Applied Mathematics Commons^™

On A Class Of Critical N-Laplacian Problems, Tsz Chung Ho

Theses and Dissertations

We establish some existence results for a class of critical N-Laplacian problems in a bounded domain in R^N. In the absence of a suitable direct sum decomposition, we use an abstract linking theorem based on the Z₂-cohomological index to obtain a nontrivial critical point.

Equichordal Tight Fusion Frames And Biangular Orthopartitionable Tight Frames, Benjamin R. Mayo

Theses and Dissertations

An equichordal tight fusion frame (ECTFF) is a sequence of equidimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound, and so is a type of optimal Grassmannian code. In the special case where its subspaces have dimension one, an ECTFF corresponds to an equiangular tight frame (ETF); such frames have minimal coherence and so are useful for compressed sensing. More generally, an ECTFF will yield a frame with minimal block coherence when its subspaces are pairwise isoclinic, namely when it is an equi-isoclinic tight fusion frame (EITFF). In this dissertation, we generalize the …

Instabilities Of Overturned Traveling Waves, Tyler B. Pierce

Theses and Dissertations

The instabilities of overturned traveling waves are determined by the use of spectral methods. Two separate numerical methods, Spectral Stability Analysis and Dynamic Stability Analysis, are used to assess the instabilities of branches of waves solved from conformally-mapped Euler equations. The branches of waves with Bond number less than two were found to be spectrally stable to super-harmonic perturbations. The branches of waves with Bond number in [2,3) had some waves that were stable and some that were unstable. All overturned waves with Bond number greater than or equal to two were unstable.

Particle Trajectories In Shallow Water Models, Diana Torres

Theses and Dissertations

In this paper we will study particle trajectories under shallow water waves. We will examine equations such as the Korteweg-de Vries and systems dealing with Boussinesq and Euler's Equations to find relationships between particles irrotational velocities. Their solutions and behavior when modeling interacting surface waves will be explored. An attempt to find approximate solutions with different parameters, such as small amplitude and long-crested waves, that will lead to new information and study will be discussed.

Dynamic Parameter Estimation From Partial Observations Of The Lorenz System, Eunice Ng

Theses and Dissertations

Recent numerical work of Carlson-Hudson-Larios leverages a nudging-based algorithm for data assimilation to asymptotically recover viscosity in the 2D Navier-Stokes equations as partial observations on the velocity are received continuously-in-time. This "on-the-fly" algorithm is studied both analytically and numerically for the Lorenz equations in this thesis.

Smooth Global Approximation For Continuous Data Assimilation, Kenneth R. Brown

Theses and Dissertations

This thesis develops the finite element method, constructs local approximation operators, and bounds their error. Global approximation operators are then constructed with a partition of unity. Finally, an application of these operators to data assimilation of the two-dimensional Navier-Stokes equations is presented, showing convergence of an algorithm in all Sobolev topologies.

The Exact Factorization Equations For One- And Two-Level Systems, Bart Rosenzweig

Theses and Dissertations

Exact Factorization is a framework for studying quantum many-body problems. This decomposes the wavefunctions of such systems into conditional and marginal components. We derive corresponding evolution equations for molecular systems whose conditional electronic subsystems are described by one or two Born-Oppenheimer levels and develop a program for their mathematical study.

A Computational Model Of Arterial Thrombus Mechanics In Stenotic Channels, Elise Kole Aspray

Theses and Dissertations

Platelet aggregation is one of the major components of blood clotting. The proximal cause of most heart attacks and many strokes is the rapid formation of a blood clot (thrombus) in response to the rupture or erosion of an arterial atherosclerotic plaque. In the context of a stenotic artery (i.e., an artery whose lumen is partially blocked by the plaque) understanding how the thrombus forms presents additional challenges because of the extremely high shear rates and stresses present as a consequence of the constriction. In this dissertation, we use a two-phase continuum model to investigate the stability of an existing …

Correlated Positron-Electron Orbital (Cpeo): A Novel Method That Models Positron-Electron Correlation In Virtual Ps At The Mean-Field Level, Kevin E. Blaine

Theses and Dissertations

The Correlated Positronic-Electronic Orbital (CPEO) method was developed and implemented to capture correlation effects at between the positron and electron in the modeling of systems that involve a bound positron. Methods that effectively model these systems require many hundred basis functions and use a mean field approach as the beginning step. CPEO builds an orbital for virtual Positronium (Ps) that contains a positron in a bound state along with an accompanying electron to the larger system. Assigning the virtual Ps orbital allows for the two particle variational optimization in conjunction with the other particles that compose the whole system. This …

