a fairly fast convergence will take place.
In addition, a variant of constructing optimal grids in the two-dimensional case is proposed.
It is based on the integer approximation of algebraic lattices. In the two-dimensional case, the grids constructed in this way will always give generalized parallelepiped grids. Moreover, there
are simple ways to assess the quality of the resulting meshes. One such method, based on the use of a hyperbolic parameter, is considered in this paper.

In this paper, the optimal parameters of the process of obtaining a heavy tungsten alloy by spark plasma sintering of the particles of the alloy VNZh 95 on the microhardness of the sintered
samples are determined by conducting a full factorial experiment of the type. The operating parameters of the spark plasma sintering unit were selected as factors: temperature, pressure,
and holding time, min. The optimal parameters of the installation operation were determined for the electroerosive material VZH 95, previously obtained in two working media: distilled water
and lighting kerosene. According to a series of experiments to determine the maximum value of the optimization parameter Y (microhardness), which amounted to: for samples obtained
from particles dispersed in water — 3498,6 MPa at 𝑇 = 1050∘C, a pressure of 𝑃 = 40 MPa and holding time 𝑡 = 10 min; for samples obtained from particles dispersed in kerosene — 2449,2 MPa at a temperature 𝑇 = 1200∘C, 𝑃 = 40 MPa and holding time 𝑡 = 5 min. The results of experimental studies of the composition, structure, and properties of heavy tungsten alloys made of electroerosion-dispersed particles of the VNJ 95 alloy obtained under optimal
conditions are presented below.

The paper considers the preconditions and the origin of the theory of nonlinear integral equations. The appearance of this theory was a natural consequence of the development of all
mathematics of the XVIII-XIX cc. At the same time, the growing interest in nonlinear problems in the late XIX and early XX centuries had a strong motivating effect. The direct investigation
of specific nonlinear integral equations was triggered by an urgent applied problem on the equilibrium figures of rotating liquid masses, which has attracted a significant number of major
mathematicians since Newton. In the first decades of the development of the theory of nonlinear integral equations, traditional approaches were cultivated, which were used to study differential and algebraic equations, according to the equation-solution scheme. That is, the foreground was the calculation and assessment of its accuracy. The complexity and originality of nonlinear problems immediately revealed the relevance of questions of the existence and uniqueness of their solutions, which made it necessary to involve other, just emerging areas of mathematics. The theory of integral equations in general was one of the origins of functional analysis. Moreover, both theories were closely intertwined and mutually stimulated each other in their evolution. This fully applies to nonlinear integral equations, for which qualitative methods have become of paramount importance. At the stage considered in this work, there was a parallel development and mixing of traditional methods for studying equations and new approaches of a qualitative
nature. In the next phase, new approaches came to the fore, merging with functional analysis and topology.

A heterogeneous in length bar with a variable cross-section is considered. The bar is compressed by a variable longitudinal force which is distributed along its axis. The article describes the case of stability loss of the straight form of equilibrium of a bar, when both, linear and curved forms are possible. The critical combination of rigidity and longitudinal force is the result of an integral representation for the solution of the given stability equation with variable coefficients by the aid of the solution of similar equation but with constant coefficients. The integral representation includes the Green function of the given equation which can be obtained by the method of perturbations. The example of compiling of the equation for critical loading
is reduced.

Secondkind equation with the variable integrable factors depending on the numerical Parametre (input
equation). The input equation Common decision is about accuracy to two Arbitrary constants by means of the integral formula, before the paper offered by the author. On the general The solution is superimposed two homogeneous conditions from which the system from two equations follows for Arbitrary constants. Demanding, that there was a nontrivial solution of an input equation, We receive the complicated nonlinear equation for numerical parametre (the spectral equation).

The article discusses spherical diffraction problem monochromatic sound wave absolutely rigid sphere. To represent the scattered field, a representation in the form of a Kirchhoff integral is used. This leads to the need to solve the Fredholm integral equation of the second kind to determine the velocity potential in the scattered wave on the surface of the scatterer. It is shown that the use of quadrature formulas based on number-theoretic grids allows you to reduce the number of calculations for the approximate calculation of integrals, when solving the integral equation and when calculating the scattered field on the surface of the sphere and in the far field. This method was compared with the simple cell method, which takes into account the mechanical formulation of the problem and has the same order of accuracy. Estimation of the accuracy of calculating the pressure on the surface of the sphere and the form-function of the scattered field based on the solution of the integral equation was carried out by comparison with the analytical solution based on the expansion in spherical wave functions.

