Mathematics Commons^™

Statistical Modeling Of Earthquake Damage, Allison Nicole Waters

Honors Program Theses

The purpose of this study was to build a statistical model of the economic damage that arises from earthquakes in order to better predict losses from future earthquakes. Though earthquakes are essentially a random event and cannot be fully anticipated, analyzing historical data and creating a statistical model can provide researchers with a more accurate estimate of future losses. The data set from which this model was built incorporated earthquakes occurring worldwide from 1915-2015 in which the total damage was recorded. The final model was a multiple linear regression model explaining total damage resulting from an earthquake through four independent …

Tessellations: An Artistic And Mathematical Look At The Work Of Maurits Cornelis Escher, Emily E. Bachmeier

Honors Program Theses

The purpose of this study was to learn more about the mathematics of tessellations and their artistic potential. Whenever I have seen tessellations, I always admire them. It is baffling how such complicated shapes can be repeated infinitely on the plane. This research aimed to increase the amount of tessellation information and activities available to secondary mathematics teachers by connecting the tessellations of Maurits Cornelis Escher with their underlying mathematics in order to use them for teaching secondary mathematics. The overarching premise of this study was to create something that would also be beneficial in my career as a secondary …

Detection Of Coherent Structures In Flows, Klodiana Shkembi

Theses, Dissertations and Culminating Projects

In this work, we have developed an experimental flow tank that can produce realistic ocean-like flows, including multi-gyre flows. By generating controllable ocean-like flow fields, we can study the flows to gain a better understanding of ocean dynamics. In particular, we use particle image velocimetry and finite-time Lyapunov exponents to determine the location of the Lagrangian Coherent Structures that determine transport in complex fluid flows. This understanding is useful for designing control algorithms and for optimizing the use of autonomous vehicles operating in the stochastic and time-dependent ocean environment.

An Exposition Of The Eisenstein Integers, Sarada Bandara

Masters Theses

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves.

An example of said ring are complex numbers of the form ℤ[ω] = {a+bω ∣ a, b ∈ ℤ} where ω² + ω + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[i] = {a + bi ∣ a, b ∈ …

Interval Edge-Colorings Of Graphs, Austin Foster

Electronic Theses and Dissertations

A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher conferences so that every person's conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and teachers, and …

Comparing The Variational Approximation And Exact Solutions Of The Straight Unstaggered And Twisted Staggered Discrete Solitons, Daniel Marulanda

Electronic Theses and Dissertations

Discrete nonlinear Schrödinger equations (DNSL) have been used to provide models of a variety of physical settings. An application of DNSL equations is provided by Bose-Einstein condensates which are trapped in deep optical-lattice potentials. These potentials effectively splits the condensate into a set of droplets held in local potential wells, which are linearly coupled across the potential barriers between them [3]. In previous works, DNLS systems have also been used for symmetric on-site-centered solitons [11]. A few works have constructed different discrete solitons via the variational approximation (VA) and have explored their regions for their solutions [11, 12]. Exact solutions …

Computational Study Of Traveling Wave Solutions And Global Stability Of Predator-Prey Models, Yi Zhu

Electronic Theses and Dissertations

In this thesis, we study two types of reaction-diffusion systems which have direct applications in understanding wide range of phenomena in chemical reaction, biological pattern formation and theoretical ecology. The first part of this thesis is on propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order $m$ without decay. The second is chemical reaction of order $m$ with a decay of order …

Weighted Low-Rank Approximation Of Matrices:Some Analytical And Numerical Aspects, Aritra Dutta

Electronic Theses and Dissertations

This dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing state-of-the-art unweighted and weighted low-rank approximation algorithms. Classical principal component analysis (PCA) is constrained to have equal weighting on the elements of the matrix, which might lead to a degraded design in some problems. To address this fundamental flaw in …

Asymptotic And Optimal Liouville Properties For Wolff Type Integral Systems, John Villavert

School of Mathematical & Statistical Sciences Faculty Publications

This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact, the two rates can be fully distinguished …

Metabolic Health Has Greater Impact On Diabetes Than Simple Overweight/Obesity In Mexican Americans, Shenghui Wu, Susan P. Fisher-Hoch, Belinda M. Reininger, Kristina Vatcheva, Joseph B. Mccormick

