Applied Mathematics Commons^™

Geometry, Greed, Games, And 'Roids, James Oxley

Dalrymple Lecture Series

A three-legged stool doesn’t wobble. But four-legged stools often teeter because the tips of their legs don’t lie in the same plane.

This phenomenon of dependent sets, first theorized 75 years ago, is the focus of the 16th Dalrymple Lecture in Mathematics, set for 5:30 p.m. Friday (May 21) at the University of Mississippi. James Oxley, who holds an alumni professorship at Louisiana State University, is to deliver the address, which is free and open to the public in the Student Union Ballroom.

“There is some beautiful and intriguing mathematics that arises from some natural problems in geometry and network …

Intrinsic Contact Geometry Of Protein Dynamics, Yosi Shibberu, Allen Holder, David Cooper

Mathematical Sciences Technical Reports (MSTR)

We introduce a new measure for comparing protein structures that is especially applicable to analysis of molecular dynamics simulation results. The new measure generalizes the widely used root-mean-squared-deviation (RMSD) measure from three dimensional to n-dimensional Euclidean space, where n equals the number of atoms in the protein molecule. The new measure shows that despite significant fluctuations in the three dimensional geometry of the estrogen receptor protein, the protein's intrinsic contact geometry is remarkably stable over nanosecond time scales. The new measure also identifies significant structural changes missed by RMSD for a residue that plays a key biological role in …

Discrete Fractional Calculus And Its Applications To Tumor Growth, Sevgi Sengul

Masters Theses & Specialist Projects

Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the "difference" of any function, from 1st order up to the n-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make n- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus …

An Algorithm To Generate Two-Dimensional Drawings Of Conway Algebraic Knots, Jen-Fu Tung

Masters Theses & Specialist Projects

The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(n) time where n is the number of crossings …

Source Optimization In Abstract Function Spaces For Maximizing Distinguishability: Applications To The Optical Tomography Inverse Problem, Bonnie Jacob

All Dissertations

The focus of this thesis is to formulate an optimal source problem for the medical imaging technique of optical tomography by maximizing certain distinguishability criteria. We extend the concept of distinguishability in electrical impedance tomography to the frequency-domain diffusion approximation model used in optical tomography.
We consider the dependence of the optimal source on the choice of appropriate function spaces, which can be chosen from certain Sobolev or Lp spaces. All of the spaces we consider are Hilbert spaces; we therefore exploit the inner product in several ways. First, we define and use throughout an inner product on the Sobolev …

On Directionally Dependent Subdifferentials, Ivan Ginchev, Boris S. Mordukhovich

Mathematics Research Reports

In this paper directionally contextual concepts of variational analysis, based on dual-space constructions similar to those in [4, 5], are introduced and studied. As an illustration of their usefulness, necessary and also sufficient optimality conditions in terms of directioual subdifferentials are established, and it is shown that they can be effective in the situations where known optimality conditions in terms of nondirectional subdifferentials fail.

Analyzing Fractals, Kara Mesznik

Renée Crown University Honors Thesis Projects - All

For my capstone project, I analyzed fractals. A fractal is a picture that is composed of smaller images of the larger picture. Each smaller picture is self- similar, meaning that each of these smaller pictures is actually the larger image just contracted in size through the use of the Contraction Mapping Theorem and shifted using linear and affine transformations.

Fractals live in something called a metric space. A metric space, denoted (X, d), is a space along with a distance formula used to measure the distance between elements in the space. When producing fractals …

Sparse Representation For Detection Of Transients Using A Multi-Resolution Representation Of The Auto-Correlation Of Wavelets, Caroline Sieger

All Theses

This thesis seeks to detect damped sinusoidal transients, specifically capacitor switching transients, buried in noise and to answer the following questions: 1.) Can the transient s(t;q) be sparsely represented from s&delta(t) = s(t;q) + &epsilon(t) using sparsity methods, where &epsilon(t) is white Gaussian noise? 2.) Does computing the local auto-correlation of the signal around the transient improve detection? 3.) How does the auto-correlation shell representation compare to the wavelet representation? 4.) Which basis is ''best''? 5.) Which method and representation is best? This thesis explores detection schemes based on classical methods and newer sparsity methods. Classical methods considered include reconstruction …

The Steiner Linear Ordering Problem: Application To Resource-Constrained Scheduling Problems, Mariah Magagnotti

All Theses

When examined through polyhedral study, the resource-constrained scheduling problems have always dealt with processes which have the same priority. With the Steiner Linear Ordering problem, we can address systems where the elements involved have different levels of priority, either high or low. This allows us greater flexibility in modeling different resource-constrained scheduling problems. In this paper, we address both the linear ordering problem and its application to scheduling problems, and provide a polyhedral study of the associated polytopes.

Increased Accuracy And Efficiency In Finite Element Computations Of The Leray-Deconvolution Model Of Turbulence, Abigail Bowers

All Theses

This thesis develops, analyzes and tests a finite element method for approximating solutions to the Leray–deconvolution regularization of the Navier–Stokes equations. The scheme combines three ideas in order to create an accurate and effective algorithm: the use of an incompressible filter, a linearization that decouples the velocity–pressure system from the filtering and deconvolution operations, and a stabilization that works well with the linearization. A rigorous and complete numerical analysis of the scheme is given, and numerical experiments are presented that show clear advantages of the scheme.