Stock Markets Performance During A Pandemic: How Contagious Is Covid-19?, Yara Abushahba

Theses and Dissertations

Background and Motivation: The coronavirus (“COVID-19”) pandemic, the subsequent policies and lockdowns have unarguably led to an unprecedented fluid circumstance worldwide. The panic and fluctuations in the stock markets were unparalleled. It is inarguable that real-time availability of news and social media platforms like Twitter played a vital role in driving the investors’ sentiment during such global shock.

Purpose:The purpose of this thesis is to study how the investor sentiment in relation to COVID-19 pandemic influenced stock markets globally and how stock markets globally are integrated and contagious. We analyze COVID-19 sentiment through the Twitter posts and investigate its …

Schur Complement Algebra And Operations With Applications In Multivariate Functions, Realizations, And Representations, Anthony Dean Stefan

Theses and Dissertations

We provide a new approach to the following multidimensional realizability problem: Can an arbitrary square matrix, whose entries are from the field of multivariate rational functions over the complex numbers, be realized as a Schur complement of a linear matrix pencil with symmetries? To answer this problem, we prove the main theorem of M. Bessmertny˘ı,“On realizations of rational matrix functions of several complex variables,” in Vol. 134 of Oper. Theory Adv. Appl., pp. 157-185, Birkh¨auser Verlag, Basel, 2002 and have included additional symmetries as an extension to his results. Furthermore, we were so thorough in our constructive approach that we …

Stability Results For Special Solutions Of Scalar-Field Equations With Variable Coeffcients, Mashael Ibrahiem Alammari

Theses and Dissertations

We study the long-time behavior of general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. In the first part of the dissertation, we take the coefficients to be uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the constant-coefficient kink. …

Optimality Of Delaunay Triangulations, Estefania A. Sierra

Theses and Dissertations

In this paper, we begin by defining and examining the properties of a Voronoi diagram and extend it to its dual, the Delaunay triangulations. We explore the algorithms that construct such structures. Furthermore, we define several optimal functionals and criterions on the set of all triangulations of points in Rd that achieve their minimum on the Delaunay triangulation. We found a new result and proved that Delaunay triangulation has lexicographically the least circumradii sequence. We discuss the CircumRadii-Area (CRA) conjecture that the circumradii raised to the power of alpha times the area of the triangulation holds true for all α …

The Brezis-Nirenberg Problem For The Generalized Kirchhoff Equation, Erisa Hasani

Theses and Dissertations

We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.

Lie Groups And Euler-Bernoulli Beam Equation, Medeu Amangeldi

Theses and Dissertations

Lie groups approach in differential equations was a breakthrough subject in the late nineteenth century. Sophus Lie, a Norwegian mathematician, introduced the systematic approach to study the solutions of differential equations. The main goal of this thesis is to study, using Lie's approach, the Euler-Bernoulli beam equation subject to swelling force, the fourth-order nonlinear differential equation used to describe the beam deflection under the swelling force. In particular, we will classify the symmetry groups of this equation, obtain several reductions, and demonstrate both analytical and numerical solutions.

A Computational Investigation Of The Biophysical Mechanisms Underlying Thermotaxis In The Afd Neurons Of Caenorhabditis Elegans, Zachary Mobille

Theses and Dissertations

Thermotaxis in the nematode Caenorhabditis elegans (C. elegans) is studied at the cellular scale of the amphid finger-like ciliated (AFD) neurons, which have previously been shown to be essential for thermoreception. The voltage and calcium signals of AFD during temperature stimuli are described with ordinary differential equations. The primary calcium model is a modified version of that published by Kuramochi and Doi in 2017 to explain the calcium responses of the chemosensitive amphid single-ciliated right (ASER) neuron to fluctuations in extracellular salt concentration. To account for the effects of temperature, changes to the stimuli conditions under which inactivation takes place …

An Analysis Of The Effects Of Technology Readiness Levels On Cost Growth, Christopher R. Bissing

