Student Presentations and Abstracts


"The Riemann Zeta Function"
Eric Laber
UCLA – California Alpha

 I will give a brief introduction to the Riemann zeta function, stating some of its basic properties.  The main result in the presentation is the meromorphic continuation to the entire complex plane.  I conclude by stating the functional equation and introducing the Riemann Hypothesis.

"Some Variations on the Tennis Ball Problem"
Nicholas P. Biller

Occidental College – California Theta

 We will study variations of the Tennis Ball Problem that generates Catalan, Motzkin and Schröder numbers. We also investigate a recursive formula for counting Motzkin paths with flaws in terms of Motzkin numbers. A combinatorial proof concerning Motzkin paths with two and three flaws will be given.

"Modeling Cell Processes with Monte Carlo Simulation"
Rita Schneider

Fairfield University – Connecticut Gamma

 Calcium is vital in many cell functions. After briefly describing calcium channel gating within the cell, we will look at how this process is modeled.

"Interactions of KdV Solitons, Positons and Negatons"
Anupama Tippabhotla
 University of South Florida – Florida Epsilon

Among integrable equations is the celebrated Korteweg-deVries (KdV) equation which is both nonlinear and dispersive.  In this talk I will discuss the various interactions of multisolitons, positons and negatons using Maple. In particular, I will show the interactions of three and four solitons, positons and negatons.

"Invariant metrics on Lie groups with Non-negative Curvature"
Nathan Brown

Northwestern University – Illinois Beta

So far, all known examples of manifolds with positive curvature rely on the fact that bi-invariant metrics on compact Lie groups have non-negative curvature. At the Williams College  SMALL REU, our group has contributed to the search for new examples by finding other invariant metrics on Lie groups with non-negative curvature.

"Counting Huffman Trees"
Erin Polsley

Elmhurst College – Illinois Iota

 Huffman trees are binary trees used to construct Huffman codes, which are minimal length encodings for messages. All non-isomorphic Huffman trees with n terminal nodes are given for n = 1, 2, …, 8.  Different properties and patterns are discussed.

"Cartesian Products of Triangles as Unit Distance Graphs"
Ryan Alexander Sternberg

Worcester Polytechnic Institute– Massachusetts Alpha

The Cartesian product of n triangles is a unit distance graph of diameter n.  It is difficult to produce a drawing of such a graph in the plane such that adjacent vertices are unit distance apart. In these graphs, the number of vertices increases exponentially while the diameter increases linearly.

"Computer Implementations of Five Important Approximations to Pi"
Nathan D. Edington
Hood College – Maryland Delta

We briefly introduce the historically significant and often surprisingly beautiful approximations to pi of Wallis, Newton, Gregory, Machin and Ramanujan. We then outline how these approximations were implemented in MATLAB and MathCAD in order to explore and compare the accuracy and rate of convergence of each approximation.

"A Mathematical Model of Tri-Trophic Interactions"
Michael Cortez
Hope College – Michigan Delta

While more difficult, the analysis of tri-trophic systems yields more insight than more commonly studied predator/prey models. Using non-linear differential equations, we modeled the interactions between a grass infected by a fungal endophyte, an herbivore, and a parasitoid. Analysis was conducted both experimentally and theoretically.

"Mathematical Biology Curriculum Development"
Henry Gould

Hope College – Michigan Delta

Mathematical Biology is an ever-expanding field that benefits greatly from its interdisciplinary nature.  At Hope College we have created a mathematical biology course co-taught to a mixed audience of biology and mathematics students. The course is based on biology research papers and includes wet labs.  We will discuss the format of the class, details of the research papers and labs, student reactions, and outcomes from the course.

"Counting Symmetric Matrices of Rank One and Two"
Andrew Wells

Hope College – Michigan Delta

Rank is one of the most important properties of a matrix. This talk focuses on counting the number of rank one and two matrices in certain vector subspaces of the space of all n n symmetric matrices.  This question is connected to the study of quadratic forms. The final results classify all possibilities for the space spanned by four 4 4 symmetric matrices.

"The Geometry of  H (Rn) : Part I"
Kristina Lund

Grand Valley State University – Michigan Iota

 The collection of all non-empty compact subsets of Rn forms a complete metric space, H (Rn, h) where h is the Hausdorff metric.  This space is an important one for several reasons. For example, this is the natural space in which to study fractals.  Applications of this metric can be found in image matching, in visual recognition by robots and in computer-aided surgery.  In this presentation  I will provide essential background information on H (Rn, h), and basic results from our efforts to understand the geometry of this space.

"Functions Concerning Distances Between Primes"
Adam Gray
University of Mississippi – Mississippi Alpha

 Many number theoretic ideas can be formulated in terms of the following functions:

 f(n) = min{a | n + a is prime}

 g(n) = min{a | n + a and na are prime} if a exists, otherwise g(n) = ¥.

 I will discuss the formulations of Bertrand’s Postulate and the Goldbach and Twin Prime Conjectures in terms of these functions.

"Sines, Cosines and Conjugates"
Caleb Hallauer

University of Mississippi – Mississippi Alpha

 A nonconstant integer polynomial f(x) is said to be irreducible if in every factorization of f(x) into a product of integer polynomials, one factor is constant.  Numbers a and b are said to be conjugates if they are roots of the same irreducible polynomial.  I seek angles whose sine and cosine are conjugates. For example sin(p/8) and cos(p/8) are roots of 8x4 – 8x2 +1.  Some interesting classes of such angles are located and studied.

"Idempotent Matrices over Commutative Principal Ideal Rings"
W. Andrew Pruett

Millsaps College – Mississippi  Delta

 I show that all idempotent matrices over a nowhere reducible commutative principal ideal ring R are diagonalizable with diagonal entries idempotent in R.

"Euclidean Problems in Spherical and Hyperbolic Geometry"
Jodi Simons

University of New Hampshire – New Hampshire Alpha

We consider several geometric problems in the Euclidean plane, including some ancient Japanese temple problems, and explore their analogs in spherical and hyperbolic geometry.  We examine their Euclidean proofs to discover how to generalize them to these other geometries. We also explain the basics of spherical and hyperbolic geometry.

"Rubik’s Cube"
Serina Alfano and Adam Kolakowski

St. Peter’s College – New Jersey Epsilon

 Our presentation is an introduction to group theory via Rubik’s Cube.  Our ultimate goal is to solve the deceptive cube with the help of algorithms and some group theory. Solving the Rubik’s Cube puzzle is separated into five steps, each a building block of functions and important mathematical content.

"A Mission to Mars with the Help of Kepler’s Laws"
Lisa Reeder

New  Mexico State University – New  Mexico Alpha

 This is an explanation of how Kepler’s laws can be used to calculate the time it would take a mission to venture from Earth to Mars and back. This has special applications for manned missions because of the return and length of time spent on Mars.

"Solving Linear Recurrence Relations Using Generating Function and Matrix Approaches"
William Neris

State University of New  York at Fredonia – New  York Pi

In this research, a single linear recurrence relation was solved using both generating function and matrix approaches. The two methods were then extended to solve a system of linear recurrence relations.  Comparison of these approaches will be discussed.

"The Geometry of  H (Rn) : Part II"
Patrick Sigmon

Wake Forest University – North Carolina Lambda

 The collection of all non-empty compact subsets of Rn forms a complete metric space, H (Rn, h) where h is the Hausdorff metric.  This space is an important one for several reasons. For example, this is the natural space in which to study fractals.  Applications of this metric can be found in image matching, in visual recognition by robots and in computer-aided surgery.

 I will share further results from our study of the geometry on  H (Rn) induced by the Hausdorff metric.

