The authors set themselves two main objectives: to characterize the main stages of life of a Lomonosov Moscow State University professor Andrei Borisovich Shydlovsky and give a brief analysis of his scientific and educational activities, which had a significant impact on the development of scientific schools in the field of the theory of transcendental numbers, where he was one of the leading national experts and organizer of international scientific conferences in this area. Among his students 15 defended their dissertations and three of them later became doctors of physical and mathematical sciences. Professor Shidlovskii studies on the theory of transcendental numbers are emphasized.

 

In present paper for n = 17 improved estimate is obtained for the minimum density of lattice coverings of the Euclidean space with equal balls. This result is directed on a solution of a problem, known in the literature as “the problem of S. S. Ryshkov concerning lattice coverings” [1, 2]. This work is a continuation of a series of author’s works. The work [3] is a basic work among them. Detailed definitions, the technique of the research and the proofs of the basic theorems are given there. We presume that the reader is acquainted with the results of the work [3]. The result based on a full description of the structure of the L -partition for the Coxeter lattice A6 17 as well as the structure of the Voronoi-Dirichlet polyhedra as polyhedra defined by their vertices is given. On the basis of this description, exact value of the covering radius and the density function are evaluated for the lattice covering corresponding to this lattice. The values of the density function of the covering proved to be better (less) than the formerly known values. Thus, for n = 17, improved estimate is obtained for the minimum density of lattice coverings of the Euclidean space with equal balls. Historically, the study of L-partitions of the Coxeter lattices Ar n was initiated by S. S. Ryshkov in [4]. There are regular simplex S relative volume 6 among L-body of the lattices Ar n (named F1 in table 1). It is well known from [4] L-body, which we use to start enumeration. Originally, we obtained L-bodies with a computer, using the well known «empty-ball method» of Delone (see [5]). As the first step of this method, we used the results of [4] for S. In the present paper, we complete the studies initiated in [4] for the form A6 17. The similar results, earlier gotten by me for the dimentions n = 11, . . . , 15, were discussed in detail by me and S. S. Ryshkov at his lattice theory special seminars at the chair of discrete mathematics at MSU Faculty of Mechanics and Mathematics. Sergey Sergeyevich gave an appreciation for those results and named them «the results of physical and mathematical PhD’s level», which was and continues to be a big stimulus for me to carry out new researches. The present result for n = 17 have surpassed all previous ones in a volume of calculations. I devote this result to the memory of my teacher — Sergey Sergeyevich Ryshkov.

 

In complex analysis an integral representations are one of the powerful tools of research. The theory of analytic functions of complex variables is largely built on the basis of Cauchy’s integral formula [1]. An important class of ill-posed problems arising in physics, engineering and other fields, are so-called inverse problems [2] — [4]. In [5] - [6] author sets an integral formula for the function f(z), holomorphic in the circle KR : |z| < R, (it is in the introduction of this article as formula (1)) is the solution of the inverse problem for Cauchy’s integral formula in the circle KR. . Equation (1), unlike the Cauchy formula for the values of the function f(z) on any circumference Cr : |z| = r (0 < r < R) lying in a circle KR, or an arbitrary closed piecewise smooth lines covering the origin and contained within a circle circle CR — the circle border KR expresses its values at all other points in the range of KR. In [ 5] the solution of inverse problems for Poisson’s formula [1] and Schwartz’ formula [7] and the formulas for derivatives of Cauchy formula [1] in [5] — [6] are obtained. The inverse problem for the Poisson integral formula is used in [8] for generalization of Poisson — Jensen’s formula [7] from which Poisson – Jensen’s formula and Jensen’s formula follow as a special cases. Similarly, [9] and the inverse problem are used for the generalization of the Schwartz — Jensen’s formula [7]. In the case of ring D: r < |z| < R to [10] the integral representation is set (in [10] is a formula (1)) for a holomorphic function in D D which, unlike the Cauchy formula for the ring, according to the values on an arbitrary closed piecewise smooth lines hugging origin and contained within the ring D, expresses its values at all other points of the rings, ie [10] the inverse problem for the Cauchy formula in the case of ring D is solved. In the article [11] for the case of the circle KR the solution of inverse problems for integral formulas is found and given in [12] (in [12] these are formulas (3) and (4)) are valid for functions holomorphic in a star domain with respect to the origin. The Cauchy formula holds in the case of several complex variables (see., Eg, [13]). In the article [14] for the case of polydisc ER = E(R1, . . . , Rn) = {z = (z1, . . . , zn) : |z1| < R1, . . . , |zn| < Rn} the inverse problems for the Cauchy formula and by deriving from its are resolved (analogous to the Poisson formulas for the case of one complex variable). The inverse problems [15] in the case of integral Temlyakov’s formulas are solved (these are formulas, see., Eg, [16]). Finally, in this article, in the case of a convex domain and circle (respectively Theorem 2 and 3) new integral representations (3) and (5) are set, of which (3) is an integral representation for holomorphic functions in a convex domain, and (5) is a solution of the inverse problem for the integral representation (3) in a circle KR.

