Omnisculptures.

<p>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 (<em>K</em>,<em>a</em>) omnisequence of length <em>n</em> is a string of letters that contains each of the <em>a^k</em> words of length <em>k</em> over [<em>A</em>]={1,2,...a} as a substring. An omnimosaic <em>O</em>(<em>n</em>,<em>k</em>,<em>a</em>) is an <em>n</em> &#215; <em>n</em> matrix, with entries from the set <em>A</em> ={1,2,...,<em>a</em>}, that contains each of the {<em>a^k²</em>} <em>k</em> &#215; <em>k</em> matrices over <em>A</em> as a submatrix. An omnisculpture is an <em>n</em> &#215; <em>n</em> &#215; <em>n</em> sculpture (a three dimensional matrix) with entries from set <em>A</em> ={1,2,...,<em>a</em>} that contains all the <em>a^k³</em> <em>k</em> &#215; <em>k</em> &#215; <em>k</em> subsculptures as an embedded submatrix of the larger sculpture. We will show that for given <em>k</em>, the existence of a minimal omnisculpture is guaranteed when <em>ka^k²/3/e &#8804; n &#8804;ka^k²/3/e</em>(1+&#949;) and <em>&#949;=&#949;_k</em> &#8594; 0 is a sufficiently small function of <em>k</em>.</p>

Name thesis:degree_name

MS (Master of Science)

Level thesis:degree_level

Thesis - unrestricted

Discipline thesis:degree_discipline

Mathematical Sciences

Year dc:date.issued

2011

Author dc:creator

• Eroglu, Cihan

• Omnisculptures
• Omnimosaics
• Graphs
• Omnisequences
• Discrete Mathematics and Combinatorics
• Mathematics
• Physical Sciences and Mathematics

Statement dc:rights

• Copyright by the authors.

Repository record dc:identifier

https://dc.etsu.edu/etd/1346

OAI identifier oai:identifier

oai:dc.etsu.edu:etd-2537