Mathematics Commons^™

Reproductive Numbers For Periodic Epidemic Systems, Christopher David Mitchell

Mathematics Dissertations - Archive

When using mathematics to study epidemics, often times the goal is to deter- mine when an infection can invade and persist within a population. This can be done in a variety of ways but the most common is to use threshold quantities called reproductive numbers. For models with only one infection, the basic reproductive number (BRN) is used to determine the stability of the disease-free equilibrium. For many years this was done solely for autonomous systems; however, many diseases exhibit seasonal behavior. If this seasonality is incorporated into models, it gives nonautonomous systems, which while more accurate in their description, …

Numerical Construction Of Diffeomorphism And The Applications To Grid Generation And Image Registration, Xi Chen

Mathematics Dissertations - Archive

Diffeomorphism is an active research topic in differential geometry. In this area, the existence and construction of diffeomorphism under certain constraints is an interesting and meaningful task. J. Moser first proved the existence of diffeomorphism under a Jacobian determinant constraint. Later, Dr. Liao along with his co-authors, proposed the deformation method to construct diffeomorphisms. A div-curl system is created in the construction of diffeomorphisms. Since the Jacobian determinant has a direct physical meaning in grid generation, i.e. the grid cell size, the deformation method was applied successfully to grid generation and adaptation problems. In this dissertation, we review the deformation …

Editorial, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Tme Volume 13, Number 3

The Mathematics Enthusiast

No abstract provided.

The Secret Life Of 1/N: A Journey Far Beyond The Decimal Point, Christopher Lyons

The Mathematics Enthusiast

The decimal expansions of the numbers 1/n (such as 1/3 = .03333..., 1/7 = 0.142857...) are most often viewed as tools for approximating quantities to a desired degree of accuracy. The aim of this exposition is to show how these modest expressions in fact have much more to offer, particularly in the case when the expansions are infinitely long. First we discuss how simply asking about the period (that is, the length of the repeating sequence of digits) of the decimal expansion of 1/n naturally leads to more sophisticated ideas from elementary number theory, as well as to …

The History Of Algorithmic Complexity, Audrey A. Nasar

The Mathematics Enthusiast

This paper provides a historical account of the development of algorithmic complexity in a form that is suitable to instructors of mathematics at the high school or undergraduate level. The study of algorithmic complexity, despite being deeply rooted in mathematics, is usually restricted to the computer science curriculum. By providing a historical account of algorithmic complexity through a mathematical lens, this paper aims to equip mathematics educators with the necessary background and framework for incorporating the analysis of algorithmic complexity into mathematics courses as early on as algebra or pre-calculus.

Integral Of Radical Trigonometric Functions Revisited, Natanael Karjanto, Binur Yermukanova

The Mathematics Enthusiast

This article revisits an integral of radical trigonometric functions. It presents several methods of integration where the integrand takes the form 1+/- sin x or 1+/- cos x. The integral has applications in Calculus where it appears as the length of cardioid represented in polar coordinates.

Mathematical Problem-Solving Via Wallas’ Four Stages Of Creativity: Implications For The Undergraduate Classroom, Milos Savic

The Mathematics Enthusiast

The central theme in this article is that certain problem-solving frameworks (e.g., Polya, 1957; Carlson & Bloom, 2005) can be viewed within Wallas’ four stages of mathematical creativity. The author attempts to justify the previous claim by breaking down each of Wallas’ four components (preparation, incubation, illumination, verification) using both mathematical creativity and problem-solving/proving literature. Since creativity seems to be important in mathematics at the undergraduate level (Schumacher & Siegel, 2015), the author then outlines three observations about the lack of fostering mathematical creativity in the classroom. Finally, conclusions and future research are discussed, with emphasis on using technological advances …

Plato On The Foundations Of Modern Theorem Provers, Ines Hipolito

The Mathematics Enthusiast

Is it possible to achieve such a proof that is independent of both acts and dispositions of the human mind? Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics, independent of the human mind. In the seventh book of his masterpiece, The Republic, Plato states “arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument” (525c). In the light of this …

Is There A Symmetric Version Of Hindman's Theorem?, Ethan Akin, Eli Glasner

Mathematics and Statistics Faculty Research & Creative Works

We show that there does not exist a symmetric version of Hindman's Theorem, or more explicitly, that the property of containing a symmetric IP-set is not divisible. We consider several related dynamics questions.

An Algorithm For The Machine Calculation Of Minimal Paths, Robert Whitinger

Electronic Theses and Dissertations

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R³, but also to the general case of finding minimal functionals on hypersurfaces in Rⁿ associated with an arbitrary metric.

