The paper gives a brief overview of the scientific activities of Professor Vladimir Anatolyevich Levin

functional system is a set of functions endowed with a set of operations on these functions.
The operations allow one to obtain new functions from the existing ones.
Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.
The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.
In our paper we consider the functional system of rational functions with rational coefficients endowed with the superposition operation. We investigate the special case of the completeness problem which is of a particular interest, namely obtaining complete systems of minimum cardinality, i.e. complete systems consisting of a single rational function (such functions are
referred to as 𝐴-functions and are analogues of Schaeffer stroke in Boolean logic).
The main results of the paper are the following:

- there exists an A-function;
- the cardinality of the set of all A-functions equals c_0;
- a number of examples of A-functions are presented explicitly.

Let 𝑂𝑋𝑌 be a Cartesian coordinate system with an integer lattice whose unit squares are staggered. The integer approximation of the segment 𝐴𝐵 is given using the cellular domain S𝐴𝐵 of (colored) cells, the interior of each of which.has a non-empty intersection with 𝐴𝐵.
If 𝑃± 𝐴𝐵 — right and left closed half-planes defined by the line 𝑙𝐴𝐵 by the point 𝐴 and 𝐵, then S± 𝐴𝐵 = S𝐴𝐵 ∩ 𝑃± 𝐴𝐵 — its right and left areas. (There are no integer points inside S𝐴𝐵.)
Polyline L±(𝐴±,𝐵±) from S± 𝐴𝐵 with ends 𝐴± and 𝐵± and whole vertices — right and left by (integer) approximations of the segment 𝐴𝐵 — the ends are selected from the vertices of the extreme cells. If 𝑙𝐴𝐵 is parallel to one of the coordinate axes, then we assume S𝐴𝐵 = ? and then approximation of the segment 𝐴𝐵 is minimum segment with integer ends containing 𝐴𝐵.
Such approximations are constructed using the algorithm ‘pulling noses", which is a geometric interpretation of the chain fraction of the angular coefficient of the straight line 𝑙𝐴𝐵. Based on this construction method, an exact formula for calculating the number of integer points inside an arbitrary triangle is obtained, and the problem of S.V. Konyagin is partially solved about chess coloring: If U(𝑡) is the set of all colored cells from a triangle cut off by a straight line 𝑓𝑡 : 𝑦 = −𝛼𝑥 + 𝑡, 𝛼, 𝑡 > 0, then the difference 𝑢(𝑡) between white and black cells from U(𝑡)for every positive irrational 𝛼 is bounded neither from below nor from above when 𝑡 → ∞. The solution is obtained for numbers of the form: 𝑒±1, tg±1, [𝑎−0 ; 𝑎−1 , 𝑎−2 , . . .]±1, [𝑎+0 ; 𝑎+1 , 𝑎+2 , . . .]±1,
[𝑎+0 ; 𝑎−1 , 𝑎+2 , 𝑙𝑑𝑜𝑡𝑠]±1, where the superscript plus (minus) indicates on the parity (odd) of the element of the continued fraction defined by 𝛼.
The method of constructing an approximation of the segment was used to solve the problem of chess coloring for the numbers
(√5+1)/2 , [𝑎+0 ; 𝑎+1 , 𝑎+2 , . . .], 𝑎−2𝑛+1 and [𝑎−0 ; 𝑎−1 , 𝑎−
2 , 𝑙𝑑𝑜𝑡𝑠], if limited 2𝑘−1𝑏3𝑏9 · · · 𝑏6(𝑘−1)+3 + · · · + 22Σ︀𝑘
𝑖1>𝑖2>𝑖3=1 𝑏6(𝑘−𝑖1)+3𝑏6(𝑘−𝑖2)+3 𝑏6(𝑘−𝑖3)+3 + 2
Σ︀𝑘 𝑖1>𝑖2=1 𝑏6(𝑘−𝑖1)+3𝑏6(𝑘−𝑖2)+3 + Σ︀𝑘 𝑖=1 𝑏6(𝑘−𝑖1)+3 + 1,
for some 𝑏𝑛 = ⌊︁𝑎−(𝑛−1)/2⌋︁ representing the whole part of 𝑎−
(𝑛−1)/2 . So for 𝑏𝑛 = 0 the chain fraction is [𝑎− 0 ; 𝑎−1 , 𝑎−2 , . . .] =
(√5+1)/2 .

