Showing papers in "Sbornik Mathematics in 1996"
TL;DR: In this paper, it was shown that and are algebraically independent and Bertrand's conjecture on algebraic independence over of the values at algebraic points of a modular function and its derivatives.
Abstract: We prove results on the transcendence degree of a field generated by numbers connected with the modular function . In particular, we show that and are algebraically independent and we prove Bertrand's conjecture on algebraic independence over of the values at algebraic points of a modular function and its derivatives.
151 citations
TL;DR: In this article, the authors studied the problem of the limiting distribution of the zeros of the polynomial extremal in the metric with respect to a measure with finitely many points of growth under the assumption that the degree of this polynomial and the number (n$ SRC=http://ej.iop.org/images/1064-5616/187/8/A04/tex_sm_153_img4.gif
Abstract: The problem of the?limiting distribution of the?zeros of the?polynomial extremal in the?-metric with respect to a?measure with finitely many points of growth is studied under the?assumption that the?degree of this polynomial and the?number (n$ SRC=http://ej.iop.org/images/1064-5616/187/8/A04/tex_sm_153_img4.gif/>) of points of growth of the?measure approach infinity so that .
100 citations
TL;DR: The concept of strongly convex R-hull of a set is introduced in this paper, and a generalization of the Krein-Mil'man theorem for strongly-convex sets is proved.
Abstract: Properties of strongly convex sets (that is, of sets that can be represented as intersections of balls of radius fixed for each particular set) are investigated. A connection between strongly convex sets and strongly convex functions is established. The concept of a strongly convex R-hull of a set (the minimal strongly convex set containing the given set) is introduced; an explicit formula for the strongly convex R-hull of a set is obtained. The behaviour of the strongly convex R-hull under the variation of R and of the sets is considered. An analogue of the Caratheodory theorem for strongly convex sets is obtained. The concept of a strongly extreme point is introduced, and a generalization of the Krein-Mil'man theorem for strongly convex sets is proved. Polyhedral approximations of convex and, in particular, of strongly convex compact sets are considered. Sharp error estimates for polyhedral and strongly convex approximations of such sets from inside and outside are established.
97 citations
TL;DR: In this article, a list of hyperbolic symmetric generalized Cartan matrices with the following properties is presented: is a?matrix of rank?3 and of elliptic or parabolic type, has a?lattice Weyl vector, and contains a?parabolic submatrix.
Abstract: Automorphic corrections for the?Lorentzian Kac-Moody algebras with the?simplest generalized Cartan matrices of rank?3, ? ?????and????are found. For this correction, which is a?generalized Kac-Moody Lie super algebra, is delivered by , the Igusa -modular form of weight 35, while for it is given by some Siegel modular form of weight?30 with respect to a 2-congruence subgroup of . Expansions of and in infinite products are obtained and the?multiplicities of all the?roots of the corresponding generalized Lorentzian Kac-Moody superalgebras are calculated. These multiplicities are determined by the Fourier coefficients of certain Jacobi forms of weight?0 and index?1. The?method adopted for constructing and leads in a?natural way to an?explicit construction (as infinite products or sums) of Siegel modular forms whose divisors are Humbert surfaces with fixed discriminants. A geometric construction of these forms was proposed by van der Geer in 1982. To show the?prospects for further studies, the?list of all hyperbolic symmetric generalized Cartan matrices with the?following properties is presented: is a?matrix of rank?3 and of elliptic or parabolic type, has a?lattice Weyl vector, and contains a?parabolic submatrix .
69 citations
TL;DR: In this paper, a topological classification of integrable Hamiltonian systems in the neighbourhood of singular leaves that contain an arbitrary number of points of rank zero is presented. But the classification is restricted to the case when the singular leaf contains one point of zero.
Abstract: One of the key problem in Hamiltonian mechanics is to describe the behaviour of integrable Hamiltonian system with two degrees of freedom in neighbourhoods of singular leaves of the Liouville foliation. In 1988 Lerman and Umanskii announced a classification theorem for such systems in the case when the singular leaf contains one point of rank zero. The proof was published in 1992 and 1993. In the solution of classical problems in physics and mechanics, integrable Hamiltonian systems arise which have a singular leaf with several points of rank zero. This paper concerns the topological classification of integrable Hamiltonian systems in the neighbourhood of singular leaves that contain an arbitrary number of points of rank zero.
60 citations
TL;DR: In this article, the concept of p-connectedness of two disjoint domains with Lipschitz boundaries and fractal contact is studied, and new examples of open periodic sets with positive effective conductivity are constructed on the basis of this analysis.