School of Mathematical & Statistical Sciences Faculty Publications

To compare the risk for diabetes in each of 4 categories of metabolic health and BMI. Methods. Participants were drawn from the Cameron County Hispanic Cohort, a randomly selected Mexican American cohort in Texas on the US-Mexico border. Subjects were divided into 4 phenotypes according to metabolic health and BMI: metabolically healthy normal weight, metabolically healthy overweight/obese, metabolically unhealthy normal weight, and metabolically unhealthy overweight/obese. Metabolic health was defined as having less than 2 metabolic abnormalities. Overweight/obese status was assessed by BMI higher than 25 kg/m2. Diabetes was defined by the 2010 ADA definition or by being on a diabetic …

Hepatitis C Virus In Mexican Americans: A Population-Based Study Reveals Relatively High Prevalence And Negative Association With Diabetes, Gordon P. Watt, Kristina Vatcheva, Laura Beretta, Jen-Jung Pan, Michael Fallon, Joseph B. Mccormick, Susan P. Fisher-Hoch

School of Mathematical & Statistical Sciences Faculty Publications

This study aimed to estimate the prevalence and risk factors for hepatitis C virus (HCV) infection in Mexican Americans living in South Texas. We tested plasma for the presence of HCV antibody from the Cameron County Hispanic Cohort (CCHC), a randomized, population-based cohort in an economically disadvantaged Mexican American community on the United States/Mexico border with high rates of chronic disease. A weighted prevalence of HCV antibody of 2·3% [n = 1131, 95% confidence interval (CI) 1·2-3·4] was found. Participants with diabetes had low rates of HCV antibody (0·4%, 95% CI 0·0-0·9) and logistic regression revealed a statistically significant negative …

Characterizations Of Rectangular (Para)-Unitary Rational Functions, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle:
(i) through the realization matrix of Schur stable systems,
(ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters,
(iii) through the (not necessarily reducible) Matrix Fraction Description (MFD).
In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the …

The H∞ Functional Calculus Based On The S-Spectrum For Quaternionic Operators And For N-Tuples Of Noncommuting Operators, Daniel Alpay, Fabrizio Colombo, Tao Qian, Irene Sabadini, Tao Qian

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend the H∞ functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz-Dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties.

The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears …

Wiener Algebra For The Quaternions, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-Lévy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.

The Spectral Theorem For Unitary Operators Based On The S-Spectrum, Daniel Alpay, Fabrizio Colombo, David P. Kimsey, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem.

In this paper we prove the quaternionic spectral theorem for unitary operators using the S-spectrum. In the case of quaternionic matrices, the S-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of S-spectrum is relatively new, see [17], and has been used for …

Math 433: Nonlinear Optimization—A Peer Review Of Teaching Project Benchmark Portfolio, Adam Larios

UNL Faculty Course Portfolios

My intention in this portfolio is to highlight various approaches to teaching higher-level mathematics with a programming component that I tried in a course on nonlinear optimization. There is a particular focus on students with little or no previous programming experience. Several case studies are done using "pre- and post-course" surveys which examined items such as student confidence in programming, and particular programming skills. Sample examples and sample homeworks are presented and discussed. Also presented are several materials I designed to lead students into programming and shed light on certain problems. These Materials assume some basic mathematical reasoning and knowledge, …

Number Knowledge And Error Types Of Elementary Portuguese Students: Implications For Instruction, Silvana Watson, Sharon Judge, João Lopes, Célia Oliveira, Ana Catarina Jesus

Communication Disorders & Special Education Faculty Publications

In the present study, we examined number knowledge skills of 697 Portuguese elementary students from first to fourth grade. Students completed three number knowledge tasks: 1) translating numbers into words, 2) symbolic magnitude (i.e., number comparison), and 3) decomposing numbers. We evaluated students’ answers by means of error analysis using a three-category coding system adopted from specific error types were computed by grade level. Results showed that there were significant differences among grades and that the prerequisite linguistic error type (i.e., pre linguistic rules or principles of the cardinal number system), particularly in the magnitude tasks, significantly contributed to students’ …