Abstract: In 1928, the Artin-Hasse Exponential was created as an analogy to the exponential function that comes from infinite products, as discussed in the paper. A introductory discussion of formal power series and their connection to the p-adic numbers is given. The fact that Qp is the completion of Q is proven, laying the grounds for a discussion of radius of convergence (for power series), where the mutual inverse isomorphism between the exponential and logarithmic functions in the p-adic number system is also shown. Intermediatary results regarding surface Topology are elucidated using metric spaces. A new proof for Dwork’s Lemma is provided via methods of induction, and is applied to prove the Integrality of the Artin Hasse Function, E(x), which is essential for further research. Extensions regarding E(x) are discussed, such as the radius of convergence and generalized images of the p-adics.
Abstract: Christiaan Huygens discovered the synchronization of coupled pendulums while considering a specific closed system. This paper furthers Huygen’s ambitions by onsidering the effects of synchronization time based on changes in string length in coupled simple pendulums designed on a moving platform. Two simple pendulums were connected through the medium of a wooden board which was then placed on cylindrical cans. String length and synchronization time seemed to display an inverse relationship based on trends of raw data. Explanations for other behaviors such as brief stops in motion and anti-phase versus in-phase synchronization are ex- plained using various laws of Classical Mechanics and are modeled with polynomial approximations. The effect of synchronization arises from the medium between the pendulums and the various dampenings of the system. The findings presented gen- erally show that synchronization can be optimized which is useful in various fields of study like the medical field where many diseases are caused by the synchronization of neurons. Finally, the equations of motion and energy are modeled with Lagrange techniques and possible extensions, like creating a model similar to the Kuramoto Model, and other applications of the problem are discussed.
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^ϵ ≡ 1 mod (n−1) where ϵ 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.