Abstract: A detailed study of the concept of p-connectedness is carried out; in particular, a criterion for the p-connectedness of two disjoint domains with Lipschitz boundaries and with fractal contact is formulated. New examples of open periodic sets with positive effective conductivity are constructed on the basis of this analysis. A new class of objects, elliptic operators in a Euclidean space with measure, is introduced; the corresponding concept of p-connectedness is introduced and a generalized theory of homogenization is developed.
54 citations
TL;DR: In this article, the authors considered two-dimensional Navier-Stokes equations and a non-linear hyperbolic equation and proved a global averaging theorem on the convergence of attractors of non-autonomous equations to the average attractor of the average autonomous equation.
Abstract: We consider two-dimensional Navier-Stokes equations and a?damped non-linear hyperbolic equation. We suppose that the?right-hand sides of these equations have the?form , . We suppose also that has an?average. The?main result of the?paper is proof of a?global averaging theorem on the?convergence of attractors of non-autonomous equations to the?attractor of the?average autonomous equation as .
39 citations
TL;DR: It is shown in this paper that a three-dimensional variety that is a conic bundle in the Mori sense has a base with at most double rational singularities of type, and a rationality criterion is proved subject to this assumption in the case when the discriminant curve is large enough, for example, for the case of.
Abstract: It is that a three-dimensional variety that is a conic bundle in the Mori sense has a base with at most double rational singularities of type . A rationality criterion is proved subject to this assumption in the case when the discriminant curve is large enough, for example, for the case when .
29 citations
TL;DR: A modification of Hermite's method for an expressly constructed Nikishin system is used in this paper to prove Apery's theorem on the irrationality of π2 is transcendental.
Abstract: A new proof of the fact that π2 is transcendental is proposed. A modification of Hermite's method for an expressly constructed Nikishin system is used. The Beukers integral, which was previously used to prove Apery's theorem on the irrationality of ζ(2) and ζ(3) is a special case of this construction.
28 citations
TL;DR: In this paper, it was shown that given a symmetric group of sufficiently large degree, every irreducible representation of it with Young diagram fitting into a square with side is of dimension at least.
Abstract: Two main results of this paper are singled out. The first one relates to the representation theory of symmetric groups. The second one deals with varieties of Lie algebras over a field of characteristic zero. The first result can be presented as follows: given a symmetric group of sufficiently large degree , every irreducible representation of it with Young diagram fitting into a square with side is of dimension at least . The second result states that there are no varieties of Lie algebras over a field of characteristic zero with lower exponent strictly less than two. At the same time, examples of varieties with exponent two are presented.
25 citations
TL;DR: In this article, the Navier-Stokes system with boundary and initial conditions is considered in a bounded domain with boundary consisting of two disjoint closed curves and such that is connected and.
Abstract: Let be a bounded domain with boundary consisting of two disjoint closed curves and such that is connected and . The Navier-Stokes system , is considered in with boundary and initial conditions and (here , , and is the outward normal to ). Let be a solution of this system such that satisfies the indicated boundary conditions on and , where . Then the existence of a control on with the following properties is proved: the solution of the Navier-Stokes system such that , and , coincides with for , that is, . In particular, if and do not depend on and is an unstable steady-state solution, then it follows from the above result that one can suppress the occurrence of turbulence by some control on . An analogous result is established in the case when and is a distributed control concentrated in an arbitrary subdomain .
TL;DR: In this article, a general theorem that establishes a relation between linear and algebraic independence of values at algebraic points of E-functions and properties of the ideal formed by all algebraic equations relating these functions over the field of rational functions is presented.
Abstract: We prove a general theorem that establishes a relation between linear and algebraic independence of values at algebraic points of E-functions and properties of the ideal formed by all algebraic equations relating these functions over the field of rational functions. Using this theorem we prove sufficient conditions for linear independence of values of E-functions as well as for algebraic independence of values of subjects of them. The main result is an assertion stating that at all algebraic points, except finitely many, the values of E-functions are linearly independent over the field of all algebraic numbers if the corresponding functions are linearly independent over the field of rational functions. The theorem is applied to concrete E-functions.
TL;DR: For a second-order elliptic equation involving a parameter, with principal part in divergence form in Lipschitz domain mixed problems (of Zaremba type) with non-homogeneous boundary conditions are considered for generalized functions in this article.
Abstract: For a second-order elliptic equation involving a parameter, with principal part in divergence form in Lipschitz domain mixed problems (of Zaremba type) with non-homogeneous boundary conditions are considered for generalized functions in . The Poincare-Steklov operators on Lipschitz piece of the domain's boundary corresponding to homogeneous mixed boundary conditions on are studied. For a homogeneous equation with separation of variables in a tube domain with Lipschitz section, the Fourier method is substantiated for homogeneous mixed boundary conditions on the lateral surface and non-homogeneous conditions on the ends.
