Abstract: Developed a novel data structure for efficiently analyzing product metrics by extending classical range tree techniques. Designed and implemented greedy range trees in Python to optimize multi-dimensional queries, improving speed and scalability for large datasets. This enables faster, more accurate analysis of product metrics, including trends, anomalies, and usage patterns.
Abstract: In 1928, Emil Artin and Helmut Hasse introduced the Artin–Hasse exponential, a p-adic analogue. After introducing the basics of p-adic analysis, including why \(\mathbb{Q}_p\) is the completion of \(\mathbb{Q}\), I show that the exponential and logarithm remain inverses, with intuition drawn from metric spaces and topology. The core of this paper is the inductive-based proof of Dwork's Lemma which is used to prove the integrality of the Artin Hasse Exponential.
Abstract: Christiaan Huygens discovered the synchronization of coupled pendulums and building on Huygens’ discovery, this paper examines how string length affects synchronization time in coupled pendulums on a moving platform. Experiments showed that shorter strings generally synchronize faster. Other behaviors, like brief pauses and in-phase vs. anti-phase motion, were explained using classical mechanics. The results show ways to optimize synchronization and this paper also models motion and energy using Lagrangian techniques.
Abstract: A compilation of selected problems and solutions from my time working with Professor Sebastian Cioba at University of Delaware’s Math Department in Sum- mer of 2023. In the process of editing the first version of, A First Course in Graph Theory and Combinatorics, our team (1 graduate student and 4 undergraduates) worked through most of the book in preparation for our research.
Abstract: Faro shuffles (or Riffle shuffles) are cards that are perfectly in- terweved that can be categorized into infaro and outfaro shuffles. Modular equations for the periods of the card’s position after per- forming a infaro or outfaro shuffle are presented, and the infaro case is proven. Breifly, basic combinatorics of faro shuffles are also eluci- dated. The main theorem presented is \(2^{\varepsilon} ≡ 1 \pmod{n−1}\) where \(\varepsilon\) is the period of a card, and this is proven through traditional and spec- tral graph theory methods using an adjacency matrix. Extensions of the problem, including flipping the cards throughout the shuffles and then finding new modular equations.