6 posts tagged with Mathematics.
TL;DR A comprehensive exploration of duality in mathematics and electrical engineering, covering reciprocal impedance duals, voltage-current parallel-series duals, and electric-magnetic capacitance-permeance duals, with detailed worked examples and magnetic circuit analysis. Duality is an approach that has been applied across countless disciplines where one takes an existing structure and transforms it into an equivalent structure, often with the intention of making it more useful for a particular context. In electronic circuits this usually means we take an existing circuit schematic and transform it in such a way that it serves a similar purpose but suited to our specific use case.
TL;DR A deep dive into the reflection coefficient ((\Gamma)): what it means physically, how to calculate SWR and load impedance from it, how feedline length affects phase, and how directional couplers measure it in practice. Today I hope to answer a rather complex question: What does the Reflection Coefficient mean exactly, how do we measure it, and what can we do with it once we do. For example if we have a reflection coefficient of \(0.
TL;DR Rotations in 4D+ space aren't just 3D rotations with extra steps — they require a fundamentally different approach using rotation planes and Householder reflections. This post walks through the math and provides a complete PDF reference. Several years ago I was writing a Machine Learning paper that required me to do rotations in an arbitrary number of dimensions. As such I had an entire section of the paper devoted to explaining how that was done before moving on to the actual algorithm.
TL;DR HAM (Hyperassociative Map) is a graph-drawing algorithm I invented in 2009 that draws graphs faster and more reliably than traditional force-directed methods — with no oscillations, no damping schedules, and guaranteed monotonic convergence. It was built for real-time distributed graph processing at massive scale. Introduction Almost 8 years ago, on Aug 15, 2009, I invented a new game-changing algorithm called the Hyperassociative Map algorithm. It was released as part of the dANN v2.
TL;DR Bayes Theorem lets you flip a conditional probability — calculate (P(A \mid B)) when you only know (P(B \mid A)), (P(A)), and (P(B)). The classic medical-test example: even with a 95% accurate TB test, if the disease is rare, a positive result means less than a 2% chance of actually being infected. I've been getting a lot of questions from friends lately about what Bayes Theorem means. The confusion is understandable because it appears in a few models that seem to be completely unrelated to each other.
TL;DR An exploration of the Verhulst logistic growth model, from unrestricted exponential growth to restricted logarithmic growth, culminating in a novel extension that adds an artificial injection term to account for advertising, conservation breeding, and other external accelerants. The Logistic Function, sometimes with modifications, has been used successfully to model a large range of natural systems. Some examples include bacterial growth, tumor growth, animal populations, neural network transfer functions, chemical reaction rates, language adoption, and diffusion of innovation, to name a few.