The article is dedicated to the anniversary of Alexander Vasilyevich Mikhalev one of the recognized specialist in mathematics and computer science. He made a significant contribution to the development of Soviet and Russian science, created one of the largest scientific mathematical schools. More than one hundred of his students became candidates and doctors of physical and mathematical sciences, they actively work in the system of higher education, carry out scientific research, and hold high administrative positions.
Alexander Vasilyevich has published about 500 works, including scientific and review articles, monographs and textbooks, translated into Russian several fundamental scientific monographs. The article provides a brief description of his major scientific achievements and presents a list of selected publications.

The article is devoted to the 70th anniversary of Alexey Lvovich Semenov, a prominent Russian mathematician and figure of Russian education. The article provides biographical information and an overview of the professional activities of academician of Russian Academy of Sciences and Russian Academy of Education, A. L. Semenov – both in the fields of mathematics and theoretical informatics, and in other important areas of A. L. Semenov’s
activity: support and development of school mathematical education, research on issues of the digital transformation of education, upgrading the content of education in primary and
secondary schools, as well as professional teacher education.

The problem being considered is the reconstruction of periodic words from a finite alphabet using multiset of fixed length subwords.
This is a special case of a more general problem of reconstruction with incomplete information and under restrictions on the words in question.
For some constraints on the multiset of subwords, conditions for possibility of reconstruction are obtained.
It is shown that a periodic word with period $$p$$ is uniquely determined by the multiset of its subwords of length $$k \geq \left\lfloor\frac{16}{7} \sqrt{p}\right\rfloor + 5$$.
For a word consisting of a non-periodic prefix of length $$q$$ and a periodic suffix with period $$p$$, repeated $$l$$ times, a similar estimate is obtained: $$k \geq \left\lfloor\frac{16}{7} \sqrt{P}\right\rfloor + 5$$, provided $$l \geq q^{\left\lfloor\tfrac{16}{7} \sqrt{P}\right\rfloor + 5}$$ where $$P = \max(p, q)$$.

We continue our researches concerning the generalization and improvement of R.T.Turganaliev’s result that states an asymptotic formula for the mean value of the Riemann zeta function in the critical strip with power factor saving in the remainder term.We find an asymptotic for the mean value of Dirichlet L-function in the critical strip. This assertion improves R.T.Turganaliev’s theorem for zeta-function in the whole interval $$(1/2 < Re 𝑠 ≤ 1)$$. Our result is based on the special use of the estimation of exponential sums by second derivative test.

The construction of decomposable curves of degree 8 with multipliers of degrees 3 and 5 is considered in this paper. Sturmfels’s modification of Viro’s patchworking method for constructing decomposable curves is used. 29 pairwise different curves were constructed

Reliability is enhancing its value in the advancement of echanical and industrial world by incorporating the repair mechanism, availability and manufacturing possibility of machines with varying working capacity in all conditions. The present paper is an initiative taken with a mechanical system operating with single repair server facility for varying nature of failures and services. Passive standby machine strategy is adopted for maintaining reliability at a gratified level in the system. The inspection process is included for filtering the machines according to its failure or to the level of repair services. The computed numerical and raphical data is proved to be beneficial for clarifying the profit and reliability behaviour with increasing/decreasing rate of repair mechanism and failure rate. The preference policy has been initiated for regular failure or to the failure requiring normal servicing charges and time period than major ones to avoid
the waiting time for normal customer

Multistable systems and their dynamic properties are interesting topics in nonlinear dynamics. Small differences in the initial conditions (for example, due to rounding errors in numerical calculations) lead to completely different results for such dynamic systems, which makes a long-term prediction of their behavior practically impossible. This happens even if such systems are deterministic, that is, their future behavior is completely determined by the choice of initial conditions without the participation of random elements. In other words, the deterministic nature of these systems does not make them predictable. The behavior of the solutions of a dynamic system depends both on the choice of their initial conditions and on the values of the system parameters. The coexistence of several attractors, or multistability, corresponds to the simultaneous existence of more than one nontrivial attractor for the same set of system parameters. This phenomenon was discovered in almost all natural sciences, including electronics, optics, biology. In recent years, the efforts of many researchers have been aimed at creating so-called megastable systems, that is, systems that, at constant values of their parameters, have a countable number of coexisting attractors. Interest in such systems is due to a wide range of applications, for example, for hiding information in communication systems and audio encryption schemes, biomedical engineering and fuzzy control. The article proposes methods for the synthesis of megastable systems using systems in Lurie form. Megastable systems containing a 1-D lattice of chaotic attractors can be obtained by replacing the nonlinearity in the original system with a periodic function. By replacing some variables with periodic functions in the original system of order n, one can construct a megastable system containing an n-D lattice of chaotic attractors. As one example, a fourth-order system with a 4-D lattice of chaotic attractors is constructed for the first time. The Lyapunov exponents and Kaplan – Yorke dimension are calculated for attractors belonging to lattices

Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-k sequence for k = 4, 5, 6, 7 was proved. In the works of Hone, representations for Somos-4 and 5 sequences were found via the Weierstrass sigma function on elliptic curves, and for k = 6 via the Klein sigma function on hyperelliptic curve of genus It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For k = 4, 5, 6, 7, the Somos sequences are Laurent polynomials of k initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables. Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free  terms are written explicitly

Covert channels allow one to transmit information using mechanisms that were not originally intended for transmission. An example is a process in which a transmitter encodes information in moves of a character of a multiplayer game, and a receiver observes the moves and decodes the original message. This channel may be noisy, since the character may fall out of the receiver’s sight, a number of network packets may be lost, etc. Thus there emerges a natural problem of ogranizing a reliable channel. We propose a formal model called a partial erasure channel that describes the interaction of a transmitter and a receiver, introduce the notion of a consistent transmission protocol, formulate and prove the consistency criterion on the transmitting side and construct the optimal receiver for the given consistent transmitter.

