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...
Physicists solve problems and communicate their work using many external representations, such as equations, words, diagrams, graphs, sketches, pictures, and more. To learn physics, then, students must learn to use external representations. In this dissertation, I present three manuscripts. Each manuscript discusses how upper-division Paradigms in Physics students use multiple...
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 provide a thorough proof of most of Bertrand's Theorem. Using Arnold's book "Mathematical methods of classical mechanics" as a backbone and calculus methods demonstrated by Jovanović in his article "A note on the proof of Bertrand's theorem," we show that for masses in a central field...
In this thesis we study mathematical and computational models for phenomena of flow and transport in porous media in the presence of changing pore scale geometries. The differential equations for the flow and transport models at Darcy scale involve the coefficients of permeability, porosity, and tortuosity which depend on the...
In 2013, Lemke Oliver created a list of all eta-quotients which are theta functions. Then in 2016, Folsom, Garthwaite, Kang, Swisher, and Treneer utilized this list of ``eta-theta'' functions along with Zwegers's construction of mock theta functions to create a set of mock modular forms which are also quantum modular...
In this work, we consider a convexity splitting scheme for a coupled phase field and energy equation, a modification of Stefan problem. The Stefan problem is a free boundary value problem that models the temperature in a homogeneous multiphase medium. Each phase is modeled using a heat diffusion parabolic partial...
Simulations of combustion and reacting flows often encounter stiffness in the equations governing chemical kinetics. Explicit solvers for these ordinary differential equations offer low computational expense, but typically cannot efficiently handle stiff systems. In contrast, implicit methods demand greater expense but offer unconditional stability—as a result, most reactive-flow solvers rely...
In this paper, we discuss two possible modifications to a numerical solution method for a model of microbiologically induced calcite precipitation (MICP). MICP provides a means to seal cracks in the surfaces of geological structures. From a mathematical and computational point of view MICP has very interesting features which make...
In this dissertation, we introduce a family of fully discrete finite difference time-domain (FDTD) methods for Maxwell’s equations in linear and nonlinear materials. Onecategory of methods is constructed using multiscale techniques involving operator splittings. We present the sequential splitting scheme, the Strang Marchuk splitting scheme,the weighted sequential splitting scheme including...