Physical Sciences and Mathematics Commons^™

An Adapative Treecode-Accelerated Boundary Integral Solver For Computing The Electrostatics Of A Biomolecule, Andrew Joseph Szatkowski

Theses and Dissertations

The Poisson-Boltzmann equation (PBE) is a widely-used model in the calculation of electrostatic potential for solvated biomolecules. PBE is an interface problem defined in the whole space with the interface being a molecular surface of a biomolecule, and has been solved numerically by finite difference, finite element, and boundary integral methods. Unlike the finite difference and finite element methods, the boundary integral method works directly over the whole space without approximating the whole space problem into an artificial boundary value problem. Hence, it is expected to solve PBE in higher accuracy. However, so far, it was only applied to a …

Estimating The Selection Gradient Of A Function-Valued Trait, Tyler John Baur

Theses and Dissertations

Kirkpatrick and Heckman initiated the study of function-valued traits in 1989. How to estimate the selection gradient of a function-valued trait is a major question asked by evolutionary biologists. In this dissertation, we give an explicit expansion of the selection gradient and construct estimators based on two different samples: one consisting of independent organisms (the independent case), and the other consisting of independent families of equally related organisms (the dependent case).

In the independent case we first construct and prove the joint consistency of sieve estimators of the mean and covariance functions of a Gaussian process, based on previous developments …

Density Estimation For Lifetime Distributions Under Semi-Parametric Random Censorship Models, Carsten Harlass

Theses and Dissertations

We derive product limit estimators of survival times and failure rates for randomly right censored data as the numerical solution of identifying Volterra integral equations by employing explicit and implicit Euler schemes. While the first approach results in some known estimators, the latter leads to a new general type of product limit estimator. Plugging in established methods to approximate the conditional probability of the censoring indicator given the observation, we introduce new semi-parametric and presmoothed Kaplan-Meier type estimators. In the case of the semi-parametric random censorship model, i.e. the latter probability belonging to some parametric family, we study the strong …

Investigation Of Sparsifying Transforms In Compressed Sensing For Magnetic Resonance Imaging With Fasttestcs, Christopher Adams Baker

Theses and Dissertations

The goal of this contribution is to achieve higher reduction factors for faster Magnetic Resonance Imaging (MRI) scans with better Image Quality (IQ) by using Compressed Sensing (CS). This can be accomplished by adopting and understanding better sparsifying transforms for CS in MRI. There is a tremendous number of transforms and optional settings potentially available. Additionally, the amount of research in CS is growing, with possible duplication and difficult practical evaluation and comparison. However, no in-depth analysis of the effectiveness of different redundant sparsifying transforms on MRI images with CS has been undertaken until this work. New theoretical sparsity bounds …

Restricting A Representation To A Principally Embedded Sl(2) Subalgebra, Hassan Lhou

Theses and Dissertations

Fix n>2. Let s be a principally embedded sl(2)-subalgebra in sl(n). A special case of work by Jeb Willenbring and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite dimensional sl(n)-representation, V, there exists an irreducible s-representation embedding in V with dimension at most b(n). We prove that the best possible value for the bound is b(n)=n.

An Exponential Time Differencing Scheme With A Real Distinct Poles Rational Function For Advection-Diffusion Reaction Equations, Emmanuel Owusu Asante-Asamani

Theses and Dissertations

A second order Exponential Time Differencing (ETD) scheme for advection-diffusion reaction systems is developed by using a real distinct poles rational function for approximating the underlying matrix exponential. The scheme is proved to be second order convergent. It is demonstrated to be robust for reaction-diffusion systems with non-smooth initial and boundary conditions, sharp solution gradients, and stiff reaction terms. In order to apply the scheme efficiently to higher dimensional problems, a dimensional splitting technique is also developed. This technique can be applied to all ETD schemes and has been found, in the test problems considered, to reduce computational time by …

Nonlocal Debye-Hückel Equations And Nonlocal Linearized Poisson-Boltzmann Equations For Electrostatics Of Electrolytes, Yi Jiang

