In the paper the formulation of problems are given and the contribution in their solution of the outstanding mathematicians of G. I. Arkhipov and S. M. Voronin are presented.
At the base of the paper two papers, which are written in the coonection with jubileum data of scientists, are put.

Data analysis is a complex and multi-dimensional concept. It is explained by both the objective complexity of the data itself and the subjective nature of the expert analyzing them.
Therefore, adequate formalization of this requires a completely new apparatus, on the one hand, capable of overcoming the objective complexity of data (irregularity and inaccuracy),
and, on the other hand, the vague nature of expert judgment. The development of Discrete Mathematical Analysis (DMA) is an important step in this direction. DMA is highly expert
oriented and occupies an intermediate position in data analysis between strict mathematical methods (statistical analysis, TFA, etc.) and soft combinatorial methods (simulation modeling,
neural networks, etc.).
This paper proposes new mathematical constructions of regression derivatives and regression integrals for discrete time series defined in general on an irregular grid. An important role in their study is played by the recently created by the authors projection method for solving systems of linear algebraic equations described at the end of this paper.
The obtained constructions of regression derivatives and regression integrals have a hierarchical character in the spirit of wavelet and fractal analysis.
The results of this work define another direction in further studies, namely, the propagation of regression differentiation and regression integration in finite mathematics under scenarios of the classical one.

In this paper we consider some questions of the theory and practice of mean first passage time quasi-metric, a generalized metric structure closely related to ergodic homogeneous Markov
chains. In particular, we consider the structure and properties of mean first passage time quasimetrics based on several graph models, including simple cycles, simple paths and their oriented
analogues.
The introduction contains the history of the problem and provides an overview of the main ideas and results presented in the article.
The second section gives basic concepts of the theory of Markov chains. In fact, a Markov chain is a mathematical model of some random process describing a sequence of possible events
in which the probability of each event depends only on the state attained in the previous event. This section collects basic definitions needed to consider the role of graph models in
the presentation and study of ergodic homogeneous Markov chains. The Markov chain can be depicted as an oriented weighted graph of transitions whose vertices correspond to the states of the chain and the arcs correspond to the transitions between them. The process will be ergodic if this weighted oriented graph is weakly connected, and the largest common divisor of the lengths of all its cycles is equal to 1. On the other hand, any connected graph can be used as a basis for building a model of the simplest Markov chain: if a vertex 𝑖 has degree 𝑘, all incident edges are converted into arcs with the weights 1/𝑘 . Moreover, in the second section the definition of the mean first passage time for an ergodic homogeneous Markov chain is given. The algorithm of finding the mean first passage time is analyzed in detail by using converging trees of the oriented graph, related to the transition matrix of the ergodic homogeneous Markov chain. At last, a mean first passage time is analyzed as the quasi-metric 𝑚 of mean first passage time
on the vertices 𝑉 = {1, 2, ..., 𝑛} of the oriented graph corresponding to the transition matrix of a given ergodic homogeneous Markov chain: 𝑚(𝑖, 𝑗) is the expected number of steps (arcs) for random wandering on the oriented graph Γ, starting at 𝑖, to reach 𝑗 for the first time. This quasi-metric has a number of important theoretical and applied properties.
The third section deals with the construction and research of mean first passage time quasimetrics for the undirected cycles 𝐶𝑛, 𝑛 ≥ 3. Examples of constructions of mean first passage
time quasi-metrics of undirected cycles for small values of 𝑛 are considered. Illustrations of the ”graphical“ procedure of building the matrix 𝑀 are given. Properties of the resulting generalized
metric structures are analyzed.
In the fourth section, similar considerations for mean first passage time quasi-metrics of the undirected paths 𝑃𝑛, 𝑛 ≥ 2, are represented.
The fifth section deals with the construction and research of mean first passage time quasimetrics for the directed (oriented) cycles 𝐶𝑛, 𝑛 ≥ 3. Examples of constructions of mean first
passage time quasi-metrics of undirected (oriented) cycles for small values of 𝑛 are considered.
Illustrations of the ”graphical“ procedure of building the matrix 𝑀 are given. Properties of the resulting generalized metric structures are analyzed.
In the sixth section, similar considerations for mean first passage time quasi-metrics of the directed (oriented) paths 𝑃𝑛, 𝑛 ≥ 2, are represented.
In the conclusion, the results of the work are summed up and possible directions of further research are outlined.

Quasigroup-based cryptoalgorithms are being actively studied in the framework of theoretic projects; besides that, a number of quasigroup-based algorithms took part in NIST contests
for selection of cryptographic standards. From the viewpoint of security it is highly desirable to use quasigroups without proper subquasigroups (otherwise transformations can degrade).
We propose algorithms that take a quasigroup specified by the Cayley table as the input and decide whether there exist proper subquasigroups or subquasigroups of the order at least 2.
Temporal complexity of the algorithms is optimized at the cost of increased spatial complexity.
We prove bounds on time and memory and analyze the efficiency of software implementations applied to quasigroups of a large order. The results were reported at the XVIII International
Conference «Algebra, Number Theory and Discrete Geometry: modern problems, applications and problems of history».