TL;DR: In this article, the authors studied the subgroups of over a finite field that comprise a concjugate in the group of upper unit-angular matrices of degree 2 over an infinite field such that it is an algebraic extension.
Abstract: We study the subgroups of over a field that comprise a conjugate in of the group of upper-unitriangular matrices of degree 2 over an infinite field such that is an algebraic extension.
TL;DR: In this article, it was proved that the first boundary value problem for a second-order system of elliptic equations has at least one solution belonging to the space of the entire cylinder.
Abstract: In the half-cylinder , , we study a second-order system of elliptic equations containing a non-linear function and right-hand side , , . If these functions satisfy certain conditions, then it is proved that the first boundary-value problem for this system has at least one solution belonging to the space , . We study the behaviour of the solutions of this system a . Along with the original system we study the family of systems obtained from it through shifting with respect to by all , . A semigroup , acts on the set of solutions of these systems of equations. It is proved that this semigroup has a trajectory attractor consisting of the solutions in that admit a bounded extension to the entire cylinder . Solutions are attracted by the attractor as . We give a number of applications and consider some questions of the theory of perturbations of the original system of equations.
TL;DR: In this paper, a generalization of the Bogolyubov-Krylov theorem on the existence of a 1-projectively invariant measure for a circle homeomorphism is presented.
Abstract: General groups of orientation-preserving homeomorphisms of R are investigated. A series of metric invariants are defined for such groups: {omega}-projectively invariant measures, where {omega} is a cardinal number. A theorem on the existence of an {omega}-projectively invariant measure is formulated, which is a natural generalization of the Bogolyubov-Krylov theorem on the existence of an invariant measure for a circle homeomorphism. For groups with an {omega}-projectively invariant measure 'obstructions' to the existence of a 1-projectively invariant measure are analysed. The approach is based on the study of the topological structure of the set of all fixed points of the elements of the group, the orbits of points in the line, minimal sets, and the combinatorial properties of groups.
TL;DR: In this article, a tree Lie algebra over a field of characteristic zero is considered and the decomposition of the homogeneous spaces into irreducible components and their multiplicities are calculated.
Abstract: In this paper we consider a tree Lie algebra over a field of characteristic zero. This algebra is a module over the full linear group, and the spaces of homogeneous elements are invariant under this action. We study the decomposition of the homogeneous spaces into irreducible components and calculate their multiplicities. One method for calculating these multiplicities involves their connection with the values of the irreducible characters of the symmetric group on conjugacy classes of elements corresponding to a product of independent cycles of the same length. In the second section we give an explicit formula for calculating such character values. This formula is analogous to the hook formula for the dimension of the irreducible modules of the symmetric group. In the second method for calculating multiplicities we make use of Witt's formula for the dimensions of the polyhomogeneous components of a free Lie algebra. The rest of this paper deal with relations between the Hilbert series of a free two-generator Lie algebra and the generating series of the multiplicities of the irreducible modules in this algebra.
TL;DR: In this article, a sequence of boundary-value problems for a second-order non-linear elliptic equation in domains is considered, and strong convergence of the solutions of the problems under consideration is proved in for ; a corrector in and a homogenized boundary value problem are constructed.
Abstract: A sequence of boundary-value problems for a second-order non-linear elliptic equation in domains and is considered. No geometric assumptions on the are made. The existence of a sequence approaching zero as is assumed such that for and for an arbitrary point . Here is the -cube with centre at and is the -capacity. The conditions imposed on the coefficients of the equation ensure that the energy space is . The strong convergence of the solutions of the problems under consideration is proved in for ; a corrector in and a homogenized boundary-value problem are constructed. These results are based on an asymptotic expansion for the sequence and on a new pointwise estimate of the solution of a certain model non-linear problem.
TL;DR: In this article, an operator algebra associated with a smooth embedding is constructed, and for elliptic elements of this algebra a finiteness theorem (the Fredholm property) is established, and the index is computed.
Abstract: An operator algebra associated with a smooth embedding is constructed. For elliptic elements of this algebra a finiteness theorem (the Fredholm property) is established, and the index is computed. A connection with Sobolev problems is shown.
TL;DR: In this paper, the -analogues of modified Bessel functions and Macdonald functions are defined, which arise in harmonic analysis on quantum symmetric spaces, based on the power series expansion.
Abstract: The -analogues of modified Bessel functions and Macdonald functions are defined in this paper. As in the case of -Bessel functions introduced by Jackson, there are two kinds of these functions. Like their classical prototypes, they arise in harmonic analysis on quantum symmetric spaces. The definition is based on the power series expansion. Recurrence relations, difference equations, and -Wronskians are obtained, as well as analogues of asymptotic expansions, which are convergent series if . Some relations for basic hypergeometric series, which result from this, are given.
