Physical Sciences and Mathematics Commons^™

Oscillation Of Quenched Slowdown Asymptotics Of Random Walks In Random Environment In Z, Sung Won Ahn

Open Access Dissertations

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed limn→∞ (Xn/) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P ω(Xn < xn) with x∈ (0,υα) decay approximately like exp{- n1-1/s} for a deterministic s > 1. More precisely, they showed that n -γ log Pω(Xn < xn) converges to 0 or -∞ depending on whether γ > 1 - 1/s or γ < 1 - 1/ s. In this paper, …

Connecting Models Of Configuration Spaces: From Double Loops To Strings, Jason M. Lucas

Open Access Dissertations

Foundational to the subject of operad theory is the notion of an En operad, that is, an operad that is quasi-isomorphic to the operad of little n-cubes Cn. They are central to the study of iterated loop spaces, and the specific case of n = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two E 2 operads, namely the little 2-cubes operad C 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of C2, which we name the operad of tethered …

Rees Algebras And Iterated Jacobian Duals, Vivek Mukundan

Open Access Dissertations

Consider the rational map Ψ : [Special characters omitted.] where the fi's are homogeneous forms of the same degree in the homogeneous coordinate ring R = k[ x1,…,xd] of [Special characters omitted.]. Assume that I = (f 1,…,fm) is a height 2 perfect ideal in the polynomial ring R. In this context, the coordinate ring of the graph of Ψ is the Rees algebra of I and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when I is almost …

Martingales, Singular Integrals, And Fourier Multipliers, Michael A. Perlmutter

Open Access Dissertations

Many probabilistic constructions have been created to study the Lp-boundedness, 1 < p < ∞, of singular integrals and Fourier multipliers. We will use a combination of analytic and probabilistic methods to study analytic properties of these constructions and obtain results which cannot be obtained using probability alone.

In particular, we will show that a large class of operators, including many that are obtained as the projection of martingale transforms with respect to the background radiation process of Gundy and Varapolous or with respect to space-time Brownian motion, satisfy the assumptions of Calderón-Zygmund theory and therefore boundedly map L1 to weak- L1.

We will also use a method of rotations to study the L p boundedness, 1 < p < ∞, of Fourier multipliers which are obtained as the projections of martingale transforms with respect to symmetric α-stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2 and therefore allows us to obtain a larger class of multipliers, indexed by a parameter, 0 < r < ∞, which are bounded on L p. As in the case of the multipliers which arise as the projection of martingale …

Extreme-Strike And Small-Time Asymptotics For Gaussian Stochastic Volatility Models, Xin Zhang

Open Access Dissertations

Asymptotic behavior of implied volatility is of our interest in this dissertation. For extreme strike, we consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first …

Applications Of The Homotopy Analysis Method To Optimal Control Problems, Shubham Singh

Open Access Theses

Traditionally, trajectory optimization for aerospace applications has been performed using either direct or indirect methods. Indirect methods produce highly accurate solutions but suer from a small convergence region, requiring initial guesses close to the optimal solution. In past two decades, a new series of analytical approximation methods have been used for solving systems of dierential equations and boundary value problems.

The Homotopy Analysis Method (HAM) is one such method which has been used to solve typical boundary value problems in nance, science, and engineering. In this investigation, a methodology is created to solve indirect trajectory optimization problems using the Homotopy …

Maximum Empirical Likelihood Estimation In U-Statistics Based General Estimating Equations, Lingnan Li

Open Access Dissertations

In the first part of this thesis, we study maximum empirical likelihood estimates (MELE's) in U-statistics based general estimating equations (UGEE's). Our technical maneuver is the jackknife empirical likelihood (JEL) approach. We give the local uniform asymptotic normality condition for the log-JEL for UGEE's. We derive the estimating equations for finding MELE's and provide their asymptotic normality. We obtain easy MELE's which have less computational burden than the usual MELE's and can be easily implemented using existing software. We investigate the use of side information of the data to improve efficiency. We exhibit that the MELE's are fully efficient, and …

Mathematical Models Of Ebola Virus Disease And Vaccine Preventable Diseases, Yinqiang Zheng

Open Access Dissertations

This thesis focuses on applying mathematical models to studies on the transmission dynamics and control interventions of infectious diseases such as Ebola virus disease and vaccine preventable diseases.

