Mathematics Dissertations and Theses
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A General Stochastic Volatility Model on VIX Options
(University of Kansas, 20191231)Abstract In this dissertation, we study a general stochastic volatility model for the VIX options (Chicago Board Options Exchange) volatility index, which is a stochastic differential equation with 8 unknown parameters. ... 
Optimal Energy Decay for the Damped KleinGordon Equation
(University of Kansas, 20190831)In this dissertation we study the long time dynamics of damped KleinGordon and damped fractional KleinGordon equations using $C_0$ Semigroup theory and its application. The $C_0$semigroups are used to solve a large ... 
Sharp time asymptotics for the quasigeostrophic equation, the Boussinesq system and near plane waves of reactiondiffusion models
(University of Kansas, 2019531)Through this dissertation we present the sharp time decay rates for three equations, namely quasigeostrophic equation (SQG), Boussinesq system (BSQ) and plane wave of general reactiondiffusion models. In addition, in ... 
A canonical form for the differential equations of curves in ndimensional space
(University of Kansas, 19300531) 
The characterizations of a class of transformations and of a class of differentiable functions
(University of Kansas, 19510531) 
Normal determinants and expansions in modified sequences
(University of Kansas, 19520531) 
Efficient Tunnel Detection with Waveform Inversion of Backscattered Surface Waves
(University of Kansas, 20190531)An efficient subsurface imaging method employing backscattered surface waves is developed to detect nearsurface underground elasticwave velocity anomalies, such as tunnels, sinkholes, fractures, faults, and abandoned ... 
An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing
(University of Kansas, 20190531)Moving sharp fronts are an important feature of many mathematical models from physical sciences and cause challenges in numerical computation. In order to obtain accurate solutions, a high resolution of mesh is necessary, ... 
Dynamics of Essentially Unstable Nonlinear Waves
(University of Kansas, 20190531)In this thesis we primarily consider the stability of traveling wave solutions to a modified KuramotoSivashinsky Equation equation modeling nanoscale pattern formation and the St. Venant equations modeling shallow water ... 
On the Existence and Stability of Normalized Ground States of the Kawahara, Fourth Order NLS and the Ostrovsky Equations
(University of Kansas, 20190531)In this dissertation we show the existence and stability of the normalized ground states for the Kawahara, fourth order nonlinear Schrödinger (NLS) and the generalized Ostrovsky equations. One of the starting points in our ... 
Surface and bulk moving mesh methods based on equidistribution and alignment
(University of Kansas, 20190531)In this dissertation, we first present a new functional for variational mesh generation and adaptation that is formulated by combining the equidistribution and alignment conditions into a single condition with only one ... 
Matroid Independence Polytopes and Their Ehrhart Theory
(University of Kansas, 20190531)A \emph{matroid} is a combinatorial structure that provides an abstract and flexible model for dependence relations between elements of a set. One way of studying matroids is via geometry: one associates a polytope to a ... 
Normal and paracompact spaces and their products
(University of Kansas, 19620831) 
Serre's Condition and Depth of StanleyReisner Rings
(University of Kansas, 20181231)The aim of this work is to garner a deeper understanding of the relationship between depth of a ring and connectivity properties of the spectrum of that ring. We examine with particular interest the case where our ring is ... 
Decompositions of Simplicial Complexes
(University of Kansas, 20180831)In this thesis we study the interplay between various combinatorial, algebraic, and topological properties of simplicial complexes. We focus on when these properties imply the existence of decompositions of the face poset. ... 
Analytical studies of standing waves in three NLS models
(University of Kansas, 20180831)In this work, we present analytical studies of standing waves in three NLS models. We first consider the spectral stability of ground states of fourth order semilinear Schrödinger and KleinGordon equations and semilinear ... 
Parameter estimation for stochastic differential equations driven by fractional Brownian motion
(University of Kansas, 20180531)This dissertation systematically considers the inference problem for stochastic differential equations (SDE) driven by fractional Brownian motion. For the volatility parameter and Hurst parameter, the estimators are ... 
Limit distributions for functionals of Gaussian processes
(University of Kansas, 20180531)This thesis is devoted to the study of the convergence in distribution of functionals of Gaussian processes. Most of the problems that we present are addressed by using an approach based on Malliavin calculus techniques. ... 
Stability of Periodic Waves in Nonlocal Dispersive Equations
(University of Kansas, 20180531)In this work consisting of joint projects with my advisor, Dr. Mathew Johnson, we study the existence and stability of periodic waves in equations that possess nonlocal dispersion, i.e. equations in which the dispersion ... 
Regularity of Stochastic Burgers’Type Equations
(University of Kansas, 20180531)In classical partial differential equations (PDEs), it is well known that the solution to Burgers' equation in one spatial dimension with positive viscosity can be solved by the so called HopfCole transformation, which ...