Filtered-Dynamic-Inversion Control For Unknown Minimum-Phase Systems With Unknown Relative Degree, 2020 University of Kentucky

#### Filtered-Dynamic-Inversion Control For Unknown Minimum-Phase Systems With Unknown Relative Degree, Sumit Suryakant Kamat

*Theses and Dissertations--Mechanical Engineering*

We present filtered-dynamic-inversion (FDI) control for unknown linear time-invariant systems that are multi-input multi-output and minimum phase with unknown-but-bounded relative degree. This FDI controller requires limited model information, specifically, knowledge of an upper bound on the relative degree and knowledge of the first nonzero Markov parameter. The FDI controller is a single-parameter high-parameter-stabilizing controller that is robust to uncertainty in the relative degree. We characterize the stability of the closed-loop system. We present numerical examples, where the FDI controller is implemented in feedback with mathematical and physical systems. The numerical examples demonstrate that the FDI controller for unknown relative degree ...

Modeling Gene Expression With Differential Equations, 2020 Arcadia University

#### Modeling Gene Expression With Differential Equations, Madison Kuduk

*Capstone Showcase*

Gene expression is the process by which the information stored in DNA is convertedinto a functional gene product, such as protein. The two main functions that makeup the process of gene expression are transcription and translation. Transcriptionand translation are controlled by the number of mRNA and protein in the cell. Geneexpression can be represented as a system of first order differential equations for the rateof change of mRNA and proteins. These equations involve transcription, translation,degradation and feedback loops. In this paper, I investigate a system of first orderdifferential equations to model gene expression proposed by Hunt, Laplace, Miller andPham ...

Phylogenetic Networks And Functions That Relate Them, 2020 The University of Akron

#### Phylogenetic Networks And Functions That Relate Them, Drew Scalzo

*Williams Honors College, Honors Research Projects*

Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate to the ...

Genetic Algorithms Used For Search And Rescue Of Vulnerable People In An Urban Setting, 2020 West Virginia University

#### Genetic Algorithms Used For Search And Rescue Of Vulnerable People In An Urban Setting, Shuhad Aljandeel

*Graduate Theses, Dissertations, and Problem Reports*

The main goal of this research is to design and develop a genetic algorithm (GA) for path planning of an Unmanned Aerial Vehicle (UAV) outfitted with a camera to efficiently search for a lost person in an area of interest. The research focuses on scenarios where the lost person is from a vulnerable population, such as someone suffering from Alzheimer or a small child who has wondered off. To solve this problem, a GA for path planning was designed and implemented in Matlab. The area of interest is considered to be a circle that encompasses the distance the person could ...

Convergence Of Approximate Solutions To Nonlinear Caputo Nabla Fractional Difference Equations With Boundary Conditions, 2020 Sun Yat-sen University

#### Convergence Of Approximate Solutions To Nonlinear Caputo Nabla Fractional Difference Equations With Boundary Conditions, Xiang Liu, Baoguo Jia, Scott Gensler, Lynn Erbe, Allan Peterson

*Faculty Publications, Department of Mathematics*

This article studies a boundary value problem for a nonlinear Ca- puto nabla fractional difference equation. We obtain quadratic convergence results for this equation using the generalized quasi-linearization method. Fur- ther, we obtain the convergence of the sequences is potentially improved by the Gauss-Seidel method. A numerical example illustrates our main results.

Simulation Based Inference In Epidemic Models, 2020 Montclair State University

#### Simulation Based Inference In Epidemic Models, Tejitha Dharmagadda

*Theses, Dissertations and Culminating Projects*

From ancient times to the modern day, public health has been an area of great interest. Studies on the nature of disease epidemics began around 400 BC and has been a continuous area of study for the well-being of individuals around the world. For over 100 years, epidemiologists and mathematicians have developed numerous mathematical models to improve our understanding of infectious disease dynamics with an eye on controlling and preventing disease outbreak and spread. In this thesis, we discuss several types of mathematical compartmental models such as the *SIR, *and *SIS *models. To capture the noise inherent in the real-world ...

A New Class Of Discontinuous Galerkin Methods For Wave Equations In Second-Order Form, 2020 Southern Methodist University

#### A New Class Of Discontinuous Galerkin Methods For Wave Equations In Second-Order Form, Lu Zhang

*Mathematics Theses and Dissertations*

Discontinuous Galerkin methods are widely used in many practical fields. In this thesis, we focus on a new class of discontinuous Galerkin methods for second-order wave equations. This thesis is constructed by three main parts. In the first part, we study the convergence properties of the energy-based discontinuous Galerkin proposed in [3] for wave equations. We improve the existing suboptimal error estimates to an optimal convergence rate in the energy norm. In the second part, we generalize the energy-based discontinuous Galerkin method proposed in [3] to the advective wave equation and semilinear wave equation in second-order form. Energy-conserving or energy-dissipating ...

