Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Albert Schueller

• TITLE: Queuing Theory
AUTHOR: Ryan Berry
ABSTRACT: This paper defines the building blocks of and derives basic queuing systems. It begins with a review of some probability theory and then defines processes used to analyze queuing systems, in particular the birth-death process. A few simple queues are analyzed in terms of steady-state derivation before the paper discusses some attempted field research on the topic.

• TITLE: Applications of Number Theory to Fermat's Last Theorem
AUTHOR: Cameron Byerley
ABSTRACT: This paper is in the form of the fifth and sixth chapters of lecture notes designed for an introductory number theory class. It uses a number of basic number theory concepts to prove three cases of Fermat's Last Theorem. Fermat's Last Theorem states there are no integral solutions to the equation $x^n+y^n=z^n$ for $n>2$. We begin with a proof of $n=4$ and use similar but more computationally and theoretically complicated ideas to prove the cases $n=3$ and $n=14$. In addition to providing mathematical details for each proof, the paper places the proofs in a historical context. Although providing a complete proof of Fermat's Last Theorem is far beyond the scope of this paper, examining three cases gives an understanding of the difficulties in generalizing the theorem and the contributions of many well-known mathematicians.

• TITLE: Maximal Subspaces of Zeros of Quadratic Forms Over Finite Fields
AUTHOR: Mark Hubenthal
ABSTRACT: This paper introduces the reader to quadratic forms defined over finite fields in the general sense. In short, a quadratic form $f \in \mathbb{F}_{q}[x_{1},\ldots,x_{n}]$ with $n$ indeterminates is a homogeneous polynomial of degree-$2$ with coefficients taken from the field $\mathbb{F}_{q}$. We will classify all quadratic forms as being one of three fundamental types and compute the number of solutions to an arbitrary quadratic form equation $f(x_{1},\ldots,x_{n}) = b$, where $f$ is one of the three types. Finally, we consider that the zeros of a quadratic form $f$ (i.e. solutions to the equation $f(x_{1},\ldots,x_{n})= 0$), can form subspaces of $\mathbb{F}_{q}^{n}$. The maximum size of such a subspace can be shown only to depend on $q$ (the field characteristic), the number of indeterminates $n$, and the particular type of $f$.

• TITLE: Parallel Programming
AUTHOR: Natalie Loebner
ABSTRACT: After years of technological advances the speed of single processors are beginning to meet their physical limitations. Thus, parallel programming has become an increasingly important tool in scientific research and commercial industries. This paper includes an exploration of how the size and number of calculations in various problems benefit from parallelization in a distributed-memory MIMD environment.

• TITLE: Chaos and Dynamics
AUTHOR: Kelsey Mace
ABSTRACT: In this paper we will study chaos through the dynamics of the quadratic family of functions. We begin with an introduction to basic dynamical notions, including orbit analysis and periodicity. Our goal is to isolate chaos within the specific example of the quadratic functions. From here, we will form a proper definition of chaos using symbolic dynamics. This material is largely a review of A First Course in Chaotic Dynamical Systems by Robert L. Devaney.
ADVISORS: Doug Hundley and Barry Balof

• TITLE: Mathematics of Evolution
AUTHOR: David Mummy
ABSTRACT: The Adaptive Landscape (AL) provides biologists with a heuristic for modeling the way the forces of evolution affect a population of organisms. We investigate the possibility of creating a computer simulation of a population moving around an AL. First, we create a simple landscape using a Gaussian distribution, and then a sample population in phenotype space. Then we explore methods of simulating the population's motion through phenotype space over time.
ADVISORS: Doug Hundley and Barry Balof

• TITLE: The Global Positioning System And Its Usage in Making Path Length Estimates
ABSTRACT: The Global Positioning System has revolutionized the way positions are found and locations are measured. This technology has impacted many areas from surveying to construction to recreational use. This paper will discuss the Global Positioning System; why it was started, the components of the system, how it is utilized, and how the system is expected to change over the years. In addition, this paper will discuss the optimal points to take measurements to maximize efficiency for several idealized shapes. This will be done by determining how to maintain the path length of the shape with a minimum number of data points.

• TITLE: Simson Lines (voted outstanding senior project)
AUTHOR: Mary Riegel
ABSTRACT: This paper is a presentation and discussion of several proofs of Simson's Theorem. Simson's Theorem is a statement about a specific type of line as related to a given triangle. The theorem has interesting implications for lines in general position, but our concern here is to examine several different methods for proving the theorem. We present one analytic proof of the theorem and the converse, three synthetic proofs of the theorem, and one synthetic proof of the converse.

• TITLE: Fractal Curves
AUTHOR: Chelle Ritzenthaler
ABSTRACT: Fractal curves are employed in many different disciplines to describe anything from the growth of a tree to measuring the length of a coastline. We define a fractal curve, and as a consequence a rectifiable curve. We explore two well known fractals: the Koch Snowflake and the space-filling Peano Curve. Additionally we describe a modified version of the Snowflake that is not a fractal itself.

• TITLE: From Fourier Transform to Wavelet Analysis: Mathematical Concepts and Examples
AUTHOR: Ly Tran
ABSTRACT: This paper studies two data analytic methods: Fourier transforms and wavelets. Fourier transforms approximate a function by decomposing it into sums of sinusoidal functions, while wavelet analysis makes use of mother wavelets. Both methods are capable of detecting dominant frequencies in the signals; however, wavelets are more efficient in dealing with time-frequency analysis. Due to the limited scope of this paper, only Fast Fourier Transform (FFT) and three families of wavelets are examined: Haar wavelet, Daub$J$, and Coif$I$ wavelets. Examples for both methods work on one dimensional data sets such as sound signals. Some simple wavelet applications include compression of audio signals, denoising problems, and multiresolution analysis. These applications are given for comparison purposes between Fourier and wavelet analysis, as well as among wavelet families.

Although wavelets are a recent discovery, their efficacy has been acknowledged in a host of fields, both theoretical and practical. Their applications can be expanded to two or higher dimensional data sets. Although they are omitted in this paper, more information is available in Primer or many other books on wavelet applications.
ADVISORS: Doug Hundley and Barry Balof

• TITLE: Triangles in Hyperbolic Geometry
AUTHOR: Laura Valaas
ABSTRACT: This paper derives the Law of Cosines, Law of Sines, and the Pythagorean Theorem for triangles in Hyperbolic Geometry. The Poincar'e model for Hyperbolic Geometry is used. In order to accomplish this the paper reviews Inversion in Hyperbolic Geometry, Radical Axes and Powers of circles and expressions for hyperbolic cosine, hyperbolic sine, and hyperbolic tangent. A brief history of the development of Non-Euclidean Geometry is also given in order to understand the importance of Euclid's Parallel Postulate and how changing it results in different geometries.

• TITLE: The Lanczos Derivative
AUTHOR: Lizzy Washburn
ABSTRACT: The Lanczos derivative is attributed to Hungarian mathematician Cornelius Lanczos who developed it in the 1950s. It is an integral based derivative derived from the least square model. The Lanczos derivative is set apart from other forms of differentiation such as the symmetric and traditional derivatives because it exists for functions where the other derivatives do not.