Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

This paper investigates the local integrals

where represents the integers of a composition algebra over a non-archimedean local field and is a non-trivial character on the units in the ring of integers of extended to by setting . The local zeta function for the trivial character is known for all composition algebras . In this paper, we show in the quaternion case that for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for equal to the quadratic character. In this latter case, for any character of order greater than .

     

On the higher delta invariants of a Gorenstein local ring

Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .

     

An existence theorem for systems of boundary value problems

We prove an existence theorem for , , in , using the shooting method. The function is supposed to be asymptotically linear.

     

On rigidity of affine surfaces

Rigidity of nondegenerate Blaschke surfaces in is studied. The rigidity criteria are given in terms of , where is the curvature of the Blaschke connection . If the rank of is 2, then the surface is rigid. If , it is nonrigid. In the case where the rank of is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.

     

Bounds for the operator norms of some Nörlund matrices

Suppose is a non-increasing sequence of non-negative numbers with , , , and is the lower triangular matrix defined by , , and , . We show that the operator norm of as a linear operator on is no greater than , for ; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the tend to a positive limit, the operator norm of on is exactly . We also give some cases when the operator norm of on is less than .

     

Every monotone graph property has a sharp threshold

In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let denote the Hamming space endowed with the probability measure defined by , where . Let be a monotone subset of . We say that is symmetric if there is a transitive permutation group on such that is invariant under . Theorem. For every symmetric monotone , if then for . ( is an absolute constant.)

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

Proof of the Simon-Ando Theorem

In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g.

where and are square blocks whose entries are all larger than those of and respectively.

Setting

partitioned as in , they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g. to and to . After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g.

where and .

The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.

     

and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

On the existence of positive solutions of nonlinear second order differential equations

Under suitable conditions on , the boundary value problem

has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.

     

Some remarks on the variation of curve length and surface area

Consider the curve , where is absolutely continuous on . Then has finite length, and if is the -neighborhood of in the uniform norm, we compare the length of the shortest path in with the length of . Our main result establishes necessary and sufficient conditions on such that the difference of these quantities is of order as . We also include a result for surfaces.

     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

A quick proof of the classification of simple real Lie algebras

Élie Cartan's classification of the simple Lie algebras over is derived quickly from some structure theory over and the classification over .

     

A theorem of Briançon-Skoda type for regular local rings containing a field

Let be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if is a minimal reduction of where is -primary, then where and is the largest ideal such that . The proof uses tight closure in characteristic and reduction to characteristic for rings containing the rationals.

     

Solution to a problem of S. Payne

A problem posed by S. Payne calls for determination of all linearized polynomials such that and are permutations of and respectively. We show that such polynomials are exactly of the form with and . In fact, we solve a -ary version of Payne's problem.

     

Parametrizing maximal compact subvarieties

For the Lie group , let be the open orbit of Lagrangian planes of signature in the generalized flag variety of Lagrangian planes in . For a suitably chosen maximal compact subgroup of and a base point we have that the orbit of is a maximal compact subvariety of . We show that for the connected component containing in the space of translates of which lie in is biholomorphic to , where denotes with the opposite complex structure.

     

Hilbert Cmodules in which all closed submodules are complemented

Let be a Calgebra. If there exists a full Hilbert module such that for each closed submodule , then is *-isomorphic to a Calgebra of (not neccesarily all) compact operators on a Hilbert space.

     

On the range of the sum of monotone operators in general Banach spaces

The purpose of this paper is to generalize the Brézis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range is topologically almost equal to the sum where is a compatible topology in as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .