The Cauchy representation is not valid for every Schwartz distribution f ∈ (D(R))′ because (t – x)−l ∉ D(R). Since for all z ∈ {z: Im z ≠ 0} the kernel (t – z)−l belongs to (R) (1 > p > ∞), the Cauchy representation of the distributions in ((R))′ seems possible.

In this paper, we prove this fact. It is also proved that every probability density defines a generalised function on the space (R)(1 < p < ∞), of test functions. Applications of these results in probability theory are discussed.

In this paper we show that the average distance constant of a general polygon which is a subset of an M-space with non-positive curvature can be expressed as the extreme value of either of two nonlinear programs and discuss the practical application of one of these nonlinear programs for the determination of the average distance constant for a polygon in general, and in particular for a planar triangle.

The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p–group is identified, and using this information it is shown that if a metacyclic p–group has Wielandt length n, its nilpotency class is n or n + 1.

The equivalence of a fixed point theorem and the Fan-Knaster-Kuratowski-Mazurkiewicz theorem in H-space has been established. The fixed point theorem is then applied to obtain a theorem on sets with H-convex sections, and also results on minimax inequalities.

We discuss the structure of some inverse semigroups and the associated C* algebras. In particular, we study the bicyclic semigroup and the free monogenic inverse semigroup, following earlier work of Conway, Duncan and Paterson. We then associate to each zero-one matrix A an inverse semigroup CA, and show that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).

This paper resulted from an attempt to answer questions like: does βP have a subspace homeomorphic to βQ and does βQ have a subspace homeomorphic to P, where P denotes the space of all irrational numbers. These questions are answered in the negative by providing the appropriate machinery which can also be applied to other examples. En route we prove that weakly Fréchet realcompact spaces have homeomorphic Stone-Čech compactifications if and only if they are homeomorphic.

Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness.

We show that a compact convex body in En cannot be partitioned by n + 1 hyperplanes into 2n+l – 1 subsets of equal measure, thus generalising a result in the plane due to R.C. and E.F. Buck.

In this paper, the topology of the Denjoy space and characterisation of precompact sets in the Denjoy space are given.

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

An associative ring R with identity is called a PS-ring if soc(RR) is projective. We construct a non- PS-ring R which is a subdirect product of two PS-rings, thus answering a question of Nicholson and Watters in the negative.

It has long been known that there is a close connection between stochastic independence of continuous functions on an interval and space-filling curves [9]. In fact any two nonconstant continuous functions on [0, 1] which are independent relative to Lebesgue measure are the coordinate functions of a space filling curve. (The results of Steinhaus [9] have apparently been overlooked in more recent work in this area [3, 5, 6].)

We give single laws for two subvarieties of Steiner quasigroups (squags); the subvariety of squags whose underlying triple systems are the affine spaces over GF[3], and the subvariety of squags whose underlying triple systems are the affine triple systems.