Exploration And Implementation Of Neural Ordinary Differential Equations, 2020 Arcadia University

#### Exploration And Implementation Of Neural Ordinary Differential Equations, Long Huu Nguyen, Andy Malinsky

*Capstone Showcase*

Neural ordinary differential equations (ODEs) have recently emerged as a novel ap- proach to deep learning, leveraging the knowledge of two previously separate domains, neural networks and differential equations. In this paper, we first examine the back- ground and lay the foundation for traditional artificial neural networks. We then present neural ODEs from a rigorous mathematical perspective, and explore their advantages and trade-offs compared to traditional neural nets.

Evaluating An Ordinal Output Using Data Modeling, Algorithmic Modeling, And Numerical Analysis, 2020 Murray State University

#### Evaluating An Ordinal Output Using Data Modeling, Algorithmic Modeling, And Numerical Analysis, Martin Keagan Wynne Brown

*Murray State Theses and Dissertations*

Data and algorithmic modeling are two diﬀerent approaches used in predictive analytics. The models discussed from these two approaches include the proportional odds logit model (POLR), the vector generalized linear model (VGLM), the classiﬁcation and regression tree model (CART), and the random forests model (RF). Patterns in the data were analyzed using trigonometric polynomial approximations and Fast Fourier Transforms. Predictive modeling is used frequently in statistics and data science to ﬁnd the relationship between the explanatory (input) variables and a response (output) variable. Both approaches prove advantageous in diﬀerent cases depending on the data set. In our case, the data ...

Symbolic Construction Of Matrix Functions In A Numerical Environment, 2020 Georgia Southern University

#### Symbolic Construction Of Matrix Functions In A Numerical Environment, Evan D. Butterworth

*University Honors Program Theses*

Within the field of Computational Science, the importance of programs and tools involving systems of differential equations cannot be overemphasized. Many industrial sites, such as nuclear power facilities, are unable to safely operate without these systems. This research explores and studies matrix differential equations and their applications to real computing structures. Through the use of software such as MatLab, I have constructed a toolbox, or collection, of programs that will allow any user to easily calculate a variety of matrix functions. The first tool in this collection is a program that computes the matrix exponential, famously studied and presented by ...

Classifying Flow-Kick Equilibria: Reactivity And Transient Behavior In The Variational Equation, 2020 Bowdoin College

#### Classifying Flow-Kick Equilibria: Reactivity And Transient Behavior In The Variational Equation, Alanna Haslam

*Honors Projects*

In light of concerns about climate change, there is interest in how sustainable management can maintain the resilience of ecosystems. We use flow-kick dynamical systems to model ecosystems subject to a constant kick occurring every τ time units. We classify the stability of flow-kick equilibria to determine which management strategies result in desirable long-term characteristics. To classify the stability of a flow-kick equilibrium, we classify the linearization of the time-τ map given by the time-τ map of the variational equation about the equilibrium trajectory. Since the variational equation is a non-autonomous linear differential equation, we conjecture that the asymptotic stability ...

Modeling The Evolution Of Barrier Islands, 2020 Virginia Commonwealth University

#### 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.

Understanding The Human Gut Microbiota: A Mathematical Approach, 2020 University of Iowa

#### Understanding The Human Gut Microbiota: A Mathematical Approach, Melissa Adrian

*Honors Theses at the University of Iowa*

In order to explore the dynamics of the human gut microbiota, we used a system of ordinary differential equations to mathematically model the biomass of three microorganism populations: *Bacteroides thetaiotaomicron*, *Eubacterium rectale*, and *Methanobrevibacter smithii*. Additionally, we modeled the concentrations of relevant nutrients necessary to sustain these populations over time. This system highlights the interactions and the competition among these species in order to further understand their dynamics. These three microorganisms were speciﬁcally chosen due to the system’s end product, butyrate, which aids in developing the intestinal barrier in the human gut. The basis of the mathematical model assumes ...

Interpretations Of Bicoherence In Space & Lab Plasma Dynamics, 2020 West Virginia University

#### Interpretations Of Bicoherence In Space & Lab Plasma Dynamics, Gregory Allen Riggs

*Graduate Theses, Dissertations, and Problem Reports*

The application of bicoherence analysis to plasma research, particularly in non-linear, coupled-wave regimes, has thus far been significantly belied by poor resolution in time, and/or outright destruction of frequency information. Though the typical power spectrum cloaks the phase-coherency between frequencies, Fourier transforms of higher-order convolutions provide an *n*-dimensional spectrum which is adept at elucidating *n*-wave phase coherence. As such, this investigation focuses on the utility of the normalized bispectrum for detection of wave-wave coupling in general, with emphasis on distinct implications within the scope of non-linear plasma physics. Interpretations of bicoherent features are given for time series ...

Elucidating The Properties And Mechanism For Cellulose Dissolution In Tetrabutylphosphonium-Based Ionic Liquids Using High Concentrations Of Water, 2020 West Virginia University

#### Elucidating The Properties And Mechanism For Cellulose Dissolution In Tetrabutylphosphonium-Based Ionic Liquids Using High Concentrations Of Water, Brad Crawford

*Graduate Theses, Dissertations, and Problem Reports*

The structural, transport, and thermodynamic properties related to cellulose dissolution by tetrabutylphosphonium chloride (TBPCl) and tetrabutylphosphonium hydroxide (TBPH)-water mixtures have been calculated via molecular dynamics simulations. For both ionic liquid (IL)-water solutions, water veins begin to form between the TBPs interlocking arms at 80 mol % water, opening a pathway for the diffusion of the anions, cations, and water. The water veins allow for a diffusion regime shift in the concentration region from 80 to 92.5 mol % water, providing a higher probability of solvent interaction with the dissolving cellulose strand. The hydrogen bonding was compared between small and ...

