Applied Mathematics Commons^™

A Review Of Random Matrix Theory With An Application To Biological Data, Jesse Aaron Marks

Masters Theses

"Random matrix theory (RMT) is an area of study that has applications in a wide variety of scientific disciplines. The foundation of RMT is based on the analysis of the eigenvalue behavior of matrices. The eigenvalues of a random matrix (a matrix with stochastic entries) will behave differently than the eigenvalues from a matrix with non-random properties. Studying this bifurcation of the eigenvalue behavior provides the means to which system-specific signals can be distinguished from randomness. In particular, RMT provides an algorithmic approach to objectively remove noise from matrices with embedded signals.

Major advances in data acquisition capabilities have changed …

Orbital Stability Results For Soliton Solutions To Nonlinear Schrödinger Equations With External Potentials, Joseph B. Lindgren

Theses and Dissertations--Mathematics

For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method.

For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.

Global Well-Posedness For The Derivative Nonlinear Schrödinger Equation Through Inverse Scattering, Jiaqi Liu

Theses and Dissertations--Mathematics

We study the Cauchy problem of the derivative nonlinear Schrodinger equation in one space dimension. Using the method of inverse scattering, we prove global well-posedness of the derivative nonlinear Schrodinger equation for initial conditions in a dense and open subset of weighted Sobolev space that can support bright solitons.

A Physics-Based Approach To Modeling Wildland Fire Spread Through Porous Fuel Beds, Tingting Tang

Theses and Dissertations--Mechanical and Aerospace Engineering

Wildfires are becoming increasingly erratic nowadays at least in part because of climate change. CFD (computational fluid dynamics)-based models with the potential of simulating extreme behaviors are gaining increasing attention as a means to predict such behavior in order to aid firefighting efforts. This dissertation describes a wildfire model based on the current understanding of wildfire physics. The model includes physics of turbulence, inhomogeneous porous fuel beds, heat release, ignition, and firebrands. A discrete dynamical system for flow in porous media is derived and incorporated into the subgrid-scale model for synthetic-velocity large-eddy simulation (LES), and a general porosity-permeability model is …

Autonomous Quadrotor Collision Avoidance And Destination Seeking In A Gps-Denied Environment, Thomas C. Kirven

Theses and Dissertations--Mechanical and Aerospace Engineering

This thesis presents a real-time autonomous guidance and control method for a quadrotor in a GPS-denied environment. The quadrotor autonomously seeks a destination while it avoids obstacles whose shape and position are initially unknown. We implement the obstacle avoidance and destination seeking methods using off-the-shelf sensors, including a vision-sensing camera. The vision-sensing camera detects the positions of points on the surface of obstacles. We use this obstacle position data and a potential-field method to generate velocity commands. We present a backstepping controller that uses the velocity commands to generate the quadrotor's control inputs. In indoor experiments, we demonstrate that the …

Triple Non-Negative Matrix Factorization Technique For Sentiment Analysis And Topic Modeling, Alexander A. Waggoner

CMC Senior Theses

Topic modeling refers to the process of algorithmically sorting documents into categories based on some common relationship between the documents. This common relationship between the documents is considered the “topic” of the documents. Sentiment analysis refers to the process of algorithmically sorting a document into a positive or negative category depending whether this document expresses a positive or negative opinion on its respective topic. In this paper, I consider the open problem of document classification into a topic category, as well as a sentiment category. This has a direct application to the retail industry where companies may want to scour …

Cyclic Codes And Cyclic Lattices, Scott Maislin

CMC Senior Theses

In this thesis, we review basic properties of linear codes and lattices with a certain focus on their interplay. In particular, we focus on the analogous con- structions of cyclic codes and cyclic lattices. We start out with a brief overview of the basic theory and properties of linear codes. We then demonstrate the construction of cyclic codes and emphasize their importance in error-correcting coding theory. Next we survey properties of lattices, focusing on algorithmic lattice problems, exhibit the construction of cyclic lattices and discuss their applications in cryptography. We emphasize the similarity and common prop- erties of the two …

Long And Short-Range Air Navigation On Spherical Earth, Nihad E. Daidzic

International Journal of Aviation, Aeronautics, and Aerospace

Global range air navigation implies non-stop flight between any two airports on Earth. Such effort would require airplanes with the operational air range of at least 12,500 NM which is about 40-60% longer than anything existing in commercial air transport today. Air transportation economy requires flying shortest distance, which in the case of spherical Earth are Orthodrome arcs. Rhumb-line navigation has little practical use in long-range flights, but has been presented for historical reasons and for comparison. Database of about 50 major international airports from every corner of the world has been designed and used in testing and route validation. …