"What’s That Remainder?"
Holly Attenborough

Miami University – Ohio Delta

 Lucas’s Theorem (1887) finds the remainder of Pascal’s Triangle entries when divided by prime numbers.  Thus, the theorem gives a computational way to find the remainders of binomial coefficients upon division by a prime.  I will illustrate the theorem with examples and if time permits, briefly discuss a proof.

"Arc Length and Surface Area – Are we on the Same Page?"
Mark Walters
Miami University – Ohio Delta

 In calculus, formulas are derived for the length of a curve and the area of a surface.  Textbooks often take two different approaches. One approach connects dots to get polygonal approximations, while the other uses tangential considerations.  This paper shows that either approach leads to the expected formulas for both curves and surfaces.

"Diophantus Meets Trigonometry"
John Filkorn

John Carroll University – Ohio Lambda

 When is the value of a certain trigonometric expression equal to the reciprocal of a square? This problem led me on quite an excursion into number theory. Let’s look at some of the highlights of this trip.

"Transforming the MAA into a Soccer Ball"
Julie Iammarino

John Carroll University – Ohio Lambda

After showing how to make a soccer ball out of the MAA, generalizations of geometric properties will be explored.

 "Intrinsic Linking of K6"
Colleen Hughes
Denison University – Ohio Iota

 Any embedding of K6, the complete graph on six vertices, will have at least one pair of linked triangles, not necessarily constructed of straight lines.  In this talk we explore the possibility of constructing straight-line embeddings of K6 with 1, 3, 5, and 7 pairs of triangles respectively.

"Coding Messages"
Carly E. Grey

John Carroll University – Ohio Lambda

We will discuss Public Key Encryption with a number of examples using small primes.

"Catch the Wave"
Stephanie S. Barille
Mount Union  College– Ohio Omicron

What do audio clips, seismographs, electrocardiograms, FBI finger print cards, and El Niño all have in common?  Come catch the “wave” and find out!

"Comparing the Eigenvalues of Products of Matrices"
Nicole Cunningham

Youngstown State University – Ohio Xi

Suppose that A and B are two matrices. Even when both products AB and BA are defined, it is seldom the case that these products are equal. In fact, if A is an n m matrix and B is an m n matrix, the products AB and BA are not even of the same type.  In this talk we consider the eigenvalues of these products and see that the products are not as dissimilar as they first appear.

 "Watch the Birdie!"
Steve Dinda

Youngstown State University – Ohio Xi

The purpose of this work is to explore two diversity indices, the Shannon-Wiener index and Simpson’s index.  These indices are specific sums of the proportion of each biological species observed and are commonly used by biologists to determine species diversity in ecological studies. Various properties of these sums are examined in detail. Comparing diversity indices requires a specialized t-test. Other more commonly used statistics are discussed and compared.

"Fun with Incircles"
Jeremy Hamilton

Youngstown State University – Ohio Xi

An interesting property regarding an incircle and three related circles will be examined.  This problem (11046) was proposed by Christoph Soland in The American Mathematical Monthly, November 2003.

"Viewing the World through the “i’s” of Complex Numbers"
Melissa Marshall

Youngstown State University – Ohio Xi

Cartographers use many different techniques to construct maps of the world.  I will explain the stereographic projection and use it to view points on the globe as points on a two dimensional map.  I will also use the stereographic projection to illustrate some surprising results from complex analysis on the Riemann sphere.

"Bivariate Normal Estimation of Digitally Imaged Data"
Theodore T. Stadnik, Jr.

Youngstown State University – Ohio Xi

Bivariate normal distributions are used to estimate the form of three-dimensional data collected from a digitally captured photograph.  Software is written to collect data and extract information to calculate parameters for a bivariate normal distribution with dependent variables. A regression curve is used to compute the major and minor axes of an ellipse. The software is then run to create a visual and statistical analysis of biological protein gels captured with digital imaging equipment.

