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...
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...
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...
Living in the Pacific Northwest, we are acutely aware of the dangers posed by wildfires. Largely due to the worsening effects of climate change, this danger is only increasing. Along with causing property and economic damage to those communities affected by wildfires, exposure to the smoke generated by wildfires can...
Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to the modular group, and Bannai, Kojima, and Miezaki's partial proof for the case of the Fricke group of level 2, we show that the zeros of certain modular functions for some low-level genus zero groups associated to...
A fully-saturated poroelastic medium is confined by the sides of a cylinder, and the regions below and above the medium are filled with fluid at respective constant pressures. The filtration flow of fluid through the poroelastic medium and the small deformations of the medium are described by a quasi-static Biot...
This arts-based autoethnography explores the experience of a graduate student of mathematics at a mid-sized research university through a collection of collage, songwriting, and personal essays. This research identifies issues in the mathematics academic pipeline associated with gender, burnout culture, perfectionism, mental health, qualifying exams, and isolation. The present research...
In this dissertation, we consider two problems in number theory, both relating to modular forms. First we consider when a given modular form can be expressed as a quotient in Dedekind's $\eta$ function. Rouse and Webb \cite{RW} have determined the integers $N \leq 500$ such that the graded ring of...
Within this dissertation, we develop tools and techniques to demonstrate the feasibility of real-time optimization of a magnetohydrodynamics generator. To ease computational complexity, we work on the kinematic magnetohydrodynamic system, prescribing the fluid-flow and model the material response of the system through an updated Generalized Ohm’s law. We focus on...
Applied problems are a necessity for a well-rounded and rigorous education in mathematics. This thesis summarizes the literature about student engagement with applied problems in order to develop a set of criteria for what makes an engaging and meaningful applied mathematics problem. Based on a review of the literature, this...
We explore one numerical method for dealing with uncertainty quantification, stochastic collocation. We adapt this method for the uncertain kinematic magnetohydrodynamic system. We then demonstrate well-posedness of the uncertain forward problem. We also describe the method in detail, and perform an error analysis of the method, describing the necessary assumptions...
We discuss an efficient numerical method for the uncertain kinematic magnetohydrodynamic system. We include aleatoric uncertainty in the parameters, and then describe a stochastic collocation method to handle this randomness. Numerical demonstrations of this method are discussed. We find that the shape of the parameter distributions affect not only the...
We discuss the well-posedness of the forward problem for the magnetohydrodynamic system with the inclusion of the ion-slip parameter. We also demonstrate the convergence of a parameter estimation scheme. Focusing on power-generation, we implement and the validate a numerical model with an engineering multi-physics software, COMSOL, using ideal-power equations. We...
Constructions using only a straightedge and compass are basic tools in any geometer's toolbox. We show how to construct an elliptic straightedge and compass in the Klein Disk model of (single) elliptic geometry, using only a Euclidean compass and straightedge.
Given an initial opinion matrix in which every member of a population gives an opinion score of all the members in a population, natural questions to ask could be regarding what happens to those opinions over time, both in forward time and in backward time. While what happens over time...
Stein's method initially introduced in 1970 by C. Stein is a powerful technique for bounding the distance between the laws of two real-valued random variables. Stein's method has been used to prove distributional convergence to many standard probability distributions such as normal, multivariate normal, Poisson and Brownian motion approximation. In...
This paper explores some optimization methods such as the gradient descent method, Gauss-Newton method, and stochastic gradient method. Some examples of minimizing objective functions are given to validate the theories. Then we introduce a simple example of artificial neural networks, define its structure, and apply the optimization methods to it....
We now have broad consensus in the mathematics education research community that active, inquiry-based classrooms provide a wealth of learning benefits for students (Freeman et al., 2014; Laursen et al., 2014; Theobald et al., 2020). Classrooms that utilize inquiry throughout the entire structure of the course, as opposed to the...
Aggregation equations have been used to model phenomena such as insect swarming and chemotaxis. Previous work on aggregation equations in the area of analysis applied to PDE has proven well-posedness of certain classes of aggregation equations in Lebesgue spaces. We will prove local existence of solutions in H^1 to an...