Forcing Optimality And Brandt's Principle, Domenico Napoletani, Marco Panza, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

We argue that many optimization methods can be viewed as representatives of “forcing”, a methodological approach that attempts to bridge the gap between data and mathematics on the basis of an a priori trust in the power of a mathematical technique, even when detailed, credible models of a phenomenon are lacking or do not justify the use of this technique. In particular, we show that forcing is implied in particle swarms optimization methods, and in modeling image processing problems through optimization. From these considerations, we extrapolate a principle for general data analysis methods, what we call ‘Brandt’s principle’, namely the …

Plithogeny, Plithogenic Set, Logic, Probability, And Statistics, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce for the first time, as generalization of dialectics and neutrosophy, the philosophical concept called plithogeny. And as its derivatives: the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), plithogenic logic (as generalization of classical, fuzzy, intuitionistic fuzzy, and neutrosophic logics), plithogenic probability (as generalization of classical, imprecise, and neutrosophic probabilities), and plithogenic statistics (as generalization of classical, and neutrosophic statistics).

Plithogeny is the genesis or origination, creation, formation, development, and evolution of new entities from dynamics and organic fusions of contradictory and/or neutrals and/or non-contradictory multiple old entities.

Plithogenic …

A Wasserstein Gradient Flow Approach To Poisson-Nernst-Planck Equations, David Kinderlehrer, Leinard Monsaingeon, Xiang Xu

Mathematics & Statistics Faculty Publications

The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of global weak solutions in a unified framework for the cases of both linear and non-linear diffusion. The proof of the main results relies on the derivation of additional estimates based on the flow interchange technique developed by Matthes et al. in [D. Matthes, R.J. McCann and G. Savare, Commun. Partial …

An Application Of Time Series Methods To Investigate Eu Dairy Commodity Price Dynamics, Dennis Bergmann

Theses

Fundamental changes to the common agricultural policy (CAP) have led to greater market orientation which in turn has resulted in significantly changed price dynamics of EU dairy prices. These changed dynamics include sharply increased price variability of EU farm gate milk prices as well EU dairy commodity prices such as butter and Skimmed Milk Powder (SMP). In addition it was expected that EU dairy prices will align to Word dairy prices. This alignment of prices would also see an alignment of volatility levels which were significantly higher for World dairy prices than EU dairy prices. The aim of this thesis …

On A Time Domain Boundary Integral Equation Formulation For Acoustic Scattering By Rigid Bodies In Uniform Mean Flow, Fang Q. Hu, Michelle E. Pizzo, Douglas M. Nark

Mathematics & Statistics Faculty Publications

It has been well-known that under the assumption of a uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation. However, the constant mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the assumed uniform flow. A customary boundary condition for rigid surfaces is that the normal acoustic velocity be zero. In this paper, a careful study of the acoustic energy conservation equation is presented that shows such a boundary condition would in fact lead to source …

Joint Hierarchical Models For Sparsely Sampled High-Dimensional Lidar And Forest Variables, Andrew O. Finley, Sudipto Banerjee, Yuzhen Zhou, Bruce D. Cook, Chad Babcock

United States National Aeronautics and Space Administration: Publications

Recent advancements in remote sensing technology, specifically Light Detection and Ranging (LiDAR) sensors, provide the data needed to quantify forest characteristics at a fine spatial resolution over large geographic domains. From an inferential standpoint, there is interest in prediction and interpolation of the often sparsely sampled and spatially misaligned LiDAR signals and forest variables. We propose a fully process-based Bayesian hierarchical model for above ground biomass (AGB) and LiDAR signals. The processbased framework offers richness in inferential capabilities, e.g., inference on the entire underlying processes instead of estimates only at pre-specified points. Key challenges we obviate include misalignment between the …

Series Solutions Of Polarized Gowdy Universes, Doniray Brusaferro

Theses and Dissertations

Einstein's field equations are a system of ten partial differential equations. For a special class of spacetimes known as Gowdy spacetimes, the number of equations is reduced due to additional structure of two dimensional isometry groups with mutually orthogonal Killing vectors. In this thesis, we focus on a particular model of Gowdy spacetimes known as the polarized T³ model, and provide an explicit solution to Einstein's equations.

