The author of this paper has shown previously how a complex representation of a finite group can be split into a virtual sum of representations induced from one-dimensional representations of subgroups in a natural way (sometimes known as explicit Brauer induction). Here the modular case is treated, yielding an analogous result at the level of Brauer characters in general, and in the Green ring for trivial source modules.

The authors prove in this paper that an involutory Hopf algebra with non-zero integral over a field of characteristic zero is cosemisimple.

Let ${\cal L}$ be a uniformly elliptic operator in divergence form on ${\bb R}^d$ , and let $p(t,x,y)$ be the fundamental solution to the heat equation for ${\cal L}$ . A new proof is given of Aronson's upper bound: \[ p(t,x,y)\le c_1t^{-d/2}\exp(-c_2|x-y|^2/t). \]

An explicit, asymptotically good, tower of curves over the field with eight elements is constructed. The genus and the number of rational points are calculated explicitly.

A criterion of isomorphism for symmetry groups of set ideals is provided in terms of ideal quotients and cardinal invariants. Furthermore, set ideals with complete symmetry group are characterized. They form a wide class, comprising, for example, all uniform dense ideals and the ideals of ‘thin’ sets in separable metric spaces. If the symmetry group of a set ideal is not complete, then its outer automorphism group is shown to be cyclic of order 2.

An example is provided of a space that does not have the homotopy type of an H-space, and yet has the same $n$ -type as a double loop space for each natural number $n$ . It is also locally homotopy equivalent to a double loop space at each prime $p$ .

This paper provides examples of closed manifolds having a Morse number different from the stable Morse number.

Suppose that H is a finite subgroup of a linear algebraic group, G. It was proved by Donkin that there exists a finite-dimensional rational representation of G whose restriction to H is free. This paper gives a short proof of this in characteristic 0. The author also studies more closely which representations of H can appear as a restriction of G.

Let p and q be odd prime numbers. It is shown that all quasitriangular Hopf algebras of dimension pq over an algebraically closed field k of characteristic zero are semisimple and therefore isomorphic to a group algebra.

Let $X$ be a non-singular algebraic curve of genus $g\ge 2,\;n\ge 2$ an integer, $\xi$ a line bundle over $X$ of degree $d>2n(g-1)$ with $(n,d)=1$ and ${\cal M}_\xi$ the moduli space of stable bundles of rank $n$ and determinant $\xi$ over $X$ . It is proved that the Picard bundle ${\cal W}_\xi$ is stable with respect to the unique polarisation of ${\cal M}_\xi$ .

The authors prove in this paper that, given any knot $k$ of genus $g$ , $k$ fails to be strongly $n$ -trivial for all $n$ with $n\ge 6g-3$ .

In this note, two geometric properties of Banach spaces are proposed and discussed, related to the validity of the lemma of Riemann–Lebesgue in spaces of weakly integrable functions. The class of Banach spaces with the analytic Riemann–Lebesgue property is shown to be precisely the class of Banach spaces for which a weak form of a theorem of Ingham holds.

Any 2-block of a finite group G with a quaternion defect group Q8 is Morita equivalent to the corresponding block of the centraliser H of the unique involution of Q8 in G; this answers positively an earlier question raised by M. Broué.

Results of the Landesman–Lazer type provide necessary and sufficient conditions for the existence of periodic solutions of certain nonlinear differential equations with forcing. Typically, they deal with scalar problems. This paper presents a discussion of possible extensions to systems. The emphasis is placed on the new phenomena produced by the increase of the dimension.

This paper solves a problem posed by Colin Adams, showing that any link $L$ in ${\bb R}^3$ is isotopic to a smooth link $\hat{L}$ that has no parallel or antiparallel tangents.

It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group Sp [rmoust ] Sw. Combined with work of J. Rickard, this proves that Broué's abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5.

This paper provides sufficient conditions that guarantee the global attractivity of the zero solution of a delay population model under impulsive perturbations.

Let $W$ be a finite-dimensional ${\bb Z}/p$ -module over a field, ${\bf k}$ , of characteristic $p$ . The maximum degree of an indecomposable element of the algebra of invariants, ${\bf k}[W]^{{\bb Z}/p}$ , is called the Noether number of the representation, and is denoted by $\beta(W)$ . A lower bound for $\beta(W)$ is derived, and it is shown that if $U$ is a ${\bb Z}/p$ submodule of $W$ , then $\beta(U)\le \beta(W)$ . A set of generators, in fact a SAGBI basis, is constructed for ${\bf k}[V_2\oplus V_3]^{{\bb Z}/p}$ , where $V_n$ is the indecomposable ${\bb Z}/p$ -module of dimension $n$ .

The author proves in this paper that every profinite group G with polynomial subgroup growth is boundedly generated; that is, it is a product of finitely many procyclic subgroups. This answers a question of P. Zalesskii. By contrast, if G is a boundedly generated group, then the subgroup growth of G is at most nclogn . As a byproduct, a short, elementary proof demonstrates that Aut(Fr) (for r [ges ] 2) and many other related groups are not boundedly generated.

For every $n\ge 6$ , examples are given of closed negatively curved riemannian $n$ -manifolds for which the natural map $PLM\to {\hbox{\it Out}}\;\pi_1M$ is not an epimorphism. $PLM$ denotes the group of piecewise linear homeomorphisms of $M$ .