Many models in studies of Ebola transmission are based on the model by Legrand et al. (2007). However, there are potential issues with the Legrand model. First, the model was originally formulated in a complex form, leading to confusion and hindering its uses in practice. To overcome the difficulty, the Legrand model is reformulated in a much simpler but equivalent form in this thesis. The reformulated model also provides an intuitive understanding …

Homological Properties Of Determinantal Arrangements, Arnold H. Yim

Open Access Dissertations

We study a certain family of hypersurface arrangements known as determinantal arrangements. Determinantal arrangements are a union of varieties defined by minors of a matrix of indeterminates. In particular, we investigate determinantal arrangements using the 2-minors of a 2 × n generic matrix (which can be thought of as natural extensions of braid arrangements), and prove certain statements about their freeness. We also study the topology of these objects. We construct a fibration for the complement of free determinantal arrangements, and use this fibration to prove statements about their homotopy groups. Furthermore, we show that the Poincaré polynomial of the …

Kernels Of Adjoints Of Composition Operators On Hilbert Spaces Of Analytic Functions, Brittney Rachele Miller

Open Access Dissertations

This thesis contains a collection of results in the study of the adjoint of a composition operator and its kernel in weighted Hardy spaces, in particular, the classical Hardy, Bergman, and Dirichlet spaces. In 2006, Cowen and Gallardo-Gutiérrez laid the groundwork for an explicit formula for the adjoint of a composition operator with rational symbol acting on the Hardy space, and in 2008, Hammond, Moorhouse, and Robbins established such a formula. In 2014, Goshabulaghi and Vaezi obtained analogous formulas for the adjoint of a composition operator in the Bergman and Dirichlet spaces. While it is known that the kernel of …

Monotonicity Formulas For Diffusion Operators On Manifolds And Carnot Groups, Heat Kernel Asymptotics And Wiener's Criterion On Heisenberg-Type Groups, Kevin L. Rotz

Open Access Dissertations

The contents of this thesis are an assortment of results in analysis and subRiemannian geometry, with a special focus on the Heisenberg group Hn, Heisenbergtype (H-type) groups, and Carnot groups.

As we wish for this thesis to be relatively self-contained, the main definitions and background are covered in Chapter 1. This includes basic information about Carnot groups, Hn, H-type groups, diffusion operators, and the curvature dimension inequality.

Chapter 2 incorporates excerpts from a paper by N. Garofalo and the author, [42]. In it, we propose a generalization of Almgren’s frequency function N : (0, 1) → R for solutions to …

Finite Dimensional Approximations And Deformations Of Group C*-Algebras, Andrew James Schneider

Open Access Dissertations

Quasidiagonality is a finite-dimensional approximation property of a C*-algebra which indicates that it has matricial approximations that capture the structure of the C*-algebra. We investigate when C*-algebras associated to discrete groups have such a property with particular emphasis on finding obstructions. In particular, we point out that groups with Kazhdan's Property (T) and only finitely many unitary equivalence classes of finite dimensional representations do not produce quasidiagonal C*-algebras. We then observe and note interactions with Property (T) and other approximation properties.

Property (QH) is a related but stronger approximation property with deep connections to E-Theory and …

Regularity Of Solutions And The Free Boundary For A Class Of Bernoulli-Type Parabolic Free Boundary Problems With Variable Coefficients, Thomas H. Backing

Open Access Dissertations

In this work the regularity of solutions and of the free boundary for a type of parabolic free boundary problem with variable coefficients is proved. After introducing the problem and its history in the introduction, we proceed in Chapter 2 to prove the optimal Lipschitz regularity of viscosity solutions under the main assumption that the free boundary is Lipschitz. In Chapter 3, we prove that Lipschitz free boundaries possess a classical normal in both space and time at each point and that this normal varies with a Hölder modulus of continuity. As a consequence, the viscosity solution is in fact …

Rank Constrained Homotopies Of Matrices And The Blackadar-Handelman Conjectures On C*-Algebras, Kaushika De Silva

Open Access Dissertations

Rank constrained homotopies of matrices:

For any n ≥ k ≥ l ∈ N, let S( n,k,l) be the set of all non-negative definite matrices a ∈ Mn(C) with l ≤ rank a ≤ k. We investigate homotopy equivalence of continuous maps from a compact Hausdorff space X into sets of the form S(n,k,l). From [37] it is known that for any n, if 4dim X ≤ k-l where dim X denote the covering dimension of X, then there is exactly one homotopy class of maps from X into S …

Ergodic Properties Of Countable Extensions, Samuel Joshua Roth

Open Access Dissertations

First, we study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to an interval map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give examples, both Markov and non-Markov, for which the criterion is violated. ^ Next, we establish a criterion for the existence of a constant slope map on the extended real line conjugate to a given countably piecewise monotone interval map. We require the given interval map to be continuous, Markov, …

Heat Trace And Heat Content Asymptotics For Schrodinger Operators Of Stable Processes/Fractional Laplacians, Luis Guillermo Acuna Valverde

