This paper is devoted to the analysis of S. B. Stechkin’s contribution to some questions
in analytic number theory. There are five areas of his research in the field of number theory.
First, the works of S. B. Stechkin on the theory of the Riemann zeta function are considered.
His results on even trigonometric polynomials played a role in these studies. Another area of
research to which S. B. Stechkin made a significant contribution together with A. Y. Popov,
relates to the asymptotic distribution of prime numbers on average. The third question, to
which one of the works of S. B. Stechkin in analytic number theory was devoted, is related
to Vinogradov’s mean value theorem, the main method for estimating Weyl sums. The fourth
area of research, where S. B. Stechkin managed to get a result that could not be strengthened
over the past 30 years, is the estimation of complete rational trigonometric sums. Finally, the
fifth direction is the study of Gauss sums. Stechkin’s result in this direction and the problem
he posed inspired followers to the present time.

The article is devoted to the life and scientific and pedagogical activity of the famous
mathematician, doctor of physical and mathematical sciences, professor Boris Maximovich
Bredikhin (1920–1994) in connection with the 100th anniversary of his birth. In first provides
brief biological information from his life. The main part of our work is devoted to the
achievements of B. M. Bredikhin in number theory. An analysis of his scientific is given.

We study the sharp Markov--Bernstein--Nikol'skii inequality of the form

$\|D^{s}u\|_{\infty}\!\le C_{p}(n;s)\times\\\times\|u\|_{p}$, $p\in [1,\infty]$ for

trigonometric and algebraic polynomials $u$ of degree at most $n$ in the

weighted space $L^{p}$ with the Gegenbauer--Dunkl differential operator $D$. In

particular cases, these inequalities are reduced to the classical inequalities

of approximation theory of the Markov, Bernstein, and Nikol'skii type, to which

numerous papers are devoted. We apply the results of V.A. Ivanov (1983, 1992),

V.V. Arestov and M.V. Deikalova (2013, 2015), F. Dai, D.V. Gorbachev and

S.Yu. Tikhonov (2020) for algebraic constants in $L^{p}$ on compact Riemannian

manifolds of rank 1 (including the Euclidean sphere) and an interval with

Gegenbauer weight, refer to the works of E. Levin and D. Lubinsky (2015),

M.I. Ganzburg (2017, 2020), a review of the classic results of

G.V. Milovanovi\'c, D.S. Mitrinovi\'c and Th.M. Rassias (1994).

 

Earlier we studied the case $s=0$. In this paper, we consider the case $s\ge

0$. Our main result is to prove the existence in the trigonometric case for even $s=2r$ of

extremal polynomials $u_{*}$ that are real, even, and

$C(n;s)=\frac{|D^{s}u_{*}(0)|}{\|u_{*}\|_{p}}$. With the help of this fact, the

relationship with the algebraic constant for the Gegenbauer weight is proved.

On the one hand, this relationship allows to automatically characterize

extremal algebraic polynomials. On the other hand, well-known algebraic results

carry over to a more general trigonometric version. The main method of proof is

the application of the Gegenbauer--Dunkl harmonic analysis constructed by

D.V. Chertova (2009). As a consequence, we give the explicit constants for

$p=2,\,\infty$ (using the results of V.A. Ivanov), we give the relations of

orthogonality and duality (proved by methods of convex analysis from

approximation theory), we establish one asymptotic result of the

Levin--Lubinsky type (due to the connection with the multidimensional

Nikol'skii constant for spherical polynomials).

