Applied Mathematics Commons^™

Equations Of The Camassa-Holm Hierarchy, Rossen Ivanov

Articles

The squared eigenfunctions of the spectral problem associated with the CamassaHolm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH …

Mathematical Aids Epidemic Model: Preferential Anti-Retroviral Therapy Distribution In Resource Constrained Countries, Nadia Abuelezam

HMC Senior Theses

HIV/AIDS is one of the largest health problems the world is currently facing. Even with anti-retroviral therapies (ART), many resource-constrained countries are unable to meet the treatment needs of their infected populations. ART-distribution methods need to be created that prevent the largest number of future HIV infections. We have developed a compartment model that tracks the spread of HIV in multiple two-sex populations over time in the presence of limited treatment. The model has been fit to represent the HIV epidemic in rural and urban areas in Uganda. With the model we examine the spread of HIV among urban and …

Biology In Mathematics At The University Of Richmond, Lester Caudill

Department of Math & Statistics Faculty Publications

In an effort to meet the needs of science students for modeling skills, three new courses have been created at the University of Richmond: Scientific Calculus I and II, and Mathematical Models in Biology and Medicine. The courses are described, and lessons learned and future directions are discussed.

Habitat Selection By Anolis Carolinensis (Green Anole) In Open Pine Forests In Eastern Texas, Richard R. Schaefer, Robert R. Fleet, D. Craig Rudolph, Nancy E. Koerth

Faculty Publications

We initiated a mark-recapture study to determine the effects of shrub density on Anolis carolinensis (Green Anole) populations. Green Anole perch site, shrub species, and shrub volume preferences were also examined. We established two study plots of different shrub densities in open pine forests on the Angelina National Forest in eastern Texas. In late spring, the Green Anole population at the higher shrub-density plot was estimated to be 16 times greater than the population at the lower shrub-density plot. Green Anoles most commonly perched on live shrubs, but exhibited very little preference or avoidance of any particular species of live …

Groups As Graphs, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Through this book, for the first time we represent every finite group in the form of a graph. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties like the subgroups of a group, normal subgroups of a group, p-sylow subgroups of a group and conjugate elements of a group …

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 …

Automated Traffic And The Finite Size Resonance, J. J. P. Veerman, Borko D. Stošić, F. M. Tangerman

Mathematics and Statistics Faculty Publications and Presentations

We investigate in detail what one might call the canonical (automated) traffic problem: A long string of N+1 cars (numbered from 0 to N) moves along a one-lane road “in formation” at a constant velocity and with a unit distance between successive cars. Each car monitors the relative velocity and position of only its neighboring cars. This information is then fed back to its own engine which decelerates (brakes) or accelerates according to the information it receives. The question is: What happens when due to an external influence—a traffic light turning green—the ‘zero’th’ car (the “leader”) accelerates?

As …

When To Spray: A Time-Scale Calculus Approach To Controlling The Impact Of West Nile Virus, Diana Thomas, Marion Weedermann, Lora Billings, Joan Hoffacker, Robert Washington-Allen

Department of Mathematics Faculty Scholarship and Creative Works

West Nile Virus (WNV) made its initial appearance in the New York City (NYC) metropolitan area in 1999 and was implicated in cases of human encephalitis and the extensive mortality in crows (Corvus sp.) and other avian species. Mosquitoes were found to be the primary vectors and NYC’s current policy on control strategies involved an eradication program that depends on the synchronicity of the summer mosquito population’s increases with the occurrence of cases in humans. The purpose of this paper is to investigate whether this is the most effective control strategy because past mathematical models assumed discrete behavior that …

Convolution Semigroups, Kevin W. Zito

LSU Doctoral Dissertations

In this dissertation we investigate, compute, and approximate convolution powers of functions (often probability densities) with compact support in the positive real numbers. Extending results of Ursula Westphal from 1974 concerning the characteristic function on the interval $[0,1]$, it is shown that positive, decreasing step functions with compact support can be embedded in a convolution semigroup in $L^1(0,infty)$ and that any decreasing, positive function $pin L^1(0,infty)$ can be embedded in a convolution semigroup of distributions. As an application to the study of evolution equations, we consider an evolutionary system that is described by a bounded, strongly continuous semigroup ${T(t)}_{tgeq0}$ in …

