Dissertation
Using Higher Dimensional Extensions of TRIP Maps for Analyzing Periodicity of Multidimensional Continued Fractions
Washington State University
Doctor of Philosophy (PhD), Washington State University
2023
DOI:
https://doi.org/10.7273/000005168
Abstract
Ordinary continued fractions nest fractions with numerators equal to 1 and positive integer denominators via addition to represent real numbers. It has been long established that such fractions will have a periodic expansion, that is, eventually have a repeating pattern in the denominators, when the number being represented is a quadratic irrational. In the 1800s, Hermite expressed interest to Jacobi to find periodic expansions for cubic irrationals. This led to the study of multidimensional continued fractions using matrices to encode algorithms for these analogs of continued fractions.
In this dissertation, we explore an extension of the TRIP map algorithm developed by Dasaratha et al. related to these multidimensional continued fractions, which relates a location of an ordered pair in a triangle. We consider the generalized version of this map, called a 3-SIMP map, which relates a location of an ordered triple in a tetrahedron. This version further expands Hermite's question as 3-SIMP maps find periodicity results that relate to quartic roots instead of cubic roots.
Additionally, we examine a connection between purely periodic multidimensional continued fractions produced by the Jacobi-Perron algorithm to the combinatorial construction of Riordan arrays. Riordan arrays provide a way to generalize Pascal's triangle and are studied for their combinatorial properties. We describe a method to relate multidimensional continued fractions to families of Riordan arrays in the case where we have purely periodic multidimensional continued fractions of period 1.
Metrics
7 File views/ downloads
20 Record Views
Details
- Title
- Using Higher Dimensional Extensions of TRIP Maps for Analyzing Periodicity of Multidimensional Continued Fractions
- Creators
- Rachel Perrier
- Contributors
- Matthew Hudelson (Advisor)Judith McDonald (Committee Member)Sheng-Chi Liu (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 119
- Identifiers
- 99901019835501842
- Language
- English
- Resource Type
- Dissertation