This work is devoted to the fifteenth edition of Chebyshevskii sbornik and output of the 60th issue of the magazine. The article questions sanctified history of the magazine. We describe the stages of formation. It talked about the contribution of various scientists in the journal work. Provides some scientometric indicators.

In this paper a one-dimensional non-stationary heat conduction problem, modeling the process of rapid local heating of the sample beam type on the lateral surface is considered. The character of heating is such that it is possible to allocate only a certain directionof the heat propagation. Temperature fields are determined by an approximate method based on the idea of the thermal front. The solution is sought in the form of a power series in the coordinate with coefficients depending on time. The boundaries of the front heat distribution as a function of time are determined by the condition of the integral satisfaction of the heat conduction equation. Considered temperature fields arise in many industrial processes, such as laser material processing, when due to large temperature gradients can arise thermal stresses, leading to microcracking inner layers or the destruction of structural elements. Analytical view of the heat conduction problem’s solution allows to obtain analytical expressions for the thermal stresses and further facilitates the results analysis. The paper presents the solution of problems with boundary conditions of the first and second kinds for two monotonic and one non-monotonic dependencies of the thermal conductivity coefficient on temperature. The approximate solution and the exact solution of the nonstationary linear problem are compared and shows the suitability of the method for future use.

The main algorithmic problems of group theory posed by M. Dehn are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group’s isomorphism problem. Among the works related to the study of the M. Dehn’s problems, the most outstanding ones are the work of P. S. Novikov who proved the undecidability of the problem of words and the conjugacy problem for finitely presented groups as well as the undecidability of the problem of isomorphism of groups. In this regard, the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups were introduced by H. S. M. Coxeter: every reflection group is a Coxeter group if its generating elements are reflections with respect to hyperplanes limiting its fundamental polyhedron. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point. In an algebraic aspect Coxeter groups are studied starting with works by J. Tits who solved the problem of words in certain Coxeter groups. The article describes the known results obtained in solving algorithmic problems in Coxeter groups; the main purpose of the paper is to analyze of the results of solving algorithmic problems in Coxeter groups that were obtained by members of the Tula algebraic school ’Algorithmic problems of theory of the groups and semigroups ’ under the supervision of V. N. Bezverkhnii. It reviews assertions and theorems proved by the authors of the article for the various classes of Coxeter groups: Coxeter groups of large and extra-large types, Coxeter groups with a tree-structure, and Coxeter groups with n-angled structure. The basic approaches and methods of evidence among which the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii concerning the introduction of R-cancellations, special R-cancellations, special ring cancellations as well as method of graphs, method of types worked out by V. N. Bezverkhnii, method of special set of words designed by V. N. Bezverkhnii on the basis of the generalization of Nielsen method for free construction of groups. Classes of group considered in the article include all Coxeter groups which may be represented as generalized tree structures of Coxeter groups formed from Coxeter groups with tree structure with replacing some vertices of the corresponding tree-graph by Coxeter groups of large or extra-large types as well as Coxeter groups with n-angled structure.

In paper we describe the way N. P. Romanoff proved his additive theorem and sufficient conditions to obtain its analogues for sets with similar distribution and arithmetic. Also the example of set with similar distribution but with different arithmetic is given. We prove that the Romanoff theorem’s analogue for this set is incorrect.

The functional independence of zeta-functions is an interesting nowadays problem. This problem comes back to D. Hilbert. In 1900, at the International Congress of Mathematicians in Paris, he conjectured that the Riemman zeta-function does not satisfy any algebraicdifferential equation. This conjecture was solved by A. Ostrowski. In 1975, S.M. Voronin proved the functional independence of the Riemann zeta-function. After that many mathematicians obtained the functional independence of certain zeta- and L-functions. In the present paper, the joint functional independence of a collection consisting of the Riemann zeta-function and several periodic Hurwitz zeta-functions with parameters algebraically independent over the field of rational numbers is obtained. Such type of functional independence is called as “mixed functional independence” since the Riemann zeta-function has Euler product expansion over primes while the periodic Hurwitz zeta-functions do not have Euler product.

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

This survey contains enlarged version of a mini-course which was read by the author in November 2015 during “Chinese - Russian workshop of exponential sums and sumsets”. This workshop was organized by professors Chaohua Jia (Institute of Mathematics, Academia Sinica) and Ke Gong (Henan University) in Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for the support and hospitality. The survey contains the Introduction, three parts and Conclusion. The basic definitions and results concerning the complete Kloosterman sums are given in the Introduction. The method of estimating of incomplete Kloosterman sums to moduli equal to the raising power of a fixed prime is described in the first part. This method is based on one idea of A. G. Postnikov which reduces the estimate of such sums to the estimate of the exponential sums with polynomial by I. M. Vinogradov’s mean value theorem. A. A. Karatsuba’s method of estimating of incomplete sums to an arbitrary moduli is described in the second part. This method is based on a very precise estimate of the number of solutions of one symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in thie problems under considering as Vinogradov’s mean value theorem in the estimating of corresponding exponential sums. The method of J. Bourgain and M. Z. Garaev is described in the third part. This method is based on very deep “sum-product estimate” and on the improvement of A. A. Karatsuba’s bound for the number of solutions of symmetric congruence. The Conclusion contains a series of recent results concerning the estimates of short Kloosterman sums.

