The lattice of quasi-orders of the universal algebra 𝐴 is the lattice of those quasi-orders on the set 𝐴 that are compatible with the operations of the algebra, the lattice of the topologies of the algebra is the lattice of those topologies with respect to which the operations of the algebra are continuous. The lattice of quasi-orders and the lattice of topologies of the algebra 𝐴, along with
the lattice of subalgebras and the lattice of congruences, are important characteristics of this algebra. It is known that a lattice of quasi-orders is isomorphically embedded in a lattice that is anti-isomorphic to a lattice of topologies, and in the case of a finite algebra, this embedding is an anti-isomorphism. A chain 𝑋𝑛 of 𝑛 elements is considered as a lattice with operations
𝑥 ∧ 𝑦 = min(𝑥, 𝑦) and 𝑥 ∨ 𝑦 = max(𝑥, 𝑦). It is proved that the lattice of quasi-orders and the lattice of topologies of the chain 𝑋𝑛 are isomorphic to the Boolean lattice of 2^(2𝑛−2) elements. A simple correspondence is found between the quasi-orders of the chain 𝑋𝑛 and words of length 𝑛 − 1 in a 4-letter alphabet. Atoms of the lattice of topologies are found. We deduce from the results on quasi-orders a well-known statement that the congruence lattice of an 𝑛-element chain is Boolean lattioce of 2^(𝑛−1) elements. The results will no longer be true if the chain is considered only with respect to one of the operations ∧,∨.

The article deals with the construction of a generalized polynomial operator necessary for finding approximate solutions of equations with fractional order of integration. Integral equations of fractional order are used in a number of problems related to the study of processes that behave discontinuously, for example, for diffusion problems, economic problems related to
the theory of sustainable development and other similar problems. At present, interest in such equations has increased, as evidenced by the publications of recent years in which the processes described by such equations are investigated. In this connection, it becomes relevant to study methods for solving such problems. Since these equations cannot be solved exactly, there is a need to develop and apply approximate methods for their solution. In this article we obtain a form of polynomial operator for some continuous functions on (0, 2𝜋) expressed through the
Lagrange interpolation polynomial on equally spaced knots. The connection of the generalized interpolation operator with the Fourier operator is also established, and the closeness value of these operators is obtained. For the interpolation polynomial operator an estimate of the error of approximation of the exact value by the metric of the space of (0, 2𝜋) continuous functions
is found. This work is a continuation of the research of the authors.

A game with nature for known state probabilities is considered. An optimality principle is proposed for decision-making for games with nature, based on efficiency and risk estimates.
In contrast to the traditional approach to the definition of a mixed strategy in game theory, this paper considers the possibility of correlation dependence of random payoff values for initial alternatives. Two variants of the implementation of the two-criteria approach to the definition of the optimality principle are suggested. The first option is to minimize the variance as a risk
estimate with a lower threshold on the mathematical expectation of the payoff. The second option is to maximize the mathematical expectation of the payoff with an upper threshold on the variance. Analytical solutions of both problems are obtained. The application of the obtained results on the example of the process of investing in the stock market is considered.
An investor, as a rule, does not form a portfolio all at once, but as a sequential process of purchasing one or another financial asset. In this case, the mixed strategy can be implemented in its immanent sense, i.e. purchases are made randomly with a distribution determined by the previously found optimal solution. If this process is long enough, then the portfolio structure
will approximately correspond to the type of mixed strategy. This approach of using the game with nature, taking into account the correlation dependence of random payoff of pure strategies, can also be applied to decision-making problems in other areas of risk management.

In harmonic analysis on a line with power weight, the unitary Dunkl transform first appeared.
It depends on only one parameter 𝑘 ⩾ 0. Then the two-parameter (𝑘, 𝑎)-generalized Fourier transform appeared, a special case of which is the Dunkl transform (𝑎 = 2). The presence of the parameter 𝑎 > 0 at 𝑎 ̸= 2 leads to the appearance of deformation properties. For
example, for functions in Schwarz space, the generalized Fourier transform may not be infinitely differentiable or decay rapidly at infinity. In the case of the sequence 𝑎=2/(2𝑟 + 1), 𝑟 ∈ Z+, the deformation properties of the generalized Fourier transform are very weak and after some change of variables they disappear. The resulting unitary transform for 𝑟 = 0 gives the usual Dunkl transform and has many of its properties. It is called the generalized Dunkl transform.
We define the intertwining operator that establishes a connection between the second-order differential-difference operator, for which the kernel of the generalized Dunkl transform is an eigenfunction, and the one-dimensional Laplace operator and allows us to write the kernel in a form convenient for its estimates. Unlike the intertwining operator for the Dunkl transform,
it has a nonzero kernel. In the paper, also on the basis of the properties of the generalized Dunkl transform, the properties of the (𝑘, 𝑎)-generalized Fourier transform for 𝑎 = 2/(2𝑟 + 1) are established.

