The explicit solution of a general three-term bilinear recurrence relation of fourth order is constructed here in terms of the Weierstrass sigma function. The construction of the elliptic curve associated to the Somos 4 sequence is presented as an example. An interpretation via the theory of integrable systems is provided, leading to a conjecture relating certain higher-order recurrences with hyperelliptic curves of higher genus.

In this paper, the theory of $\epsilon$-constants associated to tame finite group actions on arithmetic surfaces is used to define a Brauer group invariant $\mu(\X,G,V)$ associated to certain symplectic motives of weight one. The relationship between this invariant and $w_2(\pi)$ (the Galois-theoretic invariant associated to tame covers of surfaces defined by Cassou-Noguès, Erez and Taylor) is also discussed.

Define a sequence $(s_{n})$ of two-variable words in variables $x$, $y$ as follows: $s_{0}(x,y)=x$, $s_{n+1}(x,y)=[s_{n}(x,y)^{-y},s_{n}(x,y)]$ for $n\geq0$. It is shown that a finite group $G$ is soluble if and only if $s_{n}$ is a law of $G$ for all but finitely many values of $n$.

A famous result of Kostant's states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. An analogue of this result is proved for the class of special filtered algebras. This is then applied to show that the restricted Yangian and the universal enveloping algebra of the restricted current algebra, associated with the general linear Lie algebra, are both free over their centers.

Three short results are given concerning the elliptic Harnack inequality, in the context of random walks on graphs. The first is that the elliptic Harnack inequality implies polynomial growth of the number of points in balls, and the second that the elliptic Harnack inequality is equivalent to an annulus-type Harnack inequality for Green's functions. The third result uses the lamplighter group to give a counter-example concerning the relation of coupling with the elliptic Harnack inequality

Sufficient conditions are given for the global existence of solutions to the Rayleigh equation.

Consider a scalar differential equation $\dot{x}\,{=}\,f(t,x), (t,x)\,{\in}\,I\,{\times}\,\R$, where $I$ is an open interval containing $[0,\,T]$. Assume that $f(t,x)$ is continuous with a continuous derivative $f^{\prime}_{x}(t,x)$, and weakly concave (or weakly convex) in $x$ for all $t \in I$, though strictly concave (or strictly convex) for some $t\,{\in}\,[0,\,T$]. It is well known that in this case there can be either no, one or two closed solutions; that is, solutions $\varphi(t)$ for which $\varphi(0)=\varphi$$(T)$ If there are two closed solutions, then the greater has a negative characteristic exponent and the smaller has a positive one. It is easily seen that this is equivalent to a statement on localization of closed solutions. It is shown how this statement can be generalized to systems of differential equations $\dot{\underline x} = \underline{f}(t,\underline{x}), (t,\underline{x})\in I \times \R^{n}$. The requirements are that the coordinate functions ${f}_{j}(t,\underline{x})$ be continuous with continuous derivatives with respect to $x_{1},x_{2},\ldots,x_{n}$, that the $f_j$ are weakly concave (or weakly convex) in $\underline{\it x}$, and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled.

In this paper, the minimal positive entropy of pure braids is shown to be greater than $\log(2+\sqrt5)$, independent of the braid index. For pure 3-braids, the minimal entropy is shown to be equal to $\log(3+2\sqrt2)$.

The special values of multiple polylogarithms, which include multiple zeta values, appear in several fields of mathematics and physics. Many kinds of their linear relations have been investigated, as well as their algebraic relations. In this paper, from the viewpoint of a connection matrix of Fuchsian equations, two kinds of duality of these values are derived.

The optimal orders are determined for the best constants in the non-commutative Burkholder–Gundy, Doob and Stein inequalities obtained recently in non-commutative martingale theory.

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space $\varphi $ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on $\varphi$). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

A class of operators is introduced on $L_1$ that is stable under taking sums of pointwise unconditionally convergent series, contains all compact operators and does not contain isomorphic embeddings. It follows that any operator from $L_1$ into a space with an unconditional basis belongs to this class.

This paper concerns rigidity of the mapping class groups. It is shown that any homomorphism $\varphi\,{:}\,\mcg_g\,{\to}\,\mcg_h$ between mapping class groups of closed orientable surfaces with distinct genera $g\,{>}\,h$ is trivial if $g\,{\geq}\, 3$, and has finite cyclic image for all $g\,{\geq}\, 1$.

Some implications are drawn for more general homomorphs of these groups.

A proof is given that for each non-negative integer $g$, there is an infinite family of knotted surfaces of genus $g$, none of which is ambient isotopic to itself with the orientation reversed.

The Hofer–Zehnder theorem states that almost every hypersurface in a thickening of a hypersurface $S$ in a symplectic manifold $(M,\omega)$ carries a closed characteristic, provided that $S$ bounds a compact submanifold and $(M,\omega)$ has finite capacity. It is shown that it is enough to assume that the thickening of $S$ has finite capacity.

Noel Baker was born on 10 August 1932 and died, of a heart attack, on 20 May 2001. He was the only child of a farming family living near the township of Virginia north of Adelaide in South Australia, and a fourth-generation Australian. From 1938 to 1944 he attended the local school, winning a scholarship to King's College, Adelaide, and from there another scholarship to Prince Alfred College, Adelaide. Here he was inspired by an enthusiastic mathematics teacher, Mr Williams, who expected pupils to practise mathematics assiduously, even on Saturday mornings. Prince Alfred College still aims to foster a love of mathematics today, and it has recently launched a centre for excellence in mathematics, named after Noel.