It is shown that the problem of deciding whether a curve in a finite surface complex is isotopic to a point is NP-complete. This contrasts with the recursively enumerable (RE)-completeness of the corresponding homotopy problem, and exhibits surface complexes as a common framework for NP-complete and RE-complete algorithmic problems.

This paper is a survey of the current state of knowledge on groups of exponent 8. It contains a report on a first stage of an attempt to answer the Burnside questions for these groups.

This paper gives a detailed classification of the indecomposable representations of one non-trivial “commutative quiver”, gives brief details of certain generalisations, and uses methods likely to be more widely applicable.

The category of groups forms a full subcategory of the category of hypergroups. This larger category also contains other group theoretic objects, such as the conjugacy class hypergroup and character hypergroup of a finite group.

Definitions of hypergroups and hypergroup morphisms are given and related to double algebras (which are simultaneously algebras and cogebras, but not Hopf algebras in general). Quotient and orbit hypergroups are defined. Coefficients are allowed in a more or less arbitrary field.

These concepts provide a new language in which groups and their character tables can be fruitfully discussed.

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

The paper presents a necessary and sufficient condition for a given compact totally disconnected space C to be the projective limit of a given directed cone of epimorphisms onto finite discrete quotients of C. This problem is related to the question of when a compact totally disconnected algebra is profinite and some observations in this direction are recorded.

Let S be a closed convex cone in Cn, S* the polar cone, g a continuous map from Cn into itself, and e a fixed vector in S*. In this paper we prove that there is a connected set T in S of stationary points of (Dr (e), g) where Dr (e) is the set of all x in S with re(e, x) ≤ r. This extends the results of Lemke and Eaves to the complex nonlinear case and arbitrary closed convex cones in Cn. We show that if g is strictly monotone on S, then T is both unique as well as arcwise connected. This partly solves the open problems raised by Eaves in this more general setting. We also show that if x is a stationary point of (Dr (e), g) and re(e, x) < r then x is a stationary point of (S, g).

One method of finding non-parametric hypersurfaces of prescribed mean curvature which span a given curve in Rn is to find a function which minimizes a particular integral amongst all smooth functions satisfying certain boundary conditions. A new problem can be considered by changing the integral slightly and then minimizing over a larger class of functions. It is possible to show that a solution to this new problem exists under very general conditions and it is usually known as the generalized solution. In this paper we show that the two problems are equivalent in the sense that the least value for the original minimization problem and the generalized problem are the same even though no solution may exist. The case where the surfaces are constrained to lie above an obstacle is also considered.

We prove that for a large class of Segal algebras, the isometric multipliers consist of scalar multiples of translation operators.

Several new methods of approximation which assume arbitrary values on the boundary of a triangular domain are presented. All of the methods are affine invariant and have optimal algebraic precision. Nine parameter discrete interpolants which result from these methods are also given.

It is shown that the problem of deciding whether a polygonal curve c in a finite surface complex K is knotted in K is complete recursively enumerable, and hence unsolvable.

A function meromorphic in |z| < 1 is constructed such that on every curve in |z| < 1 which goes to |z| = 1 the set of limit points of the function is the entire complex plane. This example is used to prove the existence of non-normal meromorphic functions in |z| < 1 which have continuous set valued extensions. Cima and Rung had introduced a set valued extension for meromorphic functions and proved that all normal meromorphic functions have a continuous extension while all functions with a continuous extension have the Lindelöf property. For a long time it was thought that this might characterize normal meromorphic functions. This paper proves that it is not possible to determine the normality of a meromorphic function by the continuity of Cima and Rung's set valued extension. The paper closes with the open problem: do there exist non-normal analytic functions for which Cima and Rung's set valued extension is continuous?