TL;DR: In this article, a converse inequality with best possible constant is established for the products of zeros of an algebraic polynomial and its derivative lying outside the unit disc.
Abstract: Mahler has obtained an inequality for the products of zeros of an algebraic polynomial and its derivative lying outside the unit disc. In this paper a converse inequality with best possible constant is established.
TL;DR: Lower estimates in terms of all coefficients are established for polynomials and linear forms in the values of E-functions as discussed by the authors, and results for generalized hypergeometric Efunctions are indicated.
Abstract: Lower estimates in terms of all coefficients are established for polynomials and linear forms in the values of E-functions. Consequences for generalized hypergeometric E-functions are indicated.
TL;DR: In this paper, it was shown that the algebra of regular functions on quantum matrices admits a division ring of quotients and that this division ring is a twisted rational function, and that the center is a purely transcendental extension of a field of degree if both numbers and are odd.
Abstract: It is proved that the algebra of regular functions on quantum matrices admits a division ring of quotients and that this division ring is a division ring of twisted rational functions. A description is given of the field of central elements in the division ring of rational functions on quantum matrices in the one-parameter and multiparameter cases. In the one-parameter case for of a general form the center is a purely transcendental extension of a field of degree (were is the greatest common divisor of and ) if both numbers and are odd. If at least one of the numbers and is even, then the center is scalar. In the multiparameter case the answer depends upon the parameters ,, . Here the generators of the center are described and it is proved that the center is scalar for the case of even and parameters of a general form. Analogous result are obtained for the division ring of rational functions on a quantum Borel subgroup of .
TL;DR: In this paper, the supremum of a special form for rational functions and their derivatives on rectifiable curves of finite density is constructed for all open discs and certain estimates on sets that are not necessarily connected.
Abstract: Majorizing sums of special form are constructed for rational functions and their derivatives (here , ). As a consequence, several estimates of in integral metrics are obtained on rectifiable curves of finite density , where the supremum is taken over all open discs . Certain estimates on sets that are not necessarily connected are also obtained.
TL;DR: In this paper, the problem of embedding an "like" compact space in a Euclidean space is solved affirmatively under certain restrictions on the dimension of the space, and the relations between the realization problem and homotopy theory and differential topology are clarified.
Abstract: In this paper we consider questions of whether a compact space can be embedded in a Euclidean space. The problem of embedding an '-like' compact space in is solved affirmatively under certain restrictions on the dimension . We clarify the relations between the realization problem and areas of homotopy theory and differential topology.
TL;DR: In this article, a wide class of 'doubly degenerate' divergent quasilinear parabolic equations of an arbitrary order are studied, in particular the equations of non-stationary Newtonian and non-Newtonian filtration.
Abstract: Cauchy problems for a wide class of 'doubly degenerate' divergent quasilinear parabolic equations of an arbitrary order are studied. This class contains, in particular, the equations of non-stationary Newtonian and non-Newtonian filtration. For arbitrary initial functions of the lowest local regularity acceptable from the viewpoint of the theory of solubility it is proved that the rate of evolution of the supports of the generalized solutions is finite. Upper estimates of this rate are obtained which are exact both for large and small times.
TL;DR: In this paper, the authors studied groups of orientation-preserving homeomorphisms of which the complexity of the set of all fixed points of the group elements was measured. And each of the classes of groups thus distinguished a finer clasificaion was carried out in terms of the complexity of the topological structure of orbits and the composition properties of the groups.
Abstract: Groups of orientation-preserving homeomorphisms of are studied. Such metric invariants as invariant and projectively-invariant measures are investigated. The approach taken results in the classification of groups of homeomorphisms by the complexity of the set of all fixed points of the group elements. In each of the classes of groups thus distinguished a finer clasificaion is carried out in terms of the complexity of the topological structure of orbits and the combinatorial properties of the group.
TL;DR: In this paper, totally geodesic embeddings of an infinite series of 7-dimensional manifolds in 13-dimensional manifold of positive sectional curvature are constructed and the topology of these embedding is studied and the possibility of constructing new many-dimensional examples of manifolds of positive curvature is discussed.
Abstract: Totally geodesic embeddings of an infinite series of 7-dimensional manifolds in 13-dimensional manifolds of positive sectional curvature are constructed in this paper. The topology of these embeddings is studied and the possibility of constructing new many-dimensional examples of manifolds of positive curvature is discussed.
TL;DR: The homology groups of non-singular polynomial embeddings are calculated in this article for knots of arbitrary degree, and general algebraic techniques of such calculations are described.
Abstract: The homology groups of the spaces of non-singular polynomial (of degree ≤ 4) embeddings are calculated. General algebraic techniques of such calculations or spaces of polynomial knots of arbitrary degree are described.