Let $M_{n}=\sup_{P\in \mathcal{P}_{n}\setminus \{0\}} \frac{\max_{x\in

[-1,1]}|P(x)|}{\int_{-1}^{1}|P(x)|\,dx}$ be the Nikol'skii constant between the

uniform and integral norms for algebraic polynomials with complex coefficients

of degree at most $n$. D. Amir and Z. Ziegler (1976) proved that

$0.125(n+1)^{2}\le M_{n}\le 0.5(n+1)^{2}$ for $n\ge 0$. The same upper bound

was obtained by T.K. Ho (1976). F. Dai, D. Gorbachev, and S. Tikhonov

(2019--2020) refined this result by establishing that $M_{n}=Mn^{2}+o(n^{2})$

for $n\to \infty$, where $M\in (0.141,0.192)$ is the sharp Nikol'skii constant

for entire functions of exponential spherical type in the space

$L^{1}(\mathbb{R}^{2})$ and functions of exponential type in

$L^{1}(\mathbb{R})$ with weight $|x|$.

 

We prove that for arbitrary $n\ge 0$ one has $M(n+1)^{2}\le M_{n}\le

M(n+2)^{2}$, where $M\in (0.1410,0.1411)$. This statement also allows us to

refine the exact Jackson--Nikol'skii constant for polynomials on the Euclidean

sphere $\mathbb{S}^{2}$. The proof is based on the relationship between the

algebraic Nikol'skii constants and the Bernstein--Nikol'skii trigonometric

constants and our estimates of these constants (2018--2019). We also apply the

characterization of the extremal algebraic polynomial obtained by D. Amir and

Z. Ziegler (1976), V.V. Arestov and M.V. Deikalova (2015). Using this

characterization, we compose a trigonometric system for determining the zeros

of an extremal polynomial, which we solve approximately with the required

accuracy using Newton's method.

e prove algorithmic undecidability of $\exists \forall^2 \exists^3$-theory for a free  semigroup of countable rank. This strengthens the classical Quine's (1946) result [1] on algorithmic undecidability of elementary theory of an arbitrary non-cyclic free semigroup.

The author has developed a method of solving the variation problem of the density
functional theory within the framework of the orbital-free approach with the generalized gradient
approximation. The method is based on calculating the exchange -correlation potential using
an iterative procedure. Test calculations for two-atom systems have shown that our approach
allows the coupling energy of atoms and equilibrium interatomic distance in dimers to be found
with about the same accuracy as the Kohn-Sham method, but much faster.

In spaces with a Dunkl weight $v_k (x)$ of power type on $\mathbb{R}^ d$, defined by a root system and a nonnegative multiplicity function $k$ invariant with respect to a finite reflection group, a meaningful harmonic analysis is constructed. that generalizes the Fourier analysis in the Euclidean space. The classical Fourier analysis on the Euclidean space corresponds to the case $k\equiv0$. In 2012, Salem Ben Sa\v{i}d, Kobayashi, and Orsted defined the two-parameteric $(k, a)$-generalized Fourier transform, acting in spaces with weight $|x|^{a-2}v_k(x)$, $a>0$. The most interesting cases are $a =2$ and $a =1$. For $a =2$ the generalized Fourier transform coincides with the Dunkl transform and it is well studied. In case $a=1$ harmonic analysis, which is important, in particular, in problems of quantum mechanics, has not yet been sufficiently studied. One of the essential elements of harmonic analysis is the bounded translation operator, which allows one to determine the convolution and structural characteristics of func\-tions. For $a=1$, there is a translation operator $\tau^yf(x)$. Its $L^p$-boundedness was recently established by Salem Ben Sa\v{i}d and Deleaval, but only on radial functions and for $1\le p\le 2$. In this paper, a new generalized translation operator $T^tf(x)$ is proposed. It is obtained by integrating of the operator $\tau^yf(x)$ over the unit Euclidean sphere with respect to the variable $y'$, $|y'|=1$, $y=ty'$. We prove that it is positive on functions from the Schwartz space $\mathcal{S}(\mathbb{R}^d)$, for it $T^t1=1$ and it admits a representation with a probability measure. From this we deduce its $L^p$-boundedness for all $1\le p<\infty$ and boundedness on the space $C_b(\mathbb{R}^d)$ of continuous bounded functions.

