Other Applied Mathematics Commons^™

The Feasibility Of Utilizing The Open Dynamic Interaction Network (Odin) App To Assess Rema Data Across 30 Days Among Those Recovering From Alcoholism, Dennis E. Mcchargue, Bilal Khan, Jesscia Phelps, Patrick Duryea, Kimberly A. Tyler, Arthur Andrews, Ellie Reznicek, Lucy Napper, Mohamed Saad, Hsuan-Wee Lee

Department of Psychology: Faculty Publications

About ~4 million people with an alcohol use disorder receive treatment annually, and frequent relapse has produced a rapid revolving door between recovery and use. This paper presents preliminary data from a prospective micro-longitudinal study (30 days) that examines co-evolution of relapse risk processes of people with alcohol use disorders who entered a short-term residential substance use treatment. The primary goal of the current data was to assess the feasibility of using the open dynamic interaction network (ODIN) responsive ecological momentary assessment (rEMA). rEMA collected daily estimates on affect, urges, sober-support engagement, and use. The ODIN app administered twelve questions …

A Cohomological Perspective To Nonlocal Operators, Nicholas White

Honors Program: Senior Projects (Public)

Nonlocal models have experienced a large period of growth in recent years. In particular, nonlocal models centered around a finite horizon have been the subject of many novel results. In this work we consider three nonlocal operators defined via a finite horizon: a weighted averaging operator in one dimension, an averaging differential operator, and the truncated Riesz fractional gradient. We primarily explore the kernel of each of these operators when we restrict to open sets. We discuss how the topological structure of the domain can give insight into the behavior of these operators, and more specifically the structure of their …

Game-Theoretic Approaches To Optimal Resource Allocation And Defense Strategies In Herbaceous Plants, Molly R. Creagar

Department of Mathematics: Dissertations, Theses, and Student Research

Empirical evidence suggests that the attractiveness of a plant to herbivores can be affected by the investment in defense by neighboring plants, as well as investment in defense by the focal plant. Thus, allocation to defense may not only be influenced by the frequency and intensity of herbivory but also by defense strategies employed by other plants in the environment. We incorporate a neighborhood defense effect by applying spatial evolutionary game theory to optimal resource allocation in plants where cooperators are plants investing in defense and defectors are plants that do not. We use a stochastic dynamic programming model, along …

Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper

Honors Program: Senior Projects (Public)

Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …

A Mathematical Model Of Speeding, Jared Ott, Xavier Pérez Giménez

Honors Program: Senior Projects (Public)

Crime is often regarded as nonsensical, impulsive, and irrational. These conjectures are pointed, though conversation about the pros and cons of crime does not happen often. People point to harsh fines, jail times, and life restrictions as their reason for judgement, stating that the trade-offs are far too unbalanced to participate in illicit activity. Yet, everyday people commit small crimes, sometimes based on hedonistic desires, other times based on a rational thought process.

Speeding seems to be one of those that almost all people commit at least once during their life. Our work hopes to make an incremental improvement on …

Individual Based Model To Simulate The Evolution Of Insecticide Resistance, William B. Jamieson

Department of Mathematics: Dissertations, Theses, and Student Research

Insecticides play a critical role in agricultural productivity. However, insecticides impose selective pressures on insect populations, so the Darwinian principles of natural selection predict that resistance to the insecticide is likely to form in the insect populations. Insecticide resistance, in turn, severely reduces the utility of the insecticides being used. Thus there is a strong economic incentive to reduce the rate of resistance evolution. Moreover, resistance evolution represents an example of evolution under novel selective pressures, so its study contributes to the fundamental understanding of evolutionary theory.

Insecticide resistance often represents a complex interplay of multiple fitness trade-offs for individual …

Essentials Of Structural Equation Modeling, Mustafa Emre Civelek

Zea E-Books Collection

Structural Equation Modeling is a statistical method increasingly used in scientific studies in the fields of Social Sciences. It is currently a preferred analysis method, especially in doctoral dissertations and academic researches. However, since many universities do not include this method in the curriculum of undergraduate and graduate courses, students and scholars try to solve the problems they encounter by using various books and internet resources.

This book aims to guide the researcher who wants to use this method in a way that is free from math expressions. It teaches the steps of a research program using structured equality modeling …

Investigation Of Toppling Ball Flight In American Football With A Mechanical Field-Goal Kicker, Chase M Pfeifer, Timothy J. Gay, Jeff A. Hawks, Shane Farritor, Judith M. Burnfield

Department of Mechanical and Materials Engineering: Faculty Publications

A mechanical field-goal kicking machine was used to investigate toppling ball flight in American football place-kicking, eliminating a number of uncontrollable impact variables present with a human kicker. Ball flight trajectories were recorded using a triangulation-based projectile tracking system to account for the football’s 3-dimensional position during flight as well as initial launch conditions. The football flights were described using kinematic equations relating to projectile motion including stagnant air drag and were compared to measured trajectories as well as projectile motion equations that exclude stagnant air drag. Measured football flight range deviations from the non-drag equations of projectile motion corresponded …

