This article examines the scientific and pedagogical activities of Professor, Doctor of Physico-Mathematical Sciences Vladimir Nikolaevich Bezverkhnii, who made a significant contribution
to the research of algorithmic problems of the theory of groups and semigroups.
The main periods of the famous scientist’s life are highlighted.
V. N. Bezverkhnii worked for a long time at Tula State Lev Tolstoy Pedagogical University, led a scientific seminar on algorithmic problems of the theory of groups and semigroups and a postgraduate course in combinatorial group theory.
Continuing the work of Professor M. D. Grindlinger, he was the supervisor of eight PhD and scientific advisor to one doctoral dissertation.

In recent years, many researchers have focused on studying the phenomenon of extreme multistability of dynamic systems. An extremely multistable system contains an infinite number of coexisting attractors determined by different initial conditions. The latter circumstance introduces extreme uncertainty into its behavior and opens up the possibility of using such a system, for example, in cryptography and the organization of secure communication in
information transmission systems. Therefore, understanding the fundamental principle of the formation of extreme multistability is of particular interest. Only by understanding this principle we can generate systems with the desired behavior. Extreme multistability of many currently known systems can be explained by the presence of the phenomenon of offset boosting, which suggests the presence of an offset parameter in the system. As it turned out, the cancellation
of the offset parameter can lead to the presence of a continuum of coexisting attractors in the system, which are continuously located in the phase space and extend to infinity in a certain direction. This discovery can become, for example, an explanation for the occurrence
and propagation of tornadoes and turbulence. In this paper, using the dimension expansion technique, two fourth-order systems without equilibrium states containing a continuum of coexisting hidden chaotic attractors are constructed. The first system is based on the wellknown
three-dimensional Sprott system, and the second is based on the three-dimensional system proposed earlier by the authors, which has a single hidden chaotic attractor of dimension “almost 3”. The second system contains a 2D lattice, which is a union of a countable number of
strips, each of which contains a continuum of attractors.

The paper is a new edition of the authors’ previous work on this topic. A significant improvement in the results of the previous article is associated with the use of weight functions for the transition from an integral of lower to higher dimension.
Such a transition turned out to be necessary to obtain new estimates of the error of the approximate solution of the Fredholm integral equation of the second kind by the iteration method using algebraic grids.
The essence of this approach is that in the approximate calculation of the solution of the Fredholm integral equation of the second kind, a partial sum of the Neumann series consisting of integrals of different multiplicities is used. When using different algebraic grids corresponding to different purely real fields and one stretching parameter, it turns out that for a lower dimension, a smaller number of nodes of the algebraic grid will be used, and therefore the accuracy of the calculation will be lower. In order not to solve the complex problem of optimizing the number of nodes for different dimensions, this paper proposes an approach in which all integrals are reduced to one and a single algebraic grid is used for it. The second positive effect of this
approach is related to the minimization of the calculation of the values of the kernel of the Fredholm equation of the second kind due to the use of Horner’s scheme.
The paper considers two methods for choosing a purely real algebraic field. The first method is based on specifying an irreducible polynomial with integer coefficients, all of whose roots are real numbers. The second method is based on using a tower of quadratic fields.

With both methods of choosing a purely real algebraic field, we were able to use a large dimensional algebraic grid to integrate a function of a smaller number of variables. An important role in this was played by the weight function, which allows replacing the integral of a function
from the class 𝐸𝛼 𝑠 over the cube 𝐺𝑠 with the integral of a function from the class 𝐸𝛼,0 𝑠 [−1, 1] over the cube 𝐾𝑠. It is important to note that the new function goes to zero on the boundary of this cube.

In this paper we consider a number of extremal problems for nonnegative and integrable entire functions of exponential type ⩽ 𝜎 (the class ℰ+1,𝜎). The problems under consideration have the following form. Let Λ𝜌 be a translation invariant operator with a locally integrable symbol 𝜌(𝑥), 𝑥 ∈ R, such that 𝜌(𝑥) = 𝜌(−𝑥), 𝑥 ∈ R. For a fixed 𝜎 > 0, it is required to find the following constants:

This general problem reduces to an equivalent extremal problem for positive-definite functions, the solution of which is known. As consequence, we obtained exact values of 𝑀*(𝜌, 𝜎) and 𝑚*(𝜌, 𝜎) for a number of different symbols 𝜌. In particular, we consider cases where Λ𝜌 is a differential or difference operator of a special form.