The question of integrability of the Fourier transform and other integral transformations ℱ(𝑓) on classes of functions in weighted spaces 𝐿𝑝(R𝑑) is a fundamental problem of harmonic analysis. The classical Hausdorff–Young result says that if a function 𝑓 from 𝐿𝑝(R𝑑) with
𝑝 ∈ [1, 2], then its Fourier transform ℱ(𝑓) ∈ 𝐿𝑝′
(R𝑑). For 𝑝 > 2 the Fourier transform in the general situation will be a generalized function. The Fourier transform can be defined as an usual function for 𝑝 > 2 by considering the weighted spaces 𝐿𝑝(R𝑑). In particular, the classical Pitt inequality implies that if 𝑝, 𝑞 ∈ (1,∞), 𝛿 = 𝑑( 1/𝑞 − 1/𝑝′ ), 𝛾 ∈ [(𝛿)+, 𝑑/𝑞 ) and function 𝑓 is integrable in 𝐿𝑝(R𝑑) with power weight |𝑥|𝑝(𝛾−𝛿), then its Fourier transform ℱ(𝑓) belongs to the space
𝐿𝑞(R𝑑) with weight |𝑥|−𝑞𝛾. The case 𝑝 = 𝑞 corresponds to the well-known Hardy–Littlewood inequality.
The question arises of extending the conditions for the integrability of the Fourier transform under additional conditions on the functions. In the one-dimensional case, G. Hardy and J. Littlewood proved that if 𝑓 is an even nonincreasing function tending to zero and 𝑓 ∈ 𝐿𝑝(R) for 𝑝 ∈ (1,∞), then ℱ(𝑓) belongs to 𝐿𝑝(R) with weight |𝑥|𝑝−2. R. Boas (1972) suggested that for
a monotone function 𝑓 the membership | · |𝛾−𝛿𝑓 ∈ 𝐿𝑝(R) is equivalent to | · |−𝛾ℱ(𝑓) ∈ 𝐿𝑝(R) if and only if 𝛾 ∈ (−1/p′ , 1/𝑝 ). The one-dimensional Boas conjecture was proved by Y. Sagher (1976).
D. Gorbachev, E. Liflyand and S. Tikhonov (2011) proved the multidimensional Boas conjecture for radial functions, moreover, on a wider class of general monotone non-negative radial functions 𝑓: ‖| · |−𝛾ℱ(𝑓)‖𝑝 ≍ ‖| · |𝛾−𝛿𝑓‖𝑝 if and only if 𝛾 ∈ ( 𝑑/𝑝 − 𝑑+1/2 , 𝑑/𝑝 ),  where 𝛿 = 𝑑( 1/𝑝 − 1/𝑝′ ). For radial functions, the Fourier transform is expressed in terms of the Bessel
transform of half-integer order, which reduces to the classical Hankel transform and includes the cosine and sine Fourier transforms. For the latter, the Boas conjecture was proved by E. Liflyand and S. Tikhonov (2008). For the Bessel–Hankel transform with an arbitrary order, the Boas conjecture was proved by L. De Carli, D. Gorbachev and S. Tikhonov (2013). D. Gorbachev,
V. Ivanov and S. Tikhonov (2016) generalized these results to the case of (𝜅, 𝑎)-generalized Fourier transform. A. Debernardi (2019) studied the case of the Hankel transform and general monotone alternating functions.
So far, the Boas conjecture has been considered for functions on the semiaxis. In this paper, it is studied on the entire axis. To do this, we consider the integral Dunkl transform, which for even functions reduces to the Bessel–Hankel transform. It is also shown that the Boas conjecture remains valid for the (𝜅, 𝑎)-generalized Fourier transform, which gives the Dunkl transform for
𝑎 = 2. As a result, we have ‖| · |−𝛾ℱ𝜅,𝑎(𝑓)‖𝑝,𝜅,𝑎 ≍ ‖| · |𝛾−𝛿𝑓‖𝑝,𝜅,𝑎, where 𝛾 ∈ ( 𝑑𝜅,𝑎/𝑝 − (𝑑𝜅,𝑎+𝑎/2)/2 , 𝑑𝜅,𝑎
𝑝 ), 𝛿 = 𝑑𝜅,𝑎( 1/𝑝 − 1/𝑝′ ), 𝑑𝜅,𝑎 = 2𝜅 + 𝑎 − 1.

