For any irrational or transcendental number estimating of the quality of its approximation by rational fractions is one of the directions in the theory of Diophantine approximations. The quantitative characteristic of such approximation is called the measure (extent) of irrationality of the number. For almost a century and a half, scientists have developed various methods for evaluating the measure of irrationality and have obtained its values for a huge number
of irrational and transcendental numbers. Various approaches have been used to obtain the estimates and these approaches improved over time, leading to better estimates. The most commonly used method for obtaining such estimates is construction of linear forms with integer coefficients, which approximate a value, and studying of its asymptotic behavior. Approximating linear forms usually are constructed on the basis of continued fractions, Pad´e approximants,
infinite series, and integrals. Methods for studying the asymptotics of such forms are currently quite standard, but the main problem is invention of a linear form with good approximating properties. The first estimates of the values of the arctangent function were obtained by M. Huttner in 1987 on the base of integral representation of the Gausss function. In 1993 A. Heimonen, T. Matala-Aho, K. Vaananen, using, like M. Huttner, Pad´e approximants for the Gaussian
hypergeometric function, proved a general theorem for estimating of measures of irrationality of logarithms of rational numbers. Later, the same authors, using an approximating construction with Jacobi polynomials, obtained new estimates, in particular for the values of the function arctan 𝑥. Further research used various integral constructions, which allowed to get both general methods for arctan 𝑥 values and specialized methods for specific values. In the articles of E.B. Tomashevskaya, who in 2008 received a general estimate for the values of arctan 1/𝑛, 𝑛 ∈ N, was used a complex integral with the property of symmetry of integrand. This property played an
important role in obtaining the estimates, since it improved the asymptotic behavior of the coefficients of the linear form. Some integral constructions elaborated by other researchers also
had different types of symmetry. In this article, we consider the main methods for estimating the values of the arctangent function, their features, research methods, and the best estimates at the moment.

The problem of classifying integer quadratic forms has a long history, during which many mathematicians have contributed to its solution. Binary forms were comprehensively studied by Gauss. He and later researchers also outlined the main ways to solve the problem of classifying ternary forms and forms of higher dimensions. The greatest achievements of the subsequent period were the deep development of the theory of rational quadratic forms and the complete classification of indefinite forms in dimensions 3 and higher by Eichler in terms of spinor genera. The paper proposes an algorithm for calculating non-equivalent quadratic forms corresponding to the faces of the Voronoi domain of the second perfect form in many variables, and using this algorithm, all corresponding non-equivalent quadratic forms are calculated.

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

$$
\varepsilon^2\left(a^2\frac{\partial^2 u}{\partial x^2}-\frac{\partial u}{\partial t}\right)=F(u,x,t,\varepsilon),   \quad  (x,t)\in \Omega,
$$
$$
u(x,0,\varepsilon)=\varphi(x),    \quad 0\le x\le 1,
$$
$$
u(0,t,\varepsilon)=\psi_1(t), \quad u(1,t,\varepsilon)=\psi_2(t), \quad 0\le t\le T.
$$

Research is carried out under the assumption that at the corner points (𝑘, 0) of the rectangle Ω, where 𝑘 = 0 or 1, the function 𝐹(𝑢) = 𝐹(𝑢, 𝑘, 0, 0) is cubic and has the form

$$
F(u)=(u-\alpha(k))(u-\beta(k))(u-\bar u_0(k)), \quad\mbox{где}\quad  \alpha(k)\leq\beta(k)<\bar u_0(k).
$$

The nonlinear method of angular boundary functions is used, which combines the (linear) method of angular boundary functions, the method of upper and lower solutions (barriers), and the method of differential inequalities. Under the condition 𝜙(𝑘) > ¯𝑢0(𝑘), a complete asymptotic expansion of the solution for 𝜀 → 0 is constructed and its uniformity in a closed rectangle is substantiated.
Previously, the following cases of cubic nonlinearities were considered:

$$
F(u)=u^3-\bar u^3_0, \quad\mbox{where}\quad \bar u_0=\bar u_0(k)>0,
$$ 

under the assumption that the boundary value 𝜙(𝑘) > ¯𝑢0(𝑘), as well as the case

$$
F(u)=u^3-\bar u^3_0, \quad\mbox{where}\quad \bar u_0=\bar u_0(k)< 0,
$$

under the assumption that the boundary value 𝜙(𝑘) is contained in the interval 

$$
\bar u_0<\varphi(k)<\frac{\bar u_0}{2}< 0.
$$

We consider the problem on uniform estimates for an oscillatory integrals with the smooth phase functions having singularities 𝐷_∞. The estimate is sharp and analogy to estimates of the work of V.N.Karpushkin.

This paper is devoted to the summation problem for trigonometric integrals with quadratic phase. The particular cases of this problem were considered in [2],[3],[4]. We generalize the results of these papers to the multidimensional exponential integrals.

