Soft almost disjoint families

An almost disjoint family is said to be soft if there is an infinite set that meets each in a nonempty but finite set. We consider the associated cardinal invariant defined to be the minimal cardinality of an almost disjoint family that is not soft. We show that this cardinal coincides with J. Brendle's cardinal .

     

Associated primes of graded components of local cohomology modules

The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component of this local cohomology module is zero for all > 0$">. This paper is concerned with the asymptotic behaviour of as .

The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which is not finitely generated. Brodmann and Hellus have shown that is constant for all (that is, in their terminology, is asymptotically stable for ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.

Brodmann and Hellus raised various questions about such asymptotic behaviour when f$">. They noted that Singh's study of a particular example (in which ) shows that need not be asymptotically stable for . The second main aim of this paper is to determine, for Singh's example, quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.

     

Strong duals of projective limits of (LB)-spaces

We investigate the problem when the strong dual of a projective limit of (LB)-spaces coincides with the inductive limit of the strong duals. It is well-known that the answer is affirmative for spectra of Banach spaces if the projective limit is a quasinormable Fréchet space. In that case, the spectrum satisfies a certain condition which is called strong P-type. We provide an example which shows that strong P-type in general does not imply that the strong dual of the projective limit is the inductive limit of the strong duals, but on the other hand we show that this is indeed true if one deals with projective spectra of retractive (LB)-spaces. Finally, we apply our results to a question of Grothendieck about biduals of (LF)-spaces.     

A Strong Law for Mixing Random Variables

A sharp almost sure bound is derived for limit points of average sum of weakly dependent random variables, which ensures strong laws of large numbers for and -mixing random variables, without assumptions on rate of tending to zero of and -mixing parameters _n and _n.     

A uniform refinement property for congruence lattices

The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidt's construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tuma. In a previous paper, we constructed a distributive algebraic lattice with compact elements that cannot be obtained by Schmidt's construction. In this paper, we show that the same lattice cannot be obtained using the Pudlák, Tischendorf, Tuma approach.

The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.

     

Strong consistency of least squares estimates in multiple regression II

The strong consistency of least squares estimates in multiple regression models is established under minimal assumptions on the design and weak dependence and moment restrictions on the errors.     

Uniform Continuity of a Family of Weakly Regular Set Functions on Topological Spaces

Conditions for the uniform continuity of a family of weakly regular set functions defined on an algebra of subsets of a -topological space (T,) and taking values in an arbitrary topological space are found.     

Commutative Moufang Loops with Finite Conjugacy Classes of Subloops

It is proved that the following conditions are equivalent for an arbitrary commutative Moufang loop :1) the loop is finite over the center;2) every subloop of defines a finite conjugacy class of subloops;3) every associative subloop of defines a finite conjugacy class of subloops;4) every infinite associative subloop of defines a finite conjugacy class of subloops.     

Standard Envelopes of Commutative Triple Systems

Under certain constraints on the characteristic of a field , the commutative standard enveloping q-algebra >B of a commutative triple system A over is defined. It is proved that(1) if the algebra B is simple, then the system A is simple;(2) if the system A is simple, then B either is simple or decomposes into the direct sum of two isomorphic simple subalgebras (as of ideals).     

Ergodic-theoretic properties of certain Bernoulli convolutions

The aim of this paper is to prove some fixed point theorems which generalize well known basic fixed point principles of nonlinear functional analysis. Moreover, we investigate the class of mappings f: X→ X, where X is a Banach space, for which one of the main conditions in the metric fixed point theory, namely the condition (1), is satisfied. We obtain essential applications of this fact. All our results are illustrated by suitable examples. This revised version was published online in June 2006 with corrections to the Cover Date.