## Senior Capstone Projects | Mathematics

The following is a compilation of mathematics senior projects.

**2015**

#### Blake Anthony

##### Quantum State Tomography

Data rates in electronics are reaching their maximums which is causing a shift towards using light to carry information in technology such as computers and televisions. The goal of my project was to develop a technique to determine the frequencies present in the light. The method of storing information in light is to send it through a gas such as rubidium. To understand the effects of the rubidium gas on the information contained in the light, we modeled light going through the gas using Maxwell’s equations. Then we used an array of light detectors to create an intensity graph of two light beams coming onto the detectors at an angle of 12 mrad. The first beam was a high intensity beam with one frequency mode while the other beam was a low intensity beam comprised of multiple frequency modes. From the interference of the two beams, we extracted the frequency modes present with the Discrete Fourier Transform in order to see how the frequencies were affected by the gas.

#### Stephanie Carter

##### Nonlinear Model for Combined Radiopharmaceutical Therapy

This is an extension of a linear model for combined radiopharmaceutical therapy in treating neuroendocrine tumors using 131I-metaiodobenzylguanidine and 90Y-DOTA-D-Phe1-Tyr3-octreotide. Like the original linear model, this model attempts to optimize the amount of radiation delivered to the tumor without exceeding the critical dose absorbed by other organs. However, in this nonlinear model, we are modeling the effects of not just one, but two treatments of combined radiopharmaceutical therapy. The second treatment is based upon the amounts of each radiopharmaceutical used in the first treatment, and the diameter of the tumor before and after the radiation is delivered. The result is an optimization problem in four variables that will provide additional information and treatment options that the linear model was unable to predict.

#### Spencer Chandler

##### Patterns in the (a,b)-Pascal Triangle modulo a prime

The Fibonacci sequence, an important sequence of integers, is generated under the recursion F_{n}=F_{n-1}+F_{n-2}. It is well known that the Fibonacci sequence can be found as backward diagonal sums in Pascal’s triangle. In this talk, we consider a generalization of Fibonacci numbers called the (a,b)-Fibonacci numbers. We construct a modified Pascal’s triangle in which the (a,b)-Fibonacci numbers can be found and investigate patterns in this triangle modulo a prime.

#### Julie Davenport

##### Radionuclide Therapy: Optimization of a Dosing Schedule

In this paper, we propose a mathematical model that minimizes the size of a tumor after two radionuclide treatments. In this investigation, we will show that the way to treat a neuroendocrine tumor with dual-agent therapy is by giving the second dose as a fraction of the maximum dose before the tumor has recovered from the effect of the first dose. The amount of this second dose is determined by patient-specific parameters and the rebound rates of the affected critical organs. Our method use mathematical computation software to optimize the effect of the therapy. Our results give an optimal schedule for two treatment therapies.

#### Brandon Harms

##### Multimode Quantum State Measurement

In our current electronic world, we have found a need to strive for faster speeds. One way of accomplishing this task is by transmitting information via light photonics. In order to advance this technology, it is necessary to be able to characterize light that has been stopped and stored. In my project, we were able to characterize an original weak, signal beam of light by interfering it with a strong reference beam and observing the intensity pattern formed in space on our array detector. We determine the spatial frequencies or "modes" associated with this interference pattern using Fourier Analysis. With our array detection technique, we are able to plot the average photon count vs the plane wave mode angle for every mode that contributes to the pattern.

#### Victor Rielly

##### Differential Equations

We present a few rudimentary and general methods of solving first order linear and nonlinear differential equations which satisfy a specific form. The analysis is designed to present the reader with ideas more so than to present a complete and all-encompassing theory of the differential equations we study. Never the less, we present very general tests which can be run on the differential equations in question. If the equations satisfy the tests, the solutions to the differential equations can be written out directly. The analysis can be extended to a large class of differential equations, and can already tackle the general first order linear differential equation, the Bernoulli equation, separable differential equations, and a few differential equations we may be the first to solve. We end the paper with an extension to higher order differential equations and a discussion of the generality of the methods.