In this paper we consider a planar billiard bounded by an ellipse in the potential force field.
An explicit formula of the polynomial potential preserving integrability of such a billiard was found. The structure of the Liouville foliation at all non singular energy levels was studied using the method of separation of variables. Namely, an algorithm that constructs the bifurcation diagram and the Fomenko-Zieschang invariants from the values of the parameters of the potential was proposed. In addition, the topology of the isoenergetic manifold was studied and the cases of rigid body dynamics, which are Liouville equivalent to our billiard, were established.

In this paper we study properties of congruence lattices 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 a chain algebra if 𝐴 has a linearly ordered congruence lattice. 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 part which can contain arbitrary operations, and the additional part consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one are permutable with the main operations. An unar is an algebra with one unary operation. If 𝑓 is the unary operation from the signature Ω then the unar ⟨𝐴, 𝑓⟩ is called an unary reduct of algebra ⟨𝐴,Ω⟩.
A description of algebras with one operator and the main symmetric operation that have a linear ordered congruence lattice is obtained. It shown that algebra of given class is a chain algebra if and only if one is subdirectly irreducible. For algebras of given class we obtained necessary and sufficient conditions for the coincidence of their congruence lattices and congruence lattices of unary reducts these algebras.

This work is devoted to the algebraic theory of automata, which is one of the branches of mathematical cybernetics, which studies information transformation devices that arise in many applied problems. Depending on a specific problem, automata are considered, in which the main sets are equipped with additional mathematical structures consistent with the functions of an automaton. In this work, we study automata over graphs — graphic automata, that is, automata in which the set of states and the set of output signals are equipped with the mathematical structure of graphs. For graphs 𝐺 and 𝐻 universal graphic automaton Atm(𝐺,𝐻) is a universally attracting object in the category of semigroup automata. The input signal
semigroup of such automaton is 𝑆 = End 𝐺×Hom(𝐺,𝐻). Naturally, interest arises in studying the question of abstract characterization of universal graph automata: under what conditions will the abstract automaton 𝐴 be isomorphic to the universal graph automaton Atm(𝐺,𝐻) over graphs 𝐺 from the class K_1, 𝐻 from class K_2? The purpose of the work is to study the issue of elementary axiomatization of some classes of graphic automata. The impossibility of elementary
axiomatization by means of the language of restricted predicate calculus of some wide classes of such automata over reflexive graphs is proved.

In this paper a generalisation of Legendre’s three-square theorem to representations of two positive integers as sums of three squares for which the first square of each representation is the same is presented.

A polyomino is a connected figure on a plane composed from a finite number of unit squares adjacent to each other on the sides. A tiling of a plane into polyominoes is called isohedral if the symmetry group acts transitively on it, that is, if for any two polyominoes of the tiling there is a global symmetry of the tiling that moves one polyomino into the second. The paper considers the problem of counting the number of polyominoes of area 𝑛 that generate isohedral tilings of the plane. It is shown that the number of such polyominoes does not exceed 𝐶(𝜀)𝑛^4(𝜔 + 𝜀)^𝑛, where 𝜔 is the connective constant of the square lattice Z^2. It is known that 𝜔 < 2.7. Similar estimates were also obtained in the case where the perimeter rather than the area of the polyomino is fixed. In addition, a similar estimate is valid for the number of isohedral tilings of the plane themselves under the additional condition of regularity of the tilings Previously, similar results were obtained in the case of lattice tilings of the plane into
polyominoes, for the so-called 𝑝2-splits, as well as for lattice tilings into centrally symmetric polyominoes.
The proof is based on the criteria for the existence of an isohedral tiling of the plane into polyominoes obtained by Langerman and Winslow, as well as on counting the number of selfavoiding
random walks on the lattice Z2, both standard and with a given symmetry group.
In conclusion, possible directions for further research and some open problems are briefly discussed.

The article substantiates the importance of mathematical modeling in solving practice-oriented problems by students of the training direction 44.03.05 Pedagogical education of the Mathematics and Computer Science profiles. An overview of current research in the field of standardization of the content of the subject area «Mathematics and Computer Science» is provided; didactic possibilities of applied problems studied in the school course of mathematics and computer science; the potential of such tasks in revealing interdisciplinary connections of school academic disciplines; the meanings of mathematical modeling in the process of solving them. Based on the analysis of the cited sources, the complexity and versatility of the currently created mathematical models are noted, connecting elements of theory from various fields of knowledge and requiring the connection of tools from several information technologies and
technical solutions.
The authors describe in detail and illustrate with examples the main stages of constructing a mathematical model. The theoretical provisions are concretized using the example of solving a practice-oriented problem of modeling human biorhythms. The presented problem is offeredto students as part of the study of the topic «School educational tasks» of the course «Theory and Methods of Teaching Computer Science». The solution of this kind of problems by future
mathematics and computer science teachers is aimed at achieving several goals: to develop mathematical modeling skills; apply in practice previously developed information technology skills necessary to effectively complete the task; broaden your horizons in terms of related fields of science (biology, physiology). This approach to the process of training future teachers, according to the authors, allows students not only to appreciate the importance of mathematical
modeling, but also to develop the necessary skills for solving problems with practical content.

