Mathematics Commons^™

A Generalized Approach To Darboux Transformations For Differential Equations, Mehmet Unlu

Mathematics Dissertations - Archive

A Darboux transformation is a mathematical procedure to produce a solution to a differential equation when the solution to a related differential equation is known. The basic idea behind a Darboux transformation is to change the discrete spectrum of a linear differential operator in a controlled way without changing its continuous spectrum. For example, by using a Darboux transformation one can describe the change in a quantum mechanical system when some of its quantum levels are removed or some extra quantum levels are added. Darboux transformation formulas for various differential equations have been developed, but such formulas seem to be …

Properties Of The Pinched Tensor Product, Yousuf Alkhezi

Mathematics Dissertations - Archive

For complexes of modules we study a new construction, called the pinched tensor product, which was introduced by Lars Winther Christensen and David A. Jorgensen to study Tate homology. We explore properties of the pinched tensor product and their comparison to properties of the ordinary tensor product. For example; we show some isomorphisms no longer holds for the pinched tensor product. Although if we change isomorphism to quasi-isomorphism the pinched version holds. Plus if f is a map from C to D and g is a map from A to B are morphisms of complexes of R-modules with f homotopic …

Nonlinear Dispersive Partial Differential Equations Of Physical Relevance With Applications To Vortex Dynamics, Robert Vangorder

Electronic Theses and Dissertations

Nonlinear dispersive partial differential equations occur in a variety of areas within mathematical physics and engineering. We study several classes of such equations, including scalar complex partial differential equations, vector partial differential equations, and finally non-local integro-differential equations. For physically interesting families of these equations, we demonstrate the existence (and, when possible, stability) of specific solutions which are relevant for applications. While multiple application areas are considered, the primary application that runs through the work would be the nonlinear dynamics of vortex filaments under a variety of physical models. For instance, we are able to determine the structure and time …

The Use Of Trigonometry In Blood Spatter, Isela Guerra

A with Honors Projects

A common phrase to be heard in a class room setting is “When am I ever going to use trig?” Many do not value or appreciate the importance of trigonometry. However, Trigonometry holds a special place in the hearts of the blood spatter analyst. The blood spatter analyst works very hard using the known identities and trigonometric functions to find the angle of impact, that is, at what angle was the person struck, and at what angle the blood fell. Trigonometry is also used in determining the height of a person, using the angle of impact.

Horizontally Submersions Of Contact Cr-Submanifolds, Fortune Massamba, Tshikunguila Tshikuna-Matamba

Turkish Journal of Mathematics

In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures. For horizontally submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. Curvature properties between curvatures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR-submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf …

Tiling The Integers, Shasha Li

Electronic Theses and Dissertations

A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on finite Abelian groups show that Fuglede conjecture is false in high dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the finite groups ZN. Based on the Coven- Meyerowitz conjecture, it has been proved that if any spectral set in Z satisfies the the …

Stability And Controls For Stochastic Dynamic Systems, Zhixin (Harriet) Yang

Wayne State University Dissertations

This dissertation focuses on stability analysis and optimal controls for stochastic dynamic systems. It encompasses two parts. One part of our work gives an in-depth study of stability of linear jump diffusion, linear Markovian jump diffusion, multi-dimensional jump diffusion and

regime-switching jump diffusion together with the associated numerical solutions. The other part of our work is controls for stochastic dynamic systems, to be specific, we concentrated on mean variance types of control under different formulations. We obtained the nearly optimal

mean-variance controls under both two-time-scale and hidden Markov chain formulations and convergence for each case is achieved.

In Chapter 2, …

Properties Of Nonlinear Randomly Switching Dynamic Systems: Mean-Field Models And Feedback Controls For Stabilization, Guangliang Zhao

Wayne State University Dissertations

This dissertation concerns the properties of nonlinear dynamic systems hybrid with Markov switching. It contains two parts. The first part focus on the mean-field models with state-dependent regime switching, and the second part focus on the system regularization and stabilization using feedback control. Throughout this dissertation, Markov switching processes are used to describe the randomness caused by discrete events, like sudden environment change or other uncertainty.

