This article is dedicated to the memory of the famous Russian and French scientist, a mathematician Michel Deza, who tragically passed away on November 23, 2016, at the age of 77. The review of the main stages of the professional formation and growth of M. Deza (Mikhail Emovich Tylkin) in Russia in the 60-70s of the last century is given. His multilateral international scientic activity has been highlighted since he moved to France in 1973. The main directions of his fundamental mathematical and applied research with numerous co-authors are analyzed. A list of the main scientic publications of M. Deza is presented. Short review of the work of Michel Deza as a Russian poet is presented.

In the present paper we will consider the generalization of some methods for evaluation of irrationality measures for yd = pd lnppd+1 d1 and currently known results overview. The extent of irrationality for various values of Gauss hypergeometric function were estimated repeatedly, in particular for 2F(1; 1 2 ; 3 2 ; 1 d ) = p d ln p pd+1 d1 : The rst such estimates in some special cases were obtained by D. Rhinn [1], M. Huttner [2], D. Dubitskas [3]. Afterward by K. Vaananen, A. Heimonen and D. Matala-Aho [4] was elaborated the general method, which one made it possible to get upper bounds for irrationality measures of the Gauss hypergeometric
function values F(1; 1k ; 1 + 1k ; rs ); k 2 N; k > 2; rs 2 Q; (r; s) = 1; r s 2 (1; 1): This method used the Jacobi type polynomials to construct rational approach to the hypergeometric function. In [4] have been obtained many certain estimates, and some of them have not been improved till now. But for the special classes of the values of hypergeometric function later were elaborated especial methods, which allowed to get better evaluations. In the papers [5], [6] authors, worked under supervision of V.Kh.Salikhov, obtained better estimates for the extent of irrationality for some specic values d: In the basis of proofs for that results were lying symmetrized integral constructions. It should be remarked, that lately symmetrized integrals uses very broadly for researching of irrationality measures. By using such integrals were obtained new estimates for ln 2( [7]),ln 3; ln , ( [8], [9]) and other values. Here we present research and compare some of such symmetrized constructions, which earlier allowed to improve upper bounds of irrationality measure for specic values of yd.

This paper discusses the polynomials of two projectors that with any selection of these projectors have the value of the nonsingular matrix. Results of work [1] about block-triangular form pair of projectors apply to deduce equations, that the coecients of always nonsingular polynomials satisfy to. From the equations is obtained the main result, namely always nonsigular polynomial can be decomposed into a product of special polynomials. Special polynomial of two projectors P; Q is a linear binomial I + P; I + Q, or a polynomial like this I + x1(PQP  PQ) + x2(PQPQP  PQPQ) + ::: . It is proved that special polynomials are irreducible. It turns out that linear binomials can be rearranged with some other special polynomials. If in a product of special polynomials the linear binomials are rearranged as much as possible to the left, you will get a product of special polynomials, called standard. It is proved that the standard form of product by special polynomials is unigue. The obtained results have provided a description of the structure of all polynomials of two projectors that with any selection of these projectors are nilpotent matrices (nilpotent polynomials). Similar results were obtained for the involute polynomials and polynomialsprojectors.

Chebyshev polynomials are widely used in theoretical and practical studies. Recently, they have become more signicant, particularly in quantum chemistry. In research [1] their important properties are described to "provide faster convergence of expansions of functions in series of Chebyshev polynomials, compared with their expansion into a power series or in a series of other special polynomials or functions"([1], p. 6). In this paper, a result associated with an approximation theory is presented. To some extent, the analogues of this result were obtained from other studies, such as in [2] [4], respectively for the power series, as well as the series in Hermite and Faber polynomials. With regard to the denition of the signicance of the series in Chebyshev polynomials listed above, the result of this research is of particular signicance in contrast to these analogues. More precisely, we can assume that the practical solution to the particular problems, can be solved much faster with the use of Chebyshev polynomials rather than the usage of such amounts related to power series [2] and the series in Hermite polynomials [3]. In addition, it is considered the rst synthesis of the universal series for polynomials with a density of one. The concept of a universal series of functions is associated with the notion of approximation of functions by partial sums of the corresponding rows. In [2] [19] the universal property of certain functional series are reviewed. In [2] [4], [18] a generalization of this property is considered. This paper generalizes the universality series properties in Chebyshev polynomials.

This article provides a proof of the "hypothesis about of centroids", which is given in the "Experimental validation of hypotheses in GeoGebra and published in the current issue of the "Сhebyshevskiy sbornik". This hypothesis is formulated as follows: "Let are a non-degenerate triangle from each vertex held the median. Then the original triangle is split into six triangles without common interior points so that their centroids lie on the same ellipse. The proof of the hypothesis is based on symbolic computation, implemented in ve packages of computer mathematics GeoGebra, Mathcad Prime, Maxima, Maple and Mathematica [2-8]. The use of dierent systems of symbolic computation for solving a problem allows to obtain visual material for comparative assessment of these systems. In the nal part of the article oers to consider another statement: "the hypothesis about of circumcenters". It is formulated so: "Let the three cevian intersect inside acute-angled triangle in the circumcenter. Then the original triangle is split into six triangles without common interior points so that their circumcenters lie on the same ellipse. This hypothesis was proposed and conrmed experimentally, using a dynamic model constructed in GeoGebra.