The Singular Value Expansion For Compact And Non-Compact Operators, 2020 Michigan Technological University

#### The Singular Value Expansion For Compact And Non-Compact Operators, Daniel Crane

*Dissertations, Master's Theses and Master's Reports*

Given any bounded linear operator *T *: *X → Y *between separable Hilbert spaces *X *and *Y *, there exists a measure space (*M, Α**, µ*) and isometries *V *: *L*^{2}(*M*) *→* *X*, *U *: *L*^{2}(*M*) *→ **Y *and a nonnegative, bounded, measurable function *σ *: *M → *[0

*, ∞*) such that

*T *= *Um*_{σ}*V *^{†}*,*

with *m*_{σ}* *: *L*^{2}(*M** *) *→ **L*^{2}(*M** *) defined by *m** _{σ}*(

*f*

*) =*

*σf*for all

*f ∈*

*L*

^{2}(

*M*

*). The expansion*

*T*=

*Um*

_{σ}V

^{†}*is called the singular value expansion (SVE) of*

*T*.

The SVE is a useful tool for analyzing a number of problems such as ...

Sub-Sampled Matrix Approximations, 2020 Michigan Technological University

#### Sub-Sampled Matrix Approximations, Joy Azzam

*Dissertations, Master's Theses and Master's Reports*

Matrix approximations are widely used to accelerate many numerical algorithms. Current methods sample row (or column) spaces to reduce their computational footprint and approximate a matrix A with an appropriate embedding of the data sampled. This work introduces a novel family of randomized iterative algorithms which use significantly less data per iteration than current methods by sampling input and output spaces simultaneously. The data footprint of the algorithms can be tuned (independent of the underlying matrix dimension) to available hardware. Proof is given for the convergence of the algorithms, which are referred to as sub-sampled, in terms of numerically tested ...

Orthogonal Recurrent Neural Networks And Batch Normalization In Deep Neural Networks, 2020 University of Kentucky

#### Orthogonal Recurrent Neural Networks And Batch Normalization In Deep Neural Networks, Kyle Eric Helfrich

*Theses and Dissertations--Mathematics*

Despite the recent success of various machine learning techniques, there are still numerous obstacles that must be overcome. One obstacle is known as the vanishing/exploding gradient problem. This problem refers to gradients that either become zero or unbounded. This is a well known problem that commonly occurs in Recurrent Neural Networks (RNNs). In this work we describe how this problem can be mitigated, establish three different architectures that are designed to avoid this issue, and derive update schemes for each architecture. Another portion of this work focuses on the often used technique of batch normalization. Although found to be ...

Mathematical Modeling Of Diabetic Foot Ulcers Using Optimal Design And Mixed-Modeling Techniques, 2020 Western Kentucky University

#### Mathematical Modeling Of Diabetic Foot Ulcers Using Optimal Design And Mixed-Modeling Techniques, Michael Belcher

*Honors College Capstone Experience/Thesis Projects*

A mathematical model for the healing response of diabetic foot ulcers was developed using averaged data (Krishna *et al.*, 2015). The model contains four major factors in the healing of wounds using four separate differential equations with 12 parameters. The four differential equations describe the interactions between matrix metalloproteinases (MMP-1), tissue inhibitors of matrix metalloproteinases (TIMP-1), the extracellular matrix (ECM) of the skin, and the fibroblasts, which produce these proteins. Recently, our research group obtained the individual patient data that comprised the averaged data. The research group has since taken several approaches to analyze the model with the individual patient ...

Connectivity Differences Between Gulf War Illness (Gwi) Phenotypes During A Test Of Attention, 2019 Georgetown University

#### Connectivity Differences Between Gulf War Illness (Gwi) Phenotypes During A Test Of Attention, Tomas Clarke, Jessie Jamieson, Patrick Malone, Rakib U. Rayhan, Stuart Washington, John W. Vanmeter, James N. Baraniuk

*Faculty Publications, Department of Mathematics*

One quarter of veterans returning from the 1990–1991 Persian Gulf War have developed Gulf War Illness (GWI) with chronic pain, fatigue, cognitive and gastrointestinal dysfunction. Exertion leads to characteristic, delayed onset exacerbations that are not relieved by sleep. We have modeled exertional exhaustion by comparing magnetic resonance images from before and after submaximal exercise. One third of the 27 GWI participants had brain stem atrophy and developed postural tachycardia after exercise (START: Stress Test Activated Reversible Tachycardia). The remainder activated basal ganglia and anterior insulae during a cognitive task (STOPP: Stress Test Originated Phantom Perception). Here, the role of ...