# Senior Capstones

#### Fall 2015 and Spring 2016 Capstone Research

Calvin Baxter
Costas Arrays: A Musical Application

A Costas Array is a 2-dimensional, pattern-less structure computed by permuting the rows and columns of an identity matrix such that a vector between any two points is distinct. We will explore the existence, construction methods, and applications of Costas Arrays. Although common and useful applications of the Costas Array include Sonar and Radar, we focus our attention towards implementing this structure in a piece of music.

Kevin Campbell
Falling Sand

A sandpile model strives to show how sand will fall under certain predetermined perimeters. The starting conditions give the arbitrary placement of grains of sand on any subset of Z2, where any number of grains of sand can be place on any vertex. The critical degree of a vertex, denoted di, represents the amount of sand at which a vertex is considered unstable causing it to distribute grains according to predetermined rules by the model's creator. The models we will explore start with an infinite amount of sand at a single point. The models will also have two critical degrees: d1 = 5, and d2 = 9: These constraints will determine the symmetry and patterning of our model. We will also allow for our single generator to have infinite height as well, and make time finite instead of the size of our subset or the amount of sand.

Kelly Collins
Finding Sum-Free Sets: The Signed, Restricted Case

In this paper, we will discuss my findings for τ^± (G, h), defined as the maximum size of a subset in a group, G, such that 0 is not produced within the restricted h-fold span. We will discuss the values and bounds for τ^± (G, h) for exact and general h-values. Specifically, we will show a generalized proof for both a lower and an upper bound.

Kyle Furlong
Optimizing First-Year Seminar Assignment at Gettysburg College

Throughout higher education, the importance of the First-Year Seminar (FYS) has dramatically increased over the past decade. In addition to functioning as an orientation course, these classes oftentimes introduce First-Year students to a vast array of concepts that underscore the value of a liberal arts education. Although prospering, the administration of Gettysburg College is seeking to implement a number of changes to the current FYS program. To facilitate these changes, we turn to the field of operations research, a branch of applied mathematics that uses constraints to determine an optimal and feasible solution.

In this case, we formulate a model in the Xpress-Mosel programming language and soft-ware grounded in these principles and the archetypal Assignment problem found in operations research. Building on comparable models from institutions such as Dickinson College, we structure our model to assign First-Year students to seminars based on a ranked preference while simultaneously balancing other factors, such as gender, race, and socioeconomic background.

Tyler Gould
Using Statistics to Analyze Human Progression or Plateauing in Swimming

Since its international debut in the 1896 Olympics, competitive swimming has dramatically increased in popularity. While more competitors have entered the sport, more talent has been brought to the pool and world records have been broken multiple times in every event. While talent plays a large role, it is not exactly a measurable quality. This aside, the question remains: will humans get faster forever? Will world records always be broken? Along with swimming, this idea can be applied to other sports as well as warm/cold weather records being broken on a consistent basis. This paper uses statistical analysis tools such as Chebychev's inequality, as well as a recursively defined function in order to calculate probabilities to prove or disprove our central equation: If we have data on N attempts on a record, and we know how many record breaks occurred in those N attempts, how likely is it that all N attempts were independently drawn from the same distribution?

Melanie Hazlett
Wavelets and Fingerprint Identification Systems

This paper presents an introduction to the Haar wavelet transformation. It explains several approaches on how wavelets are used in fingerprint identification systems. In particular, this paper looks at how wavelets are used to extract feature vectors in multiple classifier systems as well as how they can enhance the ridges and furrows of fingerprint images so that they are better defined

Using PDEs to Model the Collapse of the Tacoma Narrows Bridge

Partial differential equations (PDEs) are among the most useful tools in modeling real life phenomena. Since a PDE relates a function to its partial derivatives, we can use PDEs in creating a model that includes multiple variables. In this paper, we will examine a historic bridge collapse, and build a model to describe the shape of the bridge to examine the mechanics of the collapse. Using partial differentials and a simplified geometry for a bridge, the model will show how the bridge responds to the introduction of an external force. After building the model, we will solve the resulting PDEs using both analytical and numerical methods to accurately describe the bridge's behavior, which can be represented graphically.

Angelica Klosky
Solving Hashiwokakero as a Constraint Satisfaction Problem

Hashiwokakero is a puzzle game where islands in a grid should be connected by a series of bridges. Though the requirement of creating a connected graph has previously inspired path-finding focused approaches, Kirchhoff's matrix allows an algorithmic test for spanning trees. Using constraint satisfaction theory, the bridges may be modeled as a series of variables in a system of equations that is easily solvable by row reduction. A Java program will iterate through these matrices to return all possible solutions for a given Hashiwokakero puzzle.

