Showing papers in "Sbornik Mathematics in 2008"
TL;DR: In this paper it was shown that a quasitoric manifold over a combinatorial n-cube admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is a Bott tower.
Abstract: A Bott tower is the total space of a tower of fibre bundles with base and fibres . Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to the fixed points. We show that a quasitoric manifold over a combinatorial n-cube admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent result of Il'inskiĭ, who showed that a smooth compact toric variety admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of Hattori's problem of semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to this product. In the case of Bott towers the homeomorphism is actually a diffeomorphism.Bibliography: 18 titles.
73 citations
TL;DR: In this paper, the boundedness and compactness of products of differentiation operators and composition operators from mixed-norm spaces to α-Bloch spaces were studied, and they were shown to be bounded and compact.
Abstract: In this paper, we study the boundedness and the compactness of products of differentiation operators and composition operators from mixed-norm spaces to α-Bloch spaces. Bibliography: 18 titles.
71 citations
TL;DR: In this paper, it was shown that a finite-dimensional matrix algebra over an algebraically closed skew field is a simple matrix algebra if and only if it is isomorphic to a matrix algebra on the skew field.
Abstract: Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. In the paper it is proved that if the characteristic of F is zero or does not divide the order of any finite subgroup of G, then R is graded simple if and only if it is isomorphic to a matrix algebra over a finite-dimensional graded skew field.Bibliography: 24 titles.
70 citations
TL;DR: In this article, the authors considered the problem of describing the irreducible maximally symmetric oriented atoms, i.e., an atom is a branched covering of another atom, with branching points at vertices of the decomposition and/or the centres of faces.
Abstract: Regular (maximally symmetric) cell decompositions of closed oriented 2-dimensional surfaces (that is, regular maps or regular abstract polyhedra) are considered. These objects are also known as maximally symmetric oriented atoms. An atom is reducible if it is a branched covering of another atom, with branching points at vertices of the decomposition and/or the centres of faces. The following two problems have arisen in the theory of integrable Hamiltonian systems: describe the irreducible maximally symmetric atoms; describe all the maximally symmetric atoms covering a fixed irreducible maximally symmetric atom. In this paper, these problems are solved in important cases. As applications, the following maximally symmetric atoms are listed: the atoms containing at most 30 edges; the atoms containing at most six faces; the atoms containing p or 2p, where p is a prime.Bibliography: 52 titles.
50 citations
TL;DR: In this paper, a criterion for embeddings between generalized Weyl-Nikol'ski'skiˇ i and generalized Lipschitz classes is proposed, based on the concept of a (λ, β)-derivative, which is a generalization of deriva-tive in the sense of Weyl.
Abstract: Necessary and sufficient conditions for the accuracy of embed- ding theorems of various function classes are obtained. The main result of the paper is a criterion for embeddings between generalized Weyl-Nikol'skiˇ i and generalized Lipschitz classes. To define the Weyl-Nikol'skiˇ i classes we use the concept of a (λ, β)-derivative, which is a generalization of the deriva- tive in the sense of Weyl. As corollaries, estimates for the norms and moduli of smoothness of transformed Fourier series are obtained. Bibliography: 59 titles.
36 citations
TL;DR: In this paper, a new compactification of the moduli scheme of Gieseker-stable vector bundles with prescribed Hilbert polynomial on a smooth projective polarized surface defined over a field of characteristic zero is constructed.
Abstract: A new compactification of the moduli scheme of Gieseker-stable vector bundles with prescribed Hilbert polynomial on a smooth projective polarized surface defined over a field of characteristic zero is constructed. The families of locally free sheaves on the surface are completed by locally free sheaves on surfaces that are certain modifications of . The new moduli space has a birational morphism onto the Gieseker-Maruyama moduli space. The case when the Gieseker-Maruyama space is a fine moduli space is considered.Bibliography: 12 titles.
27 citations
TL;DR: In this paper, several important aspects of the Nelson-Erdős-Hadwiger classical problem of combinatorial geometry are considered and new lower bounds are obtained for the chromatic numbers of the spaces and with two, three or four forbidden distances.
Abstract: Several important aspects of the Nelson-Erdős-Hadwiger classical problem of combinatorial geometry are considered. In particular, new lower bounds are obtained for the chromatic numbers of the spaces and with two, three or four forbidden distances. Bibliography: 28 titles.
27 citations
TL;DR: In this article, it was shown that the improved integrability of the gradient of the solution of a -Laplacian with variable order of nonlinearity is logarithmic rather than power-like.
