Applied Mathematics Commons^™

Zero-Inflated Longitudinal Mixture Model For Stochastic Radiographic Lung Compositional Change Following Radiotherapy Of Lung Cancer, Viviana A. Rodríguez Romero

Theses and Dissertations

Compositional data (CD) is mostly analyzed as relative data, using ratios of components, and log-ratio transformations to be able to use known multivariable statistical methods. Therefore, CD where some components equal zero represent a problem. Furthermore, when the data is measured longitudinally, observations are spatially related and appear to come from a mixture population, the analysis becomes highly complex. For this matter, a two-part model was proposed to deal with structural zeros in longitudinal CD using a mixed-effects model. Furthermore, the model has been extended to the case where the non-zero components of the vector might a two component mixture …

Modeling The Evolution Of Barrier Islands, Greg Robson

Theses and Dissertations

Barrier islands form off the shore of many coastal areas and serve as the first line of defense, protecting littoral communities against storms. To study the effects that climate change has on barrier islands, we use a cellular model of wind erosion, surface dynamics, beach dynamics, marsh dynamics, and vegetation development. We will show the inhibition of movement when vegetation is present.

Estimating Response Status In Sequential Multiple Assignment (Smar)-Like Trials, Keighly Bradbrook

Theses and Dissertations

Sequential, multiple assignment, randomized trials (SMARTs) allow investigators to develop and compare experimental treatment regimens in which individuals are successively randomized to different treatments based on some set of predetermined rules. The rules used to make decisions on when and how to switch treatments are based on a chosen set of tailoring variables. Although not always true, intermediate response is commonly used as the primary tailoring variable as it is often predictive of future treatment success. As such, successful implementation depends on identifying patients who respond to treatment, though in some situations such mechanisms may not exist. Further, patient-level covariates …

The Analysis Of Neural Heterogeneity Through Mathematical And Statistical Methods, Kyle Wendling

Theses and Dissertations

Diversity of intrinsic neural attributes and network connections is known to exist in many areas of the brain and is thought to significantly affect neural coding. Recent theoretical and experimental work has argued that in uncoupled networks, coding is most accurate at intermediate levels of heterogeneity. I explore this phenomenon through two distinct approaches: a theoretical mathematical modeling approach and a data-driven statistical modeling approach.

Through the mathematical approach, I examine firing rate heterogeneity in a feedforward network of stochastic neural oscillators utilizing a high-dimensional model. The firing rate heterogeneity stems from two sources: intrinsic (different individual cells) and network …

Some Free Boundary Problems For The Nonlinear Degenerate Multidimensional Parabolic Equations Modeling Reaction-Diffusion Processes, Amna Abu Weden

Theses and Dissertations

This dissertation presents a full classification of the short-time behavior of the interfaces or free boundaries for the nonlinear second order degenerate multidimensional parabolic partial differential equation (PDE) ut −∆u m +buβ = 0, x ∈ R N ,0 < t < T (1) with m > 0, β > 0,b ∈ R, arising in various applications in fluid mechanics, filtration of oil or gas in a porous media, plasma physics, reaction-diffusion equations in chemical kinetics, population dynamics in mathematical biology etc. as a mathematical model of nonlinear diffusion phenomena in the presence of the absorption or release of energy. Cauchy problem with compactly supported and nonnegative initial function …

Analysis Of Interfaces For The Nonlinear Degenerate Second Order Parabolic Equations Modeling Diffusion-Convection Processes, Lamees Kadhim Ali Alzaki

Theses and Dissertations

Dissertation pursues analysis of the short-time evolution of interfaces or free boundaries for the non-negative solutions of the nonlinear degenerate second order parabolic partial differential equation (PDE) ut = ( u m ) xx +b ( u γ ) x , x ∈ R,t > 0; m > 1, γ > 0,b ∈ R (1) modeling diffusion-convection processes arising in fluid or gas flow in a porous media, plasma physics, population dynamics in mathematical biology and other applications. Due to the implicit degeneration (m > 1), PDE (1) it possesses a property of the finite speed of propagation and develops interfaces or free boundaries …

