Local André-Oort conjecture for the universal abelian variety

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of , the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety over may be defined over R. We call a point R-special if is a canonical lift and ξ is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of , we show that an irreducible subvariety of containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.     

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

Some Categorical Algebraic Properties: Counter-Examples for Functor Categories

This work is a complement to the authors earlier papers, where it is shown that a functor category inherits from such properties as amalgamation, transferability and congruence extension if has either products or certain pushouts. A general scheme is given for constructing counter-examples which show that the latter condition on is essential. In particular, it is shown that the functor categories , , ( resp.) do not satisfy the amalgamation (congruence extension resp.) property in general. Moreover, one class of categories is described, where the condition of the existence of certain pushouts is not only sufficient, but also necessary for to preserve the considered properties of .Mathematics Subject Classifications (2000) 18A25, 18A32, 18B99, 08B26.Dali Zangurashvili: The support rendered by INTAS Grant 97 31961 is gratefully acknowledged.     

The Hochschild cohomology of a closed manifold

Let M be a closed orientable manifold of dimension d and be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, is a graded commutative and associative algebra. The augmentation map induces a morphism of algebras . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of , which is in general quite small. The algebra is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.     

Greedy expansions in Banach spaces

We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary . For convenience we assume that is symmetric: implies . The primary goal of this paper is to study representations of an element f∈X by a series

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group is largely determined by a linearly reductive subgroup scheme of SL(2), with the McKay quiver of relative to its standard module being the Gabriel quiver of the principal block . The graphs underlying these quivers are extended Dynkin diagrams of type or , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid , which we call ‘ -order’. This requires a theory of semicategories enriched in the quantaloid , that admit a suitable Cauchy completion. There is a quantaloid of -orders and ideal relations, and a locally ordered category of -orders and monotone maps; actually, . In particular is , with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general -orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of . Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.Mathematics Subject Classifications (2000) 06F07, 18B35, 18D05, 18D20.     

The Topology on the Space of δ-psh Functions in the Cegrell Classes

The aim of the present paper is to introduce a metric locally convex topology on the space of δ-psh functions in the Cegrell class . We prove that with this topology is a non-separable and non-reflexive Fréchet space. At the same time, we extend the Monge–Ampère operator from the class to .     

Multipliers of Fractional Cauchy Transforms

Let B_n denote the unit ball of , n ≥ 2. Given an α > 0, let denote the class of functions defined for by integrating the kernel against a complex-valued measure on the sphere . Let denote the space of holomorphic functions in the ball. A function is called a multiplier of provided that for every . In the present paper, we obtain explicit analytic conditions on which imply that g is a multiplier of . Also, we discuss the sharpness of the results obtained. This research was supported by RFBR (grant no. 08-01-00358-a), by the Russian Science Support Foundation and by the programme “Key scientific schools NS 2409.2008.1”.     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

Stability results for scattered‐data interpolation on Euclidean spheres

Let denote the unit sphere in and the geodesic distance in . A spherical‐basis function approximant is a function of the form , where are real constants, is a fixed function, and is a set of distinct points in . It is known that if is a strictly positive definite function in , then the interpolation matrix is positive definite, hence invertible, for every choice of distinct points and every positive integer M. The paper studies a salient subclass of such functions , and provides stability estimates for the associated interpolation matrices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Modified Alternating $$\vec{k}$$–generators

This paper deals with a class of pseudorandom bit generators – modified alternating –generators. This class is constructed similarly to the class of alternating step generators. Three subclasses of are distinguished, namely linear, mixed and nonlinear generators. The main attention is devoted to the subclass of linear and mixed generators generating periodic sequences with maximal period lengths. A necessary and sufficient condition for all sequences generated by the linear generators of to be with maximal period lengths is formulated. Such sequences have good statistical properties, such as distribution of zeroes and ones, and large linear complexity. Two methods of cryptanalysis of the proposed generators are given. Finally, three new classes of modified alternating –generators, designed especially to be more secure, are presented.     

The general dévissage theorem for Witt groups of schemes

Let be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex then there exists a dualizing complex of Z such that there is an isomorphism of coherent Witt groups for all . Received: 3 March 2006     

Generalized quadrangles with an ovoid that is translation with respect to opposite flags

The article [6] contains the result that if a finite generalized quadrangle of order s has an ovoid that is translation with respect to two opposite flags, but not with respect to any two non-opposite flags, then is self-polar and is the set of absolute points of a polarity. In particular, if is the classical generalized quadrangle Q(4, q) then is a Suzuki-Tits ovoid. In this article, we remove the need to assume that is Q(4, q) in order to conclude that is a Suzuki-Tits ovoid by showing that the initial assumptions in fact imply that is Q(4, q). At the same time, we also relax the requirement that have order s.Received: 14 May 2004     

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Let be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of containing a non-discrete space is generated by a class of prime spaces and if is a quotient-reflective subcategory of Top, then the assignment gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of that contain . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.Mathematics Subject Classifications (2000) 18D15, 54B30.     

Sampling Theory Associated with a Symmetric Operator with Compact Resolvent and De Branges Spaces

Let be a symmetric operator with compact resolvent defined in a Hilbert space For any fixed we consider an entire function K_a which involves the resolvent of Associated with K_a we obtain, by duality in a Hilbert space of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of regardless of the anti-linear mapping which has as its range space. There exists also a sampling formula allowing to recover any function in from its samples at the sequence of eigenvalues of This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.     

Uniform Mixed Hypergraphs: The Possible Numbers of Colors

For a mixed hypergraph , where and are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’ such that every meets some class in more than one vertex, and every has a nonempty intersection with at least two classes. The feasible set of , denoted , is the set of integers k such that admits a coloring with precisely k nonempty color classes. It was proved by Jiang et al. [Graphs and Combinatorics 18 (2002), 309–318] that a set S of natural numbers is the feasible set of some mixed hypergraph if and only if either or S is an ‘interval’ for some integer k ≥ 1. In this note we consider r-uniform mixed hypergraphs, i.e. those with |C| = |D| = r for all and all , r ≥ 3. We prove that S is the feasible set of some r-uniform mixed hypergraph with at least one edge if and only if either for some natural number k ≥ r − 1, or S is of the form where S′′ is any (possibly empty) subset of and S′ is either the empty set or {r − 1} or an ‘interval’ {k, k + 1, ..., r − 1} for some k, 2 ≤ k ≤ r − 2. We also prove that all these feasible sets can be obtained under the restriction , i.e. within the class of ‘bi-hypergraphs’. Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613.