We adjoin a point at infinity to a field with a ring topology and extend the topology by taking complements of bounded sets as neighbourhoods of infinity. We note some elementary relations between the topology on the field and the topology on the field with infinity adjoined.

It is shown that a quadratic system with a degenerate critical point has at most one limit cycle.

Maximal isotropic varieties of the singular symplectic structure on R2n are characterised in terms of generating families. The normal forms of the simplest singularities (of codimension 1) are obtained with the help of the theory of boundary singularities.

Let BO, BSO and BSpin be the classifying spaces for the infinite orthogonal, infinite special orthogonal and infinite spinor groups respectively. It is well known that their integral cohomology rings have torsion only of order 2. In this paper we present an elementary proof that for the 7-connective cover of BO, BO〈8〉, the integral cohomology ring H* (BO〈8〉; Z) too has torsion only of order 2. The method follows that of Borel and Hirzebruch and a result of Wu concerning the Steenrod reduced mod p operation for an odd prime p on the Pontrjagin classes.

The rôle of the conjugacy relation is certainly important in the structure theory of groups. Here we study this relation in a considerably more general setting, namely in the theory of loops. We first recall some basic facts about quasigroups, their multiplication groups, their inner mapping groups and the conjugacy relation. After this we estimate the size and the number of the conjugacy classes and we study the structure of loops having only two conjugacy classes. Finally, the values of the centraliser function are discussed.

We discuss permutation properties of a specific kind of binomials over finite fields. As a result, we complete Cavior's classification of binomial octic permutation polynomials over Fq with q odd.

A C*-algebra A is finite-dimensional if and only if the cone is closed in M2 ⊗ A.

A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.

In a previous paper, the author sought to classify those solutions of second order nonlinear ordinary differential equations which can be expressed as sums of solutions of related equations. In that paper one sub-class of solutions was overlooked. This paper is to remedy that defect.

We characterise continuous closed (perfect, open) functions in terms of nets. This enables us to improve some significant results, with much simpler proofs.

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

We discuss purity and pure essentialness of abelian groups in a topos of sheaves on a locale and show that purity is not a local property. We prove that is divisible if and only if it is pure in every extension, and give an example of a category in winch absolutely pure does not imply divisible. We discuss uniform abelian groups and show that each AU uniform in Ab does not imply that A is uniform in

Banaschewski showed that the pure subgroups of are exactly of the type for the different . We show that is essential in if and only if U is dense in , Finally, we characterise as complete boolean algebras the locales for which the only pure and essential subgroup of is .

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebra A of the infinite dihedral group ≤ a, b | b−1ab = a−1, b2 = 1 ≥ over the field R of real numbers: namely in the A defined by b−l ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ.

The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groups G which have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the present A. The point of the main result is that new ideas will be needed for understanding A-modules.

A number of subsidiary results are concerned with convenient generating sets for left ideals in A.

The purpose of this paper is to get some necessary conditions for a Poisson stable flow to be recurrent and to analyse the bilateral versions of positive and negative Lipschitz stability. Moreover, a characterisation of recurrent orbits is obtained in a certain flow.

Sufficient conditions are obtained for the stability of linear neutral delay-differential systems by using a delay-differential inequality.

In this paper I prove that if a semigroup S is stable then ∧L(S) and ∧R(S) (the Rhodes expansions), and ∧+(SA) (the iteration of those expansions) are also stable. I also prove that if S is stable and has a J-depth function then these expansions also have a J-depth functon. More generally, if X →→ S is a J*-surmorphism and if S is stable and has a J-depth function then X has a J-depth function. All these results are needed for the structure theory of semigroups which are stable and have a J-depth function.

The techniques used were originally developed by the author to prove that ∧+(SA) is finite if S is finite (later Rhodes found a much more direct proof of that result).

Let Mn and N2n−2 be smooth, connected manifolds of dimension n and 2n − 2 respectively with n ≡ 2 mod 4 and 6 ≤ n ≤ 26. Let f: Mn → N2n−2 be a continuous map. Under certain suitable conditions on the stable normal bundle of f, we give a direct and simpler proof that f is homotopic to an immersion. For the case 6 ≤ n ≤ 26 and n ≠ 18, the result was proved by Li-Banghe and Peterson by using non-stable obstruction theory and their earlier result.

Let X be an affine real algebraic variety. In this paper, assuming that dim X ≤ 7 and that X satisfies some other reasonable conditions, we give a characterisation of those continuous complex vector bundles on X which are topologically isomorphic to algebraic complex vector bundles on X.

We characterise relative weak compactness in σBM(∑, X), the space of sigma-additive, X-valued measures of bounded variation, where X is a Banach space.

The minimal (faithful) degree μ(G) of a finite group G is the least positive integer n such that G ≲ Sn. Clearly if H ≤ G then μ(H) ≤ μ(G). However if N ◃ G then it is possible for μ(G/N) to be greater than μ(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.