The key issue of the paper is combinatorial complexity functions of infinite words, especially factor complexity and its modifications. First of all, we present an overview of the available results for the class of words with the minimal factor complexity - Sturmian words. Special
attention is paid to the arithmetical complexity of infinite words, the study of which was initiated
by Van der Waarden Theorem on one-color arithmetic progressions. Arithmetical complexity is presented in a sense a modification of factor complexity. An overview of current results and exact values of arithmetic complexity for Sturmian words is presented. We present polynomial Van der Waerden Theorem, which gives rise to the study of a more generalized modification of
the factor complexity function - the polynomial complexity of infinite words. In conclusion, we
present open problems for further research.

For ℎ, 𝑛 ≥ 1 and 𝑒 > 0 we consider a chromatic number of the spaces R^𝑛×[0, 𝑒]^ℎ and general results in this problem. Also we consider the chromatic number of normed spaces with forbidden monochromatic arithmetic progressions. We show that for any 𝑛 there exists a two-coloring of R^𝑛 such that all long unit arithmetic progressions contain points of both colors and this coloring covers spaces of the form R^𝑛×[0,𝑒]^ℎ.

In the cone of the space of continuous functions, the method of weight metrics (analogous to Bielecki’s method) is used to prove a global theorem on the existence, uniqueness, and method of finding a nontrivial solution to the initial problem for a homogeneous 𝑛-order integro-differential equation with a difference kernel and power nonlinearity. It is shown that this solution can be found by the method of successive approximations of the Picard type and an estimate is given for the rate of their convergence to the solution in terms of the weight metric. The study is based on the reduction of the initial problem to the equivalent nonlinear Volterra integral equation.
Exact lower and upper a priori estimates for the solution are obtained, on the basis of which a complete weighted metric space is constructed that is invariant with respect to the nonlinear operator generated by this Volterra integral equation. In contrast to the linear case, it has
been established that, in addition to the trivial solution, the non-linear homogeneous Volterra integral equation can also have a non-trivial solution. An analysis of the results obtained shows that with an increase in the order of an integro-differential equation with a power nonlinearity, the exponent decreases. Examples are given to illustrate the results obtained.

Classes of monoids were identified for which the condition of the generalized Selberg lemma is satisfied, for which the strong Selberg–Bredikhin condition is satisfied, and for which the strengthened asymptotic law in Bredikhin form is satisfied. For these classes of monoids, new results on analytical continuation to the left of the abscissa of absolute convergence are obtained.
An analogue of the main lemma of S. M. Voronin is obtained from the work on the universality of the Riemann zeta function in the case of zeta functions of a monoid for which the condition of the generalized Selberg lemma or the stronger Selberg–Bredikhin condition is satisfied.
For the class of regular Selberg–Bredikhin monoids of natural numbers, we succeeded in proving the universality theorem for the zeta function of the corresponding monoid.

We study some arithmetical properties of Farey fractions by the method introduced by F. Boca, C. Cobeli and A. Zaharescu (2001). Suppose that 𝐷 ⩾ 2 is a fixed integer and denote by Φ𝑄 the classical Farey series of order 𝑄. Now let us colour to the red the fractions in Φ𝑄 with denominators divisible by 𝐷. Consider the gaps in Φ𝑄 with coloured endpoints, that do not contain the fractions 𝑎/𝑞 with 𝐷|𝑞 inside. The question is to find the limit proportions 𝜈(𝑟;𝐷) (as 𝑄 → +∞) of such gaps with precisely 𝑟 fractions inside in the whole set of the gaps under considering (𝑟 = 1, 2, 3, . . .).
In fact, the expression for this proportion can be derived from the general result obtained by C. Cobeli, M. Vˆajˆaitu and A. Zaharescu (2014). However, such formula expresses 𝜈(𝑟;𝐷) in the terms of areas of some polygons related to some geometrical transform of «Farey triangle», that is, the subdomain of unit square defined by 𝑥 + 𝑦 > 1, 0 < 𝑥, 𝑦 ⩽ 1. In the present paper, we obtain the precise formulas for 𝜈(𝑟;𝐷) (in terms of the parameter 𝑟, 𝑟 = 1, 2, 3, . . .) for the cases 𝐷 = 2, 3.