 

 

Some questions about distribution of the points with rational coordinates are natural generalizations of problems about integer points in convex domains. Upper and lower bounds for the quantity of rational points on the circle were used in the study of Hausdorff dimension of the set of the point on circle which are approximated with a given order of accuracy by the points with rational coordinates. During the last 15 years in the papers of M. Huxley, V. I. Bernik, V. V. Beresnevich, S. Velani, R. Vaughan double sided asymptotic estimates for the quantity of rational points near the smooth curves and surfaces were found. Let I = [a, b] ∈ R is an interval, y = f(x) is twice continuously differentiable function which satisfies the inequality c1 < |f ′′(x)| < c2 for c2 > c1 > 0 and for all x ∈ I. For arbitrary γ, 0 ≤ γ < 1 for sufficiently large Q we denote by AI (Q, γ) the set of rational points Γ = ( p1 q , p2 q ) , aq ≤ p1 ≤ bq, 1 ≤ q ≤ Q, for witch the following inequality holds f ( p1 q ) − p2 q < Q−1−γ . The set AI (Q, γ) consists from points lying inside the strip width of 2Q−γ near the curve y = f(x), x ∈ I. It it natural to expect that #AI (Q, γ) is a value of the order Q3−γ . It was proved using the methods of geometry of numbers and metric theory of Diophantine approximations. Recently [1] new estimates of the quantity of points α¯ = (α1, α2) ∈ R 2 , where α1, α2 are conjugate real algebraic numbers of arbitrary degree deg α1 = deg α2 = n and of the height H(α1) = H(α2) ≤ Q, in the strip width of c(n)Q−γ , 0 ≤ γ ≤ 1 2 , Q > Q0(n) near the smooth curve y = f(x) were obtained. In our paper some new results about distribution of points with conjugate real and complex algebraic coordinates were obtained. In particular generalization of result mentioned above was obtained. The main idea of the proof is based on metric theorem about diophantine approximations in the domains G of small measure µG < c2(n)Q−γ1 , 0 ≤ γ1 ≤ 1 3.

 

In the late 80ies and in the early 90ies of the past century the framework for a theory of special functions and integral transforms in several variables related with reflection groups was systematically built up in a series of papers of american mathematician C. F. Dunkl. This theory was further developed by many mathematicians. Nowadays, this theory is called Dunkl theory in the literature. Dunkl theory is widely used in probability theory, mathematical physics, approximation theory. The present paper is devoted to an application of Dunkl harmonic analysis on the Euclidean space R d and the unit Euclidean sphere S d−1 with Dunkl weight function invariant under the reflection group associated with some root system to problems of approximation theory. The problem of finding the sharp constant in Jackson’s inequality, or Jackson’s constant, between the value of best approximation of a function and its modulus of continuity in Lp-spaces is an important extremum problem of approximation theory. In the paper, the problem of Jackson’s constant in Lpspaces, 1 ≤ p < 2, on the unit circle S 1 in the Euclidean plane R 2 with Dunkl weight function invariant under the dihedral group Im, m ∈ N, is considered. Best approximation is given in terms of linear combinations of κ-spherical harmonics defined by means of the Dunkl Laplacian. We introduce the modulus of continuity using the generalized translation operator first appeared in the papers of Y. Xu. In the «weightless» case when the multiplicity function is identically equal to zero on a root system, D. V. Gorbachev proved Jackson’s inequality in Lpspaces, 1 ≤ p < 2, on the unit multidimensional Euclidean sphere S d−1 with the constant 2 1/p−1 coinciding with Jung’s constant of the Lp-spaces. He also established its sharpness. Jackson’s inequality with the same constant in Lp-spaces on the unit multidimensional Euclidean sphere S d−1 with arbitrary Dunkl weight function was established earlier by the author. Now in the paper, we obtain the lower estimate of Jackson’s constant in Lp-spaces, 1 ≤ p < 2, on the unit circle S 1 in R 2 with Dunkl weight function invariant under the dihedral group Im, m ∈ N. The dihedral groups are symmetry groups of regular m-gons in R 2 for m ≥ 3. To solve the given problem, we essentially use the method developed by V. I. Ivanov in cooperation with Liu Yongping. There are additional difficulties associated with the new modulus of continuity based on the nonsymmetric generalized translation operator in the spaces Lp[0, π], 1 ≤ p < 2, with the weight function |sin(t/2)| 2α+1| cos(t/2)| 2β+1 , α ≥ β ≥ −1/2.