In this article we consider different approaches for constructing maximal abelian extensions for local and global geometric fields. The Lubin–Tate theory plays key role in the maximal abelian extension construction for local geometric fields. In the case of global geometric fields, Drinfeld modules are of particular interest. In this paper we consider the simpliest special case
of Drinfeld modules for projective line which is called the Carlitz module.
In the introduction, we provide motivation and a brief historical background on the topics covered in the work.
In the first and second sections we provide brief information about Lubin–Tate modules and Carlitz module.
In the third section we present two main results:
• an explicit connection between the local and global field theory in the geometric case for projective line over finite field: it is proved that the extension tower of Carlitz module induces the tower of the Lubin–Tate extensions.
• a connection between Artin maps of extensions of a function field of an arbitrary projective smooth irreducible curve and extensions of completions of local rings at closed points of
this curve.
In the last section we formulate different open problems and interesting directions for further research, which include generalization first result for an arbitrary smooth projective curve over a finite field and consideration Drinfeld modules of higher rank.

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑.
Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences.
We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers.
In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case,
we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.

The full utility of economic activity is investigated in article. In the case of the Cobb- Douglass production function and the economic resource 𝐾(𝑡) = 𝐾0𝑒^(−𝜆𝑡), it is proved that the exponent of 𝜆 that delivers the maximum of total utility is in a certain interval

In extremal problems of the theory of approximation of functions an important role is played be exact inequalities of the value of the best polynomial approximation by means of averaged
values of the modules of continuity of higher orders of the derived functions. In this paper we present an inequality of type Ligun-two-sided estimate for the best weighted approximate
analytic functions in the unit disc from the Bergman space 𝐵_2,𝛾 . The resulting inequalities allow us to establish new connections between the constructive and structural properties of the
functions and for the corresponding classes of functions give an estimate from the top of the widths. The exact values of Bernstein, Kolmogorov, Gelfand, linear and projection n-widths of classes of analytic functions in unit discs defined by modules of continuity of higher orders of the derived functions in the space 𝐵_2,𝛾 averaged with positive weight are calculated

The paper considers the problem of optimizing the operation schedule for multiprocessor systems. The solution to this problem involves the formation of a rigid work schedule, which
determines the rhythm of the processes, but in practice the functioning of systems is influenced by many side factors that make the intervals of work execution random. In the work, a semi-Markov model of the formation of a stochastic schedule in conditions of pair competition is constructed. It is shown that if during the functioning of the system it is possible to execute the
items of the schedule in an arbitrary order, then the evolution of the semi-Markov process follows the Hamiltonian path. It is proved that all possible realizations of Hamiltonian paths form a
complete group of incompatible events. It is noted that, due to the imposition of restrictions on the nature of evolution, the evolution process is not strictly semi-Markov, and therefore
a method of forming a strictly semi-Markov process with a tree structure from the primary model is proposed. Dependences are obtained for calculating the distribution densities and the
probabilities of switching from states of a semi-Markov process to conjugate states, as well as the time of walking from the starting to absorbing states. Using the concept of paired discrete
competition and a distributed penalty, the effectiveness of the choice of a Hamiltonian path by one of the subjects is estimated, taking into account the fact that the algorithm of his opponent’s
behavior is known up to the construction of a semi-Markov model.