In this paper there are given necessary and sufficient conditions under which a function of fixed variables 𝜓: F^(𝑖+1)_𝑞 → F_𝑞 is bijective, where 𝑖 ∈ N ∪ {0}, F(𝑖+1)_𝑞 is the (𝑖 + 1)-ary Cartesian power of the Galois field F_𝑞 of 𝑞 = 𝑝^𝑘 elements, 𝑝 is an odd prime number and 𝑘 ∈ N. In addition, such conditions of the bijective functions 𝜓 of fixed variables are used to write a criterion for the preserving Haar measure of functions from the important class of 1-Lipschitz functions in terms of its coordinate functions on the ring of 𝑝-adic integers Z_𝑝, 𝑝 ̸= 2. In particular, the representation of 1-Lipschitz functions in terms of its coordinate functions on the ring of 2-adic integers Z_2 turned out to be a general and useful tool for obtaining mathematical results applied in cryptography. In this work, the research of such representation of 1-Lipschitz functions on the ring of 𝑝-adic integers Z_𝑝, 𝑝 ̸= 2 is being continued, with special attention to the representation of bijective 1-Lipschitz functions in terms of its coordinate functions on Z_𝑝, 𝑝 ̸= 2.

The work considers the set of all possible Dirichlet series generated by a given lattice, and studies the properties of this function space over the field of complex numbers.
A new concept of 𝐶 𝜃-power density of a Dirichlet series is introduced. A connection is established between the 𝐶 𝜃-power density of the Dirichlet series and the abscissa of its absolute convergence.
It is established that if the Dirichlet series 𝑓(𝛼|Λ) satisfies the conditions of the generalized Selberg lemma with 𝜃1 < 𝜃, then the Dirichlet series 𝑓(𝛼|Λ) extends analytically into the halfplane
with 𝑠𝑖𝑔𝑚𝑎 > 𝜃1, except for the point 𝛼 = 𝜃, at which it has a first-order pole with a subtraction of 𝐶𝜃.
A new concept 𝐶 logarithmic 𝜃-power density of the Dirichlet series is introduced. It has been established that if the Dirichlet series 𝑓(𝛼|Λ) has 𝐶 logarithmic 𝜃-power density and 𝜃 < 1, then the abscissa of absolute convergence holds the equality 𝜎𝑓 = 0 and The Dirichlet series 𝑓(𝛼|Λ) is a holomorphic function in the entire right 𝛼-half-plane with 𝜎 > 0.
It is shown that if the Dirichlet series 𝑓(𝛼|Λ) has 𝐶 logarithmic 𝜃-power density and 𝜃 < 1, then The holomorphic domain of the zeta function 𝜁(𝑀|𝛼) is 𝛼-the half-plane 𝜎 > 0.

Based on the method proposed in the article for solving the so-called (𝑟, 𝑠)-systems of linear equations proven that the orders of homogeneous invariant differential operators 𝑛 of smooth real functions of one variable take values from 𝑛 to (𝑛(𝑛+1))/2 , and the dimension of the space of all such operators does not exceed 𝑛!. A classification of invariant differential operators of order 𝑛 + 𝑠 is obtained for 𝑠 = 1, 2, 3, 4, and for 𝑛 = 4 for all orders from 4 to 10. The only, up to factors, homogeneous invariant differential operators of the smallest order 𝑛 and the largest order (𝑛(𝑛+1))/2 are given, respectively, by the product of the 𝑛 first differentials (𝑠 = 0 ) and
the Wronskian (𝑠 = (𝑛 − 1)𝑛/2). The existence of nonzero homogeneous invariant differential operators of order 𝑛 + 𝑠 for 𝑠 <((1+√5)/2)*(𝑛 − 1) is proved.