The active spread of the Internet at the beginning of the 21st century led to the unification of a large number of people on single Internet platforms, on which direct interaction between
users and entrepreneurs became possible. This served as the basis for the emergence of a new way to attract funding to risky entrepreneurial projects and startups - crowdfunding. Continuous improvement of data analysis methods makes it possible to more effectively study crowdfunding and its consequences for the global and national economic system. The results on the structure of projects that are organized by entrepreneurs on the Kickstarter crowdfunding platform to finance and implement their unique ideas provide a deeper understanding of how the crowdfunding industry needs to be improved to ensure the most effective development of innovation and small and medium-sized businesses. In this regard, the question arises of how to analyze crowdfunding projects. In this article, using the example of applied research of data on more than 100 thousand completed crowdfunding projects, it will be shown how one of the statistical methods for analyzing the relationships between variables - factor analysis, can be used.

The problem of separability of differential operators is considered for the first time in the works of V. N. Everitt and M. Hirz. Further development of this theory belongs to K. H. Boymatov,
M. Otelbaev and their students. The main part of the published papers on this theory relates to linear operators. The nonlinear case was considered mainly when studied operator was a weak perturbation of the linear one. The case when the operator under study is not a weak perturbation of the linear operator is considered only in some works. The results obtained in this paper also relate to this little-studied case. The paper studies the coercive properties of the nonlinear Laplace-Beltrami operator in the space L2(R^n) $$L[u]=-\frac{1}{\sqrt{det g(x)}}\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\left[\sqrt{det g(x)}g^{-1}(x)\frac{\partial u}{\partial x_j}\right]+V(x,u)u(x)$$, and proves its separability in this space by coercivity estimates. The operator under study is not a weak perturbation of the linear operator, i.e. it is strongly nonlinear. Based on the obtained
coercive estimates and separability, the solvability of the nonlinear Laplace-Beltrami equation in the space L2(R^n) is studied

Let L_q(M) denote the lattice of all subquasivarieties of the quasivariety M under inclusion. There is a strong correlation between the properties of the lattice L_q(M) and algebraic systems from M. A. I. Maltsev first drew attention to this fact in a report at the International Congress of Mathematicians in 1966 in Moscow.
In this paper, we obtain a characterization of the class of all distributive lattices, each of which is isomorphic to the lattice of some quasivariety of unars. A unar is an algebra with one
unary operation. Obviously, any unar can be considered as an automaton with one input signal without output signals, or as an act over a cyclic semigroup.
We construct partially ordered sets P∞ and P_s(s ∈ N0), where N0 is the set of all nonnegative
integers. It is proved that a distributive lattice is isomorphic to the lattice L_q(M) for some quasivariety of unars M if and only if it is isomorphic to some principal ideal of one of the lattices 0(Ps)(s ∈ N0) or 0c(P∞), where 0(Ps)(s ∈ N0) is the ideal lattice of the poset Ps(s ∈ N0) and 0s(P∞) is the ideal lattice with a distinguished element 𝑐 of the poset P∞.
The proof of the main theorem is based on the description of Q-critical unars. A finitely generated algebra is called Q-critical if it does not decompose into a subdirect product of its proper subalgebras. It was previously shown that each quasivariety of unars is determined by its Q-critical unars. This fact is often used to investigate quasivarieties of unars.

Subdirectly irreducible universal algebras, i.e., algebras which are not decomposable into non-trivial subdirect product, play an important role in mathematics due to well-known Birkhoff Theorem which states that any algebra is a subdirect product of subdirectly irreducible algebras (in another terminology: every algebra is approximated by the subdirectly irreducible algebras).

In view of this, it seems reasonable to study algebras with certain conditions on subdirectly irreducible algebras. One of the natural restrictions is the finiteness of all subdirectly irreducible algebras. More stronger restriction is boundness of orders of all subdirectly irreducible algebras. An act over a semigroup (it is also an automaton, and a unary algebra) is a set with an action of the given semigroup on it. The acts over a fixed semigroup form a variety whose signature coincides with self semigroup. On the other hand, it is a category whose morphisms are homomorphisms from one act into another. It is not difficult to see that the semigroups over which all subdirectly ireducible acts

are finite, are exactly the semigroups over which all subdirectly irreducible acts are finitely approximated (in another terminology: residually finite). A more narrow class form semigroups over which all acts are approximated by acts of n or less elements where n is a fixed natural number.