Some Results On Cubic Graphs, Evan Morgan

LSU Doctoral Dissertations

Pursuing a question of Oxley, we investigate whether the edge set of a graph admits a bipartition so that the contraction of either partite set produces a series-parallel graph. While Oxley's question in general remains unanswered, our investigations led to two graph operations (Chapters 2 and 4) which are of independent interest. We present some partial results toward Oxley's question in Chapter 3. The central results of the dissertation involve an operation on cubic graphs called the switch; in the literature, a similar operation is known as the edge slide. In Chapter 2, the author proves that we can transform, …

A Discrete Model Of Guided Modes And Anomalous Scattering In Periodic Structures, Natalia Grigoryevna Ptitsyna

LSU Doctoral Dissertations

We study a discrete prototype of anomalous scattering associated with the interaction of guided modes of a periodic scatterer and plane waves incident upon the scatterer. The transmission anomalies arise because of the non-robustness of a guided mode, a mode that exists only at a specific frequency and wave number pair. The simplicity of the discrete prototype allows one to make certain explicit calculations and proofs, and to examine details of important resonant phenomena of the open wave guides. The main results are (1) a formula for transmission anomalies near a non-robust guided mode with rigorous error estimates that extends …

Homological Width And Turaev Genus, Adam Lowrance

LSU Doctoral Dissertations

Khovanov homology and knot Floer homology are generalizations of the Jones polynomial and the Alexander polynomial respectively. They are bigraded Z-modules, and their underlying polynomials are recovered by taking the graded Euler characteristic. The two homologies share many characteristics, however their relationship has yet to be fully understood. In both Khovanov homology and knot Floer homology, the two gradings can be combined into a single diagonal grading. Homological width is a measure of the support of the homology with respect to the diagonal grading. In this thesis, we show that the homological width of Khovanov homology and knot Floer homology …

White Noise Methods For Anticipating Stochastic Differential Equations, Julius Esunge

LSU Doctoral Dissertations

This dissertation focuses on linear stochastic differential equations of anticipating type. Owing to the lack of a theory of differentiation for random processes, the said differential equations are appropriately understood and studied as anticipating stochastic integral equations. The unfolding work considers equations in which anticipation arises either from the initial condition or the integrand. In this regard, the techniques of white noise analysis are applied to such equations. In particular, by using the Hitsuda-Skorokhod integral which nicely extends the It integral to anticipating integrands, we then apply the S-transform from white noise analysis to study this new equation.

The Extended Picture Group, With Applications To Line Arrangement Complements, Charles Richard Egedy

LSU Doctoral Dissertations

We obtain the picture group as the quotient with a torsion subgroup, of an extended picture group, which is isomorphic to the kernel of a precrossed module homomorphism. In addition to expanding the notion of a picture group, the new formulation gives a natural way to construct homomorphisms between picture groups by describing deformations of one-vertex subpictures. The extended picture group thus provides a convenient way to describe generators for the second homotopy group of line arrangement complements as well as homomorphisms between these groups. In particular, we show that the homomorphisms relate to a lattice structure corresponding roughly to …

Function Spaces, Wavelets And Representation Theory, Jens Gerlach Christensen

LSU Doctoral Dissertations

This dissertation is concerned with the interplay between the theory of Banach spaces and representations of groups. The wavelet transform has proven to be a useful tool in characterizing and constructing Banach spaces, and we investigate a generalization of an already known technique due to H.G. Feichtinger and K. Gröchenig. This generalization is presented in Chapter 3, and in Chapters 4 and 5 we present examples of spaces which can be described using the theory. The first example clears up a question regarding a wavelet characterization of Bergman spaces related to a non-integrable representation. The second example is a wavelet …

Stochastic Navier-Stokes Equations With Fractional Brownian Motions, Liqun Fang

LSU Doctoral Dissertations

The aim of this dissertation is to study stochastic Navier-Stokes equations with a fractional Brownian motion noise. The second chapter will introduce the background results on fractional Brownian motions and some of their properties. The third chapter will focus on the Stokes operator and the semigroup generated by this operator. The Navier-Stokes equations and the evolution equation setup will be described in the next chapter. The main goal is to prove the existence and uniqueness of solutions for the stochastic Navier-Stokes equations with a fractional Brownian motion noise under suitable conditions. The proof is given with full details for two …

