Senior Capstone Projects
Marisa Allen
Global Warming: the Math Behind Climate Modeling
Global warming is a controversial topic, in part because the complex mathematical concepts that are used to explain this phenomenon are rarely made accessible to the general public. In this project, we explore the mathematics and scientific principles behind climate models, in an attempt to demystify global warming. In particular, we will look at the equations used in complex climate models, as well as a simple radiative forcing equation used in general climate models (GMC’s). We then investigate the effect of rising CO2 levels on the governing equations of climate change.
Kristen Almgren
Partitions: How many different ways can you get to the center of a
tootsie roll pop?
In this presentation we introduce the idea of partitions as well as generating functions. With these ideas in mind we examine a variety of examples in order to give a summary of the techniques that can be applied to partitions and generating functions. With these techniques in mind, we present a problem relating two sets of partitions. After an examination of the known proof of the problem, we extend the problem to a more general setting.
Jacob Artz
Counting Squares and Dominoes: What's math got to do with it?
While arranging dominoes and squares in different patterns is not necessarily esoteric in nature, its application in mathematics is. We examine a specific way in which squares and dominoes are used, namely through tiling.
We begin with a brief examination of the Fibonacci sequence and its combinatorial representation. We show that the number of ways to arrange dominoes and squares on an n-length board is equal to the nth Fibonacci number. From this basic relationship we determine a number of identities involving Fibonacci numbers and strategies to find these identities. Some methods we use are breaking an n-length board into different portions, considering the position of a specific tile, and finding correspondences between two sets of tilings.
We also explore Zeckendorf's Theorem and its application to combinatorics and specifically tiling. This allows us to produce an explicit definition for some Zeckendorf family identities.
Brittany Cuff
List Coloring and Rook Polynomials: Using chess to determine how many ways to color a graph
How many ways can you color a map? How many ways can you place r rooks on an mXn chessboard?
In graph theory, we color vertices of a graph in such a way that if two vertices are adjacent, they are colored differently. List coloring is a restriction of this coloring where not every color is available to each vertex. We now assign each vertex a list of allowed colors in which it may be colored.
A rook polynomial is a generating function that represents the number of ways we can place r non-attacking rooks on an mXn chessboard, where non-attacking is a placement such that no rook shares a column or row with any other rook.
We connect list coloring to rook polynomials by transforming the relationship between a graph's vertices and associated list assignment into a rook board. We have proved that a complete graph G that has a valid list assignment will result in a rook polynomial whose leading coefficient gives the number of proper colorings of G. We take this result further by determining the number of proper colorings of G if G is not complete via inclusion.
Zach Gantenbein
A Mathematical Sieve of the Prime Gaussian Integers
Kerensa Gimre
Answering a burning question: Analyzing methods to estimate remaining oil reserves and peak oil production
Peak oil (the time when half of all oil exploitable oil is expended) was introduced in the 1950s by M. King Hubbert who wished to predict the time of maximum oil production for both the United States and the world. Correctly following Hubbert's prediction, US oil production peaked in the 1970s. There currently exist a variety of estimates for the timing of world peak oil production. Due to the vast economic implications of running out of fuel, peak oil is a critical problem.
We investigate methods to estimate remaining amounts of untapped oil supply, specifically the Level Set Method. Developed in the 1980s, the Level Set Method has numerous applications in fluid mechanics, materials science, computer vision, computational geometry, computer-aided design, and image processing. By numerically solving the Hamilton-Jacobi equation and applying an appropriate velocity function dependent on the curvature (curvature in this instance depends on image intensity), an image can be analyzed to eliminate noise. This process assists in "cleaning up" a seismic image of the earth's subsurface. If these images are cleaner, more accurate approximations for subterranean oil can be found. This information is vital to oil companies when deciding if it cost-effective to drill a field, and is also important when predicting total remaining subterranean oil.
We conclude by estimating the timing of peak oil using the results of the Level Set Method and analyzing the popular Hubbert's Method.
Bobby Larkins
Pondering predictions of the path and pain of powerful pivoting puffs to possibly protect the population (hurricanes)
Hurricanes have been killing people and destroying places for far too long. I, how-
ever willing, can not change this fact. Improving the system in which the people are
warned was the next best thing, limiting damage and saving lives. Currently, the pub-
lic is alerted with the Saffir-Simpson Scale. I will be comparing that to the new idea
of integrated kinetic energy or IKE. We will look at some or the raw data collected,
see how that generates the IKE values, and make a conclusion as to the method that
will be better for warning the public.