In 2000, I.B.Kozhukhov proved that all non-trivial acts over a semigroup S are approximated by two-element ones if and only if S is a semilattice (a commutative idempotent semigroup).

In 2014, I.B.Kozhukhov and A.R.Haliullina proved that any semigroup with bounded orders of dubdirectly irreducible acts is uniformly locally finite, i.e., for every k, the orders of k-generated subsemigroups are bounded. In the work of I.B.Kozhukhov and A.V.Tsarev 2019, the authors described completely the abelian groups over which all acts are finitely approximated, and also abelian groups over which all acts are approximated by acts of bounded orders.

In this work, we characterize the commutative semigroups over which all acts are approximated by acts consisted of n or less elements.

A ring whose additive group coincides with an abelian group G is called a ring on G. An abelian group G is called a TI-group if every associative ring on G is filial. If every (associative) ring on an abelian group G is an SI-ring (a hamiltonian ring), then G is called an SI-group (an SI_H-group). In this article, TI-groups, SI_H-groups and SI-groups are described in the following classes of abelian groups: almost completely decomposable groups, separable torsionfree groups and non-measurable vector groups. Moreover, a complete description of non-reduced TI-groups, SI_H-groups and SI-groups is given. This allows us to only consider reduced groups when studying TI-groups.

In this paper <<The projective geometry over partially ordered skew fields, II>> the investigation of properties for partially ordered linear spaces over partially ordered skew fields is prolonged. This investigation was started in part I <<The projective geometry over partially ordered skew fields>>. Derivative lattices associated partially ordered linear spaces over partially ordered skew fields are examined. More exactly, properties of the convex projective geometry $${\cal L}$$ of a partially ordered linear space $${}_FV$$ over a partially ordered skew field $$F$$ are considered. The convexity of linear subspaces has meaning the Abelian convexity ($$ab$$-convexity), which is based on the definition of a convex subgroup for a partially ordered group. Second and third theorems of linear spaces order isomorphisms for interpolation linear spaces over partially ordered skew fields are proved. Some theorems are proved for principal linear subspaces of interpolation linear spaces over directed skew fields. The principal linear subspace $$I_a$$ of a partially ordered linear space $${}_FV$$ over a partially ordered skew field $$F$$ is the smallest $$ab$$-convex directed linear subspace of linear space $${}_FV$$ which contains the positive element $$a\in V$$. The analog for the third theorem of linear spaces order isomorphisms for principal linear subspaces is demonstrated in interpolation linear spaces over directed skew fields

Two formulas for Macdonald functions (which are a widely known in mathematics and applications particular case of cylinder functions) are obtained by using some integral bilinear functionals defined on a pair of representation spaces or a square of these spaces

In the paper investigates the possibility of homological description of Jacobson radical and locally nilpotent radical for Lie algebras, and their relation with a $$PI$$ - irreducibly represented radical, and some properties of primitive Lie algebras are studied. We prove an analog of The F. Kubo  theorem for almost locally solvable Lie algebras with a zero Jacobson radical. It is shown that the Jacobson radical of a special almost locally solvable Lie algebra $$L$$ over a field $$F$$ of characteristic zero is zero if and only if the Lie algebra $$L$$ has a Levi decomposition $$L=S\oplus Z(L)$$, where $$Z(L)$$ is the center of the algebra $$L$$, $$S$$ is a finite-dimensional subalgebra $$L$$ such that $$J(L)=0$$. For an arbitrary special Lie algebra $$L$$, the inclusion of $$IrrPI(L)\subset J(L)$$ is shown, which is generally strict. An example of a Lie algebra $$L$$ with strict inclusion  $$J(L)\subset IrrPI(L)$$ is given. It is shown that for an arbitrary special Lie algebra $$L$$ over the field $$F$$ of characteristic zero, the inclusion of $$N (L)\subset IrrPI(L)$$, which is generally strict. It is shown that most Lie algebras over a field are primitive. An example of an Abelian Lie algebra over an algebraically closed field that is not primitive is given. Examples are given showing that infinite-dimensional commutative Lie algebras are primitive over any fields; a finite-dimensional Abelian algebra of dimension greater than 1 over an algebraically closed field is not primitive; an example of a non-Cartesian noncommutative Lie algebra is primitive. It is shown that for special Lie algebras over a field of characteristic zero $$PI$$-an irreducibly represented radical coincides with a locally nilpotent one. An example of a Lie algebra whose locally nilpotent radical is neither locally nilpotent nor locally solvable is given. Sufficient conditions for the primitiveness of a Lie algebra are given, and examples of primitive Lie algebras and non-primitive Lie algebras are given.

