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Publication A General Stochastic Volatility Model on VIX Options(University of Kansas, 2019-12-31) Cui, Yanhao; Soo, Terry; Hu, Yaozhong; Tu, Xuemin; Liu, Zhipeng; Han, JieAbstract 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. It originated from a nested stochastic model based on several known models in the paper [7], stochastic volatility models and the Pricing of VIX Options. To estimate the parameters in these models from the real financial data a commonly used approach is the Generalized Method of Moments of Hansen (1982). We will study the model in more generality and we shall provide a completely different parameter estimation technique using the ergodic theory. Since our equation is more general and new and since our equation is singular in the sense it does not satisfy the global Lipschitz condition, we shall first study the existence, uniqueness and positivity of the solution of the SDE, in which Feller’s test will be used to calculate a criteria of all parameters such that the SDE has a unique and positive weak solution. The positivity property of the solution is crucial, since volatility is always positive. Then, we use the strong large law of numbers theorems given e.g. in [4] to give the region for the parameters to live in order that the model is ergodic. In important condition for the ergodicity is the positive recurrency. We give verifiable condition on the parameters so that process is positive recurrent. This results also provide ways to calculate the invariant distribution (limiting distribution). The next step is to provide a theoretical methodology of parameter estimation. Simulation process will be introduced with giving an example for each case. In the future study, I will work on testing the model using numerical schemes. Keywords: Stochastic volatility model, VIX options, Feller’s test, ergodicity, parameter estimation.Publication Optimal Energy Decay for the Damped Klein-Gordon Equation(University of Kansas, 2019-08-31) Malhi, Satbir Singh; Stanislavova, Milena; Johnson, Mathew; Mantzavinos, Dionyssios; Stefanov, Atanas; Agah, ArvinIn this dissertation we study the long time dynamics of damped Klein-Gordon and damped fractional Klein-Gordon equations using $C_0$- Semigroup theory and its application. The $C_0$-semigroups are used to solve a large class of problems commonly known as evolution equations. Such models arise from delay differential equations and partial differential equations in many disciplines including physics, chemistry, biology, engineering, and economics. Water waves, sound waves and simple harmonic motion of strings are few important models of evolution equations. The Klein-Gordon equation is a relativistic version of the Schr\"odinger equation. It was named after Oskar Klein and Walter Gordon who proposed it to describe quantum particles in the framework of relativity. It describes the motion of spinless composite particles. Indeed, one of the most fundamental questions that should be asked when studying these equations is whether the solution (if it exist) goes to equilibrium (stable) state or behaves erratically as time evolves. Understanding these properties can help determine how robust a system is, as well as provides insight on the characteristics of the corresponding phenomena it is modeling. In the first part we consider a one dimensional damped Klein-Gordon equation on the real line. It is well known fact that if there is no external force (i.e damping) acting in the system, the wave will oscillate forever in time since the energy is conserved in the system. An interesting question to ask is at what rate the energy starts leaving the system when we introduce damping force? This question was intensely studied in the last ten years. In this direction, Burq and Joly have proved that the energy decays at exponential rate if the damping force $\gamma(x)$ satisfies the geometric control condition (GCC) in a sense that there exist $T$, $\epsilon>0$, such that $\int_0^T\gamma(x(t))dt\geq \epsilon$ along every straight line unit speed trajectory. However, GCC does not provide an optimal condition to ensure exponential rate of energy decay. We address this problem in chapter 2 and provide optimal conditions on the damping coefficient $\gamma$ under which the exponential decay holds in one-dimensional setting. In addition, we derive simple to verify necessary and sufficient conditions for such exponential rate of decay. In the second part we relate the energy decay rate for the fractional damped wave equation to the order of its fractional derivative. In fact we prove that the energy decays at a polynomial rate if the order of derivative lies between $00$, such that $\int_0^T\gamma(x(t))dt\geq \epsilon$ along every straight line unit speed trajectory. However, GCC does not provide an optimal condition to ensure exponential rate of energy decay. We address this problem in chapter 2 and provide optimal conditions on the damping coefficient $\gamma$ under which the exponential decay holds in one-dimensional setting. In addition, we derive simple to verify necessary and sufficient conditions for such exponential rate of decay. In the second part we relate the energy decay rate for the fractional damped wave equation to the order of