# Senior Capstone Projects

Briana Flores

Tsunami Forecasting: Linear vs. Non-linear Models

Reflecting on tsunami events such as the Tohoku tsunami of March 2011 or the Sumatra tsunami of December 2004 reminds everyone of how destructive and devastating a tsunami can be. Mathematical models provide different types of information regarding tsunamis such as the locations that it will affect, estimated time of arrival to shore, height, and power. By relying on the results of their mathematical models, Tsunami Watch centers are able to determine what types of precautions residents should take. Models can either be linear, which are numerically more stable and faster to compute, or non-linear, which are generally more accurate. Time is of the essence for tsunami forecasting, so a faster model could save lives. So, when do linear models produce good approximations of non-linear models? To study this concept, I use software from the Pacific Tsunami Warning Center and a model based off the Shallow Water Wave Equations to compare simulations.

Hanna Landrus

Hitches Holding

A hitch is a tangle about a pole. When climbing a hitch could save a climber's life. When sailing a hitch could prevent a boat from floating away. We will present a model for determining when a hitch will hold and relate the determinant of a certain matrix to the friction required for a hitch to hold. We will consider sequences of hitches and investigate the minimum friction required for the hitches to hold. This involves examining a sequence of polynomials whose coefficients are related to Pascal's Triangle.

Mathematical Illiteracy causes Financial Collapse: Incorrect Implementation of the Black-Scholes Model

The Black-Scholes formula is a mathematical model used to find the value of a European call option in a manner that nearly eliminates risk. Upon publishing the formula, the buying and selling of options increased significantly. The Black-Scholes formula assumes ideal market conditions and derives a partial differential equation, commonly referred to as the Black-Scholes construction; the solution to the PDE is the Black-Scholes model. We investigate what is needed in order for the model to produce accurate results and find that the parameters must follow a normal distribution. It is thought that invalid assumptions of ideal conditions in the market directly contributed to the financial market collapse in 2008. We attempt to determine which parameters do not follow normal distributions and analyze the weaknesses of the Black-Scholes model. In the market situation of 2008, the stock price followed a fat-tailed distribution. Often compared to the normal distribution, fat-tails exhibit a large kurtosis which means observations far from the mean are more likely to occur as compared to normally distributed occurrences. This lead to an incorrect implementation of the Black-Scholes model and ultimately contributed to the financial market collapse in 2008.

Jack McCorkel

Gambling Partitions: Calculating Electorate and Subelectorate Voter Turnout

Social scientists have long struggled to explain the act of voting. Regardless of the statistical methods used, any model of voting becomes incredibly complex, requiring submodels describing preference, contributing socioeconomic factors, and the importance of a given vote. I begin by looking at the issues of voter turnout through the lens of economics applied to democracy. Describing the choice to vote as a gamble, I look toward econometrics in order to better understand turnout. In an attempt to reconcile data with theory, I also explore the mathematics of incorporating demographic interests into turnout. Finding the current mathematical models lacking in practicality, robustness, and depth, I offer a new formulation and investigate this model's ability to predict voter turnout and preference.

Block designs, abstract structures with applications in experiment control and data gathering, have in many cases been shown to be isomorphic to matroids, another broad family of abstract structures. We explore and discuss some of the ways a specific class of designs can also describe a matroid, and some of the ways a specific type of matroid may also constitute a design. We also consider cases where a matroid exists but not a design, and where a design exists but not a matroid. Because certain algorithms are known to produce or manipulate matroids, these may be useful in constructing certain types of designs. Meanwhile, careful study of designs may lead to useful theorems that apply to matroids.

Tony Fernandez

How Does your Triangle Like To Tile?

We investigate which triangles can tile the Euclidean or hyperbolic plane without overlapping and determine exactly which triangles tile without overlapping. Euclidean triangles and hyperbolic triangles have different and this leads to contrasting results about tilings. In particular, while the Euclidean plan can be tiled by arbitrarily small triangles, we show that there is a least area triangle which tiles the hyperbolic plane without overlapping.

Kellie Takamori

The United States vs Australia: A Comparison of High School Math Curricula

I compare the numeracy levels of public high schools in Australia and the United States by analyzing nationally implemented tests. In particular, I investigate the differences in math curricula and assessment, and correlate it to factors such as high school graduation rates. Because Australia arranges math topics in an integrated fashion as opposed to the United States' organization of math topics by subject content, I first assess the content of Australia's national numeracy test and correlate the content to U.S. math subjects such as Algebra I, Geometry, Algebra II, etc. In particular, I analyze year 9 in public high schools of the state of Oregon to represent the United States and the state of South Australia to represent Australia. I create my own scale of 'proficiency' levels for Australian students by subject and compare the data to the proficiency levels in the U.S. The tests that I use are Australia's National Assessment Program Literacy and Numeracy (NAPLAN) Test and America's Oregon Assessment of Knowledge and Skills (OAKS) Test. I then statistically analyze the benefits of different assessments and curricula, relating the results to high school graduation, first year college enrollment, and proficiency statistics.