Brunn in 1887 formulated a theorem on three parallel sections of a convex body with extreme sections of the same area, but not obtained from each other by a parallel shift, asserting that
the area of the middle section is strictly larger, and correctly proved, as Minkowski noted, that only not less. The elimination of equality, which was still considered the most difficult in the
theorem, has been proved up to the present time by many authors, using serious mathematics.
The article proposes a fundamentally different geometric approach to the proof of the theorem, due to which, for the correct completion of Brunn’s original proof, one can restrict oneself to the elementary means available to schoolchildren, bypassing the difficulties with equality. The proposed reasoning extends to all dimensions, like the theorem itself, as pointed out by Brunn. Let, in the general case, 𝑉𝑛(𝑄) be the 𝑛-dimensional volume of the body 𝑄 ⊂ R𝑛, 𝐿0, 𝐿1 be parallel hyperplanes in R𝑛+1, containing respectively convex bodies 𝑃0, 𝑃1, and 𝐿 is a parallel hyperplane, located strictly between them, and 𝑃 is the intersection of 𝐿 with the convex hull 𝑃0 ∪ 𝑃1. Brunn’s theorem states that if 𝑃1 is not obtained from 𝑃0 by parallel translation and 𝑉𝑛(𝑃1) = 𝑉𝑛(𝑃0) = 𝑣 > 0, then 𝑉𝑛(𝑃) > 𝑣. In 1887, Brunn rigorously proved that 𝑉𝑛(𝑃) > 𝑣 using the effective trick of the division of the volumes 𝑃0, 𝑃1 by a hyperplane in R𝑛+1. In
this article, this is called Brunn cuts. For the strictly inequality 𝑉𝑛(𝑃) > 𝑣, it remained a small "perturbation" go from the body 𝑃1 to another convex body ̃︀ 𝑃1, 𝑉𝑛( ̃︀ 𝑃1) = 𝑣 , so that
𝑉𝑛(𝑃) > 𝑉𝑛( ̃︀ 𝑃), where ̃︀ 𝑃 is a new section in the hyperplane 𝐿 arising after replacing 𝑃1 with ̃︀ 𝑃1. Since 𝑉𝑛( ̃︀ 𝑃) > 𝑣, then 𝑉𝑛(𝑃) > 𝑣. The easiest way is to replace 𝑃1 with ̃︀ 𝑃1 in the case
of convex polytopes 𝑃0, which can approximate convex bodies arbitrarily close. The required replacement of 𝑃1 by ̃︀ 𝑃1 is quite simple, when 𝑛-dimensional simplices act as 𝑃0, into which the
convex polytope can be split by Brunn cuts. Until now, the sufficiently naive natural geometric method outlined above has not been proposed for proving the strict inequality 𝑉𝑛(𝑃) > 𝑣,
as it were head-on, due to the fact that initially the theorem was formulated not for convex polytopes 𝑃0, 𝑃1, but for arbitrary convex bodies. The main reason, according to the author,
lies in the algebraic representation 𝑃 = (1 − 𝑡)𝑃0 + 𝑡𝑃1, where 𝑡 is the ratio of the distance from 𝐿0 to 𝐿 to the distance from 𝐿0 to 𝐿1, 0 < 𝑡 < 1. This leads to the temptation to go over in the proofs of the theorem from R𝑛+1 to R𝑛 and use the equivalent statement of the theorem, assuming 𝐿0 = 𝐿1 = R𝑛. As a result, from the general situation, when 𝐿0 ̸= 𝐿1, passed into the singularity 𝐿0 = 𝐿1, in the conditions of which the possibilities for attracting geometric intuition are significantly reduced and, as a consequence, the possibilities for simpler visual geometric justifications of the inequality 𝑉𝑛(𝑃) > 𝑣 are significantly reduced. This article shows that in the proof of the theorem in an equivalent formulation, on the contrary, the space
R𝑛 should be included in R𝑛+1 and use the original formulation of the theorem, when the main tool of the proof the elementary means are Brunn cuts. For the sake of fairness, it should be
noted that numerous applications of this theorem, obtained by Minkowski and other authors, are connected precisely with its equivalent formulation, with mixed volumes, with algebraic
representations 𝑃 = (1 − 𝑡)𝑃0 + 𝑡𝑃1, called Minkowski sums.

In this paper we consider the properties of the matrix of relative forest accessibility of the oriented cycle and the properties of the matrix of relative forest accessibility of the oriented path.
The introduction contains the history of the problem and provides an overview of the main ideas and results presented in the article.
The second section gives basic concepts of the graph theory and a ”graphical“ representation of the matrix of relative forest accessibility of the digraph Γ: F =(((𝑓𝑖𝑗))nxn)/𝑓 , 𝑖, 𝑗 = 1 . . . 𝑛, where 𝑓𝑖𝑗 is the number of spanning converging rooted forests of the digraph Γ, in which the vertices 𝑖 and 𝑗 belong to the same tree converging to 𝑗, and 𝑓 is the total number spanning converging rooted forests of the digraph Γ.
The third section deals with the construction and research of the matrix of relative forest accessibility of the oriented path 𝑃𝑛, 𝑛 ≥ 2. Examples of constructing the matrix of relative forest accessibility of oriented path for small values of 𝑛 are considered. Illustrations of the ”graphical“ procedure of building the matrix F are given. It is proved that the matrix of relative forest accessibility for the directed path 𝑃𝑛, 𝑛 ≥ 2, is related to the sequence 1, 2, 4, 8, 16, ..., 2𝑛, ... of powers of 2. In other words, the elements of 𝑓𝑖𝑗 that form the matrix are elements of the set
{1, 2, 22, ..., 2𝑛−1} filling the columns of the matrix: the first column consists of sequentially decreasing numbers 2𝑛−1, ..., 2, 1; the second column, starting at 0, contains in the second
place (the intersection with the main diagonal) the number 2𝑛−2, while the following elements are consecutively decreasing numbers 2𝑛−3, ..., 2, 1; the third column, containing zeros in two
positions above the main diagonal, contains in the third place (the intersection with the main diagonal) the number 2𝑛−2, while the following elements are sequentially decreasing numbers
2𝑛−3, ..., 2, etc. The value of 𝑓 is equal to 2𝑛−1.
In the fourth section, similar considerations for matrix of relative forest accessibility of the oriented cycle 𝐶𝑛, 𝑛 ≥ 3, are represented.
In the conclusion, the results of the work are summed up.