The paper develops the theory of matrix integral Fourier transforms based on a differential operator with piecewise constant matrix coefficients. The definition of the matrix Fourier transform is given, its properties and applications to the modeling of interrelated wave processes in piecewise homogeneous media are studied. An inversion formula for the matrix integral Fourier transform is proved. Significant differences from the scalar case are revealed. A technique for applying the matrix Fourier transform to solving interrelated mixed boundary value problems for systems of hyperbolic differential equations with matrix piecewise constant coefficients is developed. A solution is found for the vector analog of the problem of wave propagation in
an infinite string with two regions of different density. A vector analogue of the d’Alembert formula is found. A solution is obtained for a mixed initial-boundary value problem for a system of differential equations of parabolic type, which describes an 𝑛 component model of an interconnected process of heat and mass transfer in a two-layer media.

The work is devoted to some arithmetic applications to the theory of symmetric groups.
Using the properties of congruences and classes of residues from number theory, the existence in the symmetric group 𝑆_𝑛 of degree 𝑛 of cyclic, Abelian and non-Abelian subgroups respectively, of orders is establisned 𝑘, 𝜙(𝑘), and 𝑘𝜙(𝑘), where 𝑘 ≤ 𝑛, 𝜙 – Euler function, those representations jf grups (Z/𝑘Z, +), (Z/𝑘Z)* and theorem product in the form of degree substitutions 𝑘. In this case isomorphic embeddings of these groups are constructed following the proof of Cayley’s theorem,
but along with this, a linear binomial is used Z/𝑘Z residue class rings, where gcd (𝑎, 𝑘) = 1.
In addition, the result concerning the isomorphic embedding of a group (Z/𝑘Z)* in to a group (Z/𝑘Z)* in to a group 𝑆_𝑘 extends to an alternating group 𝐴_𝑘 for odd 𝑘.
The second part of the work examines some applications of prime number theory to cyclic subgroups of the symmetric group 𝑆_𝑛. In particular, applying the Euler-Maclaurin summation formula and bounds for the 𝑘 in prime, a lower bound for maximum number of prime divisors of cyclic orders in the symmetric group 𝑆_𝑛.

An asymptotic formula is obtained for the number of representations of a sufficiently large natural 𝑁 in the form 𝑏_1𝑝_1 + 𝑏_2𝑝_2 + 𝑏_3𝑝_3 = 𝑁 with the conditions

$$ |𝑏_𝑖 𝑝_𝑖 - N/3|⩽ 𝐻, 𝐻 ⩾ (𝑏_1*𝑏_2*𝑏_3)^(4/3)𝑁^2/3)(ln𝑁)&60, 𝑏_𝑖 ⩽ (ln𝑁)^(𝐵_𝑖), $$

where 𝑏_1, 𝑏_2, 𝑏_3, 𝑁 are pairwise coprime natural numbers, 𝐵_𝑖 — arbitrary fixed positive numbers

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

In the theory of the hyperbolic zeta function of lattices, a significant role is played by the Bakhvalov theorem, in which the magnitude of the zeta function of the lattice of linear comparison solutions is estimated through the hyperbolic lattice parameter.
In N. M. Korobov’s 1963 monograph, this theorem is proved by a method different from the original work of N. S. Bakhvalov. In this method, the central role is played by the lemma about the number of linear comparison solutions in a rectangular area.
In 2002, V. A. Bykovsky obtained fundamentally new estimates from below and from above, which coincided in order.
The paper gives new estimates of the number of lattice points of linear comparison solutions in rectangular regions. This allows us to prove the strengthened Bakhvalov—Korobov—Bykovsky theorem on the estimate of the hyperbolic zeta function of the lattice of linear comparison solutions.
The difference between the theorem on the number of lattice points of solutions to linear comparison in rectangular areas and the corresponding Korobov lemma is that instead of an estimate through the ratio of the volume of a rectangular area to the hyperbolic parameter, a modified Bykovsky estimate is given through minimal solutions to linear comparison.
The use of the theorem on the number of lattice points of solutions to linear comparison in rectangular domains is supplemented by the generalized Korobov lemma on estimates of the residual series and a number of other modifications in the proof of the Bakhvalov—Korobov theorem, which made it possible to prove the strengthened Bakhvalov—Korobov—Bykovsky theorem on estimates hyperbolic zeta function of the lattice of linear comparison solutions.

