All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [6] for basic definitions and elementary results.

In [1] B. J. Birch, solving in the affirmative a conjecture of Erdὅs, proved the following result:

Theorem 1. Let p and q be coprime integers greater than 1. Then every large natural number may be written as a sum of distinct terms of type paqb.

In fact Birch pointed out that, with similar arguments, one could obtain a stronger version where the exponent b of q can be bounded in terms of p and q. The proofs were entirely elementary.

Let Fn be a free group of rank n and let Out Fn be its outer automorphism group. The main result of this paper is that Out F3 has a faithful representation as a group of automorphisms of the polynomial ring in seven variables over the integers. This extends a similar result for n = 2 (see Helling [3], Horowitz [5] and Rosenberger [12]), and provides a partial answer to a conjecture attributed in [5] to W. Magnus. As an application of the special nature of the representing polynomials, we obtain our second result, that Out F3 is virtually residually torsion-free nilpotent.

In [17] the second author introduced the concept of an involutary presentation. Such presentations are useful when one wants to deal geometrically with groups which have generators of order two. Now if one wants to deal with subgroups of such groups then it is no longer adequate to stick to presentations - one must consider arbitrary 2-complexes. However, these 2-complexes will differ from those which are usually considered, in that we need to allow some edges to be equal to their inverses.

Let R be a Noetherian ring. If R is commutative then (a) every non-unit is contained in a height-1 prime ideal - this is the Krull principal ideal theorem - but (b) R can have an arbitrarily large stable range. The main aim of this paper is to show that if certain non-commutative rings satisfy condition (a) then they have stable range one. We shall prove the

Theorem. Let R be a Noetherian domain which is not commutative. Suppose that every non-unit of R lies in at least one height-1 prime ideal and that every height-1 prime is completely prime. Then the stable range of R is one.

We prove that the following conditions are equivalent: X is an F space; every ideal of C(X) is flat; every submodule of a free C(X)-module is flat.

After having investigated in [7] generic properties of compact starshaped sets in ℝd, we shall restrict here our attention to compact starshaped sets whose kernels have positive dimension. While the main results in [7] are of a topological nature and concern the whole sets, the theorems presented here describe, for kernels of codimension 0 or 1, the local aspect of the boundaries and include, for kernels of positive dimension less than d— 1, both types of statements.

We introduce a new method for proving ‘witness lemmas’ in BQO theory. As an application, we obtain witness-lemmas for arbitrary trees and for σ-scattered linear orderings. A positive answer to a question by Van Engelen, Miller and Steel follows.

In recent years, much work in algebraic topology has been devoted to stable splitting phenomena. Often the existence of these splittings has first been detected at the cohomological level in terms of modules over the Steenrod algebra.

For example, W. Richter has exhibited a decomposition of ΩSU(n) of the form

(see [7]). Not only were cohomology calculations the initial evidence for this situation, but they further suggested that each summand Gk might be the Thom complex of a suitable k-plane complex vector bundle. This possibility was also verified by Mitchell.

If PL(v,z) = Σbi(v)zi is the skein polynomial of a link L, and D = D1 * D2 is the diagram which is a planar star (Murasugi) product of D1 and D2 then bϕ(D)(v) = bϕ(D1)·bϕ(D2)(v) where ϕ(D) = n(D)– (s(D) – 1) and n(D) denotes the number of crossings of D, and s(D) is the number of Seifert circles of D.

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

A n.c. JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky for C*-algebras. Different characterizations of associativity are given.

Group algebras and crossed products have always played an important role in the theory of C*-algebras, and there has also been considerable interest in various twisted analogues, where the multiplication is twisted by a two-cocycle. Here we shall discuss a very general family of twisted actions of locally compact groups on C*-algebras, and the corresponding twisted crossed product C*-algebras. We shall then establish some of the basic properties of these algebras, motivated by the requirements of some applications we have in mind [2, 9, 10]. Some of our results will be known to others, at least in principle, but we feel that a coherent account might be useful.

We define a class of completely positive maps, closed under composition, on a von Neumann algebra. We show that when the algebra has no atomic part, the correspondences associated to this class of completely positive maps are disjoint from the identity correspondence. This enables one simultaneously to generalize the statement and simplify the proof of a theorem of A. Connes and V. F. R. Jones on factors of type II1 with property T.

A construction of Weierstrass-like functions using recurrent sets is described, and the Hausdorff dimensions of the graphs computed. An important part of the proof is the notion of a globally random recurrent set. The Hausdorff dimension of a class of such sets is calculated using techniques of random matrix products.

The family of Brownian bridge processes (ba)a∈R has a number of characterizations, the most fundamental being that ba: [0,1] → ℝ is Brownian motion conditioned to be at the point a at time 1. Equivalently, ba is a continuous process whose law Wa is that of Wiener measure conditioned on the set of paths with x1 = a. These ideas are not so easy to make precise, so that more down to earth and workable characterizations of the Brownian bridge such as the following are often used in practice (see [6] for example)

We provide a concise account of the influence of design variables on the convergence rate in an L1 regression problem. In particular, we show that the convergence rate may be characterized precisely in terms of third and fourth moments of the design variables. This result leads to necessary and sufficient conditions on the design for the Berry-Esseen rate to be achieved. We also show that a moment condition on the error distribution is necessary and sufficient for a non-uniform Berry-Esseen theorem, and that an Edgeworth expansion is possible if the design points are not too clumped.

This paper continues the study initiated in [10] of the Markov branching process (MBP) subject to an independent Poisson process of catastrophe times at which the population size is reduced by random decrements whose distribution is independent of the current population size. More generally, let (Xt:t ≥ 0) be the Feller process on the non-negative integers ℕ+ having the generator