Methods of symbolic dynamics play an essential role in the study of combinatorial properties of words, problems in number theory and the theory of dynamical systems. The paper is devoted
to the problems of combinatorics on words, its applications in algebra and dynamical systems.
Section 2.1 considers the one-dimensional case using the key example of Sturm’s words. The proof of the criterion for substitutionality of Sturm palindromes using the Rauzy induction is given, the case of one-dimensional facordynamics is considered. Section 2.2 discusses the shift of the torus and the Rauzy fractal that generates the word Tribonacci. The relationship between the periodicity of Rauzy’s schemes and the substitutionality of the word generated by this system is discussed. The implementation of the word Tribonacci through the rearrangement of line segments is given. An approach to the Pisot hypothesis is outlined. Section 2.3 talks about
unipotent torus transformations and billiards in polygons.
Chapter 3 talks about normal forms and the growth of groups and algebras. Chapter 4 is devoted to Rosie graphs, Gr¨obner bases and co-growth, and algebraic applications. Section 4.1
discusses the results in the combinatorics of multilinear words developed by V. N. Latyshev and the problems he posed. Section 4.2 talks about finitely defined objects and the problems of
controlling the relationships that define them. Section 4.3 describes some monomial algebras in terms of uniformly recurrent words.
Chapter 5 deals with the problem of height and normal forms.

statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔
∼=Ω𝑝1 ⊕. . .⊕Ω𝑝𝑛, a subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Here are
appropriate theorems for numbers not only like 𝛼 =∞Σ︀𝑗=0𝑎𝑗𝑔𝑟𝑗 where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded. But, for numbers 𝑓(𝛼), where 𝑓(𝑧) =∞Σ︀𝑗=0𝑐𝑗𝑧𝑗 ∈ Z𝑔[[𝑧]].
Furthermore, let ̂︀ Q∼=Π︀𝑝Q𝑝 be the ring of polyadic numbers, then, the article takes a look at̂︀ Ω=Π︀𝑝Ω𝑝, there are similar results for numbers like 𝑓(𝛼), where 𝑓(𝑧) =∞Σ︀𝑗=0𝑐𝑗𝑧𝑗 ∈ ̂︀ Z[[𝑧]],𝛼 =∞Σ︀𝑘=1
𝑎𝑘𝑔𝑟𝑘 , 𝑎𝑘 ∈ Z𝑔, 𝑔 = (𝑝1, . . . , 𝑝𝑛, . . .)

In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras with one operator and the main symmetric operation. A ternary operation 𝑑(𝑥, 𝑦, 𝑧) satisfying identities 𝑑(𝑥, 𝑦, 𝑦) = 𝑑(𝑦, 𝑦, 𝑥) = 𝑑(𝑦, 𝑥, 𝑦) = 𝑥 is called a minority operation. The symmetric operation is a minority operation defined by specific way. An algebra 𝐴 is called subdirectly irreducible if 𝐴 has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main one which can contain arbitrary operations, and the additional one consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one that permutable with main operations. A lattice 𝐿 with zero is called atomic if any element of 𝐿 contains some atom. A lattice 𝐿 with zero is called atomistic if any nonzero element of 𝐿 is a join of some atom set.
It shown that congruence lattices of algebras with one operator and main symmetric operation are atomic. The structure of atoms in the congruence lattices of algebras in given class is described. The full describe of subdirectly irreducible algebras and of algebras with an atomistic congruence lattice in given class is obtained.

In this paper we introduce the concept of Rees closure for subalgebras of universal algebras.
We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a
Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains
𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ on the lattice of subalgebras of universal algebra. Consequently, in the general case, a lattice
of Rees subalgebras is not a sublattice of lattice of subalgebras.
A non-one-element universal algebra 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 is trivial. We characterize Rees simple algebras in terms of Rees closure.
Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. We described Rees simple algebras in some subclasses of the class of algebras with one operator and a ternary basic operation. For algebras from these classes, the structure of lattice of Rees subalgebras is described. Necessary and sufficient conditions for the lattice of Rees subalgebras of algebras from these classes to be a chain are obtained.

The paper considers a linear Diophantine equation with six variables. Its solution is constructed as a shifted incomplete five-dimensional integer lattice in a six-dimensional space.
The basis of this lattice is constructed.
An algorithmic solution for finding all its solutions from a given six-dimensional integer parallelepiped is given. For this purpose, a new basis for this incomplete five-dimensional lattice was constructed, which allowed us to write an effective program for finding all sets that satisfy a given Diophantine equation and belong to a given rectangular parallelepiped.
As a result of the proposed algorithm implemented in the Mathcad system, it was shown that out of the total number of 10182290760 integer points lying in a given parallelepiped, only
7822045 satisfy the given Diophantine equation. Thus, the total search was reduced by 1301.7 times.
The article considers the relationship between shifted lattices and integer programming problems. It is shown how it is possible to construct bases of incomplete integer lattices, which make it possible to reduce a complete search over the points of an s-dimensional rectangular parallelepiped to a search over the points of a shifted incomplete lattice lying in this parallelepiped.
Some applications of this Diophantine equation in technical issues related to the solution of an applied mechanical engineering problem in the field of measuring tool design, in particular, sets of end length measures, are considered.
The article reflects the iterative nature of the refinement of the mathematical model of this applied problem. After the first model adjustment, the number of sets decreased by another
193.237 times, and after the second model adjustment, the total reduction of sets suitable for subsequent optimization became 581114.6 times.
In conclusion, the directions of further research and possible application of the ideas of Hopfield neural networks and machine learning for the implementation of the selection of optimal
solutions are indicated.