Qianru Wang

Delannoy Numbers: Counting Certain Restricted Paths in an Integer Lattice

Delannoy numbers count the number of paths in an integer lattice from one point to another where legal moves are to move one step up, one step right, or one step diagonally up and right. In this project, we give combinatorial proofs for formulas for the central Dellanoy numbers and general Dellanoy numbers. We proceed to use generating functions, a powerful tool from combinatorics, to find a different formula for the general Delannoy numbers. The latter process generalizes to dimensions higher than two. Finally we consider what will happen if we add an additional legal move. Can we find a formula through the same methods?

Travis White

Juggling Mathematics: Counting Juggling Sequences

It may come as a surprise that juggling and mathematics are related. But many famous mathematicians are also expert jugglers. In fact, Ronald Graham served as president of both the International Jugglers Association and the American Mathematical Society. The relationship may be because juggling is a repetitive act embedded with patterns. Jugglers describe these patterns using sequences. In this talk we will learn how to categorize juggling patterns by the number of balls being juggled, the length of the pattern and when the balls are scheduled to land. In addition, we will see how to fix a landing schedule and use generating functions to count the number of possible juggling patterns of various periods. These numbers sometimes show up in surprising places and we will investigate a few of these instances.

Lana Carter

A Cops and Robbers Game

Pursuit invasion games are used to model or explain a variety of real life situations where a team of pursers tries to capture an invader. The math that goes along with these games have been studied immensely throughout the years, tracing back to the work of Pierre Bouguer, who in 1732 studied the problem of a pirate ship pursuing a fleeing vessel. Pursuit games have been associated with many different categories and have thus been studied in a variety of ways and techniques. A discrete version of the game was introduced by Nowakowski, which eventually paved the way to the cops and robbers game. This project follows the work that was done by Tina Zhang, of Bard College, who studied the number of cops needed to catch one robber on a finite graph. Taking into consideration the rules of the game, the goal of this project is to calculate the cop number for various classes of graphs and graph minors to generalize theorems or rules that can be used to find the number of cops needed on a certain graph.

Rebecca Hoffman

Synchronous Fireflies

Fireflies in nature act in a very interesting way. When large groups of fireflies have landed together, the male fireflies begin to blink in synchrony and then continue in this behavior. I have studied the mathematical background of dynamical systems in order to understand a way to model this behavior. That is, fireflies will speed up or slow down their blinking in order to flash together with a stimulus. This behavior can be modeled using an oscillating sinusoidal function. I have manipulated this model further to incorporate a triangular piecewise function to see if it will model natural behavior better. Further, I have used Actionscript 3.0 to visually model this behavior in order to match the natural rhythm.

Kelsey Kaku

Polynomial Interpolation

Polynomials are used to define various functions in subjects such as chemistry, physics, economics, and social sciences. In the real world, a polynomial is often fit to match finitely many data points, a process known as polynomial interpolation. Polynomial interpolation is mainly practiced within the field of real numbers. It is a well-known fact that for k points and x-distinct coordinates, there exists a unique polynomial of degree k - 1. However, what happens when we replace the field of real numbers, with the ring of integers modulo n? Does the existence and uniqueness we had over the real numbers still hold for the ring of integers modulo n?

Duncan McGregor

Renormalization Group Flow on a Homogeneous Space

A geometric flow is a system of differential equations which bends a space by changing its metric (the ruler by which distance in a space is measured). The Ricci flow is a geometric flow which "bends" a space according to its curvature, essentially smoothing. This geometric flow was used by Perelman to solve the Poincaré conjecture. The Renormalization Group flow is a generalization of the Ricci flow used in quantum field theory to model changes to the metric caused by quantization a classical action. We investigate existence and uniqueness of solutions of the RGF on constant curvature and homogeneous spaces.

Jennifer Novak

Modeling Capillary Waves

I look at an equation, called the dispersion relation, which is used to related wave frequency to wave propagation speed with the forces being gravity and capillary action. I first will derive this equation by looking at Rayleigh's The Theory of Sound, and then modify the model to incorporate and arbitrary force in the vertical direction. An application to this process is for agricultural engineering where we study particle size.

Brandon Oshiro

What's Changing? The Max Flow Problem

Given a single source, single sink flow network, the maximum flow problem's goal is to find the maximum amount of flow from the source to the sink within a network. Since the 1950s, the maximum flow problem can be used for real world situations such as sewage pipes and vehicle traffic within a city. The Ford-Fulkerson algorithm was first created to find the solution to the maximum flow problem. When given a flow network G with a set of vertices V and edges E, we start with f(u,v) = 0 for all u,v contained in V, giving an initial flow of value 0. At each iteration, the flow value is increased by finding an augmenting path. Then the process is repeated until no augmenting path can be found. Throughout the years, new algorithms such as the Dinitz blocking flow algorithm and Push-relabel maximum flow algorithm, have made solving the maximum flow problem much more efficient. This project will look at these algorithms, as well as other algorithms used to solve the maximum flow problem and analyze what differences caused the change in efficiency.

