# Capstone projects

## Student work

**Leilani Leslie****Mathematical Framework for Teaching: A Lesson Analysis**

Faculty Sponsor: Sarah Hanusch

*Abstract:* Researcher and mathematics educator Deborah Loewenberg Ball has spent over 30 years researching effective practices for teaching mathematics. Her research is an expansion of research done by educational psychologist Lee S. Shulman (1896) on teacher knowledge.

Deborah Ball’s cumulative research efforts resulted in what she calls Mathematical Knowledge for Teaching. Mathematical Knowledge for Teaching “encompasses abilities such as analyzing the student thinking that led to an incorrect answer, identifying the mathematical understanding a student does not yet have, and deciding how to best represent a mathematical idea so that it can be understood by students” (Chapman, 2017). In other words, Mathematical Knowledge for Teaching acknowledges that math teachers need to be dually equipped with subject matter knowledge and pedagogical content knowledge for the most effective delivery of content (Ball, Thames & Phelps, 2008, p. 391). Ball classified both, subject matter knowledge and pedagogical content knowledge, into 3 distinct subdomains, respectively. This capstone paper will thoroughly define each of those subdomains. After defining each subdomain I will take a math lesson that I have taught and analyze it through the lenses of the Mathematical Knowledge for Teaching framework.

**Nathan CaldwellQuandles and Alexander Polynomials in Knot Theory**Faculty Sponsor: Rasika Churchill

**David CleverleyIdeals of Varieties of Five Points**

Faculty Sponsor: John Myers

*Abstract: *Let V = {p_1, · · · , p_k} be a finite set of points in P^2. Then, it is a general fact that there is a finite set S of polynomials with the following properties: every point in V lies on an intersection of all of the curves and all of the curves’ intersection points are in V . There exist many such sets fulfilling those conditions, but this paper is particularly interested in finding the one with the least amount of elements, given a set V with five points all on a line or else. The main result is that, if the five points lie on the same line, only two polynomials are needed to form an intersection of just those points. Further, when the points are not all on a line, it can be constructively shown that only three polynomials are needed to form the intersection, provided the points lie in a plane over an infinite field.

**Jeremy BrandelStudy of Students Working with White Boards**Faculty Sponsor: Preety Tripathi

**Kati BarneyChemometrics with R**Faculty Sponsor: Mark Baker

**Karl Mosbo**

Lebesgue Integration

Faculty Sponsor: Mark Elmer

Lebesgue Integration

*Abstract: *In the spring of 2020, I studied measure theory under professor Mark Elmer’s advisement. I read and did problems from ”A User-friendly Introduction to Lebesgue Measure and Integration” by Gail S. Nelson. We reviewed Riemann integration, then studied outer measure, Lebesgue measure, measurable functions, and finally Lebesgue integration.

This paper will review the concept of Lebesgue measure and Lebesgue integration as presented by Nelson, followed by my proposing a different way of setting up Lebesgue integration while simultaneously being equivalent.**Nicole WightmanLevel Progression of Proof Writing**Faculty Sponsor: Jeff Slye

**Nicolas van Kempen**

On separating systems and covers

Faculty Sponsor: Greg Churchill

On separating systems and covers

*Abstract: *Separating systems and covers deal with both graph theory and combinatorics. Taking an n-element vertex set V , we will present what is means for a pair A and B of subsets of V to separate two given vertices, and extend that definition to a family of such pairs, separating any two vertices of V , creating what we call a separating system. We will consider two optimization problems related to separating systems, first trying to minimize the number of pairs of subsets in our separating system, and then examining a weighted version, minimizing the total number of vertices in the separating system. We will then show a technique that can be used to build an optimal separating system with regards to both problems. Finally, we will extend these concepts to hypergraphs and covers, and discuss a connection between the problems of finding an optimal cover and finding a family of perfect hash functions.**Juliann GeraciConstructions of free resolutions through simplicial complexes**Faculty Sponsor: John Myers

*Abstract: *From a simplicial complex ∆ we can build a chain complex C(∆) which gives an algebraic encoding of information about ∆. We will recognize a fundamental connection between simplicial complexes and commutative algebra through C(∆), which enables us to understand a result of Bayer, Peeva, and Sturmfels that gives an effective way to describe some resolutions in terms of labeled simplicial complexes.**Brett Meerdink: Approximate and Exact Solutions to the Motion of a Simple Pendulum**Faculty Sponsor: Zheng Hao

*Abstract: *The simple pendulum is one of the most studied cases of non-linear motion. At small angles its motion appears linear, however, is non-linear at large angles. This caused by the simple pendulum’s differential equation having a non-linear term (sine of the angle). This term can be approximated using the small angle approximation which linearizes the differential equation resulting in a linear solution. This solution is accurate at describing a simple pendulum’s motion at small angles but fails at large angles, which is described by Jacobian elliptic functions. Approximate solutions for the motion of a pendulum have been developed to describe a simple pendulum’s motion at large angles. Three such approximations are were developed in Bel ́endez, A. et al., Borghi R. et al., and Johannessen, K. et al using an ansatz, Fourier series and homotropy perturbation method respectively. These methods may be used to describe the motion of a pendulum at large angles and converge for smaller angles with the exact and small angle approximation solutions.

**Sam MorleyEmpirical and Theoretical Probability and the Most Underrated Player in NBA History**Faculty Sponsor: Greg Churchill

*Abstract: *In my final semester here at SUNY Oswego, I worked with Dr. Gregory Churchill contemplating what I will call the triple-double problem, an exercise in empirical as well as theoretical probability. That is, what is the probability that an NBA player will acheive double-digit averages in points, rebounds, and assists over the course of an NBA season. With our findings I develop an argument for why Oscar Robinson is the most underrated player in NBA history.

**Kendra Walker**

**Fibonacci numbers and coin tossing distributions**

Faculty Sponsors: Ampalavanar Nanthakumar, Magdalena Mosbo

*Abstract: *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*Abstract: *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*Abstract: *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*Abstract: *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*Abstract: *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*Abstract: *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

*Abstract:*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*Abstract: *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*Abstract: *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*Abstract: *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*Abstract: *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*Abstract: *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

*Abstract:*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

*Abstract:*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

*Abstract:*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

*Abstract:*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

*Abstract:*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.