In the paper, the method of exponential sums is applied to the solution of partial differential equation. At the initial step, the authors decrease the dimension of the problem by separation of
variables. Thus the initial problem reduses to the system of the ordinary differential equations.
This allows one to use Fourier analysis.

The pseudorandom search method considered in this paper is quite universal and allows solving complex econometric problems using the discrete least squares method. The paper
considers the problem of finding the parameters of a linear combination of the Cobb-Douglas-Tinbergen function and the third production function, which is its generalization. If the choice
of parameters of the Cobb-Douglas-Tinbergen function, or the third production function after logarithmization and the application of the least squares method is reduced to a linear problem that is solved in the final form, then a linear combination of these two models requires solving an optimization problem of 10 or 11 variables with a transcendental function, which makes the problem difficult to solve.
At least 10 different types of classical number-theoretic grids are well known in the literature.
From the point of view of the organization of the pseudorandom belt, the grids and LP sequences proposed by I. M. Sobol are the most well studied. Previously, Korobov parallelepipedal grids
were used in solving problems of textural analysis in geophysics. 6-dimensional grids were used in these works.
In our work, we have to work with 10-dimensional and 11-dimensional grids with a much larger number of points in order to overcome the well-known "curse of dimensionality". It is partially possible to reduce the dimension to the 9th by using the properties of the models under consideration, which are studied in detail in this paper. As a result of the study, it was found
that three parameters cannot be determined unambiguously from the original mathematical model. An additional optimization problem arises for the least squares method if we postulate the proximity of technological coefficients.
The latter assumption requires additional economic interpretation and will be the subject of further economic research.
It would be interesting to compare the results of calculations for different regions of the country and for the country as a whole. The problem is related to the availability of data, but we expect to consider this formulation of the problem in subsequent works.

Systems of particle motion in the Hooke central potential field on a billiard book glued from flat circular billiard domains are considered. An important class of nondegenerate focal
singularities of the rank 0 of integrable systems with 2 degrees of freedom is completely realized by this class of billiards. Namely, for each semi-local focal singularity the constructed billiard system has a singularity fiberwise homeomorphic to the given one.

The classical Bernstein–Nikolskii inequalities of the form ‖𝐷𝑓‖𝑞 6 𝒞𝑝𝑞‖𝑓‖𝑝 for 𝑓 ∈ 𝑌 , give estimates for the 𝑝𝑞-norms of the differential operators 𝐷 on classes 𝑌 of polynomials and
entire functions of exponential type. These inequalities play an important role in harmonic analysis, approximation theory and find applications in number theory and metric geometry.
Both order inequalities and inequalities with sharp constants are studied. The last case is especially interesting because the extremal functions depend on the geometry of the manifold
and this fact helps in solving geometric problems.

