Summer, what a beautiful time. Everyone loves standing on the water’s edge watching the waves breaking on the shore. But how many of you have ever wondered about the extreme complexity of the movement of the water? It seems smooth and regular but as soon as a wave breaks on the sand it splits into […]
My most striking contribution to geometry is, no doubt, my problem on the number of distinct distances. This can be found in many of my papers on combinatorial and geometric problems. -Paul Erdős, On Some of My Favorite Theorems, 1996. Erdős is one of the greatest mathematicians from history, and he considered this problem one of his […]
While similar triangles can be used with polynomials of any order, this introductory post is presented in cubics to exploit the compliant architecture while highlighting the practicality and simplicity of similar triangle math. The method has particular advantages in higher order polynomials when supported by various changeable architectures, as it can alleviate onerous iterations of […]
This story will give you an introduction to how to model technical systems. It covers why modeling is so important for engineers, some basic techniques, and an example of a technical system describes via an ODE. Just to make this clear right away: Not every engineer is modeling technical systems at his job, by far. […]
The powers of two: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024… Boredom by numbers, as I like to call them. The powers of two are often the only companions of my idle mind. They kept me company through slow internet connections, bumper-to-bumper traffic, long waiting lines and practically any of the […]
Applied Mathematics In the era of big data, we frequently find ourselves manipulating high-dimensional data, whether it be image data (where the number of… 19 Jul 2020 — 16 min read Photo by Yu Kato on Unsplash In the era of big data, we frequently find ourselves manipulating high-dimensional data, whether it be image data (where the number […]
Beauty is the first test: There is no permanent place in the world for ugly mathematics. — GH Hardy in A Mathematician’s Apology Rarely does a quote provide such an apt description of the author’s personality. Still rarer for it to come from a mathematician who despised delving into commentary on his subject, believing that first-class minds should […]
Becoming a master at physics requires a lot of work, but learning the techniques and guidelines in this article is taking the first step. People often talk about physics as if it were merely a set of facts and equations: F=ma, energy can neither be created nor destroyed, E = mc², stars are giant balls of hydrogen […]
“The greatest mathematician of the 20th century” — Le Monde Mathematician Alexander Grothendieck was born in 1928 to anarchist parents who left him to spend the majority of his formative years with foster parents. His father was murdered in Auschwitz. As his mother was detained, he grew up stateless, hiding from the Gestapo in occupied France. All the while, he […]
“Mathematics is the study of structures” Mathematical structuralism is a theory in the philosophy of mathematics which argues that mathematical elements are defined solely by their place in mathematical structures. Mathematical structuralism adopts a position that’s common to most other philosophical structuralisms in that it denies that there are any “intrinsic properties” of objects. It even denies the […]
Proof Theory How Mathematical Proofs can Help Unlock the Secrets of the Brain Computational neuroscience, broadly defined, is the mathematical and physical modeling of neural processes at a chosen scale, from molecular and cellular to systems, for the purpose of understanding how the brain represents and processes information. The ultimate objective is to provide an […]
Sometimes I glance through the current Annals of Mathematics to see what types of new results are being recognized as important. If you’re unaware, Annals is widely considered to be the best journal of math research (obviously, others are in contention and I mean no disrespect to them). If your research gets accepted there, it has something important to […]
The published version of a talk by mathematician David Hilbert (1862-1943), dated the 20th of November 1915 is entitled ‘Die Grundlagen der Physik’ (“On the Foundations of Physics”). It was held five days before Einstein’s momentous paper ‘Die Feldgleichungen der Gravitation’ (“The Field Equations of General Relativity”) establishing the mathematical foundation for general relativity. Hilbert’s talk presented a theory of […]
Algorithm The Power of Binary Search Binary search is ubiquitous. Even if you are not aware of it, you have most likely used some (approximate) version of binary search one way or another Probability Theory Concentration of Measure: The Glorious Chernoff Bound ..and how to derive it Factorials Factorials and How to Compute Them I […]