A linear problem of regression analysis is considered under the assumption of the presence of noise in the output and input variables. This approximation problem may be interpreted as an improper interpolation problem, for which it is required to correct optimally the positions of the original points in the data space so that they all lie on the same hyperplane. The minimax criterion is used to estimate the measure of correction of the initial data; therefore, the proposed approach can be called the total method of Chebyshev approximation (interpolation). It leads to a nonlinear mathematical programming problem, which is reduced to solving a finite number of linear programming problems. This number depends exponentially on the number of parameters, therefore, some methods are proposed to overcome this problem. The results obtained are illustrated with practical examples based on real data, namely, the birth rate in
the Federal Districts of the Russian Federation is analyzed depending on factors such as urban population, income and investment. Linear regression dependencies for two and three features are constructed. Based on the empirical fact of statistical stability (conservation of signs of the coefficients), the possibility of reducing the enumeration of linear programming problems is demonstrated.

In the paper for an arbitrary basic monoid 𝑀(P(𝑞)) of type 𝑞 the inverse problem is solved, that is, finding the asymptotics for the distribution function of the elements of the monoid 𝑀(P(𝑞)), based on the asymptotics of the distribution of pseudo-prime numbers P(𝑞) of type 𝑞.
To solve this problem, we consider two homomorphisms of the main monoid 𝑀(P(𝑞)) of type 𝑞 and the distribution problem reduces to the additive Ingham problem.
It is shown that the concept of power density does not work for this class of monoids. A new concept of 𝐶 logarithmic 𝜃-power density is introduced.
It is shown that any monoid 𝑀(P(𝑞)) for a sequence of pseudo-simple numbers P(𝑞) of type 𝑞 has upper and lower bounds for the element distribution function of the main monoid 𝑀(P(𝑞)) of type 𝑞.
It is shown that if 𝐶 is a logarithmic 𝜃-power density for the main monoid 𝑀(P(𝑞)) of the type 𝑞 exists, then 𝜃 = 1
2 and for the constant 𝐶 the inequalities are valid 𝜋√︁1/(3ln𝑞)<= 𝐶 <= 𝜋√︁2/(3ln𝑞).
The results obtained are similar to those previously obtained by the authors when solving the inverse problem for monoids generated by an arbitrary exponential sequence of primes of type 𝑞.
For basic monoids 𝑀(P(𝑞)) of the type 𝑞, the question remains open about the existence of a 𝐶 logarithmic 1 2 -power density and the value of the constant 𝐶.

The paper introduces a new concept — a system of joint Thue polynomials for a system of integer algebraic irrationalities. A parallel presentation of the elements of the theory of Thue polynomials for one algebraic irrationality and the foundations of the theory for a system of joint Thue polynomials for a system of integer algebraic irrationalities is carried out. A hypothesis is formulated about an analogue of the theorem of M. N. Dobrovolsky (Sr.) that for each order
of 𝑗 there are two main Thue polynomials of the 𝑗th order, through which all the others are expressed. For a system of two quadratic irrationalities, for example, √2 and √3, systems of joint basic polynomials of order no lower than 0, 1 and 2 are found. A theorem is proved on the general form of a pair of basic Thue polynomials of arbitrary order 𝑛 for quadratic irrationality √𝑐, where 𝑐 is a square-free natural number.

In spaces with weight |𝑥|−1𝑣𝑘(𝑥), where 𝑣𝑘(𝑥) is the Dunkl weight, there is the (𝑘, 1)- generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, in problems of quantum mechanics. Recently, for the (𝑘, 1)-generalized Fourier transform, the
Riesz potential was defined and the (𝐿𝑝,𝐿𝑞)-inequality with radial power weights was proved for it, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential and the Dunkl–Riesz potential. In the paper, this result is generalized to the case of radial piecewise power weights. Previously, a similar inequality was proved for the Dunkl–Riesz potential.

Two extremal problems of V.A. Yudin for polynomials in a more general setting are studied. In the first problem, among polynomials with nonnegative expansion coefficients in orthogonal polynomials on a segment [−1, 1], for which several successive moments and derivatives at the point −1 are equal to zero, a polynomial with a maximum non-negativity segment is searched.
The cases of the solving of the problem are described in terms of the Krein property. In the second problem, among polynomials with zero boundary conditions and zero first two moments on the segment [−1, 1], a polynomial with a minimum segment symmetric about zero on which it is nonnegative and nonpositive outside is searched. For the second problem, a complete solution was obtained.

