In the 1954 John Nash [1] showed, through use of an iterative scheme of approximate embedding maps, that the sphere S² could be isometrically embedded into a ball of any radius by a C¹ map. In the 1980's M. Gromov [2] generalized Nash's work to the h-principal and convex integration....
As industries relating to science, technology, engineering, and mathematics in America continue to grow, employers will need more mathematicians and mathematically able workers than are currently graduating. Women are an underrepresented portion of these graduates, and researches say that this could be due to the difference between women’s and men’s...
Junior level physics students are familiar with a few types of vector field derivatives, such as divergence and curl, but are typically unfamiliar with how to take a general derivative of a vector field. Three junior-level physics students were interviewed with the open-ended prompt, “How would you think about taking...
In this dissertation, we use Fourier-analytic methods to study questions of equidistribution on the compact abelian group Zp of p-adic integers. In particu- lar, we prove a LeVeque-type Fourier analytic upper bound on the discrepancy of sequences. We establish p-adic analogues of the classical Dirichlet and Fejér kernels on R/Z,...
There are three chapters of manuscripts in this dissertation and all of them are talking about a specific theme: stochastic control, but with completely different perspectives.
In the first manuscript, we solve the optimal barrier strategy for dividend distribution in a complicated Lévy system. In this system, the capital of...
Teachers may be attracted to the use of a game in a learning activity under the presumption that students will find the game experience to be more “fun” than typical classroom activities. The use of a game in a learning activity should help students attain important learning outcomes and engage...
Mathematics Graduate Student Instructors (GSIs) have a significant impact on the teaching and learning of mathematics in post-secondary contexts through their work as instructors of record, tutors, graders, and recitation, laboratory, or discussion leaders for mathematics courses. Perhaps more importantly, GSIs are future teachers of mathematics: more than 60 percent...
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...
Consider a polygon lying in the Euclidean plane with labeled edge lengths. The moduli space of polygons is the space of all polygons with the same labeled edge lengths, modulo orientation preserving isometries. It is well known that this space is generically a smooth manifold. For certain combinations of edge...
In this work we consider a mathematical and computational model for biofilm growth and nutrient utilization. In particular, we are interested in a model appropriate at a scale of interface. The model is a system of two coupled nonlinear diffusion--reaction partial differential equations (PDEs). One of these PDEs is subject...