Shenandoah Undergraduate Mathematics and Statistics Conference
at James Madison University, October 28, 2006
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SUMS 2005
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Contact the organizers: SUMS at math dot jmu dot edu

Here are the abstracts of current contributed talks:

Parameter Influences on Instability of the Keller-Segel Aggregation Model
        Ronald C. Anderson, Thiel College
The Keller-Segel Model suggests amoebae aggregation into fruiting bodies results from an instability. This talk will introduce an instability condition for the PDE model proposed by Keller and Segel. A mathematical and biological analysis of several parameters contributing to system instability will then be presented.
         This research was conducted at Texas Tech University, under the supervision of Dr. Akif Ibraguimov.
Estimating the Eigenvalues of a Covariance Matrix
          Robert Carrico, University of  Mary Washington
The purpose of this research was to produce confidence estimates of the eigenvalues of a covariance matrix.  The confidence intervals were derived from two different confidence bands for the characteristic polynomial . Simulations were conducted for a variety of different pairs for the eigenvalues to see how well the estimators perform.
        This research was conducted at the University of Mary Washington, under the supervision of  Dr. Debra Hydorn .
A Ribbon Graph Structure on Plane Curves
             Michael Chmutov, The Ohio State University
The talk concerns the study of immersions of a circle into the plane with only transverse double point singularities. In particular, we show how to capture the information about such a plane curve using a combinatorial structure of a ribbon graph.
        This research was conducted at Oregon State University, under the supervision of  Dr. Juha Pohjanpelto.
PascGalois: Pictures and Patterns
              Julianne Coxe, JMU
Using PascGaloisJE to implement Pascal's triangle update rule to graphically represent finite space-time diagrams, I observed patterns displayed when the modular number and seed were varied.  Certain combinations of powers of 2 and prime numbers resulted in common patterns in the graphics that may be specific to those number classes.
        This research was conducted at New College of Florida, under the supervision of  Dr. Eirini Poimenidou.
The Aeroacoustics Of Turbulent Coanda Jet Flows
          Jason Fox, James Madison University
This paper extends the theory of mathematically predicting the Turbulent Mixing Noise Emitted by a plane 2D wall jet to the case of a 3D Coanda jet. The effect of key flow characteristics are discussed, extensions to the model are suggested, and comparison with experimental results are presented.
         This research was conducted at James Madison University, under the supervision of  Dr. Caroline Lubert.
The Distribution of the Number of Shared Items Between Three Randomly Ordered Lists of N Numbers
          Erin Keegan, University of Mary Washington
Suppose there are three lists of randomly ordered N numbers, and that we compare the top n numbers in each of these three lists.  The goal was to find the probability distribution for the number of shared items in each of these three sub-lists, including the mean and variance.
        This research was conducted at University of Mary Washington, under the supervision of  Dr. Debra Hydorn.
Real Polynomials, Imaginary Roots, and Enchanting Ellipses
           Liza Lawson, Randolph-Macon College
Investigation of a certain class of polynomials reveals that, as a real root varies, the nonreal roots of the derivatives of these polynomials lie on fixed ellipses in the complex plane.  Further work shows that this same configuration can be maintained for a broader class of polynomials.
        This research was conducted at Randolph-Macon College, under the supervision of  Dr. Bruce Torrence.
Population Dynamics of Two Competing Species
            Benjamin Leard, James Madison University
Using difference equations and nullclines, we model the population dynamics of two generic species competing over the same resources in a closed environment.
        This research was conducted at James Madison University, under the supervision of  Dr. Anthony Tongen and Dr. Brian Walton .
The Determination of Relative Concentrations of Hydrogen Isotopes
          Laurence A. Lewis, James Madison University
JMU produces high purity hydrogen deuteride gas for nuclear experiments. To determine the relative purity of the gas, chromatography is used, producing overlapping asymmetric gaussian peaks. Numerical algorithms using functions modeled on these peaks produce chi-squared fits which allow for extraction of the concentrations. The method and results of such fits will be discussed.
        This research was conducted at James Madison University, under the supervision of Dr. C. S. Whisnant.
Non-Destructive Recovery of Voids within a Three Dimensional Domain using Thermal Imaging
          Victor Oyeyemi, Goshen College
We develop an algorithm which recovers spherical voids in a three dimensional object.  The algorithm produces the radii and locations of each void. Our method involves the application of known heat flux to the object's boundary. The steady temperature of the  boundary is then used to image the voids.
        This research was conducted at the Rose-Hulman Institute of Technology, under the supervision of  Dr. Kurt Bryan.
Length Sets of Numerical Monoids
            Joao Paixao, Virginia Tech
We study non-unique factorization of numerical monoids. First, we determine exact solutions for length sets and then we enumerate V-sets. Then we look at the problem if equality of the V-sets implies isomorphism between the monoids. Finally, we investigate if equality of lengths sets also implies isomorphism.
        This research was conducted at the Trinity Research Experience for Undergraduates, under the supervision of  Dr.Scott Chapman.
A game of Biblical Proportions
            Joan Pharr, Wake Forest University
The game involves two players: a devil and an angel, hence the name of the game.  The players take turns moving two tokens on a line with n vertices.  The talk will include the details of the game, proposed strategies and results of our research and a sketch of a proof.
        This research was conducted at Carnegie Mellon University, under the supervision of  Dr. Andrew Beveridge and Dr. Thomas Bohman.
