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Full-Text Articles in Discrete Mathematics and Combinatorics
On The Order-Type Complexity Of Words, And Greedy Sidon Sets For Linear Forms, Yin Choi Cheng
On The Order-Type Complexity Of Words, And Greedy Sidon Sets For Linear Forms, Yin Choi Cheng
Dissertations, Theses, and Capstone Projects
This work consists of two parts. In the first part, we study the order-type complexity of right-infinite words over a finite alphabet, which is defined to be the order types of the set of shifts of said words in lexicographical order. The set of shifts of any aperiodic morphic words whose first letter in the purely-morphic pre-image occurs at least twice in the pre-image has the same order type as Q ∩ (0, 1), Q ∩ (0, 1], or Q ∩ [0, 1). This includes all aperiodic purely-morphic binary words. The order types of uniform-morphic ternary words were also studied, …
Fuglede's Conjecture In Some Finite Abelian Groups, Thomas Fallon
Fuglede's Conjecture In Some Finite Abelian Groups, Thomas Fallon
Dissertations, Theses, and Capstone Projects
This dissertation thoroughly examines Fuglede's Conjecture within some discrete settings, shedding light on its intricate details. Fuglede's Conjecture establishes a profound connection between the geometric property of being a tiling set and the analytical attribute of being a spectral set. By exploring the conjecture on various discrete settings, this thesis delves into the implications and ramifications of the conjecture, unraveling its implications within the field.
A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson
A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson
Dissertations, Theses, and Capstone Projects
We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.