The article is devoted to the Ramsey–Kass–Koopmans model. We consider an auxiliary system of differential equations, which is analogous to the system that arises in the case of constancy of the stationary rate of savings. We found that systems of this class are solved in quadrature. This allows us to find approximate solutions to the system describing the original model.

The domain of Siegel first type is not a bounded domain, but Carleman’s formulas for it play an important role in the further presentation. In this paper, the Carleman formula for the Siegel domain is found.

In the paper minimal fillings of finite metric spaces are investigated. This object appeared as a generalization of the concepts of a shortest tree and a minimal filling in the sense of Gromov. It is known that the weight of a minimal filling of a given type can be found by linear
programming and by so-called multitours technique. A relation between theses two approaches can be demonstrated using duality in linear programming, namely, rational points of the polytop
constructed by the dual problem correspond to multitours. The paper is devoted to investigation of such polytopes, It is shown that the vertices of the polytop are in one-to-one correspondence with irreducible multitours. A description of the polytop and an explicit formula for the weight of the minimal filling of the «snake–type» binary tree is obtained.

The six-minute walk test (SMWT) is one of the simplest and most widely available methods of assessing exercise tolerance. At the same time, the issues of presentation and interpretation of its results have not been fully studied yet.
It is known that in the case of unavailability of formal laws of the phenomenon, the method of dimensional analysis can be applied. This method, first proposed by Fourier and developed in the works of Rayleigh, Prandtl, Buckingham and others, is quite successfully used in physics, chemistry, engineering, economics and very rarely in biology and medicine.

The essence of the method is that the dependent variable is represented as a set of variables that are independent (or weakly dependent) on each other. This paper presents a general model of the dependence of the distance R traveled by a person on 8 variables: human parameters (mass, height, tolerance parameter), time and environmental characteristics (gravity potential, air density and viscosity, the coefficient of friction of the sole with the surface). Two
basic theorems of the dimensionality analysis method were applied to the general model: the dimensionality theorem of the quantity and Buckingham’s theorem (about finding the number of dimensionless complexes). Difficulties associated with the study of the general model do not allow us to form dependences on other quantities, which are much more difficult to vary in the experiment than others, for example, to control the density and viscosity of the medium, the gravitational potential of the Earth.
The possibility of reducing the number of model variables and forming a system of new, simpler models that allow explicit description of the motion process from all variables is shown.
A latent parameter is introduced into the model, i.e. a degree of tolerance of a person to motion which it is offered to characterize as an energy parameter. A method for its quantitative estimation and comparison of people by this parameter is proposed. In terms of assessing the dynamics of change in motion tolerance, it is necessary to further study the dependence of this parameter on time.

The article is devoted to the introduction and formation of the term and the action symbol "substitution". In mathematical research before Lagrange, it was never practiced to rearrange independent variables contained in a given function. For the first time this technique is found
in Lagrange’s work of 1771, devoted to the algebraic solution of equations.
Vandermond, who had published his work in the same year 1771, has expressed the idea of the need to introduce notations that simplify calculations and the perception of operations on root functions. However, the introduced designations were not easy to understand and became more complicated with increasing the degree of the equation.
Ruffini’s works, published from 1799 to 1813, aimed to prove the impossibility of solving the equation of the 5th degree and are, in fact, a study of the symmetric group represented by the values of the root function in the form of all possible permutations of these roots. During
these researches, he proves that the group 𝑆5 does not contain subgroups of the index 3, 4 or 8. However, just like Lagrange, Ruffini uses complex cumbersome expressions.
Cauchy, dealing with issues of combinatorial analysis, tried to generalize the result obtained by Ruffini to equations of arbitrary degree. Working on the determination of the limits that a
function of 𝑛 variables can take, Cauchy has invented a new research tool, which later became an independent theory. This was the the substitution group theory.

