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 …

Some Spectral Properties Of A Quantum Field Theoretic Hamiltonian, Devin Burnell Mcghie

Theses and Dissertations

We derive the ground-state eigenvalues and eigenvectors for a simplified version of the 1-D QED single electron-photon model that Glasgow et al recently developed [2]. This model still allows for meaningful interaction between electrons and photons while keeping only the minimum needed to do so. We investigate the interesting spectral properties of this model. We determine that the eigenvectors are orthogonal as one would expect and normalize them.

Discrete And Continuous Operational Calculus In Stochastic Games, Kenneth Ibe Iwezulu

Theses and Dissertations

First, we consider a class of antagonistic stochastic games between two players A and B. The game is specified in terms of two "hostile" stochastic processes representing mutual attacks upon random times exerting casualties of random magnitudes. This game is observed upon random epochs of time and the outcome of the game is not known in real time. The game ends at the time when the underlying fixed threshold of either player is crossed (referred to as the first passage time). The first passage time is then shifted to an epoch, i.e. upon one of the observation instants of time. …

Numerical Simulations Of Shock Waves Reflection And Interaction, Ligang Sun

Theses and Dissertations

The main objective of this dissertation is to detect and study the phenomena of reflection of one shock wave and interaction of two shock waves using numerical methods. In theory, solutions of non-linear Euler equations of compressive inviscid gas dynamics in two dimensions can display various features including shock waves and rarefaction waves. To capture the shock waves properly, highly accurate numerical schemes are designed according to second order Lax-Wendroff method. In this thesis, three numerical experiments were designed to show the reflection and interaction phenomena. Firstly, one shock was formed due to the encounter of two high speed gas …

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 …

Sparse Representation For The Isar Image Reconstruction, Mengqi Hu

Theses and Dissertations

In this paper, a sparse representation for the data form a multi-input multi-output based inverse synthetic aperture radar (ISAR) system is derived for two dimensions. The proposed sparse representation motivates the use a of a Convex Optimization directly that recovers the image without the loss information of the image with far less samples that that is required by Nyquist–Shannon sampling theorem, which increases the efficiency and decrease the cost of calculation in radar imaging.

Lie Symmetry To Second-Order Nonlinear Differential Equations And Its First Integrals, Pengfei Gu

Theses and Dissertations

There are many well-known techniques for obtaining exact solutions of differential equations, but most of them are merely special cases of a few powerful symmetry methods. In this paper, we focus our attention on a second-order nonlinear ordinary differential equation of special forms with arbitrary parameters, which is a combination of Liénard-type equation and equation with quadratic friction. With the help of Lie Symmetry methods, we identify several integrable cases of this equation. And for each case, we use the Lie Symmetry method to derive the associated determining system, and apply it further to find infinitesimal generators under …

A Comparative Study And Data Analysis For The Ultimate Fighting Championship, Victor Villalpando

Theses and Dissertations

Mixed Martial Arts is the fastest growing sport with many organizations worldwide. The biggest stage or biggest organization for Mixed Martial Arts is the Ultimate Fighting Championship (UFC). There are eight weight classes for men. The website: http://www.foxsports.com/ufc/stats provides data on fighters in all these categories. This data measures Striking Accuracy, Take downs, Reversals, Knockdowns, etc. in each category. It is interesting to understand and interpret all these numbers and study their relationships. Statistical tools like both parametric and nonparametric inference may give rise to such interpretations and provide explanations how the weight classes differ from one another. In this …

Steady State Configurations Of Cells Connected By Cadherin Sites, Jared Adam Mcbride

Theses and Dissertations

Many cells employ cadherin complexes (c-sites) on the cell membrane to attach to neighboring cells, as well as integrin complexes (i-sites) to attach to a substrate in order to accomplish cell migration. This paper analyzes a model for the motion of a group of cells connected by c-sites. We begin with two cells connected by a single c-site and analyze the resultant motion of the system. We find that the system is irrotational. We present a result for reducing the number of c-sites in a system with c-sites between pairs of cells. This greatly simplifies the general system, and provides …

Compressive Sensing And Radar Imaging, John Montalbo

Theses and Dissertations

The field of remote sensing contains many unique and practical problems. Radar imaging, in all of its many forms, lies within this field of study. One problem is the need to acquire high-resolution images and store them on-board the system acquisition vessel . For some systems this could mean storing very high amounts of data, depending on the scene in question [3]. So a very natural goal is to store only what is absolutely necessary and nothing more. We investigate methods to compress signals into their most important components so that other parties can recover the original data completely or …