its fractional derivative. In fact we prove that the energy decays at a polynomial rate if the order of derivative lies between $0Publication Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models(University of Kansas, 2019-5-31) Hadadifard, Fazel; Stefanov, Atanas; Stefanov, Atanas; Torres, Rodolfo; Stanislavova, Milena; Johnson, Mathew; Ackley, BrianThrough this dissertation we present the sharp time decay rates for three equations, namely quasi--geostrophic equation (SQG), Boussinesq system (BSQ) and plane wave of general reaction-diffusion models. In addition, in each case, we provide the dominant part of the solution which leads to the long term asymptotic profiles of each model. The first two equations, arising in fluid dynamics, model some aspect of the shallow waters with horizontal and vertical structures. Indeed, quasi--geostrophis equation models the horizontal inertia forces of a flow. As a result of that, atmospheric and oceanographic flows which take place over horizontal length scales, which are very large compare to their vertical length scales, are studied by SQG equation. On the other hand BSQ system models some vertical aspect of the flow, namely the speed, pressure and the temperature of the flow. In coastal engineering, BSQ type equations have a vast application in computer modeling. Lastly, a plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector. In order to study these equations, we made some developments in the "scaling variable" methods, so that it fits over models. In particular, we now have a good understanding of this method when it is applied to the equations with fractional dissipations.Publication A canonical form for the differential equations of curves in n-dimensional space(University of Kansas, 1930-05-31) Smith, Ronald GibsonPublication The characterizations of a class of transformations and of a class of differentiable functions(University of Kansas, 1951-05-31) Moore, Warren KeithPublication Normal determinants and expansions in modified sequences(University of Kansas, 1952-05-31) Larkin, James RichardPublication Efficient Tunnel Detection with Waveform Inversion of Back-scattered Surface Waves(University of Kansas, 2019-05-31) Wang, Yao; Tu, Xuemin; Xu, Hongguo; Tsoflias, Georgios PAn efficient subsurface imaging method employing back-scattered surface waves is developed to detect near-surface underground elastic-wave velocity anomalies, such as tunnels, sinkholes, fractures, faults, and abandoned manmade infrastructures. The back-scattered surface waves are generated by seismic waves impinging on the velocity anomalies and diffracting back toward the source. These wave events contain plentiful information of the subsurface velocity anomalies including spatial location, shape, size, and velocity of the interior medium. Studies have demonstrated that the back-scattered surface waves can be easily distinguished in the frequency-wavenumber (F-k) domain and have less interference by other wave modes. Based on these features, a near-surface velocity anomaly detection method by using waveform inversion of the back-scattered surface waves (BSWI) is proposed. The main objective of this thesis is to review the theoretical background and study the feasibility of the proposed BSWI method. The proposed BSWI method is tested with numerical and real-world examples. First, the numerical example uses the conventional full-waveform inversion (FWI) method as a benchmark to demonstrate the efficiency of BSWI method in detecting shallow velocity anomalies. Then, the BSWI method is tested with field data. In this study, 2D seismic data were acquired over a manmade concrete tunnel located on the main campus of the University of Kansas (KU). Different workflows including FWI method and BSWI method are applied to the acquired data and tested for imaging the known tunnel. The field example demonstrates that BSWI can accurately image the tunnel. Compared with FWI, BSWI is less demanding in data processing. Finally, this thesis concludes that the proposed BSWI method is capable of efficiently detecting a near-surface tunnel with the minimum amount of data processing which lends it as a method suitable for application in the field.Publication An Adaptive Moving Mesh Finite Element Method and Its Application to Mathematical Models from Physical Sciences and Image Processing(University of Kansas, 2019-05-31) Yu, Yufei; Huang, Weizhang; Huang, Weizhang; Keshmiri, Shawn; Liu, Weishi; Tu, Xuemin; Van Vleck, ErikMoving 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, which results in high computational cost if a fixed mesh is used. As a solution to this issue, an adaptive mesh method, which is called the moving mesh partial differential equation (MMPDE) method, is described in this work. The MMPDE method has the advantage of adaptively relocating the mesh points to increase the densities around sharp layers of the solutions, without increasing the mesh size. Moreover, this strategy can generate a nonsingular mesh even on non-convex and non-simply connected domains, given that the initial mesh is nonsingular. The focus of this thesis is on the application of the MMPDE method to mathematical models from physical sciences and image segmentation. In particular, this thesis includes the selection of the regularization