A characterisation is given of some of the parabolics found in Co3, and SP4 (9) using the amalgam method.

Let G be a Hausdorff topological group and μ and λ be probability measures on G. We prove necessary and sufficient conditions for the existence of a probability measure ρ such that λ * ρ * μ = ρ under certain conditions. We prove a similar result for probability measures on semigroups.

For any pair of integers p, q such that (p, q) = 1 and p ≥ 2q, the graph has vertices {0, 1, …, p − 1} and edges {ij: q ≤ |i − j| ≤ p − q}. These graphs play the same role in the study of circular chromatic number as that played by the complete graphs in the study of chromatic number. The graphs share many properties of the complete graphs. However, there are also striking differences between the graphs and the complete graphs. We shall prove in this paper that for many pairs of integers p, q, one may delete most of the edges of so that the resulting graph still has circular chromatic number p/q. To be precise, we shall prove that for any number r < 2, there exists a rational number p/q (where (p, q) = 1) which is less than r but arbitrarily close to r, such that contains a subgraph H with and . This is in sharp contrast to the fact that the complete graphs are edge critical, that is, the deletion of any edge will decrease its chromatic number and its circular chromatic number.

If U, V are the generators of a rational or irrational rotation C*-algebra then an automorphism φ of the algebra is determined by φ(U) = λUaVc and φ(V) = μUbVd where λ, μ are complex numbers of modulus 1 and a, b, c, d are integers with ad − bc = 1. If ad − bc = −1, then these formulae determine an antiautomorphsm of the algebra. The classification of such automorphisms and antiautomorphisms up to conjugacy by arbitrary automorphisms is studied and an almost complete classification is obtained.

In this paper, we extend our theories on the dynamics of transcendental semigroups. Several properties of Fatou and Julia sets of transcendental semigroups will be explored.

It is shown that meromorphic solutions of certain first-order nonlinear differential equations do not have wandering domains.

Andersson obtained weighted Lp estimates for on analytic polyhedra. We provide an example to show why they cannot be improved.

In this paper, we first prove a fixed point theorem for a family of multivalued maps defined on product spaces. We then apply our result to prove an equilibrium existence theorem for an abstract economy. We also consider a system of variational inequalities and prove the existence of its solutions by using our fixed point theorem.

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

The structure of finite quotients of “large” subgroups of the multiplicative group of a finite dimensional division algebra is interesting and is related to the Margulis-Platonov conjecture. We develop machinery to handle such quotients and we conjecture that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. The proofs rely on Wedderburn's Factorisation Theorem.

Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

In this paper we prove some coincidence point theorems for nonself single-valued and multivalued maps satisfying a nonexpansive condition. These extend fixed point theorems for multivalued maps of a number of authors.

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

We study weak convergence of tensor products of vector measures with values in nuclear spaces, such as the space of all rapidly decreasing, infinitely differentiable functions, the space of all test functions, and the strong duals of those spaces. It is shown that the weak convergence of a net of tensor products of vector measures follows from that of the corresponding net of real product measures.

A new coincidence point theorem is proved for a pair of multivalued mappings operating between G-convex spaces. From this theorem, a generalisation of the classical Fan-Glicksberg fixed point theorem is established.

A relationship is pointed out between the results in a recent paper of Shin's and those in a previously published paper by M.J.C. Gover.

In this paper, we establish some estimates for the distance between adjacent zeros of the solutions of the first order delay differential equation

and the first order advanced differential equation

where P ∈ C([t0, ∞), [0, ∞)) and τ ∈ R+. Our results improve recent results in the literature.

We prove generalised results for localised approximation by generalised translation networks. We also show the relationship between the minimum number of neurons in the generalised translation networks with one hidden layer and the desired accuracy where the target functions are in a subset V1, p ([−1, 1]s) of the Sobolev space W1, p([−1, 1]s).

According to results of Heineken and Stadelmann, a non-torsion group is a 2-Baer group if and only if it is 2-Engel, and it has all subgroups 2-subnormal if and only if it is nilpotent of class 2. We extend some of these results to values of n greater than 2. Any non-torsion group which is an n-Baer group is an n-Engel group. The converse holds for n = 3, and for all n in the case of metabelian groups. A non-torsion group without involutions having all subgroups 3-subnormal has nilpotency class 4, and this bound is sharp.

It is known that the order of the isometry group on a compact Riemannian manifold with negative Ricci curvature is finite. We show by local nilpotent structures that a bound on the orders of the isometry groups exists depending only on the Ricci curvature, the conjugate radius and the diameter.