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.

We study the Hardy inequality for a class of manifolds equipped with a singular metric that becomes infinite on the boundary; in particular, we study the best constant possible. One application of this constant has been to a certain type of domain perturbation. This technique is useful when the domain has an irregular boundary as is the case here. However, our first result shows that the class of manifolds considered has a Hardy constant that lies outside the range permitted by existing theorems. Yet we are still able to prove theorems which give information about the domain perturbation problem and, moreover, we set up a specific example which can be used to show that our results are the best possible.

2000 Mathematical Subject Classification: 35P99, 47A75,47B25,58J99.

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.

Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.

2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

We construct a family, $\{HM^{(k)}\}_{k\geq 2}$, of oriented topological equivalence invariants, for projective configurations of complex hyperplanes with fixed combinatorial pattern. We construct a family, $\{PM^{(k)}\}_{k \geq 2}$, of concordance invariants, for classical links with a fixed number of components. The constructions are carried out in the same manner, using the existence of natural additional structures on the nilpotent quotients of the fundamental groups of the complements. We give the explicit computation of the primary and secondary HM and PM invariants.

2000 Mathematical Subject Classification: primary 57M25, 32S50; secondary 20F10, 20F12.

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.

2000 Mathematical Subject Classification: 46G05, 46T20.

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.

We construct a Banach space X whose algebra of bounded linear operators has K0-group the additive group of rational numbers. We achieve this using a Gowers-Maurey space.

2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10

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.