Theses and Dissertations

Dielectric continuum models have been widely applied to the study of aqueous electrolytes since the early work done by Debye and Hückel in 1910s. Traditionally, they treat the water solvent as a simple dielectric medium with a permittivity constant without considering any correlation among water molecules. In the first part of this thesis, a nonlocal dielectric continuum model is proposed for predicting the electrostatics of electrolytes caused by any external charges. This model can be regarded as an extension of the traditional Debye Hückel equation. For this reason, it is called the nonlocal Debye-Hückel equation. As one important application, this …

Existence Of The Mandelbrot Set In The Parameter Planes Of Certain Rational Functions, Alexander Jay Mitchell

Theses and Dissertations

In complex dynamics we compose a complex valued function with itself repeatedly and

observe the orbits of values of that function. Particular interest is in the orbit of critical

points of that function (critical orbits). One famous, studied example is the quadratic

polynomial Pc(z) = z^2 +c and how changing the value of c makes a difference to the orbit of the critical point z = 0. The set of c values for which the critical orbit is bounded is called

the Mandelbrot set.

This paper studies rational functions of the form Rn;a;c(z) = z^n + a/z^n + c and …

Domain Decomposition Based Hybrid Methods Of Finite Element And Finite Difference And Applications In Biomolecule Simulations, Jinyong Ying

Theses and Dissertations

The dielectric continuum models, such as Poisson Boltzmann equation (PBE), size modified PBE (SMPBE), and nonlocal modified PBE (NMPBE), are important models in predicting the electrostatics of a biomolecule in an ionic solvent. To solve these dielectric continuum models efficiently, in this dissertation, new finite element and finite difference hybrid methods are constructed by Schwartz domain decomposition techniques based on a special seven-box partition of a cubic domain. As one important part of these methods, a finite difference optimal solver --- the preconditioned conjugate gradient method using a multigrid V-cycle preconditioner --- is described in details and proved to have …

Spline Estimation Of Principal Curves, Marcel Andreas Walther

Theses and Dissertations

Finding low-dimensional approximations to high-dimensional data is one of

the most important topics in statistics, which has also multiple applications

in economics, engineering and science. One suggestion in the literature ,based

on kernel smoothing, is a non-linear generalization of principal components.

This kernel-based approach comes with several complications. Therefore the

purpose of this thesis is to provide an alternative based on spline smoothing

which produces more reliable results.

Distance Density Analysis And Multivariate Mode Detection, Immanuel Torben Lampe

Theses and Dissertations

Finding the mode of the distribution for a sample of points is a very interesting task. In one dimensional problems this can easily be done by estimating the kernel density. Unfortunately this method does not work well in higher dimensions.

This thesis presents a new approach to solve this problem. A method is presented which finds the mode by analyzing the distribution of the distances between each point and the rest of the sample. The idea is that if the i-th sample point, x_i, is in a high-density region, most of these distances should be small, whereas if x_i is …

Statistical Contributions To Operational Risk Modeling, Daoping Yu

Theses and Dissertations

In this dissertation, we focus on statistical aspects of operational risk modeling. Specifically, we are interested in understanding the effects of model uncertainty on capital reserves due to data truncation and in developing better model selection tools for truncated and shifted parametric distributions. We first investigate the model uncertainty question which has been unanswered for many years because researchers, practitioners, and regulators could not agree on how to treat the data collection threshold in operational risk modeling. There are several approaches under consideration—the empirical approach, the “naive” approach, the shifted approach, and the truncated approach—for fitting the loss severity distribution. …

Parameter Estimation For The Spatial Ornstein-Uhlenbeck Process With Missing Observations, Sami Cheong

Theses and Dissertations

Suppose we are collecting a set of data on a rectangular sampling grid, it is reasonable to assume that observations (e.g. data that arise in weather forecasting, public health and agriculture) made on each sampling site are spatially correlated. Therefore, when building a model for this type of data, we often pair it with an underlying Gaussian process that contains different spatially dependent parameters. Here, we assume that the Gaussian process is characterized by the Ornstein-Uhlenbeck covariance function, which has the property of being both stationary and Markov under the assumption that no observations are missing. However, in reality, the …

Gaussian Process Regression For Large Data Sets, Nicolas Kuhaupt

Theses and Dissertations

Gaussian Process Regression is a non parametric approach for estimating relationships in data sets. For large data sets least square estimates are not feasible because of the covariance matrix inversion which requires O(n^3) computation. In Gaussian Process Regression a matrix inversion is also needed, but approximation methods exists for large n. Some of those approaches are studied in this thesis, among them are the random projection of the covariance matrix, Nyström method and the Johnson-Lindenstrauß Theorem. Furthermore sampling methods for Hyperparameter estimation are explored.