In this paper, we will start by deliberating at our project’s historical general view and then we will try to construct a new Poisson bracket on our simplest example 𝑠𝑙2 and then we will try
to give a universal construction based on our universal variables and then will try to construct lattice 𝑊2 algebras which will play a key role in our other constructions on lattice 𝑊3 algebras
and finally we will try to find the only nontrivial dependent generator of our lattice 𝑊4 algebras and so on for lattice 𝑊𝑛 algebras

Consider partial algebras whose equivalence relations are congruences. The problem of description of such partial algebras can be reduced to the problem of description of partial 𝑛-ary
groupoids with the similar condition. In this paper a concept of moderately partial operation is used. A description is given for the moderately partial operations preserving any equivalence
relation on a fixed set.
Let 𝐴 be a non-empty set, 𝑓 be a moderately partial operation, defined on 𝐴 (i.e. if we fix all of the arguments of 𝑓, except one of them, we obtain a new partial operation 𝜙 such that its domain dom 𝜙 satisfies the condition |dom 𝜙| > 3). Let any equivalence relation on the set 𝐴 be stable relative to 𝑓 (in the other words, the congruence lattice of the partial algebra (𝐴, {𝑓}) coinsides the equivalence relation lattice on the set 𝐴). In this paper we prove that in this case the partial operation 𝑓 can be extended to a full operation 𝑔, also defined on the set 𝐴, such that 𝑔 preserves any equivalence relation on 𝐴 too. Moreover, if the arity of the partial
operation 𝑓 is finite, then either 𝑓 is a partial constant (i.e. 𝑓(𝑥) = 𝑓(𝑦) for all 𝑥, 𝑦 ∈ dom 𝑓), or 𝑓 is a partial projection (there is an index 𝑖 such that all of the tuples 𝑥 = (𝑥1, ..., 𝑥𝑛) ∈ dom 𝑓
satisfy the condition 𝑓(𝑥1, ..., 𝑥𝑖, ..., 𝑥𝑛) = 𝑥𝑖)

The article presents results and open problems related to definability spaces (reducts) and sources of this field since the XIX century. Finiteness conditions and constraints are investigated,
including the depth of quantifier alternation and the number of arguments. Results related to the description of lattices of definability spaces for numerical and other natural structures
are described. Research methods include the study of automorphism groups of elementary extensions of the structures under consideration, application of the Svenonius theorem.

The article considers multidimensional improper integrals of functions that are the product of generalized polynomials in some degrees. Such integrals are found in many branches of
mathematics and theoretical physics. In particular, they include Feynman integrals arising in the study of various objects of quantum field theory. The exact calculation of these integrals is a
difficult and not always possible task; therefore, determining the conditions for their convergence and obtaining their asymptotic expansion in one of the parameters is of considerable practical
interest. The convergence conditions for the integrals considered in the article can still be used, for example, in the study of multiple series representing the sum of the values of a rational
function at the nodes of an integer lattice.
The article considers the problem when the integration area is R+^n, and the generalized polynomials included in the integrand are either positive everywhere except zero or have positive
coefficients. The convergence set of these integrals is described and the equivalence of the convergence condition to the condition on the Newton polytopes of polynomials in integrands
is proved.

The convergence criterion proved in the paper coincides in formulation with the corresponding result of the work of A. K. Tsikh and T. O. Ermolaeva, but it was obtained by other
methods and for a slightly wider set of integrands.
The proofs of the statements in the paper are based on the simplest properties of convex polytopes and basic facts from the theory of improper multiple integrals.

which are isomorphic to (𝑛)-groups of homomorphisms from certain 𝑛-groups to a semiabelian 𝑛-group. Such 𝑛-groups are found for infinite semicyclic 𝑛-groups.
It is known that the set 𝐻𝑜𝑚(𝐺,𝐶) of all homomorphisms from 𝑛-groups ⟨𝐺, 𝑓1⟩ to a semiabelian (abelian) 𝑛-group ⟨𝐶, 𝑓2⟩ with 𝑛-ary operation 𝑔 given by the rule 𝑔(𝜙1, . . . , 𝜙𝑛)(𝑥) = 𝑓2(𝜙1(𝑥), . . . , 𝜙𝑛(𝑥)), 𝑥 ∈ 𝐺, forms a semiabelian (abelian) 𝑛-group. It is proved that the isomorphisms 𝜓1 of 𝑛-groups ⟨𝐺, 𝑓1⟩ and ⟨𝐺′, 𝑓′1 ⟩ and 𝑝𝑠𝑖2 of semiabelian 𝑛-groups ⟨𝐶, 𝑓2⟩ and ⟨𝐶′, 𝑓′2⟩ induce an isomorphism 𝜏 of 𝑛-groups of homomorphisms ⟨𝐻𝑜𝑚(𝐺,𝐶), 𝑔⟩ and ⟨𝐻𝑜𝑚(𝐺′,𝐶′), 𝑔′⟩, which acts according to the rule 𝜏 : 𝛼 → 𝜓2 ∘ 𝛼 ∘ 𝜓−1

