Mathematics
Recent Submissions

[20160201] Studies of Coherent Synchrotron Radiation with the Discontinuous Galerkin Method
In this thesis, we present methods for integrating Maxwell's equations in FrenetSerret coordinates in several settings using discontinuous Galerkin (DG) finite element method codes in 1D, 2D, and 3D. We apply these routines ... 
[20150901] Spatial Decay of Rotating Waves and Restrictions on Finite Disks.
In this thesis, we consider the system of reactiondiffusion equations and the behavior of the solution of such a system. The focus is to concentrate on solutions which decay at infinity. Under suitable assumptions, we ... 
[20150901] Improving the Material Point Method
The material point method (MPM) was designed to solve problems in solid mechanics, and it has been used widely in research and industry. In MPM, the equations of motion are solved on a background grid and Lagrangian material ... 
[20150128] DELAUNAYLAGUERRE GEOMETRY FOR MACROMOLECULAR MODELING AND IMPLICIT SOLVATION
We develop and implement geometric methods to study threedimensional structures of proteins, the knowledge of which is critical to the understanding of the molecules and their interactions. Delaunay and Laguerre methods, ... 
[20150128] Strongly Nonlinear Phenomena and Singularities in Optical, Hydrodynamic and Biological Systems
Singularity formation is an inherent feature of equations in nonlinear physics, in many situations such as in selffocusing of light nonlinearity is essential part of the model and physical events cannot be captured ... 
[20140916] Computational Techniques for Optimal Control of Quantum System
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems ... 
[20140915] Sasakian geometry on lens space bundles over Riemann surfaces
we compute the cohomology of the join of a 3 manifold with the sphere and we see the dependence of this cohomology on one of the parameters. 
[20140712] Star Operations and Numerical Semigroup Rings
We aim to classify the star and semistar operations on conductive numerical semigroup rings which are of the form $k + x^n k[[x]]$. By classifying the star and semistar operations on conductive numerical semigroup rings ... 
[20140712] ABC THEOREMS IN THE FUNCTIONAL CASE
In this dissertation we will prove some ABC Theorems, namely for relatively prime by pairs padic entire functions in one variable, for padic meromorphic functions in several variables without common factors, under the ... 
[20140712] Arithmetic Differential Subgroups of GL_{n}
A remarkable and special Galois Theory appears from the study of the arithmetic analogue of ordinary differential equations; where functions are replaced by integers, the derivative operator replaced by the “Fermat quotient ... 
[20140712] Changes in Student Proving Skills and Attitudes Following a Cooperative Learning Seminar
This dissertation details research studies designed to explore undergraduate math students’ beliefs and attitudes about mathematical proof, ability to compose valid proofs, and ability to read and validate purported proofs ... 
[20140214] Quantification of Stability in Systems of Nonlinear Ordinary Differential Equations
A common process in ODE theory is to linearize an ODE system about an equilibrium point to determine the local stability properties of its orbits. Less common are results that quantify the domain of stability in the original ... 
[20140213] Modeling the Mechanical Response of Polycrystalline Thin Films
Microelectromechanical systems (MEMS) are part of every modern technological advance. Electrodeposited thin nickel (Ni) polycrystalline films in MEMS often show fiber texture resulting in transverse isotropic elastic ... 
[20130905] Wellposedness and Illposedness of the Nonlinear Beam Equation
The dissertation consists of two parts, Wellposedness and illposedness for the nonlinear beam equation and Strichartz estimates of the beam equation on the domains. In the first part, we will work to introduce the ... 
[20130905] Arithmetic Deformation Classes Associated to Curves via pJet Spaces
We show that for certain curves over padic rings its first pjet space admits the structure of a torsor under a line bundle. 
[20130905] A Study of the Pressure Term in the NavierStokes Equations
In this paper we consider the Cauchy problem for the 3D \NS equations for incompressible flows, and their solutions. We will discuss the results of a paper by Otto Kreiss and Jens Lorenz on the role of ... 
[20130905] Positive Sasakian Structures on Links of Weighted Complete Intersection Singularities
Links of isolated singularities defined by weighted homogeneous polynomials have a natural Sasakian structure. Since it is known that SasakiEinstein metrics have positive Ricci curvature, and since positive Sasakian ... 
[20130130] Terrestrial and extraterrestrial radiation sources that move faster than light
Maxwell's equations establish that patterns of electric charges and currents can be animated to travel faster than the speed of light in vacuo and that these superluminal distribution patterns emit tightly focused ... 
[20120828] Viscous flow past plates
I devise a numerical method of high order in space (FDMHS) to simulate flow past a finite plate and a semiinfinite plate. The method solves the incompressible NavierStokes equation in the stream functionvorticity formulation. ... 
[20120827] Numerical and Analytical Studies of Electromagnetic Waves: Hermite Methods, Supercontinuum Generation, and Multiple Poles in the SEM
The dissertation consists of three parts: Hermite methods, scattering from a lossless sphere, and analysis of supercontinuum generation. Hermite methods are a new class of arbitrary order algorithms to solve partial ...