Optimal Pairs Trading Rules, Eric Müller

Theses and Dissertations

This thesis derives an optimal trading rule for a pair of historically correlated stocks. When one stock's price increases and the other one's decreases, a trade of the pair is triggered. The idea is to short the winner and to long the loser with the hope that the prices of the two assets will converge again. In this thesis the spread of the two stocks is governed by a mean-reverting model. The objective is to trade the pair in such a way as to maximize an overall return. The same slippage cost is imposed on every trade. Furthermore, a local-time …

Longitudinal Data Models With Nonparametric Random Effect Distributions, Hartmut Jakob Stenz

Theses and Dissertations

There is the saying which says you cannot see the woods for the trees. This

thesis aims to circumvent this unfortunate situation: Longitudinal data on

tree growth, as an example of multiple observations of similar individuals

pooled together in one data set, are modeled simultaneously rather than

each individual separately. This is done under the assumption that one

model is suitable for all individuals but its parameters vary following un-

known nonparametric random effect distributions. The goal is a maximum

likelihood estimation of these distributions considering all provided data and

using basis-spline-approximations for the densities of each distribution func-

tion …

Asymptotic Estimates For Some Dispersive Equations On The Alpha-Modulation Space, Justin Trulen

Theses and Dissertations

The alpha-modulation space is a function space developed by Grobner in 1992. The alpha-modulation space is a generalization of the modulation space and Besov space. In this thesis we obtain asymptotic estimates for the Cauchy Problem for dispersive equation, a generalized half Klein-Gordon, and the Klein-Gordon equations. The wave equations will also be considered in this thesis too. These estimates were found by using standard tools from harmonic analysis. Then we use these estimates with a multiplication algebra property of the alpha-modulation space to prove that there are unique solutions locally in time for a nonlinear version of these partial …

The Root Finite Condition On Groups And Its Application To Group Rings, James Gollin

Theses and Dissertations

A group $G$ is said to satisfy the root-finite condition if for every $g \in G$, there are only finitely many $x \in G$ such that there exists a positive integer $n$ such that $x^n = g$. It is shown that groups satisfy the root-finite condition iff they satisfy three subconditions, which are shown to be independent. Free groups are root-finite. Ordered groups are shown to satisfy one of the subconditions for the root-finite condition. Finitely generated abelian groups satisfy the root-finite condition. If, in a torsion-free abelian group $G$, there exists a positive integer $r$ such that the subgroup …

The Influence Of Currents And Bathymetry On The Phytoplankton Growth Dynamics In A Deep Lake: An Application Of The Lattice Boltzmann Method, Breanna Patrice Swan

Theses and Dissertations

The invasive species, the quagga mussel, infiltrated Lake Michigan in the early 2000s and immediately began depleting the base of the aquatic food system: the lake's phytoplankton population. Today the quagga mussel covers 80% of the lake floor deeper than 10 meters, can be concentrated at 35,000 mussels per square meter, and is efficient at filtering throughout the depth of the water column. This thesis aims to contribute to the difficult task of describing the impact these mussels have on the size and preferred depth of the phytoplankton population in Lake Michigan. In a simplified model, two species of phytoplankton …

Numerical And Experimental Study Of Liquid Breakup Process In Solid Rocket Motor Nozzle, Yi-Hsin Yen

Theses and Dissertations

Rocket propulsion is an important travel method for space exploration and national defense, rockets needs to be able to withstand wide range of operation environment and also stable and precise enough to carry sophisticated payload into orbit, those engineering requirement makes rocket becomes one of the state of the art industry. The rocket family have been classified into two major group of liquid and solid rocket based on the fuel phase of liquid or solid state. The solid rocket has the advantages of simple working mechanism, less maintenance and preparing procedure and higher storage safety, those characters of solid rocket …