 

 

Ryshkov S. S. in his papers has investigated extremal forms and extremal lattices. Extremal forms and lattices are connected with hard or rigid (by M. Gromov and other) objects in mathematics. In their work with colleagues S. S. Ryshkov came also to the other hard (or rigid) objects, for instance, to rigidly connected chain. Rigid and soft methods and results already evident in the study of the classical problems in number theory. Let us dwell briefly on the interpretation in terms of hard and soft methods of binary and ternary Goldbach problems. Since the binary (respectively ternary) Goldbach problems in their present formulation there are about equalities of the type 2n = p1 + p2 (respectively 2n+1 = p1+p2+p3), where n is a natural number greater than 1 (respectively n is a natural number greater than 2), p1, p2, p3 prime numbers, then these are hard (rigid) problems; the results of their studies are also hard. However, the methods of their study include both rigid methods — the exact formula of the method of Hardy — Littlewood — Ramanujan and a combination of hard and soft methods under the investigation by the Vinogradov‘s method of trigonometric sums. A number of problems of analytic number theory allow dynamic interpretation. We note in this regard that on connection of methods of analytic number theory and the theory of dynamical systems paid attention and has developed such analogies A. G. Postnikov. The purpose of the paper is not to provide any sort of comprehensive introduction to rigidity in arithmetic and dynamics. Rather, we attempt to convey elementary methods, results and some main ideas of the theory, with a survey of some new results. We do not explore an exhaustive list of possible topics, nor do we go into details in proofs. After giving an elementary number theoretic, algebraic and algebraic geometry introduction to rigid non-Archimedean spaces in the framework of local one dimensional complete regular rings, modules over rings, trees and formal schemes follow to I. R. Shafarevich, J.-P. Serre, J. Tate, D. Mumford, we review some novel results and methods on rigidity. These include (but not exhaust) methods and results by H. Furstenberg, G. A. Margulis, G. D. Mostow, R. Zimmer, J. Bourgain, A. Furman, A. Lindenstrauss, S. Mozes, J. James, T. Koberda, K. Lindsey, C. Silva, P. Speh, A. Ioana, K. Kedlaya, J. Tuitman, and other. I am grateful to V. M. Buchstaber for helpful comments during the discussion of my talk. I thank the anonymous referee for his comments relatively the content and style of presentation and for providing suggestions for improvements. Special thanks to N. M. Dobrovolskii for help and support under the preparation of the article for publication.

 

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities α of degree n > 2, starting from some number m0 = m0(α), the sequence of residual fractions αm is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals α of degree n > 2, starting from some number m0 = m0(α), the sequence of residual fractions αm is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction αm are concentrated about fractions − Qm−2 Qm−1 is either in the interval of radius O ( 1 Q2 m−1 ) in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number m0 = m0(α), fair recurrent formula for incomplete private qm expansions of real algebraic irrationals α, Express qm using the values of the minimal polynomial fm−1(x) for residual fractions αm−1 and its derivative at the point qm−1. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of α in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − Qm−2 Qm−1 . It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number α and its limit points.

 

 

Some problems of the number theory are associated with the zeros of special functions, such as the Riemann zeta function ζ(s), Dirichlet L-functions L(s, χ) and others. The Riemann zeta function is the most famous. On the half-plane ℜs > 1, the Riemann zeta function is defined by Dirichlet series ζ(s) = ∑ +∞ n=1 n −s . In 1859, Riemann conjectured that all non-trivial zeros of the Riemann zeta function lie on the critical line ℜs = 1 2 . Hardy was the first to prove in 1914 that an infinity of zeros are on the critical line. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line [T, T + H], H = T 0.5+ε . In 1984, A. A. Karatsuba proved Selberg’s result for shorter intervals of critical line [T, T + H], H = T 27 82 +ε . For arithmetic Dirichlet series satisfying a functional equation of Riemann type but admitting no Euler product expansions, lower bounds of the correct order of magnitude for the number of their zeros on intervals of the critical line ℜs = 1/2 have not been obtained so far. The first to show that the critical line contains abnormally many zeros of an arithmetic Dirichlet series without Euler products was Voronin, who proved in 1980 that interval (0, T] of critical line contains more than cT e 1 20 √ ln ln ln ln T zeros of the Davenport–Heilbronn function. In 1989 A. A. Karatsuba developed a new method for obtaining lower bounds for the number zeros of certain Dirichlet series in intervals of critical line; by using this method, he substantially strengthened Voronin’s result. In 1991 Karatsuba solved (by his 1989 method) the problem of estimating the number zeros of linear combinations of functions which are analogous the Hardy function. In the present paper we prove a theorem similar to the theorem of A. A. Karatsuba (in 1991), but only for "almost all" intervals of the form (T, T + H), H = Xε , where ε is an arbitrary positive number, and X ≤ T ≤ 2X, X > X0(ε).