Local Behavior Of Distributions And Applications, Jasson Vindas

LSU Doctoral Dissertations

This dissertation studies local and asymptotic properties of distributions (generalized functions) in connection to several problems in harmonic analysis, approximation theory, classical real and complex function theory, tauberian theory, summability of divergent series and integrals, and number theory. In Chapter 2 we give two new proofs of the Prime Number Theory based on ideas from asymptotic analysis on spaces of distributions. Several inverse problems in Fourier analysis and summability theory are studied in detail. Chapter 3 provides a complete characterization of point values of tempered distributions and functions in terms of a generalized pointwise Fourier inversion formula. The relation of …

Impulsive Control Systems, Wei Cai

LSU Doctoral Dissertations

Impulsive control systems arose from classical control systems described by differential equations where the control functions could be unbounded. Passing to the limit of trajectories whose velocities are changing very rapidly leads to the state vector to "jump", or exhibit impulsive behavior. The mathematical model in this thesis uses a differential inclusion and a measure-driven control, and it becomes possible to deal with the discontinuity of movements happening over a small interval. We adopt the formulism of impulsive systems in which the velocities are decomposed by the slow and fast ones. The fast time velocity is expressed as the multiplication …

Unavoidable Minors In Graphs And Matroids, Carolyn Barlow Chun

LSU Doctoral Dissertations

It is well known that every sufficiently large connected graph G has either a vertex of high degree or a long path. If we require G to be more highly connected, then we ensure the presence of more highly structured minors. In particular, for all positive integers k, every 2-connected graph G has a series minor isomorphic to a k-edge cycle or K_{2,k}. In 1993, Oxley, Oporowski, and Thomas extended this result to 3- and internally 4-connected graphs identifying all unavoidable series minors of these classes. Loosely speaking, a series minor allows for arbitrary edge deletions but only allows edges …

Algorithms Related To Subgroups Of The Modular Group, Constantin Cristian Caranica

LSU Doctoral Dissertations

Classifying subgroups of the modular group PSL_2{Z} is a fundamental problem with applications to modular forms, in addition to its group-theoretic interest. While a lot of research has been done on the congruence subgroups of PSL_2{Z}, very little is known about noncongruence subgroups. The purpose of this thesis is to find and characterize small-index noncongruence subgroups of the modular group PSL_2{Z}. We use the concept of Farey symbol to describe the subgroups of PSL_2{Z}. The first part contains results concerning the geometry of subgroups of PSL_2{Z}. The second part describes a graph-theoretical approach to finding all subgroups of a given …

The Structure Of 4-Separations In 4-Connected Matroids, Jeremy M. Aikin

LSU Doctoral Dissertations

Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M. Crossing 3-separations gave rise to fundamental structures known as flowers. In this dissertation, we define generalized flower structure called a k-flower, with no assumptions on the connectivity of M. We completely classify k-flowers in terms of the local connectivity between pairs of petals. Specializing to the case of 4-connected matroids, we give a new notion of equivalence of 4-separations that we show will be needed to describe a tree decomposition for 4-connected matroids. Finally, we …

A Regularization Technique In Dynamic Optimization, Alvaro Guevara

LSU Doctoral Dissertations

In this dissertation we discuss certain aspects of a parametric regularization technique which is based on recent work by R. Goebel. For proper, lower semicontinuous, and convex functions, this regularization is self-dual with respect to convex conjugation, and a simple extension of this smoothing exhibits the same feature when applied to proper, closed, and saddle functions. In Chapter 1 we give a introduction to convex and saddle function theory, which includes new results on the convergence of saddle function values that were not previously available in the form presented. In Chapter 2, we define the regularization and extend some of …

A Covert Encryption Method For Applications In Electronic Data Interchange, Jonathan Blackledge, Dmitry Dubovitskiy

Articles

A principal weakness of all encryption systems is that the output data can be ‘seen’ to be encrypted. In other words, encrypted data provides a ‘flag’ on the potential value of the information that has been encrypted. In this paper, we provide a new approach to ‘hiding’ encrypted data in a digital image.

