In this paper, we study the twisted Alexander polynomial, due to Wada [11], for the Jones representations [6] of Artin's braid group.

In this paper, we shall establish a Kummer theory of division points over singular Drinfeld modules which is in complete analogy with the classical one in number fields.

For a commutative domain R with 1, R-module M is called a semi-Baer module if for all divisible R-modules D. We show that finitely generated modules of projective dimension at most 1 are semi-Baer modules and if R is Prüfer or Matlis, then all modules of projective dimension at most 1 are semi-Baer modules.

We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which ever infinite subset contains n mutually disjoint non-empty subsets X1,…,Xn such that X1…Xn∩Xσ(1) …xσ(n) ≠ θ for some non-identity permutation σ on the set {1,…,n}.

We consider a scale of weighted spaces a quaternion-valued functions of three real variables. This scale generalises the idea of Qp-spaces in complex function theory. The goal of this paper is to prove that the inclusions of spaces from the scale are strict inclusion. As a tool we prove some properties of special monogenic polynomials which have an importance in their own right independently of their use in the scale of Qp-spaces.

We study some properties which imply weak normal structure and thus the fixed point property. We investigate whether the latter two properties are inherited by spaces obtained by direct sum with a finite dimensional space. We exhibit a space X which satisfies Opial's condition, X ⊕ ℝ does not have weak normal structure but X ⊕ ℝ has the fixed point property.

Dedicated to Jana

The aim of this note is to check the admissibility property of the distribution attached to convolution of Hilbert modular forms.

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.

This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalise some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras et cetera). In the present paper, we introduce a new class of effect algebras, called homogeneous effect algebras. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying the Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying the Riesz decomposition property. This generalizes a recent result by Riec˘anová, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalisation of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebra E forms an orthoalgebra Es. Every block of Es is the centre of a block of E. The set of all sharp elements in the compatibility centre of E coincides with the centre of E. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space ℍ with dim (ℍ) > 1, the standard effect algebra ℰ(ℍ) of all effects in ℰ is not homogeneous.

Let p be an odd prime with p > 3. In this paper we give all positive integer solutions (x, y, m, n) of the equation x2 + p2m = yn, gcd (x, y) = 1, n > 2 satisfying 2 | n of 2 ∤ n and p ≢ (−1)(p−1)/2(mod 4n.

If Ω/F is a Galois extension with Galois G and μ(Ω) denotes the group of roots of unity in Ω, we use the group Z1 (G,μ(Ω)) of crossed homomorphisms to study radical extensions inside Ω. Furthermore, we characterise cubic radical extension, and we provide an example to show that this result can not extended for higher degree extensions.

For some of the classical groups over finite fields it is possible to express the proportion of eigenvalue-free matrices in terms of generating functions. We prove a theorem on the monotonicity of the coefficients of powers of power series and apply this to the generating functions of the general linear, symplectic and orthogonal groups. This proves a conjecture on the monotonicity of the proportions of eigenvalue-free elements in these groups.

We show that the left joint spectrum of an arbitrary n-tuple of hyponormal Hilbert space operators can be obtained from the spectral set γ introduced by McIntosh and Pryde. A dual statement for cohyponormal operators is also true. The result is a generalisation of a theorem proved by Pryde and the author for normal operators.

We study the space lp, 1 ≤ p ≤ ∞, and its natural n-norm, which can viewed as a generalisation of its usual norm. Using a derived norm equivalent to its usual norm, we show that lp is complete with respect to its natural n-norm. In addition, we also prove a fixed point theorem for lp as an n-normed space.

Dedicated to Filippo Chiarenza

The aim of this note is to prove the unique continuation property for non-negative solutions of the quasilinear elliptic equation

We allow the coefficients to belong to a generalized Kato class.

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.

Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.