parameter for the Ambrosio-Tortorelli functional, a simulation of the contact sets in the evolution of the micro-electro mechanical systems, and a numerical study of the flux selectivity in the Poisson-Nernst-Planck model. Sharp interfaces take place in all these three models, bringing interesting features and rich phenomena to study.Publication Dynamics of Essentially Unstable Nonlinear Waves(University of Kansas, 2019-05-31) Smith, Connor Yoshio; Johnson, Mathew; Stanislavova, Milena; Liu, Weishi; Mantzavinos, Dionyssios; Lamb, Jonathan PIn this thesis we primarily consider the stability of traveling wave solutions to a modified Kuramoto-Sivashinsky Equation equation modeling nanoscale pattern formation and the St. Venant equations modeling shallow water flow down an inclined plane. Numerical evidence suggests that these equations have no unstable spectrum other than λ =0, however they both have unstable essential spectrum. This unstable essential spectrum manifests as a convecting, oscillating perturbation which grows to a certain size independent on the initial perturbation — precluding stability in the regular L^2(R) space. Exponentially weighted spaces are typically used to handle such instabilities, and in Theorem 5.7 we prove asymptotic orbital linear stability in such an exponentially weighted space. We also discuss difficulties with extending this to a nonlinear stability result. In Section 5.5 we discuss another way of obtaining stability, through ad-hoc periodic wave trains. Chapter 6 concerns the general problem of creating a spectral projection to project away unstable essential spectrum. We consider this problem in the context of spatially periodic-coefficient PDE by proposing a candidate spectral projection defined via the Bloch transform and showing that initial perturbations which activate a sufficiently unstable part of the essential spectrum lead to solutions which are not Lyapunov stable. We also extend these results to dissipative systems of conservation laws. Additional chapters of interest are Chapter 3 where we address finding the spectrum and Chapter 4 where we discuss the numerics which lead to many of the figures in this thesis.Publication On the Existence and Stability of Normalized Ground States of the Kawahara, Fourth Order NLS and the Ostrovsky Equations(University of Kansas, 2019-05-31) Posukhovskyi, Iurii; Stefanov, Atanas; Stefanov, Atanas; Johnson, Mathew; Kong, Kyoungchul; Mantzavinos, Dionyssios; Torres, RodolfoIn 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 investigation were numerical stability results by S. Levandosky in [32], [31] which agree with our rigorous stability results. We show existence of the waves using variational techniques together with the concentration compactness argument. On the level of construction, we encounter certain obstacles in the form of new Gagliardo–Nirenberg–Sobolev type inequalities, which impose restrictions on the parameter space. We show stability utilizing spectral theory developed in the recent work by Z.Lin and C.Zeng in [35]. For the Kawahara model, our results provide a significant extension in the parameter space of the current rigorous results. In fact, our results rigorously establish the spectral stability for all acceptable values of the parameters. For the fourth order NLS models, we improve upon recent results on stability of, very special, explicit solutions in the one dimensional case. Our multidimensional results for the fourth order NLS equations seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second order derivatives vs pure Laplacian) has implications on the range of the existence and stability of the normalized waves. For the Ostrovsky equation, we show that all normalized waves we construct are spectrally stable. We also establish decay rates for the waves, extending the results in the paper by P. Zhang and Y. Liu [51].Publication Surface and bulk moving mesh methods based on equidistribution and alignment(University of Kansas, 2019-05-31) Kolasinski, Avary Justice; Huang, Weizhang; Gavosto, Estela; Miedlar, Agnieszka; Shontz, Suzanne; Van Vleck, ErikIn 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 dimensionless parameter. The functional is shown to be coercive which, when employed with the moving mesh partial differential equation method, allows various theoretical properties to be proved. Numerical examples for bulk meshes demonstrate that the new functional performs comparably to a similar existing functional that is known to work well but contains an additional parameter. Variational mesh adaptation for bulk meshes has been well developed however, surface moving mesh methods are limited. Here, we present a surface moving mesh method for general surfaces with or without explicit parameterization. The development starts with formulating the equidistribution and alignment conditions for surface meshes from which, we establish a meshing energy functional. The moving mesh equation is then defined as the gradient system of the energy functional, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. A selection of two-dimensional and three-dimensional examples are presented.Publication Matroid Independence Polytopes and Their Ehrhart Theory(University of Kansas, 2019-05-31) Duna, Chad Kenneth; Martin, Jeremy