In the article the problem of the acoustic radiation of a spheroid streamlined by a stationary flow of an ideal liquid is considered.
It is assumed that the velocity of the incoming flow is significantly lower than the speed of sound. Part of the surface of the spheroid makes harmonic vibrations and the rest part is absolutely rigid.
The problem is solved in a prolate spheroidal coordinate system. An approximate analytical solution of the problem was obtained with using the speed potential of the oncoming on the body flow and the speed potential of the stationary radiator acoustic field.
The results of numerical calculations of polar diagrams of the acoustic pressure distribution on the surface of a spheroid at different values of the ratio of the flow velocity to the speed of sound and different configurations of the spheroid are presented.

In this paper, we examine the 𝑤𝑡−distance characteristics over 𝑏−metric space and the conditions required to ensure the presence of the fixed point by letting 𝛽−function appropriately.
In addition, we prove some fixed point theorems.

This note discusses two applications of the asymptotic formula obtained by the authors for the number of values of the Beatty sequence in an arithmetic progression with increasing difference: asymptotic formulas are obtained for the number of elements of the Beatty sequence that are coprime to the (possibly growing) natural number 𝑎, as well as for the number of pairs of coprime elements of two Beatty sequences. Here are the main results.
Let 𝛼 > 1 be an irrational number and 𝑁 be a sufficiently large natural number. Then if the partial quotients of the continued fraction of the number 𝛼 are limited, then for the number 𝑆_𝛼,𝑎(𝑁) of elements of the Beatty sequence [𝛼𝑛], 1 ⩽ 𝑛 ⩽ 𝑁, coprime to the number 𝑎,
the following asymptotic formula holds

$$
S_{\alpha,a}(N)=N\frac{\varphi(a)}{a} + O\left(\min(\sigma(a)\ln^3 N, \sqrt{N}\tau( a)(\ln\ln N)^3)\right),
$$

where 𝜏 (𝑎) is the number of divisors of 𝑎 and 𝜎(𝑎) is the sum of the divisors of 𝑎.
Let 𝛼 > 1 and 𝛽 > 1 be irrational numbers and 𝑁 be a sufficiently large natural number. Then if the incomplete quotients of continued fractions of the numbers 𝛼 and 𝛽 are bounded, then for the number 𝑆_𝛼,𝛽(𝑁) of pairs of coprime elements of Beatty sequences [𝛼𝑚], 1 ⩽ 𝑚 ⩽ 𝑁, and [𝛽𝑛], 1 ⩽ 𝑛 ⩽ 𝑁, the following asymptotic formula holds

$$
S_{\alpha,\beta}(N)=\frac{6}{\pi^2}N^2 + O\left(N^{3/2}(\ln\ln N)^6 \right).
$$

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 ⩽ 𝐴 and 𝑌 ⩽ 𝑇 there exists an element 𝑢 ∈ ⟨𝑋, 𝑌 ⟩ such that 𝑋𝑌^𝑢 ≤ 𝐺. The notation 𝐻 ⩽ 𝐺 means that 𝐻 is a subgroup of a group 𝐺. In this paper we proved that the class of all SM-groups is closed under the product of tcc- subgroups. Here an SM-group is a group where each subnormal subgroup permutes with every maximal subgroup.

This article continues a series of papers devoted to explicit constructions of Galois extension of complete discrete valuation fields of characteristic 0 with the residue field of prime characteristic 𝑝, see [5], [6], [7], [8], [4], [10] and a survey article [9].
It is proved that any 𝑝-extension of a complete discrete valuation field containing a primitive 𝑝-th root of unity can be embedded into a tower of Artin-Schreier extensions; an estimate for the height of this tower is obtained. This result also shows that such an extension can be embedded into Inaba extension, i. e., an extension obtained by the construction from [2]; an estimate for the order of the corresponding matrix is also obtained.
Next, it is proved that any Galois 𝑝-extension of such field can be decomposed into a tower of Galois extensions of degree 𝑝 such that several upper levels have the maximal ramification jump whereas the lower ones are Artin-Schreier extensions.

In this paper we propose effective construction of simultaneous approximations for some hypergeometric functions of a special type and their derivatives with respect to parameter.
This construction is made use of for the achievement of the lower estimates of numerical linear forms of the values of such functions. Some parameters of these functions can be irrational.