Applied Mathematics Commons^™

A Restarted Homotopy Method For The Nonsymmetric Eigenvalue Problem, Brandon Hutchison

Graduate Theses and Dissertations

The eigenvalues and eigenvectors of a Hessenberg matrix, H, are computed with a combination of homotopy increments and the Arnoldi method. Given a set, Ω, of approximate eigenvalues of H, there exists a unique vector f = f(H,Ω) in Rn where λ(H-e_{1ft)=Ω. A diagonalization of the homotopy H(t)=H−(1−t)e1ft at $t=0$ provides a prediction of the eigenvalues of H(t) at later times. These predictions define a new Ω that defines a new homotopy. The correction for each eigenvalue has an O(t2) error estimate, enabling variable step size and efficient convergence tests. Computations are done primarily in real arithmetic, and …}

Global Well-Posedness For A Nonlinear Wave Equation With P-Laplacian Damping, Zahava Wilstein

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation deals with the global well-posedness of the nonlinear wave equation
u_tt − Δu − Δ_pu_t = f (u) in Ω × (0,T),
{u(0), u_t(0)} = {u₀,u₁} ∈ H¹₀ (Ω) × L ² (Ω),
u = 0 on Γ × (0, T ),
in a bounded domain Ω ⊂ ℜ ⁿ with Dirichlét boundary conditions. The nonlinearities f (u) acts as a strong source, which is allowed 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 …

K-Hyponormality Of Powers Of Multivariable Weighted Shifts, Mayra O. Martinez

Theses and Dissertations - UTB/UTPA

In mathematics, weighted shifts, also known as shift operators or translation operators, are examples of linear operators. Single and multivariable weighted shifts have played an important role in the study of (joint) k-hyponormality to LPCS. They have also played a significant role in the study of cyclicity and reflexivity, in the study of C*-algebras generated by multiplication operators on Bergman spaces, as fertile ground to test new hypotheses, and as canonical models for theories of dilation and positivity. We will first consider the hyponormality of powers of commuting multivariable weighted shifts. Specifically, if we let W(α,β) := (T1, T2) denote …

Biologically Relevant Classes Of Boolean Functions, Lori Layne

All Dissertations

A large influx of experimental data has prompted the development of
innovative computational techniques for modeling and reverse
engineering biological networks. While finite dynamical systems,
in particular Boolean networks, have gained attention as relevant
models of network dynamics, not all Boolean functions reflect the
behaviors of real biological systems. In this work, we focus on two
classes of Boolean functions and study their applicability as
biologically relevant network models: the nested and partially nested
canalyzing functions.
We begin by analyzing the nested canalyzing functions} (NCFs),
which have been proposed as gene regulatory network models due to
their stability properties. We …

The Minimum-Norm Least-Squares Solution Of A Linear System And Symmetric Rank-One Updates, Xuzhou Chen, Jun Ji

Faculty Articles

In this paper, we study the Moore-Penrose inverse of a symmetric rank-one perturbed matrix from which a finite method is proposed for the minimum-norm least-squares solution to the system of linear equations Ax = b. This method is guaranteed to produce the required result.

A Statistical Model For Long-Term Forecasting Of Strong Sand Dust Storms, Siqi Tan

UNLV Theses, Dissertations, Professional Papers, and Capstones

Dust elevated into the atmosphere by dust storms has numerous environmental consequences. These include contributing to climate change; modifying local weather conditions; producing chemical and biological changes in the oceans; and affecting soil formation, surface water, groundwater quality, crop growth, and survival (Goudie and Middleton, 1992). Societal impacts include disruptions to air, road and rail traffic; interruption of radio services; the myriad effects of static-electricity generation; property damage; and health effects on humans and animals (Warner, 2004).