Nonlocal Boundary Value Problems For Linear Hyperbolic Systems With Two Independent Variables, Afrah Almutairi

Theses and Dissertations

Nonlocal boundary value problems in a characteristic rectangle for second order linear hyperbolic systems are considered. There are established: (i) Unimprovable sufficient conditions for general boundary value problems to possess the Fredholm property; (ii) Optimal sufficient conditions of unique solvability of general boundary value problems; (iii) Effective sufficient conditions for doubly periodic problems to possess the Fredholm property; (iv) Unimprovable sufficient conditions of unique solvability of doubly periodic problems; (v) Effective sufficient conditions for boundary value problems of periodic type to possess the Fredholm property; (vi) Unimprovable sufficient conditions of unique solvability of boundary value problems of periodic type; (vii) …

Stability Analysis Of Neutral Functional Differential Equations Arising In Partial Element Equivalent Circuit Models, Howard Michael Allison

Theses and Dissertations

Neutral Functional Differential Equations (NFDEs) arise in the study of the Partial Element Equivalent Circuit (PEEC) model with time delays. We present sufficient conditions for asymptotic stability and global stability in the delays of the PEEC NFDE’s, using Lyapunov-Razumikhin function methods.. We develop, for the first time, a standard mixing-type nonlinearity for the PEEC NFDEs. Introducing time invariant and time varying nonlinear perturbation to the PEEC NFDEs, we develop sufficient conditions for stability of the nonlinear perturbed PEEC NFDEs and convergence of the nonlinear system to the original stable linear autonomous system. We also develop sufficient conditions for stability and …

Computational Models For Biological Locomotion In Gels, Hashim Mohammed Alshehri

Theses and Dissertations

We investigated Low Reynold’s Number Locomotion in two-phase biological gels. The gel is composed of two materials: a viscous fluid solvent phase and a viscoelastic polymer network phase. A novel Two-phase Immersed Boundary Method (IBM) is developed to simulate the complicated interactions between an elastic boundary and a mixture of two fluids with very different physical properties. A further extension of the method is developed for the case where fluids satisfy partial-slip and free-slip conditions on the elastic boundary. Our major conclusions are summarized as following: (i) Our numerical scheme is proved to be robust and efficient. It can successfully …

Dimension-Breaking For Traveling Waves In Interfacial Flows, Matthew W. Seiders

Theses and Dissertations

Fluid flow models in two spatial dimensions with a one-dimensional interface are known to support overturned traveling solutions. Computational methods of solving the two-dimensional problem are well developed, even in the case of overturned waves. The three-dimensional problem is harder for three prominent reasons. First, some formulations of the two-dimensional problem do not extend to three-dimensions. The technique of conformal mapping is a prime example, as it is very efficient in two dimensions but does not have a three-dimensional equivalent. Second, some three-dimensional models, such as the Transformed Field Expansion method, do not allow for overturned waves. Third, computational time …

Two-Point Boundary Value Problems For Higher Order Nonlinear Hyperbolic Equations, Audison Beaubrun

Theses and Dissertations

Two–point boundary value problems in a multidimensional box for higher order nonlinear hyperbolic equations are considered. The concepts of a strongly isolated solution, and locally and globally strong well–posedness of a nonlinear boundary value problem are introduced. For general two–point boundary value problems and periodic problems there are established: (i) Necessary and sufficient conditions of locally and globally strong well–posedness; (ii) Unimprovable Sufficient conditions of solvability. For the Dirichlet and Periodic type problems for equations of even order there are established: (i) Effective sufficient conditions of solvability and locally strong well–posedness; (ii) Unimprovable sufficient conditions of solvability for the case, …

Qualitative Analysis Of The Nonlinear Double Degenerate Parabolic Equation Of Turbulent Filtration With Absorption, Adam Prinkey