A Numerical Study Of Subgrid Artificial Viscosity Methods For The Navier-Stokes Equations, Keith Galvin

All Theses

This paper studies two artificial viscosity methods for approximating solutions to the Navier&ndashStokes Equations. Both methods that are introduced add stabilization, then remove it only on a coarse mesh. Both methods can be considered as conforming, mixed methods for 1) velocity and its gradient, and 2) velocity and vorticity. Herein we rigorously study the schemes both analytically and computationally, showing that both methods are unconditionally stable and optimally convergent. Numerical experiments show both methods provide improved results over the unstabilized Navier&ndashStokes Equations.

Compressive Sensing, Yue Mao

All Theses

Compressive sensing is a novel paradigm for acquiring signals and has a wide range of applications. The basic assumption is that one can recover a sparse or compressible signal from far fewer measurements than traditional methods. The difficulty lies in the construction of efficient recovery algorithms. In this thesis, we review two main approaches for solving the sparse recovery problem in compressive sensing: l1-minimization methods and greedy methods. Our contribution is that we look at compressive sensing from a different point of view by connecting it with sparse interpolation. We introduce a new algorithm for compressive sensing called generalized eigenvalues …

The Postsecret Phenomenon: A Contemporary Application Of Existential Psychotherapy, Dan Martin

Senior Honors Projects

In November 2004, as a whimsical break from his monotonous job, Frank Warren decided he would start a small art project in his community. This idea, which he entitled “PostSecret,” involved leaving blank post cards in various public locations that simply asked to “Share a Secret” and listed a few guidelines. Frank’s goal was to “create this non-judgmental, safe place where people could feel comfortable sharing parts of their lives that they've never told a soul.” What he expected to be a small result became a weekly blog, five published books, a traveling art gallery, and a lecture series given …

Improved Accuracy For Fluid Flow Problems Via Enhanced Physics, Michael Case

All Dissertations

This thesis is an investigation of numerical methods for approximating solutions to fluid flow problems, specifically the Navier-Stokes equations (NSE) and magnetohydrodynamic equations (MHD), with an overriding theme of enforcing more physical behavior in discrete solutions. It is well documented that numerical methods with more physical accuracy exhibit better long-time behavior than comparable methods that enforce less physics in their solutions. This work develops, analyzes and tests finite element methods that better enforce mass conservation in discrete velocity solutions to the NSE and MHD, helicity conservation for NSE, cross-helicity conservation in MHD, and magnetic field incompressibility in MHD.

Neutrosophic Physics: More Problems, More Solutions, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

When considering the laws of theoretical physics, one of the physicists says that these laws – the actual expressions of the laws of mathematics and logics being applied to physical phenomena – should be limited according to the physical meaning we attribute to the phenomena. In other word, there is an opinion that a theoretical physicist should put some limitations onto mathematics, in order to “reduce” it to the observed reality. No doubt, we can do it. However, if following this way, we would arrive at only mathematical models of already known physical phenomena. Of course, this might be useful …

The Four-Color Theorem And Chromatic Numbers Of Graphs, Sarah E. Cates

Undergraduate Theses and Capstone Projects

We study graph colorings of the form made popular by the four-color theorem. Proved by Appel and Haken in 1976, the Four-Color Theorem states that all planar graphs can be vertex-colored with at most four colors. We consider an alternate way to prove the Four-Color Theorem, introduced by Hadwiger in 1943 and commonly know as Hadwiger’s Conjecture. In addition, we examine the chromatic number of graphs which are not planar. More specifically, we explore adding edges to a planar graph to create a non-planar graph which has the same chromatic number as the planar graph which we started from.

Mathematical Modeling Of Optimal Seasonal Reproductive Strategies And A Comparison Of Long-Term Viabilities Of Annuals And Perennials, Anthony Delegge

Department of Mathematics: Dissertations, Theses, and Student Research

In 1954, Lamont Cole posed a question which has motivated much ecological work in the past 50 years: When is the life history strategy of semelparity (organisms reproduce once, then die) favored, via evolution, over iteroparity (organisms may reproduce multiple times in their lifetime)? Although common sense should dictate that iteroparity would always be favored, we can observe that this is not always the case, since annual plants are not only prevalent, but can dominate an area. Also, certain plant species may be perennial in one region, but annual in another. Thus, in these areas, certain characteristics must be present …

A Non-Autonomous Second Order Boundary Value Problem On The Half-Line, Gregory S. Spradlin

Publications

By variational arguments, the existence of a solution to a nonautonomous second-order boundary problem on the half-line is proven. The corresponding autonomous problem has no solution, revealing significant differences between the autonomous and the non-autonomous case.

Codes From Riemann-Roch Spaces For Y2 = Xp - X Over Gf(P), Darren B. Glass, David Joyner, Amy Ksir

Math Faculty Publications

Let Χ denote the hyperelliptic curve y2 = xp - x over a field F of characteristic p. The automorphism group of Χ is G = PSL(2, p). Let D be a G-invariant divisor on Χ(F). We compute explicit F-bases for the Riemann-Roch space of D in many cases as well as G-module decompositions. AG codes with good parameters and large automorphism group are constructed as a result. Numerical examples using GAP and SAGE are also given.

Switching Exponent Scaling Near Bifurcation Points For Non-Gaussian Noise, Lora Billings, Ira Schwartz, Mary Mccrary, A. Korotkrov, M. Dykman

Department of Mathematics Faculty Scholarship and Creative Works

We study noise-induced switching of a system close to bifurcation parameter values where the number of stable states changes. For non-Gaussian noise, the switching exponent, which gives the logarithm of the switching rate, displays a non-power-law dependence on the distance to the bifurcation point. This dependence is found for Poisson noise. Even weak additional Gaussian noise dominates switching sufficiently close to the bifurcation point, leading to a crossover in the behavior of the switching exponent.