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)
"I saw the angel in the marble and carved until I set him free" Michelangelo Zenodo links: Forma Potentials, Forma Means, and Forma-Variance: Coercive Polynomial Energy-Based Models, Laplace Regimes, and Universality over Finite Gaussian Mixtures (Author: Gecko) OSF links: (1) Forma Potentials, Forma Means, and Forma-Variance: Coercive Polynomial Energy-Based Models, Laplace Regimes, and Universality over … Continue reading BGecko Library
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 … 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)
There is no excerpt because this is a protected post.
I don’t fabricate storms … but I am one with the rain. You heard it coming and feared it, Worried it’d bring you pain. Don’t choose me from regret. My kindness survived battles you never saw, and there’s no winning or losing (in wars or in love) to a man who now wants to draw. … Continue reading Find Me In the Rain
"Learning digital signal processing is not something you accomplish; it's a journey you take."-- Richard G. Lyons Our last post (article III) covered radio frequency spectrum bands and their applications. However, the primary reason for starting this series is to develop both theoretical and practical knowledge of DSP. Over time, readers can expect an Applied … Continue reading Foundational — DSP Notes: On Discrete Sequences and Systems (Part I)