 

Some development early conducted investigations on the problem of Malyshev A. V. about the number of integer points lying in some areas on multidimensional hyperboloids is given in this work. The task of obtaining of asymptotic formulae for quantity of integer points in areas of the kind of De Luri on multidimensional hyperboloids is put by Malyshev A. V. [1]. De Luri [3] in case of four-dimensional hyperbolic surface p (x1, . . . , x4) = ∑ 4 k=1 akx 2 k − m = 0, m ̸= 0 in the area Ωp(L) on it by defined inequality ∑ 4 k=1 |ak| x 2 k 6 L obtained asymptotic formula (in L → ∞ and fixed a1, a2, a3, a4, and m) for value of R (Ωp(L)), equaled to the number of integer points in the area Ωp(L) on the mentioned hyperboloid, but in so doing De Luri does not value the remainder formula. Later on in [1] generalization of this value is given on multidimensional hyperboloid given by the equation p = p (x1, . . . , xs) = ∑s k=1 akx 2 k + ∑s k=1 bkxk + c = 0, where ak, bk, (k = 1, . . . , s), c ̸= 0 — integers, in addition to coefficients ak not all is one sign, but area of Ωp(L) on this hyperboloid is given by the inequality ∑s k=1 |ak| x 2 k 6 L. In development of indicated task of Malyshev A. V. we examine arbitrary quadratic form equivalent to the diagonal in the equation of hyperboloid, and the area of Ωp(L) : ∑s k=1 |ak| x 2 k 6 L is substituted for the area ∑s i=1 { Q (1) i (xi , yi) + Q (2) i (zi , ti) } 6 L, where Q (1) i и Q (2) i — binary quadratic forms, equivalent to diagonal forms. In conclusion of our asymptotic result about quantity of R (Ωp, L) the theorem about weighted number of integer points Ih(n, s) from [2] is used in n → ∞ and the complex variant of tauberian’s theorem with remainder term for the power series (see [5, 6]). Also wee will note that our obtained result is analogous to one result of Davenport [7] by generalized problem of Varing in power k = 2, but in such meaning of k our question of hyperbolic surface has several more common kind.

 

In this paper asymptotic formula for weighted number of integer points on multidimensional hyperbolic surfaces defined by direct sum of indefinite quaternary integral quadratic forms of singular kind is obtained. In doing so weighted function is chosen as a real exponent on the index of which there stands integral quadratic form being direct sum of positive binary quadratic forms with the same discriminant equal to the discriminant δF of imaginary quadratic field F = Θ (√ d ) where d is the negative without quadrate number. The choice of real kind of weighting function is conditioned by possibility application used method in investigation of question about the number of integer points lying is some fields of real kind on examining multidimensional hyperboloids. Leaning upon the method of article [7] based on the use of exact meanings of Gauss double sum we examine multidimensional problem about weighted number of integer points on hyperbolic surface of real kind. The question is about the asymptotic with remainder of series for value Ih (n, s) = ∑ p(x,y,z,t)=h e − ω(x,y,z,t) n , where n → ∞ — real parameter, p ( x, y, z,t ) = ∑s i=1 { Q (1) i (xi , yi) − Q (2) i (zi , ti) } , ω ( x, y, z,t ) = ∑s i=1 { Q (1) i (xi , yi) + Q (2) i (zi , ti) } , Q (1) i , Q(2) i — positive integral binary quadratic forms of the same discriminant δF ; h ̸= 0 — integral number. In deducing the asymptotic formula for Ih (n, s) essentially we use: 1) the formula of turning of theta-series binary quadratic form (in our case it is enough to use double theta-series instead of multidimensional); 2) formula for 1 q( ∫q+N) − 1 q(q+N) e −2πihx ( 1 n2 + 4π 2x 2 )S dx 3) estimation of sum of Kloosterman K (u, v; q) = ∑ x mod q ′ e 2πi q ( ux+vx ′ ) , where xx ′ ≡ 1 (mod q). Obtained asymptotic formula for Ih (n, s) generalises one of the results of Kurtova L. N. [7] about weighted number of integer points on four-dimensional hyperboloids for the case of multidimensional hyperboloids corresponding real kind. Besides our result in case of constant coefficients of hyperboloid equation also generalized one result of Malishev A. B. [10] for a case of some nondiagonal quadratic forms in comparison with the result of Golovizina V. V. [3] the main number in examining problem is obtained in evident kind as in our work exact meanings of Gauss double sums are used and in [3] it is expressed by way of some complex integral W(N), for which only estimation is given over in doing so in our case N = [√ n]. Later on the result about value Ih (n, s) can be applied in obtaining asymptotic formulae for the number of integer points lying in some fields of real kind on multidimensional hyperboloids.