Abstract: Elliptic equations of -Laplacian type are investigated. There is a?well-known logarithmic condition on the modulus of continuity of the nonlinearity exponent , which ensures that a?Laplacian with variable order of nonlinearity inherits many properties of the usual -Laplacian of constant order. One of these is the so-called improved integrability of the gradient of the solution. It is proved in this paper that this property holds also under a?slightly more general condition on the exponent , although then the improvement of integrability is logarithmic rather than power-like. The method put forward is based on a?new generalization of Gehring's lemma, which relies upon the reverse H?lder inequality with increased support and exponent on the right-hand side. A counterexample is constructed that reveals the extent to which the condition on the modulus of continuity obtained is sharp. Bibliography: 28 titles.
27 citations
TL;DR: In this article, the Sturm-Liouville problem on a star graph is considered and the structure and properties of the eigenvalues and eigenfunctions of this problem are investigated.
Abstract: The paper considers the Sturm-Liouville problem on a star graph and is concerned with the structure and properties of the eigenvalues and eigenfunctions of this problem. Particular emphasis has been placed on the completeness of the system of eigenfunctions in the space of square-integrable functions and on the expansion of a given function as a generalized Fourier series in terms of this system. Such problems have great value in the study of boundary-value problems for linear partial differential equations on a graph by the Fourier method, and arise, for example, in the model of oscillatory processes in an elastic mast with supporting elastic guys. Bibiliography: 8 titles.
27 citations
TL;DR: In this paper, families of submerged or surface-piercing bodies parametrized by a characteristic linear size are found that have the following property: for each and each positive integer there exists such that for the interval of the continuous spectrum of the corresponding problem contains at least eigenvalues corresponding to trapped modes, that is, to solutions of the homogeneous problem that decay exponentially at infinity and possess finite energy.
Abstract: Problems of the linearized theory of waves on the surface of an ideal fluid filling a half-space or an infinite 3D-canyon are considered. Families of submerged or surface-piercing bodies parametrized by a characteristic linear size are found that have the following property: for each and each positive integer there exists such that for the interval of the continuous spectrum of the corresponding problem contains at least eigenvalues corresponding to trapped modes, that is, to solutions of the homogeneous problem that decay exponentially at infinity and possess finite energy. Bibliography: 38 titles.
24 citations
TL;DR: In this paper, the capacity functor defines a monad in the category of compacta containing the monad of inclusion hyperspaces as a submonad, and a metrization of the space of capacities of a compact metric space is defined.
Abstract: Spaces of upper-semicontinuous capacities on compacta are studied. It is proved that the capacity functor defines a monad in the category of compacta containing the monad of inclusion hyperspaces as a submonad. In addition, a metrization of the space of capacities of a compact metric space is defined. It is also proved that the capacity functor is open. Bibliography: 17 titles.
TL;DR: In this paper, the best approximations of the Besov classes of periodic functions of several variables in the spaces and by trigonometric polynomials whose harmonic indices lie in step hyperbolic crosses are obtained.
Abstract: Order estimates are obtained for the best approximations of the Besov classes of periodic functions of several variables in the spaces and by trigonometric polynomials whose harmonic indices lie in step hyperbolic crosses. The orders of the orthoprojection widths of the classes and the linear widths of the classes and in the space are found. Bibliography: 22 titles.
TL;DR: It is established that for any computably enumerable ( c.e.) set there exists a family that is -c.e. if and only if the set is not -computable.
Abstract: An almost computably enumerable family that is not -computably enumerable is constructed. Moreover, it is established that for any computably enumerable (c.e.) set there exists a family that is -c.e. if and only if the set is not -computable.Bibliography: 5 titles.
TL;DR: Analogues of theorems on a central point, a central transversal and also of Tverberg's theorem are proved in the context when arrangements of hyperplanes or planes of fixed dimension are considered in place of point sets as mentioned in this paper.
Abstract: Analogues of theorems on a central point, a central transversal and also of Tverberg's theorem are proved in the context when arrangements of hyperplanes or planes of fixed dimension are considered in place of point sets.Bibliography: 20 titles.
TL;DR: In this article, the equivalence of the global and local Pompeiu properties for a compact subset of without any assumptions on the Heisenberg group has been established for subspaces of invariant under shifts and unitary transformations.
