Mathematics Commons^™

Optimizing A Game Of Chinese Checkers, Nicholas Fonseca

Honors Program Theses and Projects

Chinese Checkers is a multi-player strategy game in which game play can become surprising complex as the game progresses. In spite of this game's complexity, questions involving games with multiple players have received little research attention. This paper considers the three player case and discusses how to describe short games. By utilizing these tendencies for short games, a heuristic function can be defined which associates a player's possible move with a heuristic value. These heuristic values guide a search algorithm which searches through all the possible moves made in a game. To guide this discussion for three player games, the …

A Critical Analysis Of Random Response Techniques, Emanuel Zanzerkia

Honors Program Theses and Projects

In order to understand and make informed decision on sensitive topics such as domestic violence and drug use, interviews have been used to collect data. However it is difficult to assess how truthful respondents are since they may not feel at ease revealing the truth to an interviewer. Surveyors of sensitive issues face the problem that respondents may be reluctant to answer truthfully since the respondent may feel pressured socially or may fear the repercussions of their truthful answer. Processes known as random response techniques have been introduced to allow interviewers the ability to extract information they need for a …

Speedups And Orbit Equivalence Of Finite Extensions Of Ergodic Zᵈ-Actions, Aimee S.A. Johnson, D. M. Mcclendon

Mathematics & Statistics Faculty Works

We classify n-point extensions of ergodic Zᵈ-actions up to relative orbit equivalence and establish criteria under which one n-point extension of an ergodic Zᵈ-action can be sped up to be relatively isomorphic to an n-point extension of another ergodic Zᵈ-action. Both results are characterized in terms of an algebraic object associated to each n-point extension which is a conjugacy class of subgroups of the symmetric group on n elements.

Towards A Theory Of Recursive Function Complexity: Sigma Matrices And Inverse Complexity Measures, Bradford M. Fournier

LSU New Orleans Theses and Dissertations

This paper develops a data structure based on preimage sets of functions on a finite set. This structure, called the sigma matrix, is shown to be particularly well-suited for exploring the structural characteristics of recursive functions relevant to investigations of complexity. The matrix is easy to compute by hand, defined for any finite function, reflects intrinsic properties of its generating function, and the map taking functions to sigma matrices admits a simple polynomial-time algorithm . Finally, we develop a flexible measure of preimage complexity using the aforementioned matrix. This measure naturally partitions all functions on a finite set by characteristics …

Postural Responses To Perturbations Of The Vestibular System During Walking In Healthy Young And Older Adults, Jung Hung Chien

Theses & Dissertations

It has been shown that approximate one-third of US adults aged 40 years and older (69 million US citizens) have some type of vestibular problems. These declining abilities of the vestibular system affect quality of life. Difficulties in performing daily activities (dressing, bathing, getting in and out of the bed and etc.) have been highly correlated to loss of balance due to vestibular disorders. The exact number of people affected by vestibular disorders is still difficult to quantify. This might be because symptoms are difficult to describe and differences exist in the qualifying criteria within and across studies. Thus, it …

Calculus: An Integrated Approach, Matthew E. Cathey, Joseph A. Spivey

Faculty Books

This is a self-published digital textbook “Calculus: An Integrated Approach” by Matt Cathey and Joseph Spivey. The textbook covers a two-semester calculus sequence for Wofford College. The text is freely available for use under the terms of the creative commons copyright, found in the textbook file. The free software application Wolfram Player is required to read the textbook; it can be downloaded here: https://www.wolfram.com/player/.

Here is a video showing important features of the textbook: https://www.youtube.com/watch?v=lAPxqZP_LQg .

Here is a video helping Mac users troubleshoot the most common issues in installing and opening the textbook: https://www.youtube.com/watch?v=lAPxqZP_LQg .