Over the past 30 years a meaningful harmonic analysis has been construc\-ted in the spaces with Dunkl weights of power type on $\mathbb{R}^d$. The classical Fourier analysis on Euclidean space corresponds to the weightless case. The Dunkl--Riesz potential and the Dunkl--Riesz transforms defined by Thangavelu and Xu play an important role in the Dunkl harmonic analysis.
In particular, they allow one to prove the Sobolev inequalities for the Dunkl gradient. Particular results were obtained here by Amri and Sifi, Abdelkefi and Rachdi, Veliku. Based on the weighted inequalities for the Dunkl--Riesz potential and the Dunkl--Riesz transforms, we prove the general $(L^q,L^p)$ Sobolev inequalities for the Dunkl gradient with radial power weights. The weighted inequalities for the Dunkl--Riesz potential were established earlier.The $L^p$-inequalities for the Dunkl--Riesz transforms with radial power weights are established in this paper. A weightless version of these inequalities was proved by Amri and Sifi.

The paper considers only finite groups. A class of groups $\mathfrak F$ is called a Fitting class if it is closed under normal subgroups and products of normal $\mathfrak F$-subgroups; formation, if it is closed with respect to factor groups and subdirect products; Fitting formation if $\mathfrak F$ is a formation and Fitting class at the same time.

For a nonempty subset $\omega$ of the set of primes $\mathbb P$ and the partition $\sigma =\{\sigma_i\mid i\in I\}$, where $\mathbb P=\cup_{i\in I}\sigma _i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\not =j$, we introduce the
$\omega\sigma R$-function $f$ and $\omega\sigma FR$-function $\varphi$. The domain of these functions is the set
$\omega\sigma\cup\{\omega'\}$, where
$\omega\sigma=\{ \omega\cap\sigma_i\mid\omega\cap\sigma_i\not =\varnothing\}$,
$\omega'=\mathbb P\setminus\omega$. The range of function values is the set of Fitting classes and the set of nonempty Fitting formations, respectively. The functions $f$ and $\varphi$ are used to determine the $\omega\sigma$-fibered Fitting class
$\mathfrak F=\omega\sigma R(f,\varphi)=(G: O^{\omega} (G)\in f(\omega' )$ and
$G^{\varphi (\omega\cap\sigma_i )} \in f(\omega\cap\sigma_i )$ for all $\omega\cap\sigma_i \in\omega\sigma (G))$ with the $\omega\sigma$-satellite $f$ and the $\omega\sigma$-direction $\varphi$.

The paper gives examples of $\omega\sigma$-fibered Fitting classes. Two types of $\omega\sigma$-fibered Fitting classes are distinguished: $\omega\sigma$-complete and $\omega\sigma$-local Fitting classes. Their directions are indicated by
$\varphi_0$ and $\varphi_1$, respectively. It is shown that each nonempty nonidentity Fitting class is an $\omega\sigma$-complete Fitting class for some nonempty set $\omega\subseteq\mathbb P$ and any partition $\sigma$. A number of properties of $\omega\sigma$-fibered Fitting classes are obtained. In particular, a definition of an internal $\omega\sigma$-satellite is given and it is shown that each $\omega\sigma$-fibered Fitting class always has an internal $\omega\sigma$-satellite. For $\omega=\mathbb P$, the concept of a $\sigma$-fibered Fitting class is introduced. The connection between
$\omega\sigma$-fibered and $\sigma$-fibered Fitting classes is shown.

The paper considers the problem of calculating the parameters of the plane of a spatial
triangle from its central projection. Under certain conditions, the existence theorem for a
solution to this problem and its uniqueness are proved. Examples of conditions under which a
solution does not exist or is not unique are given. An algorithm for the approximate search of
all possible solutions to the problem under certain conditions is also proposed. The problem
considered in the article arises when constructing three-dimensional models of objects from their
photograph.

Let ${\mathbb F}$ be an arbitrary field. We consider a  commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit  over ${\mathbb F}$ and we prove that the image over ${\mathfrak M}$ of every non-associative multilinear polynomial over ${\mathbb F}$ is a vector space. The same question we consider for two subalgebras: an algebra of the rock, the paper and the scissors without unit, and an algebra of trace zero elements with zero scalar part. Moreover in this paper we consider the questions of possible eveluations of homogeneous polynomials on these algebras.