**2014**

**Alex Caroline**

**Periodic out billiards in the hyperbolic plane**

The topic of outer billiards is a relatively new field in geometry with many results waiting to be discovered. In the broadest sense, outer billiards can be thought of as a crude model on how objects will orbit around a body. Two common questions about orbits usually arise: which orbits will eventually repeat, or be periodic, and which orbits will eventually become unbounded, getting farther and farther away from the body. We focus on which orbits in hyperbolic space will be periodic. We define a particular type of orbit called a special orbit. We show that if a convex polygon has a special orbit, then it all of its orbits are periodic orbits.

**Liam Dalton**

**Iterative 3D Scanning with Structured Light**

3-dimensional models of real-world objects can be acquired using scanning techniques of varying cost and quality. In this research, we reproduced the cheapest of such techniques: structured-light scanning. We improve on preexisting methods by computing the error in the model as well as the model, and then using this error to determine where to execute further scanning.

**Eva Forrester**

**The Mathematics of Decision-Making**

Linear programming has been widely recognized as one of the most important scientific advances of the mid-20th century. This remarkable field in mathematics has given rise to the tools that have become industry standard in the field of Operations Research. These tools have application to a broad class of decision-making problems pertinent to nearly all industries. The most common type of application involves the general problem of allocating limited resources among competing activities in the best possible (i.e. optimal) way. Our research examines the mathematical theory behind linear programming as well as the many forms these problems can assume. The logic and methods for solving such optimization problems are then discussed in the context of applications to business and manufacturing.

**Taylor Matyasz**

**On the Number of Minimal Solutions to Diophantine Inequalities**

Diophantine equations, those in which only integer solutions are of interest, have been studied since the 3^{rd} century. Also of great interest are Diophantine inequalities. Indeed, these inequalities arose even in senior capstone projects. In 2013, Evan Cooper encountered a Diophantine inequality in his work to improve upon a college football ranking system. Cooper needed to determine the so-called minimal solutions to his inequality. Inspired by his work, we seek to determine the number of minimal solutions to a general linear Diophantine inequality. We begin by investigating the case of a two dimensional, linear, Diophantine inequality, and find a simple closed expression for the number of minimal solutions. For higher dimensional problems, we are able to determine a recursive formula for the number of minimal solutions. Finally, we make use of Ehrhart polynomials to find a closed form for both an upper and lower bound on the number of minimal solutions to a general linear Diophantine inequality in n variables.

**Jeff Mintz**

**Synthetic Wavefront Reconstruction**

Traditional 3D displays create the illusion of depth by providing separate images for both eyes, however there is currently no method to produce imagery with the proper focal distance. Focusing light requires rays come together at the correct angles, creating a wavefront that converges: a wavefront with curvature. In this project we investigate the use of lenslet arrays to add curvature to the wavefront created by an ordinary display screen. We show that by selection of the proper pixels, rays can be made to converge at a point in front of the screen. A vector version of Snell’s law is derived and used to predict the angle at which light from an arbitrary pixel will leave the lens-screen system. The resulting pixel-to-angle map will facilitate the development of future content for lenslet based display systems.

**Matthew Morse**

**Modeling the One-Dimensional Propagation of the Caveolae-Inclusive Cardiac Action Potential**

Two models (based off of the Luo-Rudy 1 guinea-pig ventricular model) were produced to analyze the effects of caveolar sodium current on a single cardiac action potential and the one-dimensional propagation of a cardiac action potential in a line of cardiomyocytes. Evidence suggests that the opening of caveolae recruits additional sodium channels on the cardiomyocyte membrane that can affect both the peak voltage overshoot and the maximum upstroke velocity of the cardiac action potential—the maximum upstroke velocity in turn can alter the conduction velocity of an electrical signal. We examined two opening mechanisms of caveolae. The first opening mechanism simulated a 1-cm^{2} patch of membrane perfused with a β-adrenergic agonist that opened a certain number of caveolae on the membrane. The second opening mechanism simulated a 1-cm^{2} patch of membrane with stochastically opening caveolae that open according to a Poisson process. The effects of these two opening mechanisms of caveolae on a single cardiac action potential using the LR1 model were compared to previous computational results using the Pandit et al rat ventricular model. Our simulations (which incorporated varying capacitance) revealed a 4.1% increase in peak voltage overshoot and a 19.1% increase in the maximum upstroke velocity for a 42% increase in sodium current due to β-adrenergic stimulation. Incorporating stochastically opening caveolae, we observed a delay in ventricular repolarization, a characteristic of a serious heart condition called Long QT Syndrome.