This paper comes to compare four different approximations of the solution to a layered linear elastic plate bending problem, obtained by the structural functions method. This method is in representation of a nonhomogeneous body displacement field as a weighted
sum of spatial derivatives of the so-called concomitant body displacements, the weighting coefficients are named structural functions of the nonhomogeneous body; the concomitant
body is a homogeneous one, subjected to the same loadings and boundary conditions, as the nonhomogeneous body; we come through the basic steps of structural functions method in this
paper. For the concomitant plate displacements, we consider two well-known approximations: the classical plate theory and the first-order shear deformation theory. We obtain the first- and the second-order structural functions of a layered plate. We derive direct formulae for the firstand second-order structural functions method approximations of the nonhomogeneous plate displacements, using both concomitant plate displacements approximations. For a set of sample plates, we compute the obtained structural functions method approximations, and compare the computation results with a known Pagano solution to the nonhomogeneous plate bending problem. The approximation, based on the first-order shear deformation theory approach to
the concomitant body displacements computation, gives an acceptable result in the considered cases.

The main stages of the conception of metallurgy and the formation of the metallurgical industry in the world are disclosed in the article. The locations of large metal-containing wealth at different times and their influence on the division of peoples into metal producers and their
consumers are shown. The close relationship between the development of metallurgy and the economy of individual countries as a whole is shown. The development of metallurgy depending
on human culture and life at a particular time and at a particular location is explained. The close relationship between the development of the metallurgical industry and military affairs in various countries is illustrated. The stage-by-stage development of metallurgy from private manufactories to large iron ore plants among the major countries producing metals and alloysis considered. The influence of the development and spread of metallurgical processes on the
modernization of the economic aspects of the industry is shown. The first developments in the field of alloying alloys, thermal and mechanical processing of finished metal products, as well as the processing of waste from metallurgical production are illustrated. The main factors for determining the optimal location for the construction of metallurgical production are given in the article.

In the article, for the case of large deformations, an exact analytical solution of the problem of the stress-strain state of a composite slab is presented, which is built by connecting two predeformed layers. Each layer is obtained by straightening a cylindrical panel, initially shaped like a sector of a hollow circular cylinder. The cylinders are made of incompressible non-linear-elastic materials — Treloar’s, or neo-Hookean materials. The axes of cylinders before deformation are
orthogonal. After connection, the plate is subjected to biaxial tension or compression in its plane. The problem is formulated on the basis of the theory of superimposed large strains.
An important role in solving the problem is played by the fact that the plate material is incompressible. When solving the problem, as well as when conducting numerical studies, nonlinear effects are investigated. The resulting solution can be used to verify software that is designed to numerically solve problems of the stress-strain state of structural elements made by connecting pre-deformed parts. For the obtained solution of the problem, numerical studies were carried out, the results of which - the dependence of the stress at the ends of the plates
on various deformation parameters - are presented in the work.

The work is devoted to one of the problems of the theory of superimposed large deformations.
An algorithm for the exact solution of the problem of forming an infinite circular compound cylinder from a certain finite number of hypoelastic layers is presented. The problem is formulated in a quasi-static statement. The hypoelasticity model corresponding to the material of the cylindrical layers is described by the equations of state with the participation of the corotational Dienes derivative. When attached, each successive layer undergoes two phases of deformation over some time intervals. The first phase of deformation is the radial expansion or contraction of the cylindrical layer. The second phase of deformation is torsion. Each successive layer is attached to the composite hypoelastic cylindrical body after the deformation of the
previous layer is completed. At the same time, the deformation of each hypoelastic layer affects the general state of the composite cylinder, that is, all internal layers. It is required to determine the stress field in a composite nonlinearly elastic cylinder. The paper describes the notation and coordinate systems used in solving the problem. All the main steps for solving the problem are described, including the calculation of the stress tensor components. The formulas for the axial
force and torque of a compound cylinder are also given. Numerical studies have been carried out. The results of numerical studies - graphs of the dependence of the axial force and torque on the deformation parameters - are presented at the end of the work.

This paper considers convergence estimation of the spectral element method implemented in CAE Fidesys. It was based on exact analytical solutions of the Lame problems in small deformations in the elastic and elastic-perfectly plastic obeying Huber-von Mises yield criterion
formulations. Due to the symmetry, we consider quarters of the models. Numerical results were obtained in the CAE Fidesys strength analysis system using the finite element method for the first and second orders and the spectral element method for the third to ninth orders. Based on the results obtained, an analysis was carried out to determine the nature of the decrease in the errors of the CAE Fidesys spectral element method with an increase in the order of the
elements. The study was conducted using a specialized automated testing system. The results of the work can be useful in making a decision on the use of the spectral element method in industrial calculations.

