The article is devoted to the life and activities of academician Oleg Borisovich Lupanov. Provides information on all periods of the biography of O. B. Lupanov.
The first Appendix contains the list of scientic works of O. B. Lupanov, and the second is the list of important dates of life and activities of O. B. Lupanov.

A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. In the case of zero characteristic of the main field all the information about the variety is contained in multilinear parts of relatively free algebra of the variety. We can study the identities of variety by means of investigations of multilinear part of degree n as module of the symmetric group Sn. Using the language of Lie algebras we say that an algebra is metabelian if it satisfies the identity (xy)(zt) ≡ 0.

In this paper we study the identities of non-associative one-generated free metabelian algebra and its factors. In particular, the infinite set of the varieties with different fractional exponents between one and two was constructed. Note that the sequence of codimensions of these varieties asymptotically formed by using colength, and not by using the dimension of some irreducible module of the symmetric group what was for all known before examples.

A large part of number theory deals with arithmetic properties of numbers with “missing digits” (that is numbers which digits in a number system with a fixed base belong to a given set). The present paper explores the analog of such a similar problem in the finite field.

We consider the linear vector space formed by the elements of the finite field Fq with q = p r over Fp. Let {a1, . . . , ar} be a basis of this space. Then every element x ∈ Fq has a unique representation in the form Pr j=1 cjaj with cj ∈ Fp; the coefficients cj may be called “digits”. Let us fix the set D ⊂ Fp and let WD be the set of all elements x ∈ Fq such that all its digits belong to the set D. In this connection the elements of Fp \ D may be called “missing digits”. In a recent paper of C.Dartyge, C.Mauduit, A.S´ark¨ozy it has been shown that if the set D is quite large then there are squares in the set WD. In this paper more common problem is considered.

Let us fix subsets D1, . . . , Dr ⊂ Fp and consider the set W = W(D1, . . . , Dr) of all elements x ∈ Fq such that cj ∈ Dj for all 1 ≤ j ≤ r. We prove an estimate for the number of squares in the set W, which implies the following assertions:

1) if Qr i=1 |Di | ≥ (2r − 1)rp r(1/2+ε) for some ε > 0, then the asymptotic formula |W ∩ Q| = = |W| 1 2 + O(p −ε/2 ) is valid;

2) if Qr i=1 |Di | ≥ 8(2r − 1)rp r/2 , then there exist nonzero squares in the set W.

In this paper we select general classes of finite order formal solutions of an algebraic (polynomial) ordinary differential equation (ODE), that can be calculated by the methods of the plane power geometry based on the method of determining leading terms of the equation by Newton-Bruno polygon.

Beside that in this paper we prove the theorem that if a formal solution of the selected class exists than the first approximation (the truncation) of this solution is the (formal) solution of the first approximation of the initial equation (that is called the truncated equation). Calculated formal solutions by means of these methods relate to much more general classes of the formal solutions that are called grid-based series and transseries in the foreign papers. Grid-based series and transseries are fairly new objects and in spite of the large number of publications they are slightly studied. They appear among formal solutions of the differential equations including equations that are important in physics. Other general methods of the calculation of such series do not exist yet. Therefore it is important to select the classes of the formal solutions that can be calculated algorithmically by the methods of the plane power geometry.

Generalized Fibonacci numbers { F (g) i } are defined by the recurrence relation F (g) i+2 = gF(g) i+1 + F (g) i with the initial conditions F (g) 0 = 1, F (g) 1 = g. These numbers generater representations of natural numbers as a greedy expansions n = ∑k i=0 εi(n)F (g) i , with natural conditions on εi(n). In particular, when g = 1 we obtain the well-known Fibonacci numeration system. The expansions obtained by g > 1 are called representations of natural numbers in generalized Fibonacci numeration systems.

This paper is devoted to studying the sets F (g) (ε0, . . . , εl), consisting of natural numbers with a fixed end of their representation in the generalized Fibonacci numeration system. The main result is a following geometrization theorem that describe the sets F (g) (ε0, . . . , εl) in terms of the fractional parts of the form {nτg}, τg = √ g 2+4−g 2 . More precisely, for any admissible ending (ε0, . . . , εl) there exist effectively computable a, b ∈ Z such that n ∈ F (g) (ε0, . . . , εl) if and only if the fractional part {(n + 1)τg} belongs to the segment [{−aτg}; {−bτg}]. Earlier, a similar theorem was proved by authors in the case of classical Fibonacci numeration system.

As an application some analogues of classic number-theoretic problems for the sets F (g) (ε0, . . . , εl) are considered. In particular asymptotic formulaes for the quantity of numbers from considered sets belonging to a given arithmetic progression, for the number of primes from considered sets, for the number of representations of a natural number as a sum of a predetermined number of summands from considered sets, and for the number of solutions of Lagrange, Goldbach and Hua Loken problem in the numbers of from considered sets are established.

