A 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; this problem 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 polynomials with real coefficients endowed with the superposition operation for this system we study the problem of closed classes (structure, basis, number of finite and infinite closed classes).
Importance of the problem of closed classes is ensured by the fact that completeness problem can frequently be solved with the help of (maximal) closed classes.
The main results concerning the functional system of polynomials with real coefficients presented in our paper are the following:
1. all finite closed classes are described explicitly ;
2. the number of finite closed classes, infinite closed classes and all closed classes is found ;
3. the problem of bases of closed classes is studied, namely, it is established that there exist
closed classes with a finite basis, there exist closed classes with an infinite basis, and there
exist closed classes without a basis; explicit examples of the corresponding closed classes
are given;
4. the number of closed classes with a finite basis, the number of closed classes with an infinite
basis and the number of closed classes without a basis are established.

We obtain a complete description of all congruences of a free unar with arbitrary set of free generators. Namely, every congruence is characterized uniquely by a collection of parameters which are either non-negative integers or the symbol ∞; the restrictions on the parameters are formulated.

In the rectangle Ω = {(𝑥, 𝑡) | 0 < 𝑥 < 1, 0 < 𝑡 < 𝑇} we consider an initial-boundary value
problem for a singularly perturbed parabolic equation

$$𝜀2(︂𝑎^2((𝜕^2)𝑢/𝜕𝑥^2)−𝜕𝑢/𝜕𝑡)︂= 𝐹(𝑢, 𝑥, 𝑡, 𝜀), (𝑥, 𝑡) ∈ Ω,$$
$$𝑢(𝑥, 0, 𝜀) = 𝜙(𝑥), 0 ⩽ 𝑥 ⩽ 1,$$
$$𝑢(0, 𝑡, 𝜀) = 𝜓1(𝑡), 𝑢(1, 𝑡, 𝜀) = 𝜓2(𝑡), 0 ⩽ 𝑡 ⩽ 𝑇.$$

It is assumed that at the corner points of the rectangle the function 𝐹 with respect to the variable 𝑢 is cubic. To construct the asymptotics of the solution to the problem, the nonlinear method of angular boundary functions is used, which involves the following steps:
1) splitting the area into parts;
2) construction in each subdomain of lower and upper solutions of the problem;
3) continuous joining of the lower and upper solutions on the common boundaries of the subdomains;
4) subsequent smoothing of piecewise continuous lower and upper solutions.
In the present work, we succeeded in constructing barrier functions suitable for the entire region at once. The form of barrier functions is determined using boundary-layer functions that are solutions of ordinary differential equations, as well as taking into account the necessary properties of the desired solutions. As a result, a complete asymptotic expansion of the solution for 𝜀 → 0 is constructed and its uniformity in a closed rectangle is justified.

In paper proved that it is impossible to build an algorithm that allows you to determine from an arbitrary finite task of the group whether it is solvable her positive theory. The specified group property is not Markov, so the fundamental Adyan-Rabin theorem does not apply to it.

In the proposed work we construct a regularized asymptotics for the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis arising from a semiclassical transition in the Schrodinger equation in the coordinate representation. The potential energy profile chosen in the paper leads to a singularity in the spectrum of the limit operator in the form strong the turning point. Based on the ideas of asymptotic integration of problems with an unstable spectrum by S.A. Lomov and A.G. Eliseev, it is indicated how and from what considerations regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for the problem posed is described in detail, and justification of this algorithm and an asymptotic solution of any order with respect to a small parameter is constructed.

In paper, properties of an integrable pseudo-Euclidean analogue of the Kovalevskaya top are studied for the zero level of the additional first Kovalevskaya integral. The class of motions of a classical top under the same condition is also called the first Appelrot class or the Delaunay class. We describe the homeomorphism class of each fiber, the fiberwise homeomorphism classes of the foliation in a neighborhood of each bifurcation fiber (i.e. analogues of Fomenko 2-atoms) and on the two-dimensional intersection of the level 𝐾 = 0 and each nondegenerate symplectic leaf of the Poisson bracket. It is proved that non-compact one-dimensional Liouville fibers, noncritical
bifurcations of compact and non-compact fibers appear in this integrable system. The non-degeneracy problem (in the Bott sense) for all points of the 𝐾 = 0 level is also studied, and it is proved that the critical sets of the of classical Kovalevskaya top and its pseudo-Euclidean analogue coincides.

