Degree Name
MS (Master of Science)
Program
Mathematical Sciences
Date of Award
8-2011
Committee Chair or Co-Chairs
Anant P. Godbole
Committee Members
Michel Helfgott, Teresa W. Haynes
Abstract
In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (K,a) omnisequence of length n is a string of letters that contains each of the ak words of length k over [A]={1,2,...a} as a substring. An omnimosaic O(n,k,a) is an n × n matrix, with entries from the set A ={1,2,...,a}, that contains each of the {ak2} k × k matrices over A as a submatrix. An omnisculpture is an n × n × n sculpture (a three dimensional matrix) with entries from set A ={1,2,...,a} that contains all the ak3 k × k × k subsculptures as an embedded submatrix of the larger sculpture. We will show that for given k, the existence of a minimal omnisculpture is guaranteed when kak2/3/e ≤ n ≤kak2/3/e(1+ε) and ε=εk → 0 is a sufficiently small function of k.
Document Type
Thesis - unrestricted
Recommended Citation
Eroglu, Cihan, "Omnisculptures." (2011). Electronic Theses and Dissertations. Paper 1346. https://dc.etsu.edu/etd/1346
Copyright
Copyright by the authors.