In this thesis, we extend the idea of empirical recurrence rate (ERR), developed by Ho (2008), to model the temporal trend of the …

“Drawing” Upon Your Students’ Creativity: Teaching (Your Subject Here) With Comic Books, Bruce Kessler

Mathematics Faculty Publications

During Spring 2009, Dr. Kessler created and published a comic book series that embedded math content into the story for 4th-6th grade students. The comics were well received in the classrooms at Cumberland Trace Elementary. Dr. Kessler contends that this approach to teaching and learning can be used in any content area, and is useful for engaging students who might not be as interested otherwise. This session will explore ways of utilizing the skills of your students to construct learning comics in your classes, regardless of the funds, technology, and artistic experience at your disposal. The session will include a …

Being Seen: An Art Historical And Statistical Analysis Of Feminized Worship In Early Modern Rome, Olivia J. Belote

History Honors Projects

Female saints in early Christianity found their place in public veneration often through violent means, martyrdom.  These saints, while publicly suffering in the imitation of Christ, were the original agents to navigate the gendered hierarchy within the religion.  Female saints created an avenue for later female worshippers to understand Christianity on a strictly feminine level.  Through the frescoed depictions of these female saints in 18 churches throughout Rome, this paper historically and statistically analyzes how the artistic representations of female saints added to or created a space for feminized worship.

Unlv Enrollment Forecasting, Sabrina Beckman, Stefan Cline, Monika Neda

Festival of Communities: UG Symposium (Posters)

Our project investigates the future enrollment of undergraduates at UNLV in the entire university, the College of Science, and the Department of Mathematical Sciences. The method used for the forecast, is the well-known least-squares method, for which a mathematical description will be presented. Studies for the numerical error are pursued too. The study will include graphs that describe the past and future behavior for different parameter settings. Mathematical results obtained show that the university will continue to grow given the current trends of enrollment.

Mathematical Analysis And Applications Of Logistic Differential Equation, Eva Arnold, Monika Neda

Festival of Communities: UG Symposium (Posters)

Logistic differential equation has a way to measure the proportionality of various resources with respect to time. This equation has been used in many research areas, such as, biology, medicine, psychology, economics, etc. A mathematical description, analysis and solution of the logistic type differential equation is studied. Besides the mathematical part, the poster will contain biological examples, graphs of the direction fields for different parameter settings and logistic plots for specific species population. The logistic growth function will also be applied to learning curves in area of psychology, as a rate at which performance improves.

Oral Presentation: The Universe In A Box, Jason Jaacks

Festival of Communities: UG Symposium (Posters)

When and how galaxies formed throughout the history of the Universe is one of the most fundamental questions of astronomy and astrophysics. As technology improves, astronomers are able to push the frontier of galaxy observation to a period when the Universe was less than 1 billion years old. This is when the first galaxies are beginning to form. However, beyond the limits of observational technology lies data fundamental to our complete understanding of these processes. Using state-of-the-art cosmological hydrodynamic computer codes combined with access to the nation’s largest and fastest supercomputers, we are able to simulate the formation and evolution …

Multiple Solutions For An Elliptic Problem Related To Vortex Pairs, Yi Li, Shuangjie Peng

Mathematics and Statistics Faculty Publications

Let Ω be a bounded domain in RN (N⩾2), φ a harmonic function in Ω¯. In this paper we study the existence of solutions to the following problem arising in the study of vortex pairs(Pλ){−Δu=λ(u−φ)+p−1,x∈Ω,u=0,x∈∂Ω. The set Ωp={x∈Ω,u(x)>φ} is called “vortex core”. Existence of solutions whose “vortex core” consists of one component and asymptotic behavior of “vortex core” were studied by many authors for large λ recently. Under the condition that φ has k strictly local minimum points on the boundary ∂Ω, we obtain in this paper that for λ large enough, (Pλ) has a solution with “vortex core” …

Dynamic Decision Making And Race Games, Shipra De

Calvert Undergraduate Research Awards

Frequent criticism in dynamic decision making research pertains to the overly complex nature of the decision tasks used in experimentation. To address such concerns we study dynamic decision making with respect to the simple race game Hog, which has a computable optimal decision strategy. In the two-player game of Hog, individuals compete to be the first to reach a designated threshold of points. Players alternate rolling a desired quantity of dice. If the number one appears on any of the dice, the player receives no points for his turn; otherwise, the sum of the numbers appearing on the dice is …