Historically, Bernstein’s inequalities are referred to the case 𝑝 = 𝑞, and Nikolskii’s inequalities to the estimate of the identity operator for 𝑝 < 𝑞. For the first time, an estimate for the derivative of a trigonometric polynomial for 𝑝 = ∞ was given by S.N. Bernstein (1912), although earlier A.A. Markov (1889) gave its algebraic version. Bernstein’s inequality was refined by E. Landau,
M. Riess, and A. Sigmund (1933) proved it for all 𝑝 > 1. For 𝑝 < 1, the Bernstein order inequality was found by V.I. Ivanov (1975), E.A. Storozhenko, V.G. Krotov and P. Oswald (1975), and the sharp inequality by V.V. Arestov (1981). For entire functions of exponential type, the sharp Bernstein inequality was proved by N.I. Akhiezer, B.Ya. Levin (𝑝 > 1, 1957), Q.I. Rahman and G. Schmeisser (𝑝 < 1, 1990).
The first one-dimensional Nikolskii inequalities for 𝑞 = ∞ were established by D. Jackson (1933) for trigonometric polynomials and J. Korevaar (1949) for entire functions of exponential
type. In all generality for 𝑞 6 ∞ and 𝑑-dimensional space, this was done by S.M. Nikolskii (1951). The estimates of Nikolskii constants were refined by I.I. Ibragimov (1959), D. Amir and
Z. Ziegler (1976), R.J. Nessel and G. Wilmes (1978), and many others. Bernstein–Nikolskii order inequalities for different intervals were studied by N.K. Bari (1954). Variants of inequalities for general multiplier differential operators and weighted manifolds can be found in the works of P.I. Lizorkin (1965), A.I. Kamzolov (1984), A.G. Babenko (1992), A.I. Kozko (1998),
K.V. Runovsky and H.-J. Schmeisser (2001), F. Dai and Y. Xu (2013), V.V. Arestov and P.Yu. Glazyrina (2014) and other authors.
For a long time, the theory of Bernstein–Nikolskii inequalities for polynomials and entire functions of exponential type developed in parallel until E. Levin and D. Lubinsky (2015) established that for all 𝑝 > 0 the Nikolskii constant for functions is the limit of trigonometric constants. For the Bernstein–Nikolskii constants, this fact was proved by M.I. Ganzburg and S.Yu. Tikhonov (2017) and refined by the author together with I.A. Martyanov (2018, 2019).
Multidimensional results of the Levin–Lyubinsky type were proved by the author together with F. Dai and S.Yu. Tikhonov (the sphere, 2020), M.I. Ganzburg (the torus, 2019 and the
cube, 2021).
Until now, the sharp Nikolskii constants are known only for (𝑝, 𝑞) = (2,∞). The case of the Nikolskii constant for 𝑝 = 1 is intriguing. Advancement in this area was obtained by Ya.L. Geronimus (1938), S.B. Stechkin (1961), L.V. Taikov (1965), L. H¨ormander and
B. Bernhardsson (1993), N.N. Andreev, S.V. Konyagin and A.Yu. Popov (1996), author (2005), author and I.A. Martyanov (2018), I.E. Simonov and P.Yu. Glazyrina (2015). E. Carneiro,
M.B. Milinovich and K. Soundararajan (2019) pointed out applications in the theory of the Riemann zeta function. V.V. Arestov, M.V. Deikalova et al (2016, 2018) characterized extremal
polynomials for general weighted Nikolskii constants using duality. Here, S.N. Bernshtein, L.V. Taikov (1965, 1993) and others stood at the origins.
A new direction is the proof of Nikolskii’s sharp inequalities on classes of functions with constraints. It reveals a connection with the extremal problems of harmonic analysis of Turan, Delsarte, the uncertainty principle by J. Bourgain, L. Clozel and J.-P. Kahane (2010) and others. For example, the author and coauthors (2020) showed that the sharp Nikolskii constant for nonnegative spherical polynomials gives an estimate for spherical designs by P. Delsarte, J.M. Goethals and J.J. Seidel (1977). Variants of problems for functions lead to famous estimates for the density of spherical packing, and order results are closely related to Fourier inequalities.
These results are presented in the framework of the general theory of Bernstein–Nikolskii inequalities, applications in approximation theory, number theory, metric geometry are presented, open problems are proposed.

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 a simple graph model, so-called  caterpillar,  and its oriented analogues. The other simple graph models, including oriented and non-oriented simple cycles  and simple paths, where considered before.

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.
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-oriented graphs) and a spanning converging forest (for oriented graphs) are given. Some examples are represented (simple path, simple cycle, caterpillar and their oriented analogues).
In the same section a number of properties of Fibonacci numbers necessary to obtain the main results of the article for the undirected case is formulated.

In the third 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 caterpillar  and the number of spanning rooted trees for the non-oriented caterpillar 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  $2^k$, $k\geq 0$; results for the non-oriented case are formulated in terms of Fibonacci numbers $u_k$, $k\geq 1$. The proofs are based on elementary methods of enumerating Combinatorics.

