Showing papers in "Izvestiya: Mathematics in 2007"
TL;DR: In this paper, the equivalence of the modulus of smoothness and the -functional is established for functions of the Nikol'skii-Besov type and function spaces of the type are defined.
Abstract: We study problems of approximation of functions on? in the metric of? with power weight using generalized Bessel shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Bessel shifts. We establish the equivalence of the modulus of smoothness and the -functional. We define function spaces of Nikol'skii-Besov type and describe them in terms?of best approximations. As a?tool for approximation, we use a?certain class of entire functions of exponential type. In this class, we prove analogues of Bernstein's inequality and others for the Bessel differential operator and its fractional powers. The main tool we use to solve these problems is Bessel harmonic analysis.
37 citations
TL;DR: In this article, the authors extended the Khovanov homology theory over an arbitrary coefficient ring to the case of virtual knots and introduced a complex which is well-defined in the virtual case and is homotopy equivalent to the original Khovans complex in the classical case.
Abstract: The Khovanov homology theory over an arbitrary coefficient ring is extended to the case of virtual knots. We introduce a complex which is well-defined in the virtual case and is homotopy equivalent to the original Khovanov complex in the classical case. Unlike Khovanov's original construction, our definition of the complex does not use any additional prescription of signs to the edges of a cube. Moreover, our method enables us to construct a Khovanov homology theory for `twisted virtual knots' in the sense of Bourgoin and Viro (including knots in three-dimensional projective space). We generalize a number of results of Khovanov homology theory (the Wehrli complex, minimality problems, Frobenius extensions) to virtual knots with non-orientable atoms.
32 citations
TL;DR: In this paper, a family of Fibonacci tilings of the unit interval consisting of short and long elementary intervals with the ratio of the lengths equal to the golden section was constructed.
Abstract: We use the -operator to construct a family of Fibonacci tilings of the unit interval consisting of short and long elementary intervals with the ratio of the lengths equal to the golden section . We prove that the tilings satisfy a recurrence relation similar to the relation for the Fibonacci numbers. The ends of the elementary intervals in the tilings form a sequence of points whose derivatives are sequences similar to the sequence . We compute the direct and inverse renormalizations for the sequences . We establish a connection between our tilings and the Sturm sequence, and give some applications of the tilings in the theory of numbers.
24 citations
TL;DR: In this article, Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the Hodge operator, conjecture on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture on Kunneth components of the diagonal for smooth complex projective varieties are reduced to the case of rational varieties.
Abstract: We consider the following conjectures: , (over a perfect finitely generated field), Grothendieck's standard conjecture of Lefschetz type on the algebraicity of the Hodge operator , conjecture on the coincidence of the numerical and homological equivalences of algebraic cycles and conjecture on the algebraicity of Kunneth components of the diagonal for smooth complex projective varieties. We show that they are compatible with monoidal transformations: if one of them holds for a smooth projective variety and a smooth closed subvariety , then it holds for , where is the blow up of along . All of these conjectures are reduced to the case of rational varieties.
17 citations
TL;DR: In this paper, the extremal problem of finding the minimum of all such that there is an -uniform (each edge contains exactly vertices) hypergraph with exactly edges that does not possess property is studied.
Abstract: We study a classical problem (first posed by Erdős) in the extremal theory of hypergraphs. According to Erdős, a hypergraph possesses property if its set of vertices admits a 2-colouring such that no edge of the hypergraph is monochromatic. The problem is to find the minimum of all such that there is an -uniform (each edge contains exactly vertices) hypergraph with exactly edges that does not possess property . We consider more general problems (including the case of polychromatic colourings) and introduce a number of parametric properties of hypergraphs. We obtain estimates for analogues of for extremal problems on various classes of hypergraphs.
16 citations
TL;DR: In this article, the authors prove a theorem on the approximation of a trigonometric sum by a shorter one with the constants in the remainder calculated concretely, and prove a similar result for the case where the constants are constant constants.
Abstract: We prove a theorem on the approximation of a trigonometric sum by a shorter one with the constants in the remainder calculated concretely.
16 citations
TL;DR: In this article, the authors considered the homogenization of initial-boundary-value problems for parabolic equations with asymptotically degenerate rapidly oscillating periodic coefficients, which are models for diffusion processes in a strongly inhomogeneous medium.