The HyperLogLog (HLL) algorithm is used to estimate the cardinality of large sets. This thesis gives a novel analysis of the HyperLogLog algorithm by using techniques from statistics and probability. Initially, closed form bounds for the mean and variance of the max of n independent and identically distributed geometric random...
Many geophysical phenomena exhibit complicated dynamics that, due to a variety of factors, diverge quickly from physical models. The arrival of new observations allows researchers to combine the model estimate with measurements in a statistical process called data assimilation to produce a revised estimate of the phenomenon. This assimilation of...
In this dissertation we consider two application specific flow and transport models in porous media at multiple scales: 1) methane gas transport models for hydrate formation and dissociation in the subsurface under two-phase conditions, and 2) coupled flow and biomass-nutrient model for biofilm growth in complex geometries with biofilm, and...
The aim of this dissertation is to construct a virtual element method (VEM) for models in magneto-hydrodynamics (MHD), an area that studies the behavior and properties of electrically conducting fluids such as a plasma. MHD models are a coupling of the Maxwell’s equations for electromagnetics and models for fluid flow....
In this paper we explain at the theoretical level how discrete Morse theory can provide us a more efficient approach to compute persistent homologies. In achieving so we also provide a framework for discrete Morse theory to be applied to persistent homology for other purposes.
An important problem in computer graphics is to determine where contour lines and ridges appear in a surface constructed from a triangle mesh. In this presentation we will investigate a new answer to this problem – the horizon measure. The horizon measure determines the likelihood of contour lines to appear...
This dissertation explores mathematical theory of the 3 dimensional incompressible Navier-Stokes equations that consists a set of partial differential equations which govern the motion of Newtonian fluids and can be seen as Newton's second law of motion for fluids. The main interest of this work focuses on how local perturbation...
In 1941, J.H.C. Whitehead posed the question of whether asphericity is a hereditary property for 2-dimensional CW complexes. This question remains unanswered, but has led to the development of several algebraic and topological properties that are sufficient (but not necessary) for the asphericity of presentation 2-complexes. While many of the...
Institutions of higher education have experienced a rapid increase in international student enrollment within a short period of time, especially in mathematics classrooms. It is therefore important for instructors to have the knowledge and skills to support international students in the learning of mathematics. There are few research studies of...
The topic of statistical mechanics has been studied for over a century, and it is one of the pillars of modern physics. This theory can be applied to the study of the thermodynamic behavior of large systems of interacting particles, in which case it is referred to as equilibrium statistical...
There is an underrepresentation of minority groups and women in the field of mathematics. With growing interest in equity and inclusion within the mathematics education community, it is important to look at how such concepts can be implemented in the classroom. Equitable and inclusive teaching practices have been shown to...
Enumerative combinatorics is an area of mathematics that is both highly accessible for students and widely applicable to other sciences and areas of mathematics (Kapur, 1970; Lockwood, Wasserman, & Tillema, 2020). One important class of problems in combinatorics is combinatorial proofs of binomial identities, which is a type of proof...
Mapper is a tool designed by Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson for Topological Data Analysis (TDA) that constructs a simplicial complex from a finite subset of a metric space, which has been met with great success. Due to the many moving parts of Mapper and the potential it...
In this dissertation, we use Fourier-analytic and spectral theory methods to analyze the behavior of solutions of the incompressible Navier-Stokes equations in 2D and 3D (with an eye towards better understanding turbulence). In particular, we investigate the possible existence of so-called ghost solutions to the Navier-Stokes Equations. Such solutions, if...
Let H be a cyclically-presented group on n generators with a single defining relator. Attempts have been made to classify such groups by their order, their status as a 3-manifold group, and the asphericity status of their presentations. For groups with a defining relator of length 3 these classifications are...
This is a program that searches for reduced spherical pictures over 1-generator, 1-relator relative group presentations with finite cyclic coefficient group. The search is depth-first and builds pictures face-by-face using a prescribed order. This program was written as part of Matthias Merzenich's PhD research. More details are available in the...
The theory behind magnetohydrodynamics (MHD) is utilized to present a 3-D solution to how the induced magnetic field changes with respect to time. Several MHD-based assump- tions are made to simplify the coupling of Maxwell’s equations with two constitutive laws and Ohm’s law. The velocity field is assumed to be...