At the macroscopic scale, we have pumps that use the classical laws of physics to move liquids at a well defined rate. In the microscopic world, physicists are exploring pumps that make use of quantum mechanical behavior to build analogous pumps for quantum particles. The importance of such “quantum pumps”...
Facial recognition has become increasingly important in recent years, due to the wide range of applications it has in fields such as security, surveillance, and human-computer interaction. Three popular methods for facial recognition are the Principal Component Analysis (PCA), Karhunen-Loeve Expansions, which is fundamentally a continuous form of PCA but...
Combinatorics is a field of mathematics that concerns enumeration and existence, and its most notable applications are in computer science and statistics. Most students are introduced to combinatorics through counting problems, where they are tasked with determining the cardinality of a set of outcomes. Such problems are well-known for being...
We define an inner product on a vector space of adelic measures over a number field $K$. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational map is in this vector space, and...
The case of Mongolia’s democratization remains unexplained by the various theories of democratization and democratic consolidation. The purpose of this thesis is to consider the unique case of Mongolian democratization, how well this case is described by various theories of democratization and reach both a description of how Mongolia’s democratization...
We introduce a numerical criterion which allows us to bound the degree of any algebraic integer having all of Galois conjugates in an interval of length less than 4. Using this criterion, we study two arithmetic dynamical questions with local rationality conditions. First, we classify all unicritical polynomials defined over...
Intuitively, it seems as though natural language processing tasks might benefit from explicit representations of the syntactic and semantic properties of text. Ontonotes is a dataset which attempts to annotate texts, to represent as much as possible of the meaning of the text explicitly within the annotation. Many tools exist...
Loop quantum gravity is a theoretical framework which aims to quantize general relativity. One of the unique aspects of this theory is that it imposes a discrete structure on space-time. One toy model of loop quantum gravity is loop quantum cosmology, which maintains the idea of a discrete spatial structure...
Loop quantum gravity is a framework for quantizing general relativity which imposes a discrete structure on space. In exploring the structure of loop quantum gravity, researchers have investigated loop quantum cosmology, a toy model with reduced degrees of freedom while maintaining a discrete space. Polymer quantum mechanics is a framework...
There is high projected growth for STEM-based job fields; however, a significant barrier to students entering those fields is the requirement to take Introductory Calculus. This thesis summarizes the current research done to improve university level math education, specifically calculus courses. It then uses the MAA Instructional Practices Guide to...
In "Level Number Sequences for Trees" Flagotet and Prodinger investigate the problem of counting the number of level number sequences associated to binary trees of $n$ binary nodes. I convert this problem into terms of exterior nodes or "leaves" and leaf number sequences. "Polynomial representation" is then defined to address...
In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on the unit circle tightly associated with this group. In general, their fundamental domain enjoys what has since been...
In 1956, Alder conjectured an integer partition inequality which generalized Euler’s partition identity, the first Rogers-Ramanujan identity, and a partition identity of Schur. Alder’s conjecture, proved in part by Andrews in 1971, followed by Yee in 2008, and finally completed by Alfes, Jameson, and Lemke Oliver in 2010 states that...
In biological models, advection is inherently a non-local process. Coupled with diffusion, it typically models chemotaxis, which is the response of bacteria to the presence of some chemo attractant. For example, E. coli cells use their flagella to probe their surroundings to determine where they should move. The advection-diffusion equation...
In this work we introduce some basic concepts within homotopy type theory (HoTT), a proposed alternative mathematical foundation to classical set theory. In particular, our discussion revolves around the Axiom of Choice (AC). In Part I, we introduce the classical AC and some of its most important equivalents. In Part...
Mathematics outreach typically consists of community events that show the exciting applications of mathematics, particularly to K-12 students. The goal of mathematics outreach events is to increase student interest and involvement in mathematics-related activities. Students start to develop a stigma against mathematics by the end of elementary school. Outreach events...
Machine learning models are powerful tools which may aid in the prediction of survival outcomes of cancer patients. This study evaluated eight classification models and eight regression models on their ability to predict survival outcomes on breast cancer and prostate cancer data sets from the SEER database. The most accurate...
Machine learning models are powerful tools which may aid in the prediction of survival outcomes of cancer patients. This study evaluated eight classification models and eight regression models on their ability to predict survival outcomes on breast cancer and prostate cancer data sets from the SEER database. The most accurate...