Abstract: Local versions of the Brown-Schreiber-Taylor theorem on spectral analysis in are obtained under most general assumptions. This has made it possible, in particular, to prove the equivalence of the global and the local Pompeiu properties for a compact subset of without any assumptions on . Perfect analogues of these results are established for systems of convolution equations on the Heisenberg group . As an application, for subspaces of invariant under shifts and unitary transformations a spectral synthesis theorem is proved, analogues of which were known before only for functions of slow growth.Bibliography: 20 titles.
TL;DR: In this paper, a method for deriving new pointwise estimates for the solution and integral estimates of the gradient of the solution with a point singularity was developed. But this method is not suitable for solutions with a singularity.
Abstract: This paper is concerned with the investigation of solutions with a point singularity of the general elliptic equation A method for deriving new pointwise estimates for the solution and integral estimates for the gradient of the solution is developed. Precise conditions are established on the behaviour of the term characterizing the absorption to ensure the non-existence of solutions with a point singularity.Bibliography: 11 titles.
TL;DR: In this article, the authors considered the problem of the absence of global solutions of initial-boundary value problems for the Kuramoto-Sivashinsky equation and provided sufficient conditions for the absence.
Abstract: The problem of the absence of global solutions of initial-boundary value problems for the Kuramoto-Sivashinsky equation is considered. Sufficient conditions for the absence of global solutions of the problems under consideration are obtained both for bounded and unbounded domains. These conditions imply a?priori the blow-up of the solution of the corresponding initial-boundary value problem. The proof uses a?generalization of the method of non-linear capacity based on the choice of asymptotically optimal test functions.Bibliography: 20 titles.
TL;DR: In this paper, it was shown that a homogeneous elliptic operator with constant coefficients and locally bounded fundamental solution can be uniformly approximated on the plane by solutions of the same equation having singularities outside the plane.
Abstract: Let X be an arbitrary compact subset of the plane. It is proved that if L is a homogeneous elliptic operator with constant coefficients and locally bounded fundamental solution, then each function f that is continuous on X and satisfies the equation Lf = 0 at all interior points of X can be uniformly approximated on X by solutions of the same equation having singularities outside X. A theorem on uniform piecemeal approximation of a function is also established under weaker constraints than in the standard Vitushkin scheme. Bibliography: 24 titles.
TL;DR: In this article, the convergence of sequences of solutions of non-linear equations and variational inequalities for obstacle problems is proved, and sufficient conditions for the uniqueness of solutions are presented.
Abstract: Results on the convergence of sequences of solutions of non-linear equations and variational inequalities for obstacle problems are proved. The variational inequalities and equations are defined by a non-linear, pseudomonotone operator of the second order with periodic, rapidly oscillating coefficients and by sequences of functions characterizing the obstacles and the boundary conditions. Two-scale and macroscale (homogenized) limiting problems for such variational inequalities and equations are obtained. Results on the relationship between solutions of these limiting problems are established and sufficient conditions for the uniqueness of solutions are presented. Bibliography: 25 titles.
TL;DR: In this paper, the non-rationality of double covers of branched over a hypersurface of degree with isolated singularities was proved for the case where the multiplicity of each singular point of does not exceed a certain threshold and the projectivization of its tangent cone is smooth.
Abstract: The non-rationality is proved for double covers of branched over a hypersurface of degree with isolated singularities such that the multiplicity of each singular point of does not exceed and the projectivization of its tangent cone is smooth. Bibliography: 15 titles.
TL;DR: In this article, it was shown that every skew-Hermitian element of any properly infinite von Neumann algebra can be represented in the form of a finite sum of commutators of projections in this algebra.
Abstract: It is proved that every skew-Hermitian element of any properly infinite von Neumann algebra can be represented in the form of a finite sum of commutators of projections in this algebra. A new commutation condition for projections in terms of their upper (lower) bound in the lattice of all projections of the algebra is obtained. For the full matrix algebra the set of operators with canonical trace zero is described in terms of finite sums of commutators of projections and the domain in which the trace is positive is described in terms of finite sums of pairwise products of projections. Applications to AF-algebras are obtained. Bibliography: 33 titles.
TL;DR: In this article, the maximal ideal spaces of the C*-algebras generated by irreversible dynamical systems and corresponding reversible extensions of endomorphisms are calculated, and connections between the objects that arise and dynamical system of Smale horseshoe and other types are revealed.
Abstract: A construction of a reversible extension of irreversible dynamical systems is presented. It is based on calculating the maximal ideal spaces of the C*-algebras generated by these systems and the corresponding reversible extensions of endomorphisms. Connections between the objects that arise and dynamical systems of Smale horseshoe and other types are revealed. Bibliography: 20 titles.
