Introduction to Operator Theory (Essay I)
Course Introduction This course is based primarily on John B. Conway’s “A Course in Operator Theory” book. The goal is to develop a rigorous foundation suitable for graduate-level pure mathematics, while presenting the material in a way that remains meaningful to students in the applied sciences. $latex x^{2};bg=ffffff&fg=000000$ Operator Theory sits naturally at the…
Notes on CEP
Measurement is fundamental to virtually all human endeavors of practical importance. However, many measurements require accounting for error, and Circular Error Probable (CEP) is one method of quantifying deviations from a best estimate, commonly applied in the context of weapons systems and GPS. This article will derive CEP and explore its practical applications. Deriving CEP…
Advanced –A Treatise on Parametric Forms of Multimodal Distributions
This document is part of a developing theoretical framework authored by Christopher Lee Burgess. Abstract Classical variance is insufficient for characterizing the geometric structure of multimodal data. In this work, we will define a new quantity called the pseudovariance, and demonstrate how it captures shape, modality, and dispersion through a general class of functions we…
Developmental – Welcoming The Gamma Function
“The art of doing mathematics consists in finding that special case which contains all germs of generality.” — David Hilbert. The Gamma function appears in many areas of mathematics, physics, and engineering. This post will not only provide motivation for studying the Gamma function but also present several methods for deriving it, including an introduction…
Developmental – DMD Meets Space City (Part I)
Imagine being able to detect subtle patterns in Earth’s landscape using mathematical tools. With NASA’s API for obtaining LANDSAT Imagery and the Dynamic Mode Decomposition (DMD) technique, this becomes not only possible but accessible to anyone with enough curiosity and a reasonable understanding of python. Here are the highlights: LANDSAT Summary Before diving into advanced…
Developmental – What About Those Pesky Integrals?
So what about those pesky definite integrals? I mean the ones integrating over a high number of dimensions. Many ML problems, especially in Bayesian statistics, involve computing probabilities or expectations over high-dimensional spaces. For this reason, we’re gonna need a clever way to compute these integrals. Enter Monte Carlo Integration! This technique isn’t just a…
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