Talking about the Diophantine analysis’ history, namely, the problem of rational solutions of Diophantine equations, we should note the longevity of the algebraic approach, which goes
back to Diophantus’ “Arithmetica”. Indeed, after the European mathematicians of the second half of the XVI century became acquainted with Diophantus’ oeuvre, algebraic apparatus
of variable changes, substitutions and transformations turned into the main tool of finding rational solutions of Diophantine equations. Despite the limitations of this apparatus, there were
obtained important results on rational solutions of quadratic, cubic and quartic indeterminate equations in two unknowns. Detailed historico-mathematical analysis of these results was done, inter alia, by I. G. Bashmakova and her pupils. The paper examines the departure from this algebraic treatment of Diophantine equations, typical for most of the research up to the end of XIX century, towards a more general viewpoint on this subject, characterized also by radical expansion of the tools used in the Diophantine equations’ investigations. The works
of A. L. Cauchy, C. G. J. Jacobi and ´E. Lucas, where this more general approach was developed, are analyzed. Special attention is paid to the works of J. J. Sylvester on Diophantine equations
and the paper “On the Theory of Rational Derivation on a Cubic Curve” by W. Story, which were not in the focus of the research on history of the Diophantine analysis and where apparatus
of algebraic curves was used in a pioneering way.

On May 4 (16 new style), P.L. Chebyshev was born - a great Russian scientist who represents the pride and glory of our Fatherland. In connection with the 200th anniversary of P.L. Chebyshev, a retrospective look at his contemporary era is offered and an attempt is made to comprehend the role of a genius personality not only in the history of science and
technology, but also in the history of Russia, which he served with all his talents, including scientific and pedagogical. The report provides information about the tradition of serving the
Fatherland in the Chebyshev family. A brief overview of the work of P.L. Chebyshev for 48 years at the Imperial St. Petersburg Academy of Sciences is presented. The memoirs of contemporaries about the work of P.L. Chebyshev for 35 years at the Imperial St. Petersburg University are discussed. The work of P.L. Chebyshev for 40 years in the Artillery Department of the Military Scientific Committee is considered. The work of P.L. Chebyshev for 17 years in the Scientific Committee of the Ministry of Public Education is analyzed. Its role in the educational reform (1863-1864) initiated by Emperor Alexander II is clarified, in particular in the development of the general university charter of 1863, in the compilation of recommendatory bibliography, in the organization of teaching mathematics in primary schools in Russia, in the systematic training of engineers in Russia. The role of P.L. Chebyshev in the creation of the Russian mathematical school is briefly highlighted.

In paper the problem of the reflection and refraction of a plane harmonic sound wave by a homogeneous isotropic elastic plate with an continuously inhomogeneous anisotropic elastic
coating is considered. It is assumed that the plate is adjoin to ideal fluids.
The propagation of small perturbations in a ideal fluid in the case of steady state oscillations is described by Helmholtz’s equation. The propagation of elastic waves in a uniform isotropic
elastic plate is described by scalar and vector Helmholtz’s equations for longitudinal and transverse waves. Oscillations of an inhomogeneous anisotropic elastic coating are described
by general motion equations of the continuous medium.
The boundary-value problem for the system of ordinary second order differential equations is constructed for determination of the displacement field in inhomogeneous anisotropic coating.
An analytical description of the reflected and transmitted through the plate acoustic fields is obtained.
Particular cases when the material of an inhomogeneous coating is transversely isotropic and isotropic are considered.
The results of numerical calculations of dependence of reflection coefficient of a homogeneous isotropic plate with a transversely isotropic coating from the incidence angle of plane wave are
presented.

A layered composite made using 3D technology from copper and stainless steel powders is considered. Each layer has a porosity due to the technological process of sintering powder materials. The effective elastic characteristics of such a composite are calculated, taking into account the layering and porosity in the components.

The maximum value of the vibration decrement of a porous metal composite made using 3D technology is determined. The influence of porosity on the damping and stiffness properties of the composite is studied. The optimal porosity value is obtained, which provides a maximum of the vibration decrement at a significant load level on the sample. The results of numerical
calculation of the decrement for a composite made of chromium-nickel porous steel are presented.

