This thesis is composed of three main parts. Each chapter is concerned with
characterizing some properties of a random ensemble or stochastic process. The
properties of interest and the methods for investigating them di er between chapters.
We begin by establishing some asymptotic results regarding zeros of random
harmonic mappings, a topic of much interest to mathematicians and astrophysicists
alike. We introduce a new model of harmonic polynomials based on the so-called
"Weyl ensemble" of random analytic polynomials. Building on the work of Li and
Wei [28] we obtain precise asymptotics for the average number of zeros of this model.
The primary tools used in this section are the famous Kac-Rice formula as well as
classical methods in the asymptotic analysis of integrals such as the Laplace method.
Continuing, we characterize several topological properties of this model of
harmonic polynomials. In chapter 3 we obtain experimental results concerning the
number of connected components of the orientation-reversing region as well as the geometry
of the distribution of zeros. The tools used in this section are primarily Monte
Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology
software known as Perseus. While the results in this chapter are empirical rather
than formal proofs, they lead to several enticing conjectures and open problems.
Finally, in chapter 4 we address an industry problem in applied mathematics
and machine learning. The analysis in this chapter implements similar techniques to
those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or
Control Area Network) is a network for allowing micro-controllers inside of vehicles
to communicate with each other. We propose and demonstrate the e ectiveness of an
algorithm for detecting malicious tra c using an approach that discovers and exploits
the natural geometry of the CAN surface and its relationship to random walk Markov
chains. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection
| Identifer | oai:union.ndltd.org:fau.edu/oai:fau.digital.flvc.org:fau_40891 |
| Contributors | Tyree, Zachariah (author), Lundberg, Erik (Thesis advisor), Long, Hongwei (Thesis advisor), Florida Atlantic University (Degree grantor), Charles E. Schmidt College of Science, Department of Mathematical Sciences |
| Publisher | Florida Atlantic University |
| Source Sets | Florida Atlantic University |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
| Format | 83 p., application/pdf |
| Rights | Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder., http://rightsstatements.org/vocab/InC/1.0/ |
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