Abstract: We deal with the homogenization of initial-boundary-value problems for parabolic equations with asymptotically degenerate rapidly oscillating periodic coefficients, which are models for diffusion processes in a strongly inhomogeneous medium. The solutions of these problems depend on a finite positive parameter and two small positive parameters. We obtain homogenized initial-boundary-value problems (whose solutions determine approximate asymptotics for solutions of the problems under consideration) and prove estimates for the accuracy of these approximations. The homogenized problems are initial-boundary-value problems for integro-differential equations whose solutions depend on additional positive parameters: the intensity of diffusion exchange and the impulse exchange. In the general case, the homogenized equations form a system of equations coupled through the exchange coefficients and define multiphase mathematical models of diffusion for a homogenized (limiting) medium. We consider the spectral properties of some homogenized problems. We also prove assertions on asymptotic reductions of the homogenized problems under additional hypothesis on the limiting behaviour of the exchange parameters.
15 citations
TL;DR: In this article, the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces was proved in dimension 2 and in dimension 3 by Agol.
Abstract: After results of the author (1980, 1981) and Vinberg (1981), the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces remained unknown in dimensions only. It was proved recently (2005) in dimension 2 by Long, Maclachlan and Reid and in dimension 3 by Agol. Here we use the results in dimensions 2 and 3 to prove the finiteness in all remaining dimensions . The methods of the author (1980, 1981) are more than sufficient for this using a very short and very simple argument.
15 citations
TL;DR: In this article, exact asymptotics of the distributions of integral functionals of Bessel processes in the -norm were derived for continuous-time Markov processes with continuous time.
Abstract: We prove theorems on exact asymptotics of the distributions of integral functionals of the occupation time of Bessel processes. Using these results, we obtain exact asymptotics of small deviations for Bessel processes in the -norm. We use Laplace's method for the occupation times of Markov processes with continuous time. Computations are carried out for and . We also solve extremal problems for the action functional.
13 citations
TL;DR: In this article, the authors classify birational transformations into elliptic fibrations of a general quasi-smooth hypersurface in of degree that has terminal singularities, and classify them into two classes.
Abstract: We classify birational transformations into elliptic fibrations of a general quasi-smooth hypersurface in of degree that has terminal singularities.
12 citations
TL;DR: In this paper, it was shown that it is impossible to construct unconditional bases of exponentials in the Bergman space in the case when there is a bounded convex domain on the plane such that at some point of the boundary the curvature exists and is different from zero.
Abstract: We prove that it is impossible to construct unconditional bases of exponentials in the Bergman space in the case when is a bounded convex domain on the plane such that at some point of the boundary the curvature exists and is different from zero.
TL;DR: In this paper, the authors considered dynamical systems of the form, where is a monic irreducible polynomial with coefficients in the ring of integers of a -adic field.
Abstract: We consider dynamical systems of the form , where is a monic irreducible polynomial with coefficients in the ring of integers of a -adic field . We also study 2-periodic points of some simple polynomials of this form in the case when .
TL;DR: In this paper, the authors studied numerical characteristics of Lie superalgebras and the growth of codimensions and constructed an insoluble variety of almost polynomial growth.
Abstract: We study numerical characteristics of varieties of Lie superalgebras and, in particular, the growth of codimensions. An example of an insoluble variety of almost polynomial growth is constructed. We prove that the exponent of this variety is equal to three and calculate the growth exponents for two earlier known soluble varieties.
TL;DR: In this article, the authors establish formulae for isometric embeddings and immersions of Mobius bands with a locally Euclidean metric and study extrinsic geometric properties of these surfaces.
Abstract: We establish formulae for isometric embeddings and immersions of Mobius bands with a locally Euclidean metric and study extrinsic geometric properties of these surfaces. We consider both standard Mobius bands corresponding to embeddings of a rectangular Mobius strip and general Mobius bands, in particular, those with generators orthogonal to the directrix.
TL;DR: The envelope of holomorphy of model manifolds of order 4 has been studied in this article, where it is shown to be a domain of bounded type whose distinguished boundary coincides with the initial manifold.
Abstract: We construct envelopes of holomorphy for model manifolds of order 4 and describe a class of such manifolds whose envelope is a cylindrical domain (with respect to certain variables) based on a Siegel domain of the second kind. This enables us to prove the holomorphic rigidity of model manifolds of this class. We also study the envelope of holomorphy of a special model manifold of type (1,4) and show it to be a domain of bounded type whose distinguished boundary coincides with the initial manifold. The holomorphic automorphism group of this domain coincides with that of the initial manifold. The envelope of holomorphy is fibred into orbits of this group. The generic orbits are 8-dimensional homogeneous non-spherical completely non-degenerate manifolds in .
TL;DR: In this paper, an intersection theory for subvarieties of a torus is developed, which takes into account the product of these points as elements of the ambient torus and yields Bernshtein's formula for the number of roots of a system as well as Khovanskii's formulas for their product.