A Markov Decision Process Approach To Adaptive Contact Strategies, Artur Grygorian

College of Graduate Studies: Theses & Dissertations

In the field of survey methodology, optimizing contact strategies helps organizations increase response rates using their allocated budget. Markov Decision Processes (MDP) are widely used to model decision-making strategies in situations where the outcomes have a random component. In this research, we use MDPs and adaptive sampling techniques to construct a strategy that, based on target audience characteristics, suggests the best contact policy. The data we use comes from the First Destination Survey conducted by the Office of Career Services at Georgia Southern University. The constructed model is quite flexible and can be used by other organizations to optimize their …

The Effects Of Disturbance And Species Specific Interactions On Diversity In An Agent Based Forest Simulation, Matthew E. Mills

Theses and Dissertations

In ecology literature, there is much data which suggests that conspecific negative density dependence (CNDD) and abiotic disturbances increase biodiversity in forests. This thesis elucidates the notion that not only do these two forces increase diversity, but they may also interact with one another in order to achieve higher levels of biodiversity. Abiotic disturbances, like fires and hurricanes, can indirectly impact conspecific effects because when these forces remove individuals from the landscape, the role of the conspecific effects will change. The interaction of these two factors in biodiversity are explored in an agent based forest simulation through a resource surface. …

Computational Fluid Dynamics In A Terminal Alveolated Bronchiole Duct With Expanding Walls: Proof-Of-Concept In Openfoam, Jeremy Myers

Theses and Dissertations

Mathematical Biology has found recent success applying Computational Fluid Dynamics (CFD) to model airflow in the human lung. Detailed modeling of flow patterns in the alveoli, where the oxygen-carbon dioxide gas exchange occurs, has provided data that is useful in treating illnesses and designing drug-delivery systems. Unfortunately, many CFD software packages have high licensing fees that are out of reach for independent researchers. This thesis uses three open-source software packages, Gmsh, OpenFOAM, and ParaView, to design a mesh, create a simulation, and visualize the results of an idealized terminal alveolar sac model. This model successfully demonstrates that OpenFOAM can be …

Network Analytics For The Mirna Regulome And Mirna-Disease Interactions, Joseph Jayakar Nalluri

Theses and Dissertations

miRNAs are non-coding RNAs of approx. 22 nucleotides in length that inhibit gene expression at the post-transcriptional level. By virtue of this gene regulation mechanism, miRNAs play a critical role in several biological processes and patho-physiological conditions, including cancers. miRNA behavior is a result of a multi-level complex interaction network involving miRNA-mRNA, TF-miRNA-gene, and miRNA-chemical interactions; hence the precise patterns through which a miRNA regulates a certain disease(s) are still elusive. Herein, I have developed an integrative genomics methods/pipeline to (i) build a miRNA regulomics and data analytics repository, (ii) create/model these interactions into networks and use optimization techniques, motif …

Interaction Graphs Derived From Activation Functions And Their Application To Gene Regulation, Simon Joyce

Graduate Dissertations and Theses

Interaction graphs are graphic representations of complex networks of mutually interacting components. Their main application is in the field of gene regulatory networks, where they are used to visualize how the expression levels of genes activate or inhibit the expression levels of other genes.

First we develop a natural transformation of activation functions and their derived interaction graphs, called conjugation, that is related to a natural transformation of signed digraphs called switching isomorphism. This is a useful tool for the analysis of interaction graphs used throughout the rest of the dissertation.

We then discuss the question of what restrictions, if …

Compressive Vector Reconstruction: Hypothesis For Blind Image Deconvolution, Alonso Orea Amador

Open Access Theses & Dissertations

Alternative imaging devices propose to acquire and compress images simultaneously. These devices are based on the compressive sensing (CS) theory. A reduction in the measurement required for reconstruction without a post-compression sub-system allows imaging devices to become simpler, smaller, and cheaper. In this research, we propose a new algorithm to compress and reconstruct blurred images for CS imaging devices. Blur effect in images is common due to relative motion, lens, limited aperture dimensions, lack of focus, and/or atmospheric turbulence. Our intention is to compress a blurred image with CS techniques and then reconstruct a blur-free version using the proposed algorithm. …

Disease Models With Immigration, Reem Almarashi

Theses and Dissertations (Comprehensive)

In this thesis we focus first on studying the susceptible, exposed, and infected ($SEI$) disease model without immigration. We determine the basic reproduction number $\mathcal{R}_0$, which can be interpreted as the expected number of new cases that can be produced by a single infection in a completely susceptible population. Further, by using the Jacobian matrix, we determine the local stability of the disease model. Then we have the result that when $\mathcal{R}_0<1$ the DFE point is locally asymptotically stable(L.A.S). In contrast, when $\mathcal{R}_0>1$ we find that the endemic equilibrium is L.A.S. After that, we analyze the $SEI$ model with immigration of infected individuals.