Open Access Dissertations

Let V be a bounded and integrable potential over R^d and 0 < α ≤ 2. We show the existence of an asymptotic expansion by means of Fourier Transform techniques and probabilistic methods for the following quantities [special characters omitted] and [special characters omitted] as t ↓ 0. These quantities are called the heat trace and heat content in R^d with respect to V, respectively. Here, p((α)/ t)(x, y) and p( ^H_V/t)(x, y) denote, respectively, the heat kernels of the heat semigroups with infinitesimal generators given by (-Δ)(α/2) and H_V = (-Δ)(α/2) + V. The former operator is known as the fractional Laplacian whereas the latter one is known as the fractional Schrödinger Operator. ^ The study …

Orderability And Rigidity In Contact Geometry, Peter Weigel

Open Access Dissertations

We study the existence of positive loops of contactomorphisms on a Liouville-fillable contact manifold (&Sgr;, ξ = ker(α)). Previous results (see [1]) show that a large class of Liouville-fillable contact manifolds admit contractible positive loops. In contrast, we show that for any Liouville-fillable (&Sgr;, α) with dim(&Sgr;) ≥ 7, there exists a Liouville-fillable contact structure ξ' on &Sgr; which admits no positive loop at all. Further, ξ' can be chosen to agree with ξ' on the complement of a Darboux ball. We then define a relative version of orderability for a Legendrian submanifold, and discuss the relationship between the two …

Permutohedra, Configuration Spaces And Spineless Cacti, Yongheng Zhang

Open Access Dissertations

It has been known that the configuration space F(R², n) of n distinct ordered points in R² deformation retracts to a regular CW complex with n!permutohedra P_n as the top dimensional cells. In this paper, we show that there exists a similar but different permutohedral structure of the spaceCact(n) of spineless cacti with n lobes. Based on these structures, direct homotopy equivalences between F (R², n) and Cact(n) are then given. It is well known that the little 2-discs space D₂(n) is homotopy equivalent toF(R², n). …

Uncertainty Quantification And Calibration Of Physical Models, Xian He

Open Access Dissertations

An ecosystem model is a representation of a real complex ecological system, and is usually described by sophisticated mathematical models. Terrestrial Ecosystem Model (TEM) is one of the ecosystem models, that describes the dynamics of car- bon, nitrogen, water and other vegetation related variables. There are uncertainties in the TEM which are attributed to inaccurate input data, insufficient knowledge of the parameters, inherent randomness and simplification of the physical model. Quantification of uncertainty of such an ecosystem model is computationally very heavy. Bayesian calibration method has been used as an efficient way to calibrate and quantify uncertainties of the computer …

Parallel Symmetric Eigenvalue Problem Solvers, Alicia Marie Klinvex

Open Access Dissertations

Sparse symmetric eigenvalue problems arise in many computational science and engineering applications: in structural mechanics, nanoelectronics, and spectral reordering, for example. Often, the large size of these problems requires the development of eigensolvers that scale well on parallel computing platforms. In this dissertation, we describe two such eigensolvers, TraceMin and TraceMin-Davidson. These methods are different from many other eigensolvers in that they do not require accurate linear solves to be performed at each iteration in order to find the smallest eigenvalues and their associated eigenvectors. After introducing these closely related eigensolvers, we discuss alternative methods for solving the saddle point …

Investigating Synergy: Mathematical Models For The Coupled Dynamics Of Hiv And Hsv-2 And Other Endemic Diseases, Christina M Alvey

Open Access Dissertations

This dissertation presents epidemiological models that investigate synergy: synergy between HIV and HSV-2 or between humans and mosquitoes in a malaria study. Each of the three coupled disease models addresses different epidemiological questions with regard to gender or disease structure in the context of sexually-transmitted diseases (STDs), while the malaria model focuses on age-structure of the human population. ^ Mounting evidence indicates that HSV-2 infection may increase susceptibility to HIV infection and that co-infection may increase infectiousness. Accordingly, antiviral treatment of people with HSV-2 may mitigate the incidence of HIV in populations where both pathogens occur. To better understand the …

Applications Of Microlocal Analysis To Some Hyperbolic Inverse Problems, Andrew J Homan

Open Access Dissertations

This thesis compiles my work on three inverse problems: ultrasound recovery in thermoacoustic tomography, cancellation of singularities in synthetic aperture radar, and the injectivity and stability of some generalized Radon transforms. Each problem is approached using microlocal methods. In the context of thermoacoustic tomography under the damped wave equation, I show uniqueness and stability of the problem with complete data, provide a reconstruction algorithm for small attenuation with complete data, and obtain stability estimates for visible singularities with partial data. The chapter on synthetic aperture radar constructs microlocally several infinite-dimensional families of ground reflectivity functions which appear microlocally regular when …

Functional Inequalities And The Curvature Dimension Inequality On Totally Geodesic Foliations, Bumsik Kim