Modeling the Oscillations of Acoustically Coupled Bubbles
           Joseph Roberts, Pennsylvania State University
The Rayleigh-Plesset equation is a well known ordinary differential equation that describes the acoustic oscillations of a single spherical bubble.  We have derived a model for a two bubble system using a similar approach.  Our numerical simulations have indicated complex dynamics.  Poincare plots are analyzed and Lyapunov exponents are calculated.
        This research was conducted at the W.G. Pritchard Labs, Pennsylvania State University, under the supervision of  Professor Andrew Belmonte.
An Algorithm for Producing Arrow Diagram Formulas for the Coefficients of the Conway Polynomial
           Alfred Rossi, Ohio State University
The coefficients of the Conway Polynomial are Vassiliev invariants of links. We give an algorithm for producing the arrow diagram formula for the n-th coefficient from the corresponding formula of the n-1 coefficient.
        This research was conducted at Ohio State University, under the supervision of Dr. Sergei Chmutov.
Computations of Quantum Entanglement
            Robert Schaeffer, Lebanon Valley College
The purpose of this research was to understand the mathematical notion of quantum bits (qubits) and their entanglement types. Hence I developed several code segments to test qubits for entanglement properties. These tests yielded counterexamples and helped in various proofs. This research aided in the exploration of 'Irreducible Quantum Entanglement'.
        This research was conducted at Lebanon Valley College, under the supervision of Dr. David Lyons.
Tweakable Block Ciphers under Exponential Attacks
           Hakan Seyalioglu, The College of William and Mary
A tweakable block cipher is a block cipher in which an additional input, the tweak, is used to construct an essentially different instance of the block cipher. Motivated by Patarinâ's recent results on exponential security, we explore the notion of exponential security for tweakable blockciphers.
        This research was conducted at The College of William and Mary, under the supervision of  Professor Moses Liskov.
Hamiltonicity of <2,4,t> Cayley Graphs
           James Sharpnack, The Ohio State University
Which Cayley graphs have Hamilton cycles remains unsolved.  By considering certain types of Euler paths in a reduced ribbon graph of the original graph, we know that Cayley graphs with <2,4,t> presentations have Hamilton cycles.
        This research was conducted at The Ohio State University, under the supervision of  Dr. Sergei Chmutov.
Classifying Hand-written Digits With Persistent Barcodes
          Mandy Smith, Centre College
We discuss a new approach to shape recognition using persistent homology, which incorporates both topological and geometric features of objects into "shape descriptors" that enable a computer to match new objects with others of similar shape. We focus on classifying scanned-in numbers from the MNIST database of hand-written digits.
        This research was conducted at Centre College, under the supervision of  Dr. Anne Collins.
Interpolation of linear subspaces by P-matrices
           Christian Sykes, University of North Carolina, Greensboro
A $P$-matrix is a matrix whose principal minors are positive. Let $X$, $Y$ be $n \times k$ real matrices of rank $k$. We consider the problem of what conditions are necessary and sufficient for the existence of  a $P$-matrix $A$ such that $AX = Y$.
        This research was conducted at the 2006 REU in Matrix Analysis at The College of William & Mary, under the supervision of  Dr. Charles R. Johnson.
Tuning in: a variation of radio labeling
             Desmond Torkornoo, University of Richmond
An L(3,2,1)-labeling of graph G is f:V(G)~W>N such that for x,y in V(G): d(x,y)=1 implies |f(x)-f(y)|>=3; d(x,y)=2 implies |f(x)-f(y)|>=2; and d(x,y)=3 implies |f(x)-f(y)|>=1. The L(3,2,1)-number of G is the smallest k such that G has L(3,2,1)-labeling f with k=max{f(V(G))}. We have determined the L(3,2,1)-number for simple graphs.
        This research was conducted at the Valparaiso University Summer REU 2006, under the supervision of  Dr. Zsuzsanna Szaniszlo.
Classification of Ice Crystals through Fractal Analysis
             Heather Umberger, Shenandoah University
Using methods of fractal analysis, a new classification system for ice crystals can be established.  The current analysis involves the use of two-dimensional, digital images and the use of FracTop software.  Consideration to practical use of the classification system will be given.
        This research was conducted at Shenandoah University, under the supervision of  Dr. Elaine Magee.
Thistlethwaite's Theorem for Virtual Links
            Jeremy Voltz, Ohio State University
 Thistlethwaite's theorem relates the Jones polynomial of a link to the Tutte polynomial of a corresponding planar graph.  We give a generalization of this theorem to virtual links by way of ribbon graphs and the Bollobás-Riordan polynomial. 
        This research was conducted at The Ohio State University, under the supervision of  Dr. Sergei Chmutov.
Minimizing Ropelength of Composite Knots and Links
             Rachel Whitaker, University of Georgia
Through developing computer programs and exploiting link symmetry, we created a library of composite links by connected summing the well-understood prime links.  By tightening links to their minimum ropelength configuration we hope to demonstrate the correlation of ropelength to the behavior of a subatomic particle, the glueball.
        This research was conducted at University of Georgia, under the supervision of Dr. Jason Cantarella .
The Hamiltonicity of Cayley Graphs
               Justin Wiser, Ohio State University
This talk will examine the famous conjecture that all Cayley graphs are Hamiltonian from an algebraic perspective.  We will use basic representation theory to present a new geometric method that might be used to eventually prove the conjecture.
        This research was conducted at Ohio State University, under the supervision of Dr. Sergei Chmutov .