The fourth 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 caterpillar and its oriented analogue. Total number of spanning converging forests (for oriented caterpillar) and total number of spanning rooted forests (for non-oriented caterpillar) 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 caterpillar), and enumeration of the spanning rooted forests, in which a vertex $i$ belongs to a tree with a root $j$ (for the non-oriented caterpillar) are represented. As before, results for the oriented case are formulated in terms of values  $2^k$, $ k\geq 0$; results for the non-oriented case are formulated in terms of Fibonacci numbers $u_k$, $ k\geq 1$.

The fifth section contains the examples of the the matrix of relative connectivity via  forests.

The sixth section  (conclusion) presents the main conclusions of the article, outlines the ideas of further studies.

When studying questions of the asymptotic distribution of integer points over domains on hyperboloids, as well as integer matrices of the second and third orders, it becomes necessary
to use primitive unassociated matrices of the second and third orders of a given determinant.
Counting the number of integer matrices of the same order and a given determinant requires the selection of pairwise unassociated matrices among them. Non-associated second-order matrices appear when considering preliminary ergodic theorems for flows of integer points on hyperboloids when applying the discrete ergodic method to the problem of representing integers by ternary quadratic forms. The number of unassociated second-order matrices is also used to express the number of binary quadratic forms, the arithmetic minimum of which is divisible. In addition, formulas for the number of primitive unassociated matrices of the second and third orders make it possible to determine the orders of the principal terms in asymptotic formulas for the number of integer matrices of large norm(determinant). In this paper, based on the canonical triangular form of the third-order integer matrices, a formula is obtained for the number of primitive unassociated third-order matrices represented by the canonical decomposition. A formula is also obtained for the number of primitive unassociated matrices of the third order of a given determinant, divisible by a given matrix. The main results related to the question of the number of non-associated integer matrices of a given determinant belong to Yu. V. Linnik, B. F. Skubenko, A.V. Malyshev and the authors of this work, the results of which can be further transferred to integer matrices of any order.

Given a class $F$ of metric spaces and a family of transformations $T$ of a metric, one has to describe a family of transformations $ T'\subset T$ that transfer $F$ into itself and preserve some types of minimal fillings. % We denote  $\rho_{ij}=\rho(p_i,p_j)$ and\\ $\r=(\rho_{12}, \rho_{13}, \ldots, \rho_{n-1,n})$ for any metric space $(M,\r)$ where $M=\{p_1,\ldots,p_n\}$.
%We considered the case
The article considers two cases.
%\iffalse
%\begin{itemize}
    %\item
    First, when $F$ is the class of all finite pseudometric spaces, the class $T$ consists of the maps $M\mapsto AM+\tau$, where the matrices $A$ and $\tau$ define the mapping of pseudometric matrix $M$, and the elements of $T'$ preserve any type $G$ of minimal fillings of pseudometric spaces whose points correspond to vertices of degree 1 of the graph G, and we prove that $A=\lambda E$ for some $\lambda\ge 0$, and $\tau$ is a pseudometric matrix, one of the minimal fillings of which is a star.
    Second when $F$ is the class of all finite pseudometric spaces, the class $T$ consists of the maps $\r\to A\r$, where $A$ is a diagonalizable matrix with two eigenvalues $\lambda_{max}> \lambda_{min} \ge 0$, the largest eigenvalue $\lambda_{max}$ of which has multiplicity 1, the eigenspace corresponding to the value $ \lambda_{min} $, does not contain nonzero pseudometrics, and the elements of $T'$ preserve the types $G$ of minimal fillings of the pseudometric space, whose points correspond to vertices of degree $1$ of graphs $G$. And we prove that for any mapping matrix from $T'$ there is a pseudometrics that is an eigenvector with the eigenvalue $\lambda_{max}$, among the minimum fillings of which there is a filling of the star type.
    Second, when $F$ is the class of all finite metric spaces, the class $T$ consists of the maps $\r\to N\r$, where the matrix $N$ is the sum of a positive diagonal matrix $A$ and a matrix with the same rows of non-negative elements. The elements of $T'$ preserve all minimal fillings of the type of non-degenerate stars. %we proved
    It has been proven that $T'$ consists of maps $\r\to N\r$, where $A$ is scalar. %;
    Third, when $F$ is the class of all finite additive metric spaces, $T$ is the class of all linear mappings given by matrices, and the elements of $T'$ preserve all non-degenerate types of minimal fillings, and we proved that for metric spaces consisting of at least four points $T'$  is the set of transformations given by scalar matrices. %;
    Fourth, when $F$ is the class of all finite ultrametric spaces, $T$ is the class of all linear mappings given by matrices, and we proved that for three-point spaces the matrices have the form $A=R(B+\lambda E)$, where $B$ is a matrix of identical rows of positive elements, and $R$ is a permutation of the points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$.

