Michael Higley and Dr. Peter Bates, Mathematics Laplace’s equation is a differential equations that describes many physical properties in steady state systems. It can be used to describe heat distributions, displacements in elastic media, or electrostatic fields. My interest is in solving an electrostatics problem where the domain—the area in which the equation has physical […]

## Ratios of Positive Definite Matrices

Dena Plant and Dr. Wayne Barrett, Mathematics As our research began we focused on proving the hypothesis that for a 4 x 4 Positive Definite matrix A, the ratio where |123| is the determinant of the submatrix of A defined by the rows and columns 1, 2 and 3 (others are similar). Although we were […]

## Length Minimizing Network for n points on the k-torus

W. Lauritz Petersen and Dr. Denise Halverson, Mathematics Our project consisted of several components with the end goal being: given n points on the ktorus (a donut with k number of holes), what would the shortest path network be to connect all the n points (network meaning a network of paths connecting each of the […]

## Optimal Energy Recovery in Finite Time for Linearly Dissipative Systems

Mark Meilstrup and Dr. Scott Glasgow, Mathematics Many physical processes involve the loss or dissipation of energy. For example, when pushing a box across the floor, some of the energy that is spent will be lost to frictional forces. Our research is concerned with the study of the energy which is lost, and also the […]

## Orca Research Report

Jeanna Chilton and Dr. Gary Lawlor, Mathematics The work I was able to do for the research on the minimal surface area of the octahedron was time consuming, but unfortunately did not bring about any profound knowledge for anyone besides myself. Through correspondence with fax machines, my professor sent me many different problems for me […]

## A Comparison of Computational Efficiency: Cholesky Decomposition and Spectral Methods for the Generation of Fractional Brownian Motion

Micah S. Allred and Dr. David Clark, Mathematics Since its popularization by Mandelbrot and Von Ness in the early 60’s, Fractional Brownian Motion (fBm), has found a great many applications in such fields as Option Pricing, Signal Processing, Internet Traffic, Hydrology, and Geology. This process is an extension of Brownian Motion which allows for a […]

## Computer Aided Complex Analysis

Jared Whitehead and Adam Rich with Dr. Michael Dorff, Mathematics Department Complex analysis is a fundamental course in mathematics required by some engineering and physical science disciplines. Since complex numbers are two dimensional – they have a real and imaginary part – visualizing their graphs, a very important part of practical and theoretical analysis, can […]

## Traveling Waves

Keith Rudd and Dr. Jeffery Humpherys, Mathematics Traveling waves are, mathematically speaking, partial differential equations which can be expresses as U(x-ct), where x is the spatial variable, t is time, and c is the speed of the wave. Traveling waves have applications in fluid dynamics, optics, and many other areas. In our research we continued […]

## Foundations of Tropical Geometry Classification of Polynomials

Nathan Manwaring and Dr. Tyler Jarvis, Mathematics Tropical Mathematics is a relatively new field of mathematics that is getting a lot of attention recently. It is applicable in the field of Combinatorics as well as Phylogenetics, and more uses are being found each day. Our goal was to find a way to classify Tropical Polynomials […]

## Finding a Galois Representation Corresponding to a Hecke Eigenclass

Meghan De Witt and Dr. Darrin Doud, Mathematics Department My project was to finish a thirteen year old unsolved problem that provided an important computational example of a conjecture in the field of Algebraic Number Theory. Originally, we worked with the techniques of elliptic curves and class field theory to try and isolate the objects […]