Abstract: We develop an intersection theory for subvarieties of a torus. Besides the number of intersection points for a generic pair of subvarieties of complementary dimensions, this theory takes into account the product of these points as elements of the ambient torus. In the case of a complete intersection of divisors, our intersection theory yields Bernshtein's formula for the number of roots of a system as well as Khovanskii's formula for their product. When constructing this theory, we naturally encounter `piecewise-linear' subsets of the torus which are referred to as complex tropical varieties.
TL;DR: In this article, the problem of diffraction of an electromagnetic wave on a scatterer in a hollow waveguide was considered and the solubility of the boundary value problem for Maxwell's equations in a cylinder was studied.
Abstract: The paper deals with two problems of waveguide theory: the problem of radiation of electromagnetic waves in a regular waveguide with a filling variable in the cross-sections, and the problem of diffraction of an electromagnetic wave on a scatterer in a hollow waveguide. We consider radiation conditions and the solubility of the boundary-value problem for Maxwell's equations in a cylinder. We study several spectral problems connected with radiation conditions.
TL;DR: In this article, the authors consider Walsh functions on the binary group and study uniqueness sets for -fold multiple Walsh series under convergence over cubes (in other words, -sets), and prove that every finite set is a -set, construct examples of countable -sets and non-empty perfect -sets.
Abstract: We consider Walsh functions on the binary group and study uniqueness sets for -fold multiple Walsh series under convergence over cubes (in other words, -sets). We prove that every finite set is a -set, construct examples of countable -sets and non-empty perfect -sets, and give an example of a -set having the maximum possible Hausdorff dimension.
TL;DR: In this article, the authors dealt with non-negative solutions of the elliptic inequalities in, where and are functions and is an unbounded open subset of, and dealt with the problem of finding a nonnegative solution.
Abstract: This paper deals with non-negative solutions of the elliptic inequalities in , where and are functions and is an unbounded open subset of ,
TL;DR: The Lubin-Tate extension is an Abelian extension whose Galois group is isomorphic to for arbitrary fields that have Henselian discrete valuation rings with finite residue fields.
Abstract: We give an elementary proof of the assertion that the Lubin-Tate extension is an Abelian extension whose Galois group is isomorphic to for arbitrary fields that have Henselian discrete valuation rings with finite residue fields. The term `elementary' only means that the proofs are algebraic (that is, no transcedental methods are used [1], pp. 327, 332).
TL;DR: In this article, the authors studied the variation of the Mumford quotient by the action of a?maximal torus? on a?flag variety as they change the projective embedding, where the -linearization is induced by?the standard linearization.
Abstract: We study the variation of the Mumford quotient by the action of a?maximal torus? on a?flag variety as we change the projective embedding , where the -linearization is induced by?the standard -linearization. To do this, we describe the linear spans of the supports of the semistable orbits. This enables us to calculate the rank of?the Picard group of the quotient in the case when contains no simple components of type?.
TL;DR: In this paper, Kushnirenko and Bernshtein this paper showed that for a Lie group with a compact isotropy group, there is a Ricci-flat holomorphic invariant metric on the complexification of at least one of the irreducible components of the standard -simplex.
Abstract: To every homogeneous space of a Lie group with a compact isotropy group , where the isotropy representation consists of irreducible components of multiplicity , we assign a compact convex polytope in , namely, the Newton polytope of the rational function defined to be the scalar curvature of the invariant metric on . If is a compact semisimple group, then the ratio of the volume of to the volume of the standard -simplex is a positive integer . We note that in many cases, coincides with the number of isolated invariant holomorphic Einstein metrics (up to homothety) on . We deduce from results of Kushnirenko and Bernshtein that in all cases, . To every proper face of we assign a non-compact homogeneous space with Newton polytope that is a contraction of . The appearance of a "defect" is explained by the fact that there is a Ricci-flat holomorphic invariant metric on the complexification of at least one of the .
TL;DR: In this paper, the authors consider surfaces whose points are the lines on the real three-dimensional varieties of degree 3 and prove that the equivariant topological type of a complex Fano surface with the involution of complex conjugation determines the rigid isotopy class of the corresponding real 3-dimensional cubic.
Abstract: We consider surfaces whose points are the lines on the real three-dimensional varieties of degree 3. These surfaces are called Fano surfaces. This paper deals with finding the topological types, that is, a topological classification, of real Fano surfaces. Moreover, we prove that the equivariant topological type of the corresponding complex Fano surface with the involution of complex conjugation determines the rigid isotopy class of the corresponding real three-dimensional cubic.