1 .On the additive group of integers 𝑍 we construct an abelian 𝑛-group ⟨𝑍, 𝑓1⟩ with 𝑛-ary operation 𝑓1(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 𝑙, where 𝑙 is any integer. For a nonidentical automorphism 𝜙(𝑧) = −𝑧 on 𝑍, we can specify semiabelian 𝑛-group ⟨𝑍, 𝑓2⟩ for 𝑛 = 2𝑘 + 1, 𝑘 ∈ 𝑁, with the 𝑛-ary operation 𝑓2(𝑧1, . . . , 𝑧𝑛) = 𝑧1 − 𝑧2 + . . . + 𝑧2𝑘−1 − 𝑧2𝑘 + 𝑧2𝑘+1. Any infinite semicyclic 𝑛-group is isomorphic to either the 𝑛-group ⟨𝑍, 𝑓1⟩, where 0 ≤ 𝑙 ≤ [𝑛−1
2 ], or the 𝑛-group ⟨𝑍, 𝑓2⟩ for odd 𝑛. In the first case we will say that such 𝑛-group has type (∞, 1, 𝑙), and in the second case, it has type (∞,−1, 0).
In studying the 𝑛-groups of homomorphisms ⟨𝐻𝑜𝑚(𝑍,𝐶), 𝑔⟩ from an infinite abelian semicyclic 𝑛-group ⟨𝑍, 𝑓1⟩ (0 ≤ 𝑙 ≤ 𝑛−1
2 ) to a semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩ we construct on the 𝑛-group ⟨𝐶, 𝑓2⟩ an abelian group 𝐶 with the addition operation 𝑎 + 𝑏 = 𝑓2(𝑎, (𝑛−3) 𝑐 , ¯𝑐, 𝑏), in which there is an element 𝑑2 = 𝑓2( (𝑛)
𝑐 ) and an automorphism 𝜙2(𝑥) = 𝑓2(𝑐, 𝑥, (𝑛−3) 𝑐 , ¯𝑐). Choose a
set 𝑃1 of such ordered pairs (𝑎, 𝑢) of elements from 𝐶 that satisfy the equality 𝑙𝑎 = 𝑑2 + ∼𝜙2(𝑢), where ∼𝜙2(𝑥) = 𝑥 + 𝜙2(𝑥) + . . . + 𝜙𝑛−2 2 (𝑥), 𝑥 ∈ 𝐶 is an endomorphism of the group 𝐶, and for
the first component of these pairs the equality is true 𝜙2(𝑎) = 𝑎. On this set, we define a 𝑛-ary operation ℎ1 by the rule ℎ1((𝑎1, 𝑢1), . . . , (𝑎𝑛, 𝑢𝑛)) = (𝑎1 + . . . + 𝑎𝑛, 𝑓2(𝑢1, . . . , 𝑢𝑛)). It is proved that ⟨𝑃1, ℎ1⟩ is a semiabelian 𝑛-group, which is isomorphic to the 𝑛-group of homomorphisms from an infinite abelian semicyclic 𝑛-group ⟨𝑍, 𝑓1⟩ (0 ≤ 𝑙 ≤ 𝑛−1 2 ) to an 𝑛-group ⟨𝐶, 𝑓2⟩. The consequence of this isomorphism is an isomorphism of 𝑛-groups of ⟨𝑃1, ℎ1⟩ and 𝑛-groups of homomorphisms from an infinite abelian semicyclic 𝑛-group of type (∞, 1, 𝑙) to a semiabelian
𝑛-group ⟨𝐶, 𝑓2⟩. When studying the 𝑛-group of homomorphisms ⟨𝐻𝑜𝑚(𝑍,𝐶), 𝑔⟩ from the infinite semicyclic 𝑛-group ⟨𝑍, 𝑓′1 ⟩ to the semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩ in the abelian group 𝐶 choose the subgroup 𝐻 = {𝑎 ∈ 𝐶 | 𝜙2(𝑎) = −𝑎}. On 𝐻 we define a semiabelian 𝑛-group ⟨𝐻, ℎ⟩, where ℎ acts according to the rule ℎ(𝑎1, 𝑎2, . . . , 𝑎𝑛−1, 𝑎𝑛) = 𝑎1 +𝜙2(𝑎2)+. . .+𝜙𝑛−2 2 (𝑎𝑛−1)+𝑎𝑛. Then in the
𝑛-group ⟨𝐶, 𝑓2⟩ we select the subgroup ⟨𝑇, 𝑓2⟩ of all dempotents, if 𝑇 ̸= ∅. It is proved that foran odd number 𝑛 > 1 a direct product of semiabelian 𝑛-groups ⟨𝐻, ℎ⟩ × ⟨𝑇, 𝑓2⟩ is isomorphic
to 𝑛-group of homomorphisms from infinite semicyclic 𝑛-groups of ⟨𝑍, 𝑓′ 1⟩ to a semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩ with a non empty set of idempotents 𝑇. The consequence of this isomorphism
is the isomorphism of the 𝑛-group ⟨𝐻, ℎ⟩ × ⟨𝑇, 𝑓2⟩ and 𝑛-groups of homomorphisms from an infinite semicyclic 𝑛-group of type (∞,−1, 0) to the semiabelian 𝑛-group ⟨𝐶, 𝑓2⟩. Similar facts were obtained when studying the 𝑛-group of homomorphisms ⟨𝐻𝑜𝑚(𝑍,𝐶), 𝑔⟩ from 𝑛-groups ⟨𝑍, 𝑓1⟩ and ⟨𝑍, 𝑓′ 1⟩ to an abelian 𝑛-group ⟨𝐶, 𝑓2⟩