In Chapter 2, the mean-field models we studied are formulated by nonlinear stochastic differential equations hybrid with state-dependent regime switching. It originates from the phase transition problem in statistical physics. The mean-field term is …

Subfunctors Of Extension Functors, Furuzan Ozbek

Theses and Dissertations--Mathematics

This dissertation examines subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enveloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalø and Enochs and has proven to be beneficial for module theory as well as for representation theory. The first few chapters examine the subfunctors of Ext and their properties. It is showed how the class of precoverings give us subfunctors of Ext. Furthermore, the characterization of these subfunctors and some examples are given. In the latter chapters ideals, the subfunctors of Hom, are investigated. The definition of cover and …

Comparative Analysis Of Gps Data, Kiel Von Lindenberg

Undergraduate Journal of Mathematical Modeling: One + Two

The goal of this project is to calculate the distance traveled during a bike ride around the University of South Florida using information gathered by a GPS (Global Positioning System) application on a smartphone. We calculate the route distance in two ways: from latitudes and longitudes using the haversine formula and 2) from velocities and times using the trapezoidal rule. These computed distances were compared to the distance given by the smartphone application.

Voltage Smoothing With A Capacitor, Martin Copello

Undergraduate Journal of Mathematical Modeling: One + Two

A rectifier is an electrical circuit that converts alternating current to direct current, e.g., a half wave rectifier removes lower valleys of the AC voltage wave. If a capacitor is placed in parallel with a half wave rectifier, the transition from voltage peak to peak is smoothed to closer resemble constant DC voltage. The objective of this project was to model the voltage changes of a half wave rectifier with a parallel capacitor. It was found that the smoothness of the voltage waves increased as the capacitance of the capacitor increased.

Calculating The Solar Energy Of A Flat Plate Collector, Ariane Rosario

Undergraduate Journal of Mathematical Modeling: One + Two

The amount of solar energy that could be obtained by a flat plate solar collector of one square meter dimension is calculated in three different locations: Tampa FL, Fairbanks AL, and Pontianak Indonesia, considering the varying sunset time for each day of the year. The results show that if the collectors are placed near the equator, more total energy could be obtained. In fact, by placing a solar collector in Pontianak, Indonesia 12.42% more solar energy can be obtained than by placing it in Tampa and 96.9% more solar energy than Alaska.

Florida Sinkholes And Grout Injection Stabilization, Charles Hunt Griffith Ii

Undergraduate Journal of Mathematical Modeling: One + Two

Florida has a major problem when it comes to sinkholes. These sinkholes can become very hazardous to people, homes, and to the landscape as a whole. Florida sits on a carbonate platform which is highly indicative of sinkholes. There are three main types of sinkholes which occur in Florida: dissolution, cover subsidence, and cover collapse. I will compare these types of sinkholes to the underlying formation beneath Florida to see if there is a connection between the types of sinkholes that occur. I will also create a 3D model of grout injection stabilization and calculate its volume to compare to …

Rocket Flight Path, Jamie Waters

Undergraduate Journal of Mathematical Modeling: One + Two

This project uses Newton’s Second Law of Motion, Euler’s method, basic physics, and basic calculus to model the flight path of a rocket. From this, one can find the height and velocity at any point from launch to the maximum altitude, or apogee. This can then be compared to the actual values to see if the method of estimation is a plausible. The rocket used for this project is modeled after Bullistic-1 which was launched by the Society of Aeronautics and Rocketry at the University of South Florida.

Fukushima: A Model Study, Melinda Kathryn Mckell

Undergraduate Journal of Mathematical Modeling: One + Two

This article investigates the radiation dispersion modeling system available to Japanese authorities during the response phase of the Fukushima Dai-Ichi incident.