We study infinite linear independence of polyadic numbers

$$𝑓0(𝜆) =
∞Σ︁
𝑛=0
(𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) =
∞Σ︁
𝑛=0
(𝜆 + 1)𝑛𝜆𝑛$$,

where 𝜆 is a certain polyadic Liouville number. The series considered converge in any field
Q𝑝 . Here (𝛾)𝑛 denotes Pochhammer symbol, i.e. (𝛾)0 = 1 , and for 𝑛 ≥ 1 we have(𝛾)𝑛 =
𝛾(𝛾 + 1)...(𝛾 + 𝑛 − 1). The result extends the previous author’s result on the polyadic numbers

$$𝑓0(1) =
∞Σ︁
𝑛=0
(𝜆)𝑛, 𝑓1(1) =
∞Σ︁
𝑛=0
(𝜆 + 1)𝑛$$, 

The values of generalized hypergeometric series are the subject of numerous studies. If the parameters of the series are rational numbers, then they come either in the class of 𝐸 (if these
series are entire functions) or the class of 𝐺 functions (if they have a finite non-zero radius of convergence) or to the class of 𝐹− series ( in the case of zero radius of convergence in the
field of complex numbers, however, they converge in the fields of 𝑝− adic numbers). In all these cases, the Siegel-Shidlovsky method and its generalizations are applicable. If the parameters of the series contain algebraic irrational numbers, then the study of their arithmetic properties is based on the Hermite-Pade approximations.
In this case, the parameter is a transcendental number. It should be noted that earlier A. I.
Galochkin proved the algebraic independence of the values of 𝐸−functions at a point that is a real Liouville number. We also mention the published works of E. Yu. Yudenkova on the values
of 𝐹− series in polyadic Liouville points. We especially note that in this paper we consider the values in the polyadic transcendental point of hypergeometric series, the parameter of which is the polyadic transcendental (Liouville) number.
Note that earlier A.I. Galochkin proved the algebraic independence of values of 𝐸−functions at points which are real Liouville numbers.We also mention submitted papers (E.Yu.Yudenkova) about the arithmetic properties of values of 𝐹−series at polyadic Liouville numbers. It should be specially mentioned that here we study the values of hypergeometric series with a parameter which is a polyadic Liouville number.

Rauzy introduced a fractal set associated with the two-dimensional toric shift by the vector (𝛽−1, 𝛽−2), where 𝛽 is the real root of the equation 𝛽3 = 𝛽2 + 𝛽 + 1 and showed that this
fractal is divided into three fractals that are bounded remainder sets with respect to a given toric shift. The introduced set was named as Rauzy fractal. It obtains many applications in the
combinatorics of words, geometry, theory of dynamical systems and number theory. Later, an infinite sequence of tilings of 𝑑 − 1-dimensional Rauzy fractals associated with algebraic Pisot units of the degree 𝑑 into fractal sets of 𝑑 types were introduced. Each subsequent tiling is a subdivision of the previous one. These tilings are closely related to some irrational toric shifts and allowed to obtain new examples of bounded remainder sets for these shifts, and also to get some results on self-similarity of shift orbits.
In this paper, we continue the study of generalized Rauzy tilings related to Pisot numbers. A new approach to definition of Rauzy fractals and Rauzy tilings based on expansions of natural
numbers on linear recurrence sequences is proposed. This allows to improve the results on the connection of Rauzy tilings and bounded remainder sets and to show that the corresponding
estimate of the remainder term is independent on the tiling order. The geometrization theorem for linear recurrence sequences is proved. It states that the natural number has a given endpoint of the greedy expansion on the linear recurrence sequence if and only if the corresponding point of the orbit of toric shift belongs to some set, which is the union of the tiles of the Rauzy tiling. Some number-theoretic applications of this result is obtained.
In conclusion, some open problems related to generalized Rauzy tilings are formulated.

This paper proves infinite linear and algebraic independence of the values of 𝐹-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. 𝐹-
series have form 𝑓𝑛 =Σ︀∞𝑛=0 𝑎𝑛𝑛!𝑧𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties.
These series converge in the field Q𝑝 of 𝑝-adic numbers and their algebraic extensions K𝑣.
Polyadic number is a series of the form Σ︀∞𝑛=0 𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞|< 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼−𝐴|𝑝 < 𝐴−𝐷. Infinite linear (algebraic) independence means that for any nonzero linear form
(any nonzero polynomial) there are infinitely many primes 𝑝 and valuations 𝑣 extending 𝑝-adic valuation to an algebraic number field K with the following property: the result of substitution
in the considered linear form (polynomial) of the values of 𝐹 - of series instead of variables is a nonzero element of the field.
Previously, only the existence of at least one prime number 𝑝 with the properties listed above was proved.

