A note on meromorphic operators

Let be a complex Banach space and a bounded linear operator on . is called meromorphic if the spectrum of is a countable set, with the only possible point of accumulation, such that all the nonzero points of are poles of . By means of the analytical core we give a spectral theory of meromorphic operators. Our results are a generalization of some results obtained by Gong and Wang (2003).

     

On the algebra of functions -extendable for each finite

For each positive integer we construct a -function of one real variable, the graph of which has the following property: there exists a real function on which is -extendable to , for each finite, but it is not -extendable.

     

Stable rank of corner rings

B. Blackadar recently proved that any full corner in a unital C*-algebra has K-theoretic stable rank greater than or equal to the stable rank of . (Here is a projection in , and fullness means that .) This result is extended to arbitrary (unital) rings in the present paper: If is a full idempotent in , then . The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if where is a finitely generated projective generator, and can be generated by elements, then .

     

Spaces on which every pointwise convergent series of continuous functions converges pseudo-normally

A topological space is a -space provided that, for every sequence of continuous functions from to , if the series converges pointwise, then it converges pseudo-normally. We show that every regular Lindelöf -space has the Rothberger property. We also construct, under the continuum hypothesis, a -subset of of cardinality continuum.

     

On the Betti numbers of sign conditions

Let be a real closed field and let and be finite subsets of such that the set has elements, the algebraic set defined by has dimension and the elements of and have degree at most . For each we denote the sum of the -th Betti numbers over the realizations of all sign conditions of on by . We prove that

This generalizes to all the higher Betti numbers the bound on . We also prove, using similar methods, that the sum of the Betti numbers of the intersection of with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form or for , is bounded by

making the bound more precise.

     

Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings

Let be a real Banach space with a uniformly Gâteaux differentiable norm possessing uniform normal structure, be a nonempty closed convex and bounded subset of , be an asymptotically nonexpansive mapping with sequence . Let be fixed, be such that , , and . Define the sequence iteratively by , n= 0, 1, 2, ..._. $"> It is proved that, for each integer , there is a unique such that If, in addition, and , then converges strongly to a fixed point of .

     

Small prime solutions of quadratic equations II

Let be non-zero integers and any integer. Suppose that and for . In this paper we prove that (i) if the are not all of the same sign, then the above quadratic equation has prime solutions satisfying and (ii) if all the are positive and , then the quadratic equation is soluble in primes Our previous results are and in place of and above, respectively.

     

Self-adjointness of the perturbed wave operator on

We give classes of unbounded real-valued for which is self-adjoint on , , where is the wave operator defined on .

     

A row removal theorem for the Ext quiver of symmetric groups and Schur algebras

In 1981, G. D. James proved two theorems about the decomposition matrices of Schur algebras involving the removal of the first row or column from a Young diagram. He established corresponding results for the symmetric group using the Schur functor. We apply James' techniques to prove that row removal induces an injection on the corresponding between simple modules for the Schur algebra.

We then give a new proof of James' symmetric group result for partitions with the first part less than . This proof lets us demonstrate that first-row removal induces an injection on Ext spaces between these simple modules for the symmetric group. We conjecture that our theorem holds for arbitrary partitions. This conjecture implies the Kleshchev-Martin conjecture that for any simple module in characteristic . The proof makes use of an interesting fixed-point functor from -modules to -modules about which little seems to be known.

     

Concentration of mass and central limit properties of isotropic convex bodies

We discuss the following question: Do there exist an absolute constant 0$"> and a sequence tending to infinity with , such that for every isotropic convex body in and every the inequality holds true? Under the additional assumption that is 1-unconditional, Bobkov and Nazarov have proved that this is true with . The question is related to the central limit properties of isotropic convex bodies. Consider the spherical average . We prove that for every and every isotropic convex body in , the statements (A) ``for every , " and (B) ``for every , , where " are equivalent.

     

On functions whose graph is a Hamel basis

We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.

     

Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .

     

Coverings by convex bodies and inscribed balls

Let be a Hilbert space. For a closed convex body denote by the supremum of the radiuses of balls contained in . We prove that for every covering of a convex closed body by a sequence of convex closed bodies , . It looks like this fact is new even for triangles in a 2-dimensional space.

     

Infinite time blow-up for superlinear parabolic problems with localized reaction

We consider the nonlocal diffusion equation

on the space interval , with Dirichlet boundary conditions. It is known that if the curve remains in a compact subset of for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on is sharp: for a large class of functions approaching the boundary as , blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered.

     

Remark about heat diffusion on periodic spaces

Let be a complete Riemannian manifold with a free cocompact -action. Let be the heat kernel on . We compute the asymptotics of in the limit in which and . We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on coming from the Hodge inner product on .

     

Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Let be a Noetherian homogeneous ring with one-dimensional local base ring . Let be an -primary ideal, let be a finitely generated graded -module and let . Let denote the -th local cohomology module of with respect to the irrelevant ideal 0} R_n$"> of . We show that the first Hilbert-Samuel coefficient of the -th graded component of with respect to is antipolynomial of degree in . In addition, we prove that the postulation numbers of the components with respect to have a common upper bound.

     

Tietze extension theorem for Hilbert -modules

We prove the following generalization of the noncommutative Tietze extension theorem: if is a countably generated Hilbert -module over a -unital -algebra, then the canonical extension of a surjective morphism of Hilbert -modules to extended (multiplier) modules, , is also surjective.

     

Joins of projective varieties and multisecant spaces

Let , , be integral varieties. For any integers 0$">, , and set and . Let be the set of all linear -spaces contained in a linear -space spanned by points of , points of , ..., points of . Here we study some cases where has the expected dimension. The case was recently considered by Chiantini and Coppens and we follow their ideas. The two main results of the paper consider cases where each is a surface, more particularly:

or

     

``Beurling type' subspaces of and

In this note we extend the ``Beurling type' characterizations of subspaces of and to and , respectively.

     

A character of the gradient estimate for diffusion semigroups

Let be the semigroup of the diffusion process generated by on . It is proved that there exists and an -valued function such that holds for all 0$"> and all if and only if satisfies the formula for all