In this paper we consider an extremal problem related to a set of continuous positive definite functions on R whose support is contained in the closed interval [−𝜎, 𝜎], 𝜎 > 0 and the value
at the origin is fixed (the class F𝜎).
We consider the following problem. Let 𝜇 be a linear locally bounded functional on the set of continuous functions which have compact support, i.e. 𝐶𝑐(R) and suppose that 𝜇 is real-valued functional on the sets F𝜎, 𝜎 > 0. For a fixed 𝜎 > 0, it is required to find the following constants:
𝑀(𝜇, 𝜎) := sup {𝜇(𝜙) : 𝜙 ∈ F𝜎} , 𝑚(𝜇, 𝜎) := inf {𝜇(𝜙) : 𝜙 ∈ F𝜎} .
We have obtained a general solution to this problem for functionals of the following form 𝜇(𝜙) = ∫︀ R 𝜙(𝑥)𝜌(𝑥)𝑑𝑥, 𝜙 ∈ 𝐶𝑐(R), where 𝜌 ∈ 𝐿𝑙𝑜𝑐(R) and 𝜌(𝑥) = 𝜌(−𝑥) a.e. on 𝑥 ∈ R.
For 𝜌(𝑥) ≡ 1, the value of 𝑀(𝜇, 𝜎) was obtained by Siegel in 1935 and, independently, by Boas and Kac in 1945. In this article, we have obtained explicit solution to the problem under consideration in cases of 𝜌(𝑥) = 𝑖𝑥, 𝜌(𝑥) = 𝑥2 and 𝜌(𝑥) = 𝑖 sign 𝑥, 𝑥 ∈ R.
In addition, in this paper we study the connection between the problem under consideration and pointwise inequalities for entire functions of exponential type 6 𝜎 whose restrictions on R
are in 𝐿1(R). In particular, sharp inequalities are obtained for the first and second derivatives of such functions.

The theory of operational calculus is developed on the basis of a differential operator with piecewise constant coefficients. A formula for the generalized Laplace transform is proposed.
An inversion formula of Mellin-Laplace type is proved. The concept of a generalized originalfunction and a generalized image is proposed. A theorem on the isomorphism of the spaces of originals and generalized originals is proved. Using transmutation operators, it is established that the generalized Laplace transform of the generalized original coincides with the Laplace transform of the corresponding original-function. Theorems on differentiation and integration of the generalized original, theorems on homogeneity, similarity, exponential scaling, first and second shifting theorems, and others are proved. In terms of the transmutation operator, a
connection between the convolution of generalized original-functions and the corresponding convolution of original-functions is established. An algorithm for solving linear differential equations with piecewise constant coefficients is presented. A solution to the heat equation with a piecewise constant coefficient at the time derivative on the real axis is found. A mixed boundary value problem for the heat equation with a piecewise constant coefficient at the time derivative on the real semiaxis is solved.

For integrable Hamiltonian systems with two degrees of freedom we investigate the topology of the Liouville foliation in a 3-dimensional non-compact invariant neighborhood of a singular
leaf. All the singularities of the system are supposed to be non-degenerate. In the case when all the leaves of the Liouville foliation are compact, this problem is already solved: the well-known A. T. Fomenko theorem states that any non-degenerate 3-dimensional singularity (3-atom) is an 𝑆1-fibration of the special type (Seifert fibration) over a 2-dimensional singularity (2-atom).
Thus, the problem of the topological classification of 3-atoms is reduced to the significantly more simple classification problem for 2-atoms (i. e. singularities of foliations determined by Morse
functions on 2-surfaces). The latter problem is well-studied in the framework of the Fomenko classification theory for integrable systems.
In the non-compact case, the set of all 3-atoms becomes much richer. That is why we consider only 3-atoms satisfying the following conditions: completeness of the Hamiltonian
flows generated by the first integrals of the system, finiteness of the number of orbits of the Hamiltonian R2-action on the singular leaf, and existence among these orbits of a noncontractible one. Under these restrictions, we proof that the 3-atom admits a Hamiltonian locally free 𝑆1-action preserving the leaves of the Liouville foliation. As a corollary, we obtain the
analogue of the Fomenko theorem and thus reduce the classification problem for non-compact 3-atoms satisfying the above conditions to the similar classification problem for non-compact 2-atoms that we solved earlier.

Assuming the validity of the extended Riemann hypothesis for the average values of Chebyshev functions over all characters modulo 𝑞, the following estimate holds

$$𝑡(𝑥; 𝑞) =Σ︁𝜒mod𝑞max𝑦≤𝑥|𝜓(𝑦, 𝜒)| ≪ 𝑥 + 𝑥1/2𝑞L2, L = ln 𝑥𝑞$$.
When solving a number of problems in prime number theory, it is sufficient that 𝑡(𝑥; 𝑞) admits an estimate close to this one. The best known estimates for 𝑡(𝑥; 𝑞) previously belonged to
G. Montgomery, R. Vaughn, and Z. Kh. Rakhmonov. In this paper we obtain a new estimate of the form
$$𝑡(𝑥; 𝑞) = Σ︁ 𝜒mod𝑞 max 𝑦≤𝑥 |𝜓(𝑦, 𝜒)| ≪ 𝑥L^28 + 𝑥^(4/5) 𝑞^(1/2)L^31 + 𝑥^(1/2)𝑞L^32$$,
using which for a linear exponential sum with primes we prove a stronger estimate $$𝑆(𝛼, 𝑥) ≪ 𝑥𝑞^(−1/2)L^33 + 𝑥^(4/5)L^32 + 𝑥^(1/2)𝑞^(1/2)L^33$$, when $$|𝑎-a/q|<1/q^2,   (a,q)=1$$.
We also study the distribution of Hardy-Littlewood numbers of
the form 𝑝+𝑛2 in short arithmetic progressions in the case when the difference of the progression is a power of the prime number.

In paper the problem of scattering of a plane sound wave by an absolutely solid sphere with a continuously inhomogeneous anisotropic elastic coating in the presence of a flat surface is
considered. It is believed that the body is placed in an ideal fluid, the spreading flat surface is absolutely rigid and absolutely soft, the laws of heterogeneity of the coating material are described by differentiable radial coordinate functions.
The approximate analytical solution to the problem is obtained for the case when the material of the sphere coating is radially inhomogeneous and transversally isotropic. In this case the reflection from the plane of the waves scattered by the body is not taken into account, but scattering by the sphere of the wave arising from the reflection of the incident wave from the plane is taken into account.
By virtue of the linear formulation of the problem, the velocity potential of the total acoustic field is represented as the sum of the potentials of the incident plane wave; wave arising from
the reflection of the incident plane waves from the plane; wave arising from the scattering of an incident plane wave by sphere; wave arising from scattering by a sphere reflected from plane of
the wave.
Wave fields in a containing medium are described by expansions in spherical wave functions.
A boundary value problem is constructed for a system of ordinary differential equations of the second order for finding displacement fields in an inhomogeneous anisotropic coating of sphere.