Theses and Dissertations

This research seeks to evaluate the effects of Technology Readiness Levels (TRL) on Cost Growth. It makes use of data from Technology Readiness Assessments (TRA) and Selected Acquisition Reports (SAR) to explore relationships between TRLs at Milestone B and cost growth in Major Defense Acquisition Programs (MDAP) and Major Automated Information Systems (MAIS). Programs using higher proportions of critical technologies rated below TRL 7 tend to experience greater cost growth than programs that use more mature technologies. Current DoD doctrine requires TRL 6 to enter Milestone B. The results of this research seek to evaluate the merit of this requirement. …

The Reemergence Of Eradicated Disease Due To Ecological Impact Of Climate Change, Claudia Kolakowski

Theses and Dissertations

Global warming is radically changing aspects of the Earth. As scientists continue to research the effects, the ramifications of melting permafrost is coming to light. We build off of a previously existing Anthrax model in the hopes to include climate change as a factor in Anthrax spread. Chapter II develops a simplified version of an Anthrax model. Parameters for the model are found by using previous research and eigenvalues are analyzed in order to find thresholds and equilibria. Chapter III consider the general solutions of the model through eigenvalues and eigenvectors. This model is then extended to include a parameter …

Mathematical Modeling Of Lung Inflammation: Macrophage Polarization And Ventilator-Induced Lung Injury With Methods For Predicting Outcome, Sarah B. Minucci

Theses and Dissertations

Lung insults, such as respiratory infections and lung injuries, can damage the pulmonary epithelium, with the most severe cases needing mechanical ventilation for effective breathing and survival. Furthermore, despite the benefits of mechanical ventilators, prolonged or misuse of ventilators may lead to ventilation-associated/ventilation-induced lung injury (VILI). Damaged epithelial cells within the alveoli trigger a local immune response. A key immune cell is the macrophage, which can differentiate into a spectrum of phenotypes ranging from pro- to anti-inflammatory. To gain a greater understanding of the mechanisms of the immune response in the lungs and possible outcomes, we developed several mathematical models …

Bivariate Markov Chain Model Of Irritable Bowel Syndrome (Ibs) Subtypes And Abdominal Pain, Ricardo Reyna Jr.

Theses and Dissertations

Researchers use stochastic models like continuous-time Markov chains (CTMC) to model progression of morbidities of public health impact, like HIV and Hepatitis C. Most of the research in that area is done for a single disease. In this research, we use a bivariate continuous-time Markov chain (CTMC) to model progression of co-morbidities. In particular, we use a bivariate CTMC to model the joint progression of Irritable Bowel Syndrome (IBS) and abdominal pain. Symptoms of IBS are known to change throughout the duration of the disorder. Hence, patients are normally asked to make a journal of the stool type, symptoms, and …

Optimal Control Of Multiphase Free Boundary Problems For Nonlinear Parabolic Equations, Evan Cosgrove

Theses and Dissertations

Dissertation research is on the optimal control of systems with distributed parameters described by singular nonlinear partial differential equations (PDE) modeling multi-phase Stefan type second order parabolic free boundary problems. This type of free boundary problems arise in various applications, such as biomedical engineering problem on the laser ablation of biological tissues, aerospace engineering problem on the ice accretion in aircrafts mid-flight, biomedical problem on the growth of cancerous tumor, and many other phase transition processes in thermophysics and fluid mechanics. The aim of the optimal control of distributed free boundary systems is two fold: identification of functional parameters of …

Complete Integrability And Discretization Of Euler Top And Manakov Top, Austin Marstaller

Theses and Dissertations

The Euler top is a completely integrable system with physical system implications and the Manakov top is its four-dimensional extension. We are concerned about their complete integrability and the preservation of this property under a specific discretization known as the Hirota-Kimura Discretization. Surprisingly, it is not guaranteed that under any discretization the conserved quantities are preserved and therefore they must be discovered. In this work we construct the Poisson bracket and Lax pair for each system and provide the Lie algebra background needed to do such such constructions.