The article discusses the problem of teaching computer science, mathematics and logic in modern education, in particular, in a secondary general education school as inherited from the problems of the existing relationships between the sciences of computer science and mathematics, as well as generated by the prevailing forms and practice of teaching in the system
of general education.
The methodological aspects of teaching computer science, mathematics and logic are considered inherent in them and applied to them in a differentiated form:
- a meaningful approach inherent in computer science and applied in mathematics and logic;
- a formal approach inherent in mathematics and used in computer science and logic;
- a socio-cultural approach characteristic of the triad ”computer science-mathematics-logic”.
Against the background of the existing correspondence of the selected approaches (or the measure of this correspondence) to each local and independent element of the triad, it is assumed
that there is an identification of the regular relations between the subject and the subject of education, the principles of organizing the processes of implementing methodological knowledge
based on these regularities.
For our study, it is important that the methodology and methodology of each differentiated system of the triad should contain everything that is ”necessary and sufficient” for the
functioning of the educational system of the corresponding level. In this aspect, the article notes that mathematics studies form, computer science - form and content. The study of objects only
in form leads to formalism in the information sphere (which is unacceptable), therefore, possible integration must be combined with the necessary differentiation. Since the processes of the development of knowledge of science, its methodology and teachingmethods are interrelated and interdependent, it is unconditionally interesting to reveal the potential (possibilities) of intersubject and metasubject connections and relations of the triad under consideration as close, but not homogeneous educational systems that have their own subject. specificity of the content, methods and forms of training. In this perspective, possible ways of solving the problems that have arisen are proposed in the context of changes in the school curricula of the content of information and mathematics education under the influence of the requirements of the new federal state educational standards and the structure of subject teaching in computer science and mathematics, ensuring their efficiency and effectiveness.

setting up a complete factor experiment on the average particle size of the obtained electroerosive materials. The parameters of the EED system operation were selected as factors:
the voltage at the electrodes, the capacity of the discharge capacitors, and the pulse repetition rate. The optimal parameters of the installation were determined for two working media: distilled water and lighting kerosene. According to a series of experiments to determine the maximum value of the parameter optimization in EDM average size of the particles that made up: for water — 51,38 𝜇m when capacity bit of 65.5 𝜇f capacitors, the voltage across the electrodes 210, the pulse repetition frequency of 230 Hz; for kerosene — 61,73 𝜇m when capacity bit of 65.5 𝜇f
capacitors, the voltage across the electrodes 160 and the pulse repetition frequency of 205 Hz.

In calculating microwaves - amplifiers and generators based on the emission of highly relativistic electronic beams in limited plasma, you have to face a number of difficulties, one of
which is the correct setting of radiation conditions. Since there is no one-size-fits-all algorithm to overcome these challenges, you have to use a variety of simplistic assumptions and corresponding models. For example, plasma generators typically assumed that the width of the spectrum of vibrations generated was small and the central frequency corresponded to the frequency of accurate Cherenkov resonance. However, these assumptions were justified only for beams with currents of smaller maximum vacuum current. It was for such beams, using the method slowly- changing amplitude and introducing a constant ratio of plasma wave reflection from the radiating mouthpiece, it was possible to create a non-stationary theory of plasma microwave - a generator. However, the possibility of applying this approach is very limited, as it does not use
a strict form of radiation conditions. This is due to the fact that known boundary conditions of radiation were developed to describe only established vibrational processes. Currently, there are various options for generalizing these boundary conditions for a non-stationary case, but all of them are not without certain shortcomings. One of the most successful variants of the radiation boundary conditions for the complete non-stationary system of Maxwell - Vlasova is, in our view, the unsteady analogue of the partial radiation conditions. However, the practical implementation of these conditions also faces serious mathematical difficulties. The question of the feasibity and
effectiveness of these radiation conditions in relation to a specific electrodynamic system is being considered in this paper.

Aim. The aim of the work is to study the history of ideas about coarseness (structural stability), which is not only one of the most important concepts of the theory of nonlinear systems, but lies at the heart of our worldview. To nowаdays, structural stability has been considered in historical terms only fragmentarily (mainly in connection with the Andronov school) and has not been the subject of a consistent historical study. Method. The study is
based on an analysis of original works, historical and scientific literature with the involvement of the memories of participants in the events described. Results. In Andronov’s school, in the
context of applied problems, two-dimensional systems for which structural stability is a typical property have been exhaustively studied. Since the late 1950s there is a shift in research on
structural stability in the context of applied problems towards the theory of dynamical systems.
M. Peixoto studied structural stability on closed two-dimensional manifolds and proved the density of such systems. S. Smale hypothesized the existence of structurally stable systems in
the multidimensional case (𝑛 > 3) . Such systems exist (Morse-Smale systems), but he himself established their atypicality, they do not constitute a dense set. Multidimensional systems are
characterized by complex behavior; an example of such a system (Smale’s horseshoe) was built. The study of systems with complex behavior stimulated the development of hyperbolic theory.
Discussion. Structural stability was an important factor in the discovery of the complex behavior of dynamical systems already in the three-dimensional case; it continues to play a
significant role in the modern theory of dynamical systems. Structural stability is of general scientific importance, played a key role in the construction of catastrophe theory, it went beyond the framework of the theory of dynamical systems and mathematics itself, penetrates into other areas of science, including the humanitarian sphere.

