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)
Here is a collection of documents that have been pushed to OSF: Pseudovariance via Forma Potentials, Laplace Parametric Multimodal Densities, and Polynomial Universality over Gaussian Mixtures
Imagine we're all on a ship. It's an old interstellar cruiser with a thousand blinking panels, a thousand petty arguments, and a thousand little routines that keep the air flowing. Somewhere deep in the hull, a warning light has been on for as long as anyone can remember... impact is inevitable. No date, no coordinates, … Continue reading The Ship
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
Recently, I opened a new brokerage account and created a new investment strategy in order to feel and observe the weight of mindless spending. This plan will be called the "Mirror" plan, and it should go as follows: Any drip or incidental spending must be reflected in my investments. Even after buying groceries, whenever an … Continue reading 镜像投资 (Part One)
(To see Part III, click here) (To see Part IV, click here) https://youtu.be/zHv-iLlZ66A?si=EVw7ccen9mqVRefh After weeks of late-night calls and soft, thoughtful texts, I came to know the mystery woman as something enduring. Nothing had ever truly been lost. As I stepped onto the bricked walkway outside the Sweet Paris Crêperie & Café, I saw her … Continue reading Operation Gradient: My 15A Experience (Part V)
(To see Part II, click here) (To see Part III, click here) At this point, I made many friends in the 15A program. On a beautiful Saturday afternoon, we had just completed one of the escape rooms at Escape This on Beauvoir Road... and like a bowling ball, hunger suddenly struck me and the small … Continue reading Operation Gradient: My 15A Experience (Part IV)
(To see Part I, click here) (To see Part II, click here) Part III and IV of Operation Gradient took me a while to write... not from lack of effort, but because they demanded deep introspection. At the start of this year, so much was unfolding that to offer anything less than my full focus … Continue reading Operation Gradient: My 15A Experience (Part III)
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