Many real dynamical systems are characterized by the presence of a coexisting attractors set.
This property of systems is called multistability. In multistable systems, a sudden transition to unwanted or unknown attractors can occur. Such a transition can lead to catastrophic events. It turned out that multistability is also associated with the emergence of unpredictable attractors, which are called hidden attractors. One of the defining reasons for studying multistable chaotic systems with different characteristics is a wide range of their potential engineering applications - synchronization of the receiver and transmitter, masking and recovery of messages, noise filtering, recovery of information signals, as well as the development of decoding and coding
algorithms that allow you to present an arbitrary digital message through the symbolic dynamics of a chaotic system.
This paper proposes not only a mathematical model of a secure communication scheme based on adaptive synchronization between a pair of identical megastable systems with a 2-D band of hidden chaotic attractors, but also its numerical simulation using the MATLAB & Simulink environment. The use of synchronization in communication systems is of fundamental importance, since it forces systems to simultaneously output the same output data and, in turn, leads to accurate restoration of information signals. Meanwhile, on the receiver side, information can be successfully recovered using adaptive technology. The presented method is stable with respect to various levels of additive white Gaussian noise. The scheme used for synchronization made it possible to overcome the well-known difficulties presented in the works of a number of specialists that arise in the problem of synchronizing in the case of multistability and coexistence of hidden oscillations, with the wrong choice of the form of the control signal.
Numerical simulations are given to verify the feasibility of proposed synchronization and better performance of image encryption technique in terms of histogram, robustness to noise and visual imperceptibility. Three types of masked messages (text, grayscale image and audio signal) are considered as test examples.

In paper, an algorithm is proposed for constructing isospectral and partially-isospectral Sturm–Liouville boundary value problems on the finite segment.

It has been proved that there is left-invariant contact metric structure (𝜂, 𝜉, 𝜙, 𝑔) whose Riemannian metric is different from the metric of the direct product on the group model of the real extension of the Lobachevsky plane H^2 × R. The restriction of the metric 𝑔 to the contact distribution is the metric of the Lobachevsky plane and, together with a completely nonholonomic contact distribution, defines a sub-Riemann structure on H^2 × R.
The found almost contact metric structure is normal and therefore Sasakian. The lie group of automorphisms of this structure has maximum dimension. The basis vector fields of its Lie algebra are found. In addition to the Levi-Civita connection ∇, we consider a contact metric
connection ˜∇ with skew-symmetric torsion, which, like the Levi-Civita connection, is also invariant under the automorphism group. The structure tensors 𝜂, 𝜉, 𝜙, 𝑔, the torsion tensor˜ 𝑆 and the curvature tensor ˜𝑅
of a given connection are covariantly constant. The curvature tensor ˜𝑅 of the connection ˜∇ has the necessary properties to introduce the concept of sectional curvature. It is established that the sectional curvature ˜𝑘 belongs to the numerical segment [−2, 0]. Using the field of orthonormal frames adapted to the contact distribution, the coefficients of the truncated connection and the differential equations of its geodesics are found. It has been proved that the contact geodesics of the connections ∇ and ˜∇ coincide with the geodesics of truncated connection, that is, both connections are compatible with the contact distribution.
This means that there is only one contact geodesic through each point in each contact direction.

We consider entire functions that are even canonical products of zero genus, all roots of which are located on the real axis.We study the question of lower bound the minimum modulus of such functions on the circle in terms of some negative power of the maximum modulus on the same circle, when the radius of the circle runs through segments with a constant ratio of ends. In 2002 A. M. Gaisin, correcting the erroneous reasoning of M. A. Evgrafov from the book «Asymptotic estimates and entire functions», proved that for each function of the class under consideration there exists a sequence of circles, whose radii tend to infinity, the ratio of the subsequent radius
to the previous one is less than 4, and these circles are such that on each of them the minimum modulus of the function exceeds the −20-th power of the maximum of its modulus. This result is strengthened by us in three directions. First, the exponent −20 has been replaced by −2.
Secondly, we proved that the radii of the circles on which the minimum modulus of the function exceeds the −2-th power maximum of its modulus occur on every interval whose end ratio is 3. Thirdly, we found out that the discussed inequality is true for the functions of the class under study «on average». The latter means that if we take the logarithm of the product of the minimum modulus of a function on a circle and the square of its maximum modulus, divide by the cube of the radius and integrate over all radii belonging to an arbitrary segment with an end ratio of 3, it will be a positive value.