The existence of the zeros of the Riemann zeta-function in the short segments of the critical line
(or the real zeros of Hardy's function $Z(t)$, that is the same) is one of the topical problems in the theory of the Riemann zeta-function.
The study of the zeros of Hardy function's derivatives $Z^{(j)}(t)$ is the generalization of such problem.
Let $T>0$. Let us define the quantity $H_j(T)$, the distance from $T$ to the nearest real zero not less than $T$ of the $j$-th derivative of the Hardy function. In the paper, an upper bound for $H_j(T)$ is proved.

We study here polyadic Liouville numbers, which are involved in a series of recent papers. 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.
Let 𝑘 ≥ 2 be a positive integer. We denote for a positive integer 𝑚

$$Φ(𝑘,𝑚) = 𝑘^𝑘^(...)^𝑘$$

Let

$$𝑛𝑚 = Φ(𝑘,𝑚)$$

and let

$$𝛼 =∞Σ︁𝑚=0(𝑛𝑚)!.$$

Theorem 1. For any positive integer 𝑘 ≥ 2 and any prime number 𝑝 the series 𝛼 converges to a transcendental element of the ring Z𝑝. In other words, the polyadic number 𝛼 is globally transcendental.

The paper presents the results of optimizing the fusion of an electroerosive cobalt-chromium charge, which was carried out by setting up a complete factor experiment and the method
of steep ascent of Box and Wilson. The optimal parameters of the process of obtaining a cobalt-chromium alloy by spark plasma sintering of particles of the KHMS alloy in hardness
have been determined. The parameters of operation of the spark plasma fusion plant were selected as factors: temperature, pressure and holding time. The optimal parameters of the
installation were determined for the electroerosive material KHMS, previously obtained in butyl alcohol. According to the conducted series of experiments, the limiting values of the optimization parameter (hardness) for the process of fusing an electroerosive cobalt-chromium charge were determined, which amounted to: 223.8 HB at a pressure of 30 MPa, a temperature of 560 ∘ C and a holding time of 3 minutes.

The note is devoted to the arrival in June 1921 of a large group of Moscow mathematicians, mainly students of N. N. Luzin, to Petrograd to participate in memorial meetings on the occasion
of the 100th anniversary of the birth of P. L. Chebyshev. This historic visit became the first mathematical conference in Soviet history, as well as the first step of Muscovites in overcoming
the conflict situation that arose in the 1880s in the relationship between mathematicians of the two capitals.

In the fall of 1938, the Tula State Pedagogical Institute (later renamed the L. N. Tolstoy Tula State Pedagogical University) was organized. The staff of the institute consisted of invited
teachers from other cities. This circumstance determined the entire educational process, since the disciplines were read in compressed blocks and in a short time. One of the three faculties
was physics and mathematics with the only department of mathematics and physics. In the fall of 1939, the faculty managed to invite young scientists-mathematicians, specialists in the
field of mathematical analysis and differential equations. They were the spouses P. V. Soloviev and V. M. Gushchina. Both of them - natives of the Tula region, received their education
and defended their PhD theses in Moscow. P. V. Soloviev had good scientific results and as a scientist could have created a school of mathematical analysis at the faculty, but in 1941
the war broke out, and he volunteered for the front, where he died in 1943. In 1950, two mathematical departments were separated from the Department of Mathematics and Physics,
one of which was the Department of Mathematical Analysis. The first head of the Department of Mathematical Analysis was Professor S. P. Sluginov, who was in Tula on visits from Moscow,
and the staff of the department consisted of 8 people. In June 1951, the department was headed by a well-known scientist in the field of the theory of functions, Doctor of Physical
and Mathematical Sciences, Professor V. I. Levin. The formation of postgraduate studies at the department and faculty is associated with his name. The first graduate of the department’s
postgraduate studies, who successfully defended her thesis for the degree of candidate of physical and mathematical sciences, was S. N. Levina, a student of Professor V. I. Levin. The students
of Professor V. I. Levin were many graduates of TSPU named after L. N. Tolstoy, who later worked at the department. In the 1960s, the scientific and pedagogical component of the Department of Mathematical Analysis was strengthened: associate professors V. I. Antropova (1964), V. I. Rybakov (1969) and A. S. Simonov (1971). Over the years, among the teachers of the department there were first-class specialists who left a noticeable mark in mathematics, but the department achieved the greatest success in the 1970s-80s. This was due to the provision of highly qualified personnel. The article traces the development of mathematical analysis at the faculty through the history of the department, as well as through the activities of its heads and teachers.