The article is devoted to the limit plastic state and localization of plastic deformations along shear bands in dilating media with a non-associative flow rule. The equations of the characteristics of systems of equations for stresses and velocities in a plane strain state for an
arbitrary function of the yield surface dependenе on the first two invariants in the rigid-plastic framework are obtained. Equations for stresses along the characteristics for the limit state and the condition of their hyperbolicity are obtained. A numerical model of the solution of the elastic-plastic problem by Galerkin equations on high-order spectral elements is presented.
Numerical experiments have been carried out for the linear function of the yield surface in order to establish the boundaries of the range of possible slopes of the shear bands and to test the theoretical results.

This paper presents and implements an algorithm that performs topological optimization of the mass distribution of a two-dimensional body under load. The ultimate goal of the algorithm is to minimize body weight under stress constraints at the points of the body. The approach is based on the idea of variable density and the BESO algorithm that adds and deletes elements depending on stresses.
The algorithm uses the finite element method and is an iterative process. At each iteration the stresses in the body are calculated using CAE Fidesys, and then the calculation results are analyzed. According to the analysis, Young’s moduli at the nodes of the finite element mesh are changed to reflect new mass distribution adjusted for better compliance with loads.
The specific feature of the used approach is utilization of objective function with the special term. This term is the sum of the squares of the differential derivatives of density in four directions. This feature permits one to avoid sharp changes in density and the appearance
of lattice structures in the early iterations. The Adam gradient method is used to determine densities at each iteration.
The implemented algorithm is verified on a number of test cases for plane static problems of the theory of elasticity. The results of computations are presented. A comparison is made with the results obtained by other authors. For one of the problems, the results of calculations on different grids are given. These results allows one to conclude about the grid convergence of the algorithm.

The aim of the work is to study the evolution of the concept of stability, which is a structureforming concept in all areas of science and technology, and even beyond them. The stages of this long evolution corresponded to the dominant trends in the mathematics of their time.
By the end of the XIX century. the complexity of the concept of stability was realized, the question arose of a mathematically rigorous approach to the problem. A general theory of motion stability was built on a solid mathematical foundation. This became a milestone not
only in the development of the subject itself, but was one of the foundations for constructing a qualitative theory. Subsequently, the theory of stability was divided into two branches: one - the expansion of the theory in breadth based on old ideas, strengthening the links with applications; the other is stability in the context of the theory of dynamical systems. In the latter case, stable movements are considered in the series of all movements; in the stability-instability dichotomy both poles are equal and meaningful. Instability also turns out to be a complex concept that has a variety of forms. Instability has acquired a constructive meaning; it ensures innovation and development. Typical is the coexistence of stability and instability with a complex topology of such a structure. Diverse types of instability demonstrate the phenomenon of turbulence. The study of this phenomenon at the modern level requires the use of mathematics according to the canons of rigor adopted in mathematics itself. One can raise the question of the limits of applicability of the possibilities of the most qualitative description and the concept of stability.
In this regard, there are first results, new ideas are required.

The problem of determining the type of inhomogeneity of the external anisotropic layer of an elastic ball from the scattered field of a plane sound wave is considered. It is assumed that the density and elastic moduli of the outer layer material are linear functions of the distance
from the center of the ball. It is believed that the laws of dependency of all moduli of elasticity are identical. According to the acoustic pressure in the vicinity of the ball, it is required to determine the coefficients in the dependences for the density and elastic moduli. The problem of sound diffraction by a ball is solved by a numerical-analytical method. The scattered acoustic field and the field of elastic oscillations in the homogeneous part of the ball is represented by an
expansion in terms of spherical harmonics. For the displacement and stress vector components in an inhomogeneous layer, a boundary value problem is numerically solved based on the equations
of motion and boundary conditions on the layer surfaces. To determine the desired coefficients in the dependences of the density and elastic moduli of the outer layer, the observed pressure
values are compared at a certain set of points on a spherical surface centered at the center of the ball and the calculated pressure values at these points. A variant of forming an indicator of the proximity of observed and calculated pressure values based on the division of observation points into groups is proposed. It is proposed to use the proximity indicator to identify the coefficients in the laws of density inhomogeneity and elastic moduli in the layer.

