Senior Capstone Projects
Dale Blem
Logarithmic Differentiation and Higher Order Newton's Method
The Logarithmic Derivative is the derivative of the natural log of a function D(ln(f(x)). This is more commonly expressed as the ratio f'(x)/f(x). It is used throughout mathematics in the fields of Differential Equations and Number Theory. We investigate properties of the Logarithmic Derivative, first strictly as an operator, and then as it is applied to Newton's Method. We illustrate a Modified Newton's Method that uses a higher order logarithmic derivative. We show both algebraically and geometrically that this Modified Newton's Method can converge more quickly than the classical Newton's Method.
Tara Fechter
Exploring the Derivative of a Natural Number Using the Logarithmic Derivative
We give an expository overview of the concept of the derivative of a natural number. Several examples of the concept are illustrated. We relate the concept to the logarithmic derivative, n'/n. We examine this concept in an unfamiliar setting, a function that, at first, appears to have no discernable pattern. We then determine the limit of the average values of this logarithmic derivative by bounding its values between two generating functions.
Matthew Rose
Wavelets and Filters
Wavelets, which are localized waves, are an exciting new alternative to Fourier series to analyze mathematical signals. Some emerging applications of wavelets include data compression and feature extraction in sound and image processing, including the new JPEG2000 standard. We will specifically explore the Haar Wavelet, which is a pulse of 1 and -1 over a finite range. The Haar Wavelet is just one of many wavelets to have the important property of orthogonality and we will show how this can be used to establish a basis from a collection of wavelets.
Heather Helmandollar (LuBean)
Length Minimizing Paths in the Hyperbolic Plane: Proof Via Paired Subcalibrations
Minimization proofs using paired calibrations have in the past been done with vector fields of divergence zero. In this paper we explore the possibility of using vector fields of nonzero divergence and their applications in paired calibration proofs in the hyperbolic plane. We will explore Steiner's Problem with three and four points spaced evenly around a circle.
Joint work with Keith Penrod during a summer REU at BYU
Tim Prins
Scheduling a Bridge Club: A Case Study in Discrete Optimization
We consider a scheduling problem posed in 1992 to two mathematicians at the University of Michigan by a local bridge club. The club wanted a schedule which would allow each player to play against every other player an equal number of times over their eight yearly meetings. To find such a schedule, we first determine its characteristics and define a function which gives us a numerical representation of how close a given schedule is to optimal. This operations research problem does not submit well to linear programming, so we try other algorithms. Using computer programs we wrote, we attempt to find an optimal schedule using several well known algorithms: exhaustive search, greedy, steepest descent, annealed search, and tabu search. Using the greedy algorithm, we find a schedule which is close to optimal while we find optimal schedules using steepest descent, annealed search, and tabu search. We also compare the run times of these algorithms.
Kalei Titcomb
Periodicity In Dynamical Systems
Dynamical systems arise in the study of many physical phenomena including the motion of heavenly bodies, variation in weather, and the rise and fall of populations. Often, these phenomena exhibit periodic behavior: planets and solar systems maintain fairly stable orbits; temperature and rainfall display annual patterns; predator populations cycle with prey populations. The mathematics of dynamical systems helps analyze this periodic behavior. In this talk, we investigate theory that guarantees, in such systems, the existence of periodicity and that allows us to numerically estimate periodic points. We apply this theory to the logistic family of functions, a family that arises in population dynamics. We visualize the dynamics through the use of web and bifurcation diagrams..
Kristine Callan
The Mathematics Behind Nonlinear and Chaotic Dynamics
Jennifer Gadd
Mathematics in Electrochemistry
Joe Green
Exploring Sound Waves with Fourier Analysis
Jeremy LeCrone
Matroids and Finite Groups
Julie Thomas
Exploring Lie Symmetries of the Heat Equation
Christopher Ventura
Conservation Laws of the Nonlinear Shrödinger Equations