# Honors Theses & Senior Projects

## Sample Honors Thesis

#### Hamming Codes and the McEliece Cryptosystem

Ben Dill - Mathematics with a Concentration in Physics

In this paper, we seek to understand some basic principles of coding theory. Specifically, we define and explore binary codes, Hamming codes, and a special application of coding theory known as the McEliece cryptosystem. We describe the meaning and usage of generator matrices and parity check matrices and present bounds on the number of code words in codes of a given minimum distance and length. We present and prove several known results about Hamming codes, including the fact that they are necessarily perfect codes. We show how to construct a cryptosystem using any linear code and discuss the strength of the McEliece cryptosystem in a future where quantum computers are a reality. Finally, we present one program which automatically decodes strings encoded with a Hamming code and another which encrypts and decrypts messages using the McEliece cryptosystem.

#### A Look into Rail Gun Physics and Design

Scott Harmon - Mathematics with a Concentration in Engineering

Rail guns promise to make space travel and weapons systems much more efficient for the future. I conducted a project to skim the surface of rail gun physics and design. I first started by constructing a small rail gun as a proof-of concept. I then delved deeper into the physics of operation by measuring and modeling the magnetic field associated with the operation of the device.

The model rail gun had all of the basic components of much larger, more complicated devices. It consisted of a pair of rails, a bank of powerful capacitors, a charging circuit, and an AC power source. When the capacitors were charged to 60 volts, I estimated the force on the projectile to be 9.12×10^(-6) Newtons. Unfortunately, this was not enough force to overcome friction forces as well as tertiary "welding" of the projectile to the rails.

In order to understand the basic physics principles behind the operation of the rail gun, I measured the magnetic field produced by the rails. The field produced by a single rail was very non-uniform, revealing weaknesses in my initial rail design. I then conducted the same experiment, but with a
solid wire. This produced the expected magnetic field, which was relatively small on the ends, with the maximum in the center of the length of wire. This taught me that a solid wire produces a desirable magnetic field, rather than my initial design that had many air-filled voids within the rails.
With the knowledge gleaned from this immersion into rail gun physics, I will be able to better design similar devices in the future with a greater understanding of magnetic field production.

#### Bayesian Statistics in Epidemiology

Brittany Harris - Mathematics with a Concentration in Biology

Epidemiologists and biostatisticians depend on high-quality statistical analysis to determine correlative and predictive information relating to health states. Due to the nature of public health data collection, many data sets include missing data points and sparse information. This issue has created an emerging interest in the use of Bayesian statistics for its ability to incorporate prior studies and expert belief into the analysis. It is difficult, however, to access the relative accuracy of Bayesian methods to traditional, or "frequentist", methods. This study examines both Bayesian and frequentist methods in general and analyses deaths in vehicle collisions in Montana between 2004 and 2009 using each method.
Two specific questions of correlation will be addressed:

1. Is a given age group more likely to die in a vehicle collision than average?
2. Is a given season more likely to see more vehicle collision deaths than average?

The results will compare predictive Bayesian and frequentist estimates for earlier years, 2004-2008, with each other and with actual vehicle death occurrences in 2009.

#### The Separation of a Two-Nanosatellite System via Differential Drag

Ian Lyon-Mathematics with a Concentration in Engineering

Small nanosatellites (1 to 10 kg), with their low production costs and unique mission strategy possibilities, can be inexpensively deployed in groups to maximize scientific returns. The Space Science and Engineering Laboratory (Montana State University) has demonstrated this with their FIRE BIRD satellites, a two-satellite system set to observe relativistic electron bursts in the Earth's magnetosphere. FIRE BIRD separation strategy scenarios that use differential drag to influence relative satellite velocities are studied here. Relevant orbital mechanics concepts are discussed, and equations of motion are derived and incorporated into an orbital simulation model written in MATLAB.

It is found that a springless separation strategy utilizing differential drag induced by differing satellite masses creates a near-constant acceleration throughout the FIRE BIRD mission duration. This method is found to be a feasible alternative to the traditional springed-separation strategy. At an altitude of 700 km, a mass difference of five grams will separate the FIRE BIRD satellites to a maximum allowed relative distance of 100 km in roughly a year, plus or minus eight months.
Additional physics concepts which could improve the accuracy of the model are also discussed.

#### Analysis of Business Operations of Hometown Auto & Ag Using SIMPROCES

Amber Nuxoll - Mathematics with a Concentration in Business

The flow of vehicles through an auto parts and repair business, Hometown Auto & Ag, was modeled using SIMPROCES software. 100 replications of a week of business were used to analyze the amount of time customers waited for a mechanic to begin repair on their vehicles. These results were used to provide insight on whether or not the addition of another mechanic would significantly improve these waiting times. This model showed that the addition of another mechanic would reduce waiting times for resources and would most likely be beneficial to the business, especially if business was likely to increase. The sensitivity of the results to certain parts of the model, like the percentages representing the probability of the occurrence of certain types of repairs and the distributions used to represent repair times, was also analyzed. The results were particularly sensitive to both the percentages and distributions. This indicated a need for further study of the frequency of the types of repairs and the nature of the probability distributions that would most accurately portray repair times.

#### Football Betting Trends

Sam Schaefer - Mathematics with a Concentration in Physics

This thesis examines multiple aspects of sports betting within the NFL. The purpose of this thesis is to determine whether or not a statistical advantage existed between general betting trends and whether or not these trends yielded favorable outcomes in the long run. In this thesis all of the NFL games from the 2009 season are placed into three ranges according to their corresponding point spreads. This thesis then uses a binomial test to check whether advantages exist by betting on the favorite or underdog. Money-line bets are also examined and the competitiveness of these games is based on the previously created point spread ranges. Within these money-line bets this thesis tests for statistical advantages by betting on either the favorite or the underdog from one point spread range to another. It also tests whether or not any of these bets would prove to be beneficial in the long run. In addition, this thesis examines the probabilities of coming out even by making a select few bets in a particular range. From these tests the results show that there was very little difference from one betting trend to another, and that most of them resulted in a net loss. Therefore, the results suggest that in order to turn a profit gambling, more sophisticated trends must be used as well as focusing time and attention on the less obvious aspects of sports betting.

## Independent Explorations with Faculty Members

One of Carroll's requirements to graduate with Honors is the writing and presentation of a thesis during the student's senior year. Within the Mathematics, Engineering, and Computer Science Department, students have the opportunity to work with faculty of varied backgrounds and interests. At right are thesis abstracts of our past Honors graduates.