In the work, asymptotic formulas with a remainder term are obtained for the number of representations of a pair of integers 𝑚 and 𝑛, respectively, as a sum of 𝑠 ⩾ 5 variables, and each solution of such a Diophantine system satisfies the congruential a condition of a special type, associated in a certain way with a linear form.
Asymptotic formulas with a remainder term for the number of solutions of such a Diophantine system are derived for 𝑁 → ∞, where 𝑁 = Δ𝑚 − 𝑛2 and Δ equals the sum of the squares on the coefficients on the linear form. In addition, two-sided lower and upper bounds are obtained for a special series of Diophantine system under study based on the upper bound based on formulas for the number of solutions of
a congruence of the second degree modulo the power 𝑥21+ . . . + 𝑥2𝑠 ≡ 𝑎 (mod 𝑝𝑘) of the prime number, where 𝑎 is natural number.
This work is a continuation of a previous study, relating to the case of an even number of variables.

This paper is devoted to the description of unars satisfying various conditions that are close to flatness. I.A.Sakharov showed that projective unars coincide with free ones and are a coproduct of rays. Previously, the author obtained a complete description of flat unars. This paper continues this line of research and provides a complete description of unars that are close to flat ones, namely: pullback flat, equalizer flat, weakly flat, principally weakly flat, torsion free, unars with conditions (P) or (E), faithful, strongly faithful and regular ones. It is proved that pullback flat and equalizer flat unars coincide and are a coproduct of lines and rays. Unars satisfying condition (P), flat, weakly flat, principally weakly flat and torsion free ones coincide and are a coproduct of lines, rays and cycles. Faithful, strongly faithful, regular unars and unars satisfying condition (E) are exactly unars that do not contain a cycle.

In 1963, relying on estimates of special trigonometric sums, Hawley first proved an asymptotic formula for the average number of divisors of a quadratic polynomial with a powerlaw reduction in the remainder term compared to the principal term. Later, these estimates were improved.
The paper proves new stronger results in this area of research in analytical number theory.

In the previous work of the authors, the foundations of the theory of smooth manifolds of number-theoretic lattices were laid. The case of arbitrary multidimensional lattices was considered.
This article considers the general case of shifted multidimensional lattices. Note that the geometry of the metric spaces of multidimensional lattices is much more complicated than the geometry of ordinary Euclidean space. This is evident from the paradox
of non-additivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox it follows that the problem of describing geodesic lines in spaces of multidimensional lattices remains open, as well as finding a formula for the length of arcs of lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts. A further direction of research may be the study of the analytic continuation of the
hyperbolic zeta function on the spaces of shifted multidimensional lattices. As is known, the analytic continuation of the hyperbolic zeta functions of lattices are constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytical continuations in the left half-plane on the space of lattices has not been studied. All of these, in our opinion, are relevant directions for further research.

The paper studies extremal problems related to the best polynomial approximation of functions that are analytic in the unit disk and belong to the Bergman space 𝐵2 with a finite norm

Let

An exact theorem is proved between the value of the best approximation 𝐸𝑛−1(𝑓)2 and the value of the modulus of continuity of the mth order sin(𝜋𝑡/ℎ) (0 < ℎ ⩽ 𝜋/𝑛) of functions 𝜔𝑚(𝑓(𝑟), 𝑡)2 averaged with the weight 𝑓 ∈ 𝐵(𝑟)2 . The connection between the proven theorem and the behavior of exact constants in the Jackson-Stechkin inequality for moduli of continuity 𝜔𝑚(𝑓(𝑟), 𝑡)2 is clarified. For the class of functions 𝑊(𝑟) 𝑚 (Φ)2, given a given monotonically increasing moharant Φ, satisfying some restrictions, the exact values of various 𝑛-widths in 𝐵2 space are calculated.

