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Undergraduate Research

## 2014-2015

### The Elements of Special Relativity

Einstein’s Special Theory of Relativity is founded upon the curious fact that the speed of light as measured by an observer is independent of that observer’s relative motion to the source of the light. Amanda Maxwell discussed some of the surprising and counter-intuitive predictions of this theory, in particular the prediction that different observers can, and often do, measure different time intervals between the exact same two events.Student: Amanda Maxwell

Mentor: Dr. Mike Crumley

### To Infinity & Beyond: A Study of 3-D Kaleidoscopes

Student: Bethany Nye

Mentor: Dr. Pam Warton

### Predicting the Draft and Career Success of Quarterbacks in the NFL

Mentor: Dr. Pam Warton

### Encrypting and Decrypting Text Messages

Inspired by different encryption methods used in the Cryptology Mathematics course, research was planned to decrypt shorthand text message conversations, since words would be missing numerous vowels. To explore this idea, standard text messages that included many abbreviated words and phrases were encrypted, and then decrypted using the same methods. After deciphering the messages, the object of the research was to figure out how much longer it took to decrypt shorthand text messages compared to normal messages where all of the words were spelled out. Different encryption methods were compared to decide whether certain methods worked better than others with shorthand messages.Mentor: Dr. Aaron Blodgett

### The Magic of Elusive Triangular Numbers

With collaborators across the country, these researchers showed the existence of an intersection between triangular numbers, Sierpinski numbers and Riesel numbers. In recent work, the aim is to show that each integer within the representative Sierpinski and Riesel number sequences has at least two distinct prime divisors. A further goal is to prove that infinitely many triangular numbers exist that cannot be written as the sum or difference of two prime powers.Mentor: Dr. Dan Baczkowski

## 2013-2014

### Scrambling Rubik's Cube with Purpose

Rubik's cubism has been an art form since the 80's. A digital photograph was pixelated and a mathematical algorithm was written to output the required movements needed to transform a solved Rubik's Cube to match a portion of the pixelated photograph. The pixelated photo was then reproduced with over 150 Rubik's Cubes.

### Polygonal Numbers, Riesel Numbers, Sierpinski Numbers and More

Polygonal numbers are those that can be expressed geometrically by an arrangement of equally spaced points. Such examples include: triangular, pentagonal and hexagonal numbers, which all can be expressed with the corresponding geometric shape. Through the use of coverings, one can find an intersection with these polygonal number sequences and the Riesel numbers or Sierpinski numbers.

## 2012-2013

### Mathematics Madness: Insight on Picking March Madness Brackets

In this presentation, students Alexandria Bishop and Dan Brooks showed the mathematics behind picking the winning teams based on the teams' performance throughout the season. The method is that of paired comparisons and the AHP (Analytical Hierarchy Process) that uses matrices and eigenvectors.

### Avoiding Crime with Mathematics

The use of computers' algorithms has a tight connection with mathematics. This project demonstrates that the statement through the use of concepts in graph theory integrated into a Google Maps application. The application aims to construct the safest route from point A to B using local crime statistics. In order to build a path with the least amount of crime, the principles of the shortest path algorithm are implemented except using crime as the driving force. The application incorporates many concepts within computer science as well as mathematics, and looks at how the integration of the two can result in new powerful and innovative applications.

## 2011-2012

### Attack on Lake Erie: Invasion of the Carp?

Eugene Braig, IV identified Lake Erie as being the most habitable environment for Asian Carp in the Great Lakes. Given the frequency with which the electric fences near the Chicago seaway goes down, and the ability of the carp to bypass them by jumping, is it probable that two carp could make their way through Lake Michigan, Lake Huron, Lake Erie and then find the Maumee River and, thus, start a colony of reproducing Asian Carp? If so, then we could lend credibility to those who wish to close the seaway and could help provide evidence that could be used to protect the Great Lakes fisheries. Our program attempts to answer the question utilizing random variable generators, combined with their lifespan and swim speed.

### Written Work for Online Homework

A qualitative study of student attitudes toward turning in written work for online homework in math was conducted. Students in Math 133, Applied Calculus, Spring 2011, were required to turn in written work for the online homework assigned in the course. At the completion of the course, student participants were asked if they would have kept written work to the degree that they had, if it had not been required. Additionally, responses to other questions were collected to judge student study habits. Background on online homework systems, including pros and cons of online home work in general, and specific attributes of WeBWorK, the system used in the course also will be presented. The researchers concluded that students know what they should do in class to help themselves study, but they do not always do it without some sort of incentive or penalty.

## 2010-2011

### Quarantining Worms with a Game of Cops and Robbers

Graph theory plays an important role in information security. Threats are prevalent on networks, and in particular, worms are an example of a malicious mobile code, which can debilitate a network's functionality. By modeling a network with a mathematical graph, playing virtual games of cops and robbers on this graph can provide methods of quarantining the worm in the shortest amount of time.

### When I Was Your Age .....

In the media, it is commonly reported that mathematics students in the U.S. do not compare favorably academically with students from other countries. This situation needs to be addressed; however, to fix the problem, the source needs to be identified. Many argue that the problem is with the educators, the standards or the schools, but the cause has not yet been definitively pinpointed. The presenters will determine if the changes to the textbooks and delivery methods could be a source of the issue. Textbooks in math have changed over the last 50 years in both the content and delivery methods. Could these changes be one of the problems in the education system today?

## 2009-2010

### The Mathematics of Counter-Terrorism

This project involved using graph theory and discrete mathematics to analyze the terrorist cell network in the United States. The project pinpointed the city that would have the greatest detrimental effect on the national network if the terrorist cells in the city would be shut down. This information would be useful to Homeland Security in order to know which city they should concentrate their resources and focus.

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The Chaotic World of Complex Fractals

This project involved investigating the relationship between chaos theory, complexity theory, and the beautiful fractals. Even systems acting chaotically eventually start to form patterns that are actually fractals. This project then focused on understanding the famous Mandelbrot set.

## 2008-2009

### Winning Strategies of Monopoly

This project involved setting up a Markov chain to model all of the nuances of the game of Monopoly which resulted in a large matrix. Techniques from linear algebra were then applied to determine which properties would be landed on the most. This project was invited to be written up and submitted for publication to the Pi Mu Epsilon Journal.

### Modeling the Population of the Round Goby in Lake Erie

### This project involved gathering data from several state and federal organizations on the invasive species Round Goby in Lake Erie, as well as data on the more desirable species in the lake. Techniques from differential equations were used to develop two different models - one showed that the Round Goby could live in harmony with the other species, and the other showed that the Round Goby would take over the lake habitat and force the desirable fish species out.

## 2007-2008

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Bridging the Gap Between Algebra and Abstract Algebra

This project investigates the connection between typical high school algebra and group theory. It takes a historical look at how many topics in group theory were created to determine if a general quantic equation had a general solution.

### Symmetries of a Cube

After investigating symmetries of many two-dimensional objects, it is natural to explore the group created by the symmetries of a cube. This group along with its cyclic and non-cyclic subgroups are explored, and an isomorphic group is discovered.