Mathematics Commons^™

Partial Orders For Representing Uncertainty, Causality And Decision Making: General Properties, Operations, And Algorithms, Francisco Adolfo Zapata

Open Access Theses & Dissertations

One of the main objectives of science and engineering is to help people select the most beneficial decisions. To make these decisions, we must know people's preferences, we must have the information about different possible consequences of different decisions. Since information is never absolutely accurate and precise, we must also have information about the degree of certainty of different parts on information. All these types of information naturally lead to partial orders:

- For preferences, a <= b means that b is preferable to a. This relation is used in decision theory.

- For events, a <= b means that a can influence b. This causality relation is one of the fundamental notions of physics, especially of physics of space-time.

* For uncertain statements, a <= b means that a is less certain than b. This relation is used in logics describing uncertainty, such as fuzzy logic.

In each of these areas, there is abundant research about studying the corresponding partial orders. …

Question 1: Google And Pac-Man; Question 2: Magnetic Levitation, Larry Weinstein

Physics Faculty Publications

On the 30th anniversary of the introduction of the video game Pac-Man, Google created a miniature version of the game and placed it as its logo on its home page. How much total time was wasted in the United States playing this game?

(Thanks to Daniel Drill of Old Dominion University for suggesting the question.)

Animal tissue is diamagnetic. How large a magnetic field gradient is needed to levitate frogs or people?

On K-Trees And Special Classes Of K-Trees, John Wheless Estes

Electronic Theses and Dissertations

The class of k-trees is defined recursively as follows: the smallest k-tree is the k-clique. If G is a graph obtained by attaching a vertex v to a k-clique in a k-tree, then G is also a k-tree. Trees, connected acyclic graphs, are k-trees for k = 1. We introduce a new parameter known as the shell of a k-tree, and from the shell special subclasses of k-trees, tree-like k-trees, are classified. Tree-like k-trees are generalizations of paths, maximal outerplanar graphs, and chordal planar graphs with toughness exceeding one. Let fs = fs( G) be the number of independent sets …

A Note On Acoustic Propagation In Power-Law Fluids: Compact Kinks, Mild Discontinuities, And A Connection To Finite-Scale Theory, Dongming Wei

Mathematics Faculty Publications

Acoustic traveling waves in a class of viscous, power-lawfluids are investigated. Both bi-directional and unidirectional versions of the one-dimensional (1D), weakly-nonlinear equation of motion are derived; traveling wave solutions (TWS)s, special cases of which take the form of compact and algebraic kinks, are determined; and the impact of the bulk viscosity on the structure/nature of the kinks is examined. Most significantly, we point out a connection that exists between the power-law model considered here and the recently introduced theory of finite-scale equations.

Flow Over A Traveling Wavy Foil With A Passively Flapping Flat Plate, Nansheng Liu, Yan Peng, Youwen Liang, Xiyun Lu

Mathematics & Statistics Faculty Publications

Flow over a traveling wavy foil with a passively flapping flat plate has been investigated using a multiblock lattice Boltzmann equation and the immersed boundary method. The foil undergoes prescribed undulations in the lateral direction and the rigid flat plate has passive motion determined by the fluid structure interaction. This simplified model is used to study the effect of the fish caudal fin and its flexibility on the locomotion of swimming animals. The flexibility of the caudal fin is modeled by a torsion spring acting about the pivot at the conjuncture of the wavy foil and the flat plate. The …

The New Publishing Scene And The Tenure Case: An Administrator’S View, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

No abstract provided.

Upper Bounds In The Ohtsuki-Riley-Sakuma Partial Order On 2-Bridge Knots, Scott M. Garrabrant, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K we characterize all other 2-bridge knots J such that {K, J} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.

Congrats You’Re Abd! Now What?, Alissa Crans

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

Characterization Of Radially Symmetric Finite Time Blowup In Multidimensional Aggregation Equations, Andrea L. Bertozzi, John B. Garnett, Thomas Laurent

Mathematics, Statistics and Data Science Faculty Works

This paper studies the transport of a mass $\mu$ in $\mathbb{R}^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent p we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.

On Levi-Civita’S Alternating Symbol, Schouten’S Alternating Unit Tensors, Cpt, And Quantization, Evert Jan Post, Stan Sholar, Hooman Rahimizadeh, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

The purpose of the present article is to demonstrate that by adopting a unifying differential geometric perspective on certain themes in physics one reaps remarkable new dividends in both microscopic and macroscopic domains. By replacing algebraic objects by tensor-transforming objects and introducing methods from the theory of differentiable manifolds at a very fundamental level we obtain a Kottler-Cartan metric-independent general invariance of the Maxwell field, which in turn makes for a global quantum superstructure for Gauss-Amp`ere and Aharonov-Bohm “quantum integrals.” Beyond this, our approach shows that postulating a Riemannian metric at the quantum level is an unnecessary concept and our …

Spatial Graphs With Local Knots, Erica Flapan, Blake Mellor, Ramin Naimi

Mathematics, Statistics and Data Science Faculty Works

It is shown that for any locally knotted edge of a 3-connected graph in S³, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S³.