Theses and Dissertations

The goal of the dissertation is to pursue qualitative analysis of the mathematical model of turbulent polytropic filtration of a gas in a porous media with reaction or absorption described by the second order nonlinear double degenerate parabolic equation ∂u ∂t − ∂ ∂x F [ ∂u m ∂x ] + Q(u) = 0, (1) where F(y) = |y| p−1 y, Q(u) = buβ , m, p, β > 0, b ∈ R. In the absence of the reaction term there is a finite speed of propagation with an expanding interface in the case of slow diffusion (mp > 1), and infinite …

Generalized Random Measures On Topological Spaces, Ali Hussein Mahmood Al-Obaidi

Theses and Dissertations

Our work deals with classes of random measures on -compact Hausdorff spaces perturbed by stochastic processes. We render a rigorous construction of the stochastic integral of functions of two variables and show that such an integral is a random measure. We establish a new Campbell-type formula that, along with a rigorous construction of modulation, leads to the intensity of a modulated random measure. We further introduce and study a marked Poisson random measure on a - compact Hausdorff space. The underlying parameters of this measure are changing in accordance with the evolution of a stochastic process. This generalized random measure …

Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Courtney A. Schmitt

Theses and Dissertations

An equiangular tight frame (ETF) is a sequence of equal-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs, which are formed by restricting the characters of a finite abelian group to a difference set. Recently, it was shown that some harmonic ETFs are themselves comprised of regular simplices. In this thesis, we continue the …

Modeling The Distribution Of Lightning Strike Distances Outside A Preexisting Lightning Area, Dawn L. Sanderson

Theses and Dissertations

Air Force Instruction 91-203 (AFI 91-203) directs that a lightning warning be issued when lightning is occurring or imminent within a 5 nautical mile (NM) radius of a predetermined location or activity. The 45 Weather Squadron (WS), located on the central eastern coast of Florida, balances the safety of personnel and space launch vehicles with lost productivity of taking shelter from lightning. The primary objective of this study investigates if this 5 NM safety radius can be reduced while maintaining a desired level of safety. The research uses processed Lightning Detection and Ranging (LDAR) data to map the movement of …

An Imputation Approach To Developing Alternative Futures Of Country Conflict, Zachary J. Kane

Theses and Dissertations

Understanding what causes countries to be in a state of violent conflict is of vital importance to developing realistic national strategies on both a regional and global scale. Given these causes, it is important to understand the effects of missing data, how to impute that data, and the interrelation between data elements. Utilizing both open source data and previously generated equations that predict a country’s likelihood to transition conflict statuses, this research projects data into the future and predicts each nations’ subsequent conflict statuses. Future data is populated using a novel approach inspired by stochastic regression imputation. The replicated future …

Determination Of Optimal Parameter Estimates For Medical Interventions In Human Metabolism And Inflammation, Marcella Torres

Theses and Dissertations

In this work we have developed three ordinary differential equation models of biological systems: body mass change in response to exercise, immune system response to a general inflammatory stimulus, and the immune system response in atherosclerosis. The purpose of developing such computational tools is to test hypotheses about the underlying biological processes that drive system outcomes as well as possible real medical interventions. Therefore, we focus our analysis on understanding key interactions between model parameters and outcomes to deepen our understanding of these complex processes as a means to developing effective treatments in obesity, sarcopenia, and inflammatory diseases.

We develop …

A Statistical Investigation Of Stock Market Activity And Instances Of Financial Crime, Elizabeth G. Biggs

Theses and Dissertations

This project aims to identify a possible statistical relationship between the stock market and incidences of financial crime in Illinois, Texas, and Utah through big data analysis techniques using data from the FBI’s Federal Bureau of Investigation’s National Incident-Based Reporting System (NIBRS) database. By analyzing the occurrences of financial crime by location, year, and type in relation to the S&P 500’s percent change in close price on the first day of the calendar year, it will be possible to determine if a statistical relationship exists between the two. In addition, regression analysis was performed in order to predict financial crime …

Kings In The Direct Product Of Digraphs, Morgan Norge

Theses and Dissertations

A k-king in a digraph D is a vertex that can reach every other vertex in D by a directed path of length at most k. A king is a vertex that is a k-king for some k. We will look at kings in the direct product of digraphs and characterize a relationship between kings in the product and kings in the factors. This is a continuation of a project in which a similar characterization is found for the cartesian product of digraphs, the strong product of digraphs, and the lexicographic product of digraphs.