The paper proves the strong compactness of the sequence $\{\tilde{c}^{\,\varepsilon}(\boldsymbol{x},t)\}$ in $\mathbb{L}_{2}(\Omega_{T})$,
$\Omega_{T}=\Omega\times(0,\\T)$, $\Omega\subset \mathbb{R}^{3}$, bounded in the space $\mathbb{W}^{1,0}_{2}(\Omega_{T})$ with
the sequence of time derivatives
$\Big\{ \displaystyle \frac{\partial}{\partial t}\big(\chi(\boldsymbol{x},t,\frac{\boldsymbol{x}}{\varepsilon})\Big.$
$\Big.\tilde{c}^{\,\varepsilon}(\boldsymbol{x},t)\big) \Big\}$ bounded in the space $\mathbb{L}_{2}\big((0,T);\mathbb{W}^{-1}_{2}(\Omega)\big)$,
where characteristic function $\chi(\boldsymbol{x},t,\boldsymbol{y})$ is 1-periodic in a variable $\displaystyle \boldsymbol{y}\in Y= \left(-\frac{1}{2},\frac{1}{2} \right)^{3}\subset \mathbb{R}^{3}$.

As an application we consider the homogenization of a diffusion-convection equation in non-periodic structure, given by 1-periodic in $\boldsymbol{y}$ characteristic function $\chi(\boldsymbol{x},t,\boldsymbol{y})$ with a sequence of divergent-free velocities $\{\boldsymbol{v}^{\varepsilon}(\boldsymbol{x},t)\}$ weakly convergent in $\mathbb{L}_{2}(\Omega_{T})$.

In the present note we prove a reduction theorem for subgroups of the general linear group ${\operatorname{GL}}(n,T)$ over a skew-field $T$,
generated by a pair of microweight tori of the same type. It turns out, that any pair of tori of residue $m$ is conjugate to such
a pair in ${\operatorname{GL}}(3m,T)$, and the pairs that cannot be further reduced to ${\operatorname{GL}}(3m-1,T)$ form a single ${\operatorname{GL}}(3m,T)$-orbit.
For the case $m=1$ this leaves us with the analysis of ${\operatorname{GL}}(2,T)$, that was carried through some two decades ago by the second author,
Cohen, Cuypers and Sterk. For the next case $m=2$ this means that the only cases to be considered are ${\operatorname{GL}}(4,T)$ and ${\operatorname{GL}}(5,T)$.
In these cases the problem can be fully resolved by (direct but rather lengthy) matrix calculations, which are relegated to a
forthcoming paper by the authors.

The paper proves the regularity in a neighborhood of zero of the Laplace transform of the
Fourier transform of an even function obtained from an odd function regular in a neighborhood
of the real axis by changing the parity. This fact implies that the sine and cosine of the Fourier
transforms are commutable up to the sign.

This paper consists of 3 sections. In the first section, we will give a brief introduction to the ”Feigin’s homomorphisms” and will see how they will help us to prove our main and fundamental theorems related to quantum Serre relations and screening operators.

In the second section, we will introduce Local integral of motions as the space of invariants of nilpotent part of quantum affine Lie algebras and will find two and three-point invariants in the case of 𝑈𝑞(𝑠^𝑙2) by using Volkov’s scheme.
In the third section, we will introduce lattice Virasoro algebras as the space of invariants of Borel part 𝑈𝑞(𝐵+) of 𝑈𝑞(𝑔) for simple Lie algebra 𝑔 and will find the set of generators of Lattice Virasoro algebra connected to 𝑠𝑙2 and 𝑈𝑞(𝑠𝑙2)
And as a new result, we found the set of some generators of lattice Virasoro algebra.