Enhancements Of Rack Counting Invariants Via Dynamical Cocycles, Alissa S. Crans, Sam Nelson, Aparna Sarkar

Mathematics, Statistics and Data Science Faculty Works

We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Alexander polynomial.

Convergence Of A Steepest Descent Algorithm For Ratio Cut Clustering, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht

Mathematics, Statistics and Data Science Faculty Works

Unsupervised clustering of scattered, noisy and high-dimensional data points is an important and difficult problem. Tight continuous relaxations of balanced cut problems have recently been shown to provide excellent clustering results. In this paper, we present an explicit-implicit gradient flow scheme for the relaxed ratio cut problem, and prove that the algorithm converges to a critical point of the energy. We also show the efficiency of the proposed algorithm on the two moons dataset.

How Well Do The Nsf Funded Elementary Mathematics Curricula Align With The Gaise Report Recommendations?, Anna E. Bargagliotti

Mathematics, Statistics and Data Science Faculty Works

Statistics and probability have become an integral part of mathematics education. Therefore it is important to understand whether curricular materials adequately represent statistical ideas. The Guidelines for Assessment and Instruction in Statistics Education (GAISE) report (Franklin, Kader, Mewborn, Moreno, Peck, Perry, & Scheaffer, 2007), endorsed by the American Statistical Association, provides a two-dimensional (process and level) framework for statistical learning. This paper examines whether the statistics content contained in the NSF funded elementary curricula Investigations in Number, Data, and Space, Math Trailblazers, and Everyday Mathematics aligns with the GAISE recommendations. Results indicate that there are differences in the approaches used …

Characterizing Conflict In Wikipedia, Nathaniel Miller

Mathematics, Statistics, and Computer Science Honors Projects

Wikipedia serves as the Internet's most widely viewed reference. In order to ensure its success, editors who create and maintain articles must resolve conflicts over appropriate article content. Previous research has measured Wikipedia conflict at two levels: single articles and categories of pages. I observe conflicts within small groups of articles, identifying their frequency, size, and intensity. Additionally, I identify individual conflicts spanning multiple articles and effects of conflict upon users' editing habits. I analyze cross-article conflict in three stages. First, I cluster a group of 1.4 million Wikipedia articles. Next, I find individual user conflicts within each article cluster …

The Best Mixing Time For Trees, Jeanmarie Youngblood

Mathematics, Statistics, and Computer Science Honors Projects

A graph G consists of a set of vertices connected in pairs by edges. Two vertices connected by an edge are called neighbors. A random walk on G is a sequence of vertices, where the next vertex is chosen randomly from the neighbors of the current vertex. Under mild assumptions, the distribution of a walk's current location converges to the stationary distribution. In this distribution, the probability of being at a given vertex is proportional to the size of its neighbor set. The expected time to convergence is called a mixing measure of G. There are a variety …

Reducibility, Degree Spectra, And Lowness In Algebraic Structures, Rebecca M. Steiner

Dissertations, Theses, and Capstone Projects

This dissertation addresses questions in computable structure theory, which is a branch of mathematical logic hybridizing computability theory and the study of familiar mathematical structures. We focus on algebraic structures, which are standard topics of discussion among model theorists. The structures examined here are fields, graphs, trees under a predecessor function, and Boolean algebras.

For a computable field F, the splitting set S_F of F is the set of polynomials in F[X] which factor over F, and the root set R_F of F is the set of polynomials in F[X] which have a root in F …

On Some Order 6 Non-Symplectic Automorphisms Of Elliptic K3 Surfaces, Jimmy J. Dillies

Mathematical Sciences: Faculty Publications

We classify primitive non-symplectic automorphisms of order 6 on K3 surfaces. We show how their study can be reduced to the study of non-symplectic automorphisms of order 3 and to a local analysis of the fixed loci. In particular, we determine the possible fixed loci and show that when the Picard lattice is fixed, K3 surfaces come in mirror pairs.

R0 Analysis Of A Spatiotemporal Model For A Stream Population, H. W. Mckenzie, Y. Jin, J. Jacobsen, M. A. Lewis

Department of Mathematics: Faculty Publications

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem im- pacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R₀ for advection-diffusion-reaction equations and on related measures. We apply the measures to popula- tion persistence in rivers under various flow regimes. This …

The Distribution Of Individual Stock Returns In A Modified Black-Scholes Option Pricing Model, Daniel Lee Richey

College of Graduate Studies: Theses & Dissertations

Author's abstract: There have been many attempts to find a model that can accurately price options. These models are built on many assumptions, including which probability distribution stock returns follow. In this paper, we test several distributions to see which best fit the log returns of 20 different companies over a period between November 1, 2006 to October 31, 2011. If a "best" distribution is found, a modified Black-Scholes model will be defined by modifying the Weiner process. We use Monte Carlo simulations to generate estimated prices under specified parameters, and compare these prices to those simulated by the model …