Bioinformatic Game Theory And Its Application To Cluster Multi-Domain Proteins, Brittney Keel

Department of Mathematics: Dissertations, Theses, and Student Research

The exact evolutionary history of any set of biological sequences is unknown, and all phylogenetic reconstructions are approximations. The problem becomes harder when one must consider a mix of vertical and lateral phylogenetic signals. In this dissertation we propose a game-theoretic approach to clustering biological sequences and analyzing their evolutionary histories. In this context we use the term evolution as a broad descriptor for the entire set of mechanisms driving the inherited characteristics of a population. The key assumption in our development is that evolution tries to accommodate the competing forces of selection, of which the conservation force seeks to …

Clique Topology Reveals Intrinsic Geometric Structure In Neural Correlations, Chad Giusti, Eva Pastalkova, Carina Curto, Vladimir Itskov

Department of Mathematics: Faculty Publications

Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a 2D environment, and confirmed that our method is able to detect geometric organization using …

The Neural Ring: Using Algebraic Geometry To Analyze Neural Codes, Nora Youngs

Department of Mathematics: Dissertations, Theses, and Student Research

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects …

Combinatorial And Algebraic Coding Techniques For Flash Memory Storage, Kathryn A. Haymaker

Department of Mathematics: Dissertations, Theses, and Student Research

Error-correcting codes are used to achieve reliable and efficient transmission when storing or sending information across a noisy channel. This thesis investigates a mathematical approach to coding techniques for storage devices such as flash memory storage, although many of the resulting codes and coding schemes can be applied in other contexts. The main contributions of this work include the design of efficient codes and decoding algorithms using discrete structures such as graphs and finite geometries, and developing a variety of strategies for adapting codes to a multi-level setting.

Information storage devices are prone to errors over time, and the frequency …

A Non-Erasable Magnetic Memory Based On The Magnetic Permeability, J R. Petrie, K A. Wieland, R A. Burke, G A. Newburgh, J E. Burnette, G A. Fisher, A S. Edelstein

United States Army: Publications

A non-erasable memory based on using differences in the magnetic permeability is demonstrated. The method can potentially store information indefinitely. Initially the high permeability bits were 10–50 μm wide lines of sputtered permalloy (Ni₈₁Fe₁₉) on a glass substrate. In a second writing technique a continuous film of amorphous, high permeability ferromagnetic Metglas (Fe78Si13Bg) was sputtered onto a similar glass substrate. Low permeability, crystalline 50 Jlm wide lines were then written in the film by laser heating. Both types of written media were read by applying an external probe field that is locally modified by the permeability …

Random Search Models Of Foraging Behavior: Theory, Simulation, And Observation, Ben C. Nolting

Department of Mathematics: Dissertations, Theses, and Student Research

Many organisms, from bacteria to primates, use stochastic movement patterns to find food. These movement patterns, known as search strategies, have recently be- come a focus of ecologists interested in identifying universal properties of optimal foraging behavior. In this dissertation, I describe three contributions to this field. First, I propose a way to extend Charnov's Marginal Value Theorem to the spatially explicit framework of stochastic search strategies. Next, I describe simulations that compare the efficiencies of sensory and memory-based composite search strategies, which involve switching between different behavioral modes. Finally, I explain a new behavioral analysis protocol for identifying the …

Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

Department of Mathematics: Dissertations, Theses, and Student Research

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …

On Morrey Spaces In The Calculus Of Variations, Kyle Fey

Department of Mathematics: Dissertations, Theses, and Student Research

We prove some global Morrey regularity results for almost minimizers of functionals of the form u → ∫_Ω f(x, u, ∇u)dx. This regularity is valid up to the boundary, provided the boundary data are sufficiently regular. The main assumption on f is that for each x and u, the function f(x, u, ·) behaves asymptotically like the function h(|·|)^α(x), where h is an N-function.

Following this, we provide a characterization of the class of Young measures that can be generated by a sequence …

Neural Spike Renormalization. Part I — Universal Number 1, Bo Deng

Department of Mathematics: Faculty Publications

For a class of circuit models for neurons, it has been shown that the transmembrane electrical potentials in spike bursts have an inverse correlation with the intra-cellular energy conversion: the fewer spikes per burst the more energetic each spike is. Here we demonstrate that as the per-spike energy goes down to zero, a universal constant to the bifurcation of spike-bursts emerges in a similar way as Feigenbaum’s constant does to the period-doubling bifurcation to chaos generation, and the new universal constant is the first natural number 1.

Neural Spike Renormalization. Part Ii — Multiversal Chaos, Bo Deng

Department of Mathematics: Faculty Publications

Reported here for the first time is a chaotic infinite-dimensional system which contains infinitely many copies of every deterministic and stochastic dynamical system of all finite dimensions. The system is the renormalizing operator of spike maps that was used in a previous paper to show that the first natural number 1 is a universal constant in the generation of metastable and plastic spike-bursts of a class of circuit models of neurons.

