This paper proves that a triangulated graph is an outline triple system of even index if and only if it is an amalgamated triple system of even index.

2000 Mathematical Subject Classification: primary 05B07; secondary 05C99.

A semilinear space S is ultrahomogeneous if each isomorphism between the semilinear structures induced on two finite subsets can be extended to an automorphism of S. We give a complete classification of all finite ultrahomogeneous semilinear spaces. Our theorem extends a result of A. Gardiner on graphs and a result of A. Devillers and J. Doyen on linear spaces.

2000 Mathematical Subject Classification: 05B25, 51E14, 20B25.

In this paper we study the diophantine equation of the title, which was first introduced by Nagell and Ljunggren during the first half of the twentieth century. We describe a method which allows us, on the one hand when n is fixed, to obtain an upper bound for q, and on the other hand when n and q are fixed, to obtain upper bounds for x and y which are far sharper than those derived from the theory of linear forms in logarithms. We also show how these bounds can be used even when they seem too large for a straightforward enumeration of the remaining possible values of x. By combining all these techniques, we are able to solve the equation in many cases, including the case when n has a prime divisor less than 13, or the case when n has a prime divisor which is less than or equal to 23 and distinct from q.

2000 Mathematical Subject Classification: primary 11D41; secondary 11J86, 11Y50.

In this paper the authors prove the following result.

Let $\alpha$ be an irrational number. Then for any $\varepsilon > 0$, there are infinitely many prime numbers p such that $\| \alpha p \| < p^{-16/49 +\varepsilon}$.

The exponent $\frac{16}{49}$ improves on $\frac{9}{28}$, which was obtained recently by the second author [Sci. China Ser. A 43 (2000) 703-721]. The result is very close to the exponent $\frac{1}{3}$, which can be obtained under the Generalized Riemann Hypothesis.

Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non-trivial for the exponent $\frac{1}{3}$. This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis.

2000 Mathematical Subject Classification: 11N36, 11J71.

This series of papers presents and rigorously analyzes a probabilistic algorithm for finding small prime factors of an integer. The algorithm uses the Jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. This second paper in the series is concerned with the order of the group of rational points on the Jacobian of a curve of genus 2 defined over a finite field. We prove a result on the distribution of these orders.

2000 Mathematical Subject Classification: 11Y05, 11G10, 11M20, 11N25.

This paper continues the study of quantised function algebras $\mathcal{O}_{\epsilon}[G]$ of a semisimple group $G$ at an $\ell$th root of unity $\epsilon$. These algebras were introduced by De Concini and Lyubashenko in 1994, and studied further by De Concini and Procesi and by Gordon, amongst others. Our main purpose here is to increase understanding of the finite-dimensional factor algebras $\mathcal{O}_{\epsilon}[G](g)$, for $g \in G$. We determine the representation type and block structure of these factors, and (for many $g$) describe them up to isomorphism. A series of parallel results is obtained for the quantised Borel algebras $U_{\epsilon}^{\geq 0}$ and $U_{\epsilon}^{\leq 0}$.

2000 Mathematical Subject Classification: 16W35, 17B37.

A theorem of K. W. Roggenkamp and L. L. Scott shows that for a finite group G with a normal p-subgroup containing its own centralizer, any two group bases of the integral group ring $\mathbb{Z} G$ are conjugate in the units of $\mathbb{Z}_{p}G$. Though the theorem presents itself in the work of others and appears to be needed, there is no published account. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the survey [K. W. Roggenkamp, ‘The isomorphism problem for integral group rings of finite groups’, Progress in Mathematics 95 (Birkhäuser, Basel, 1991), pp. 193--220], where the main ingredients of a proof are given, is false. In this paper, it is shown how to close this gap, at least if one is only interested in the conclusion mentioned above. Therefore, some consequences of the results of A. Weiss on permutation modules are stated. The basic steps of which any proof should consist are discussed in some detail. In doing so, a complete, yet short, proof of the theorem is given in the case that G has a normal Sylow p-subgroup.

2000 Mathematical Subject Classification: primary 16U60; secondary 20C05.

Let $F$ be an algebraically closed field of characteristic $p \geq 0$ and $A_n$ be the alternating group on $n$ letters. The main goal of this paper is to describe the pairs $(G, E)$, where $E$ is an irreducible $FA_n$-module and $G < A_n$ is a proper subgroup of $A_n$ such that the restriction $E{\downarrow_G}$ is irreducible. We are able to give a list of all such pairs, provided $p > 3$. The case $p = 0$ has been treated by J. Saxl. The problem is important for the classification of maximal subgroups in finite classical groups.

2000 Mathematical Subject Classification: 20C20, 20C30, 20B35, 20B20.

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4\pi$, the first non-zero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard $S^1$-action on $S^2$, one can restrict the Laplace operator to the subspace of $S^1$-invariant functions and consider its spectrum there. The corresponding eigenvalues will be called invariant eigenvalues, and the purpose of this paper is to analyse their possible values.

We first show that there is no general analogue of Hersch's theorem, by exhibiting explicit families of $S^1$-invariant metrics with total area $4\pi$, where the first invariant eigenvalue ranges through any value between 0 and $\infty$. We then restrict ourselves to $S^1$-invariant metrics that can be embedded in $\mathbb{R}^3$ as surfaces of revolution. For this subclass we are able to provide optimal upper bounds for all invariant eigenvalues. As a consequence, we obtain an analogue of Hersch's theorem with an optimal upper bound (greater than 2 and geometrically interesting). This subclass of metrics on $S^2$ includes all $S^1$-invariant metrics with non-negative Gauss curvature.

One of the key ideas in the proofs of these results comes from symplectic geometry, and amounts to the use of the moment map of the $S^1$-action as a coordinate function on $S^2$.

2000 Mathematical Subject Classification: primary 35P15; secondary 58J50, 53D20.

In this paper we expand the theory of connected components in non-archimedean discrete dynamical systems. We define two types of components and discuss their uses and applications in the study of dynamics of a rational function $\phi\in K(z)$ defined over a non-archimedean field K. We prove that some fundamental conjectures, including the No Wandering Domains conjecture, are equivalent, regardless of which definition of 'component' is used. We derive several results on the geometry of our components and the existence of periodic points within them. We also give a number of examples of p-adic maps with interesting or pathological dynamics.

2000 Mathematical Subject Classification: primary 37B99; secondary 11S99, 30D05.