The complete, up-to-date …

Method Of Lines Transpose: An Efficient A-Stable Solver For Wave Propagation, Matthew Causley, Andrew Christlieb, Eric Wolf

Mathematics Publications

Building upon recent results obtained in [7,8,9], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOLT), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [7], A-stable schemes of high order were derived, and in [9] a high order, fast O(N) spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow …

Fast Methods For Variable-Coefficient Peridynamic And Non-Local Diffusion Models, Che Wang

Theses and Dissertations

In previous studies, scientists developed the classical solid mechanic theory. The model has been widely used in scientific research and practical production. The main assumption of the classical theory of solid mechanics is that all internal forces act through zero distance. Because of this assumption, the mathematical model always leads to partial differential equations, which meet with problems when describing the spontaneous formation of discontinuities and other singularities. A peridynamic model was proposed as a reformation of solid mechanics [40, 41, 43, 44, 45], which leads to a non-local framework that does not explicitly involve the notion of deformation gradients, …

A Survey Of The Kinetic Monte Carlo Algorithm As Applied To A Multicellular System, Michael Richard Laughlin

Theses and Dissertations

We explore the origins and implementation of the Kinetic Monte Carlo method on a system of cells suspended in a liquid media. The situation presented herein has applications in the emerging field of biofabrication, which may have lasting impacts in medical science. The theory behind the method is explained in detail, starting with its emergence in the 1960s, and two major improvements to the scaling of the method are presented, along with a restriction to a special case. Finally, we give the results of several simulations.

Modeling, Simulation, And Applications Of Fractional Partial Differential Equations, Wilson Cheung

Theses and Dissertations

The Black-Scholes model is commonly used to track the price of European options with respect to maturity in many financial markets. This model degenerates into a partial differential equation that relates the European-style option price to the underlying price and time of expiry. Black-Scholes assumes that underlying prices satisfy a geometric Brownian motion.

After the U.S. stock market crash of 1987, this assumption becomes inaccurate as it fails to represent the behavior of S&P 500 European vanilla option prices. Specifically, under the measure of moneyness, the volatility smirk does not flatten out and the resulting conditional return distribution does not …

The Definitions Of Three-Dimensional Landmarks On The Human Face: An Interdisciplinary View, Stanislav Katina, Kathryn Mcneil, Ashraf Ayoub, Brendan Guilfoyle, Balvinder Khambay, Paul Siebert, Federico Sukno, Mario Rojas, Liberty Vittert, John Waddington, Paul F. Whelan, Adrian W. Bowman

Publications

The analysis of shape is a key part of anatomical research and in the large majority of cases landmarks provide a standard starting point. However, while the technology of image capture has developed rapidly and in particular three-dimensional imaging is widely available, the definitions of anatomical landmarks remain rooted in their two-dimensional origins. In the important case of the human face, standard definitions often require careful orientation of the subject. This paper considers the definitions of facial landmarks from an interdisciplinary perspective, including biological and clinical motivations, issues associated with imaging and subsequent analysis, and the mathematical definition of surface …

On The Construction Of Simply Connected Solvable Lie Groups, Mark E. Fels

Research Vignettes

This worksheet contains the implementation of Theorems 4.2, 5.4 and 5.7 in the paper On the Construction of Solvable Lie Groups. All the examples in the paper are demonstrated here, along with one in Section 6 that was too long to include in the article.

Direct Phenotypic Screening In Mice: Identification Of Individual, Novel Antinociceptive Compounds From A Library Of 734 821 Pyrrolidine Bis-Piperazines, Richard A. Houghten, Michelle L. Ganno, Jay P. Mclaughlin, Colette T. Dooley, Shainnel O. Eans Torrey Pines Institute For Molecular Studies, Radleigh Santos, Travis Lavoi, Adel Nefzi, Greg Welmaker, Marc A. Giulianotti, Lawrence Toll

Mathematics Faculty Articles

The hypothesis in the current study is that the simultaneous direct in vivo testing of thousands to millions of systematically arranged mixture-based libraries will facilitate the identification of enhanced individual compounds. Individual compounds identified from such libraries may have increased specificity and decreased side effects early in the discovery phase. Testing began by screening ten diverse scaffolds as single mixtures (ranging from 17 340 to 4 879 681 compounds) for analgesia directly in the mouse tail withdrawal model. The “all X” mixture representing the library TPI-1954 was found to produce significant antinociception and lacked respiratory depression and hyperlocomotor effects using …

