# Capstone projects

## Student work

**Kendra Walker****Fibonacci numbers and coin tossing distributions**

Faculty Sponsors: Ampalavanar Nanthakumar, Magdalena Mosbo

A scenario in which an unbiased coin is tossed until two consecutive heads are achieved results in a probability distribution containing Fibonacci numbers in its numerator. The probability for a sequence of n spaces and with specific place holders, called strings, of heads and tails patterns will be derived. Also, formulas to explicitly and mathematically provide reasoning for the appearance of the Fibonacci numbers will be investigated, even extending the scenario to three consecutive heads to look for patterns. Exploring the expected value, variance, skewness, kurtosis, and moment generating functions for both the two consecutive heads scenario and the three consecutive heads scenario will give insight on the distributions’ characteristics.

We find that this situation applies to four and five heads and then extends to n consecutive head scenarios as well. We will use an R program to confirm the statistical properties derived from each coin tossing scenario and also provide a basis to look at how further “n-acci” sequences compare to our two and three consecutive head scenarios.

**Casey Stone****A primal-dual method to solving the obstacle problem
**

Faculty Sponsor: Zheng Hao

This semester I worked with Dr. Zheng Hao studying the obstacle problem. Throughout the semester I studied a numeric method to solve this problem and explored a solution to the obstacle problem using the Primal-Dual Method and MATLAB code to run this algorithm. I then compared the work Junwei Lu did on approximating the 2D obstacle problem with the Finite Difference Method, under the advisement of Dr. Hao, with my work on the 2D obstacle problem.

**Andrew Smith****Cubic spline interpolation
**

Faculty Sponsor: Elizabeth Wilcox

A cubic spline is a piecewise smooth cubic polynomial that interpolates a set of ordered data points. Cubic spline interpolation is often chosen over polynomial interpolation because of better behavior controls and often less computational overhead. While cubic spline interpolation is often viewed as a way to interpolate data points, it can also be used to model the curve of natural or man-made objects. This was the premise of Roel J. Stroeker’s paper “On the Shape of a Violin,” in which Stroker derived a cubic spline to describe the shape of a violin, which could then be made.[3] Likewise, other interpolation methods such as thin plate spline interpolation appear to be useful modeling tools for real life objects, as seen in fields such as morphometrics. The purpose of this project though was to learn about cubic spline interpolation, and write a program that could model real world inputs, and produce graphics to potentially fabricate a model.

**Colin Beshures****On the Hilbert series of a graded ring**

Faculty Sponsor: John Myers

In this paper we will compute the dimension (or size) of rings. For this, we split a ring into an infinite sequence of vector spaces, which yields an infinite sequence of dimensions. We then use the growth of this sequence as the dimension of the ring. We quantify this growth using objects called Hilbert series, and our main tool to compute Hilbert series is an advanced form of linear algebra.

**Elizabeth Andrews****Julia Robinson and the J.R. Hypothesis**

Faculty Sponsor: Sarah Hanusch

Julia Robinson is one of the most renowned mathematicians of the twentieth century. The first woman elected president of the National Academy of Sciences, Robinson’s career in mathematics spanned over thirty years. Her love for both number theory and recursion as a young college student led her to study Hilbert’s tenth problem, commencing her life’s work. It is her conjecture, namely the J.R. Hypothesis, that ultimately led to the groundbreaking solution to Hilbert’s tenth problem, proving her invaluable to the math community. In this paper, I will discuss not only her personal life, but I will also expound upon her famous hypothesis, discussing the key terms and theorems in her work.

**Junwei Lu****A Finite Difference Method Approach to Numerical Solutions to Obstacle Problem with Constant Boundary Values
**

Faculty Sponsor: Zheng Hao

This paper is description of the Obstacle problem. The obstacle problem is one of the main motivations for the development of the theory of variational inequalities and the problematic of free boundary problems.

**Olivia Peel****Investigation of the Distribution of the Collison Data**Faculty Sponsor: Ampalavanar Nanthakumar

This project aims to study the probability distributions that are involved in analyzing collision related data. The monthly data collected over a period of three years by a local body shop was used in this study. The data showed that the amount of collisions related damages is normally distributed while the number of parts needed to repair these vehicles (which is nested within the number of damaged vehicles) is Poisson distributed. Goodness-of-fit tests were performed to confirm the distributional patterns.

**Kamani Marchant & Kevon Cambridge****Joint distribution of Directional Data**

Faculty Sponsor: Ampalavanar Nanthakumar

The directional data is very common in meteorology, spatial modeling, geology, dentistry etc. For example, wind speed and wind direction form a bivariate directional data. The project considered the descriptive and inferential aspects of bivariate directional data.

**Darryl Gomes-Lewis****NCAA Probability Analysis**

Faculty Sponsor: Ampalavanar Nanthakumar

The project deals with a probability analysis to see whether a low ranked team could beat a top seeded team in the NCAA basketball tournaments.

**Lennisha John****A Study of Benford’s Law**

Faculty Sponsor: Ampalavanar Nanthakumar

The project aims to verify the Benford’s Law empirically by looking at some published numerical tables.

**Jonathan Edwards****Euro Jackpot Lottery**

Faculty Sponsor: Ampalavanar Nanthakumar

The project uses a probabilistic analysis to study the pattern of winning lottery numbers in European Jackpot Lotteries.

**Victoria Nguyen****Regular Calculus vs Stochastic Calculus**

Faculty Sponsor: Ampalavanar Nanthakumar

The project was on explaining the similarities and dissimilarities between the regular Calculus and the Stochastic Calculus. Some examples were given to show the differences.

**Erika Wilson****Investigation of Cepheid Period-Luminosity Relationship**Faculty Sponsor: A. Nanthakumar

This research encompasses the use of many statistical software programs to fit four different copula models on data that measures the Cepheid Period-Luminosity Relation. We first discuss basic copula theory and its applications. Next, we talk about the techniques of data classification used and the results from this procedure. We then move on to fitting copula models and how to do so. We conclude with a discussion of how we analyzed and proved a suspected break in data at the point of X=1.

**Kyle Buscaglia & Sean Crowder****Hidden Markov Chain in Ice-hockey**Faculty Sponsor: A. Nanthakumar

This research focuses at how the concept of Markov Chains and Hidden Markov Chains can be applied in analyzing Ice-hockey games from the videos.

**Laura Murtha****Building a Foundation for Calculus**Faculty Sponsor: Terry Tiballi

It took approximately 100 years from the development of calculus for it to become rigorous and logically sound. Unfortunately, a majority of the calculus students never see the rigor behind it. I will take several facts of calculus that are taken for granted and prove they are logically sound through real analysis. I will walk through a set of problems from the book Real Analysis: A First Course by Russell A. Gordon to shed light on the underlying connections in the proof of major calculus facts and theorems.

**Nicholas Powers****The Use of Mathematics in Nuclear Physics**Faculty Sponsor: Sue Fettes

Problems in nuclear physics will be discussed and examined from a mathematical standpoint. Specific examples will be looked at in which nuclear physics problems are solved using algebra and calculus. The derivation of certain equations used in nuclear physics will also be briefly examined.

**Jacob Gallagher****Linear Fractional Transformations in the Complex Plane**Faculty Sponsor: Terry Tiballi

I will be talking about what a linear fractional transformation is and the different situations that might arise. I will being giving a few simple examples of some of these situations. Then I will talk about a special case of linear fractional transformation with why and how circles are mapped to circles.