In the paper, the general problem of extremal functional interpolation in the mean for real functions that have derivative of order 𝑛 almost everywhere is formulated on an arbitrary partition Δ = {𝑥𝑘}∞𝑘
=−∞ of the real axis. It is required to find the smallest value of the 𝐿∞-
norm of the 𝑛-derivative among functions that interpolate in the mean (with averaging intervals of length 2ℎ) any sequence of real numbers 𝑦 = {𝑦𝑘}∞𝑘 =−∞ from a class 𝑌 of sequences whose divided differences of order 𝑛 are bounded from above on such a grid. In this paper, the problem is considered in the case of 𝑛 = 2. We give the above and below estimates for the 𝐿∞-norm of the second derivative in terms of grid steps ℎ𝑘 = 𝑥𝑘+1−𝑥𝑘 provided that 2ℎ ⩽ ℎ = inf𝑘 ℎ𝑘. The
obtained results are developments is research of Yu. N. Subbotin, the author and S. I. Novikov in the well-known Yanenko—Stechkin problem of extremal functional interpolation. This problem was put in the early 60-s years of the last century for the case of the uniform grid.

In 2013, Andriy Bondarenko constructed a two-distance set on the unit sphere 𝑆^64 ⊂ R^65, consisting of 416 points that cannot be partitioned into 83 parts of smaller diameter. In this paper, we show that this construction works not only for the Euclidean space but for all ℓ𝑝-spaces.

The extension of the method of forming norms for structural elements of a complex system based on the methodology of Bayesian intelligent measurements and econometric modeling in small samples is presented. The implementation of the method is demonstrated using the example of agriculture in the Tula region.

The paper studies the properties of the hyperbolic zeta function of diagonal two-dimensional unimodular lattices. A theorem on the analytic continuation of this function is proved.

The problem of scattering of a plane harmonic sound wave by an obstacle in the form of a liquid body with a non-canonical shape and a piecewise-smooth surface, approximated by a polygonal mesh, is considered. The process model is based on the equations of hydrodynamics for an ideal fluid. Two numerical-analytical approaches to solving the problem are compared: the finite element method (FEM) and the boundary element method (BEM). In the first approach,
the obstacle is enclosed within a sphere, and the domain inside, taking into account the surface of the obstacle, is divided into spatial (3D) finite elements. In this domain, the problem is solved using FEM, which provides the potential values on the sphere. These values are then used to determine the coefficients of the spherical expansion of the scattered wave potential. In the second approach, using the spatial Green’s function for the Helmholtz equation, the problem is
reduced to a system of integral equations over the surface of the obstacle. The Burton-Miller method is also applied to eliminate the non-uniqueness of the solution, and singular integrals are regularized using identities for the static Green’s function. In the BEM, it is sufficient to divide the surface into boundary (2D) elements. The main equations for applying the numerical methods and the results of solving the sound scattering problem for a liquid body in the form of two spheres of equal radius are presented. It is established that to achieve acceptable accuracy in calculating the scattered field, the BEM requires significantly fewer computational resources compared to the FEM.

For hypoelastic media with initial stresses, the propagation of acoustic waves is considered
from the point of view of superposition of small perturbations on finite strains. The initial
state of the medium is characterized by homogeneous fields of finite strains and stresses, wave
propagation is described by small perturbations of the displacement field. In the article, the
formulation of the theorem on the change in the kinetic energy of the medium, linearized in
the vicinity of the initial state, and, as a consequence, the formulation of the acoustic Poynting
theorem for a hypoelastic medium are obtained. An expression for the Umov – Poynting vector
for a hypoelastic medium is written in terms of a generalized true stress tensor.
For plane monochromatic waves, the change in the stress tensor associated with the
passage of a wave in a medium with initial stresses is determined, and an expression for the
Umov–Poynting vector is obtained through the second Christoffel tensor and the initial stresses
acting in the medium. An expression for the radial velocity vector that takes into account the
initial stresses acting in the medium is obtained. It is shown that under the action of initial
stresses, the Umov – Poynting vector deviates from the radial velocity vector. This result does
not allow to use the radial velocity vector to determine the direction of energy flows during the
propagation of acoustic waves in hypoelastic media with initial stresses.