Mathematical Modeling Of Optimal Seasonal Reproductive Strategies And A Comparison Of Long-Term Viabilities Of Annuals And Perennials, Anthony Delegge

Department of Mathematics: Dissertations, Theses, and Student Research

In 1954, Lamont Cole posed a question which has motivated much ecological work in the past 50 years: When is the life history strategy of semelparity (organisms reproduce once, then die) favored, via evolution, over iteroparity (organisms may reproduce multiple times in their lifetime)? Although common sense should dictate that iteroparity would always be favored, we can observe that this is not always the case, since annual plants are not only prevalent, but can dominate an area. Also, certain plant species may be perennial in one region, but annual in another. Thus, in these areas, certain characteristics must be present …

Metastability And Plasticity In A Conceptual Model Of Neurons, Bo Deng

Department of Mathematics: Faculty Publications

For a new class of neuron models we demonstrate here that typical membrane action potentials and spike-bursts are only transient states but appear to be asymptotically stable; and yet such metastable states are plastic — being able to dynamically change from one action potential to another with different pulse frequencies and from one spike-burst to another with different spike-per-burst numbers. The pulse and spike-burst frequencies change with individual ions’ pump currents while their corresponding metastable-plastic states maintain the same transmembrane voltage and current profiles in range. It is also demonstrated that the plasticity requires two one-way ion pumps operating in …

From Energy Gradient And Natural Selection To Biodiversity And Stability Of Ecosystems, Bo Deng

Department of Mathematics: Faculty Publications

The purpose of this paper is to incorporate well-established ecological principles into a foodweb model consisting of four trophic levels --- abiotic resources, plants, herbivores, and carnivores. The underlining principles include Kimura's neutral theory of genetic evolution, Liebig's Law of the Minimum for plant growth, Holling's functionals for herbivore foraging and carnivore predation, the One-Life Rule for all organisms, and Lotka-Volterra's model for intraand interspecific competitions. Numerical simulations of the model led to the following statistical findings: (a) particular foodwebs can give contradicting observations on biodiversity and productivity, in particular, all known functional forms -- - positive, negative, sigmoidal, and …

A Computational Study Of The Effects Of Temperature Variation On Turtle Egg Development, Sex Determination, And Population Dynamics, Amy L. Parrott

Department of Mathematics: Dissertations, Theses, and Student Research

Climate change and its effects on ecosystems is a major concern. For certain animal species, especially those that exhibit what is known as temperature-dependent sex determination (TSD), temperature variations pose a possibly serious threat (Valenzuela and Lance, 2004). In these species, temperature, and not chromosomes, determines the sex of the animal (Valenzuela and Lance, 2004). It is conceivable therefore, that if the temperature changes to favor only one sex, then dire consequences for their populations could occur. In this dissertation, we examine possible effects that climate change may have upon Painted Turtles (Chrysemys picta), a species with TSD. We investigate …

Modeling And Analysis Of Biological Populations, Joan Lubben

Department of Mathematics: Dissertations, Theses, and Student Research

Asymptotic and transient dynamics are both important when considering the future population trajectory of a species. Asymptotic dynamics are often used to determine whether the long-term trend results in a stable, declining or increasing population and even provide possible directions for management actions. Transient dynamics are important for estimating invasion speed of non-indigenous species, population establishment after releasing biocontrol agents, or population management after a disturbance like fire. We briefly describe here the results in this thesis.

(1) We consider asymptotic dynamics using discrete time linear population models of the form n(t + 1) = An(t) where …

Conceptual Circuit Models Of Neurons, Bo Deng

Department of Mathematics: Faculty Publications

A systematic circuit approach tomodel neurons with ion pump is presented here by which the voltage-gated current channels are modeled as conductors, the diffusion-induced current channels are modeled as negative resistors, and the one-way ion pumps are modeled as one-way inductors. The newly synthesized models are different from the type of models based on Hodgkin-Huxley (HH) approach which aggregates the electro, the diffusive, and the pump channels of each ion into one conductance channel. We show that our new models not only recover many known properties of the HH type models but also exhibit some new that cannot be extracted …

The Origin Of 2 Sexes Through Optimization Of Recombination Entropy Against Time And Energy, Bo Deng

Department of Mathematics: Faculty Publications

Sexual reproduction in nature requires two sexes, which raises the question why the reproductive scheme did not evolve to have three or more sexes. Here we construct a constrained optimization model based on the communication theory to analyze trade-offs among reproductive schemes with arbitrary number of sexes. More sexes on one hand lead to higher reproductive diversity, but on the other hand incur greater cost in time and energy for reproductive success. Our model shows that the two-sexes reproduction scheme maximizes the recombination entropy-to-cost ratio, and hence is the optimal solution to the problem.

Why Is The Number Of Dna Bases 4?, Bo Deng

Department of Mathematics: Faculty Publications

In this paper we construct a mathematical model for DNA replication based on Shannon’s mathematical theory for communication. We treatDNAreplication as a communication channel. We show that the mean replication rate is maximal with four nucleotide bases under the primary assumption that the pairing time of the G–C bases is between 1.65 and 3 times the pairing time of the A–T bases.