On The Qualitative Theory Of The Nonlinear Parabolic P-Laplacian Type Reaction-Diffusion Equations, Roqia Abdullah Jeli

Theses and Dissertations

This dissertation presents full classification of the evolution of the interfaces and asymptotics of the local solution near the interfaces and at infinity for the nonlinear second order parabolic p-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration ut − ( |ux| p−2 ux ) x +buβ = 0, p > 1, β > 0. (1) Nonlinear partial differential equation (1) is a key model example expressing competition between nonlinear diffusion with gradient dependent diffusivity in either slow (p > 2) or fast (1 < p < 2) regime and nonlinear state dependent reaction (b > 0) or absorption (b < 0) forces. If interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β,sign b, and asymptotics of the initial function near its support. In the fast diffusion regime strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are generaliii ization of the methods developed in U.G. Abdulla & J. King, SIAM J. Math. Anal., 32, 2(2000), 235-260; U.G. Abdulla, Nonlinear Analysis, 50, 4(2002), 541-560 and based on rescaling laws for the nonlinear PDE and blow-up techniques for the identification of the asymptotics of the solution near the interfaces, construction of barriers using special comparison theorems in irregular domains with characteristic boundary curves.

Compressive Sampling For Phenotype Classification, Eric L. Brooks

Theses and Dissertations

Phenotype classification has become an increasingly important genomic research method for disease identification and treatment. Phenotype classification is the investigation into the genetic information concerned with locating biomarkers (features) in order to identify an observed effect. The primary challenge associated with phenotype classification is with analyzing the data due to the inherent high-dimensionality of DNA data. As a result, phenotype classification faces challenges with feature selection, and consequently, classification accuracy. This research developed a methodology to alleviate these challenges while improving classification performance. The methodology leverages concepts of compressive sampling, to arrive at a process that identifies features most relevant …

Statistical Inference To Evaluate And Compare The Performance Of Correlated Multi-State Classification Systems, Beau A. Nunnally

Theses and Dissertations

The current emphasis on including correlation when comparing diagnostic test performance is quite important, however, there are cases in which correlation effects may be negligible with respect to inference. This proposed work examines the impact of including correlation between classification systems with continuous features by comparing the optimal performance of two diagnostic tests with multiple outcomes as well as providing inference for a sequence of tests. We define the optimal point using Bayes Cost, a metric that sums the weighted misclassifications within a diagnostic test using a cost/benefit structure. Through simulation, we quantify the impact of correlation on standard errors …

Numerical Simulation Of High Energy Laser Propagation, Dana F. Morrill

Theses and Dissertations

High energy lasers have many applications, such as in aerospace, weapons, wireless power transfer, and manufacturing. Fluid-laser interaction is important to predicting power at receiver, and other measures of laser beam quality. Typically the carrying medium of the laser is modeled statistically. This dissertation describes a novel method of coupling fluid dynamics to beam propagation in free space. The coupled laser-fluid solver captures dynamic interaction of fluid temperature and beam intensity. Ultimately, the model captures the effects of fluid convection in the laser intensity-field. Boundary conditions play an important role for fluid dynamics, more so than for beam dynamics. Simulation …

Lower Order Perturbations Of Critical Fractional Laplacian Equations, Khalid Fanoukh Al Oweidi

Theses and Dissertations

We give sufficient conditions for the existence of nontrivial solutions to a class of critical nonlocal problems of the Brezis-Nirenberg type. Our result extends some results in the literature for the local case to the nonlocal setting. It also complements the known results for the nonlocal case.