The paper presents many unknown facts from the history of the Tula school of number
theory. It is shown that the main role in the revival of the Tula school of number theory was
played by professors N. M. Korobov, V. I. Nechaev, S. B. Stechkin, and N. M. Dobrovolsky.
The role of these main participants in the revival of the scientific school of number theory
in Tula is revealed. Various details of the relationship between these participants in the process
of reviving the Tula school of number theory are given.
To characterize the activities of the Tula scientific school on number theory after its revival,
an overview of the main directions of its work over the past 45 years and a brief description of the
work done is provided. The article provides a bibliography of the main scientific publications on
the number-theoretic method in approximate analysis by N. M. Korobov and N. M. Dobrovolsky,
on which the revival of the Tula school of number theory and its subsequent functioning over
the past 45 years were based.

The paper is devoted to the approximation of periodic functions of high smoothness by arithmetic means of Fourier sums. The simplest and natural example of a linear process of approximation of continuous periodic functions of a real variable is the approximation of these functions by partial sums of the Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the entire class of continuous $2\pi$-periodic functions. In connection with this, a significant number of papers is devoted to the study of the approximative properties of other approximation methods, which are generated by certain transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for each function $f \in C$. In particular, over the past decades, de la Vallee Poussin sums and Fejer sums have been widely studied. Today, publications have accumulated a large amount of factual material. One of the most important directions in this field is the study of the asymptotic behavior of upper bounds of deviations of arithmetic means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M.~Nikolsky, S.B.~Stechkin, N.P.~Korneichuk, V.K.~Dzadyk and others.

The aim of the work is to systematize known results related to the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums and to present new facts obtained for particular cases.

and to present new approximative properties of Fejer sums on the classes of periodic functions that can be regularly extended into the corresponding strip of the complex plane. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations in the uniform metric of Fejer sums on Poisson integrals classes. The deduced formula provides a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

The article considers the transcendence and algebraic independence problems,
introduce statements and proofs of theorems for some kinds of elements
from direct product of $p$-adic fields and polynomial estimation theorem.
Let $\mathbb{Q}_p$ be the $p$-adic completion of $\mathbb{Q}$, $\Omega_{p}$ be the
completion of the algebraic closure of $\mathbb{Q}_p$,
$g=p_1p_2\ldots p_n$ be a composition of separate prime numbers,
$\mathbb{Q}_g$ be the $g$-adic completion of $\mathbb{Q}$,
in other words $\mathbb{Q}_{p_1}\oplus\ldots\oplus\mathbb{Q}_{p_n}$.
The ring $\Omega_g\cong\Omega_{p_1}\oplus\ldots\oplus\Omega_{p_n}$,
contains a subring $\mathbb{Q}_g$. The transcendence and algebraic independence over $\mathbb{Q}_g$
are under consideration. Here are appropriate theorems for numbers like
$\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}},\;
\text{where}\;a_{j}\in \mathbb Z_g,$ and non-negative rational numbers $r_{j}$ increase
to strictly unbounded.

The paper is devoted to numerical modeling of the process of propagation of a low-intensity
shock wave from a pure gas into an inhomogeneous medium, which is a gas suspension
of solid particles. Computational experiments considered both electric neutral and charged
suspensions of solid particles. In the mathematical model used in the work, the conservation
of the momentum components of the carrier medium was described by the system of Navier-
Stokes equations for a compressible gas in a two-dimensional formulation. When describing
the interaction of the carrier and the dispersed phase of the gas suspension, the Stokes law,
Archimedes’ principle, the virtual masses force were considered, interphase heat transfer was
also taken into account. For the dispersed component of the mixture, a complete hydrodynamic
system of equations of motion was solved. It included the equation of continuity, the equation
of conservation of momentum and energy. The system of equations of the mathematical model,
supplemented by boundary conditions, was solved by an explicit finite-difference method of
the second order of accuracy. In the numerical model, an algorithm for suppressing numerical
oscillations was also used. Numerical modeling showed that the presence of an electric charge in
the dispersed component of the mixture affects the movement of the dispersed component and,
due to interfacial interaction, the gas flow. As a result of numerical calculations, it was found
that an increase in particle size leads to a significant increase in interfacial velocity slip. It was
determined that the intensity of the velocity slip between the carrier and the dispersed phases in
an electrically charged dusty medium occurs in the direction of increasing the specific Coulomb
force. While in an electrically neutral gas suspension, the growth of velocity slip occurs in the
direction of motion of the shock wave.