 

 

In this paper we consider binary additive problem of the form n1 +n2 = N with n1 ∈ N(α, I1), N2 ∈ N(β, I2), where N(α, I) = {n ∈ N : {nα} ∈ I}. Main examples of such sets are the sets of natural numbers with specified ending of greedy expansion of the number by linear recurrence sequences associated with Pisot numbers. Besides that, the sets N(α, I) are special cases of quasilattices. Previously additive problems on the sets of this type are considered only for the case α = β. In this case was obtained asymptotic formulaes for the number of solutions of the additive problem with an arbitrary number of terms, and for number of solutions in analogues of ternary Goldbach problem, Hua-Loken problem, Waring problems, and Lagrange problem about the representation number of natural numbers as a sum of four squares. Wherein, Gritsenko and Motkina discovered that in the case of linear problems we have the following nontrivial effect: apprearence of a rather complicated function in the main term of the asymptotics for the number of solutions. For nonlinear problems corrsponding effect is missing and the form of the main term can be obtained by the density considerations. In our problem, we show that the behavior of the main term of the asymptotic formula for the number of solutions significantly depends on the arithmetic of α and β. If 1, α and β are linearly independent over the ring of integers Z, then the main term of the asymptotic has the "density"form, i.e. it is equal to |I1||I2|N. In the case of linear dependence of 1, α and β we have the Gritsenko-Motkina effect, i.e. the main term is ρ({Nβ})N, where ρ is a rather complicated efficiently computable piecewise linear function of the fractional part {Nβ}. we obtain an algorithm for computation of the function ρ, and study basic properties of this function. In particular, we obtain sufficient conditions for its non-vanishing. Also we give a numerical example of the computation of this function for some concrete sets N(α, I1), N(β, I2). In the final part of the paper we discuss some open problems in this area.

 

A dialgebra is a vector space equipped with two binary operations ⊣ and ⊢ satisfying the following axioms: (D1) (x ⊣ y) ⊣ z = x ⊣ (y ⊣ z), (D2) (x ⊣ y) ⊣ z = x ⊣ (y ⊢ z), (D3) (x ⊢ y) ⊣ z = x ⊢ (y ⊣ z), (D4) (x ⊣ y) ⊢ z = x ⊢ (y ⊢ z), (D5) (x ⊢ y) ⊢ z = x ⊢ (y ⊢ z). This notion was introduced by Loday while studying periodicity phenomena in algebraic K-theory. Leibniz algebras are a non-commutative variation of Lie algebras and dialgebras are a variation of associative algebras. Recall that any associative algebra gives rise to a Lie algebra by [x, y] = xy−yx. Dialgebras are related to Leibniz algebras in a way similar to the relationship between associative algebras and Lie algebras. A dialgebra is just a linear analog of a dimonoid. If operations of a dimonoid coincide, the dimonoid becomes a semigroup. So, dimonoids are a generalization of semigroups. Pozhidaev and Kolesnikov considered the notion of a 0-dialgebra, that is, a vector space equipped with two binary operations ⊣ and ⊢ satisfying the axioms (D2) and (D4). This notion have relationships with Rota-Baxter algebras, namely, the structure of Rota-Baxter algebras appearing on 0-dialgebras is known. The notion of an associative 0-dialgebra, that is, a 0-dialgebra with two binary operations ⊣ and ⊢ satisfying the axioms (D1) and (D5), is a linear analog of the notion of a g-dimonoid. In order to obtain a g-dimonoid, we should omit the axiom (D3) of inner associativity in the definition of a dimonoid. Axioms of a dimonoid and of a g-dimonoid appear in defining identities of trialgebras and of trioids introduced by Loday and Ronco. The class of all g-dimonoids forms a variety. In the paper of the second author the structure of free g-dimonoids and free n-nilpotent g-dimonoids was given. The class of all commutative g-dimonoids, that is, g-dimonoids with commutative operations, forms a subvariety of the variety of g-dimonoids. The free dimonoid in the variety of commutative dimonoids was constructed in the paper of the first author. In this paper we construct a free commutative g-dimonoid and describe the least commutative congruence on a free g-dimonoid.

 