Wavelet-Based Analysis Of Neutron-Induced Photon Spectral Data, Bruce Kessler, Alexander Barzilov, Phillip Womble

Mathematics Faculty Publications

Neutron-based methods of non-destructive inter- rogation of objects for the purpose of their characterization are well-established techniques, employed in the field of bulk material analysis, contraband detection, unexploded ordnance, etc. The characteristic gamma rays produced in nuclear reactions initiated by neutrons in the volume of the irradiated object (inelastic neutron scattering, thermal neutron capture, and activation) are used for the elemental identification. In many real-world applications, an automated spectral analysis is needed, and many algorithms are used for that purpose. The Applied Physics Institute at Western Kentucky University has recently started to employ a mathematical spectrum analysis technique based on …

A Nonnegative Analysis Of Politics, Tim Chartier, Charles D. Wessell

Math Faculty Publications

The article investigates how linear algebra can recover mathematical information from the electronic messages using the Enron Email Sets in Pennsylvania. It states that a term-by-email matrix has been created to cluster algorithm, which allows one to mine through data and discover meaning. Moreover, nonnegative matrix factorization (NMF) enables one to interpret the resulting factorization in terms of the original problem.

A Numerical Method For Solving The Elliptic And Elasticity Interface Problems, Liqun Wang

Doctoral Dissertations

Interface problems arise when dealing with physical problems composed of different materials or of the same material at different states. Because of the irregularity along interfaces, many common numerical methods do not work, or work poorly, for interface problems. Matrix-coefficient elliptic and elasticity equations with oscillatory solutions and sharp-edged interfaces are especially complicated and challenging for most existing methods. An accurate and efficient method is desired.

In 1999, the boundary condition capturing method was proposed to deal with Poisson equations with interfaces whose variable coefficients and solutions may be discontinuous. In 2003, a weak formulation was derived. Built on previous …

Shape Reconstruction And Classification Using The Response Matrix, Wei Wang

Doctoral Dissertations

This dissertation presents a novel method for the inverse scattering problem for extended target. The acoustic or electromagnetic wave is scattered by the target and received by all the transducers around the target. The scattered field on all the transducers forms the response matrix which contains the information of the geometry of the target. The objective of the inverse scattering problem is to reconstruct the shape of the scatter using the Response Matrix.

There are two types of numerical methods for solving the inverse problem: the direct imaging method and the iterative method. Two direct imaging methods, MUSIC method and …

Results In Lattices, Ortholattices, And Graphs, Jianning Su

Doctoral Dissertations

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce "π-versions" of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (b ✶ c) ≤ (a ✶ b) ✶ c for all 3-element antichains { a, b, c}. We consider …

Quantitative Stability Of Linear Infinite Inequality Systems Under Block Perturbations With Applications To Convex Systems, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra

Mathematics Research Reports

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is loo(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn …

Preliminary Analysis Of An Agent-Based Model For A Tick-Borne Disease, Holly Gaff

Biological Sciences Faculty Publications

Ticks have a unique life history including a distinct set of life stages and a single blood meal per life stage. This makes tick-host interactions more complex from a mathematical perspective. In addition, any model of these interactions must involve a significant degree of stochasticity on the individual tick level. In an attempt to quantify these relationships, I have developed an individual-based model of the interactions between ticks and their hosts as well as the transmission of tick-borne disease between the two populations. The results from this model are compared with those from previously published differential equation based population models. …

A Least Squares Closure Approximation For Liquid Crystalline Polymers, Traci Ann Sievenpiper

Mathematics & Statistics Theses & Dissertations

An introduction to existing closure schemes for the Doi-Hess kinetic theory of liquid crystalline polymers is provided. A new closure scheme is devised based on a least squares fit of a linear combination of the Doi, Tsuji-Rey, Hinch-Leal I, and Hinch-Leal II closure schemes. The orientation tensor and rate-of-strain tensor are fit separately using data generated from the kinetic solution of the Smoluchowski equation. The known behavior of the kinetic solution and existing closure schemes at equilibrium is compared with that of the new closure scheme. The performance of the proposed closure scheme in simple shear flow for a variety …

An Extensible Mathematical Model Of Glucose Metabolism, Caleb L. Adams

Mathematics & Statistics Theses & Dissertations

The American Diabetes Association reports that diabetes is the fifth leading cause of death by disease in the United States. An estimated 23.6 million individuals, or seven percent of the population, have diabetes. Nearly one-third are unaware that they have the disease. The total of the direct and indirect medical costs associated with diabetes in 2007 was projected to be $174 billion, or approximately one out of every ten health care dollars.

