A description is given of methods that have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. It transpires that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.

A counterexample has been constructed to show that a conjectured global solution structure for bifurcation of non-trivial solutions from a simple eigenvalue of the linearization at zero really can occur. In addition, new results and counterexamples have been obtained for bifurcation from an eigenvalue of geometric multiplicity 1 and odd algebraic multiplicity.

This paper proves that for every Lipschitz function $f:{\bb R}^n\longrightarrow {\bb R}^m,\;m < n$ , there exists at least one point of $\varepsilon$ -differentiability of $f$ which is in the union of all $m$ -dimensional affine subspaces of the form $q_0+{\rm span}\{q_1,q_2,\ldots,q_m\},\;{\rm where}\;q_j(j=0,1,\ldots,m)$ are points in ${\bb R}^n$ with rational coordinates.

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.

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 ${\cal M}_\xi$ .

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.

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 $M$ .

A surprising relationship is established in this paper, between the behaviour modulo a prime $p$ of the number $s_n({\cal G})$ of index $n$ subgroups in a group ${\cal G}$ , and that of the corresponding subgroup numbers for a normal subgroup in ${\cal G}$ with cyclic quotient of $p$ –power order. The proof relies, among other things, on a twisted version due to Philip Hall of Frobenius' theorem concerning the equation $x^m=1$ in finite groups. One of the applications of this result, presented here, concerns the explicit determination modulo $p$ of $s_n({\cal G})$ in the case when ${\cal G}$ is the fundamental group of a tree of groups all of whose vertex groups are cyclic of $p$ –power order. Furthermore, a criterion is established (by a different technique) for the function $S_n({\cal G})$ to be periodic modulo $p$ .

In the paper, three lower bounds are given for the Morse index of a constant mean curvature torus in Euclidean $3$ -space, in terms of its spectral genus $g$ . The first two lower bounds grow linearly in $g$ and are stronger for smaller values of $g$ , while the third grows quadratically in $g$ but is weaker for smaller values of $g$ .

An example is given of a random time $\rho$ associated with Brownian motion such that $\rho$ is not a stopping time but ${\bf E}M_{\rho}={\bf E}M_0$ for every uniformly integrable martingale $M$ .