Machine Learning For Disease Prediction, Abraham Jacob Frandsen

Theses and Dissertations

Millions of people in the United States alone suffer from undiagnosed or late-diagnosed chronic diseases such as Chronic Kidney Disease and Type II Diabetes. Catching these diseases earlier facilitates preventive healthcare interventions, which in turn can lead to tremendous cost savings and improved health outcomes. We develop algorithms for predicting disease occurrence by drawing from ideas and techniques in the field of machine learning. We explore standard classification methods such as logistic regression and random forest, as well as more sophisticated sequence models, including recurrent neural networks. We focus especially on the use of medical code data for disease prediction, …

American Spread Option Pricing With Stochastic Interest Rate, An Jiang

Theses and Dissertations

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR …

Weak Cayley Table Groups Of Wallpaper Groups, Rebeca Ann Paulsen

Theses and Dissertations

Let G be a group. A Weak Cayley Table mapping ϕ : G → G is a bijection such that ϕ(g1g2) is conjugate to ϕ(g1)ϕ(g2) for all g1, g2 in G. The set of all such mappings forms a group W(G) under composition. We study W(G) for the seventeen wallpaper groups G.

Weakly Holomorphic Modular Forms In Prime Power Levels Of Genus Zero, David Joshua Thornton

Theses and Dissertations

Let N ∈ {8,9,16,25} and let M^#₀(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M^#₀(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results …

Existence Of A Periodic Brake Orbit In The Fully Symmetricplanar Four Body Problem, Ammon Si-Yuen Lam

Theses and Dissertations

We investigate the existence of a symmetric singular periodic brake orbit in the equal mass, fully symmetric planar four body problem. Using regularized coordinates, we remove the singularity of binary collision for each symmetric pair. We use topological and symmetry tools in our investigation.

Schur Rings Over Projective Special Linear Groups, David R. Wagner

Theses and Dissertations

This thesis presents an introduction to Schur rings (S-rings) and their various properties. Special attention is given to S-rings that are commutative. A number of original results are proved, including a complete classification of the central S-rings over the simple groups PSL(2,q), where q is any prime power. A discussion is made of the counting of symmetric S-rings over cyclic groups of prime power order. An appendix is included that gives all S-rings over the symmetric group over 4 elements with basic structural properties, along with code that can be used, for groups of comparatively small order, to enumerate all …

Periodic Points And Surfaces Given By Trace Maps, Kevin Gregory Johnston

Theses and Dissertations

In this thesis, we consider the properties of diffeomorphisms of R3 called trace maps. We begin by introducing the definition of the trace map. The group B3 acts by trace maps on R3. The first two chapters deal with the action of a specific element of B3,called αn. In particular, we study the fixed points of αn lying on a topological subspace contained in R3, called T . We investigate the duality of the fixed points of the action ofαn, which will be defined later in the thesis.Chapter 3 involves the study of the fixed points of an element called …

Partitions Of Finite Frames, James Michael Rosado

Theses and Dissertations

An open question stated by Marcus, Spielman, and Srivastava [10] asks "whether one can design an efficient algorithm to find the partitions guaranteed by Corollary 1.5." This corollary states that given a set of vectors in C whose outer products sum to the identity there exists a partition of these vectors such that norms of the outer-product sums of each subset satisfy an inequality bound. Here particular types of vector sets called finite frames are analyzed and constructed to satisfy the inequality described in Corollary 1.5. In this thesis, rigorous proofs and formulations of outer-product norms are utilized to find …

Decoding Of Non-Binary Multiple Insertion/Deletion Error Correcting Codes, Tuan Anh Le

Theses and Dissertations

Data that is either transmitted over a communication channel or stored in memory is not always completely error free. Many communication channels are subject to noise, and thus errors may occur during transmission from transmitter to receiver. For example, DRAM memory cell contents can change spuriously due to electromagnetic interference while magnetic flux can cause one or more bits to flip in magnetic storage devices. To combat these errors, codes capable of correcting insertion/deletion errors have been investigated.

Levenshtein codes are the foundation of this thesis. His codes, first constructed by Varshamov-Tenengol’ts, are capable of correcting one insertion/deletion error. Helberg …

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 …