are isomorphic to (𝑛, 2)-nearrings of endomorphisms of certain semiabelian 𝑛-groups. Such almost (𝑛, 2)-nearrings are found for semicyclic 𝑛-groups.
On the additive group of integers 𝑍 we construct an abelian 𝑛-group ⟨𝑍, 𝑓1⟩ with 𝑛- ary operation 𝑓1(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 𝑙, where 𝑙 is any integer. For a nonidentical automorphism 𝜙(𝑧) = −𝑧 on 𝑍, we can specify semiabelian 𝑛-group ⟨𝑍, 𝑓2⟩ for 𝑛 = 2𝑘 + 1, 𝑘 ∈ 𝑁, with the 𝑛-ary operation 𝑓2(𝑧1, . . . , 𝑧𝑛) = 𝑧1 − 𝑧2 + . . . + 𝑧2𝑘−1 − 𝑧2𝑘 + 𝑧2𝑘+1. Any infinite semicyclic 𝑛-group is isomorphic to either the 𝑛-group ⟨𝑍, 𝑓1⟩, where 0 ≤ 𝑙 ≤ [𝑛−1
2 ], or the 𝑛-group ⟨𝑍, 𝑓2⟩ for odd 𝑛. In the first case we will say that such 𝑛-group has type (∞, 1, 𝑙), and in the second case, it has type (∞,−1, 0).
In 𝑍 we select the set 𝑃 = {𝑚|𝑚𝑙 ≡ 𝑙 (mod 𝑛 − 1)} and define an 𝑛-ary operation ℎ by the rule ℎ(𝑚1, . . . ,𝑚𝑛) = 𝑚1 + . . . + 𝑚𝑛 on this set. Then the algebra ⟨𝑃, ℎ, ·⟩, where · is the multiplication of integers, is a (𝑛, 2)-ring. It is proved that ⟨𝑃, ℎ, ·⟩ is isomorphic to (𝑛, 2)-ring of endomorphisms of semicyclic 𝑛-group of type (∞, 1, 𝑙).

In the 𝑛-group ⟨𝑍 × 𝑍, ℎ⟩ = ⟨𝑍, 𝑓2⟩ × ⟨𝑍, 𝑓2⟩ we define the binary operation ◇ by the rule (𝑚1, 𝑢1) ◇ (𝑚2, 𝑢2) = (𝑚1𝑚2,𝑚1𝑢2 +𝑢1). Then ⟨𝑍 ×𝑍, ℎ, ◇⟩ is an (𝑛, 2)-nearringsg. It is proved that ⟨𝑍 ×𝑍, ℎ, ◇⟩ is isomorphic to (𝑛, 2)-nearrings of endomorphisms of a semicyclic 𝑛-group of type (∞,−1, 0).
It is proved that (𝑛, 2)-ring ⟨𝑍, 𝑓, *⟩, where 𝑓(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + . . . + 𝑧𝑛 + 1 and 𝑧1 * 𝑧2 = 𝑧1𝑧2(𝑛 − 1) + 𝑧1 + 𝑧2, is isomorphic to (𝑛, 2)-rings of endomorphisms of infinite cyclic 𝑛-group.
On additive group of the ring of residue classes of 𝑍𝑘 we define 𝑛-group ⟨𝑍𝑘, 𝑓3⟩, where the 𝑛- ary operation 𝑓3 operates according to the rule 𝑓3(𝑧1, . . . , 𝑧𝑛) = 𝑧1+𝑚𝑧2+. . .+𝑚𝑛−2𝑧𝑛−1+𝑧𝑛+𝑙, 1 ≤ 𝑚 < 𝑘 and 𝑚 is relatively prime to 𝑘. In addition, 𝑚 satisfies the congruence 𝑙𝑚 ≡ 𝑙 (mod 𝑘) and multiplicative order of 𝑚 modulo 𝑘 divides 𝑛 − 1. Any finite semicyclic 𝑛-group of order 𝑘
is isomorphic to 𝑛-group ⟨𝑍𝑘, 𝑓3⟩, where 𝑙 | gcd(𝑛 − 1, 𝑘) for 𝑚 = 1 and 𝑙 | gcd(𝑚𝑛−1−1 𝑚−1 , 𝑘) for 𝑚 ̸= 1. We will say that such 𝑛-group has type (𝑘, 𝑚, 𝑙).

In the 𝑛-group ⟨𝑃, ℎ⟩ = ⟨𝑍𝑘, 𝑓3⟩ × ⟨𝑍𝑙, 𝑓4⟩, 𝑓4(𝑧1, . . . , 𝑧𝑛) = 𝑧1 + 𝑟𝑧2 + . . . + 𝑟𝑛−2𝑧𝑛−1 + 𝑧𝑛, where 𝑟 is the remainder of dividing 𝑚 by 𝑙, we define the binary operation ◇ by the rule