**Scott Perez**

**Chop it up: An outline of Geometric Dissections**

Geometric dissections involve the cutting of one or more figures into pieces that can be rearranged to form other figures. Dissections can be used to provide visual proofs of mathematical identities, such as the Pythagorean Theorem, and to reveal previously unknown identities. They can help those not familiar with advanced mathematics to understand certain fundamental results. In this presentation we introduce the method of dissections and illustrate several interesting examples. Finally we discuss the Wallace-Bolyai-Gerwien Theorem, which provides the foundation for the method of dissections.

**Daniel Pitluck**

**A Study of DeBruijn Graphs and Sequences**

At the crossroads of mathematics and computer science lies graph theory. Graphs are composed of vertices, or nodes, and edges, which link vertices. Common applications include network models and computer algorithms, and graph theory offers solutions and insights to many practical problems. A relatively new area of graph theory, deBruijn graphs and sequences originally “solved the problem of finding a minimum-length binary string that contains as a (continuous) substring every binary string of a prescribed length-k.” These graphs have length-k bits as vertices composed of an n-letter alphabet. Character shifts indicate the direction of travel from one vertex to another. Unique characteristics make them strong candidates for modeling peer-to-peer computer networks. Properties, characteristics, and applications of deBruijn and similar graphs are presented.

**Connor Ryan**

**Dead Ends in Groups**

Geometric group theory relates algebraic objects to geometric objects and provides a way to use geometry to study algebra. Given a finitely presented algebraic group, one can define its Cayley graph and study the geometric properties of this graph. Certain vertices of a Cayley graph are called dead ends and the corresponding elements in the associated group are algebraically important. We examine dead ends in finite lamplighter groups classify these elements with respect to the natural generating set of the group. We also investigate the ratio of the number of dead end elements to the order of the group.

**Joshua Siva**

**Moore Than You Needed To Know About Kautz and De Bruijn Graphs**

Numbers and words sometimes fall short when tackling particular problems in mathematics and computer science. For instance, say we want to represent a collection of computers and the connections between them. We could start describing them in words, but instead we put pencil to paper and draw dots (dots, of course, being the galactic standard for representing random objects) to represent the computers and lines connecting the dots to represent the connections. We now have a clear picture of what is going on with what we call a graph! Now let’s say we are faced with the problem of trying to find a path through this graph that hits every dot and returns to the first dot we started at. What if we complicate the graph by forcing the connections to have directions? A logical question is how could these computers be better connected so as to allow for higher connectivity? What if we have a cable shortage and we need to connect the maximum amount of computers without sacrificing connectivity too much? Kautz and De Bruijn graphs model these situations and Moore. These two graphs are used to structure networks such as parallel computing architectures and Peer to Peer networks because they come very close to the Moore Bound. This means the graph has high connectivity in the network in addition to the guarantee on the maximum distance a message must travel before getting to its destination. The Kautz graph is better than the De Bruijn graph in this respect. On the other hand, the initial inspiration for the De Bruijn graph, the De Bruijn sequence, illustrates that there is more going on in these graphs that there appears to be. We construct these graphs, show what questions they can answer, and we tie together the concepts of sequences, necklaces, and Moore graphs.

### 2013

#### Evan Cooper

##### College Football Ranking Systems: Improvements of Two Models

Over the years, the Bowl Championship Series has been widely criticized. Many fans felt that their teams were poorly represented in the rankings and thus left out of more prestigious postseason games. As a result of this dissatisfaction, the NCAA is changing the way the national champion in Division I college football will be determined. Starting in the 2013-14 season, the national champion will be the winner of a four team, two-round playoff. A committee of experts, presumably using a ranking system, will select the teams to participate in the playoff. In this talk we explore two existing mathematical models that can be used to rank college football teams. We discuss methods that improve the accuracies of these models including the use of Diophantine equations in weighting margin of victory and using least squares regression to determine parameters involved in predicting the margin of victory of a game.

#### 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 Holdin

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.

#### Kaci Linton

##### 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.

### 2012

#### Jesse Amano

##### Connecting Block Designs and Matroids

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 Takamor

##### 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 Lattic

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.

### 2011

#### 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.

### 2010

#### 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 graphKn, 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

Programmers 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.

### 2009

#### 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.

### 2008

#### 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.

### 2007

#### 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.

### 2006

#### 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.

#### 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..