This paper presents the results of a study of the decay of the Fe3C phase during thermal cycling near the temperature of the phase transition of the second kind, which is 210 C (Curie point of cementite). As a result of the conducted studies, it was found that structurally free tertiary cementite, present in low-carbon steels, is primarily susceptible to decay. In preeutectoid and eutectoid steels, cementite as an independent phase may be present in the composition of granular perlite. It was revealed that in the structure of steel U10 with the initial state of granular perlite, the maximum volume fraction (about 0,8).

The authors of the article set themselves the following tasks: to briefly describe the twenty-year history of the Department of Mathematics and Informatics of the Tula branch of the Financial University and to give a brief analysis of the scientific activities at the department that influenced the development of mathematics, computer science, mathematical and information education in the Tula region. Teachers of the department conducted and
continue to conduct research in the field of economic and mathematical methods and models, methods of mathematics, the use of information technologies in mathematics and teaching
mathematics.
Research work is carried out by all teachers of chair within the limits of state budgetary subjects and realised in participation in scientific, scientifically-methodical and scientificallypractical
conferences of different levels from regional to the international. Results are reflected in set of the publications most appreciable of which are monographies and articles in the largest magazines. The article presents the results obtained by the teachers of the department in different periods of its 20-year history, including the results of research in recent years.

The article considers the influence of the tempering temperature on the structure and mechanical properties of low-carbon reinforcing steels of grades St3 and St5. It is established
that steel of the St3 brand in bars 14 mm is effectively processed by the VTMO method, being strengthened to the level of the 3rd class. The mechanical properties recorded directly
after the VTMO are steadily preserved after the release by electric heating to tempera-tures of 350 . . . 370∘C. It was found that after tempering at a temperature of 500 . . . 550∘C, the hardening
in St5 steel remains at the level of the 6th strength class only if the heating is carried out according to the speed mode (28∘ C/sec). It is shown that the substructure created during
VTMO is de-stroyed during accelerated tempering to a lesser (to a lesser) extent than after furnace tempering. All this determines a higher level of steel hardening as a result of VTMO
and high-speed tempering.

The article considers the question of the behavior of the values of 𝐶𝑛 with increasing 𝑛, where 𝐶𝑛 is the constant of the best joint diophantine approximations. Shows the differences in this question for 𝑙2 and 𝑚𝑎𝑥 norms are shown.

The article discusses the issue of improving one algorithm for finding the best joint Diophantine approximations and estimating its complexity.

In this paper the version of the Stirling formula is found. It is the useful and siutable for applications. A deduction of this formula is based on two the Euler’s statements: the expansion of the Gamma-function into the infinite product and the Euler–MacLauren summation formula of values of the smooth function over integers.

We continue the study of the sharp Bernstein–Nikolskii constants for spherical polynomials in the space 𝐿𝑝(S𝑑) with the Dunkl weight. We consider the model case of the octahedral
reflection group Z𝑑+1 2 and weight Π︀𝑑+1 𝑗=1 |𝑥𝑗 |2𝜅𝑗 when the explicit form of the Dunkl intertwining operator is known. We show that for min 𝜅 = 0 the multidimensional problem is reduced to the one-dimensional problem for the Gegenbauer weight, otherwise not.

The hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations is studied. A functional equation is found for the hyperbolic zeta function of a two-dimensional
lattice of Dirichlet approximations in the case of rational 𝛽, which sets an analytical continuation on the entire complex plane, except for the point 𝛼 = 1, in which the pole is of the first order.
The found functional equation allows us to raise the question of continuity for the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations in the case of rational 𝛽.

