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 .

     

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 .

     

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.

     

-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 .

     

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 .

     

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.

     

Yang index of the deleted product

For any a -dimensional polyhedron is constructed such that the Yang index of its deleted product equals . This answers a question of Izydorek and Jaworowski (1995). For any a -dimensional closed manifold with involution is constructed such that , but can be mapped into a -dimensional polyhedron without antipodal coincidence.

     

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 .

     

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.

     

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 , .

     

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.

     

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let be a polynomial ring over a field. For a graded -module generated in degree at most , the Castelnuovo-Mumford regularity of each of (i) its symmetric power, (ii) its torsion-free symmetric power and (iii) the integral closure of its torsion-free symmetric power is bounded above by a linear function in with leading coefficient at most . For a graded ideal of , the regularity of is given by a linear function of for all sufficiently large . The leading coefficient of this function is identified.

     

Injective modules and linear growth of primary decompositions

The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal in a commutative Noetherian ring has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , there exists a minimal primary decomposition with for all . The generalization involves a finitely generated -module and several ideals; the short proof is based on the theory of injective -modules.

     

Abstract blowing down

Assume that is a surface over an algebraically closed field . Let be obtained from by blowing up a smooth point and let be the exceptional curve. Let be the category of coherent sheaves on . In this note we show how to recover from , if we know the object .

     

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 .

     

P-continuity on classical Banach spaces

Given a Banach space and an integer , the existence of an -homogeneous polynomial which is not uniformly continuous with respect to the polynomial topology on is investigated. We provide a complete characterization for some classical Banach spaces, while for others a surprising unresolved difficulty is encountered for a certain value of (depending on ).

     

Every -regular

We prove the following: Theorem A. If is a -regular ultrafilter, then either

(a)

is -regular, or

(b)

the cofinality of the linear order is , and is -regular for all .

Corollary B. Suppose that is singular, and either is regular, or . Then every -regular ultrafilter is -regular.

We also discuss some consequences and variations.

     

The sixth, eighth, ninth, and tenth powers of Ramanujan's theta function

In his Lost Notebook, Ramanujan claimed that the ``circular' summation of the -th powers of the symmetric theta function satisfies a factorization of the form . Moreover, Ramanujan recorded identities expressing , , , , and in terms of his theta functions , , and . Ramanujan's claims were proved by Rangachari, and later (via elementary methods) by Son. In this paper we obtain similar identities for , , , and .

     

A generalization of the Lefschetz fixed point theorem and detection of chaos

We consider the problem of existence of fixed points of a continuous map in (possibly) noninvariant subsets. A pair of subsets of induces a map given by if and elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If is metrizable, and are compact ANRs, and is continuous, then has a fixed point in provided the Lefschetz number of is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.

     

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 .