Augmenting The Immersed Boundary Method With Radial Basis Functions (Rbfs) For The Modeling Of Platelets In Hemodynamic Flows, Varun Shankar, Grady B. Wright, Robert M. Kirby, Aaron L. Fogelson

Mathematics Faculty Publications and Presentations

We present a new computational method by extending the Immersed Boundary (IB) method with a geometric model based on parametric Radial Basis Function (RBF) interpolation of the Lagrangian structures. Our specific motivation is the modeling of platelets in hemodynamic flows, though we anticipate that our method will be useful in other applications involving surface elasticity. The efficacy of our new RBF-IB method is shown through a series of numerical experiments. Specifically, we test the convergence of our method and compare our method with the traditional IB method in terms of computational cost, maximum stable time-step size and volume loss. We …

Finite Groups In Which Pronomality And 𝔉-Pronormality Coincide, Adolfo Ballester-Bolinches, James C. Beidleman, Arnold D. Feldman, Matthew F. Ragland

Mathematics Faculty Publications

For a formation 𝔉, a subgroup U of a finite group G is said to be 𝔉-pronormal in G if for each g ∈ G, there exists x ∈ ⟨U, U^g⟩ ^𝔉 such that U^x = U^g. If 𝔉 contains 𝔑, the formation of nilpotent groups, then every 𝔉-pronormal subgroup is pronormal and, in fact, 𝔑-pronormality is just classical pronormality. The main aim of this paper is to study classes of finite soluble groups in which pronormality and 𝔉-pronormality coincide.

Efficient High-Order Methods For Solving Fractional Differential Equations Of Order Α ∈ (0, 1) Using Fast Convolution And Applications In Wave Propagation, Matthew F. Causley, Peter G. Petropoulus

Mathematics Publications

In this work we develop a means to rapidly and accurately compute the Caputo fractional derivative of a function, using fast convolution. The key element to this approach is the compression of the fractional kernel into a sum of M decaying exponentials, where M is minimal. Specifically, after N time steps we find M= O (log N) leading to a scheme with O (N log N) complexity. We illustrate our method by solving the fractional differential equation representing the Kelvin-Voigt model of viscoelasticity, and the partial differential equations that model the propagation of electromagnetic pulses in the Cole-Cole model of …

Hypercube Unfoldings That Tile R3 And R2, Giovanna Diaz, Joseph O'Rourke

Computer Science: Faculty Publications

We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane.

Comparison Theorems And Asymptotic Behavior Of Solutions Of Discrete Fractional Equations, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

Consider the following n-th order nabla and delta fractional difference equations

rn r (a)x(t) = c(t)x(t), t 2 Na+1, x(a) > 0.

and

Va+v-1x(t) = c(t)x(t + v - 1), t 2 Na, x(a + n - 1) > 0

We establish comparison theorems by which we compare the solutions x(t) of (*) and (**) with the solutions of the equations rn r(a)x(t) = bx(t) and Dn a+v-1x(t) = bx(t + v -1), respectively, where b is a constant. We obtain four asymptotic results, one of them extends the recent result [F. M. Atici, P. W. Eloe, Rocky Mountain J. Math. 41(2011), …

Stochastic Optimal Control For Online Seller Under Reputational Mechanisms, Milan Bradonij', Matthew Causley, Albert Cohen

Mathematics Publications

In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller’s reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which …

The Minimum And Other Free Energies For Non-Linear Materials With Memory., John Murrough Golden

Articles

Expressions are obtained for free energies of materials with a certain type of non-linear constitutive relation. In particular, the minimum and related free energies are considered in some detail. Minimal states are defined for these materials, and it is shown that any free energy yielding a linear constitutive equation that is a functional of the minimal state has a counterpart in the non-linear case which is also a minimal state functional in this more general context. These results are explored for simple examples, including discrete spectrum materials.