Stephanie Murayama
Is it a really a small world after all? A study of small world networks
Six degrees of separation is a theory that everyone is separated by at most six people. You might not know everyone, but you probably know someone who knows someone who knows that person. What is a small world network? A small world network is this type of social network. One way to display this situation is by modeling the network with a graph.
Graphs of social networks have taken many forms over the years. Sometimes, these graphs are completely random, with the thought that anyone in the world could possibly know any other person in the world. In this project, we consider more structured networks. Graphs such as complete graphs and the 1-lattice graph take into account that it is more likely that you will know your neighbors than know a random person. For example, a complete graph could model a very close group of friends, where everyone in the group knows everyone else in the group, while the 1-lattice could model a neighborhood where a person knows his neighbor and his neighbor’s neighbor, but he doesn’t know anyone else in the neighborhood. Of course, your neighbor might know someone across town, and that person might know someone around the world, making it a small world. Notice there is still a bit of randomness amid the structure. By looking at graphs such as these, we are able to investigate the more traditional, completely random models, and the more modern, structured models.
In this project, we will focus on the more structured, 1-lattice model that Watts and Strogatz created in 1998. We investigate this graph, along with the complete graph and some others models in detail. Which graph models which real-world situation best? We investigate different properties such as clustering and path length. We also attempt to extend generating functions to help us calculate these properties.
Meagan Potter
Breakdown! Discovering the Relationships Between Designs and Matroids
Introduced in the early 1930s, design theory and matroid theory are two distinct areas of discrete mathematics. Are they related? How? Some matroids are designs; some designs are matroids. A matroidal design occurs when the blocks of a design are the hyperplanes of a matroid; a base design occurs when the blocks of a design are the bases of a matroid. Both designs and matroids have connections to geometry, and we can use geometries to link the two structures. Affine planes give rise to (n^2+n, n^2, n+1, n, 1)-designs, projective planes give rise to (n^2+n+1, n+1, 1)-designs, and all projective geometries are matroids.
The most well-known example that connects designs and matroids is the Fano plane, which is also the Fano matroid and the (7, 3, 1)-design. We use this to investigate the relationship between specific components of the three structures. Then, we look at several different classes of matroids -- uniform, cycle, transversal, etc -- and develop a method of translating among matroids and designs. We find designs given a specific matroid and find matroids given a specific design. Can we generalize this to an entire class of matroids? an entire class of designs? We consider the correlations from these specific examples and seek to make general conclusions about the nature of one structure from the other.
Marissa Utterberg
Thinking Inside the Box: Consequential Partition Variations
Maria Walters
Patching the Holes in Quasiderivations
"Number derivatives'' or "quasiderivations'' $\Delta(x)$ were first mentioned over 40 years ago in the Putnam Competition as a map from the integers to the integers that would satisfy the product rule. This definition was later expanded to all nonzero rational numbers. In 2007, this allowed Emmons to define the quasiderivation of a function f(x) for any "number quasiderivation'' $\Delta$. However, this definition was found to have a large number of ``holes'' whenever $\Delta(x)$ = 0. This motivated us to incorporate the limit of an infinitely-dimensional vector in order to patch these holes and define a continuous quasiderivation $f^{\Delta}(x)$. We then touch on a few examples where this quasiderivation $f^{\Delta}(x)$ resembles our standard derivative f'(x), and many more cases where they differ widely.
Karsten Gimre
A Numerical Analysis of Nonlinear Partial Differential Equations Related to Electrochemistry
In this project, we set up a system of nonlinear partial differential equations to model an electrochemical reaction. Our results answer a long-standing conjecture as to whether current is influenced by a certain reaction parameter. After manipulating the domain of the functions involved, we could solve them numerically with MATLAB. With Laplace Transforms, we were able to convert the system of equations into a single equation. We determined that an analytical solution to the equation seems unknown, and likely not possible. Using existence and uniqueness theorems, together with the Lax Equivalence Theorem, we showed that the numerical solution is indeed the correct solution.
Alexis Sakaida-Diaz
An Examination of the Effect of Standards Based Curriculum on AP-Calculus Test Scores
The National Council of Teachers of Mathematics has published standards for teaching mathematics to children Pre-K through 12th grade. While the standards appear to be reasonable, there is still controversy as to whether these standards lead to appropriate levels of grade-school learning in mathematics. In this investigation, we examine one measure of the success of the standards by determining if a standards based high school curriculum leads to higher scores on the AP-Calculus exam.
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