Simultaneous Translational and Multiplicative Tiling

A ug 2 00 6 SIMULTANEOUS TRANSLATIONAL AND MULTIPLICATIVE TILING AND WAVELET SETS IN R 2 EUGEN

Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider the following simultaneous tiling problem: Give...

The Structure of Multiplicative Tilings of the Real Line

Tilings have long fascinated mathematicians [6]. The case where one moves a single object by translation in an abelian group (translational tiling) has proved both challenging and full of connections to Functional Analysis [8], such as connections to the so-called Fuglede Conjecture or Spectral Set Conjecture [5, 18, 10]. Simultaneous tiling of a set by both translation and multiplication (with...

Rhombellanic Crystals and Quasicrystals

Design of some crystal and quasicrystal networks, based on rhombellane tiling,is presented. [1,1,1]Propellane,is a synthesized organic molecule; its hydrogenated form, the bicyclo[1.1.1]pentane,may be represented by the complete bipartite graph K2,3 which is the smallest rhombellane. Topology of translational and radial structures involving rhombellanes is described in terms of vertex symbol, c...

Thermodynamics of a Homogeneous Limit-Periodic System

Limit-periodicity describes structures without translational symmetry but which are the union of infinitely many patterns with translational symmetry on increasing scales. Because of its relative simplicity, the limit-perodic Socolar-Taylor tiling offers an ideal system to probe the physical consequences of this distinct type of order. In this undergraduate thesis, we construct a two-deminsiona...

The Study of Translational Tiling with Fourier Analysis