We study the behaviour of the transfinite diameter of regular polygons of fixed diameter, as a function of the number of their vertices.

In this paper, using the concept of w-distance, we first prove common fixed point theorems in a complete metric space. Then these theorems are used to improve Kannan's fixed point theorem, Ćirić's fixed point theorem, Kada, Suzuki and Takahashi's fixed point theorem and Ume's fixed point theorem.

In this paper, we find the dual relationship between solidness and the angle property of cones, which is characteristic of reflexivity for locally convex spaces.

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

We derive explicit formulae for the height of the Leech lattice and give evidence supporting the conjecture that the Leech lattice has minimum height and minimum values of the Epstein zeta function among all 24 dimensional lattices of determinant one.

We show that any Cuntz-Krieger algebra A over an infinite 0–1 matrix A may be realised as a Cuntz-Pimsner algebra X for a Hilbert bimodule X over a suitable Abelian C*-algebra with totally disconnected spectrum. Using Pimsner's six-term exact sequence for the KK-groups we calculate the K-groups of A. We also give a description of the corresponding Toeplitz algebra X in terms of generators and relations.

Dedicated to Professor K. Doerk on his 60th Birthday.

In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

We consider the singular limit of small solutions of some singularly perturbed problems where the nonlinearity changes sign.

Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of SV and ŜV, their restrictions to SW and ŜU, their dual coactions, and their full and reduced crossed products. If N (A) denotes the imprimitivity bimodule associated to a coaction δ of SV on a C*-algebra A by Ng's imprimitivity theorem, we prove that for a suitably nondegenerate injective right-Hilbert bimodule coaction of SV on AXB, the balanced tensor products and are isomorphic right-Hilbert A×ŜV×rSU − B × ŜW bimodules. This can be interpreted as a natural equivalence between certain crossed-product functors.

We prove that if an operator A is a finite sum of finite products of Toeplitz operators on the Bergman space of the unit ball Bn, then A is compact if and only if its Berezin transform vanishes at the boundary. For n = 1 the result was obtained by Axler and Zheng in 1997.

Coset enumeration, based on the methods described by Todd and Coxeter, is one of the basic tools for investigating finitely presented groups. The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems. Here we consider one such family of presentations proposed by B.H. Neumann. We show that the problems are much easier than they first appear, albeit at the expense of considerable preliminary ‘experimentation’. This demonstrates how far the range of applicability of coset enumeration has improved.

As a limiting case of the Sobolev imbedding theorem, the Moser-Trudinger inequality was obtained for functions in with resulting exponential class integrability. Here we prove this inequality again and at the same time get sharper information for the bound. We also generalise the Linearised Moser inequality to higher dimensions, which was first introduced by Beckner for functions on the unit disc. Both of our results are obtained by using the method of Carleson and Chang. The last section introduces an analogue of each inequality for the Laplacian instead of the gradient under some restricted conditions.

A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that .

It is known that l2 and l∞ have the ADP, whereas lp fails to have the ADP if 1 ≤ p < 2. We show that lp also fails to have the ADP for 3 ≤ p ≤ ∞. Our method seems to be able to decide also the case 2 < p < 3, but the computational difficulties increase as p comes closer to 2.

Let n ≥ 2 be an integer. Let P be the set of all integers in [1,n + 1] and let σ be a cyclic permutation on P. Assume that f is the linearisation of σ on P. Then we show that f has rational Artin-Mazur zeta function which is closely related to the characteristic polynomial of some n × n matrix with entries either zero or one. Some examples of non-conjugate maps with the same Artin-Mazur zeta function are also given.

Any linear operator A on 2 is shown to have two real quadratic form on S1 associated with it. They represent the expansion and rotation of the map and the eigenvalues of A can be described in a geometrically intrinsic way in therms of the eigenvalues of these two quadratic forms via the formula .

A number of theorems concerning these quadṙatic forms are presented.

For a hereditary torsion theory τ, we show by examples that the concepts of τ-Artinian τ-simple, τ-Artinian τ-primitive, τ-Artinian τ-prime, τ-Artinian τ-semiprimitive, and τ-Artinian τ-semiprime rings are different from each other and thus answer a question raised by Bland in his book [Topics in Torsion Theory, (Mathematical Research, 103, Wiley-VCH, 1998]. The example of a τ-Artinian τ-primitive ring which is not τ-simple here appears to be a counter-example to a result of Bland in the same publication.

To Bernhard Neumann on his 90th Birthday

We prove an inequality involving the number of relations and of relations and of generators of any group, and derive some consequences. The proof applies the theory of pro-p groups.

We characterise the complex measures μ on the open unit disk D such that for all f in the Hardy space H2. The characterisation involves Carleson measures, the duality between H1 and BMOA, and Hankel operators.

Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:

(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.

(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.

(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.

(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.