Numerical Simulation Of Low Reynolds Number Locomotion In Viscoelastic Media, Nesreen Abdulrahim Althobaiti

Theses and Dissertations

We use computational models to investigate 2D swimmers within various fluid media with low Reynolds Number. Extensions of the standard Immersed Boundary (IB) Method are proposed so that the fluid media may satisfy no slip, partial slip or free-slip conditions on the moving boundary. The fluid equations are solved through a Multigrid preconditioned GMRES solver. Our numerical results indicate that slip may lead to substantial speed enhancement for swimmers in a viscoelastic fluid, as well as in a viscoelastic two-fluid mixture. Under the slip conditions, the speed of locomotion is dependent in a nontrivial way on both the viscosity and …

An Investigation Of Gene Regulatory Network State Space Variability, Sara Faye Liesman

Theses and Dissertations

Genes are segments of DNA that provide a blueprint for cells and organisms to effectively control processes and regulations within individuals. There have been many attempts to quantify these processes, as a greater understanding of how genes operate could have large impacts on both personalized and precision medicine. Gene interactions are of particular interest, however, current biological methods can not easily reveal the details of these interactions. Therefore, we infer networks of interactions from gene expression data which we call a gene regulatory network, or GRN. Due to the robust behavior of genes and the inherent variability within interactions, models …

A Study Of The Efficacy Of Machine Learning For Diagnosing Obstructive Coronary Artery Disease In Non-Diabetic Patients, Demond Larae Handley

Theses and Dissertations

According to the Centers for Disease Control and Prevention, about 18.2 million adults age 20 and older have Coronary Artery Disease in the United States. Early diagnosis is therefore of crucial importance to help prevent debilitating consequences, and principally death for many patients. In this study we use data containing gene expression values from peripheral blood samples in 198 non-diabetic patients, with the goal of developing an age and sex gene expression model for diagnosis of Coronary Artery Disease. We employ machine learning methods to obtain a classification based on genetic information, age and sex. Our implementation uses feed forward …

A Mathematical Development Of Minimal Surface Theory: From Soap Films To Black Holes, Timothy Pitts

Theses and Dissertations

Minimal surfaces are a special subset of surfaces that have gone through a long and extensive development and have also led to many fruitful findings in mathematics. Several periods that are key to the progression of the theory are coined as Golden Ages for the field’s development. Here, a historical and mathematical development of minimal surface theory is presented that spans from its inception in the late 18th century to the present day. Along with the development, there is an emphasis on showing connections of minimal surfaces to various natural phenomena that occur such as soap films, black holes, biological …

Critical Elliptic Boundary Value Problems With Singular Trudinger-Moser Nonlinearities, Shiqiu Fu

Theses and Dissertations

In this dissertation, we prove the existence of solutions for two classes of eliptic problems that are critical with respect to singular Trudinger-Moser embedding. The proofs are based on compactness and regularity arguments.

A Computational Investigation Of The Biomechanics For Platelets Aggregation, Ghadah Mohammed Alhawael

Theses and Dissertations

The proximal cause of most heart attacks and many strokes is the rapid formation of a blood clot (thrombus) in response to the rupture or erosion of an arterial atherosclerotic plaque. The formation of a thrombus in arteries is a very complex process whose workings are subjects of intense research. In this dissertation, we investigate the biomechanics of platelet aggregation in large arteries using a two-phase continuum computational model. The model tracks the number densities of various platelet populations, the concentration of one platelet-activating chemical, as well as the number densities of inter-platelet bonds. Through the formation of elastic bonds, …

Optimal Control Of Coefficients For The Second Order Parabolic Free Boundary Problems, Ali Hagverdiyev

Theses and Dissertations

Dissertation aims to analyze inverse Stefan type free boundary problem for the second order parabolic PDE with unknown parameters based on the additional information given in the form of the distribution of the solution of the PDE and the position of the free boundary at the final moment. This type of ill-posed inverse free boundary problems arise in many applications such as biomedical engineering problem about the laser ablation of biomedical tissues, in-flight ice accretion modeling in aerospace industry, and various phase transition processes in thermophysics and fluid mechanics. The set of unknown parameters include a space-time dependent diffusion, convection …

Modeling Nonlinear Heat Transfer For A Pin-On-Disc Sliding System, Brian A. Boardman

Theses and Dissertations

The objective of this research is to develop a numerical method to characterize heat transfer and wear rates for samples of Vascomax® 300, or Maraging 300, steel. A pin-on-disc experiment was conducted in which samples were exposed to a high-pressure, high-speed, sliding contact environment. This sliding contact generates frictional heating that influences the temperature distribution and wear characteristics of the test samples. A two-dimensional nonlinear heat transfer equation is discretized and solved via a second-order explicit finite difference scheme to predict the transient temperature distribution of the pin. This schematic is used to predict the removal of material from the …