A proof is given of the following theorem, which characterizes full automorphism groups of ordered abelian groups: a group$H$is the automorphism group of some ordered abelian group if and only if$H$is right-orderable.

In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.

It is shown that the classical result of Pringsheim on the convergence of complex continued fractions extends to all dimensions, both geometrically and as a result on continuedfractions whose coefficients lie in a Clifford algebra.

It is proved that the number of ${\bb F}_q$-rational points of an irreducible projective smooth 3-dimensional geometrically unirational variety defined over the finite field ${\bb F}_q$ with $q$ elements is congruent to 1 modulo $q$. Some Fermat 3-folds, some classes of rationally connected 3-folds and some weighted projective $d$-folds also having this property are given.

This paper introduces baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter $t=0$ (the so-called rational Cherednik algebras at parameter $t=0$), and presents their most basic properties. Baby Verma modules are then used to answer several problems posed by Etingof and Ginzburg, and to give an elementary proof of a theorem of Finkelberg and Ginzburg.

Minimal complex surfaces of general type with $p_g = 0$ and $K^2 = 7$ or $8$ whose bicanonical map is not birational are studied. It is shown that if $S$ is such a surface, then the bicanonical map has degree 2 (see Bulletin of the London Mathematical Society 33 (2001) 1–10) and there is a fibration $f : S \rightarrow {\bb P}^1$ such that (i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; (ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$.

Furthermore, if $K^2_S = 8$, then $f$ is an isotrivial fibration with six double fibres, and if $K^2_S = 7$, then $f$ has five double fibres and it has precisely one fibre with reducible support, consisting of two components.

This paper presents the rigidity of $\lambda_2$-eigenmaps from $S^{4}$ to itself, and the nonexistence of some $\lambda_2$-eigenmaps, as well as new examples of eigenmaps between spheres.

A relatively long-standing problem concerning the best constants of approximation by entire functions of exponential type on the class $W^rH_{\omega}$ is solved. The central tool is a limit relation between the best constants of approximation by trigonometric polynomials and entire functions of exponential type on $W^rH_{\omega}$ and its periodic analogue.

In a beautiful result, Herrero (D. A. Herrero, ‘Normal limits of nilpotent operators’, Indiana Univ. Math. J. 23 (1973/74) 1097–1108) showed that a normal operator on $l^2$ lies in the closure of the set of nilpotent operators if and only if its spectrum is connected and contains zero. In the quest for an automatic continuity result for algebra homomorphisms between ${\rm C}^*$-algebras, Dales showed that, if a discontinuous algebra homomorphism $\theta :{\cal A} \rightarrow {\cal U}$ exists between ${\rm C}^*$-algebras ${\cal A}$ and ${\cal U}$, and if $\theta({\cal A})$ is dense in ${\cal U}$, then there is a ${\rm C}^*$-algebra ${\cal U}_2$ with a dense subalgebra ${\cal N} \subset {\cal U}_2$ such that every $x \in {\cal N}$ is quasinilpotent (see p. 685 of H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs 24, Oxford University Press, 2001). (A discontinuous homomorphism $\theta_2 :{\cal A}_2 \rightarrow {\cal U}_2$ can be defined with the same basic properties as $\theta$, but the revised target space ${\cal U}_2$ has a dense subalgebra consisting of quasinilpotent elements.) As remarked by Dales, no such ${\rm C}^*$-algebra was then known; but here we present one. Indeed, using the full power of Herrero's result, one may arrange that every $x \in {\cal N}$ is nilpotent. The ${\rm C}^*$-algebra is constructed in a ‘neat’ way; it is most naturally constructed as a non-separable, concrete ${\rm C}^*$-algebra of operators on a separable Hilbert space ${\cal K}$; but one can arrange that the algebra ${\cal U}$ itself be separable if desired.

Let $B$ be a unital commutative Banach algebra, and let $A$ be an arbitrary subalgebra of $B$. Suppose that an ideal $I\subset A$ consists of elements that are non-invertible in $B$. Then there exists an ideal $J$ in $A$ that: (i) contains $I$, (ii) also consists of elements non-invertible in $B$, and (iii) is of codimension one in $A$. This result is used in the study of a class of subspectra on $A$.

This paper considers the cone multiplier operator which is defined by $\[\widehat{S^\mu f}(\xi,\tau)=m_\mu(\xi,\tau)\widehat f(\xi,\tau)$, $\qquad (\xi,\tau)\in \mathbb R^2\times \mathbb R\]$ where $m_\mu(\xi,\tau)=\phi(\tau)(1-|\xi|^2/\tau^2)_+^\mu/\Gamma(\mu+1)$ and $\phi\in C_0^\infty(1,2)$. For $-3/2<\mu<-3/14$, sharp $L^p-L^q$ estimates and endpoint estimates for $S^{\mu}$ are obtained.

In this paper, the behaviour of the positive eigenfunction $\phi$ of $\Lp u=\la |u|^{p-2}u$ in $\Om$, $u_{|\p \Om} =0$, $p>1$, is studied near its critical points. Under some convexity and symmetry assumptions on $\Om$, $\phi$ is seen to have a unique critical point at $x=0$; also, the behaviour of both $\phi$ and $\nabla\phi$ is determined nearby. Positive solutions $u$ to some general problems $\Lp u=f(u)$ in $\Om$, $u_{|\p \Om} =0$, are also considered, with some convexity restrictions on $u$.

Star complements and associated quadratic functions are used to obtain a sharp upper bound for the order of a graph with an eigenspace of prescribed codimension. It is shown that for regular graphs the bound can be reduced by 1, and that this reduced bound is attained by a regular graph $G$ if and only if $G$ is an extremal strongly regular graph.

Let $X$ be the Shimura curve corresponding to the quaternion algebra over $\Q$ ramified only at 3 and 13. B. Jordan showed that $X_{\Q(\sqrt{-13})}$ is a counterexample to the Hasse principle. Using an equation of $X$ found by A. Kurihara, it is shown here, by elementary means, that $X$ has no $\Q(\sqrt{-13})$-rational divisor classes of odd degree. A corollary of this is the fact that this counterexample is explained by the Manin obstruction.

Jeffrey Dennis Weston was born on 15 September 1919 in London. He died suddenly on 10 March 2000, the 55th anniversary of his marriage to Mary.