Numerical Study For Non-Newtonian Fluid-Structure Interaction Problems, Shuhan Xu

All Dissertations

In this work, we consider non-Newtonian fluid structure problems, which have significant applications in biology and industry. Numerical approximation schemes are developed based on the Arbitrary Lagrangian-Eulerian (ALE) formulation of the flow equations. A spatial discretization is accomplished by the finite element method, and the time discretization is carried out by the implicit Euler method. We first consider a fluid-structure interaction problem that consists of a two-dimensional viscoelastic flow and a one-dimensional structure equation. We show how the system can be decoupled and how each subproblem can be solved using interface conditions. Numerical results of different algorithms are presented, showing …

Ε-Kernel Coresets For Stochastic Points, Haitao Wang, Lingxiao Huang, Jian Li, Jeff Mark Phillips

Computer Science Faculty and Staff Publications

With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the …

An Investigation Into Lagrangian Coherent Structure Detection For Low Reynolds Number Flows, Kyle Fitzsimmons

Theses, Dissertations and Culminating Projects

In order to better understand and make use of complex fluid systems, regions of similar behavior can be categorized, thus reducing the problem to one of identifying the important structures within the fluid system, and examining their interaction. An important category of fluid structures are those known as Lagrangian Coherent Structures. An experimental setup has been constructed in order to study methods for improving the detection of Lagrangian Coherent Structures, and experimental data has been analyzed in order to verify the experimental setup. Several modifications to the basic gyre flow have been analyzed in order to provide direction, and a …

Applications Of The Homotopy Analysis Method To Optimal Control Problems, Shubham Singh

Open Access Theses

Traditionally, trajectory optimization for aerospace applications has been performed using either direct or indirect methods. Indirect methods produce highly accurate solutions but suer from a small convergence region, requiring initial guesses close to the optimal solution. In past two decades, a new series of analytical approximation methods have been used for solving systems of dierential equations and boundary value problems.

The Homotopy Analysis Method (HAM) is one such method which has been used to solve typical boundary value problems in nance, science, and engineering. In this investigation, a methodology is created to solve indirect trajectory optimization problems using the Homotopy …

Oscillation Of Quenched Slowdown Asymptotics Of Random Walks In Random Environment In Z, Sung Won Ahn

Open Access Dissertations

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed limn→∞ (Xn/) = υα > 0. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities P ω(Xn < xn) with x∈ (0,υα) decay approximately like exp{- n1-1/s} for a deterministic s > 1. More precisely, they showed that n -γ log Pω(Xn < xn) converges to 0 or -∞ depending on whether γ > 1 - 1/s or γ < 1 - 1/ s. In this paper, …

Maximum Empirical Likelihood Estimation In U-Statistics Based General Estimating Equations, Lingnan Li

Open Access Dissertations

In the first part of this thesis, we study maximum empirical likelihood estimates (MELE's) in U-statistics based general estimating equations (UGEE's). Our technical maneuver is the jackknife empirical likelihood (JEL) approach. We give the local uniform asymptotic normality condition for the log-JEL for UGEE's. We derive the estimating equations for finding MELE's and provide their asymptotic normality. We obtain easy MELE's which have less computational burden than the usual MELE's and can be easily implemented using existing software. We investigate the use of side information of the data to improve efficiency. We exhibit that the MELE's are fully efficient, and …

Rees Algebras And Iterated Jacobian Duals, Vivek Mukundan

Open Access Dissertations

Consider the rational map Ψ : [Special characters omitted.] where the fi's are homogeneous forms of the same degree in the homogeneous coordinate ring R = k[ x1,…,xd] of [Special characters omitted.]. Assume that I = (f 1,…,fm) is a height 2 perfect ideal in the polynomial ring R. In this context, the coordinate ring of the graph of Ψ is the Rees algebra of I and the co-ordinate ring of the image of Ψ is the special fiber ring. We study two settings. The first setting is when I is almost …

Martingales, Singular Integrals, And Fourier Multipliers, Michael A. Perlmutter

Open Access Dissertations

Many probabilistic constructions have been created to study the Lp-boundedness, 1 < p < ∞, of singular integrals and Fourier multipliers. We will use a combination of analytic and probabilistic methods to study analytic properties of these constructions and obtain results which cannot be obtained using probability alone.

In particular, we will show that a large class of operators, including many that are obtained as the projection of martingale transforms with respect to the background radiation process of Gundy and Varapolous or with respect to space-time Brownian motion, satisfy the assumptions of Calderón-Zygmund theory and therefore boundedly map L1 to weak- L1.

We will also use a method of rotations to study the L p boundedness, 1 < p < ∞, of Fourier multipliers which are obtained as the projections of martingale transforms with respect to symmetric α-stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2 and therefore allows us to obtain a larger class of multipliers, indexed by a parameter, 0 < r < ∞, which are bounded on L p. As in the case of the multipliers which arise as the projection of martingale …

Extreme-Strike And Small-Time Asymptotics For Gaussian Stochastic Volatility Models, Xin Zhang

Open Access Dissertations

Asymptotic behavior of implied volatility is of our interest in this dissertation. For extreme strike, we consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first …

Mathematical Models Of Ebola Virus Disease And Vaccine Preventable Diseases, Yinqiang Zheng

Open Access Dissertations

This thesis focuses on applying mathematical models to studies on the transmission dynamics and control interventions of infectious diseases such as Ebola virus disease and vaccine preventable diseases.

