Rigidity of Circle Packings with Crosscuts

Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a bounded, simply connected domain. We introduce the concept of crosscuts and investigate the rigidity of circle packings with respect to maximal crosscuts. The main result is a discrete version of an indentity theorem for analytic functions (in the spirit of Schwarz' Lemma), which has implications to uniqueness statements for discrete conformal mappings.Comment: 38 pages, 22 figures. keywords: circle packing, crosscut, prime ends, conformal mapping, Schwarz's lemma, Apollonian packin

Relaxation procedures on graphs

Abstract: The procedures studied in this paper originate from a problem posed at the International Mathematical Olympiad in 1986. We present several approaches to the IMO problem and its generalizations. In this context we introduce a "signed mean value procedure" and study "relaxation procedures on graphs". We prove that these processes are always finite, thus confirming a conjecture of Akiyama, Hosono and Urabe MSC: 05C85, 68W01, 68P10, 91A4

Exploring complex functions using phase plots

Non UBCUnreviewedAuthor affiliation: Technische Universität Bergakademie FreibergFacult

Relaxation procedures on graphs

AbstractThe procedures studied in this paper originate from a problem posed at the International Mathematical Olympiad in 1986. We present several approaches to the IMO problem and its generalizations. In this context we introduce a “signed mean value procedure” and study “relaxation procedures on graphs”. We prove that these processes are always finite, thus confirming a conjecture of Akiyama, Hosono and Urabe [J. Akiyama, K. Hosono, M. Urabe, Some combinatorial problems. Discrete Mathematics 116 (1993) 291–298]. Moreover, we indicate relations to sorting and to an iterative method used in circle packing

From the Buffon Needle Problem to the Kreiss Matrix Theorem

In this paper we present a theorem concerning the arc length on the Riemann sphere of the image of the unit circle under a rational function. But our larger purpose is to tell a story. We thought at first that the story began in 1962 with the Kreiss matrix theorem, the application that originally motivated us. However, our arc length question turns out to be more interesting than that. The story goes back to the famous "Buffon needle problem" of 1777