The following question of V. Stakhovskii was passed to us by N. Dolbilin . 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)
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.
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)
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. Inspired by work of Ford, we describe a geometric representation of real and complex continued fractions by… (More)
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)
Let f be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under f of a radial segment in the unit disc. Our methods are based on the hyperbolic geometry of plane domains, and we address some new questions that follow naturally from this approach.