TL;DR: In this paper, a homotopy classification of elliptic operators on a stratified manifold is given, and an explicit formula for the obstruction of Atiyah-Bott type to the existence of Fredholm problems in the case of stratified manifolds.
Abstract: We give a homotopy classification of elliptic operators on a stratified manifold. Namely, we establish an isomorphism between the set of elliptic operators modulo stable homotopy and the K-homology group of the manifold. By way of application, we obtain an explicit formula for the obstruction of Atiyah-Bott type to the existence of Fredholm problems in the case of stratified manifolds.
TL;DR: In this paper, the authors obtained upper and lower bounds for the best approximation in Stechkin's problem for the differentiation operator and in the problem of the reconstruction of the derivative from the values of the function at a given number of points for Nikol'skii and Besov classes of functions satisfying mixed Holder's conditions.
Abstract: We obtain upper and lower bounds for the best accuracy of approximation in Stechkin's problem for the differentiation operator and in the problem of the reconstruction of the derivative from the values of the function at a given number of points for Nikol'skii and Besov classes of functions satisfying mixed Holder's conditions. These estimates give the order of these quantities for almost all values of the parameters involved.
TL;DR: In this article, the authors considered the stability of solutions of non-linear second-order differential equations of a special form and considered the problem of whether two solutions can coincide given supplementary information in terms of the values of the functionals defined on the solutions.
Abstract: The paper deals with the approximation of functions belonging to the Sobolev spaces? and by functions of the form . The results obtained are applied to the study of the stability of solutions of non-linear second-order differential equations of a special form. We consider the problem of whether two solutions can coincide given supplementary information in terms of the values of the functionals , , defined on the solutions.
TL;DR: In this paper, it was shown that the set-valued e-projection on a weakly convex set and the unit outer normal vector to a solidly smooth set satisfy, as set functions, the Lipschitz condition and the Holder condition with exponent 1/2, respectively, relative to the Hausdorff metric.
Abstract: We continue the investigations started in [1]-[4], where weakly convex sets and set-valued maps with weakly convex images were studied. Sufficient conditions are found for the existence of a Lipschitz parametrization for a set-valued map with solidly smooth (generally, non-convex) images. It is also shown that the set-valued e-projection on a weakly convex set and the unit outer normal vector to a solidly smooth set satisfy, as set functions, the Lipschitz condition and the Holder condition with exponent 1/2, respectively, relative to the Hausdorff metric.
TL;DR: In this article, the authors established conditions guaranteeing that a group possesses the following property: there is a number such that if elements, of generate a finite subgroup then lies in the normalizer of, these conditions are of a quite special form.
Abstract: We establish conditions guaranteeing that a group possesses the following property: there is a number such that if elements , of generate a finite subgroup then lies in the normalizer of . These conditions are of a quite special form. They hold for groups with relations of the form which appear as approximating groups for the free Burnside groups of sufficiently large even exponent . We extract an algebraic assertion which plays an important role in all known approaches to substantial results on the groups of large even exponent, in particular, to proving their infiniteness. The main theorem asserts that when is divisible by 16, has the above property with .
TL;DR: In this article, the authors consider stable smooth maps from closed smooth manifolds to smooth manifold of higher dimensions and find universal linear relations between the Euler characteristics of the manifolds of singularities in their images.
Abstract: We consider stable smooth maps from closed smooth manifolds to smooth manifolds of higher dimension. For maps with corank 1 singularities, we find universal linear relations between the Euler characteristics of the manifolds of singularities in their images. The calculations are based on resolving the singularities by a construction that generalizes the iteration principle from algebraic geometry.
TL;DR: In this paper, the authors established results on the asymptotic behavior of solutions of non-stationary linearized equations of hydrodynamics with a small viscosity coefficient and periodic data oscillating rapidly with respect to the spatial variables.
Abstract: We establish results on the asymptotic behaviour of solutions of non-stationary linearized equations of hydrodynamics with a small viscosity coefficient and periodic data oscillating rapidly with respect to the spatial variables. We obtain boundary-layer terms, homogenized (limiting) equations and cell problems (whose solutions determine approximate asymptotics of solutions of the equations under consideration) and obtain estimates for the accuracy of the asymptotics. The form of the asymptotics depends strongly on the mutual asymptotic behaviour of the viscosity coefficient and the periodicity parameter that characterizes rapid oscillations of the data. When the viscosity coefficient is very small, the asymptotics can contain rapidly oscillating terms that increase linearly with respect to the time variable. Similar theorems are proved for non-stationary Stokes equations and partial results are obtained for non-stationary Navier-Stokes equations.