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

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

We prove a characterisation, in terms of regularity and cancellation conditions, of the convolution kernels of Marcinkiewicz multiplier operators m (𝔏1,…,𝔏n, iT) on the Heisenberg group ℍn, where 𝔏1,…,𝔏n are the n partial sub-Laplacians. The necessity of these regularity and cancellation conditions was established by Fraser (2001); here, we prove their sufficiency.

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.

We study the following questions. Which finite Blaschke products are eigenvectors of the composition operators Tu: f ↦ f ∘ u, what are the possible eigenvalues, and which pairs (B, C) of finite Blaschke products commute (that is, satisfy B ∘ C = C ∘ B).

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.

Chen (1999) established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimension. Matsumoto (to appear) dealt with similar problems for sub-manifolds in complex space forms.

In this article we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in (K, μ)-contact space forms.

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.

In this paper we study a nonlinear hemivariational inequality involving the p-Laplacian. Our approach is variational and uses a recent nonsmooth Linking Theorem, due to Kourogenis and Papageorgiou (2000). The use of the Linking Theorem instead of the Mountain Pass Theorem allows us to assume an asymptotic behaviour of the generalised potential function which goes beyond the principal eigenvalue of the negative p-Laplacian with Dirichlet boundary conditions.

In this paper we study the validity of several convergence theorems for measures defined on an effect algebra and taking values in a Hausdorff commutative topological group. We establish the Brooks-Jewett theorem and the Nikodým convergence theorem, giving as a corollary a result, due to Aarnes, about the convergence of a sequence of normal linear functionals on a von Neumann algebra. We prove two new convergence theorems concerning completely additive and τ-smooth measures, and we obtain also a convergence theorem for regular measures.

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}.

In this paper, we give several inequalities concerning the genus and the degree of the Jones Polynomial of an adequate link and of a homogeneous link and their applications.

We study the semigroups l and r of left and right invertible operators modulo an operator ideal , respectively. We show that these semigroups allow us to obtain useful characterisations of the radical rad of  For example, rad; is the perturbation class for l and r.

We consider radical classes specified by an associating polynomial in two variables which have a similar form to the polynomial defining the class of quasiregular rings. In particular, the relationships of these classes to the classes of semiprime, nil and quasiregular rings are explored.

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.

Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist distinct elements x, y ∈ X and integers t0, t1…, tk such that , where xi, ∈ {x, y}, i = 0, 1,…,k, x0 ≠ x1. If the integers t0, t1,…,tk are the same for any subset X of G, we say that G is in the class k(n). The class k (n) is defined exactly as Ωk(n) with the additional conditions . Let t2, t3,…,tk be fixed integers. We denote by the class of all groups G such that for any infinite subsets X and Y there exist x ∈ X, y ∈ Y such that , where xi ∈ {x, y}, x0 ≠ x1, i = 2, 3, …, k. Here we prove that (1) If G ∈ k(2) is a finitely generated soluble group, then G is nilpotent.

(2) If G ∈ Ωk(∞) is a finitely generated soluble group, then G is nilpotentby-finite.

(3) If G ∈ k(n), n a positive integer, is a finitely generated residually finite group, then G is nilpotent-by-finite.

(4) If G is an infinite -group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient, then G is nilpotent-by-finite.

In this paper we study the centre of a simply connected analytic group. We present two structure theorems on the centres of such groups. We also study maximal tori and maximal compact subgroups of an analytic group, and as a result, we extend a theorem by Goto.

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.

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.

Dedicated to Jana

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