In this paper we prove in detail a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension 4. We also present a new proof of the criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry in dimension 4.
As a multidimensional generalization of continued fractions, we consider Klein polyhedra.

Exact inequalities are found between the best polynomial approximation of functions analytics in the disk 𝑈𝑅 :={︀𝑧 ∈ C, |𝑧| < 𝑅}︀, 𝑅 ⩾ 1 and the averaged modulus of continuity angular boundary values of the 𝑚th order derivatives. For the class 𝑊(𝑚) 𝑞,𝑅 (𝑚 ∈ Z+, 1 ⩽ 𝑞 ⩽ ∞, 𝑅 ⩾ 1) of functions 𝑓 ∈ 𝐻(𝑚) 𝑞,𝑅 whose 𝑚-order derivatives 𝑓(𝑚) belong to the Hardy space 𝐻𝑞,𝑅 and satisfy the condition ‖𝑓(𝑚)‖𝑞,𝑅 ⩽ 1, the exact values of the upper bounds of the best approximations are calculated. Moreover, for the class 𝑊(𝑚) 𝑞,𝑅 (Φ), consisting of all functions 𝑓 ∈ 𝐻(𝑚) 𝑞,𝑅 , for which any 𝑘 ∈ N, 𝑚 ∈ Z+, 𝑘 > 𝑚 the averaged moduli of continuity of the boundary values of the 𝑚th order derivative 𝑓(𝑚), dominated in the system of points {𝜋/𝑘}𝑘∈N
by the given function Φ, satisfy the condition

$$∫︁(0, 𝜋/𝑘) 𝜔(︀𝑓^(𝑚), 𝑡)︀_(𝑞,𝑅) 𝑑𝑡 ⩽ Φ(𝜋/𝑘),$$

the exact values of the Kolmogorov and Bernstein 𝑛-widths are calculated in the norm of the space 𝐻𝑞 (1 ⩽ 𝑞 ⩽ ∞).
The results obtained generalize some results of L.V.Taikov on classes of analytic functions in a circle of radius 𝑅 ⩾ 1.

The article presents a new empirical mathematical model of the dynamics of change in the coefficient of friction of a polymer on steel in a vacuum during ion bombardment, which contains a number of new tribological characteristics that allow a more detailed characterization of the frictional interaction. The validity of the developed mathematical model for describing the friction of the AMAN material on steel under a variable set of external conditions is shown.

The article deals with the problem of stress concentration in an elastic layered plane with an elliptical cutout. The phenomenon is investigated using the concept of stress concentration tensor. Two levels of concentration are studied: because of the layering and because of the cutout.
Formulas for the stress concentration tensor components are given separately in the case of an infinite layered plane (first level), as well as in the case of a homogeneous anisotropic plane with an elliptical cutout (second level). Stress concentration tensor in a layered plane with It is represented as a product of concentration tensors at the first and second levels. Approximate formulas for the components of the concentration tensor are given. The case of the coincidence of the orientation of the layers and the main axes of the elliptical hole is considered in detail.
In this case, the concentration coefficients at characteristic points are calculated, graphs of the dependence of these coefficients on the ratio of the elastic modulus of the layers are given. In addition, a numerical solution of the problem was carried out using a finite element analysis
package. The obtained analytical and numerical results are consistent with good accuracy.

The article focuses on the history of mathematical education in Russia in the second half of the 18th century. The authors’ main research topic is the analysis of the four-volume “Theoretical and Practical Course of Pure Mathematics” by E.D. Voityakhovsky and its place in Russian mathematical education. This book is interesting because the author attempted to create the most comprehensive textbook, in which the integrity of theory and practice played a special role. Although this textbook was primarily intended for military schools, it presented the basic concepts and formulations of general mathematics as a science, as well as solutions to a large number of applied problems. Along with the textbooks by L.F. Magnitsky, S.Ya. Rumovsky, M.E. Golovin, Voityakhovsky’s “Course of Pure Mathematics” became a starting point and an ideal methodological platform for the formation of subsequent mathematics courses. Thanks to these works, an educational basis was created for serious mathematical education, essentially combining the experience of creating all the textbooks of the 18th century.

