Computer Science A story on reproduction in biology, machines, and AI 25 Nov 2019 — 13 min read Wikipedia (https://en.wikipedia.org/wiki/Self-replicating_machine) A story on reproduction in biology, machines, and AI We are already used to see the impact of AI everywhere around us not just in digital life (from recommendations on Netflix to voice recognition by […]
Using no more than high school algebra, here’s how to solve the infamous question 6 from the 1988 International Mathematics Olympiad. Motivation This problem has a reputation for being one of the hardest, and perhaps the hardest, IMO problem of all time. And you can solve it only using high school algebra. Struggling through and understanding […]
Quantum Mechanics A Layperson’s Guide To Quantum Effects You expect the physical world around you to behave a certain way. If you knock a glass off the table, it will fall to the floor and likely shatter. How do you know this? Primarily from everyday experience, which your brain distills into your intuition. Once you’ve […]
Before we begin, be aware that talking about control theory in any meaningful way means talking about linear algebra. Please don’t be intimidated, even if you don’t have the first idea what any of the symbols or terms mean. Linear Algebra is hard but its core ideas are intuitive and I will explain everything as […]
“Most mathematicians prove what they can, von Neumann proves what he wants” It is indeed supremely difficult to effectively refute the claim that John von Neumann is likely the most intelligent person who has ever lived. By the time of his death in 1957 at the modest age of 53, the Hungarian polymath had not […]
Our setting is the open unit disk, D and T the unit circle, in the complex plane, C. By H² is meant the standard Hardy space, the Hilbert space of holomorphic functions in D having square-summable Taylor coefficients at the origin. As usual, H² will be identified with its space of boundary functions, the subspace […]
Applied Mathematics Principal Component Analysis: The Basics of Dimensionality Reduction In the era of big data, we frequently find ourselves manipulating high-dimensional data, whether it be image data (where the number of… Graph Theory Dijkstra’s Shortest Path Algorithm in Python From GPS navigation to network-layer link-state routing, Dijkstra’s Algorithm powers some of the most taken-for-granted […]
Last time I talked about how the Yoneda Lemma allows us to think about non-traditional spaces. Today we’ll look at a practical application of this to music theory. Here’s the motivational problem: Can we construct a geometric space that has chords as its points and also encodes useful music theory in it somehow? I know that’s vague, […]
A visual proof of the Gauss Bonnet Theorem for triangles on spheres! Spherical geometry is a beautiful, and very visual, area of mathematics, with weird properties (such as that the angles of triangles don’t sum to 180 !!!). Understanding a little bit of Spherical Geometry The Gauss-Bonnet Theorem for triangles on spheres is a special, […]
Philosophy Is ‘Philosophy of Mathematics’ somehow useful? What the philosophy of mathematics is useful for? Max Black, the author of The Nature of Mathematics (1933), thought the main task of the foundation of mathematics (and, consequently the main task of any philosophy of mathematics) would be to elucidate “and analyze the notion of integer or […]
Before we begin, you should know that I’m not actually going to present a proof of the Riemann Hypothesis. This article is about a fictional object known as the field with one element, sometimes denoted Fᵤₙ. You can probably guess why: F is for field and “un” is the French word for 1. When I first […]
The year 2020 might be the year in which I’ve received the most math problems from friends and seen the most math tricks shared on social media. It turns out that sharing math tricks with one another is one of the things people do most when they’re under lockdown. That fact is tempting me to […]
This article will introduce a powerful class of numerical analysis techniques called relaxation methods in the context of electrostatics and fluid dynamics. Let ϕ be a function of x and y and let A, B, …, G also be functions of x and y. Any second-order partial differential equation with ϕ as its solution may be written in the form: The subscripts […]
A few days ago, I explained the content of my previous article on Benford’s Law to a friend, who’s one of the few people who would humor my enthusiasm with maths. And, being the brilliant person she is, she asked how we might observe Benford’s Law and see it in action. It’s one thing to create charts […]