Let G be a finitely generated Coxeter group with presentation G =< a1, . . . , an; (aiaj ) mij = 1, i, j = 1, n >, where mij — are the elements of the symmetric Coxeter matrix: ∀i, j ∈ 1, n, mii = 1, mij ≥ 2, i 6= j.

If mij ≥ 3(mij > 3), i 6= j, then G is a Coxeter group of large (extra-large) type. These groups introduced by K. Appel and P. Schupp.

If the group G corresponds to a finite tree-graph Γ such that if the vertices of some edge e of the graph Γ correspond to generators ai , aj , then the edge e corresponds to the ratio of the species (aiaj ) mij = 1, then G is a Coxeter group with a tree-structure.

Coxeter groups with a tree-structure introduced by V. N. Bezverkhnii, algorithmic problems in them was considered by V. N. Bezverkhnii and O. V. Inchenko.

The group G can be represented as tree product 2-generated of Coxeter groups, amalgamated by cyclic subgroups. Thus from the graph Γ of G will move to the graph Γ in the following way: the vertices of the graph Γ we will put in line Coxeter group on two generators Gij =< ai , aj ; a 2 i = a 2 j = 1,(aiaj ) mij = 1 > and Gjk =< aj , ak; a 2 j = a 2 k = 1,(ajak) mjk = 1 >, to every edge e joining the vertices corresponding to Gij and Gjk is a cyclic subgroup < aj ; a 2 j = 1 >.

In this paper we prove the following theorem: normalizer of finitely generated subgroup of Coxeter group with tree-structure G = Gij∗Gjk, Gij =< ai , aj ; a 2 i = a 2 j = 1,(aiaj ) mij = 1 >, Gjk =< aj , ak; a 2 j = a 2 k = 1,(ajak) mjk = 1 > finitely generated and exist algorithm for generating. 

P. S. Novikov in 1955–1956 showed the unsolvability of the main algorithmic problems in class of finite defined groups. In this connection there was important task of consideration of these problems in specific classes of finite defined groups. Thus, class of finite defined groups of Coxeter represents scientific interest. The class of groups of Coxeter was defined by H. S. M. Coxeter in 1934.

The classe of finitely generated groups of Coxeter with tree structure was defined by V. N. Bezverkhnii in 2003.

Let finitely generated group of Coxeter with tree structure is defined by presentation

G = ha1, ...an; (ai) 2 ,(aiaj ) mij , i, j ∈ 1, n, i 6= ji

where mij — number which corresponds to a symmetric matrix of Coxeter. At that, if i 6= j, that mij = mji, mij > 2. If mij = ∞, that between ai and aj relation does not exist . The group matches finite coherent tree-graph Γ such that: if tops of some edge -e of graf Г are elements ai and aj , that the edge e matches relation (aiaj ) mij = 1.

On the other hand group G may be represented as wood product of the two-generated groups of Coxeter, which are united by final cyclic subgroups. In this case, we will pass from graf Г of group G to graf Γ as follows: we associate tops of some edge e of graf Γ groups of Coxeter with two generating elements Gji = haj , ai ; (aj ) 2 ,(ai) 2 ,(ajai) mji i and Gik = hai , ak; (ai) 2 ,(ak) 2 ,(aiak) mik i, and edge e — cyclic subgroup hai ; (ai) 2 i.

The problem of intersection of the adjacency classes of finitely generated subgroups is that you need to find an algorithm that will help determine empty or not intersection w1H1 ∩ w2H2, where H1 and H2 any subgroup of group G and w1, w2 ∈ G.

Previously, the author proved the solvability of this problem for free product with association of two Coxeter’s groups with two generating element.

In the article author shows solvability of a problem of intersection of the adjacency classes of finite number of finitely generated subgroups of Coxeter’s group with tree structure. For this purpose group G was presented as wood product of n two-generated groups of Coxeter, which are united by finite cyclic subgroups.

To prove of this result, the author used the method of special sets and method of types. These methods were defined V. N. Bezverkhnii. He used these methods for research of various algorithmic problems in free constructions of groups.

In this paper we study the problem of analytical behavior of Dirichlet series with a bounded summatory function on its axis of convergence, σ = 0. This problem was also considered in the authors’ earlier works in case of Dirichlet series with coefficients determined by finite-valued numerical characters, which, in turn, was connected with a solution for a well-known Chudakov hypothesis.

The Chudakov hypothesis suggests that generalized characters, which do not vanish on almost all prime numbers p and asymptotic behavior of whose summatory functions is linear, are Dirichlet characters. This hypothesis was proposed in 1950 and was not completely proven until now. A partial proof based on the behavior of a corresponding Dirichlet series when it approaches to the imaginary axis was obtained in one of authors’ works. There are reasons to anticipate that this approach may eventually lead to a full proof of the Chudakov hypothesis.