This paper discusses issues related to the rate of convergence of the Bubnov–Galerkin method in numerical calculation of stress-strain state of geometrically nonlinear shells in the dynamic case. To address these issues involved the unit strongly continuous semigroups of limited operators. Methods of functional semigroups of operators was applied effectively in the theory of boundary value problems since the 60s XX-th century. It should be noted author E. Hill, R. Phillips, S. G. Krein, S. Mizohata and others. So, using the methods of strongly continuous semigroups of operators S. G. Krein proved a new theorem on the existence and uniqueness of solutions of linear equations of mechanics in late 60s. In 2000, V. N. Kuznetsov and T. A. Kuznetsova first used the methods limited semigroups of operators to solution of linear equations of shallow shells, which solved the problem of smoothness of solutions of linear systems of equations of shells. At the same time V. N. Kuznetsov and T. A. Kuznetsova have developed a method called a linear approximation in separated parameters, which allow to solve the problem of smoothness of solutions of nonlinear equations of the theory of plates and shells. This made it possible to determine the speed of convergence of the Bubnov–Galerkin method the numerical solution of nonlinear boundary value problems for the geometrically nonlinear shells in the area of sustainability in the parameters. In this paper, we complete the proof of the result of the rate of convergence of the Bubnov–Galerkin method in the case of an arbitrary configuration shell borders.

In this paper we consider a class of Dirichlet series with multiplicative coefficients which define functions holomorphic in the right half of the complex plane, and for which there are sequences of Dirichlet polynomials that converge uniformly to these functions in any rectangle within the critical strip. We call such polynomials approximating Dirichlet polynomials. We study the properties of the approximating polynomials, in particular, for those Dirichlet series, whose coefficients are determined by nonprincipal generalized characters, i.e. finite-valued numerical characters which do not vanish on almost all prime numbers and whose summatory function is bounded. These developments are interesting in connection with the problem of the analytical continuation of such Dirichlet series to the entire complex plane, which in turn is tied with the solution of a well-known Chudakov hypothesis about every generalized character being a Dirichlet character.

We prove the completeness of the list of closed convex polyhedra in E3, that are strongly symmetric with respect to the rotation of the faces . Polyhedron is called symmetric if it has at least one non-trivial rotation axis. All axes intersect at a single point called the center of the polyhedron. All considered polyhedra are polyhedra with the center. A convex polyhedron is called a strongly symmetrical with respect to the rotation of the faces, if each of its faces Fhas an rotation axis L, intersects the relative interior of F, and Lis the rotation axis of the polyhedron. It is obvious that the order of rotation axis of Ldoes not necessarily coincide with the order of this axis, if the face of Fregarded as a figure separated from the polyhedron. It has previously been shown, that the requirement of global symmetry of the polyhedron faces the rotation axis can be replaced by the weaker condition of symmetry of the star of each face of the polyhedron: to polyhedron was symmetrical with respect to the rotation of the faces, it is necessary and sufficient that some nontrivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. Under the star of face Fis understood face itself and all faces have at least one common vertex with F. Given this condition, the definition of the polyhedron strongly symmetric with respect to the rotation of the faces is equivalent to the following: the polyhedron is called a strongly symmetrical with respect to the rotation of the faces , if some non-trivial rotation axis of each face, regarded as a figure separated from the polyhedron, is the rotation axis of the star of face. In the proof of the main theorem on the completeness of the list of this class of polyhedra using the result of the complete listing of the so- called polyhedra of 1st and 2nd class [1]. In this paper we show that in addition to the polyhedra of the 1st and 2nd class, listed in [1], only 8 types of polyhedra belongs to the class of polyhedra stronghly symmetric with respect to the rotation of faces. Seven of this eighteen types are not combinatorially equivalent regular or semi-regular (Archimedean). One type of eight is combinatorially equivalent Archimedean polyhedra, but does not belong to polyhedra of 1st or 2nd class. Turning to the polyhedra, dual strongly symmetrical about the rotation of faces, that is, to the polyhedra, stronghly symmetric about the rotation of polyhedral angles, we get their complete listing. It follows that there are 7 types of polyhedra, highly symmetric with respect to the rotation of polyhedral angles which are not combinatorially equivalent to Gessel bodies. Class of polyhedra stronghly symmetric with respect to the rotation of faces, as well as polyhedra 1st and 2nd class mentioned above can be viewed as a generalization of the class of regular (Platonic) polyhedra. Other generalizations of regular polyhedra can be found in [3],[4],[12]-[15].

