This document is a guide that can be used by undergraduate physics students alongside Richard J. Jacob and Professor Emeritus’s Tutorials in the Mathematical Methods of Physics to aid in their understanding of the key mathematical concepts from PHY201 and PHY302. This guide can stand on its own and be used in other upper division physics courses as a handbook for common special functions. Additionally, we have created several Mathematica notebooks that showcase and visualize some of the topics discussed (available from the GitHub link in the introduction of the guide).

The classical double copy maps exact solutions of general relativity to exact solutions of U(1) Yang-Mills theory and suggests a hitherto unknown connection between gravity and gauge theory. In this thesis I study three problems using the Kerr-Schild and Weyl formulations of the classical double copy. Using the Kerr-Schild double copy, I analyze the single copy of a rotating nonsingular black hole and analyze its horizon structure to probe the relationship between the presence of horizons on the gravity side and the single copy field on the gauge theory side. In the second problem I describe the mapping between the surface gravity of static spherically symmetric black holes and the force on a test particle due to the single copy field of the black hole. I also describe potential routes to extending this map to rotating black holes. Finally, inspired by the extended Weyl double copy for spacetimes possessing sources, I reinterpret the single copy of the Taub- NUT metric as being comprised of two terms each being sourced by a separate parameter (the mass and the NUT charge).

This qualitative study sought to investigate the potential reaction between the 3,3',5,5'-tetramethylbenzidine (TMB) radical and LAF-1 RGG, the N-terminus domain of an RNA helicase which functions as a coacervating intrinsically disordered protein. The study was performed by adding horseradish peroxidase to a solution containing TMB and either LAF-1 or tyrosine in various concentrations, and monitoring the output through UV-Vis spectroscopy. The reacted species was also analyzed via MALDI-TOF mass spectrometry. UV-Vis spectroscopic monitoring showed that in the presence of LAF-1 or tyrosine, the reaction between HRP and TMB occurred more quickly than the control, as well as in the highest concentration of LAF-1, the evolution of a peak at 482 nm. The analysis through MALDI-TOF spectrometry showed the development of a second peak likely due to the reaction between LAF-1 and TMB, as the Δ between the peaks is 229 Da and the size of the TMB species is 240 Da.

Regulation in the insurance market has increased greatly over the past four decades, and recent regulatory frameworks such as Solvency II have made simulations increasingly important. Monte Carlo simulations are often too inefficient to be used by themselves, and these Monte Carlo simulations begin to struggle when the complexity of insurance contracts increases. For that reason, there have been numerous suggested improvements to traditional MC methods such as the sample recycling method and a neural network method. This thesis will review various risk measures, the methods used to calculate them, and a detailed analysis of the neural network method and the sample recycling method. The sample recycling method and the neural network method will then be analyzed in detail, and a comparative analysis of the sample recycling method and the neural network method will be given. It was discovered that both the sample recycling method and the neural network method provide a large improvement in computational cost and overall run time with minor impacts on the accuracy. Thus, it was concluded that the sample recycling method is best suited for contracts where the inner loop estimations are particularly complex and the neural network is a general method that pairs well with complex input portfolios.

The reionization of the Universe is thought to have completed by redshift z~5.5. To probe this era, galaxy observations in the Subaru Deep Field (SDF) have identified more than 100 galaxies at z~6, many spectroscopically confirmed through follow-up observations. Using available optical/IR data, we model with CIGALE the spectral energy distributions (SEDs) of 43 SDF galaxies, including newly acquired data from the UKIRT WFCAM K-band for seven previously studied objects. In particular, modeling deep IR photometry is sensitive to the galaxy's Lyman continuum (LyC) escape fraction (fesc). We find the median implied fesc value as ~0.4+/-0.1 (mean error). Significant uncertainties in data and fitting result in a large range of fesc for individual objects, but analysis suggests that fesc is likely high enough for galaxies to finish reionization by z~6. More importantly, we find trends between the CIGALE UV slope b, fesc, and dust extinction E(B-V): for a given E(B-V), b appear steeper by ~0.4 than at z=0. Lower fesc values appear to be associated with bluer b and lower E(B-V), but only weakly. This suggests that LyC could have escaped through holes with sufficiently wide opening angles surrounding the ISM from outflows of supernovae and/or weak AGN to escape, but resulting in a large range of implied fesc values depending on the orientation of each galaxy. The current HST, Spitzer and ground-based photometric and model errors for the 43 galaxies are large, so IR spectroscopic observations with the James Webb Space Telescope are needed to better constrain this possibility.

The purpose of this paper is to provide an analysis of entanglement and the particular problems it poses for some physicists. In addition to looking at the history of entanglement and non-locality, this paper will use the Bell Test as a means for demonstrating how entanglement works, which measures the behavior of electrons whose combined internal angular momentum is zero. This paper will go over Dr. Bell's famous inequality, which shows why the process of entanglement cannot be explained by traditional means of local processes. Entanglement will be viewed initially through the Copenhagen Interpretation, but this paper will also look at two particular models of quantum mechanics, de-Broglie Bohm theory and Everett's Many-Worlds Interpretation, and observe how they explain the behavior of spin and entangled particles compared to the Copenhagen Interpretation.

The goal of this project was to develop a prototype for an educational tool that will help users understand how the voting system deployed by a government can affect the outcomes of elections. This tool was developed in Java SE, consisting of a model for the simulation of elections capable of supporting various voting systems, along with a variety of fairness measures, and educational and explanatory material. While a completed version of this tool would ideally be fully self-contained, easily accessible in-browser, and provide detailed visualizations of the simulated elections, the current prototype version consists of a GitHub repository containing the code, with the educational material and explanations contained within the thesis paper. Ultimately, the goal of this project was to be a stepping stone on the path to create a tool that will instill a measure of systemic skepticism in the user; to give them cause to question why our systems are built the way they are, and reasons to believe that they could be changed for the better. In undertaking this project, I hope to help in providing people with the political education needed to make informed decisions about how they want the government to function. The GitHub repository containing all the code can be found at, https://github.com/SpencerDiamond/Votes_that_Count

In thesis we will build up our operator theory for finite and infinite dimensional systems. We then prove that Heisenberg and Schrodinger representations are equivalent for systems with finite degrees of freedom. We will then make a case to switch to a C*-algebra formulation of quantum mechanics as we will prove that the Schrodinger and Heisenberg pictures become inadequate to full describe systems with infinitely many degrees of freedom because of inequivalent representations. This becomes important as we shift from single particle systems to quantum field theory, statistical mechanics, and many other areas of study. The goal of this thesis is to introduce these mathematical topics rigorously and prove that they are necessary for further study in particle physics.