Series An Amazing Formula Generalizing Leibniz’ formula for π, the alternating harmonic series, and the Basel problem Complex Analysis Beyond Infinity Analytic Continuation and its Applications in Quantum Physics Number Theory Transcendental Numbers Into the Unknown Differential Equations Numerical Solvers for ODEs An introduction to Euler methods Number Theory The Euler-Mascheroni Constant Remember the harmonic […]
Want to win a million bucks? Just solve one of these problems. No strings attached. Ok maybe one string: the problems are somewhat hard. Scratch that, really hard. The Gauntlet is Thrown At the start of the millenium, the Clay Mathematics Institute put forward these seven problems which are deemed as some of the most […]
The terms machine learning or artificial intelligence are thrown around a lot these days. It has become fashionable to say that such and such a device has artificial intelligence. But what does this mean? My objective is to give a non-mathematical description of how AI algorithms work. Many papers on the topic quickly get bogged down in heavy duty mathematics, […]
The Indonesian Caves of Borneo Island offer an insight to the most primitive form of recorded communication. Around some 40,000 years ago, before the evolution of written language, physical illustrations on cave walls were the most precise method of recorded communication. As time progressed, the methodology of record-taking evolved from cave paintings to complex alphabets — offering complete expression […]
Captivated by the motion of waves & mesmerized by the perfectly-imperfect symmetry of leaves, it’s crystal-clear we’re a pattern-seeking species. Externally, studies have proven that we use patterns to weigh our environment of danger — a disruption in our daily routine (particularly back in hunter-&-gather societies) signals to our conscious that something is off. Internally, patterns are inscribed in […]
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, […]