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The following question of V. Stakhovskii was passed to us by N. Dolbilin [4]. Take the barycentric subdivision of a triangle to obtain six triangles, then take the barycentric subdivision of each of these six triangles and so on; is it true that the resulting collection of triangles is dense (up to similarities) in the space of all triangles? We shall show… (More)

- Alan F. Beardon
- Australasian J. Combinatorics
- 2004

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.

- Alan F. Beardon, Kathy Driver
- Journal of Approximation Theory
- 2005

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)

We show that if a pair of meromorphic functions parametrize an algebraic curve then they have a common right factor, and we use this to derive a variety of results on algebraic curves.

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.

- Alan F. Beardon
- Discrete Applied Mathematics
- 2013

Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online's data policy on reuse of materials please consult the policies page. Abstract We unify and extend three classical theorems in continued fraction theory, namely the Stern-Stolz Theorem,… (More)

- Alan F. Beardon
- Australasian J. Combinatorics
- 2004

- A. F. Beardon
- 1996

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)