Generalized non-uniform Korobov grids are considered in the paper.
Three new constructions are considered: the product of non-uniform grids by mutually simple modules; modified non-uniform grids; the product of an uneven grid and a parallelepipedal grid by a mutually simple module.
A paradoxical result is established about the value of the mathematical expectation of the error of approximate integration over modified non-uniform grids.
It is shown that the algorithm of approximate integration using the product of an uneven grid and a parallelepipedal grid in a mutually simple module is unsaturated with the order 𝛼/2 .

The article presents the results of the development of a methodology for predicting the process of corrosion cracking of underground pipelines using the COMSOL Multiphysics 5.6:
Corrosion Module software product. Using the developed technique, it was found that at small  values of longitudinal deformation (1.75 and 2.75 mm), a uniform distribution of stresses, corrosion potential, density of anode and cathode current is observed along the entire length of the crack. With an increase in the degree of deformation of the order of 3.75 and 4 mm, the distribution of stresses, corrosion potential, density of the anode and cathode current is more uneven: the maximum values of these values are reached at the top of the corrosion crack, and their more uniform distribution is characteristic along its edges. It is shown that the effect of local elastic deformation on the corrosion crack does not contribute to the strengthening of the mechanical-electrochemical interaction, which contributes to an increase in corrosion activity.

Gas laser processing of metal alloys is considered on the example of gas laser cutting with different power, speed, auxiliary gas pressure and focal length. As a result of the comprehensive
studies carried out, it was found that the use of gas laser cutting in the manufacture of parts from metal alloys of the iron –carbon system opens up the possibility of combining in one operation and obtaining and hardening the part without the use of additional thermal and chemicalthermal treatment. Adequate mathematical nonlinear regression models have been developed
for the extent of the gas-laser thermal influence zone (L); orthogonality of the cutting surface (a); surface roughness of the gas-laser cut (Rz), which link the quality indicators of the gaslaser
cutting surface with the parameters of laser processing, the carbon content in steel and the thickness of the steel sheet.

The article deals with the problem of detecting attacks on computer networks. A method of proactive counteraction based on the use of detectors built in the form of hidden Markov models is proposed.

An abelian group 𝐴 is called 𝜋-bounded for a set of prime numbers 𝜋, if all 𝑝-primary components 𝑡𝑝(𝐴/𝐵) are finite for every subgroup 𝐵 ⊂ 𝐴 and for every 𝑝 ∈ 𝜋. E. V. Sokolov has
introduced the class of 𝜋-bounded groups investigating ℱ𝜋-separable and 𝜋′-isolated subgroups in the general group theory. The description of torsion 𝜋-bounded groups is trivial. E. V. Sokolov has proved that the description of mixed 𝜋-bounded groups can be reduced to the case of torsion free groups. We consider the class of 𝜋-bounded torsion free groups in the present paper and we prove that this class of groups coincides with the class of 𝜋-local torsion free abelian groups of finite rank.
We consider also abelian groups satisfying the condition (*), that is such groups that their quotient groups don’t contain subgroups of the form Z𝑝∞ for all prime numbers 𝑝 ∈ 𝜋, where 𝜋
is a fixed set of prime numbers. It is clear that all 𝜋-bounded groups satisfy the condition (*).
We prove that an abelian group 𝐴 satisfies the condition (*) if and only if both groups 𝑡(𝐴) and 𝐴/𝑡(𝐴) satisfy the condition (*). We construct also an example of a non-splitting mixed group
of rank 1, satisfying the condition (*), for every infinite set 𝜋 of prime numbers.

This work is dedicated to the twentieth anniversary of the Chebyshev sbornik and the release of the 80th issue of the magazine.
The article highlights the history of the magazine. We describe the stages of formation. It talked about the contribution of various scientists in the journal work.
Provides some scientometric indicators.