$$(𝑢1, 𝑣1) ◇ (𝑢2, 𝑣2) = (𝑢2𝑠1 + 𝑢1, 𝑣2𝑠1 + 𝑣1)$$

where 𝑠1 ∈ 𝑍𝑘, (𝑠1 − 1 = 𝑠0 + 𝑣1)/𝑘
𝑙 , and 𝑠0 is solution of congruence 𝑥 ≡ (𝑛−1)𝑢1
𝑙 (mod 𝑘/𝑙 ) for 𝑚 = 1 and 𝑥 ≡ 𝑚𝑛−1−1 𝑚−1 𝑢1 𝑙 (mod 𝑘/𝑙 ) for 𝑚 ̸= 1. It is proved that the algebra ⟨𝑃, ℎ, ◇⟩ is (𝑛, 2)-ring for 𝑚 = 1 and (𝑛, 2)-nearring for 𝑚 ̸= 1, which is isomorphic to (𝑛, 2)-ring of endomorphisms of abelian semicyclic 𝑛-group of type (𝑘, 1, 𝑙) with 𝑚 = 1 and (𝑛, 2)-nearring of endomorphisms of semicyclic 𝑛-groups of type (𝑘, 𝑚, 𝑙) for 𝑚 ̸= 1.
It is proved that (𝑛, 2)-ring ⟨𝑍𝑘, 𝑓, *⟩, where 𝑓(𝑧1, 𝑙𝑑𝑜𝑡𝑠, 𝑧𝑛) = 𝑧1 + 𝑙𝑑𝑜𝑡𝑠 + 𝑧𝑛 + 1 and 𝑢1 *𝑢2 = 𝑢1 𝑐𝑑𝑜𝑡𝑢2 𝑐𝑑𝑜𝑡(𝑛−1)+𝑢1+𝑢2, is isomorphic to (𝑛, 2)-ring of endomorphisms of finite cyclic 𝑛-group of order 𝑘.

The paper proposes the development of the mathematical variational Hashin-Strickman method, which was previously used to determine the maximum possible values of effective elastic
characteristics. In this case, the effective plasticity characteristics of two-component composites are determined. In particular, the fork of possible values of the effective yield strength of such
composite materials is determined

The paper presents the results of a study of the sliding friction process of a porous material based on iron impregnated with lubricating oil with dispersed particles of fluorinated graphene.
It is established that the regularities of the kinetics of external sliding friction have a sigmoidal and sigmoidal-linear character. Experimental results have been obtained showing that with an
increase in the concentration of aggregates from flakes of fluorinated graphene in the lubricating oil, the average force and coefficient of friction decrease, while a good anti-friction effect is
observed.

The paper presents the results of a study of the sliding friction processes of a porous copper-based material impregnated with lubricating oil with dispersed particles of fluorinated graphene. Mathematical regularities of changes in the characteristics of the friction interaction are established. It is shown that the regularities of changes in the average friction force have a
sigmoid-step character. Experimental results have been obtained showing that with an increase in the concentration of aggregates from flakes of fluorinated graphene in the lubricating oil,
the average friction force and coefficient of friction decrease, while a good anti-friction effect is observed. It is shown that the average work of the friction force, and consequently the energy
losses due to friction, when adding 0.01% of aggregates from fluorinated graphene flakes to the lubricating oil decreases by 3721 j, and when adding 0.1% — by 4098 j. It was found that the
average coefficient of friction when adding 0.01% of fluorinated graphene flake aggregates to the lubricating oil decreases by 27%, and when adding 0.1% — by 30%.

Nikolai Nikolaevich Luzin’s life (1883 – 1950) and work of this outstanding Russian mathematician coincided with a very difficult period in Russian history: two World Wars, the 1917 revolutions, the civil war, the construction of a new type of state – the Union of Soviet Socialist Republics, which included collectivization and industrialization, accompanied by the mass terror that without exception affected all the strata of Soviet society. Against
the background of these dramatic events took place the process of formation and flourishing of Luzin the scientist, the creator of one of the leading mathematical schools of the XXth century – the Moscow school of function theory, which became one of the cornerstones in the foundation of the Soviet mathematical school. Luzin’s work could be divided into two periods:
the first one comprises the problems regarding the metric theory of functions, culminating in his famous dissertation "Integral and Trigonometric Series"(1915), and the second one which is mainly devoted to the development of problems arising from the theory of analytic sets. The underlying idea of Luzin’s research was the problem of the structure of the arithmetic continuum, which became the super task of his work.

The article examines the reasons why mathematical schools (mathschools) have become an important and very productive phenomenon in Russian education in recent decades. A
brief description of the modern model of productive education traced back to the established tradition of teaching in mathschools, which can lead to the construction of a reproducible model of productive education not only for specialized teaching of mathematics, but also for other areas
of Russian general education

The history of the development of topological education in Nizhny Novgorod is described — from the first lecture on topology for schoolchildren, given in 1939 by Professor A.G. Mayer, to the present. The need for knowledge of topology for further research was first understood in Nizhny Novgorod by representatives of the school of academician A.A. Andronov on the theory of nonlinear oscillations and the qualitative theory of differential equations. An important moment was organization by S.I. Alber in 1964 Gorky topological school, in which many outstanding mathematicians took part (D.V. Anosov, M.L. Gromov, S.P. Novikov, Y.G. Sinai, etc.) However, the specialist in real algebraic geometry, Professor D.A. Gudkov, became the "motor" for introducing topology into the educational process. This activity took place in close cooperation with Leningrad Professor V.A. Rokhlin and his students O.Ya. Viro and V.M. Kharlamov