James (Alex) Patton

Derivations of the Zeta Function

Dirichelet series are infinite series of the form sum(a(n)/n^5, n=1..infinity) when a(n) is an arithmetic function. We note that the Rieman-Zeta function-ζ(s)-is a Dirichelet series with a(n)=1 for all n. Function convolution allows for a useful arithmetic involving Dirichlet series, which Emmons used along with log derivations to create a new representation of ζ(s). By cleverly defining our log derivations, we can create new representations of ζ(s) and show their absolute convergence.

Summer Steenberg

Blocks and Bases and BIBDs, Oh My!

When does a matroid constitute a block design? Matroids can be found throughout mathematics, constructed of a family of independent sets defined on a ground set. They are essential in our understanding of combinatorial optimization and bridge various fields of discrete mathematics. Block designs allow us to divide a set of varieties into different groups, called blocks. When each block does not contain every element, they are all the same size, and each pair of varieties appears the same number of times we have a BIBD (balanced incomplete block design). Creating the categorization of sets into equal sections, block designs are utilized in practical studies such as shampoo and fertilizer testing. In this talk we explore which types of graphic matroids constitute block designs and what properties are necessary for this to occur. When do the bases of the matroid form the blocks of a design? Is there another way to relate the two? Previous knowledge of matroids or block designs is not required, simply an interest in some graphic material!

Catherine Tardif

The Real Beyond the 15-Puzzle

The 15-Puzzle has been an object of mathematical study for over 100 years. One can classify which initial positions are solvable, and bound the number of moves needed to solve a puzzle depending on where the pieces lie on the puzzle. We are working to determine how much the puzzle would be simplified, if at all, were another piece to be removed. This removal would make the (n^2-1)-Puzzle (where n = 4, for the 15-Puzzle) into an (n^2-2)-Puzzle. By appropriately modifying the proofs for the (n^2-1)-Puzzle, we work to explicitly quantify how much simpler the (n^2-2)-Puzzle is.

Michaela Balkus

Shuffle Your Way to Order

Throughout time, millions of people have played card games. Since most card games begin with a deck in random order, methods of shuffling cards have been a subject of particular interest. In this talk we will explain and explore the “pinch” shuffle introduced in __Number Theory__ by George Andrews. Our main focus will be the following, how many shuffles does it take to get particular deck of cards back to its original order? We will answer this question for a deck of any size and continue further by discussing upper and lower bounds for the necessary number of “pinch” shuffles to return a deck back to its original order.

Stephanie Lowery

Tiling Fibonacci Theorems And Generalizing The Theorems To Include

M-Ominoes

A common childhood game is dominoes. What if you wanted to construct a row that was eight inches long using dominoes that are two inches long and squares that are one inch long? How many different ways could you combine squares and dominoes to construct such a row? We begin this talk by answering this question in relation to the Fibonacci numbers. We then look at some known theorems related to Fibonacci numbers and discuss how our tilings can be used to understand why they are true. This idea is then expanded using m-ominoes, dominoes of length m, to generalize the previous results.

Cody Stein

Measuring Symmetry: The Distinguishing Number

What makes a graph symmetric? If a graph can be rotated, reflected, or is in some other way symmetric, what special properties are implied about the graph? Distinguishing graphs is the process of observing these inherent symmetries and assigning a coloring to the vertices such that these symmetries no longer exist. The distinguishing number of a graph, denoted *D(G)*, is the minimum number of colors, *r*, required to distinguish the graph *G*. We define an algorithm that will determine if a specific coloring of a graph is distinguished by systematically determining if each similar vertex and set of vertices are distinguished. With these properties, we will formally prove the distinguishing number of the cycle graph, *Cn*, the wheel graph *Wn*, the complete graph *Kn*, the complete bipartite graph *Kn,m* and the star graph *Sn*. A greedy algorithm for finding *D(G)* of a tree graph has been found. We seek to elaborate on or create a new algorithm with which to find the distinguishing number of a more general graph.

Chris Upshaw

Datatypes As Mathematical Objects

Programers use datatypes to describe what values a particular program uses. This allows the computer to know how to store and work with those values. By investigating this practice mathematically we can discover fascinating and useful new mathematical objects, and find that some familiar ideas have much broader applications then one would first think. Unfortunately there exists very few introductions to this field. I attempt to provide a overview of the basic ideas and show some examples of the types of results this investigation has yielded, in particular showing how to do algebra and calculus on datatypes.

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*