2004 MathFest - Providence, Rhode Island

The Council on Undergraduate Research Award ($150) for the best research presentation at the Pi Mu Epsilon sessions went to Nicole Cunningham.

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 x m matrix and B is an m x 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.

The SIAM Award ($150) for the best presentation on environmental issues at the Pi Mu Epsilon sessions went to Michael Cortex.

A Mathematical Model of Tri-Trophic Interactions

**Michael Cortex**

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.

** The Pi Mu Epsilon Student Speaker Awards funded by the American Mathematical
Society (in alphabetical order):**

Catch the Wave

**Stephanie Barille**

Mount Union College (Ohio Omicron)

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

Computer Implementations of Five Important Approximations to Pi

**Nathan 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.

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.

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.

Bivariate Normal Estimation of Digitally Imaged Data

**Theodore Stadnik**

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.

Cartesian Products of Triangles as Unit Distance Graphs

**Ryan 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.

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.