The Filippov’s article discusses a possible definition of the solution of differential equation with discontinuous right-hand side. The lemma on the structure of the set defining differential
inclusion given by Filippov implies an equivalent solution definition, which allows us to expand possible domains and codomains of the function, that is in the right-hand side of the equation. In this paper we find a generalization of this lemma to the case of general topologic and measure spaces. Proofs of corresponding theorems are given here.

Based on recently proved estimates for the 𝐿1-Nikolskii constants for S𝑑 and R𝑑, effective bounds for the constant 𝐾 are given in the following inequality of the type Brown–Lucier for functions 𝑓 ∈ 𝐿𝑝(S𝑑), 0 < 𝑝 < 1:
‖𝑓 − 𝐸1𝑓‖𝑝 6 (1 + 2𝐾)1/𝑝 inf 𝑢∈Π𝑑 𝑛 ‖𝑓 − 𝑢‖𝑝, where Π𝑑
𝑛 is the subspace of spherical polynomials, 𝐸1𝑓 is a best approximant of 𝑓 from Π𝑑𝑛 in the metric 𝐿1(S𝑑). The results are generalized to the case of the Dunkl weight.

A subgroup 𝐴 of a group 𝐺 is called 𝑂𝑆-propermutable in 𝐺 if there is a subgroup 𝐵 such that 𝐺 = 𝑁𝐺(𝐴)𝐵, 𝐴𝐵 is a subgroup of 𝐺 and the subgroup 𝐴 permutes with all Schmidt subgroups of 𝐵. In this situation, the subgroup 𝐵 is called 𝑂𝑆-prosupplement to 𝐴 in 𝐺.
In this paper, we proved the 𝑝-solubility of a finite group 𝐺 such that a Sylow 𝑝-subgroup of 𝐺 is 𝑂𝑆-propermutable in 𝐺, where 𝑝 > 5.

We suggest an extremely short proof of Ball’s lemma by means of harmonic analysis only.

for a nonlinear partial differential equation of Sobolev type that is not resolved with respect to the time derivative of the first order, the so-called Cahn-Hillard equation, in the Banach space of continuous bounded functions on the entire number axis, for which there are limits by minus and plus infinity. The existence of a classical solution is proved (by which we mean a sufficiently smooth function that has all continuous derivatives of the required order and satisfies the equation at each point of the domain of the considered Cauchy problem) on an arbitrary
time interval. A priori estimates are obtained that ensure the existence of a global solution to the Cauchy problem for the pseudoparabolic Cahn-Hillard equation, since the classical solution 𝑣 (𝑥, 𝑡) from the interval [0, 𝑡*], taking 𝑣 (𝑥, 𝑡*) as a new initial function, continues to the classical solution 𝑣 (𝑥, 𝑡) on the interval [0, 𝑡* + 𝛿], where the value of 𝛿 depends only on the norm of the initial function and the parameters of the Cahn-Hillard equation. Repeating this process, a sufficiently large number of times, we obtain the classical solution of the considered Cauchy problem on an arbitrary time interval.

We consider the class of functions Φ: R → [0,+∞], which are lower semicontinuous, even, convex and Φ(0) = 0. The Fenchel transform Ψ from Φ also belongs to this class of functions. We will define functions that play the role of derivatives for all functions from our class and prove that these functions are mutually inverse in a generalized sense.

On April 11, 2021, Alexander Ivanovich Nizhnikov turned seventy-five years old, a wellknown Russian mathematician, Professor, Doctor of Pedagogical Sciences, Head of the Department of Technological and Information Systems of the Institute of Physics, Technology and Information Systems of the Moscow Pedagogical State University (MSPU), Vice-Rector of MSPU.
Alexander Ivanovich Nizhnikov has made a significant contribution to development of Russian scientific school on Functional Analysis. Moreover, the scientific and educational
activities of Professor A. I. Nizhikov have been very influential in the development of higher education in Russia. Alexander Ivanovich Nizhnikov published more than 150 scientific and
methodical works. For many years he is a member of the editorial board of the journal “Chebyshevskii Sbornick”. For achievements in the development of Russian Science and Education A. I. Nizhnikov was awarded several medals and honorary signs.