Let Q be the space of all rational numbers and (X, τ) be a topological space where X is countably infinite. Here we prove that (1) τ is the join of two topologies on X both homeomorphic to Q if and only if τ is non-compact and metrizable, and (2) τ is the join of topologies on X each homeomorphic to Q if and only if τ is Tychonoff and noncompact.

Given the equations for the n + 1 hyperplanes that bound an n-simplex in Rn, simple formulae are derived for the contents of the n − r simplices (0 ≦ r < n) embedded in it. For example, when n = 3, the formulae include the volume of the tetrahedron, the areas of its faces and the lengths of its edges.

The k-profile of an Hadamard matrix of order n is a function defined on the integers 0, 1,…,n. If k is even, k-profile of an Hadamard matrix of order n (k even) has non-zero terms only in every eighth position. If k is divisible by 4, the non-zero positions are those congruent to n (modulo 8).

We say that a positive integer d has property (A) if for all positive integers m there is an integer x, depending on m, such that, setting n = m + d, x lies between m and n and x is co-prime to mn. We show that infinitely many even d and infinitely many odd d have property (A) and that infinitely many even d do not have property (A). We conjecture and provide supporting evidence that all odd d have property (A).

Following A. R. Woods [3] we then describe conditions (Au) (for each u) asserting, for a given d, the existence of a chain of at most u + 2 integers, each co-prime to its neighbours, which start with m and increase, finishing at n = m + d. Property (A) is equivalent to condition (A1), and it is easily shown that property (Ai) implies property (Ai+1). Woods showed that for some u all d have property (Au), and we conjecture and provide supporting evidence that the least such u is 2.

Let X be a compact affine real algebraic variety of dimension 4. We compute the Witt group of symplectic bilinear forms over the ring of regular functions from X to C. The Witt group is expressed in terms of some subgroups of the cohomology groups .

We prove a version of the Feller-Miyadera-Phillips theorem characterizing the infinitesimal generators of positive C0-semigroups on ordered Banach spaces with normal cones. This is done in terms of N(A) as well as the canonical half-norms of Arendt Chernoff and Kato defined by N(a) = inf{‖b‖ |b ≥ a}, where N(A) = sup{N(Aa) |N(a) ≶ 1} for operator A. A corresponding result on –semigroups is also given.

In this note the L2-angle between two concentric rings and between the ring and the exterior of the disc in the complex plane are calculated. In the second part we prove that the L2-angles between domains A and B and between A × C and B × C are equal. We give also some examples of discontinuity of the L2-angle between domains.

A number of sufficient conditions for stability or strong stability, as used in the context of Hamiltonian systems, are found for the differential equation where the continuous function f(t) is periodic ω in t, D = d/dt and p(s), q(s) are real monic polynomials having special properties which allow the differential equation to be transformed into a canonical system of k second order equations.

We investigate a family consisting of functions whose convolution with is starlike of order α 0 ≤ α < 1. We determine extreme points, inclusion relations, and show how this family acts under various linear operators.

A construction for Fitting formations given by the author and C. L. Kanes is generalised. The original examples were based on the use of Fitting families of modules over algebraically closed fields. An example of Haberl and Heineken in 1984 suggested that the methods should work with modules over arbitrary fields. We show that this is indeed the case, provided we restrictthe class of groups considered.

A.McIntosh and A. Pryde introduced and gave some applications of notion of “spectral set”, γ(T), associated with each finite, commuting family of continuous linear operators T in a Banach space. Unlike most concepts of joint spectrum, the set γ(T) is part of real Euclidean space. It is shown that γ(T) is always non-empty whenver there are at least two operators in T.

It is shown that every boolean right near-ring R is weakly commutative, that is, that xyz = xzy for each x, y, z ∈ R. In addition, an elementary proof is given of a theorem due to S. Ligh which states that a d.g. boolean near-ring is a boolean ring. Finally, a characterization theorem is given for a boolean near-ring to be isomorphic to a particular collection of functions which form a boolean near-ring with respect to the customary operations of addition and composition of mappings.

Using a new proof technique of the first author (by adding a new vertex to a graph and creating a total colouring of the old graph from an edge colouring of the new graph), we prove that the TCC (Total Colouring Conjecture) is true for any graph G of order n having maximum degree at least n - 4. These results together with some earlier results of M. Rosenfeld and N. Vijayaditya (who proved that the TCC is true for graphs having maximum degree at most 3), and A. V. Kostochka (who proved that the TCC is true for graphs having maximum degree 4) confirm that the TCC is true for graphs whose maximum degree is either very small or very big.

All known Banach spaces have an infinite-dimensional separable quotient and so do all nonnormable Fréchet spaces, although the general question for Banach spaces is still open. A properly separable topological vector space is defined, in such a way that separable and properly separable are equivalent for an infinite-dimensional complete metrisable space. The main result of this paper is that the strict inductive limit of a sequence of non-normable Fréchet spaces has a properly separable quotient.

A presentation is given for the cohomology ring of a finitely presented combinatorially aspherical group with trivial coefficients in an integral domain. Cohomological periodicity is characterized in terms of the cup product.

We study real hypersurfaces of a complex projection space and show that there are no such hypersurfaces with harmonic curvature on which the structure vector is principal.

A theory of generalized gradients is presented for a class of Lipschitz vector-valued mappings from a Banach space to a locally convex order complete vector lattice. Necessary optimality conditions are obtained for nonconvex programming problems on Banach spaces with vector- valued operator constraints and/or an arbitrary set constraint. Sufficient optimality conditions are also obtained under mild convexity assumptions.

It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram Γ can be classified by the following data: a partially ordered set U with maximum element 1 and a map λ: U → 2Γ with λ(1) = Γ and with the property that for all J1, J2, J3 ∈ U with J1 > J2 > J3, any connected component of λ(J2) is contained in either λ(J1) or λ(J3). In this paper we show that λ comes from a regular monoid if and only if the following conditions are satisfied: (1) U is a ∧-semilattice; (2) If J1, J2 ∈ U, then λ(J1)∧ λ(J2) λ(J1 ∧ J2); (3) If θ ∈ Γ, J ∈ U, then max{J1 ∈ U|J1 > J, θ ∈ λ (J1)} exists; (4) If J1, J2 ∈ U with J1 > J2 and if X is a two element discrete subset of λ(J1) ∪ λ(J2), then X λ(J) for some J ∈ UJ with J1 > J > J2.

We extend in different ways the class of null sequences of real numbers that are of bounded variation and study the Walsh-Fourier series of integrable functions on the interval [(0, 1) with such coefficients. We prove almost everywhere convergence as well as convergence in the pseu dometric of Lr(0, 1) for 0 < r < 1.

The first and last papers of Harald Bohr deal with ordinary Dirichlet series and their order (or Lindelöf) function μ(σ) (= inf{α;f(σ + it) + 0(|t|α)}). The Lindelöf hypothesis is μ(σ) = inf(0, ½ − t) when an = (−1)n. Are there ordinary Dirichlet series with −l < μ′ (σ) < 0 for some σ? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harald Bohr. This paper gives (1) a review on Bohr's results on ordinary Dinchlet series; (2) a review on results of the author and of Queffelec on “almost sure” and “quasi sure” properties of series with the solution of a previous problem of Bohr; (3) the following answer to the last problem: μ′(σ) can approach − ½, and necessarily μ(σ + μ(σ) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.