Alan F. Beardon

Learn More
There are many results on edge-magic, and vertex-magic, labellings of finite graphs. Here we consider magic labellings of countably infinite graphs over abelian groups. We also give an example of a finite connected graph that is edge-magic over one, but not over all, abelian groups of the appropriate order.
This paper contains some tentative steps towards describing the structure of non-discrete subgroups of SL(2; R). The main idea is that if a one-parameter family of groups G z varies analytically with the parameter z , then, using analytic continuation, certain results about discrete groups can be analytically continued to those groups in the family that are(More)
In an earlier paper [Journal of Mathematical Economics, 37 (2002) 17–38], we proved that if a preference relation on a commodity space is non-representable by a real-valued function then that chain is necessarily a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain. In this paper, we study the class of planar chains, the simplest example(More)
By using the theory of elliptic integrals we give an exact formula for the hyperbolic density of a rectangle at its centre. We compare this to the hyperbolic density of an infinite strip and obtain (in this special case) a quantitative version of the Carathéodory Kernel Theorem.