A group is covered by a collection of subgroups if it is the union of the collection. The intersection of an irredundant cover of n subgroups is known to have index bounded by a function of n, though in general the precise bound is not known. Here we confirm a claim of Tompkinson that the correct bound is 16 when n is 5. The proof depends on determining all the ‘minimal’ groups with an irredundant cover of five maximal subgroups.

An Abelian group G is called an SI-group if for every ring R with additive group R+ = G, every subring S of R is an ideal in R. A complete description is given of the torsion SI-groups, and the completely decomposable torsion free SI-groups. Results are obtained in other cases as well.

For any ring R graded by a finite group, we give a bound on the classical Krull dimension of R in terms of the dimension of the initial component Re. It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too.

We characterise tree sign pattern matrices that require at least k zero eigenvalues, and exactly k zero eigenvalues.

We examine a 3-manifold Γ whose fundamental group is known to be non-subgroup-separable (non-LERF). We show that this manifold Γ is a graph manifold and that the subgroup known to be non-separable is not geometric. On the other hand, there are incompressible surfaces immersed in the manifold which do not lift to embeddings in any finite-degree covering space. We then prove that these bad incompressible surfaces must have non-empty boundary.

Quasi reflexive Banach spaces are characterised among the weakly countably determined Asplund spaces, in terms of the cardinality of the sets of linearly independent bounded linear functionals each of which does not attain its supremum on the unit sphere.

We find the convex planar set of maximal area which is partitioned into subsets of equal area by a given propellor. The argument depends on the characterisation of an n-sided polygon which is circumscribed about a regula n-gon Pn, and inscribed in a regula n-gon Qn, where Pn and Qn are homothetic about their common centre.

Let p be a prime number and let k be a field of characteristic not equal to p. Assuming k contains the appropriate roots of unity, we characterise the non-cyclic Galois extensions of k of degree p3. Concrete examples of such extensions are given for each possible case which can occur, up to isomorphism.

Given a fixed curve C0 in Rn of length L0 and a variable curve C of fixed length L ≤ L0, the thread problem seeks a least-area surface bounded by C0 + C. We show that an extreme case is a circular arc and its chord. We provide some counterexamples and generalisations to higher dimensions.

We obtain automatic continuity results for finite-rank perturbations of causal sequence space operators, and provide examples to illustrate cases where automatic continuity does not hold.

For a field k of characteristic not two, it is known that k is algebraically closed in the function field of any (non-degenerate) quadratic form in three or more variables. In this note we consider fields of characteristic two and decide when k is algebraically closed in a function field of a quadratic k-form. For quadratic forms in three variables this has recently been done by Ohm.

In this short note we give a new characterisation of weak compactness.

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

In this paper we obtain characterisations of first and second countability and separability for GPO-spaces, a class of topological spaces that include LOTS and GO-spaces. Some additional results concerning the transmission of these properties to weaker/finer topologies are derived in this framework.

In this paper we introduce some new mappings associated with the weighted geometric mean. These are used to derive structural results linking weighted geometric and arithmetic means.

We study the mod 2 homology of the moduli space of instantons associated with the prinicpal Sp (n) bundle over the four-sphere and the classifying space of the gauge group using the Serre spectral sequence and the homology operations.

We prove a Schwarz Lemma for conformal mappings between two complete Riemannian manifolds when the domain manifold has Ricci curvature bounded below in terms of its distance function. This gives a partial result to a conjecture of Chua.

It is shown that, if G is a totally disconnected, compactly generated and nilpotent locally compact group, then it has a base of neighbourhoods of the identity consisting of compact, open, normal subgroups. An example is given showing that the hypothesis that G be compactly generated is necessary.

We show the instability of solutions of the Dirichlet problem for Hamilton-Jacobi equations under quite general conditions.

We describe a virtually Abelian group G generated by a finite set X such that there is no regular language of geodesic X-words that surjects to G by evaluation.