"Drawing Graphs from Degree Sequences: A Computer Based Approach to Recursive Algorithms"
Brian Black

Providence College – Rhode Island Gamma

We describe a computer based approach to the Havel-Hakimi recursive algorithm based on Euler’s handshaking lemma for determining whether a sequence of non-negative integers is graphical. The program determines if a sequence is graphical and draws any resulting graphs. This presentation includes discussion of the problems arising from the computerization of the innate logic humans use to draw graphs.

 "Knot Your Usual Talk About Celtic Art"
Angela Brown
Sam Houston State University – Texas Epsilon

 Two knots are equivalent if their projections can be transformed into one another through a sequence of Reidemeister moves or planar isotopies.  The classification of knots is an open question. This talk will apply known and well-developed methods to the classification of some examples of Celtic knots.

"The Impact of Additional Data Values on Standard Statistical Estimators"
Ashley Moses

Sam Houston State University – Texas Epsilon

In this talk we will discuss the impact on statistical estimators resulting from the availability of additional sample data. We will include the impact of single and multiple new values on the sample mean, variance, standard deviation and correlation coefficient, illustrating the results with a specific example.

"Oscillating Patterns in Langton’s Ant"
Dakota Blair

Texas A&M University – Texas Eta

 It is known through the Cohen-Kung Theorem that using the Langton’s Ant algorithm, a single ant cannot oscillate.  However certain patterns with multiple ants can oscillate. We present a way to create oscillating patterns with exactly two ants, examples of oscillators, and patterns resembling gliders in Conway’s game of Life.

"New Results in Wavelet Set Theory"
Ryan Westbrook

Texas A&M University – Texas Eta

 We will present surprising new discoveries in wavelet theory that show there’s more under the surface.

"Intuition vs. Formalism in Mathematics"
Paul Dawkins
Angelo State University – Texas Zeta

In a 1986 paper, Chris Freiling used an intuitive argument to prove the Axiom of Choice and the Continuum Hypothesis false. This is impossible formally since they are independent statements.  I discuss the conflict between formal and intuitive mathematics in this context and reconcile the conflict Freiling’s paper sets up.

"Partitioning and Power Series"
Melanie Antos
St. Norbert College – Wisconsin Delta

 Problem 6A of the 2003 Putnam Exam called for finding a partition of the non-negative integers having certain properties.  We will present the solution to this problem and then use power series to prove that the partition is unique.

"The Mathematics of Polarized Helium"
Brian Hahn

St. Norbert  College – Wisconsin Delta

 The neutron can be a very complicated item to study because it has no charge and is unstable. By using Polarized Helium 3 we are able to study the neutron by appealing to some quantum mechanics and a little bit of math.

"The Accuracy of Three-dimensional Bone Models Constructed from Computed Tomography Scans"
Jill Schmidt

St. Norbert  College – Wisconsin Delta

 Three-dimensional models are critical when performing finite element analysis to assess stress and strain distribution in bone, particularly around an implanted prosthesis. The purpose of this study was to quantify the error of these models. Three-dimensional models of the carpal (wrist) bones created from computed tomography (CT) scans were compared to those made from laser scan data of the prepared cadaveric bones. Point cloud data were then extracted and the error was quantified. In addition, both inter-user and inter-software variability was tested. Research on this project was done with Maarten Beek and Heidi Ploeg.

"Mathematics in Adaptive Education"
Sarah Van Asten

St. Norbert College – Wisconsin Delta

 Adaptive education refers to teaching students with disabilities.  We will discuss various methods for teaching elementary mathematics to students with certain disabilities.

"Mathematical Espionage:  Breaking the “Unbreakable” Enigma Code"
Alyssa Wood

St. Norbert College – Wisconsin Delta

We will discuss the mathematical methods by which the Allies broke the Enigma Code during World War II. We will also highlight some of the influential men and women who worked for the Allied forces to develop methods of decrypting. A short history of the cryptanalytic bombe will also be discussed.