Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 3 of 3
Full-Text Articles in Number Theory
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, …
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
Dissertations, Theses, and Capstone Projects
In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …
On The Second Case Of Fermat's Last Theorem Over Cyclotomic Fields, Owen Sweeney
On The Second Case Of Fermat's Last Theorem Over Cyclotomic Fields, Owen Sweeney
Dissertations, Theses, and Capstone Projects
We obtain a new simpler sufficient condition for Kolyvagin's criteria, regarding the second case of Fermat's last theorem with prime exponent p over the p-th cyclotomic field, to hold. It covers cases when the existing simpler sufficient conditions do not hold and is important for the theoretical study of the criteria.