In this paper we propose several hypotheses related to cevias of triangle and the conic sections passing through the grounds of these cevians or via other points. To formulate these hypotheses and implement their experimental test have been used dynamical mathematics environment GeoGebra. Check each of hypotheses <1-<9 was carried out on a specially built for her dynamic model. In all cases, it was experimentally managed conrm the validity of the proposed hypothesis. Search of mathematical proofs of these hypotheses we did not make, and here is something to think about for the reader. Here is the wording of three of the nine hypotheses. Hypothesis <3. In an arbitrary non-degenerate acute-angled triangle, the grounds of the three altitudes and the grounds of three medians drawn from dierent vertices lie on the same circle. Hypothesis <6. Let from each vertex a non-degenerate triangle held the median. Then this triangle is splited into six triangles without common interior points so that their centroids lie on the same ellipse. Hypothesis <9. Let the rst point of the Fermat is inside an arbitrary non-degenerate triangle, and through this point from each vertex held cevian. Then the original triangle is splited into six triangles without common interior points so that their second points of Napoleon lie on the same hyperbola.

In this paper we are considering questions about the possibility of existence of invariant nontrivial pseudocharacters on free groups. It is proved that nontrivial pseudocharacters exist on a certain type of HNN-expansions in complex cases. We got some results about the width of verbal subgroups generated by words from commutator subgroup and non-triviality of the second group of bounded cohomologies for considered HNN-expansions. Thus, partial answer to the question, formulated R. I. Grigorchuk, is received. Pseudocharacter is the real functions f from group G to R such that f(xy)-f(x)-f(y) for some " > 0 and for any x; y 2 G and f(xn) = nf(x) 8n 2 Z, 8x 2 G. A pseudocharacter is called non-trivial if '(ab)

Well known properties of numerical series P+1 n=1 an in the course of analysis, which have asymptotic growth of powers of n at innity. Relevant tests of convergence was laid in the works of Gauss. We study the necessary and sucient conditions for the positive (and constant sign) a sequence of numbers fang+1 n=1 with the rate of decrease (growth) in logarithmic scale for the convergence of the series P+1 n=1 an. Examples of the use of the criteria of convergence, as in the case of constant sign of series, and in the case of alternating series. The importance of a logarithmic scale due to the fact that it is found in various sections of the analysis and, in particular, the problem of nding the spectrum of the operator of SturmLiouville on the half-line for the fast growing potentials. On a logarithmic scale arise and the relevant questions on the presence of regularized sums, for the special potentials of the operator of SturmLiouville on the half-line.

This article deals with the issues of the structural theory of Lie algebras. The construction of the structural theory of algebraic systems implies the existence of certain structures of a special form, which are simpler than the base system. The important tool to study algebraic systems is the radical. The development of the structural theory of Lie algebras led to the emergence of various radicals. There are many radicals of Lie algebras in numerous publications. For example, the Killing radical, the Parfenov radical, the Jacobson radical and the prime radical are considered in various articles. The important area of research is the study of radicals of innite-dimensional Lie algebras. The article is devoted to proving properties of prime radical of a weakly artinian Lie algebra. A Lie algebra is said to be a weakly artinian if the Lie algebra satises the descending chain condition on ideals. In the rst section of the paper we introduced the concept of the prime radical in the following way. A Lie algebra L is said to be prime if [U; V ] = 0 implies U = 0 or V = 0 for any ideals U and V of L. We say that the ideal P of a Lie algebra L is prime if the factor algebra L=P is prime. The intersection of all prime ideals is called the prime radical P(L) of a Lie algebra L. In the second section it is shown that any nite set of elements of the prime radical of a weakly artinian Lie algebra generates the nilpotent subalgebra. This means that the prime radical is locally nilpotent. The third section is devoted to the solvability of the prime radical of a weakly artinian Lie algebra. There is a history of solving Mikhalev' s problem about the prime radical of a weakly artinian Lie algebra in this section also.

The paper is survey of recent results of investigations on varieties of Leibniz-Poisson algebras. We show that a variety of Leibniz-Poisson algebras has either polynomial growth or growth with exponential not less than 2, the eld being arbitrary. We show that every variety of Leibniz-Poisson algebras of polynomial growth over a eld of characteristic zero has a nite basis for its polynomial identities.We construct a variety of Leibniz-Poisson algebras with almost polynomial growth. We give equivalent conditions of the polynomial codimension growth of a variety of Leibniz-Poisson algebras over a eld of characteristic zero. We show all varieties of Leibniz-Poisson algebras with almost polynomial growth in one class of varieties. We study varieties of Leibniz-Poisson algebras, whose ideals of identities contain the identity fx; yg fz; tg = 0, we study an interrelation between such varieties and varieties of Leibniz algebras. We show that from any Leibniz algebra L one can construct the Leibniz-Poisson algebra A and the properties of L are close to the properties of A. We show that if the ideal of identities of a Leibniz-Poisson variety V does not contain any Leibniz polynomial identity then V has overexponential growth of the codimensions. We construct a variety of Leibniz-Poisson algebras with almost exponential growth. Let f n(V)gn1 be the sequence of proper codimension growth of a variety of Leibniz-Poisson algebras V. We give one class of minimal varieties of Leibniz-Poisson algebras of polynomial growth of the sequence f n(V)gn1, i.e. the sequence of proper codimensions of any such variety grows as a polynomial of some degree k, but the sequence of proper codimensions of any proper subvariety grows as a polynomial of degree strictly less than k.

This paper is dedicated to the 85th anniversary of the founder of the Tula algebraic school on algorithmic problems of the groups and semigroups theory, Doctor of Physical and Mathematical Sciences, Professor Martin Davidovich Grindlinger. It gives biographical data and a brief overview of his scientic, pedagogical, organizational and publishing activities.