Kenneth Lewis
Exploring What Defines Chaos through Sharkovsky’s Theorem

This paper explores dynamical systems through Sharkovsky's Theorem, which proves that a dynamical system that has a least period n will also have periodic orbits of numbers succeeding n in Sharkovsky's Ordering. This ordering implies that if the dynamical system defined by the function has a periodic orbit of 3, it contains all other periodic orbits. There are other papers that discuss properties of period-3 functions based off this theorem, and how it may imply chaotic behavior. There are other fields of study that also look at the converse of this theorem, to construct periodic functions which have a period n but do not contain the period preceding n. These properties give us insight into the field of chaos and dynamical systems and allow us to create formulas and properties of certain periodic functions.

Beth Matys
Flipping a Coin over the Phone

Flipping a coin is often used as a way to decide between two equally likely options. Consider two parties, who do not necessarily trust each other, that want to flip a coin over the telephone for decision making. Since the two parties cannot watch the coin flip happen simultaneously, it becomes very easy to lie about the outcome. We explore previously proposed protocols based in cryptographic techniques for securely flipping a coin over the phone. We also discuss how to securely perform this ‘coin flip' for more than two parties and differing odds.

Kodie McNamara
A Day at the Races: Optimizing the Pari-mutuel Betting System

Horse races like the Kentucky Derby have been running since the 1800s. With a long history and an abundance of data, these races have become popular in the world of betting. To an experienced bettor, wagers and odds dictate betting behavior. But what about when these odds are consistently changing up until race time, such as in horse racing? In the pari-mutuel betting system, odds and payouts are dictated by the crowd's betting behavior, which are swayed by external forces beyond the track. In this paper, we aim to maximize our growth of wealth by applying the Kelly Criterion along with utilizing techniques of operations research. In addition, we will apply our algorithm to historical race data in order to get a better sense of a real world application for our techniques.

Timothy Meinert
Moving in Circles: The Mathematics of Periodic Motion

An oscillator is a differential equation that has periodic solutions. Different kinds of synchronization exist in nature. We will first explore the case of a single oscillator, and prove that a specific equation has a periodic solution. In our second section, we have three oscillating quantities whose behavior influences each other. With three-dimensional systems, we will explore three unique behaviors of solutions in our exploration of periodic motion.

Zachary Miller
Max-Plus Algebra and its Applications

In this paper, we will define the set R ∪ {¿∞}, using the maximum operation as the additive operation and the addition operation as the multiplicative operation, to be the max-plus algebra. We will prove that the set, under these operations, is an example of an abelian, idempotent semiring. We will explain a system of linear algebra that closely resembles the linear algebra over the reals with which we are familiar. We will explain the matrix operations and solve the matrix equation Ax = b in this system. Additionally, we will make connections to graph theory to establish the existence of eigenvectors and eigenvalues in this max-plus linear algebra. We will conclude by using this system of solving linear max-plus systems to solve two operations research problems: the shortest route problem, and the project scheduling problem.

Sarah Scott
Going Viral: The Social Media Bug

What do “the Dress," Gangnam Style, and Kim Kardashian have in common? Each has spread at some point through the sphere of social media like a contagious disease does through a human population. Much work has been done in different academic areas to attempt to understand how pieces of information move through different social applications. One such work, published in 2011, studied “Patterns of Temporal Variation in Online Media" and develops an algorithm in order to study patterns that exist between online content and how its popularity fluctuates over time. By incorporating the Twitter hashtag data published with this paper into the Susceptible-Infected-Susceptible disease model, this paper redefines how we think

Casey Stickney
Dealing with Pai Gow Poker

Pai Gow Poker is one of the newer casino games. The hand counts, deck size and cards dealt have all increased. The competition has changed and it plays by a different set of house rules depending on the casino. If all the casinos have different sets of house rules then all of the sets cannot be the best option for casino income. So what is the best set of house rules for a Pai Gow dealer? We will use the choose function in its entirety to make a casino profitable and end up in the black.

Tessa Thorsen
Edge Detection with Wavelets

Much work has been done on developing wavelet-based methods of edge detection. Many different methods exist, ranging from the simple to the complex. We explore four different methods, two very simple and two more advanced, and compare the accuracy of these methods in detecting edges in various images. We find that, although the more sophisticated methods are computationally expensive, they outperform the simple methods and are more robust to noise. Additionally, the sophisticated algorithms are able to distinguish between different types of edges, unlike the simple methods.

Xiaotong Zou
How to Construct a Regular Polygon?

Euclidean Constructions consist of straightedge and compass. In this paper we will prove the famous Gauss Theorem which states that for a regular n-gon to be constructible, n has to be the product of powers of two and distinct Fermat primes. The first direction of the prove involves ideas in group theory and number theory such as field extension, primitive roots, and roots of complex numbers. The other direction uses the Galois Theory.