The investigation of the arithmetic nature of the values of differentiated with respect to parameter generalized hypergeometric functions was carried out in many works; see [1]–[7] and also corresponding chapters of the books [8] and [9]. Primarily the method of Siegel was used for these purposes. This method can be applied for the investigation of hypergeometric functions with rational parameters and the results concerning transcendence and algebraic independence of the values of such functions and corresponding quantitative results (for example estimates of the measures of algebraic independence) were obtained by means of it. The possibilities of application of Siegel’s method in case of hypergeometric functions with irrational parameters are restricted. In its classic form Siegel’s method cannot be applied in this situation and here were required some new considerations. But it must be noted that the most general results concerning the arithmetic nature of the values of hypergeometric functions with irrational parameters were obtained exactly by Siegel’s method (by modified form of it, see [10] and [11]). In this case it’s impossible to say of the results of transcendence or algebraic independence and one must restrict oneself by the results concerning linear independence of the corresponding values. In Siegel’s method reasoning begins with the construction of functional linear approximating form which has a sufficiently high order of zero at the origin of coordinates. Such a form is constructed by means of the Dirichlet principle. The impossibility to realize the corresponding reasoning for the functions with irrational parameters is an obstacle for the attempts to apply Siegel’s method in case of irrational parameters. It was noted long ago that in some cases the linear approximating form can be constructed effectively and explicit formulae can be pointed out for its coefficients. This method is inferior to Siegel’s one in the sense of the generality of the results obtained. But by means of the method based on the effective construction of linear approximating form the most precise low estimates of the modules of linear forms in the values of hypergeometric functions were obtained and in many cases were established linear independence of the values of functions with irrational parameters (see for example [12]). The effective construction of linear approximating form for the function (2) was proposed in the work [13]. In this work the construction was based on a contour integral which was earlier used for the achievement of results concerning the estimates of linear forms of the values of hypergeometric functions with different parameters; see [14]. In this paper we propose a new approach for the construction of linear approximating form for functions (2). Here we make use of a connection between hypergeometric functions of different types which makes it possible to reduce above mentioned constructing of linear approximating form to less difficult task. In the conclusion we give short directions concerning possible applications.

 

In this paper we consider hypergeometric functions and their derivatives (see (2) and (3)). One begins the investigation of arithmetic nature of the values of such functions with the construction of functional linear approximating form having sufficiently high order of zero at the origin. If the parameters of functions under consideration (in our case these are numbers (1)) are rational the construction of such a form can be fulfilled by means of the Dirichlet principle. Further reasoning is based on the employment of the constructed form and the whole scheme is called Siegel’s method, see [1] and [2]. If some of the numbers (1) are irrational the functions (2) and (3) cannot be reduced to the so called E-functions and it is impossible to use Siegel’s method (in its classic form) for such functions: the scheme doesn’t work at the very beginning of reasoning for we cannot use the Dirichlet principle for the construction of the first approximating linear functional form (in the process of reasoning by Siegel’s method we get several such forms). It was noticed that in some cases the first approximating form can be constructed effectively (see for example [3] and [4]). Having at one’s disposal such a form one can reason as in Siegel’s method (or it is possible in some cases to use special properties of the effectively constructed linear form) and receive required results. These results are not so general as those received by Siegel’s method but the method based on effectively constructed approximating form has its own advantages. One of them consists in the possibility of its application also in case when some of the parameters (1) are irrational. The other advantage is the more precise estimates (if we consider for instance the measure of linear independence) that can be obtained by this method. The above concerns the case when the functions under consideration are not differentiated with respect to parameter. Application of Siegel’s method for the differentiated with respect to parameter functions (for example such functions as (4) and (5)) is possible also and it has been in fact fulfilled in a series of works; see the remarks to chapter 7 of the book by A.B.Shidlovskii [5]. But as before the parameters of the functions under consideration must be rational and the obtained results are not sufficiently precise. The performed investigations show that the employment of simultaneous approximations instead of construction of linear approximating form almost always gives better results. For that reason the main new results concerning differentiated with respect to parameter hypergeometric functions have been obtained exactly by means of the effective constructions of simultaneous approximations although the appearance (comparatively recently) of effective constructions of linear approximating forms for such functions did make it possible to solve some related problems. In this paper we propose a new effective construction of simultaneous approximations for the differentiated with respect to parameter hypergeometric functions in homogeneous case. On possible applications of this construction we give only brief instructions: one can obtain some results on linear independence of the values of functions of the type (5) in case of irrationality of some of the numbers (1); it is possible also to improve some of the related quantitative results.

 

Ideal designs, made of rigid bars (levers), inextensible cables and incompressible struts are considered. In English such constructions are called "tensegrity frameworks". In the particular case of structures composed of only the levers, — this is a bar and joint frameworks. In recent times the tensegrity frameworks are increasingly used in architecture and construction, for example, the construction of bridges. In English mathematical literature geometric properties of such structures were studied since the seventies of the last century. This article is apparently the first in Russian mathematical literature devoted to this topic. It is a breath survey to the theory of tensegrity frameworks. It introduces mathematical formalization of tensegrity frameworks in the spirit of the work of the author on hinge mechanisms. This formalization includes original Russian terminology, not reducible to the borrowing of English words. Only not pinned tensegrity frameworks are investigated. We call a tensegrity frameworks, allowing the internal stress, and not allowing a continuous deformation with a change of form, — a truss. A truss that can’not be assembled in a different way to be not congruent to initial one is called Globally Rigid. If a tensegrity frameworks is Globally Rigid in Rn and also Globally Rigid in every RN for N > n it is called Universally Rigid. We focus on the problem — when a given tensegrity framework is Globally Rigid? We consider an effective method for solving this problem, based on investigation of particular function – the potential energy of the structure. We search a tensegrity frameworks for which this potential energy is minimal. The method is described in detail in the article. The main theorem, giving a sufficient condition of Universal Rigidity of tensegrity framework is proved in details. The study of internal stresses of a tensegrity framework and its stress matrix, by means of which the potential energy is written, is of fundamental importance. Examples of applications of this theorem to planar and spatial tensegrity frameworks are presented. In general, this subject is not yet sufficiently developed, and is currently actively investigated. At the end of the article some open questions are formulated.

