"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 *n*
– *a*
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 8*x*4
– 8*x*2
+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.