The article explains the term "powder metallurgy"and defines the main differences from the classical methods of metallurgy. Historical aspects of the development of powder metallurgy
technologies since ancient times are described. The main technological methods of powder metallurgy and their application for the production of modern products are presented. The main areas of application of various types of products of modern powder metallurgy are listed

The article considers the influence of the quality of the initial charge on the structure and physical and mechanical properties of the structural medium-alloy steel 30KhGSA. It was found
that the 30KhGSA steel (smelters No. 3-4) cast in cast-iron molds (ingot weight 2.6 t) has a significant chemical heterogeneity and liquation in carbon and the main alloying elements (Cr
and Mn), and in the melting on the semi-product of the boiling slag layer, the liquation is more pronounced. Starting from the tempering temperature of 500∘C, the structure of both melts (the
fluidized slag layer and the ordinary metallized charge) becomes more uniform and the mechanical properties are aligned. Microstructural studies have established that with an increase in
the annealing temperature at a single exposure, the chemical inhomogeneity decreases, but not significantly. Starting from 1000 ∘C, when the annealing temperature increases, the steel
grain diversity and the growth of austenitic grains appear, especially in the melting on the semi-product of the boiling slag layer. Different grain sizes adversely affect the mechanical
properties of steel: grain enlargement reduces the resistance of steel to brittle fracture, worsens the service characteristics of the metal. Elimination of chemical and structural heterogeneity
can be achieved by conducting a ho-mogenizing annealing (annealing at 950 ∘C for 8 hours).
An increase in the annealing temperature leads to an intensive grain growth, especially in steel smelted on the original charge. It is shown that 30XGSA steel, which is melted on a semi-product
of a boiling slag layer, in an improved state with equal hardness and strength characteristics, has increased plastic properties and especially impact strength in comparison with steel on a
conventional metallized charge. Apparently, this should also determine the higher operating characteristics of steel obtained with the use of pure primary charge.

In article the problem of the interaction of two flat cord charges at explosion on the ground surface is considered. The impulse-hydrodynamic solid-liquid model is used. The action of
charges on the medium by the pressure pulse is determined. The boundary of the ejection crater is a surface with some constant speed module value.
It is believed that in general the charges have different widths and pressure pulses characterizing the effect of charges on the medium may be different.
The exact solution to the problem is built by displaying the areas of change complex potential and complex speed per area of change auxiliary parametric variable.
Detailed parametric analysis done for the case of equal charges The behavior of the solution when changing the main dimensionless parameters is learned. Limitations on values
of determining parameters are specified. Limiting cases is investigated The results of calculations of the shape of the ejection crater for various sets of values of defining parameters are given.

Let Λ be a 𝑛-dimensional lattice, and 𝑐1, . . . , 𝑐𝑛−1 be any 𝑛 − 1 vectors in 𝑛-dimensional real Euclidean space. We show that there exists a basis 𝛼1, . . . ,𝛼𝑛 of Λ such that

$$|𝛼𝑖 − 𝑁𝑐𝑖| = 𝑂(log^2 𝑁), (1 <= 𝑖 <= 𝑛 − 1)$$

holds for any real number 𝑁 > 2, where the constant implied by the 𝑂 symbol depends only on Λ and 𝑐1, . . . , 𝑐𝑛−1.

In the theory of Diophantine Approximations one considers a question on approximation of Real Numbers by rational fractions with one and the same denominators. Among intensively
studied questions of this theory a special place occupy Metric questions. Here such questions of the theory are considered which take place for almost all real numbers from given interval.
For the first time similar questions have been studied by Khintchine for approximation of independent quantities. It had been investigated by him conditions at which for almost all
real numbers specified accuracy of approximation is reached. Very important in the technical plan Khintchince’s transference principle allows us to connect a simultaneous approximations
of dependent quantities with approximations of linear forms with integral coefficients.
In 1932 Mahler K. has entered classification of transcendental numbers into consideration.
He showed that almost all transcendental numbers are 𝑆 -numbers. Moreover, Mahler had proved an existence of a constant 𝛾 > 0 such that for almost all 𝜔

$$|𝑃(𝜔)| > ℎ^(−𝑛𝛾)$$,

for all polynomials 𝑃 of degree no more 𝑛 and height ℎ > ℎ0(𝜔, 𝑛, 𝛾). Mahler showed that it is possible to take

$$𝛾 = 4 + 𝜀$$.

In the same work Mahler made an assumption that it is possible to take 𝛾 = 1 + 𝜀 for almost all real numbers.
This hypothesis was proved by Sprindzhuk V. G by a method of essential and nonessential domains. Simultaneously, Sprindzhuk V. G. advanced some hypotheses generalising and improving Mahler’s results. Further these investigations of Sprindzuk led to the development of a new direction in the theory of Diophantine Approximations – to the research of extremality of manifolds.
In the present article we develop a new approach to these questions and offer a new proof for extremality of algebraic varieties. This method allows to establish extremality of affine image of topological product of some varieties. Considering one particular case, we prove that the extremality for these varieties is possible to deduct from theorems on the convergence exponent
of a special integral of Terry’s problem, using E.I. Kovalevskaya’s lemma. Further, we derive from the getting result particular case of the Sprindzuk’s hypothesis on extremality of varieties,
induced by monomials of a polynomial in two variables.

