Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

     

A variant of the diamond principle for combinatorial ideals

We use a variant of the diamond principle to show many ideals on are not -saturated if is large. For instance, the -indescribable ideal is not -saturated if is almost ineffable.

     

-sequences

A sequence of positive integers is called a -sequence if every integer has at most representations with all in and . A -sequence is also called a -sequence or Sidon sequence. The main result is the following

Theorem. Let be a -sequence and for an integer . Then there is a -sequence of size , where .

Corollary. Let . The interval then contains a -sequence of size , when .

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

Conjugate -connections and holonomy groups

In this article we show that when the structure group of the reducible principal bundle is and is an -subbundle of , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to , and use the estimate to show that for all odd prime , if the holonomy group of the irreducible connection as above is simple and is not isomorphic to , , or , then it is isomorphic to .

     

Completely continuous multilinear operators on spaces

Given a -linear operator from a product of spaces into a Banach space , our main result proves the equivalence between being completely continuous, having an -valued separately continuous extension to the product of the biduals and having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to being weakly compact, and that, for , being weakly compact implies the conditions above but the converse fails.

     

Fonctions qui operent sur les espaces de Besov

Soient , et trois réels tels que , , et et soit une fonction appartenant à l'espace de Besov . Nous montrons que si est une fonction, de la variable réelle, nulle à l'origine, lipschitzienne et appartenant à l'espace on a alors . La preuve est essentiellement basée sur des résultats d'approximation par des fonctions splines de degré .

     

Congruences between the coefficients of the Tate curve via formal groups

Let be the Tate curve with canonical differential, . If the characteristic is , then the Hasse invariant, , of the pair should equal one. If , then calculation of leads to a nontrivial separable relation between the coefficients and . If or , Thakur related and via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of . Our analysis was motivated by the study of the invariant which is an infinite Witt vector generalizing the Hasse invariant.

     

Kuttner's problem and a Pólya type criterion for characteristic functions

Let be a continuous function with and . If is convex, then , , is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya's theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner's function , , which gives the minimal value of such that is a characteristic function. Specifically, . Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.

     

Matrices over orders in algebraic number fields as sums of -th powers

David R. Richman proved that for every integral matrix is a sum of seven -th powers. In this paper, in light of a question proposed earlier by M. Newman for the ring of integers of an algebraic number field, we obtain a discriminant criterion for every matrix over an order of an algebraic number field to be a sum of (seven) -th powers.

     

On subdiagonal algebras for subfactors

We show that if are type factors with finite index (and common identity) and is the trace preserving conditional expectation, then there are no subdiagonal algebras in with respect to unless .

     

Sampling sequences for Hardy spaces of the ball

We show that a sequence in the unit ball of is sampling for the Hardy spaces , , if and only if the admissible accumulation set of in the unit sphere has full measure. For the situation is quite different. While this condition is still sufficient, when (in contrast to the one dimensional situation) there exist sampling sequences for whose admissible accumulation set has measure 0. We also consider the sequence obtained by applying to each a random rotation, and give a necessary and sufficient condition on so that, with probability one, is of sampling for , .

     

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

Type factors generated by purely infinite simple C*-algebras associated with free groups

Let be a free product of at least two but at most countably many cyclic groups. With each such group we associate a family of C*-algebras, denoted and generated by the reduced group C*-algebra and a collection of projections onto the -spaces over certain subsets of . We determine , the weak closure of in , and use this result to show that many of the C*-algebras in question are non-nuclear.

     

Lipschitz continuity of oblique projections

Let and be complementary spaces of a finite dimensional unitary space and let denote the projection of on parallel to . Estimates for the norm of are derived which involve the norm of the restriction of to or the gap between and .

     

Relative Brauer groups of discrete valued fields

Let be a non-trivial finite Galois extension of a field . In this paper we investigate the role that valuation-theoretic properties of play in determining the non-triviality of the relative Brauer group, , of over . In particular, we show that when is finitely generated of transcendence degree 1 over a -adic field and is a prime dividing , then the following conditions are equivalent: (i) the -primary component, , is non-trivial, (ii) is infinite, and (iii) there exists a valuation of trivial on such that divides the order of the decomposition group of at .

     

An obstruction to 3-dimensional thickenings

In this paper we give a characterization of those locally finite -dimensional simplicial complexes that have an orientable -manifold thickening. This leads to an obstruction for a fake surface to admit such a thickening. The obstruction is defined in , where is the subgraph consisting of all the -simplexes of order three.

     

Reflection and uniqueness theorems for harmonic functions

Suppose that is harmonic on an open half-ball in such that the origin 0 is the centre of the flat part of the boundary . If has non-negative lower limit at each point of and tends to 0 sufficiently rapidly on the normal to at 0, then has a harmonic continuation by reflection across . Under somewhat stronger hypotheses, the conclusion is that . These results strengthen recent theorems of Baouendi and Rothschild. While the flat boundary set can be replaced by a spherical surface, it cannot in general be replaced by a smooth -dimensional manifold.

     

A characterization of total reflection orders

Let be a Coxeter system with set of reflections . It is known that if is a total reflection order for , then, for each , and its complement are stable under conjugation by . Moreover the upper and lower -conjugates of are still total reflection orders. For any total order on , say that is stable if is stable under conjugation by for each . We prove that if and all orders obtained from by successive lower or upper -conjugations are stable, then is a total reflection order.