L; Katz, Dan; Bayer, Margaret; Witt, Emily; Nutting, EileenA \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 matroid, then uses both combinatorics and geometry to understand the polytope and thereby the original matroid. By a \emph{polytope}, we mean a bounded convex set in Euclidean space $\mathbb{R}^n$ defined by a finite list of linear equations and inequalities, or equivalently as the convex hull of a finite set of points. The best-known polytope associated with a matroid $M$ is its \emph{base polytope} $P(M)$, first introduced by Gel'fand, Goresky, Macpherson and Serganova in 1987~\cite{GGMS}. This dissertation focuses on a closely related construction, the \emph{independence polytope} $Q(M)$, whose combinatorics is much less well understood. Both $P(M)$ and $Q(M)$ are defined as convex hulls of points corresponding to the bases or independence sets, respectively; defining equations and inequalities were given for $P(M)$ by Feichtner and Sturmfels~\cite{Feichtner_Sturmfels} in terms of the ``flacets'' of $M$, and for $Q(M)$ by Schrijver~\cite{Schrijver_B}. One significant difference between the two constructions is that matroid basis polytopes are \emph{generalized permutahedra} as introduced by Postnikov \cite{Beyond}, but independence polytopes do not \emph{a priori} share this structure, so that fewer tools are available in their study. One of the fundamental questions about a polytope is to determine its combinatorial structure as a cell complex: what are its faces of each dimension and which faces contain others? In general it is a difficult problem to extract this combinatorial structure from a geometric description. For matroid base polytopes, the edges (one-dimensional faces) have a simple combinatorial descriptions in terms of the defining matroid, but faces of higher dimension are not understood in general. In Chapter~2 we give an exact combinatorial and geometric description of all the one- and two-dimensional faces of a matroid independence polytope (Theorems~\ref{theorem: Edge Theorem} and ~\ref{theorem: 2 Faces}). One consequence (Proposition~\ref{prop: is a gen perm}) is that matroid independence polytopes can be transformed into generalized permutahedra with no loss of combinatorial structure (at the cost of making the geometry slightly more complicated), which may be of future use. In Chapter~3 we consider polytopes arising from \emph{shifted matroids}, which were first studied by Klivans~\cite{Klivans_Thesis, Klivans_paper}. We describe additional combinatorial structures in shifted matroids, including their circuits, inseparable flats, and flacets, leading to an extremely concrete description of the defining equations and inequalities for both the base and independence polytopes (Theorem~\ref{theorem: pièce de résistance}). As a side note, we observe that shifted matroids are in fact \emph{positroids} in the sense of Postnikov~\cite{Positroid_Postnikov}, although we do not pursue this point of view further. Chapter~4 considers an even more special class of matroids, the \emph{uniform matroids} $U(r,n)$, whose independence polytopes $TC(r,n)=Q_U(r,n)$ are hypercubes in $\Rr^n$ truncated at ``height''~$r$. These polytopes are strongly enough constrained that we can study them from the point of view of Ehrhart theory. For a polytope $P$ whose vertices have integer coordinates, the function $i(P,t) = |tP\cap\Zz^n|$ (that is, the number of integer points in the $t^{th}$ dilate) is a polynomial in $t$ \cite{OG_Ehrhart}, called the \emph{Ehrhart polynomial}. We give two purely combinatorial formulas for the Ehrhart polynomial of $TC(r,n)$, one a reasonably simple summation formula (Theorem~\ref{thm: Ehrhart_Polynomial_Truncated_Cube}) and one a cruder recursive version (Theorem \ref{theorem: Gross_Formula}) that was nonetheless useful in conjecturing and proving the ``nicer'' Theorem~\ref{thm: Ehrhart_Polynomial_Truncated_Cube}. We observe that another fundamental Ehrhart-theoretic invariant, the \emph{$h^*$-polynomial} of $TC(r,n)$, can easily be obtained from work of Li~\cite{Nan_Li} on closely related polytopes called \emph{hyperslabs}. Having computed these Ehrhart polynomials, we consider the location of their complex roots. The integer roots of $i(Q_M,t)$ can be determined exactly even for arbitrary matroids (Theorem \ref{thm: Integer_Roots}), and extensive experimentation using Sage leads us to the conjecture that for all $r$ and $n$, all roots of $TC(r,n)$ have negative real parts. We prove this conjecture for the case $r=2$ (Theorem \ref{thm: Conj_r=2}), where the algebra is manageable, and present Sage data for other values in the form of plots at the end of Chapter~4.Publication Normal and paracompact spaces and their products(University of Kansas, 1962-08-31) Joseph, Albert F.Publication Serre's Condition and Depth of Stanley-Reisner Rings(University of Kansas, 2018-12-31) Holmes, Brent; Dao, Hailong; Witt, Emily; Katz, Daniel; Martin, Jeremy; Agah, ArvinThe 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 a Stanley- Reisner ring. In this circumstance, we consider the simplicial complex that corresponds to the spectrum of R. We examine properties of simplicial complexes whose Stanley-Reisner rings satisfy depth conditions such as Cohen-Macaulay and Serre's condition (S_l). We leverage these properties