 

Some well-known classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z and p-adic integers Zp and maximal orders of local fields and fields of algebraic numbers go back to classical papers by S. S. Ryshkov, P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the representations, and one of the possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n. The basic idea underlying a geometric approach was given in Ryshkov’s papers on the computation of the finite subgroups of GLn(Z) and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached. In the present paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of the group G. A combinatorial construction of the series of these representations is given explicitly. In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions is constructed. We consider certain related questions including the embedding problem in Galois theory for local faithful primitive representations of supersolvable groups and integral representations arising from elliptic curves.

 

We study the arithmetic properties of almost polyadic numbers ∑∞ n=1 ai (ai + bi). . .(ai + (n − 1) bi), i = 1, ..., m, where the numbers ai , bi ∈ Z, (ai , bi) = 1.

The concepts Rickart rings and Baer rings have their roots in the theory of linear operators in Hilbert space. The concept of Baer rings was introduced by I. Kaplansky in 1955 and the concept of Rickart rings was introduced by Maeda in 1960. In recent years, many authors have been actively studied the module theoretic analogue of these rings. In this paper, we introduce the concept of essentially Baer modules, essentially quasi-Baer modules and study their properties. We prove that, every direct summand of an essentially Baer module is also an essentially Baer module. We also prove that, every free module over essentially quasi-Baer ring is an essentially quasi-Baer module and each finitely generated free module over dual essentially quasi-Baer ring is a dually essentially quasi-Baer module; if M is CS-Rickart and M has the SSIP-CS then M is essentially Baer. The converse is true if SocM ✂M; if M is d-CS-Rickart and M has the SSSP-d-CS then M is dual essentially Baer. The converse is true if RadM ≪ M; if R is a right semi-artinian ring, then M is an essentially Baer module if and only if M is CS-Rickart and M has the SSIP-CS; if R is a right max ring, then M is a dual essentially Baer module if and only if M is d-CS-Rickart and M has the SSSP-d-CS; if M be a projective module and P(M) = 0, then M is a quasiBaer module if and only if every fully invariant submodule of M is essential in a fully invariant direct summand of M, if and only if the right annihilator in M of every ideal of S is essential in a fully invariant direct summand of M. We also give some characterizations of projective quasi-Baer modules. The presented results yield the known results related to Baer modules and dual Baer modules.

 

A hyperbolic plane Hb of positive curvature is realized on the external domain whith respect to the oval curve of the projective plane P2, i.e. on the ideal domain of the Lobachevskii plane. In works of the author the first partitions of the plane Hb are constructed. Among them there are series of the normal, but not monohedral partitions and the series of the monohedral partitions which are not the normal. In this work the series of the first normal monohedral partitions of the plane Hb are constructed. One of topological differences of the plane Hb from the Lobachevskii plane is in the following fact. No line of the plane Hb partitions the plane (the set of Betti numbers for the plane Hb: β0 = 1, β1 = 1, for the plane Λ 2 : β0 = 1, β1 = 0). Therefore the main known methods of a construction of partitions of the Lobachevskii plane can not be applied in partitions of the plane Hb. As an exception it is possible to consider the tiling scheme of the plane Λ 2 offered by the Hungarian mathematician K. Beretsky. In present work Beretsky’s scheme is adapted for the plane Hb. On the basis of this scheme the normal monohedral partitions the plane Hb with one remote parabolic line are constructed. The cells of the constructed partitions are the correct horocyclic n-trapezes. They are in detail investigated in this work. The correct horocyclic n-trapeze called the (n+3)-hedral which contain two congruent edges on the parallel hyperbolic lines. The other edges of (n + 3)-hedral are the congruent elliptic segments. One of them serves as an internal chord of some horocycle ω, and other n segments are the internal chords of the concentric with ω horocycle. For research of the cells of partitions in present work the orthogonal horocyclic coordinate system is entered. Auxiliary formulas of the areas of some figures of the plane Hb are received. It is proved that the area of the correct horocyclic n-trapeze can be expressed by means of the function αe of a quasiparallelism angle entered by the author on the plane Hb. The length of the side edge no depend from the length of elliptic edges and is equal to ρ ln n, where ρ is the radius of curvature of the plane Hb.

Studied groups with restrictions on subgroups.Considered locally solvable groups with the maximal condition for subgroups. Selected subgroup of a locally solvable group, such that satisfiability for the conditions maximal condition should the feasibility of the conditions throughout the group.

