We study the class of groups having no nontrivial linear representations over certain fields. After showing the class to be closed under perfect extensions with locally soluble kernel, we expand considerably the number of acyclic groups known to be in the class, by application to both binate groups and the acyclic automorphism groups of de la Harpe and McDuff.

Let f(z) be a meromorphic function, let P(z) and Q(z) be two polynomials. We shall investigate the asymptotic behaviour of the ratio T(r, f(P))/T(r, f(Q)), and discuss the growth of the meromorphic solutions of some functional equations.

A finitely presented group G is quasi-simply-filtrated (abbreviated qsf) if, given a finite complex with fundamental group G, the universal cover of the complex can be “approximated” by simply connected finite complexes. This notion is a generalisation of a concept of Casson's used in the study of three-manifolds.

In this paper we show that any extension of a finitely presented infinite group by a finitely presented infinite group is qsf.

In this paper we study the symmetric powers of quantum GLn. These are comodules and we give a complete description of the subcomodule structure.

We generalise the notion of proper Hughes subgroup in a group, asking that, for some prime p, the set-complement of the union of two proper subgroups consist solely of elements of order p. A structure theorem for non-p–groups of this type is proved.

Let R be a left linearly compact ring with left ideals I ⊆ J such that RJ is finitely generated and R(J/I) is Artinian. We prove that if J is nil over I then J is nilpotent over I.

We prove a conjecture of Willis saying that for each f in the Volterra algebra the sequence (‖fn‖)n is a weight sequence which is regulated at 1.

We give sufficient conditions for a homogeneous superposition operator to be a continuous mapping between Orlicz-Sobolev spaces. This extends a result of Marcus and Mizel concerning mappings between Sobolev spaces.

In this paper, we establish a new coincidence theorem for a Browder type set-valued mapping and an upper semi-continuous set-valued mapping with compact acyclic values in an H-space which generalises some recent results in the literature. As applications we obtain two H a type coincidence theorems and existence theorems of maximal elements for preference correspondences.

We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise denned polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is denned and shown to correspond to function composition.

A semigroup S is called E-solid if and only if for all idempotents e, f, g ∈ S such that e L f R g there exists an idempotent h ∈ S such that e R h L g. Each completely regular semigroup is E-solid. We characterise the idempotents of an arbitrary E-solid regular semigroup as a set with a binary operation on it satisfying a given finite set of identities.

It is shown that, in Pettis's criterion for Bochner measurability of a vector-valued function f: S → X, scalar measurability of f can be weakened to requiring that u of be measurable for u in some subset of the dual X* separating the points of X. Even then, the separability hypotheses in Pettis's Theorem can be weakened as well.

An identity for two generalised Tribonacci sequences is obtained. Earlier identities by Pethe and Horadam follow as special cases.

The second–order behaviour of a nonsmooth convex function f is reflected by the so–called second–order subdifferential mapping ∂2f. This mathematical object has been intensively studied in recent years. Here we study ∂2f in connection with the geometric concept of “second-order normal vector” to the epigraph of f.

The concept of superspace is fundamental for some recent physical theories, notably supersymmetry, and a mathematical feedback for it is provided by superanalysis and supergeometry. We survey the state of affairs in superanalysis, shifting our attention from supermanifold theory to “plain” superspaces. The two principal existing approaches to superspaces are sketched and links between them discussed. We examine a problem by Manin of representing even geometry (analysis) as a collective effect in infinite-dimensional purely odd geometry (analysis), by applying the technique of nonstandard (infinitesimal) analysis.

Sufficient conditions involving uniform multisplittings are established for the convergence of relaxed and AOR versions of asynchronous or chaotic parallel iterative methods for solving a large scale nonsingular system of linear equations Ax = b.