In paper discusses a methodical approach to the study of the elements of the artificial intelligence system in the school course of informatics. This topic is relevant in today’s world.
The article presents the structure and content of intelligent systems and technologies as a new section of the general education course in informatics. AI is explored in various ways, including a detailed description of AI as a subject and as a learning tool. The formation of
students’ digital skills occurs through the creation and use of intelligent algorithms. The use of virtual and augmented reality technologies in the educational process is also considered.
Particular attention is paid to the use of augmented reality technology on platforms such as Argin and Metaverse Studio. The process of teaching the elements of AI in the school course of computer science is considered variably, corresponding to the direction of the profile training of students. A system of tasks with different levels of difficulty is used as a means to develop digital skills. A three-level complexity of tasks is proposed, where students, depending on the level of
preparedness, choose the task that suits them. The article provides sample tasks with solutions for the profile class of the IT direction. They are given from the field of an expert system, a database analysis system, modeling, as well as a search task for a suitable piece of land that meets the required conditions on a geographic information systems platform. The paper notes that in the school course of computer science it is necessary to include a separate section on
intellectual technology and systems, to modernize the criteria for achieving educational results in teaching computer science, corresponding to profile training, and to introduce new criteria
for the effectiveness of training.

In the article the problem of a symmetric stationary cavitation flow around a wedge by an infinite flow of ideal incompressible weightless fluid in the presence of a given intensity point effluent located at the top of the wedge is considered.
To schematize the flow in the aft part of the cavity the Efros scheme with a return stream going to the second sheet of the Riemannian surface is used.
The exact solution of the problem is constructed by displaying the areas of change complex potential and complex flow velocity per area change of the auxiliary parametric variable.
A complete parametric analysis of the problem has been carried out.
For a wide range values of the cavitation number, dimensionless flow rate and angle wedge solution, the shape and dimensions of the cavitation cavity are found, and See also the values of the drag coefficient.
The shape and dimensions of the cavitation cavity and also the values of the resistance coefficient are found for a wide range of cavitation number values, dimensionless consumption of effluent and opening angle of the wedge.

Since the beginning of the XIX century in European countries (Germany, France, Belgium, Austria) astronomers, engineers, mechanics invented, created and improved gyroscopes. The practical demand for gyroscope devices has been significant, but there has not yet been a specific theory of the gyroscope. The foundation of the theory was laid by Euler, developed by Lagrange, and continued by Poisson. On the other hand, in the 19th century in the works of
Jacobi, Abel, Weierstrass, the theory of elliptic functions was created and began to develop. Based on this theory, K. Jacobi and O.I. Somov created a special theory of the gyroscope.

Two-dimensional spectral problems for hyperbolic equations are well studied, and their multidimensional analogs, as far as the author knows, have been little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of
a fundamental nature, since the very attractive and convenient method of singular integral equations used for two-dimensional problems cannot be used here due to the absence of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. These functions have an important application in mathematical and theoretical physics, and in the theory of multidimensional singular equations. In the cylindrical domain of Euclidean space for a class of multidimensional hyperbolic equations, the Poincar? spectral problem is considered. The solution is sought as an expansion in multidimensional spherical functions. The existence and uniqueness theorems are proved. The conditions for the unique solvability of the problem, which significantly depend on the height of the cylinder, are obtained.

We consider the algebra 𝐴0(𝑋) of polynomial functions on a simplicial complex 𝑋. The algebra 𝐴0(𝑋) is the 0th component of Sullivan’s dg-algebra 𝐴∙(𝑋) of polynomial forms on 𝑋.
All algebras are over an arbitrary field 𝑘 of characteristic 0.
Our main interest lies in computing the de Rham cohomology of the algebra 𝐴0(𝑋), that is, the cohomology of the universal dg-algebra Ω∙
𝐴0(𝑋). There is a canonical morphism of dgalgebras 𝑃 : Ω∙
𝐴0(𝑋) → 𝐴∙(𝑋). We prove that 𝑃 is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra 𝐴0(𝑋) is canonically isomorphic to the cohomology of the simplicial complex 𝑋 with coefficients in 𝑘. Moreover, for 𝑘 = Q the dg-algebra Ω∙
𝐴0(𝑋) is a model of the simplicial complex 𝑋 in the sense of rational homotopy theory. Our result shows that for the algebra 𝐴0(𝑋) the statement of Grothendieck’s comparison theorem holds (proved
by him for smooth algebras).
In order to prove the statement we consider ˇCech resolution associated to the cover of the simplicial complex by the stars of the vertices.
Earlier, Kan–Miller proved that the morphism 𝑃 is surjective and gave a description of its kernel. Another description of the kernel was given by Sullivan and F´elix–Jessup–Parent.

