A linear function associated to asymptotic prime divisors

Let be a Noetherian standard -graded ring and finitely generated, -graded -modules. Let be finitely many homogeneous ideals of . We show that there exist linear functions such that the associated primes over of and are stable whenever satisfies and , respectively.

     

Matrix presentations of braids and applications

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries or and the free monoid generated by generators and relation and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.

     

Zero distribution of Müntz extremal polynomials in

Let be a sequence of distinct positive numbers. Let and let denote the extremal Müntz polynomial in with exponents . We investigate the zero distribution of . In particular, we show that if

then the normalized zero counting measure of converges weakly as to

while if or , the limiting measure is a Dirac delta at or , respectively.

     

Non-commutative disc algebras and their representations

It is shown that the smallest closed subalgebra

generated by any sequence of isometries on a Hilbert space such that is completely isometrically isomorphic to the non-commutative ``disc' algebra introduced in Math. Scand. 68 (1991), 292--304. We also prove that for the Banach algebras and are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the ``disc' algebras on a Hilbert space are characterized. In particular, we prove that a sequence of operators is simultaneously similar to a contractive sequence (i.e., ) if and only if it is completely polynomially bounded. The first cohomology group of with coefficients in is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras introduced in Math. Scand. 68 (1991), 292--304. The right joint spectrum of the left creation operators on the full Fock space is also determined.

     

The -indices of Tsirelson type spaces

If and are countable ordinals such that , denote by the completion of with respect to the implicitly defined norm

where the supremum is taken over all finite subsets of such that and . It is shown that the Bourgain -index of is . In particular, if \alpha =\omega^{\alpha_{1}}\cdot m_{1}+\dots+\omega^{\alpha_{n}}\cdot m_{n}$"> in Cantor normal form and is not a limit ordinal, then there exists a Banach space whose -index is .

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

``Lebesgue measure' on , II

Let be the set of real numbers, and define . We construct a complete measure space where the -algebra contains the Borel subsets of , and is a translation-invariant measure such that for any measurable rectangle , if , then , where is Lebesgue measure on . The measure is not -finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure , we construct, via selfadjoint operators on , a ``Schrödinger model' of the canonical commutation relations: , , .

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

Invariant subspaces of the harmonic Dirichlet space with large co-dimension

In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift ) of the harmonic Dirichlet space . Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces with , . We will also generalize this to the Dirichlet classes , , as well as the Besov classes , , .

     

A closed model category for ()-connected spaces

For each integer , we give a distinct closed model category structure to the category of pointed spaces such that the corresponding localized category is equivalent to the standard homotopy category of -connected CW-complexes. The structure of closed model category given by Quillen to is based on maps which induce isomorphisms on all homotopy group functors and for any choice of base point. For each , the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on for .

     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

Paracompact subspaces in the box product topology

In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers .

We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.

     

A formula for -hyponormality of backstep extensions of subnormal weighted shifts

Let be a weight sequence of positive real numbers and let be a subnormal weighted shift with a weight sequence . Consider an extended weight sequence with and let 0: W_{\alpha (x)} \text{is} k \text{-hyponormal}\}$">for , where is the set of natural numbers. We obtain a formula to find the interval , which provides several examples to distinguish the classes of -hyponormal operators from one another.

     

Inversion formula and singularities of the solution for the backprojection operator in tomography

Let , , be the backprojection operator. The range of this operator as an operator on non-smooth functions is described and formulas for are derived. It is proved that the operator is not injective on but is injective on the subspace of which consists of even functions . Singularities of the function are studied. Here is a piecewise-smooth compactly supported function. Conditions for to have compact support are given. Some applications are considered.

     

Trudinger type inequalities in and their best exponents

We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

     

Extensions and extremality of recursively generated weighted shifts

Given an -step extension of a recursively generated weight sequence , and if denotes the associated unilateral weighted shift, we prove that

1).\end{cases}\end{displaymath}">

In particular, the subnormality of an extension of a recursively generated weighted shift is independent of its length if the length is bigger than 1. As a consequence we see that if is a canonical rank-one perturbation of the recursive weight sequence , then subnormality and -hyponormality for eventually coincide. We then examine a converse--an ``extremality" problem: Let be a canonical rank-one perturbation of a weight sequence and assume that -hyponormality and -hyponormality for coincide. We show that is recursively generated, i.e., is recursive subnormal.

     

Parametric decomposition of powers of ideals versus regularity of sequences

Let be an ideal in a Noetherian local ring . Then the sequence is -regular if every is a non-zerodivisor in and if for all integers , where runs over the elements of the set .

     

Purely infinite, simple -algebras arising from free product constructions. III

In the reduced free product of C-algebras, with respect to faithful states and , is purely infinite and simple if is a reduced crossed product for an infinite group, if is well behaved with respect to this crossed product decomposition, if and if is not a trace.     

The classical Banach spaces

In this paper we study some structural and geometric properties of the quotient Banach spaces , where is an arbitrary set, is an Orlicz function, is the corresponding Orlicz space on and , being the ideal of elements with finite support. The results we obtain here extend and complete the ones obtained by Leonard and Whitfield (Rocky Mountain J. Math. 13 (1983), 531-539). We show that is not a dual space, that , if for every , that has no smooth points, that it cannot be renormed equivalently with a strictly convex or smooth norm, that is a Grothendieck space, etc.