In conventional (symmetric) encryption, the plaintext is usually represented as a binary stream and encrypted using an XOR type operation with a binary cipher. The algorithm used is ideally designed to: (i) generate a maximum entropy cipher so that there is no bias with regard to any …

The Segal-Bargmann Transform On Inductive Limits Of Compact Symmetric Spaces, Keng Wiboonton

LSU Doctoral Dissertations

We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces $\{L^2(M_n)^{K_n}\}_n$ where $\{M_n = U_n/K_n\}_n$ is a propagating sequence of symmetric spaces of compact type with the assumption that $U_n$ is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on $L^2(M_n)^{K_n}, \ n = 1,2,...$. For each $n$, let $\widehat{U_n}$ be the set of equivalence classes of irreducible unitary representations of $U_n$ and let $\widehat{U_n/K_n} \subseteq \widehat{U_n}$ be the set of $K_n$-spherical representations. The definition of the propagation gives a nice property allowing us to embed $\widehat{U_n/K_n}$ …

Cedar Rapids Flood Of 2008: Manmade Or Statistical Anomaly?, Beth Kolsrud

Honors Program Theses

One of the main research questions for this thesis is what factors caused the flooding last summer and why was it so extreme? This is an important question to answer because it is vital to understand what causes a flood of this magnitude so communities can take steps to avoid or minimize damage that may result from floods in the future. The second question is what impact does this flood have on flood insurance? Flood insurance is one tool used to protect property owners from substantial loss due to flooding and plays a vital role in recovery after a flood, …

Managing Risk In Life Insurance With Your Genetic Information: The Issues, The Options, And The Speed Of Progress, Michelle Gogerty

Honors Program Theses

legislation is already in place to restrict the use of genetic information in the health insurance industry (Genetics and Public Policy Center 2008) and several states have similar laws restricting its use in the life insurance industry as well (National Conference of State legislatures 2008}. It is only a matter of time before national legislation applicable to life insurance is enacted on the matter and I expect it to come within the next five years. Unless both insurers and consumers actively protect their rights to privacy, equality, and profitability, the result of that legislation may be very harmful to one …

Bifurcation Analysis Of A Kaldor-Kalecki Model Of Business Cycle With Time Delay, Liancheng Wang, Xiaoqin P. Wu

Faculty Articles

In this paper, we investigate a Kaldor-Kalecki model of business cycle with delay in both the gross product and the capital stock. Stability analysis for the equilibrium point is carried out. We show that Hopf bifurcation occurs and periodic solutions emerge as the delay crosses some critical values. By deriving the normal forms for the system, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established. Examples are presented to confirm our results.

Positive Solutions For The (N, P) Boundary Value Problem, Bo Yang

Faculty Articles

We consider the (n, p) boundary value problem in this paper. Some new upper estimates to positive solutions for the problem are obtained. Existence and nonexistence results for positive solutions of the problem are obtained by using the Krasnosel'skii fixed point theorem. An example is included to illustrate the results.

On The Estimation Of Averages Over Infinite Intervals With An Application To Average Persistence In Population Models, Sean F. Ellermeyer

Faculty Articles

We establish a general result for estimating the upper average of a continuous and bounded function over an infinite interval. As an application, we show that a previously studied model of microbial growth in a chemostat with time–varying nutrient input admits solutions (populations) that exhibit weak persistence but not weak average persistence.

Heritability Of Cognition Subscales In The Cache County Memory Study Cohort And Methods For Estimating Heritability, Colette Kelly Childs

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The rapid improvement of genotyping technology holds the promise of better understanding the genetic causes of complex disease. While traits of interest often include the presence or absence of disease, there is growing interest in intermediate phenotypes (or so-called endophenotypes) that may yield more information about disease onset or course. In particular, changes over time with respect to an investigative ordinal measure often contain significant predictive power, and rate-of-change phenotypes are becoming important in their own right when studying genetic association. Initial steps in assessing the potential genetic determinants of a continuous trait often involve estimating the degree of the …