We study the dependence of the functions of capital (resource) and consumption in the Ramsey–Kass–Koopmans economic model in the case when saving is an identical constant.
The system of differential equations describing the evolution of the economic model under consideration is solved in quadratures under the assumptions made. Upper estimates of the consumption function are found based on the obtained solution.

Starting from a real analytic surface ℳ with a real analytic conformal Cartan connection A. Bor´owka constructed a minitwistor space of an asymptotically hyperbolic Einstein–Weyl manifold with ℳ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on ℳ can be extended to symmetries of the obtained Einstein–Weyl manifold.

Generalized hypergeometric series are of the form

$$𝑓(𝑧) =∞Σ︁𝑛=0((𝑎1)𝑛 . . . (𝑎𝑙)𝑛)/((𝑏1)𝑛 . . . (𝑏𝑚)𝑛)𝑧𝑛$$

If 𝑙 < 𝑚 and if the parameters are rational, they are closely related to Siegel’s 𝐸-functions. If 𝑙 = 𝑚 and if the parameters are rational, they are 𝐺-functions. For 𝑙 > 𝑚 and if the parameters
are rational, they are 𝐹-series.
The arithmetic properties values of generalized hypergeometric series is an actual problem with a long history. We shall only mention Siegel C. L., Shidlovskii A. B., Salikhov V. Kh.,
Beukers F., Brownawell W. D., Heckman G., Galochkin A. I., Oleinikov V. A., Ivankov P. L., Gorelov V. A., Chirskii V. G., Zudilin W., Matala–Aho T. etc. We consider the so–called 𝐹-series. Chirskii V.G. proved the infinitу algebraic independence
of the corresponding values.
Here we obtain lower estimates of polynomials and linear forms in the values of these series and their derivatives in a concrete 𝑝-adic field.

The article takes a look at transcendence and algebraic independence problems, introduces statements and proofs of theorems for some kinds of elements from direct product of 𝑝-adic fields and polynomial estimation theorem. Let Q𝑝 be the 𝑝-adic completion of Q, Ω𝑝 be the completion of the algebraic closure of Q𝑝, 𝑔 = 𝑝1𝑝2 . . . 𝑝𝑛 be a composition of separate prime numbers, Q𝑔 be the 𝑔-adic completion of Q, in other words Q𝑝1 ⊕. . .⊕Q𝑝𝑛. The ring Ω𝑔 ∼=Ω𝑝1⊕...⊕Ω𝑝𝑛, a
subring Q𝑔, transcendence and algebraic independence over Q𝑔 are under consideration. Also, hypergeometric series

$$𝑓(𝑧) =∞Σ︁𝑗=0((𝛾1)𝑗 . . . (𝛾𝑟)𝑗)/((𝛽1)𝑗 . . . (𝛽𝑠)𝑗)(𝑧𝑡)^𝑡𝑗 $$,

and their formal derivatives are under consideration. Sufficient conditions are obtained under which the values of the series 𝑓(𝛼) and formal derivatives satisfy global relation of algebraic
independence, if 𝛼 =∞Σ︀𝑗=0 𝑎_𝑗𝑔^(𝑟_𝑗), where 𝑎𝑗 ∈ Z𝑔, and non-negative rationals 𝑟𝑗 increase strictly unbounded.

This paper proves infinite algebraic independence of the values of hypergeometric 𝐹 – series at polyadic Liouville points. Hypergeometric functions are defined for |𝑧| < 1 by the power series:

$$Σ︁𝑛=0((𝛼1)𝑛 · · · (𝛼𝑟)𝑛)/((𝛽1)𝑛 . . . (𝛽𝑠)𝑛 𝑛!)𝑧^𝑛$$.

𝐹 – series have form 𝑓𝑛 = Σ︀∞ 𝑛=0 𝑎_𝑛𝑛!𝑧^𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties.
These series converge in the field Q𝑝 of 𝑝 – adic numbers and their algebraic extensions K𝑣. Polyadic number is a series of the form Σ︀∞𝑛=0 𝑎_𝑛𝑛!, 𝑎_𝑛 ∈ Z. Liouville number is a real number x
with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞| < 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼 − 𝐴|𝑝 < 𝐴^(-𝐷).

In the article dedicated to the 75th anniversary of Alexander Olshanskii, colleagues, friends and students reflected biographical information about the hero of the day, information about
his student life, studies in graduate school, gave brief information about his scientific and pedagogical activities, about participation in the mathematical life of the world community.

In the article dedicated to the 80th anniversary of Yuri Filippovich Golovnev, colleagues, friends and students reflected biographical information about the hero of the day, information about studies in graduate school, gave brief information about his scientific and pedagogical activities, about participation in scientific conferences.