The paper solves the problem of entering into the Artin group with a woody structure.

Voronoi obtained three results for perfect forms. First, he proved that the form corresponding to the closest packing is perfect. Secondly, he established that there are a finite number of perfect forms from a given number of variables. And most importantly, thirdly, Voronoi proposed a method for finding all perfect forms. This method relies on the so-called perfect polyhedron, a highly complex multidimensional polyhedron introduced by Voronoi. In principle, having found all perfect forms by the Voronoi method, one can calculate the densities for a finite number of corresponding packings and single out those that correspond to the maximum value.
The classical Voronoi problem of finding perfect forms, closely related to Hermite’s well-known problem of arithmetic minima of positive quadratic forms. They also appeared in the works of S.L. Sobolev and Kh.M. Shadimetov in connection with the construction of lattice optimal cubature formulas. In this paper, we propose an improved Voronoi algorithm for calculating the Voronoi neighborhood of a perfect form in many variables, and using this algorithm, the
Voronoi neighborhood of the main perfect form in five variables is calculated.

By using a functional defined on a pair of the assorted represention spaces of the connected subgroup of the proper Lorentz group, a formula for the Buschman–Erdelyi transform of the Legendre function (up to a factor) is derived. Also a formula for the Mehler–Fock transform of the Legendre function of an inverse argument is obtained. Moreover, a generalization of one known formula for the Mehler–Fock transform is derived.

The article examines the following problem. Let there be two subsets of the set of natural numbers, which we denote as 𝐴 and 𝐵. Let it also be additionally known that the asymptotic density of these sets 𝐴,𝐵 is 1. We define the set of natural numbers that are representable
as the product of these sets 𝐴𝐵, that is, such elements 𝑎𝑏, where 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵. We study the properties of this subset of products in the set of all natural numbers. The authors S.
Bettin, D. Koukoulopoulos and C. Sanna in the article [1] proved, among other things, that the density of the set 𝐴𝐵 is also equal to 1. Moreover, a quantitative version of this statement was derived, namely, an estimate was obtained for the set N ∖ 𝐴𝐵, which we will denote by 𝐴𝐵. Namely, by these authors, in the case when quantitative upper bounds are known for 𝐴 ∩ [1, 𝑥] = 𝛼(𝑥)𝑥,𝐵 ∩ [1, 𝑥] = 𝛽(𝑥)𝑥, 𝛼(𝑥), 𝛽(𝑥) = 𝑂(1/(log 𝑥)𝑎), 𝑥 → ∞ the upper bound on the set 𝐴𝐵 ∩ [1, 𝑥] is also derived. In this paper, we study the case when 𝛼, 𝛽 tend to zero
slower than in the above case and somewhat refine the upper bound on the set 𝐴𝐵 ∩ [1, 𝑥]. In this paper we consider the case of 𝛼(𝑥), 𝛽(𝑥) = 𝑂(︀ 1/(log log 𝑥)𝑎)︀ for some fixed 𝑎 > 1. We borrow approaches, arguments and proof scheme from the mentioned work of three authors S. Bettin, D. Koukoulopoulos and C. Sanna[1].

The outstanding Russian mathematician Alexei Nikolaevich Parshin (1942–2022) was also a deep thinker and an original historian of science. He summed up his ideas in a number of articles published in various journals and collections, the main of which formed the content of two books: «The Way. Mathematics and Other Worlds» (2002), «Ladder of Reflections« (2022).
An attempt is made to give an analysis of the main ideas of A.N.Parshin.

The article gives some recollections of the authors about the initial stage of mathematical education of two remarkable graduates of the Boarding School of the Moscow State University 18: Misha Boshernitsan and Igor Krichever, some facts related to the anniversary of the Boarding School FMSH 18 at Moscow State University are given. Some photos from the photo archives of I. N. Naida and Yu. V. Matiyasevich are also given. The authors did not touch on the scientific work of I. M. Krichever and M. D. Boshernitsan, since there are excellent reviews on this topic in the literature.

The article is devoted to the 80-th anniversary of Sergey Sergeevich Demidov – the outstanding Russian historian of mathematics, president of the International Academy of the History of Science.