One must understand the glucose regulatory system of the healthy body to understand diabetes. Blood glucose concentration returns to a constant level after eating and is maintained during …

Set-Based Corral Control In Stochastic Dynamical Systems: Making Almost Invariant Sets More Invariant, Eric Forgoston, Lora Billings, Philip Yecko, Ira Schwartz

Department of Mathematics Faculty Scholarship and Creative Works

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times …

A “Peak” At The Algorithm Behind “Peaklet Analysis” Software, Bruce Kessler

Mathematics Faculty Publications

In response to a problem posed by faculty at the Applied Physics Institute at Western Kentucky University, the speaker has developed an algorithm for providing an automated analysis of spectrum data for the purpose of determining the elemental composition of the item generating the data. A full, non-provisional patent application has been filed on the idea, and a full marketing campaign has started to license software implementing the algorithm. This presentation will give a brief explanation of the mathematics in use in the algorithm, and will give some examples of the software in action.

Quantum Mechanical Model Of The Hydrogen Atom, Andres Ortiz^, Snehanshu Saha*

COURI Symposium Abstracts, Spring 2011

The quantum mechanical wave function of an object under a certain potential is determined by the Schrödinger equation. It is known that the latter equation can be solved analytically only for a finite number of distinct potentials, being a Coulomb potential one of them. For an electron in a hydrogen atom, we analytically find the angular parts of its wave function by the usual method. Under the appropriate change of variable, the radial part of the equation is turned into a Legendre differential equation whose solutions are Legendre polynomials. Using finite difference method and shooting method taking into account physically …

Modeling Of Bacillus Spores: Inactivation And Outgrowth, Alexis X. Hurst

Theses and Dissertations

This research models and analyzes the thermochemical damage produced in Bacillus spores by short, high-temperature exposures as well the repair process within damaged Bacillus spores. Thermochemical damage in spores is significantly due to reaction with water, hydrolysis reactions. Applying heat to the spore causes absorbed and chemically bound water molecules become mobile within the spore. These mobile water molecules react by hydrolysis reactions to degrade DNA and enzyme molecules in the spore. In order to survive the thermal inactivation, the spore must repair the damaged DNA during spore germination. The DNA repair process, as well as other germination functions, is …

Quantitative Interpretation Of A Genetic Model Of Carcinogenesis Using Computer Simulations, Donghai Dai, Brandon Beck, Xiaofang Wang, Cory Howk, Yi Li

Mathematics and Statistics Faculty Publications

The genetic model of tumorigenesis by Vogelstein et al. (V theory) and the molecular definition of cancer hallmarks by Hanahan and Weinberg (W theory) represent two of the most comprehensive and systemic understandings of cancer. Here, we develop a mathematical model that quantitatively interprets these seminal cancer theories, starting from a set of equations describing the short life cycle of an individual cell in uterine epithelium during tissue regeneration. The process of malignant transformation of an individual cell is followed and the tissue (or tumor) is described as a composite of individual cells in order to quantitatively account for intra-tumor …

A Primer On Chaos And Fractals, Bruce Kessler

Mathematics Faculty Publications

This is a prelude to a performance of the play "Arcadia" exclusively for the science and math majors of Lipscomb University. One of the main characters of the story is a mathematical genius, and has realized the power and limitations of iterations in generating mathematical models and structures, although she is living in the early 1800's. This talk gives an introduction to the ideas of chaos theory, fractals, and randomness.

Maximal Sensitive Dependence And The Optimal Path To Epidemic Extinction, Eric Forgoston, Simone Bianco, Leah B. Shaw, Ira B. Schwartz

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that …