In the previous essays, we discussed operators and the hierarchy of Topological spaces on which they may act. Now we will begin connecting fundamental concepts from linear algebra to operator theory to build deeper intuition. We will derive the Rank-Nullity Theorem and show how it connects naturally to the Fredholm index. Assuming we have a … Continue reading On the Fredholm Index (Essay III)
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. Operator Theory sits naturally at the intersection of analysis, … Continue reading Introduction to Operator Theory (Essay I)
As I develop the BGecko course Lectures on Operator Theory and Operator Algebras, it is essential to establish a rigorous overview on the hierarchy of topological spaces. In our previous essay, we defined the operator, and how they might be interpreted over both finite and infinite dimensional spaces. In Linear Algebra, we study the geometry … Continue reading On the Hierarchy of Topological Spaces (Essay II)
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 … Continue reading Notes on CEP
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 … Continue reading Advanced –A Treatise on Parametric Forms of Multimodal Distributions
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"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 … Continue reading Developmental – Welcoming The Gamma Function
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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 … Continue reading Developmental – What About Those Pesky Integrals?
Welcome to the final module of our comprehensive study of Ridge Regression! In Module 1, we uncovered various facets of Ridge Regression, starting with SVD (Singular Value Decomposition) approach. We carefully dissected the formula for the Ridge estimator: $latex w_{Ridge} = (X^{T}X + \lambda I)^{-1}X^{T}y$, unraveling its intricacies through calculus. Our previous discussions in Module … Continue reading Advanced – Ridge Regression Notes (Module 3)
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