The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.

A class of rings in which each member is the extension of a nilpotent torsion ring by a semisimple semiartinian ring is presented.

In this paper, we study periodic braid representations of periodic links. It is shown that no 2-bridge non-fibred knot has a periodic braid representation and conditions under which periodic links are fibred are given. We give a construction of all periodic links over a fixed factor link.

In this paper a full classification of piecewise monotonic maps from the point of view of the topological sequence entropy is established.

It is shown that if a bounded linear operator A commutes with a prespectral operator T of class Γ, then A commutes with the resolution of the identity of class Γ for T, say P (·), if and only if A*(Γ) ⊆ [PC*Γ. Here A* is the dual operator of A and [Pc]*Γ is the linear span of the set {U*ξ U ε P (·) c, ξ ε Γ} where P (·) c denotes the commutant of the range of P (·).

A ring R is called right FSG if every finitely generated right R-subgenerator is a generator. In this note we consider the question of when a ring extension of a given right FSG ring is right FSG and the converse. As a consequence we obtain some results about right FSG group rings.

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

The Magnus matrix is an algebraic invariant assigned to each homology cobordism of a surface. We show that this matrix satisfies an equality which can be regarded as a non-commutative symplectic relation.

Integral mean and coefficient bounds for some Bazilevic functions are determined.

Let G be an analytic group and let Aut G denote the group of all topological group automorphisms of G. We investigate when Aut G is almost algebraic. We provide various conditions, some of which are known, under which Aut G is almost algebraic. We also provide examples showing that there does not seem to be a clear and concise way to characterise G so that Aut G is almost algebraic in terms of the maximal central torus in G.

Let f be a function meromorphic in D = {|z| < 1} and let X be the chordal distance on the Riemann sphere. Then f satisfies the Lipschitz condition

in D if and only if |f′(z)|/(1 + |f(z)|2) = O((1 – |z|)α−1) and |z| → 1.

In this note we obtain a convolution identity for the coefficients Bn(α, θ, q) defined by

using Ramanujan's 1Ψ1 summation. The identity contains as special cases convolution identities of Kung-Wei Yang and a few more interesting analogue.

For a noncompact riemannian manifold R, let MP (R) be the P-algebra, and R*P the P-compactification, with the assumption that ∫RPdV = ∞. If s is the P-singular point of the P-harmonic boundary ΔP, and Δ is the harmonic boundary of Royden's compactification R*, we construct a continuous mapping π: R* → R*P such that π(Δ) = Δp or π(Δ) = ΔP – s.

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.

A theorem of T. Ando, R. Nagel and H. Uhlig on the positivity of generators of some positive semigroups in Banach lattices can not be generalized to general ordered Banach spaces.

In the basic Economic Lot Scheduling problem, a production schedule is required to manufacture sequentially a number of products on a single machine, with the schedule chosen to minimize set-up and inventory costs. The products suffer continuous demand, and no shortfall is allowed. A recent approach involves repetitions of a production cycle (such as ABCBC for three products A, B and C, with manufacturing times chosen to prevent shortage occurring); an exhaustive search is performed over a large set of possible cycles to discover the optimal schedule.

This paper discusses the question “How many such sycles need to be examined?”, Since the answer is very relevant to practical application of the method. The case of three products is considered. Complete information is obtained for cycles up to length 12 (that is, 12 production switch overs), and partial results for longer ones. An estimate, apparently reasonable, is obtained for cycles of any length. The major trend to emerge is that surprisingly few cycles are involved.

Certain Hilbert spaces of functions with known smoothness are considered. The weights and mesh of the quadrature formulae which have least estimate of error with respect to the norm for functions in these spaces are found and their properties are discussed.