TL;DR: In this article, it was shown that the Leech dimension of a free partially commutative monoid is equal to the least upper bound of the cardinalities of finite subsets of pairwise commuting generators of the monoid.
Abstract: The paper is devoted to problems arising when applying homological algebra to computer science. It is proved that the Leech dimension of a free partially commutative monoid is equal to the least upper bound of the cardinalities of finite subsets of pairwise commuting generators of the monoid. For an arbitrary free partially commutative monoid in which every subset of pairwise commuting generators is finite and for any contravariant natural system on we construct a semicubical set with a homological system on this set such that the Leech homology groups are isomorphic to the cubical homology groups . Complexes of Abelian groups are also constructed enabling one to obtain (under additional finiteness conditions) algorithms for computing the Leech homology groups and homology groups with coefficients in right -modules. Bibliography: 16 titles.
TL;DR: A generalization to several variables of the classical Poincare theorem on the asymptotic behaviour of solutions of a linear difference equation is presented in this article, where the authors consider the case of linear difference equations.
Abstract: A generalization to several variables of the classical Poincare theorem on the asymptotic behaviour of solutions of a linear difference equations is presented.
TL;DR: In this paper, it was shown that the inequality holds for an arbitrary sequence of independent functions,,,, if and only if has the Kruglov property, which is necessary and sufficient for a version of Maurey's well-known inequality for vector-valued Rademacher series with independent coefficients to hold in.
Abstract: Let be a separable or maximal rearrangement invariant space on [0,1]. It is shown that the inequality holds for an arbitrary sequence of independent functions , , , if and only if has the Kruglov property. As a consequence, it is proved that the same property is necessary and sufficient for a version of Maurey's well-known inequality for vector-valued Rademacher series with independent coefficients to hold in .Bibliography: 24 titles.
TL;DR: The existence of a function for which the greedy algorithm in the Faber-Schauder system is divergent in measure on is established in this paper, where it is shown that for each,, there exists a measurable subset of of measure such that each one can find a function coinciding with on.
Abstract: The existence of a function for which the greedy algorithm in the Faber-Schauder system is divergent in measure on is established It is shown that for each , , there exists a measurable subset of of measure such that for each one can find a function coinciding with on , whose greedy algorithm in the Faber-Schauder system converges uniformly on Bibliography: 33 titles
TL;DR: In this article, a linear system of differential equations describing the joint motion of a thermoelastic porous body and an incompressible thermofluid occupying a porous space is considered.
Abstract: A linear system of differential equations describing the joint motion of a thermoelastic porous body and an incompressible thermofluid occupying a porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve non-smooth rapidly oscillating coefficients, inside the differentiatial operators. A rigorous substantiation based on Nguetseng's two-scale convergence method is carried out for the procedure of the derivation of homogenized equations (not containing rapidly oscillating coefficients), which for different combinations of the physical parameters can represent Biot's system of equations of thermo-poroelasticity, the system consisting of Lame's non-isotropic equations of thermoelasticity for the solid component and the acoustic equations for the fluid component of a two-temperature two-velocity continuum, or Lame's non-isotropic thermoelastic system for a two-temperature one-velocity continuum. Bibliography: 16 titles.
TL;DR: In this article, the authors investigate the asymptotic behavior of the largest minimal weighted pairwise distance between two points restricted to a rectifiable compact set embedded in Euclidean space.
Abstract: We investigate the asymptotic behaviour, as N grows, of the largest minimal weighted pairwise distance between N points restricted to a rectifiable compact set embedded in Euclidean space, and we find the limit distribution of asymptotically optimal configurations. Bibliography: 23 titles.
TL;DR: In this article, precisely characterizations of an admissible rate of decrease of a non-trivial function having zero integrals over all balls of fixed radius are established, and the case of essentially anisotropic behaviour of the function at infinity is considered for the first time.
Abstract: Precise characterizations of an admissible rate of decrease of a non-trivial function having zero integrals over all balls of fixed radius are established. The case of an essentially anisotropic behaviour of the function at infinity is considered for the first time. In particular, the function is even allowed to grow exponentially in one variable, which is compensated in a certain sense by its rapid decrease in other variables. Bibliography: 17 titles.
TL;DR: The divisorial canonicity of Fano double hypersurfaces of general position was proved in this article, where the authors proved that the Fano hypersurface is divisorially canonicity.
Abstract: The divisorial canonicity of Fano double hypersurfaces of general position is proved.Bibliography: 19 titles.