Weak Galerkin Finite Element Method With Mixed Boundary Conditions, Saqib Hussain

Theses and Dissertations

Mixed boundary conditions are practical situations that appear in most potential and physics problems. In this dissertation, a weak Galerkin (WG) finite element method is proposed and analyzed for the general elliptic equation with mixed boundary conditions. Furthermore, we developed a modified weak Galerkin (MWG) method for the second order elliptic problem with mixed boundary conditions. While keeping the same accuracy, the modified weak Galerkin method reduces the degree of freedom of the original weak Galerkin method by eliminating unknowns associated with element boundaries. Optimal order error estimates are established in both a discrete $H^1$ norm and the standard $L^2$ …

On The Qualitative Theory Of The Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes, Habeeb Abed Kadhim Aal-Rkhais

Theses and Dissertations

We consider nonlinear second order degenerate or singular parabolic equation ut − a(um)xx + buβ + c(up)x = 0, a, m, β, p > 0, b, c ∈ R describing reaction-diffusion-convection processes arising in many areas of science and engineering, such as filtration of oil or gas in porous media, transport of thermal energy in plasma physics, flow of chemically reacting fluid, evolution of populations in mathematical biology etc. We apply the methods developed in U.G. Abdulla, Journal of Differential Equations, 164, 2(2000), 321-354 for the reaction-diffusion equation (c = 0) and prove the existence, uniqueness, boundary regularity and comparison theorems …

Boundary Value Problems In A Multidimensional Box For Higher Order Linear And Quasi-Linear Hyperbolic Equations, Noha Aljaber

Theses and Dissertations

Boundary value problems in a multidimensional box for higher order linear hyperbolic equations are considered. The concept of associated problems are introduced. For general boundary value problems there are established: (i) Necessary and sufficient conditions for a linear problem to have the Fredholm property in two–dimensional case; (ii) Necessary and sufficient conditions of well–posedness in two–dimensional case; (iii) Unimprovable sufficient conditions for a linear problem to have the Fredholm property; (iv) Unimprovable sufficient conditions of well–posedness and α–well–posedness; (v) Effective sufficient conditions of unqie solvability of two–point, periodic and Dirichlet type problems. (iv) Unimprovable conditions of unique solvability of two …

Radial Basis Function Generated Finite Differences For The Nonlinear Schrodinger Equation, Justin Ng

Theses and Dissertations

Solutions to the one-dimensional and two-dimensional nonlinear Schrodinger (NLS) equation are obtained numerically using methods based on radial basis functions (RBFs). Periodic boundary conditions are enforced with a non-periodic initial condition over varying domain sizes. The spatial structure of the solutions is represented using RBFs while several explicit and implicit iterative methods for solving ordinary differential equations (ODEs) are used in temporal discretization for the approximate solutions to the NLS equation. Splitting schemes, integration factors and hyperviscosity are used to stabilize the time-stepping schemes and are compared with one another in terms of computational efficiency and accuracy. This thesis shows …

An Analysis Of The Estimate At Complete For Department Of Defense Contracts, Deborah B. Kim

Theses and Dissertations

When contractors provide timely and reliable information on the status of a contract, both contractors and government program offices can provide an accurate estimate of a contract’s completion costs. This research shows that the cumulative cost performance indices provided by contractors and program offices are high and less accurate than those of previous years and/or that a significant amount of ACWP is being documented in the final portion of a contract. The high performance indices resulted in EACs that were low-balled during the majority of a contract’s life which shows a need to improve the use of EVM metrics for …

Improving Annual Fixed Wing Maintenance Cost Estimates Through Cost Estimating Relationships, Kirsten Bunecke

Theses and Dissertations

The Air Force executes its mission primarily through the use of fixed-wing aircrafts. Maintaining these aircrafts represents approximately one-third of annual Operating and Support costs; which in turn, make up the majority of Life Cycle Costs. The current approach to estimating aircraft maintenance costs does not take into consideration programmatic data available in the Air Force Total Ownership Cost (AFTOC) database. The four maintenance cost subcategories researched in this thesis are Consumable Materials and Repair Parts, Depot Level Repairables (DLRs), Depot Maintenance, and Contractor Logistic Support. Each of these cost categories must first be standardized to be able to compare …