The paper presents generalization of incidence algebras, which includes the case of generalized matrix rings. Constructions similar to partial ordering and quasi-ordering were introduced - η-poset and η-qoset respectively. The question of invertibility of elements of generalized incidence algebras was studied. The algorithm of finding inverse element and clear formula were found. This formula holds for incidence algebras, in particular. The case of generalized incidence algebra over a field was examined explicitly. In this case we can introduce equivalence relation on the underlying set, under which generalized incidence algebra would have block structure. As with incidence algebras, there is close connection between algebras over η-posets and η-qosets. For example, if we know sizes of equivalence classes, then we can reconstruct algebra over η-qoset by corresponding algebra over η-poset. It was shown that generalized incidence algebras can be viewed as subalgebras of some formal matrix rings of the same size as the underlying set. The problem of isomorphism was studied and it was shown that it can be reduced to the problem of isomorphism of generalized incidence algebras over η-posets. Partial solution to this problem was found. The paper introduces Mobius function of generalized incidence algebra. Analogue of Mobius inversion formula was found and it was shown that basic properties of classical Mobius function are remain to be true. Generalized incidence algebras with so-called {0, 1}-multiplicative system are of peculiar interest. There is good reason to believe that all generalized incidence algebras over a field are isomorphic to algebras with {0, 1}-multiplicative system.

 

In the paper the estimate of the complete rational arithmetical sum from a polynomial is found. It’s is a correct on the power of a denominator with the estimate of the constant depending of the degree of a polynomial.

 

The article provides a series of custom recursive functions for a variety of tasks of searching and replacing elements in nested arrays. The latter are defined recursively, as is done in the system of engineering and scientific calculations, PTC Mathcad Prime, then there are matrices whose elements can be scalars, strings, and again the nesting arrays. Some problems of the search are considered in [1-4]. Our goal was the development of existing and creation of new means associated with both simple and generalized searching and replacing elements in nested arrays. Let A be a scalar, a string, or a nested array, B — nested array. The problem with generalized occurrences of A in B (generalized searching A in B) and replaced of such occurrences arise when A or B to be included by special elements associated with any scalar, string or nested array. In the article are formulated 10 problems. For each suggested one or more solutions as functions in a programming language PTC Mathcad Prime. All the features are tested on a large number of examples, but the tests are not completely.

 

Problem solving with data presented nested arrays, is difficult because they of the rather is unpredictаble in their structure. And here, in many cases helps recursion. Its use allows linearly according to the same scheme to implement a run on all the elements of each nesting level of any of the array, regardless of its structure and the depth of nesting. Nested array can be interpreted by a tree, whose root is the array itself, from its go arc to the array elements, etc. The leaves of this tree are scalars or strings — finite elements that are not referenced in the following arrays. In an article for the solution of several problems of a General nature with nested arrays is offered appropriate recursive program-functions. Examples of such tasks: calculate the total number of leaves of the array; to form an array of transposed elements of the original array at all levels of nesting; determine whether a given object (scalar, string, a simple array, nested array) of a element of this array to any level of nesting; count the number of occurrences of an object in the array at all levels of nesting; collect all the leaves of the array into the vector, replacing the leaves a given array on components the vector, etc. In all cases, recursive triad is as follows: the parameter of recursion — nest array; decomposition — transitions at all levels of nesting of arrays to their elements, and so on until the leaves; recursive base, i.e. the trivial cases in recursion — lists of arrays [1]. Offer laconic recursive programs-features of the solution are listed and some other tasks are implemented on a simple and intuitive programming language system engineering and scientific computing PTC Mathcad Prime (version 3.1) [2,3]. Note that in this system all nesting arrays are nested matrix.

In PTC Mathcad, and in previous versions of Mathcad, for numeric and symbolic computation proposed special vectorization operation, which can be used to perform many built-in and some custom functions of one variable over each scalar element or string element simple or nested arrays. This operator looks in the form of directed arrows from left to right over expression. The operation of the vectorization can be applied to built-in functions of several variables, but only over a simple array with a scalar or string elements. So, we emphasize that for built-in functions of one or several variables vectorization operation in the case of nested arrays can be implemented not always. And for user-defined functions, it is usually not implemented even for simple arrays. In the article removed all these constraints and are constructed analogues operation vectorization for any built-in or user-defined functions from one or more variables over simple or nested arrays. There are proposed compact recursive functions that perform the role of the vectorization operator. We considered two possible approaches to solving this problem. When you first approach for functions g of n variables are constructed separate recursive programs-functions F1, F2, F3, ..., implement vectorization respectively for n = 1, 2, 3, ... . The second approach for a function g from n variables creates a single for any n = 1, 2, ... program-function F that performs the role of the vectorization operator. In connection with the problem of vectorization for nested arrays formulated some auxiliary problem and were proposed solutions in a form of recursive functions.