The paper is based on the Binomial theorem and its generalizations to the polynomials of
binomial type. Thus, we give some applications to the generalized Waring problemm (Loo-Keng
Hua) and Hilbert-Kamke problem (G.I.Arkhipov). We also prove Taylor-Maclaurin formula
for the polynomials and smooth functions and give its applications to the numerical analysis
(Newton’s root-finding algorithm, Hensel lemma in full non-archimedian fields, approximate
evaluaion of the function at given point). Next, we prove an analogue of Binomial theorem
for Bernoulli polynomials, Euler-Maclaurin summation formula over integers and Poisson
summation formula for the lattice and consider some examples of binomial-type polynomials
(monomials, rising and falling factorials, Abel and Laguerre polynomials). We prove some
binomial properties op Appel and Euler polynomials and establish the multidimensional Taylor
formula and the analogues of Euler-Maclaurin and Poisson summation formulas over the lattices.
Finally, we consider the multidimensional analogues of these formulas for the multidimensional
complex space and prove some properties of binomial-type polynomials of several variables.

The paper presents the implementation of the technology of electroerosive grinding on the
example of solid alloy waste. It is shown that the power of electric contact heat sources can be
sufficient for the implementation of the process of electroerosive dispersion. Dependences are
obtained that allow us to perform a calculated estimation of the fractional composition of the
powder material obtained under the action of electric contact heat sources.

The paper provides information about the evolution of mathematical models of sliding
friction of solids. It is Shown that taking into account deviations from the Leonardo da Vinci-
Amonton-Coulomb law, it is necessary to Refine it using the correction function of the normal
force. A mathematical model of the generalized sliding friction law has been created that takes
into account the abrupt changes in the linear dependence of the friction force on the normal
force.

The article illustrates the role of mathematics in research in the field of technical Sciences,
devoted to the study of the properties of metallic materials on the example of titanium.

The work deals with the topic of the formation of religious views among representatives of
the Moscow Mathematical School at the turn of the 19th - 20th centuries and the influence of
the world outlook on their scientific creativity. The main core of this group of scientists included
N.D. Brashman, N.V. Bugaev, P.A. Nekrasov, D.F. Yegorov, N.N. Luzin, P.A. Florensky.
The general tendency can be distinguished in the evolution of ideas of Moscow mathematicians-
thinkers of the 19th — early 20th centuries: they went all the way from Mathematics
to Philosophy and came back again to Mathematics. Moscow mathematical society
(N.D. Brashman, N.V. Bugaev etc.) cultivated Moscow Philosophical and Mathematical School
(N.V. Bugaev, P.A. Nekrasov, P.A. Florensky etc.) and the latter one gave an impulse to creating
Moscow School of Function Theory (D.F. Yegorov, N.N. Luzin etc.).
The work reveals philosophic sources of forming Moscow Mathematical School for the first
time. Philosophic preferences of representatives of this school are close to Slavophilism (negative
attitude to the development of Russia according to the Western patterns, the doctrine of spirit
integrity (which denies cognition only through reason or through senses not including spirit);
the doctrine of collegiality as obtaining freedom through the dissolution of the individual in the
church, society, state; Orthodox worldview; love to Motherland). These ideas also influenced
the nature of the mathematical creative work of Moscow mathematicians. It got some specific
features : 1) collective character, generating new directions in science and a strong wish to share
them with other scientists; 2) concentration on seeking general methods and regularities; 3) a
tendency to contemplation, a preference for theoretical research over practical (the sphere of
scientific interests included number theory, set theory, function theory etc.)