to use algebraic tools to examine combinatorial problems. For example, the gluing lemma in (Hol18) allows us to construct bounds on the diameter of a class of graphs acting as a generalization of the 1-skeleton of polytopes. Throughout, we give special consideration to Serre's condition (S_l). We create a generalized Serre's condition (S_l^j) and prove equivalent homological, topological, and combinatorial properties for this condition. We generalize many well-known results pertaining to (S_l) to apply to (S_l^j). This work also explores a generalization of the nerve complex and considers the correlation between the homologies of the nerve complex of a Stanley-Reisner ring and depth properties of that ring. Finally we explore rank selection theorems for simplicial complexes. We prove many results on depth properties of simplicial complexes. In particular, we prove that rank selected subcomplexes of balanced (S_l) simplicial complexes retain (S_l). The primary focus of this work is on Stanley-Reisner rings, however, other commutative, Noetherian rings are also considered.Publication Decompositions of Simplicial Complexes(University of Kansas, 2018-08-31) Goeckner, Bennet; Martin, Jeremy L; Bayer, Margaret; Talata, Zsolt; Witt, Emily; Slusky, DavidIn 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. In Chapter 2, we present a counterexample to Stanley's partitionability conjecture, we give a characterization of the h-vectors of Cohen-Macaulay relative complexes, and we construct a family of disconnected partitionable complexes. In Chapter 3, we introduce colorated cohomology, which aims to combine the theories of color shifting and iterated homology. Colorated cohomology gives rise to certain decompositions of balanced complexes that preserve the balanced structure. We give conditions that would guarantee the existence of a weaker form of Stanley's partitionability conjecture for balanced Cohen-Macaulay complexes. We consider Stanley's conjecture on k-fold acyclic complexes in Chapter 4, and we show that a relaxation of this conjecture holds in general. We also show that the conjecture holds in the case when k is the dimension of a given complex, and we present a framework that may lead to a counterexample to the original version of this conjecture.Publication Analytical studies of standing waves in three NLS models(University of Kansas, 2018-08-31) Feng, Wen; Stanislavova, Milena; Stanislavova, Milena; Chen, Geng; Johnson, Mat; Kong, Man; Stefanov, AtanasIn 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 semi-linear Schrödinger and Klein-Gordon equations and semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. We use Hamiltonian index counting theory, together with the information from a variational construction to develop sharp conditions for spectral stability for these waves. The second case is about the existence and the stability of the vortices for the NLS in higher dimensions. We extend the existence and stability results of Mizumachi from two-space dimensions to $n$ space dimensions. Finally, the third equation we consider is a nonlocal NLS which comes from modeling nonlinear waves in Parity-time symmetric systems. Here again, we investigate the spectral stability of standing waves of its $\mathcal{PT}$ symmetric solutions.Publication Parameter estimation for stochastic differential equations driven by fractional Brownian motion(University of Kansas, 2018-05-31) ZHOU, HONGJUAN; Nualart, David; Hu, Yaozhong; Liu, Zhipeng; Soo, Terry; Zhang, JianboThis 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 constructed using iterated power variations. To prove the strong consistency and the central limit thoerems of the estimators, the asymptotics of the power variatons are studied, which include the strong consistency, central limit theorem, and the convergence rate for the iterated power variations of the Skorohod integrals with respect to fractional Brownian motion. The iterated logarithm law of the power variations of fractional Brownian motion is proved. The joint convergence along different subsequence of power variations of Skorohod integrals is also studied in order to derive the central limit theorem for the estimators. Another important topic considered in this dissertation is the estimation of drift parameters of the SDEs. A least squares estimator (LSE) is proposed and the strong consistency is proved for the fractional Ornstein-Uhlenbeck process that is the solution to the linear SDE. The fourth moment theorem is applied to obtain the central limit theorems. Then the LSE is considered for the drift parameter of the multi-dimensional nonlinear SDE. While proving the strong consistency of LSE, the regularity structure of the SDE’s solution is explored and the maximal inequality for the Skorohod integrals is derived. The main tools used in this research are Malliavin calculus and some Gaussian analysis elements.Publication Limit distributions for functionals of Gaussian processes(University of Kansas, 2018-05-31) Jaramillo, Arturo; Nualart, David; Feng, Jin; Liu, Zhipeng; Soo, Terry; Zhang, JianboThis 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. Our main contributions are the following: First, we study the asymptotic law of the approximate derivative of the self-intersection local time (SILT) in $[0,T]$ for the fractional Brownian motion. In order to do this, we describe the asymptotic behavior of the associated chaotic components and show that the first chaos approximates the SILT in $L^2$. Secondly, we examine the asymptotic law of the approximate self-intersection local time process for the fractional Brownian motion. We achieve this in two steps: the first part consists on proving the convergence of the finite dimensional distributions by using the `multidimensional fourth moment theorem'. The second part consists on proving the tightness property, for which we follow an approach based on Malliavin calculus techniques. The third problem consists on proving a non-central limit theorem for the process of weak symmetric Riemann sums for a wide variety of self-similar Gaussian processes. We address this problem by using the so-called small blocks-big blocks methodology and a central limit theorem for the power variations of self-similar Gaussian processes. Finally, we address the problem of determining conditions under which the eigenvalues of an Hermitian matrix-valued Gaussian process collide with positive probability.Publication Stability of Periodic Waves in Nonlocal Dispersive Equations(University of Kansas, 2018-05-31) Claassen, Kyle Matthew; Johnson, Mathew A; Mantzavinos, Dionyssios; Miedlar, Agnieszka; Stanislavova, Milena; Jackson, TimothyIn 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 relation between the temporal frequency, omega, and wavenumber, k, of a plane wave is not polynomial in ik. In models that involve only classical derivative operators (known as local equations), the behavior of the system at a point is influenced solely by the behavior in an arbitrarily small neighborhood. In contrast, equations involving nonlocal operators incorporate long-range interactions as well. Such operators appear in numerous applications, including water wave theory and mathematical biology. Specifically, we establish the existence and nonlinear stability of a special class of periodic bound state solutions of the Fractional Nonlinear Schrodinger Equation, where the nonlocality of the fractional Laplacian presents formidable analytical challenges and elicits the development of functional-analytic tools to complement the absence of more-understood techniques commonly used to analyze local equations. Further, we use numerical methods to survey the existence and spectral stability of small- and large-amplitude periodic wavetrains in Bidirectional Whitham water wave models, which implement the exact (nonlocal) dispersion relation of the incompressible Euler equations and are thus expected to better capture high-frequency phenomena than the unidirectional Whitham and Korteweg-de Vries (KdV) equations.Publication Regularity of Stochastic Burgers’-Type Equations(University of Kansas, 2018-05-31) Lewis, Peter; Nualart, David; Alexander, Perry; Feng, Jin; Johnson, Mat; Soo, TerryIn 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 Hopf-Cole transformation, which linearizes the PDE. In particular, this converts Burgers' equation to the linear heat equation, which can be solved explicitly. On the other hand, the Feynman-Kac formula is a tool that can be used to solve the heat equation probabilistically. An interesting and perhaps surprising result which we prove is that one can still make sense of these approaches to Burgers' equation in the presence of space-time white noise, which is very rough. After proving that a suitable Feynman-Kac representation solves stochastic Burgers' equation under a Hopf-Cole transformation, we study some regularity properties of this solution. In particular, we prove moment estimates and Holder continuity, which can be thought of as how ``big'' the solution gets in time and space, and how ``rough'' this solution can be. From this, we then obtain sub-exponential moments and bounds on the tails of the probability distribution for the solution. Prior to this work, no results about any kinds of moment estimates or tails of distributions for stochastic Burgers'-type equations had been established. Furthermore, only one publication on Burgers' equation contains a discussion of Holder regularity. Given the solution to a stochastic partial differential equation (SPDE), it is natural to ask whether this stochastic process has a well-behaved probability law. For example, does the solution have a smooth probability density function or just an absolutely continuous one? Using some powerful tools from Malliavin calculus, we answer this question for stochastic Burgers' equation with our Hopf-Cole solution. Finally, we study regularity of the probability law of the solution to a more general class of semilinear SPDEs which contain Burgers' equation as an example. These results take a less tangible approach since there is no explicit representation for solutions to these equations. However, as we will see, there are some clever techniques and interesting results that can be used to establish such properties. For example, we prove a comparison theorem for this class of SPDEs which, interestingly enough, will be instrumental in obtaining regularity of the probability density function of the solution at fixed points in time and space. The projects in this thesis are joint work of the author and David Nualart.