Modeling Rocket Flight In The Low-Friction Approximation, Logan White

Undergraduate Journal of Mathematical Modeling: One + Two

In a realistic model for rocket dynamics, in the presence of atmospheric drag and altitude-dependent gravity, the exact kinematic equation cannot be integrated in closed form; even when neglecting friction, the exact solution is a combination of elliptic functions of Jacobi type, which are not easy to use in a computational sense. This project provides a precise analysis of the various terms in the full equation (such as gravity, drag, and exhaust momentum), and the numerical ranges for which various approximations are accurate to within 1%. The analysis leads to optimal approximations expressed through elementary functions, which can be implemented …

Survival Analysis - Breast Cancer, Minh Hoang Pham

Undergraduate Journal of Mathematical Modeling: One + Two

In this study we used the parametric survival approach to analyze the survival time of African American breast cancer patients. In research about the survival time of patients, the Cox Proportional Hazard model, a semi-parametric method, is primarily used, but this approach does not rely on the distributional assumptions. The parametric method is more consistent with a theoretical approach compared to a semi-parametric approach. We observe that the survival time of African American breast cancer patients follows the Weibull Probability Distribution. First, we fitted the model to represent the survival ability of the general population (African American women diagnosed with …

Decay Estimates On Trace Norms Of Localized Functions Of Schrödinger Operators, Aaron Saxton

Theses and Dissertations--Mathematics

In 1973, Combes and Thomas discovered a general technique for showing exponential decay of eigenfunctions. The technique involved proving the exponential decay of the resolvent of the Schrödinger operator localized between two distant regions. Since then, the technique has been been applied to several types of Schrödinger operators. This dissertation will show that the Combes--Thomas method works well with trace, Hilbert--Schmidt and other trace-type norms. The first result we prove shows exponential decay on trace-type norms of a resolvent of a Schrödinger operator localized between two distant regions. We build on this result by applying the Combes--Thomas method again to …

Representation Theory Of Totally Reflexive Modules Over Non-Gorenstein Rings, Denise Amanda Rangel

Mathematics Dissertations - Archive

In the late 1960's Auslander and Bridger published Stable Module Theory, in whichthe idea of totally reflexive modules first appeared. These modules have been studiedby many. However, a bulk of the information known about them is when they are overa Gorenstein ring, since in that case they are exactly the maximal Cohen-Macaulaymodules. Much is already known about maximal Cohen-Macaulay modules, that is,totally reflexive modules over a Gorenstein ring. Therefore, we investigate the existence and abundance of totally reflexive modules over non-Gorenstein rings.It is known that if there exist one non-trivial totally reflexive module over a non-Gorenstein ring, then there exists …

An Isometric Dynamics For A Causal Set Approach To Discrete Quantum Gravity, S. Gudder

Mathematics Preprint Series

We consider a covariant causal set approach to discrete quantum gravity. We first review the microscopic picture of this approach. In this picture a universe grows one element at a time and its geometry is determined by a sequence of integers called the shell sequence. We next present the macroscopic picture which is described by a sequential growth process. We introduce a model in which the dynamics is governed by a quantum transition amplitude. The amplitude satisfies a stochastic and unitary condition and the resulting dynamics becomes isometric. We show that the dynamics preserves stochastic states. By “doubling down” on …

The Prevalence Of Antibodies Against Wheat And Milk Proteins In Blood Donors And Their Contribution To Neuroimmune Reactivities, Aristo Vojdani, Datis Kharrazian, Partha Sarathi Mukherjee

Mathematics Faculty Publications and Presentations

The aim of this study was to look for the presence of IgG, IgM, and IgA antibodies against two widely consumed foods, wheat and milk, in a relatively large number of specimens. As wheat, milk, and their antigens have been found to be involved in neuroimmune disorders, we measured the co-occurrence of their antibodies against various neural antigens. We assessed the reactivity of sera from 400 donors to wheat and milk proteins, GAD-65, cerebellar, MBP, and MOG. Statistical analysis showed significant clustering when certain wheat and milk protein antibodies were cross-referenced with neural antibodies. Approximately half of the sera with …

On Constructions Of Generalized Skein Modules, Uwe Kaiser

Mathematics Faculty Publications and Presentations

Jozef Przytycki introduced skein modules of 3-manifolds and skein deformation initiating algebraic topology based on knots. We discuss the generalized skein modules of Walker, defined by fields and local relations. Some results by Przytycki are proven in a more general setting of fields defined by decorated cell-complexes in manifolds. A construction of skein theory from embedded TQFT-functors is given, and the corresponding background is developed. The possible coloring of fields by elements of TQFT-modules is discussed for those generalized skein modules. Also an approach of defining skein modules from studying compressions of fields is described.

