Miscellaneous Works

Here, I consider a surface wave in the presence of a bottom-mounting, vertical cylinder in water of finite depth. Assuming water to be incompressible and inviscid with irrotational motion, I obtain the velocity potential function that satisfies the Laplace equation. Using the cylindrical coordinate system and separation of variables method, I obtain the solution for the velocity potential function. The wave load is obtained from the velocity potential function by integrating the pressure over the surface of the cylinder, and the analytical solution is presented. 

Click here to read the manuscript and click here to see the slides.

It's not every semester that you get to take a course entirely dedicated to games! I studied the computational complexity of games ranging from games you can draw like Tic-Tac-Toe, to board games such as Celtic, and even video games. In particular, I focused on the computational complexity of popular Nintendo games like Mario and Donkey Kong, along with my partner Victoria. Together, we explored how the 2015 work, Aloupis et. all, presents proofs regarding the reachability decision problem presented in the most popular Nintendo games, proposing that all belong to the NP-Hard complexity space, and that certain special cases in the worlds of Legend of Zelda and Donkey Kong Country, belong to the PSPACE-hard complexity space. We provided a basic proof outline and gadgets used in the construction. 

First, we prepared slides to educate our classmates. Many of them had questions after our talk, so we addressed them subsequently in this manuscript. Also, we created a game, which is the reason why we called ourselves the FROG group. 

As a 2021 Gilman Scholar, I had the opportunity to participate in an international internship, facilitated by IES Abroad. Early on, I got matched with a spectacular company, Praxis EMR, that focuses on building human-centric electronic medical recording software that benefits both the patient and the practice. Throughout the semester, I evaluated the performance of 75+ medical practice specialty pages, wrote 10+ articles about selecting EMRs, and prepared biographical media relations article for a leading online business magazine within a 24-hour deadline. 

You can read my thoughts about the experience by clicking here. 

Reading Fermat's Enigma by Simon Singh for one of my courses reminded me of all the reasons why I love - and sometimes resent - the study of mathematics. This work pulls together all my favorite quotes from the book paired with personal commentary. Click here to read the report. 

In this work, we introduce a history of Euclid, the father of geometry, to build a historical context to how the study of properties and relations of points, lines, surfaces, solids, and higher dimensional analogs began. We provide the foundation and notation necessary to understand the theorems of Ceva and Menelaus. After outlining these proofs, we discuss how Euclidean geometry segways into other branches of geometry, as well as other interesting corollaries and where modern geometry is headed. 

Click here to read the full manuscript. 

This work presents a brief history of the connections of network security with mathematics, with an emphasis on RSA encryption. Read more here or watch on YouTube. 

Reporting a broader gender pay gap at a company could impact the rate in which prospective employees apply to work there. IT companies seek to be the best option for their prospective employees and have incentive to offer more equitable pay across genders. Our research goal is to determine whether IT professionals have significantly large difference between base pay that is earned by a male, compared to base pay that is earned by a female. We sourced data from Glassdoor that collected a sample of 36 males and 36 females with the IT job title. Statistical methods employed in this work include the use of descriptive and summary statistics, hypothesis testing through use of a two-sample t-test, and regression analysis. The results of these tests found that the difference in base pay for college-educated male IT professionals, compared to college-educated female IT professionals, was insignificant. Potential future research is discussed. Click here to view the findings in a report or watch on YouTube. 

This paper will outline the definitions required to identify surface as manifolds, and subsequently submanifolds, as in a way that extends to infinitely spanning dimensions. A formal proof will summarize how to prove the existence of a manifold and submanifold structure. Concluding remarks will suggest further areas of study and potential applications to general relativity.  Click here to read the manuscript.

In the past few decades no other field of mathematical study has seen more growth than the field of combinatorics. To advance mathematical knowledge past current boundaries, undergraduate students will benefit greatly by familiarizing themselves with the study that observes how number, place, and combination intersect. Consider this hypothetical situation: someone shuffles a deck of cards, records the card order, shuffles the deck again, and receives the same exact order. What are the odds? This academic paper intends to define the concept of a permutation and prove its usefulness in solving the question posed. The exact odds of receiving the same shuffling order twice is the factorial of 52 (52!), or simply 8.0658175e+67. Comparisons will be made to bring scale to this astronomical number, complicated exceptions to these odds will be addressed, and further topics to study will be suggested. Click here to read the manuscript. 

Chip ring is essentially a game played on a graph. The objects we work with are called chips, which occupy the vertices (or sites) of a graph G. We are given a two-player game with the following rules: We have n boxes placed in a line (for example, let’s choose n = 9). Players take turns placing chips into boxes. If a player puts their chip into a box that already has one, one of those chips is red to the box to the left, and the other to the box to the right. Which player has a winning strategy? We apply the "IDEA" method of problem solving proposed by George Pólya's How to Solve It (1945) to make conclusions about gameplay. Click here to view the findings.