In 2023, the 100th anniversary of the birth of the famous Russian scientist Fyodor Andreevich Medvedev (1923-1993), who made an invaluable contribution to the development of the world history of mathematics, was celebrated. For many decades, he was one of the leading researchers at the Institute of History, Natural Science and Technology (IIET) of the Russian Academy of Sciences. He wrote five monographs, each of which was innovative for historical and
mathematical literature, both in terms of volume and depth, and in terms of the significance of the material covered. His scientific works were highly appreciated abroad, for several years he was a member of the editorial board of the international journal "Historia Mathematica". The works he wrote on the history of mathematics have not lost their relevance at the present time.
The reconstruction of the scientific path of F.A. Medvedev is proposed, the characteristics of his fundamental works are given and their role in the development of the history of mathematics in Russia is revealed.

This article provides a brief overview of the largest research areas of the scientific school on applied probabilistic analysis and queuing theory, which has been formed over the past 50 years at the Department of Probability Theory and Mathematical Statistics of the National Research
Tomsk State University. The background of the formation of the department and the scientific interests of its employees are briefly described. Among the main areas of research are such as: management and adaptation in queuing systems; study of queuing systems with variable parameters; study of time series trends; study of doubly stochastic arrivals; identification of models and smoothing of experimental data by splines; search for a moving signal in a
multichannel system; study of mathematical models of communication networks with random multiple access protocols; development of methods of asymptotic analysis, limit decomposition, dynamic sieving; development of research methods for non-markovian systems and queuing networks with non-Poisson arrivals; development of research methods for queuing systems with random volume of resource requirements; development of new mathematical models of data transmission processes in the transport connection of telecommunication networks and their analysis in order to optimize their reliability and performance. For each direction of research, the article contains the names of researchers who have made a significant contribution to the development of the corresponding direction and received the most significant results.
The publications of some of them and the most fundamental works are referenced in the list of references. The article presents the main research topics being conducted at the moment, mentions the annual international conference held and founded by representatives of the
scientific school on the applied queuing theory, and bearing the name of its founder Alexander Fedorovich Terpugov.

The article deals with two problems of approximating a given positive number 𝑁 by the sum of two primes, and by the sum of a prime and two squares of primes.
In 2001, R. Baker, G. Harman, and J. Pintz proved for the number of solutions of the inequality |𝑝 − 𝑁| ⩽ 𝐻 in primes 𝑝 a lower bound for 𝐻 ⩾ 𝑁^(21/40+𝜀), where 𝜀 is an arbitrarily small positive number. Using this result and the density technique, in this paper we prove a lower bound for the number of solutions of the inequality |𝑝_1 + 𝑝_2 − 𝑁| ⩽ 𝐻 in prime numbers 𝑝_1, 𝑝_2 for 𝐻 ⩾ 𝑁^(7/80+𝜀.)
Also based on the density technique, we prove a lower bound for the number of solutions of the inequality |𝑝^2_1+ 𝑝^2_2+ 𝑝_3 − 𝑁|⩽ 𝐻 in prime numbers 𝑝_1, 𝑝_2 and 𝑝_3 for 𝐻 ⩾ 𝑁^(7/72+𝜀).

The paper obtained a new estimate for the error of quadrature formulas with optimal parallelepipedal meshes modulo 𝑁.

In this paper, we show that factor complexity of the infinite word F𝑏 is defined by concatenating base-𝑏 representations of the 𝑛! is full. Then we show that the arithmetic complexity of this word is full as well. On the other hand, F𝑏 is a disjunctive word. In number theory, this kind of words is called rich numbers.

The paper obtained a new estimate for the error of quadrature formulas with optimal parallelepipedal meshes modulo 𝑁.

The article studies the following problem. Let two finite subsets from the set of natural numbers be given, which will be denoted throughout the text as 𝐴 and 𝐵. We will assume that they belong to a finite interval of numbers [1,𝑄]. By definition, we define a set of fractions 𝐴/𝐵 whose elements are representable as a quotient of these sets 𝐴,𝐵, in other words such elements 𝑎/𝑏, where 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵. The article investigates the properties of this subset of quotients. In
the article [13], a non-trivial lower bound on the size of the set 𝐴/𝐵 for such sets 𝐴,𝐵 was obtained without any additional conditions on these sets. In this article, we in details consider an extreme case, which is as follows. Let it be known that the size of the set of products 𝐴𝐵 is
asymptotically the smallest possible. We deduce from this that the size of the set of quotients 𝐴/𝐵 is the asymptotically largest possible value.