In article the problem of the diffraction of cylindrical monochromatic sound waves by an homogeneous elastic cylinder with a radially inhomogeneous elastic covering is considered. It is believed that the body is placed in an endless space filled with ideal fluid. The analytical solution of the problem is received.
Wave fields in a containing medium and homogeneous elastic cylinder are found in the form of expansions in wave cylindrical functions. The boundary-value problem for the system of
ordinary second order differential equations is constructed for determination of the displacement field in inhomogeneous coating.
Numerical calculations of angular and frequency characteristics of the scattered field for elastic cylinders with homogeneous and inhomogeneous coatings are performed. Influence of
continuously inhomogeneous elastic coatings on sound-reflecting properties of elastic cylindrical bodies are revealed

Basic equations and determining relations are given, which determine stressed-deformed plastic state of metal materials taking into account their physical and structural parameters.
The approach to the formulation of defining relationships is based on the inclusion in the number of criterion, along with traditional macro-mechanical, physical and structural parameters. These include, first of all, the parameter of material damage by defects of deformation origin. On the basis of experiments, a connection was established between the stress necessary for the
movement of the blocked dislocation and the measure of damage by deformation microfects, necessary for determining the yield strength and, further, the evolution of the loading surface taking into account the factual factors affecting it. The two-stage stretching tests of the AlMg3 alloy samples showed a significant effect of strain resistance on the stress state.

Let 𝑃(𝑛) be a polynomial, having an irrational coefficient of the highest degree. A word 𝑤 (𝑤 = (𝑤𝑛), 𝑛 ∈ N) consists of a sequence of first binary numbers of {𝑃(𝑛)} i.e. 𝑤𝑛 = [2{𝑃(𝑛)}].
Denote by 𝑇(𝑘) the number of different subwords of 𝑤 of length 𝑘 . We’ll formulate the main result of this paper.

Theorem. There exists a polynomial 𝑄(𝑘), depending only on the power of the polynomial 𝑃, such that 𝑇(𝑘) = 𝑄(𝑘) for sufficiently great 𝑘.

In paper describes an approach to building a formal model of information security based on the use of predicate algebra. The model is represented as a decision tree. The algorithm of its construction based on the deductive method of searching for answers is developed and investigated.

A subgroup 𝐴 of a group 𝐺 is called tcc-subgroup in 𝐺, if there is a subgroup 𝑇 of 𝐺 such that 𝐺 = 𝐴𝑇 and for any 𝑋 6 𝐴 and 𝑌 6 𝑇 there exists an element 𝑢 ∈ ⟨𝑋, 𝑌 ⟩ such that 𝑋𝑌 𝑢 ≤ 𝐺. The notation 𝐻 6 𝐺 means that 𝐻 is a subgroup of a group 𝐺. In this paper we consider a group 𝐺 = 𝐴𝐵 such that 𝐴 and 𝐵 are tcc-subgroups in 𝐺. We prove that 𝐺 belongs to F, when 𝐴 and 𝐵 belong to F and F is a saturated formation such that U ⊆ F. Here U is the formation of all supersoluble groups.

35 years ago, a new subject informatics was appeared in the schools of the Soviet Union. Today it is believed that this was a landmark event that subsequently changed the entire system of education. But his path to school was not easy and quite long. The analysis is carried out which sciences have formed the methodological basis of informatics, which concepts have
determined its content, and which of the scientists stood at its origins. It is considered what factors influenced the content of the first textbook of informatics and subsequent ones, how
the domestic informatization of education was born, how a new pedagogical specialty "teacher of informatics"was created. It is noted which works of scientists, authors and research teams
formed the basis of modern theory and methods of teaching information and informatization of education. It is shown how the content of the subject changed, what factors influenced, as well as what problems are facing school informatics and in what direction its transformation will take place in the near future.

This article is based on a report made by V. N. Chubarikov at the International Scientific and Practical Conference "Informatization of Education — 2020" in the city of Orel, October 26–30, 2020. The conference was dedicated to the 115th anniversary of the birth of Patriarch of Russian education, the great teacher, and mathematician, academician S. M. Nikol’skii (1905- 2012 gg.). The event was organized by the Academy of education Informatization and Orel state University. I. S. Turgenev, with the financial support of RFBR.
The article deals with various aspects of the informatization of society, primarily from the point of view of mathematicians and scientists associated with various areas of use of computer
technology and computer technology.
The following questions are briefly discussed: 1. Programming — the basis of computer science; 2. Models of computing systems, computers, programming languages; 3. Artificial
Intelligence systems; 4. What is TeX and LaTeX? 5. Computational complexity; 6. Search for computer science literature; 7. Albert Rubenovich Yesayan Scientific School and Computer
Science at the Tolstoy State Pedagogical University.
The article is dedicated to the memory of the outstanding mathematician and teacher, Professor Albert Rubenovich Yesayan. 50 years of his work at the Tula State Pedagogical
University named after L. N. Tolstoy were devoted to the introduction of advanced approaches in the field of teaching mathematics into the pedagogical process of training teachers of
mathematics and computer science. He made a significant contribution to the development of practical approaches to the teaching of computer science in pedagogical universities.