Algebraic correspondences on the Riemann sphere generalise both Kleinian groups and rational maps. We classify quadratic correspondences satisfying certain "diagram conditions" and derive canonical forms for them. We exhibit examples for which the regular set and limit set are both nontrivial, and show that such examples can simultaneously exhibit Kleinian-group-like and rational-map-like behaviour. We also state some general theorems and conjectures.

We study the bifurcation set in $(b,\phi,\alpha)$-space of the equation $\dot{z} = e^{i\alpha} z +e^{i\phi} z \,|z|^2 + b \bar{z}^3$. This $\mZ_4$-equivariant planar vector field is equivalent to the model equation that has been considered in the study of the 1:4 resonance problem.

We present a three-dimensional model of the bifurcation set that describes the known properties of the system in a condensed way, and, under certain assumptions for which there is strong numerical evidence, is topologically correct and complete. In this model, the bifurcation set consists of surfaces of codimension-one bifurcations that divide $(b,\phi,\alpha)$-space into fifteen regions of generic phase portraits. The model also offers further insight into the question of versality of the system. All bifurcation phenomena seem to unfold generically for $\phi \neq \pi/2, \, 3\pi/2$.

The ring of integers $\OO_N$ of a tame Galois extension $N/\Qdb$ with Galois group $G$ is $\Zdb[G]$-projective. There exists a necessary and sufficient condition for $\OO_N\oplus\Zdb[G]$ and $\Zdb[G]^2$ to be isomorphic. We give a first example where this condition does not imply that $\OO_N$ is free over $\Zdb[G]$.

L'anneau des entiers $\OO_N$ d'une extension galoisienne $N/\Qdb$ modérément ramifiée de groupe de Galois $G$ est $\Zdb[G]$-projectif. Il existe une condition nécessaire et suffisante pour que la somme $\OO_N\oplus\Zdb[G]$ soit isomorphe à $\Zdb[G]^2$. On donne un premier exemple où cette condition n'implique pas que $\OO_N$ soit $\Zdb[G]$ libre.

We show how to construct $133 \times 133$ matrices over $\GF (5)$ generating the Harada--Norton group. We also obtain generators for its automorphism group. For many purposes this permits much faster calculations in the group than the alternative of permutations on 1,140,000 points. More importantly, it reduces storage requirements by a factor of around 500.

We describe an exhaustive search that proves the nonexistence of homometric Golomb ruler pairs with 7 to 12 marks. We also give a two-dimensional geometric model for the known family of six-mark homometric ruler pairs.

The theory of automatic groups (groups that can be treated by means of finite-state automata), and in particular the techniques of automatic coset enumeration developed by Redfern, are applied to the problem of computing limit sets of a large class of Kleinian groups. We discuss in detail the case of groups in the Maskit embedding of the \tspace/ of the punctured torus.