Performance Modeling And Optimization Techniques For Heterogeneous Computing, Supada Laosooksathit

Doctoral Dissertations

Since Graphics Processing Units (CPUs) have increasingly gained popularity amoung non-graphic and computational applications, known as General-Purpose computation on GPU (GPGPU), CPUs have been deployed in many clusters, including the world's fastest supercomputer. However, to make the most efficiency from a GPU system, one should consider both performance and reliability of the system.

This dissertation makes four major contributions. First, the two-level checkpoint/restart protocol that aims to reduce the checkpoint and recovery costs with a latency hiding strategy in a system between a CPU (Central Processing Unit) and a GPU is proposed. The experimental results and analysis reveals some benefits, …

Generalized Least-Squares Regressions Iii: Further Theory And Classification, Nataniel Greene

Publications and Research

This paper continues the work of this series with two results. The first is an exponential equivalence theorem which states that every generalized least-squares regression line can be generated by an equivalent exponential regression. It follows that every generalized least-squares line has an effective normalized exponential parameter between 0 and 1 which classifies the line on the spectrum between ordinary least-squares and the extremal line for a given set of data. The second result is the presentation of fundamental formulas for the generalized least-squares slope and y-intercept.

Lecture-Free Calculus For Science And Engineering, Benjamin Wiles

IMPACT Symposium

Plane Analytic Geometry and Calculus I (MA 16100) is a historically difficult, required course for engineering and science majors. The traditional configuration consists of 250 students meeting in a large lecture 3 times per week and twice per week in smaller recitations of size 40. Additionally, those who repeat the course often continue to encounter difficulty. A scalable re-design has been implemented to attempt to address the needs of students that are not being met in the traditional configuration by diverting resources from lecture to problem sessions and from traditional Q&A recitations to student-driven presentation/collaboration-based recitations. The students work in …

Eigenvalue Multiplicites Of The Hodge Laplacian On Coexact 2-Forms For Generic Metrics On 5-Manifolds, Megan E. Gier

Theses and Dissertations--Mathematics

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δ_g are all simple for a residual set of C^r metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of C^r metrics such that the nonzero eigenvalues of the Hodge Laplacian Δ_g^(k) on k-forms are all …

Boij-Söderberg Decompositions, Cellular Resolutions, And Polytopes, Stephen Sturgeon

Theses and Dissertations--Mathematics

Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. In chapter 2 using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules. These results are published in the Journal of Algebra, see [25].

In chapter 3 we provide some further results about Boij-Söderberg decompositions. We …

The Mathematics Of Spot It, Marcus Heemstra

The Journal of Undergraduate Research

Spot It™ is a fun, fast paced game for people of all ages. The game appears to have a rich mathematical structure, which we investigated, using combinatorial analysis. We constructed some similar Spot It type games with a different number of cards, and displayed that some games are impossible to complete. In particular, the original Spot It game has eight objects on each card; however, it is impossible to construct a complete game (a game in which every object appears with every other object) in which every card has seven objects. We have investigated the mathematics behind the game.

Titchmarsh-Weyl Theory For Canonical Systems, Keshav R. Acharya

Publications

The main purpose of this paper is to develop Titchmarsh- Weyl theory of canonical systems. To this end, we first observe the fact that Schrodinger and Jacobi equations can be written into canonical systems. We then discuss the theory of Weyl m-function for canonical systems and establish the relation between the Weyl m-functions of Schrodinger equations and that of canonical systems which involve Schrodinger equations.

Delayed Rejection Algorithm To Estimate Bayesian Social Networks, Alberto Caimo, Antonietta Mira

Articles

Statistical social network analysis has become a very active and fertile area of research in the recent past. Recent developments in Bayesian computational methods have been successfully applied to estimate social network models. The Delayed rejection (DR) strategy is a modification of the Metropolis-Hastings (MH) algorithms that reduces the variance of the resulting Markov chain Monte Carlo estimators and allows partial adaptation of the proposal distribution. In this paper we show how the DR strategy can be exploited to estimate dyadic independence social network models leading to an average 40% variance reduction relative to the competing MH algorithm, confirming that …