It is known that many of the problems of mathematical physics, reduced to a differential equation with partial derivatives written in cylindrical and spherical coordinates, by using method of separation of variables, in particular, leads to the Bessel differential equation and Bessel functions. In practice, especially in problems of electrodynamics, celestial mechanics and modern applied mathematics most commonly used Fourier series in orthogonal systems of special functions. Given this, it is required to determine the conditions of expansion of functions in series into these special functions, forming in a given interval a complete orthogonal system. The work is devoted to obtaining accurate estimates of convergence rate of Fourier series by Bessel system of functions for some classes of functions in a Hilbert space L2 := L2([0, 1], xdx) of square summable functions f: [0, 1] → R with the weight x. The exact inequalities of Jackson–Stechkin type on the sets of L2(r) 2 (D), linking En−1(f)2 — the best approximation of function fby partial sums of order n−1 of the Fourier–Bessel series with the averaged positive weight of generalized modulus of continuity of morder Ωm(

The concept of Rees congruence was originally introduced for semigroups. R. Tichy generalized this concept to universal algebras. Let Abe an universal algebra. Denote by △ the identity relation on A. Any congruence of the form B2 ∪△ on Afor some subalgebra Bof Ais called a Rees congruence. Subalgebra Bof Ais called a Rees subalgebra whenever B2 ∪△ is a congruence on A. An algebra Ais called a Rees algebra if its every subalgebra is a Rees one. In this paper we introduce concepts of Rees simple algebra and Rees congruence algebra. A non-one-element universal algebra Ais called Rees simple algebra if any Rees congruence on A is trivial. An universal algebra Ais called Rees congruence algebra if any congruence on Ais Rees congruence. 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. For algebras with one operator and an arbitrary basic signature some conditions to be Rees algebra are obtained. Necessary condition under which algebra of the same class is Rees congruence algebra is given. For algebras with one operator and a connected unary reduct that has a loop element and does not contain the nodal elements, except, perhaps, the loop element necessary condition for their Rees simplicity are obtained. A n-ary operation (n> 3) is called near-unanimity operation if it satisfies the identities (x, . . . , x, y) = (x, . . . , x, y, x) = . . . = (y, x, . . . , x) = x. If n= 3 then operation is called a majority operation. Rees algebras and Rees congruence algebras of class algebras with one operator and basic near-unanimity operation g(n)which defined as follows g(3)(x1, x2, x3) = m(x1, x2, x3), g(n)(x1, x2, . . . , xn) = m(g(n−1)(x1, x2, . . . , xn−1), xn−1, xn) (n > 3) are fully described. Under m(x1, x2, x3) we mean here a majority operation which permutable with unary operation and which was defined by the author on arbitrary unar according to the approach offered by V.K. Kartashov.

In the article drl-semirings are studied. The obtained results are true for drl-semigroups, because a drl-semigroup with zero multiplication is drl-semiring. This algebras are connected with the two problems: 1) there exists common abstraction which includes Boolean algebras and lattice ordered groups as special cases? (G. Birkhoff); 2) consider lattice ordered semirings (L. Fuchs). A possible construction obeying of the first problem is drl-semigroup, which was defined by K. L. N. Swamy in 1965. As a solution to the second problem, Rango Rao introduced the concept of l-semiring in 1981. We have proposed the name drl-semiring for this algebra. In the present paper the drl-semiring is the main object. Results of K. L. N. Swamy for drl-semigroups are extended and are improved in some case. It is known that any drl-semiring is the direct sum S = L(S) ⊕ R(S) of the positive to drl-semiring L(S) and l-ring R(S). We show the condition in which L(S) contains the least and greatest elements (theorem 2). The necessary and sufficient conditions of decomposition of drl-semiring to direct sum of l-ring and Brouwerian lattice (Boolean algebra) are founded at theorem 3 (resp. theorem 4). Theorems 5 an 6 characterize l-ring and cancellative drl-semiring by using symmetric difference. Finally, we proof that a congruence on drl-semiring is Bourne relation.

In this article we prove that, if integer polynomial Psatisfies |P(w)|p< H−w, then for w > 2n− 2 and sufficiently large H the root belongs to the field of p-adic numbers.

We present A. V. Malyshev‘s approach to Minkowski‘s conjecture (in Davis‘s amendment) concerning the critical determinant of the region |x|p+ |y|p< 1 for p > 1 and Malyshev‘s method. In the sequel of this article we use these approach and method to obtain the main result.

This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V.I. Bernik.