In paper the problem of the diffraction of a spherical monochromatic sound wave on a homogeneous isotropic elastic cylinder with a radially non-uniform anisotropic elastic coating.
It is assumed that the body is located in a free space filled with an ideal liquid. The analytical solution of the problem is obtained.
In the case of steady-state oscillations, the propagation of small perturbations in an ideal fluid is described by the Helmholtz scalar equation. The radiation field of a spherical source is written as a series expansion in cylindrical wave functions. The propagation of elastic waves in an isotropic elastic cylinder is described by the scalar and vector Helmholtz equations. The oscillations of an non-uniform anisotropic elastic cylindrical layer are described by the general
equations of motion of a continuous medium.
The asymptotic formula for the far field zone is obtained by the steepest descent method.
Numerical calculations of the frequency characteristics of the scattered field are carried out for elastic cylinders with homogeneous and inhomogeneous transversally isotropic coatings, as well as for the case of a homogeneous isotropic coating. A significant influence and mutual influence of the inhomogeneity and anisotropy of the coating material on the acoustic properties of the scattering cylindrical body is revealed.

In this article, a method and some results of the numerical estimation of effective Biot’s coefficient of a porous material are described. The estimation is made by carrying out the calculation on a representative volume element (RVE) of the material. The outer boundary of the RVE is rigidly fixed, and pressure is applied to the surface of the pores. The elasticity boundary value problem with given boundary conditions is solved numerically, using the Russian CAE-system "Fidesys". The stress field obtained as a result of the solution is averaged over the
volume. Effective Biot’s coefficient is calculated from the averaged stress tensor.
Comparison of the numerical results for the effective Biot coefficient with the analytical solution is carried out on the simplest model - a cube with a spherical pore. For different porosities and for different Poisson’s ratios of the material, it is shown that the error is no
more than 5%. Effective Biot’s coefficient is also estimated for models with a more complex pore shape. Numerical results show the presence of significant anisotropy in obtained Biot’s coefficients.

We prove the weighted Carleman inequality for the fractional gradient
‖𝑒−𝑡⟨𝑎, · ⟩| · |−𝛾𝑓‖𝑞 <= 𝐶‖𝑒−𝑡⟨𝑎, · ⟩| · |¯𝛾−¯𝛿∇𝛼𝑓‖𝑝, 𝑓 ∈ 𝐶∞
0 (R𝑑), 𝑡 > 0.
For 𝛼 = 1, it was proved by L. De Carli, D. Gorbachev, and S. Tikhonov (2020). An application of the Carleman inequality is given to prove the weak unique continuation property of a solution of the differential inequality with the potential |∇𝛼𝑓| <= 𝑉 |𝑓| in a weighted Sobolev space.

The extremal Delsarte problem 𝐴(𝑑, 𝑠) for spherical 𝑠-designs allows us to estimate from below the minimum number of nodes 𝑁(𝑑, 𝑠) of a weighted quadrature formula on the sphere S𝑑.
We prove that 𝐴(3, 4) = 14.560317967882 . . . .
Hence 𝑁(3, 4) > 15. Our open conjecture is that 𝑁(3, 4) = 16.

The new error estimation of the error of the approximate solution of the Fredholm integral equation of the second kind by iteration using algebraic grids are obtained.

This paper is devoted to obtaining estimates of the deviation of a parallelepipedal grid, which is a rational grid approximating the algebraic grid of a quadratic field.
New tasks have been set for further research.

This work is devoted to obtaining estimates of the type of Bykovsky estimates for a measure of the quality of optimal coefficients.
The ways to obtain analogs of the Bykovsky estimate for the finite deviation of the parallelepipedal grid are outlined.

This work is dedicated to the 85th anniversary of Professor Ashot Enofovich Ustyan.
The paper presents facts from the biography of the hero of the day.
Information about the scientific work of A. E. Ustyan is given.
Sufficient space is given to the organizational and pedagogical activities of Professor A. E. Ustyan.
The information about awards and incentives of A. E. Ustyan for his fruitful scientific and pedagogical activity is given.

Alexander Evgenievich Gvozdev studied a fundamental scientific direction of applied importance related to the development of the theory of superplasticity of metal alloys during phase transformations and its application for the production of cutting tool blanks from
powder high-speed steel. Based on the proposed integrated approach, he revealed the nature of extreme effects and causes of changes in strength and plasticity in heterophase metal systems of traditional metallurgical processing and powder production method with different dispersion of carbide phases, forging alloys based on aluminum, copper-zinc metal alloy, etc. under thermomechanical influences and in pre-transition states before phase transformations of the I and II kind.