2023 COES Design and Research Conference
Mathematics and Statistics Senior Projects
Integrated Engineering and Science Building 205.
Team Member: Haynes Mandino
Advisor: Dr. Jonathan Walters
There are close to 1.2 million career and volunteer firefighters across the United States. In the year 2020 alone, 62 of these firefighters died and 64,875 were injured. I performed the following research to determine whether the firefighter profession has become safer due to new standards and regulations. Each year the National Fire Protection Agency (NFPA) and the Federal Emergency Management Agency (FEMA) collect data on the number of firefighter deaths and injuries to determine whether the standards and regulations are keeping firefighters safe. Statistical hypothesis testing and linear regression were performed on the data to show if, in fact, the profession is safer. In this paper, I use a t-distribution hypothesis to test the correlation coefficient and the slope parameter and determine that over the last 40 years, firefighter deaths and injuries have significantly decreased.
Predicting State GDP Using Time Series Forecasting
Team Member: Austin Neitfeld
Advisor: Mr. Stan McCaa
I conducted this research to determine the weight certain taxes and expenditures have over state US Gross Domestic Product (GDP) as well as how accurately these data points can predict future GDP. The motivation behind this project comes from a desire to find the most efficient way to increase the GDP of states with poorer economies. This will improve the quality of life of citizens of these states. To come to a consensus as to what data points are most influential, hierarchical clustering will be used to split the states into four groups. The average of each tax, expenditure, and GDP from 2015-2020 will be calculated for each group. This data along with Time Series Forecasting Analysis will indicate which data point is most influential for each group. The Time Series Forecasting will result in an equation that can be used to predict the GDP of 2021 utilizing the GDP of 2020. I will then compare the predicted GDP to the actual GDP for each group. The results will lead to future works in which I will allow for more data points and groupings to make more accurate predictions.
Does the Three-Point Shot Affect Winning Percentage?
Team Member: Marcamus Winn
Advisor: Dr. Xiyuan Liu
The three-point shot, introduced in the late 1970s, is a shot that occurs typically 24 feet away from the basket at the professional level. Strategically, the game of basketball was originally based on two-point field shots within 10 feet of the basket. Recently, there has been a noticeable trend in the popularity of the three-point shot among professional teams. Nowadays, three-point shot attempts account for more than a third of the average NBA shot selection. Statistical analysis is becoming an integral part of athletics. Multiple studies use linear and logistic regression models to form a prediction algorithm for the outcome of games. Many have statistically characterized the increase in three-point shooting on a year-to-year basis. The purpose of this research is to analyze this statistical jump and determine if three-point shooting is statistically significant to winning probability before and after the jump. By progressing from linear to logistic regression, the three-point shot point was found to be statistically insignificant to the winning probability. However, the performance of the model is better than the interpretability.
Extending the Radius of Convergence of Drinfeld Logarithms
Team Member: Ethan Clapp
Advisor: Dr. Nathan Green
Drinfeld modules are, in essence, a way to create a function field analog to complex multiplication. With these modules, we can define a logarithm function analogous to the natural logarithm by virtue of a power series. Similarly to how we extend the radius of convergence of the natural logarithm with complex numbers, we can extend the radius of convergence for these Drinfeld module logarithms. While there are proofs for extending the radius of convergence of the Carlitz module, the simplest form of a Drinfeld module, there is no proof for a generalized Drinfeld. In this paper, I examine a method of extending the Carlitz logarithm’s radius of convergence using Newton polygons. Afterward, a way to apply this proof to a general Drinfeld logarithm function will be walked through.
Regression Analysis of Injuries on NFL Quarterbacks
Team Member: Julie Weems
Advisor: Mr. Stan McCaa
Risk assessment is important in many careers such as those of first responders and the military. Its importance for these fields is no different for people who play sports, especially people who are engaged in contact sports such as football. These players’ lives can be changed forever with one bad hit. This research is designed to analyze the risk of an injury for the National Football League’s (NFL) quarterbacks. It is hard to predict when, what, and where an injury will occur. Because of this, very little work has been done on the subject in a general form. This paper is designed to research what variables play a role in factoring into a player being injured. The data was collected using NFL combine data as well as historical injury reports. A binary logistic regression analysis with the variables and using a success if the player was injured and a failure if not. This general model can be used for quarterbacks within the NFL to determine whether they should keep playing or end their careers based on their data during that time.
Four Colorful Methods for Finding the Chromatic Polynomial
Team Member: Emerson Statom
Advisor: Dr. Galen Turner
Mathematicians apply algebraic graph theory to address and interpret problems arising out of data structures and optimization. For example, graph coloring problems have intrigued mathematicians and computer scientists alike. The chromatic polynomial was created to help solve such problems. This research is designed to create a greater understanding of the underlying structures behind chromatic polynomials by analyzing the algebraic, deletion-contraction, and color partition methods. Each method has differing applications, but by viewing them all together, this research hopes to show the depth of this most interesting subject in a condensed manner.
Effects of Topography on Tornado Paths
Team Member: Kayleigh Smith
Advisor: Dr. Brian Barron
Tornadoes are relatively common in Louisiana, which has an average of 55 tornadoes per year. Predicting tornado paths has been extremely challenging due to the many factors that play into the formation of a tornado, such as speed, pressure, and atmospheric conditions. According to the Smithsonian Astrophysical Observatory’s NASA-funded Astrophysics Data System, topography can have a significant influence on tornado direction. For this research, I analyze tornado patterns to determine if local topography has an effect on tornadic activity. Navier-Stokes partial differential equations will be used to model data collected from the National Weather Service and the finite difference method will be used to solve the partial differential equations. This method will analyze the trends in the tornadic activity around Mount Driskill by using Microsoft Excel.
Time-Frequency Analysis of Dispersive Systems
Team Member: Jordan Savoie
Advisor: Dr. Jonathan Walters
The signal processing technique of time reversal is used for various purposes in electromagnetic interference testing, radar, communication, sonar, and medicine. This paper models the impulse response of reverberant chambers that might be used for a time-reversal cavity (TRC) using time-frequency methods. Reverberant chambers have chaotic delaying phase responses that cause the majority of the distortion seen in their impulse response. Time reversal cancels this, creating a focused signal from a dispersed one. Most research modeling the behavior of TRCs is more concerned with their physics, so there is relatively little comment on the form of their signals. This paper is focused on how the chaotic phase response of the impulse response corresponds to the short-time Fourier transform, which appears to be an exponential decay across its bandwidth. Data of a real impulse response was measured using an acoustic reverberant chamber.