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.

This paper deals with better-than-uniform approximation by superharmonic functions on non-compact sets. An application of the main result improves recent work of GowriSankaran and Nersessian concerning approximation with an error bound that decays arbitrarily quickly at infinity.

If $\Gamma$ is a discrete group, conditions are given that ensure that the full group C*-algebra $C^*(\Gamma)$ is primitive, thereby partially answering a question posed by P. de la Harpe.

Crispin St John Alvah Nash-Williams was born in Cardiff on 19 December 1932. His father was Keeper of Archaeology at the National Museum of Wales and also Senior Lecturer in Archaeology at University College, Cardiff. War broke out when Crispin was six; his father joined the army and his consequent absence changed the family's life.

Let $f$ be meromorphic of finite order in the plane, such that the second derivative $f^{\prime\prime}$ has finitely many zeros. Then $f$ has finitely many poles. This result was conjectured by the author in 1996, and an example shows that the theorem is sharp.

As well as the arc-sine laws for the time spent positive before time 1 and the time at which the maximum over [0, 1] occurs, Paul Lévy also established a third arc-sine law for Brownian motion; this relates to the time of the last visit to zero before time 1. For random walks, analogues of the first two of the these results are well known. Here it is shown that for any random walk satisfying Spitzer's condition there is an analogue of the third, but care has to be taken in its formulation. An associated conditional uniform law and some applications to stable processes are also given.

Let $A_n=K\langle x_1,\ldots,x_n \rangle$ be a free associative algebra over a field $K$. In this paper, examples are given of elements $u \in A_n$, $n \ge 3$, such that the factor algebra of $A_n$ over the ideal generated by $u$ is isomorphic to $A_{n-1}$, and yet $u$ is not a primitive element of $A_n$ (that is, it cannot be taken to $x_1$ by an automorphism of $A_n$). If the characteristic of the ground field $K$ is $0$, this yields a negative answer to a question of G. Bergman. On the other hand, by a result of Drensky and Yu, there is no such example for $n=2$. It should be noted that a similar question for polynomial algebras, known as the embedding conjecture of Abhyankar and Sathaye, is a major open problem in affine algebraic geometry.

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.

Let $D\subset H^n(-k^2)$ be a convex compact subset of the hyperbolic space $H^n(-k^2)$ with non-empty interior and smooth boundary. It is shown that the volume of D can be estimated by the total curvature of $\partial D$. More precisely, $(n-1)k^n{\rm Vol}(D)+ {\rm Vol}(S^{n-1})\leq \int_{\partial D}K$, where K denotes the Gauss–Kronecker curvature of $\partial D$ and Vol$(S^{n-1})$?> denotes the Euclidean volume of the sphere.

Let $\phi\,{:}\,G \to G$ be a group endomorphism where G is a finitely generated group of exponential growth, and denote by $R(\phi)$ the number of twisted ϕ-conjugacy classes. Fel'shtyn and Hill (K-theory 8 (1994) 367–393) conjectured that if ϕ is injective, then R(ϕ) is infinite. This paper shows that this conjecture does not hold in general. In fact, R(ϕ) can be finite for some automorphism ϕ. Furthermore, for a certain family of polycyclic groups, there is no injective endomorphism ϕ with $R({\phi}^n)\,{<}\,\infty$ for all n.

Let $V$ be a commutative valuation domain of arbitrary Krull-dimension, with quotient field $F$, let $K$ be a finite Galois extension of $F$ with group $G$, and let $S$ be the integral closure of $V$ in $K$. Suppose that one has a 2-cocycle on $G$ that takes values in the group of units of $S$. Then one can form the crossed product of $G$ over $S$, $S\ast G$, which is a $V$-order in the central simple $F$-algebra $K\ast G$. If $S\ast G$ is assumed to be a Dubrovin valuation ring of $K\ast G$, then the main result of this paper is that, given a suitable definition of tameness for central simple algebras, $K\ast G$ is tamely ramified and defectless over $F$ if and only if $K$ is tamely ramified and defectless over $F$. The residue structure of $S\ast G$ is also considered in the paper, as well as its behaviour upon passage to Henselization.

This paper gives a necessary criterion in terms of Hodge theory for representability by quaternion algebras of certain 2-torsion classes in the unramified Brauer group of a complex function field. This criterion is used to give examples of threefolds with unramified Brauer group elements which are the classes of biquaternion division algebras.

It is proved here that if $f : {\rm C\!\!\!I} \rightarrow \bar{{\rm C\!\!\!I}}$ is an elliptic function and $q$ is the maximal multiplicity of all poles of $f$, then the Hausdorff dimension of the Julia set of $f$ is greater than $2q/(q+1)$, and the Hausdorff dimension of the set of points that escape to infinity is less than or equal to $2q/(q+1)$. In particular, the area of this latter set is equal to 0.

A quantum deformation of the classical conjugation action of ${\rm GL}(N,{\bb C})$ on the space of $N\times N$ matrices $M(N,{\bb C})$ is defined via a coaction of the quantum general linear group ${\cal O}({\rm GL}_q(N,{\bb C}))$ on the algebra of quantum matrices ${\cal O}(M_q(N,{\bb C}))$ . The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of ${\cal O}(M_q(N,{\bb C}))$ generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.

Jensen's operator inequality and Jensen's trace inequality for real functions defined on an interval are established in what might be called their definitive versions. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions considered in earlier versions of the theory by the authors, and by Brown and Kosaki. As a consequence, one no longer needs to impose conditions on the interval of definition. It is shown how this relates to the pinching inequality of Davis, and how Jensen's trace inequality generalizes to C*-algebras.