The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

Throughout this article p denotes a fixed number such that 1 ≤ p < ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.

We give a simpler proof to a theorem of L. A. Karlovitz that the dual of a flat Banach space is flat, and also study some geometric properties of the dual space.

In this paper the congruence (f ∘ g)(n) = 0 (mod n) and the functional equation f ∘ f ∘ … ∘ f = g, are studied, where ∘ is an exponential regular convolution.

We obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

For two subsets Z and Y of a metric space (X, d) the set Z is said to be a bisector in Y iff Z ⊂ Y and there exist two distinct points y1, y2 ∊ Y such that Z = {z: d(z, y1) = d(z, y2) and z ∊ Y}. Considering chains of consecutive bisectors X ⊃ X1 ⊃ … ⊃ Xk we denote by b(X, d) the maximum of their length. The topological invariant b(X) is defined as the minimum of b(X, d) taken over the set of all metrizations of X. It is proved that if X is compact then dim(X) ≤ b(X) ≤ 2 dim(X) + 1, b(X) = 0 iff dim(X) = 0 and b(X) = n implies dim(X) = n for n = 1 and ∞. The sharp result b(En) = n for n = 1, 2, … is obtained for Euclidean space En.

Efforts to determine the orders of the sublattices of an arbitrary finite lattice date back at least to the early 1930's, and notably, in the work of Fritz Klein-Barmen [3], [4]. Nevertheless, very little that is new has appeared in the literature since that time.

Initial and final value theorems for Hardy transformations and of a suitably chosen function f(x) under a certain set of conditions on v and p where

1

Jv(x) and Yv(x) being Bessel functions of the first and second kind, and

2

su, v(x) being Lommel's function, are proved.

I wish to present here some of the results of a research in the Theory of Fibrations initiated some time ago by Peter Booth, Philip Heath, and myself. The philosophy behind the work is to approach certain aspects of the Theory of Fibrations in a unified way through the systematic use of the sections of suitable fibrations; this yields general theorems, of which some well-known results are eventually particular cases.

In this note we continue the study of Abian's order for reduced rings initiated in papers such as [1], [5], [3], [4]. A simple proof is given of Abian's result that taking suprema commutes with ring multiplication. The properties of orthogonally complete rings and of rings satisfying chain conditions with respect to Abian's order are investigated.

The first general integrability criterion is due to Riemann. He observed that a necessary and sufficient condition for a function f(x) on [a, b] to be integrable in his sense is that for each ε > 0 there exists a δ > 0 so that

where {x0, x1, …, xn} is a partition of [a, b] with diameter less than δ (and ω(f, I) denotes the oscillation of f on the interval I).

This paper is concerned with the infinite geometric products

and their generalizations to higher dimensions. Some new expressions and identities are derived for these products by using stochastic theory. The function is tabulated for p = 0(0.01)1.

In 1927, Jackson [5] obtained a transformation connecting a

where N is any integer, with a

viz.,

1

where | q | > l and |qγ-α-βN| > l.

This paper presents variants of the Holder inequality for integrals of functions (as well as for sums of real numbers) and its inverses. In these contexts, all possible transliterations and some extensions to more than two functions are also mentioned.

‖ A ‖ désignera le cardinal de A. Soient T une théorie complète du premier ordre dans un langage dénombrable, M un modèle de T, X un sous-ensemble de M; soit T(X) l'ensemble des formules closes à paramètres dans X satisfaites par M; on appelle n-type de T sur X un ensemble consistant avec T(X) et maximal de formules à paramètres dans X à n variables libres.

A geometric proof is presented that, under certain restrictions, the product of an h-cobordism with a closed manifold of Euler characteristic zero is a product cobordism. The results utilize open book decompositions and round handle decompositions of manifolds.

The purpose of this note is to unify Theorem 4 of G. Baumslag [2] and a result of D. M. Smirnov in [6] in a more general setting. We prove the following result.

In the first part of this note we give a very simple and elegant proof of the theorem on the order of the error function of the (k, r)-integers, which the authors proved earlier using elaborate calculations. We also obtain an improvement of an earlier result on the order of the same error function on the basis of the Riemann hypothesis.

In their paper [1], Buseman and Glassco state that N(C, 7, 3) = 5 has been claimed but questioned. Schouten in [2] provides a classification of the orbits of ∧3V under the action of the group of automorphisms of ∧3V induced by automorphisms of V when dim V = 7.