Open Access Dissertations

We discover following analytic / geometric properties on Riemannian foliations with bundle-like metric and totally geodesic leaves, or shortly, totally geodesic foliations. Under a certain curvature condition, we obtain (1) Sobolev-isoperimetric inequalities, global Poincar\'e inqualities, and a lower bound for Cheeger's isoperimetric constant, (2) Poincar\'e inequalities on balls and uniqueness of positive(or $L^p,p\geq 1$) solutions for the subelliptic heat equation, (3) A lower bound for the first non-zero eigenvalue of sub-Laplacians (Lichnerowicz theorem), and Obata's sphere theorem. In this context, the curvature condition is a sub-Riemannian analogue of lower bounds for Ricci curvature tensor. Earlier, it is given by Baudoin-Garofalo's …

A Pure-Jump Market-Making Model For High-Frequency Trading, Chi Wai Law

Open Access Dissertations

We propose a new market-making model which incorporates a number of realistic features relevant for high-frequency trading. In particular, we model the dependency structure of prices and order arrivals with novel self- and cross-exciting point processes. Furthermore, instead of assuming the bid and ask prices can be adjusted continuously by the market maker, we formulate the market maker's decisions as an optimal switching problem. Moreover, the risk of overtrading has been taken into consideration by allowing each order to have different size, and the market maker can make use of market orders, which are treated as impulse control, to get …

G-Frobenius Manifolds, Byeongho Lee

Open Access Dissertations

The goal of this dissertation is to introduce the notion of G-Frobenius manifolds for any finite group G. This work is motivated by the fact that any G-Frobenius algebra yields an ordinary Frobenius algebra by taking its G-invariants. We generalize this on the level of Frobenius manifolds. To define a G-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of G-braided spaces. These are defined as G-graded G-modules with certain braided-commutative "rings of functions", generalizing the commutative rings of power series on ordinary vector spaces. As the genus zero part of any ordinary …

A P-Adic Spectral Triple, Sumedha Hemamalee Rathnayake

Open Access Dissertations

We construct a spectral triple for the C*-algebra of continuous functions on the space of p-adic integers. On the technical level we utilize a weighted rooted tree obtained from a coarse grained approximation of the space combined with the forward derivative D on the tree. Our spectral triple satisfies the properties of a compact spectral metric space and the metric on the space of p-adic integers induced by the spectral triple is equivalent to the usual p-adic metric. Furthermore, we show that the spectrum of the operator D*D is closely related to the roots of a certain …

Modeling, Optimization, And Sensitivity Analysis Of A Continuous Multi-Segment Crystallizer For Production Of Active Pharmaceutical Ingredients, Bradley James Ridder

Open Access Dissertations

We have investigated the simulation-based, steady-state optimization of a new type of crystallizer for the production of pharmaceuticals. The multi-segment, multi-addition plug-flow crystallizer (MSMA-PFC) offers better control over supersaturation in one dimension compared to a batch or stirred-tank crystallizer. Through use of a population balance framework, we have written the governing model equations of population balance and mass balance on the crystallizer segments. The solution of these equations was accomplished through either the method of moments or the finite volume method. The goal was to optimize the performance of the crystallizer with respect to certain quantities, such as maximizing the …

A 2-Categorical Extension Of The Reshetikhin--Turaev Theory, Yu Tsumara

Open Access Dissertations

We concretely construct a 2-categorically extended topological quantum field theory that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.

Nonparametric Variable Selection And Dimension Reduction Methods And Their Applications In Pharmacogenomics, Jingyi Zhu

Open Access Dissertations

Nowadays it is common to collect large volumes of data in many fields with an extensive amount of variables, but often a small or moderate number of samples. For example, in the analysis of genomic data, the number of genes can be very large, varying from tens of thousands to several millions, whereas the number of samples is several hundreds to thousands. Pharmacogenomics is an example of genomics data analysis that we are considering here. Pharmacogenomics research uses whole-genome genetic information to predict individuals' drug response. Because whole-genome data are high dimensional and their relationships to drug response are complicated, …

Probabilistic Uncertainty Quantification And Experiment Design For Nonlinear Models: Applications In Systems Biology, Vu Cao Duy Thien Dinh

Open Access Dissertations

Despite the ever-increasing interest in understanding biology at the system level, there are several factors that hinder studies and analyses of biological systems. First, unlike systems from other applied fields whose parameters can be effectively identified, biological systems are usually unidentifiable, even in the ideal case when all possible system outputs are known with high accuracy. Second, the presence of multivariate bifurcations often leads the system to behaviors that are completely different in nature. In such cases, system outputs (as function of parameters/inputs) are usually discontinuous or have sharp transitions across domains with different behaviors. Finally, models from systems biology …