The article considers the inverse problem on determination of the inhomogeneity laws of an
elastic coating of an absolutely rigid cylinder located in a plane waveguide, one boundary of
which is absolutely hard and the other is acoustically soft.
It is believed that the waveguide filled by ideal fluid. The harmonic sound pressure wave
excited by a given distribution sources on the section of the waveguide located on the final
distance from the axis of the cylinder is propagated along the walls of the waveguide on normal
to the surface of the cylindrical body. The inhomogeneity parameters for providing minimum
sound reflection are determined.
The solution of the inverse problem is obtained based on the solution of the direct problem
diffraction.
Dependences of the density and elastic moduli of the coating material from the radial
coordinate are approximated by polynomials of the third degrees.
Functionals defined on the class of cubic functions and expressing the average intensity of
sound scattering in a given section of the waveguide at a fixed frequency or at some frequency
range are built.
The minimization of the functionals is done with using the genetic algorithm. Analytical
description of the optimal inhomogeneity laws of an cylinder coating are received to ensure
minimal sound reflection.

In article the problem of the scattering of an obliquely incident plane monochromatic sound
wave by an elastic cylinder with a radially non-uniform elastic coating in presence of a flat surface
(absolutely rigid and acoustically soft) is considered. The analytical solution of the problem
by the method of imaginary sources using addition theorems for cylindrical wave functions is
received. Wave fields in a containing medium and homogeneous elastic cylinder are found in the
form of expansions in wave cylindrical functions. The boundary-value problem for the system of
ordinary second order differential equations is constructed for determination of the displacement
fields in inhomogeneous coatings.
Numerical calculations of frequency and angular characteristics of the scattered field for
elastic homogeneous cylinders with and without coating located near the underlying plane
are performed. Influence of continuously inhomogeneous elastic coatings on sound-reflecting
properties of elastic cylindrical bodies are revealed.

In this article, based on the classical works of G. Kirsch, K. Inglis, G.~V.~Kolosov, and N.~I.~Muskhelishvili, we continue to develop a mathematical apparatus that allows us to obtain solutions to a number of three-dimensional problems of fracture mechanics in a hardened metal medium.

Based on the work of G.~R.~Irwin, G.~I.~Barenblatt, Westergaard, L.~D.~Landau, and E.~M.~Livshits, the authors performed mathematical modeling of the stress-strain state in the volume of a loaded steel sample in the vicinity of pores of various morphologies resulting from operational loads and aggressive environmental influences. An algorithm for determining the components of the stress tensor near concentrators in the form of pores of various shapes is proposed for understanding the force lines of the stress field in a metallic medium. A stationary case with a fixed ratio of external stress and yield strength was considered.

This paper is devoted to the approximation of a quadratic algebraic lattice by an integer
lattice. It calculates the distances between a quadratic algebraic lattice and an integer lattice
when they are given by the numerator and denominator of a suitable fraction to the square root
of the discriminant d — of a square-free natural number.
The results of this work allow us to study questions about the best approximations of
quadratic algebraic lattices by integer lattices.

We study the sharp Nikol'skii--Bernstein constants for spherical polynomials in
the space $L^{p}(\mathbb{S}^{d})$ with the Dunkl weight. An interrelationship
with one-dimensional constants for algebraic polynomials in the space
$L^{p}[-1,1]$ with the Gegenbauer weight is established.

The present essay tells about the life and the scientific creativity of the brilliant Russian
mathematician Dmitry Aleksandrovich Popov, who has celebrated his 80th anniversary in
August, 2019. D.A. Popov made a considerable contribution to the mathematical foundations
of the X-ray, ultrasonic and acoustic tomography, to the theory of the oscillating integrals
estimations and to the problems concerning the distribution of lattice points in the plain circle
and in the bodies of rotation. A special attention is paid to the recent results of the anniversarian
that demonstrate a connection between the spectrum of Laplace operator on the fundamental
domain of modular group with the distribution of primes.