In our case this problem is particularly interesting in connection with finding analytical conditions of almost periodic behavior of a bounded number sequence, different from those obtained before by various authors, for example, by Szego.

Our study is based on a so called method of reduction to power series. This method was developed by Prof. V. N. Kuznetsov in the 1980s and it consists in studying the relation between the analytical properties of Dirichlet series and the boundary behavior of the corresponding (i.e. with the same coefficients) power series.

In our case this method of reduction to power series allowed us to show that such Dirichlet series are continuous in the wide sense on the entire imaginary axis. Moreover, this method also helped to construct a sequence of Dirichlet polynomials which converge to a function determined by a Dirichlet series in any rectangle inside the critical strip.

The work concerns the questions of approximation of periodic differentiable functions of high smoothness by repeated arithmetic means of Fourier sums. One of the classifications of periodic functions nowadays is the classification suggested by A. Stepanets which is based on the concept of (ψ, β)–differentiation. The given classification allows to distinguish all classes of summable periodic functions from the functions where the Fourier series can deviate to infinitely differentiable functions including analytical and entire ones. When choosing the parameters properly, classes of (ψ, β)–differentiable functions exactly coincide with the well-known classes of Vail differentiable functions, Sobolev classes Wl p and classes of convolutions with integral kernels.

De la Vallee Poussin sums and their special cases (Fourier sums and Fejer sums) were extensively studied for many decades by many prominent experts in the theory of functions. At present, the large amount of factual material is accumulated in numerous publications. One of the most important directions in this field is the investigation of approximation properties of these sums for various classes of functions. The aim of the present paper is to systematize the known results related to the approximation properties of de la Vallee Poussin sums on classes of Poisson integrals and to present new facts obtained for their generalizations.

In certain cases asymptotic equalities are found for upper bounds of deviations in the uniform metric of the repeated de la Vallee Poussin sums on the classes C ψ β,∞ and C ψ β Hω which are generated by multiplicators ψ(k) and shifts on argument β provided that sequences ψ(k) which define the specified classes tend to zero with the rate of geometrical progression. In doing so classes C ψ β,∞ and C ψ β Hω consist of analytic functions which can be regularly extended in the corresponding strip.

We introduce generalized de la Vallee Poussin sums and study their approximation properties for the classes of analytic periodic functions. We obtain asymptotic equalities for upper bounds of deviations of the repeated de la Vallee Poussin sums on classes of Poisson integrals. Under certain conditions, these equalities guarantee the solvability of the Kolmogorov–Nikol’skiy problem for the repeated de la Vallee Poussin sums and classes of Poisson integrals. We indicate conditions under which the repeated sums guarantee a better order of approximation than ordinary de la Vallee Poussin sums.

The asymptotically best circuit synthesis methods produced by academician Lupanov are considered in their historical evolution. An attempt to explain the large influence of Lupanov’s results over the circuit synthesis theory is made. This influence appears as a straightforward adaptation of the Lupanov methods by other scientists in their researches and also as an outline of the most important problems of this branch of mathematics. Brief summary of other Lupanov’s results is given.

Lyapunov matrix equations and their generalizations — linear matrix Sylvester equation widely used in the theory of stability of motion, control theory, as well as the solution of differential Riccati and Bernoulli equations, partial differential equations and signal processing. If the structure of the general solution of the homogeneous part of the Lyapunov equation is well studied, the solution of the inhomogeneous equation Sylvester and, in particular, the Lyapunov equation is quite cumbersome. By using the theory of generalized inverse operators, A. A. Boichuk and S. A. Krivosheya establish a criterion of the solvability of the Lyapunov-type matrix equations AX − XB = D and X −AXB = D and investigate the structure of the set of their solutions. The article A.A. Boichuk and S.A. Krivosheya based on pseudo-inverse linear matrix operator L, corresponding to the homogeneous part of the Lyapunov type equation.

Using the technique of Moore–Penrose pseudo inverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of the bilinear matrix equation and, in particular, the Sylvester matrix equation in general case when the linear matrix operator L, corresponding to the homogeneous part of the bilinear matrix equation, has no inverse. We find an expression for family of linearly independent solutions of the bilinear matrix equation and, in particular, the Sylvester matrix equation in terms of projectors and Moore-Penrose pseudo inverse matrices. This result is a generalization of the result article A. A. Boichuk and S. A. Krivosheya to the case of bilinear matrix equation.

The suggested the solvability conditions and formula for constructing a particular solution of the inhomogeneous bilinear matrix equation is illustrated by an examples.