Furthermore, we investigate the direction that the …

Mathematical Models Of The Inflammatory Response In The Lungs, Sarah B. Minucci

Theses and Dissertations

Inflammation in the lungs can occur for many reasons, from bacterial infections to stretch by mechanical ventilation. In this work we compare and contrast various mathematical models for lung injuries in the categories of acute infection, latent versus active infection, and particulate inhalation. We focus on systems of ordinary differential equations (ODEs), agent-based models (ABMs), and Boolean networks. Each type of model provides different insight into the immune response to damage in the lungs. This knowledge includes a better understanding of the complex dynamics of immune cells, proteins, and cytokines, recommendations for treatment with antibiotics, and a foundation for more …

Fitting A Linear Regression Model And Forecasting In R In The Presence Of Heteroskedascity With Particular Reference To Advanced Regression Technique Dataset On Kaggle.Com., Samuel Mbah Nde

All Student Theses and Dissertations

Since ancient times, men have built and sold houses. But just how much is a house worth? The challenge is to be able to use information about a house such as its location, and the area on which it is built to predict its price. Such predicted prices can be of great importance to any participant in the real estate business be it an agent, a buyer, seller or a bank to make intelligent decisions and the profit that come with such decisions. Since every company’s success depends on its ability to accurately predict financial outcomes, its profitability will depend …

An Introduction To The Lebesgue Integral, Ikhlas Adi

All Student Theses and Dissertations

The Riemann integral is the simplest integral to define, and it allows one to integrate every continuous function. It is really important to have a definition of the integral that allows a wider class of functions to be integrated. However, there are many other types of integrals, the most important of which is the Lebesgue integral. The Lebesgue integral allows one to integrate unbounded or discontinuous functions whose Riemann integral does not exist, and it has mathematical properties that the Riemann integral does not. The definition of the Lebesgue integral requires the use of measure theory since picking out a …

Multi-Scale Cardiovascular Flow Analysis By An Integrated Meshless-Lumped Parameter Model, Leonardo A. Bueno, Eduardo A. Divo, Alain J. Kassab

Publications

A computational tool that integrates a Radial basis function (RBF)-based Meshless solver with a Lumped Parameter model (LPM) is developed to analyze the multi-scale and multi-physics interaction between the cardiovascular flow hemodynamics, the cardiac function, and the peripheral circulation. The Meshless solver is based on localized RBF collocations at scattered data points which allows for automation of the model generation via CAD integration. The time-accurate incompressible flow hemodynamics are addressed via a pressure-velocity correction scheme where the ensuing Poisson equations are accurately and efficiently solved at each time step by a Dual-Reciprocity Boundary Element method (DRBEM) formulation that takes advantage …

The Bessel Function, The Hankel Transform And An Application To Differential Equations, Isaac C. Voegtle

College of Graduate Studies: Theses & Dissertations

In this thesis we explore the properties of Bessel functions. Of interest is how they can be applied to partial differential equations using the Hankel transform. We use an example in two dimensions to demonstrate the properties at work as well as formulate thoughts on how to take the results further.

A New Formulation Of Time Boundary Integral Equation For Acoustic Wave Scattering In The Presence Of A Uniform Mean Flow, Fang Q. Hu, Michelle E. Pizzo, Douglas M. Nark

Mathematics & Statistics Faculty Publications

It has been well-known that under the assumption of a constant uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation, in both the time domain and the frequency domain. Compared with solving partial differential equations, numerical methods based on the boundary integral equation have the advantage of a reduced spatial dimension and, hence, requiring only a surface mesh. However, the constant uniform mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the uniform mean flow. In …

Fire, Ice, Water, And Dirt: A Simple Climate Model, John Kroll

Mathematics & Statistics Faculty Publications

A simple paleoclimate model was developed as a modeling exercise. The model is a lumped parameter system consisting of an ocean (water), land (dirt), glacier, and sea ice (ice) and driven by the sun (fire). In comparison with other such models, its uniqueness lies in its relative simplicity yet yielding good results. For nominal values of parameters, the system is very sensitive to small changes in the parameters, yielding equilibrium, steady oscillations, and catastrophes such as freezing or boiling oceans. However, stable solutions can be found, especially naturally oscillating solutions. For nominally realistic conditions, natural periods of order 100kyrs are …

Paving The Randomized Gauss-Seidel, Wei Wu

Scripps Senior Theses

The Randomized Gauss-Seidel Method (RGS) is an iterative algorithm that solves overdetermined systems of linear equations Ax = b. This paper studies an update on the RGS method, the Randomized Block Gauss-Seidel Method. At each step, the algorithm greedily minimizes the objective function L(x) = kAx bk2 with respect to a subset of coordinates. This paper describes a Randomized Block Gauss-Seidel Method (RBGS) which uses a randomized control method to choose a subset at each step. This algorithm is the first block RGS method with an expected linear convergence rate which can be described by the properties of the matrix …