The article considers the problem of diffraction of a harmonic cylindrical sound wave on a multilayered spheroid is considered. The multilayered spheroid consist from an absolutely rigid spheroid and surrounding homogeneous spheroidal layers of an ideal compressible liquid. It is assumed that the spheroid is in an infinite ideal liquid. A cylindrical wave is emitted by an infinitely long linear source parallel to the axis of rotation of the spheroid.
The problem is solved in an elongated spheroidal coordinate system. An analytical solution of the problem has been obtained. A special case of a two-layer spheroid is considered.

This title describes the development of a mathematical control system for a small fixedwing aircraft with unconventional aerodynamic surfaces. The research process involved an
analysis of the complex factors influencing the dynamics of aircraft, represented by a set of differential equations. The study focused on identifying and selecting the parameters necessary for creating a mathematical model of a non-standard control system. This system is designed to generate and deliver commands to actuators responsible for the aerodynamic surfaces, which are arranged in a unique configuration. The developed mathematical model incorporates the control of innovative, domestically produced brushless electric motors, taking into account their specific technological characteristics. The efficiency of the mathematical model was validated in the MATLAB software environment using the Simulink toolbox, considering various operational conditions. The culmination of the work was the experimental testing of the control system and the mathematical model on a physical prototype, enabling the control of unmanned aerial vehicles with innovative, non-standard aerodynamic configurations.

A method for analyzing the ideal contour of an aircraft, which is necessary for designing aircraft with non-standard aerodynamic surfaces, is described. The aerodynamic surfaces of the aircraft can be considered a control channel, which is influenced by the command output unit. To create an ideal control loop, the frequency and phase characteristics were calculated.
The mathematical model was tested in the MATLAB software package using the SIMULIINK add-on, taking into account various operating conditions.
In the course of the work, the transient characteristic of the process of testing the control action of the system was optimized using the developed mathematical model of the aircraft in the MATLAB SIMULINK environment.
The result of the work was the testing of an ideal control loop on a real device, while maintaining a margin of stability and flight stability with various delays of the computing module.

An original method for calculating in situ characteristics of inhomogeneity of anisotropic
materials based on machine vision technology (photogrammetry) is proposed. The objects were
samples of powder alloys 316L and Inconel 718, manufactured using selective laser fusion (SLM)
technology and subjected to static stretching.
The technique is based on the joint use of hardening curves in the calculation of parameters, based on the results of in situ determination of the intensities of true stresses and deformations in individual micro-volumes of samples. Cross sections of a dividing grid applied to the surface of the samples were used as such micro-volumes.
Information about the geometry of the cells of the dividing grid, which changed during stretching, was obtained in the measuring unit from the results of photogrammetry - measurements of digital images of the dividing grid obtained during photofixation of the
sample during stretching. In the calculation block, the equations of dependence of the material heterogeneity index on mechanical properties and the characteristics of the unevenness of its plastic deformation on the intensity of acting stresses and deformations are obtained using the technique.
The use of mathematical algorithms for optimizing photogrammetry and programming the calculation block in Python JAVA will automate the process of calculating the obtained equations and their parameters and, as a result, create an extensive database of anisotropy characteristics and heterogeneity of properties of products manufactured by the SLM method.
This will ensure the creation of theoretical foundations for in-depth analysis and reasonable forecasting of the impact of technological anisotropy and heterogeneity of the properties of SLM products on their in situ performance.

This article examines the scientific and pedagogical activities of Professor, Doctor of Physical and Mathematical Sciences Urusbi Mukhamedovich Pachev, who made a significant contribution to research on analytical number theory and geometry of numbers. The main periods of the famous scientist’s life are covered.
U. M. Pachev has been working for a long time at the Kabardino-Balkarian State University named after Kh. M. Berbekov, and heads a scientific seminar on number theory.
Continuing the work of Professor A. V. Malyshev, he was the scientific supervisor of four postgraduate students on candidate dissertations.