Many models in studies of Ebola transmission are based on the model by Legrand et al. (2007). However, there are potential issues with the Legrand model. First, the model was originally formulated in a complex form, leading to confusion and hindering its uses in practice. To overcome the difficulty, the Legrand model is reformulated in a much simpler but equivalent form in this thesis. The reformulated model also provides an intuitive understanding …

Accuracy And Precision Of Occlusal Contacts Of Stereolithographic Casts Mounted By Digital Interocclusal Registrations, Jason T. Krahenbuhl, Seok-Hwan Cho, Jon Patrick Irelan, Naveen K. Bansal

Mathematics, Statistics and Computer Science Faculty Research and Publications

Statement of problem

Little peer-reviewed information is available regarding the accuracy and precision of the occlusal contact reproduction of digitally mounted stereolithographic casts.

Purpose

The purpose of this in vitro study was to evaluate the accuracy and precision of occlusal contacts among stereolithographic casts mounted by digital occlusal registrations.

Material and methods

Four complete anatomic dentoforms were arbitrarily mounted on a semi-adjustable articulator in maximal intercuspal position and served as the 4 different simulated patients (SP). A total of 60 digital impressions and digital interocclusal registrations were made with a digital intraoral scanner to fabricate 15 sets of mounted stereolithographic …

Biomarkers For Radiation Pneumonitis Using Noninvasive Molecular Imaging, Meetha Medhora, Steven Haworth, Yu Liu, Jayashree Narayanan, Feng Gao, Ming Zhao, Said H. Audi, Elizabeth R. Jacobs, Brian L. Fish, Anne V. Clough

Mathematics, Statistics and Computer Science Faculty Research and Publications

Our goal is to develop minimally invasive biomarkers for predicting radiation-induced lung injury before symptoms develop. Currently, there are no biomarkers that can predict radiation pneumonitis. Radiation damage to the whole lung is a serious risk in nuclear accidents or in radiologic terrorism. Our previous studies have shown that a single dose of 15 Gy of x-rays to the thorax causes severe pneumonitis in rats by 6–8 wk. We have also developed a mitigator for radiation pneumonitis and fibrosis that can be started as late as 5 wk after radiation. Methods: We used 2 functional SPECT probes in vivo in …

Characterizations Of Pareto, Weibull And Power Function Distributions Based On Generalized Order Statistics, M. Ahsanullah, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Characterizations of probability distributions by different regression conditions on generalized order statistics has attracted the attention of many researchers. We present here, characterization of Pareto and Weibull distributions based on the conditional expectation of generalized order statistics extending the characterization results reported by Jin and Lee (2014). We also present a characterization of the power function distribution based on the conditional expectation of lower generalized order statistics.

Optimal Recovery Of Integral Operators And Its Applications, Vladyslav Babenko, Yuliya Babenko, Nataliya Parfinovych, Dmytro Skorokhodov

Faculty Articles

In this paper we present the solution to a problem of recovering a rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE’s.

Breaking Spaces And Forms For The Dpg Method And Applications Including Maxwell Equations, Carsten Carstensen, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. …

Applications Of Discrete Mathematics For Understanding Dynamics Of Synapses And Networks In Neuroscience, Caitlyn Parmelee

Department of Mathematics: Dissertations, Theses, and Student Research

Mathematical modeling has broad applications in neuroscience whether we are modeling the dynamics of a single synapse or the dynamics of an entire network of neurons. In Part I, we model vesicle replenishment and release at the photoreceptor synapse to better understand how visual information is processed. In Part II, we explore a simple model of neural networks with the goal of discovering how network structure shapes the behavior of the network.

Vision plays an important role in how we interact with our environments. To fully understand how visual information is processed requires an understanding of the way signals are …

Support And Rank Varieties Of Totally Acyclic Complexes, Nathan Thomas Steele

Mathematics Dissertations - Archive

Support and rank varieties of modules over a group algebra of an elementary abelian p-group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this dissertation we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent. We also show that any homogeneous affine variety is realizable as the support of some …

The Density Topology On The Reals With Analogues On Other Spaces, Stuart Nygard

Boise State University Theses and Dissertations

A point x is a density point of a set A if all of the points except a measure zero set near to x are contained in A. In the usual topology on ℝ, a set is open if shrinking intervals around each point are eventually contained in the set. The density topology relaxes this requirement. A set is open in the density topology if for each point, the limit of the measure of A contained in shirking intervals to the measure of the shrinking intervals themselves is one. That is, for any point x and a small enough …

Realizing Spaces As Classifying Spaces, Gregory Lupton, Samuel Bruce Smith

Mathematics and Statistics Faculty Publications

Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of and are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.