Physical Sciences and Mathematics Commons^™

A Complete Characterization Of Near Outer-Planar Graphs, Tanya Allen Lueder Genannt Luehr

Doctoral Dissertations

A graph is outer-planar (OP) if it has a plane embedding in which all of the vertices lie on the boundary of the outer face. A graph is near outer-planar (NOP) if it is edgeless or has an edge whose deletion results in an outer-planar graph. An edge of a non outer-planar graph whose removal results in an outer-planar graph is a vulnerable edge. This dissertation focuses on near outer-planar (NOP) graphs. We describe the class of all such graphs in terms of a finite list of excluded graphs, in a manner similar to the well-known Kuratowski Theorem for planar …

Global Well-Posedness And Scattering For The Defocusing Quintic Nonlinear Schrödinger Equation In Two Dimensions, Xueying Yu

Doctoral Dissertations

In this thesis we consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in two dimensions. We take general data in the critical homogeneous Sobolev space dot H^1/2. We show that if a solution remains bounded in dot H^1/2 in its maximal time interval of existence, then the time interval is infinite and the solution scatters.

Well-Posedness For The Cubic Nonlinear Schrödinger Equations On Tori, Haitian Yue

Doctoral Dissertations

This thesis studies the cubic nonlinear Sch\"rodinger equation (NLS) on tori both from the deterministic and probabilistic viewpoints. In Part I of this thesis, we prove global-in-time well-posedness of the Cauchy initial value problem for the defocusing cubic NLS on 4-dimensional tori and with initial data in the energy-critical space $H^1$. Furthermore, in the focusing case we prove that if a maximal-lifespan solution of the cubic NLS \, $u: I\times\mathbb{T}^4\to \mathbb{C}$\, satisfies $\sup_{t\in I}\|u(t)\|_{\dot{H}^1(\mathbb{T}^4)}

A Relation Between Mirkovic-Vilonen Cycles And Modules Over Preprojective Algebra Of Dynkin Quiver Of Type Ade, Zhijie Dong

Doctoral Dissertations

The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian to a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. The relation between this conjecture and the reduceness …

On The Growth Of Sobolev Norms For The Nonlinear Schrödinger Equation On Tori And Boundary Unique Continuation For Elliptic Pde, Michael Boratko

Doctoral Dissertations

This dissertation is composed of two parts. The first part applies techniques from Harmonic and nonlinear Fourier Analysis to the nonlinear Schrödinger equation, and therefore tools from the study of Dispersive Partial Differential Equations (PDEs) will also be employed. The dissertation will apply the $\ell^2$ decoupling conjecture, proved recently by Bourgain and Demeter, to prove polynomial bounds on the growth of Sobolev norms of solutions to polynomial nonlinear Schrödinger equations. The first bound which is obtained applies to the cubic nonlinear Schrödinger equation and yields an improved bound for irrational tori in dimensions 2 and 3. For the 4 dimensional …

Asymptotic Behavior Of The Random Logistic Model And Of Parallel Bayesian Logspline Density Estimators, Konstandinos Kotsiopoulos

Doctoral Dissertations

This dissertation is comprised of two separate projects. The first concerns a Markov chain called the Random Logistic Model. For r in (0,4] and x in [0,1] the logistic map f_r(x) = rx(1 - x) defines, for positive integer t, the dynamical system x_r(t + 1) = f(x_r(t)) on [0,1], where x_r(1) = x. The interplay between this dynamical system and the Markov chain x_r,N(t) defined by perturbing the logistic map by truncated Gaussian noise scaled by N^-1/2, where N -> infinity, is studied. A natural question is …

Framed Sheaves On A Quadric Surface, Nguyen Thuc Huy Le

Doctoral Dissertations

We study framed sheaves on a smooth quadric surface and conjecture that the moduli of such framed sheaves admits a twistor deformation similar to one studied in the paper "Brill-Noether duality for moduli spaces of sheaves on K3 surfaces" by Markman.

Equidimensional Adic Eigenvarieties For Groups With Discrete Series, Daniel Robert Gulotta '03

Doctoral Dissertations

We extend Urban's construction of eigenvarieties for reductive groups G such that G(R) has discrete series to include characteristic p points at the boundary of weight space. In order to perform this construction, we define a notion of "locally analytic" functions and distributions on a locally Q_p-analytic manifold taking values in a complete Tate Z_p-algebra in which p is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on p-adic Lie groups given by Johansson and Newton.

On Modeling Quantities For Insurer Solvency Against Catastrophe Under Some Markovian Assumptions, Daniel Jefferson Geiger

Doctoral Dissertations

"Insurance companies sometimes face catastrophic losses, yet they must remain solvent enough to meet the legal obligation of covering all claims. Catastrophes can result in large damages to the policyholders, causing the arrival of numerous claims to insurance companies at once. Furthermore, the severity of an event could impact the time until the next occurrence. An insurer needs certain levels of startup capital to meet all claims, and then must have adequate reserves on a continual basis, even more so when catastrophes occur. This work examines two facets of these matters: for an infinite time horizon, we extend and develop …

Incremental Proper Orthogonal Decomposition For Pde Simulation Data: Algorithms And Analysis, Hiba Fareed

Doctoral Dissertations

"We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. We introduce an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD). The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. We also modify the algorithm to initialize and incrementally update both the SVDand an error bound during the …

Cox-Type Model Validation With Recurrent Event Data, Muna Mohamed Hammuda

Doctoral Dissertations

"Recurrent event data occurs in many disciplines such as actuarial science, biomedical studies, sociology, and environment to name a few. It is therefore important to develop models that describe the dynamic evolution of the event occurrences. One major problem of interest to researchers with these types of data is models for the distribution function of the time between events occurrences, especially in the presence of covariates that play a major role in having a better understanding of time to events.

This work pertains to statistical inference of the regression parameter and the baseline hazard function in a Cox-type model for …

Hdg Methods For Dirichlet Boundary Control Of Pdes, Yangwen Zhang

Doctoral Dissertations

"We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems for PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. In this thesis, we use an existing HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control in the high regularity case. We also propose a new HDG method to approximate the solution of a Dirichlet boundary …