211
What is an elliptic curve? (2019)
The informal definition of an elliptic curve is simple. The formal definition is more involved.
New
The latest blog posts to take off on Hacker News. 2,851 posts.
212
Erdős Problem #1026
Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. The purpose of this blog post is to try…
213
From Zero to QED: An informal introduction to formality with Lean 4
An informal introduction to formality in Lean 4
214
50 years of proof assistants
Crackpots ranging from billionaire Peter Thiel to random YouTube influencers claim that science has been stagnating for the past 50 years. They admit that computing is an exception: they don’t pretend that my personal 32GB laptop is not an advance over the 16MB mainframe that served the whole Caltech community when I…
215
Using secondary school maths to demystify AI
Educators can show in secondary school maths that AI systems don’t think, making maths more interesting while teaching core concepts of AI.
216
When would you ever want bubblesort? (2023)
There are very few universal rules in software engineering, but there are are a lot of near-universal principles. Things like "prefer composition to...
217
Beautiful Abelian Sandpiles
I can't remember where I first saw them, but ever since, I have been unable to forget them: abelian sandpiles. I'm far from the only one . They're remarkably simple, yet produce lovely symmetric patterns. I loved them so much that I adorned the title banner of this blog with an animation of an abelian sandpile. But…
218
My favourite small hash table
I'm the kind of person who thinks about the design and implementation of hash tables. One design which I find particularly cute, and I think deserves a bit more publicity, is Robin Hood open-addressing with linear probing and power-of-two table size. If you're not familiar with hash table terminology, that might look…
219
Quanta to publish popular math and physics books by Terence Tao and David Tong
Quanta Books is delighted to announce two new upcoming books by mathematician Terence Tao and theoretical physicist David Tong. Six Math Essentials will be Tao’s first math book written for a popular…
220
Learning Feynman's Trick for Integrals
Among a few other integral tricks and techniques, Feynman's trick was a strong reason that made me love evaluating integrals, and although the technique itself goes back to Leibniz being commonly known as the Leibniz integral rule, it was Richard Feynman who popularized it, which is why it is also referred to as…
221
Zero knowlege proof of compositeness
Testing whether a number is composite as an example of a zero knowledge proof. How ZKPs are used more generally.
222
A triangle whose interior angles sum to zero
In hyperbolic geometry, the sum of a the interior angles of a triangle can be any value less than π. If the sum is zero the triangle has area π.
223
The inefficiency of RL, and implications for RLVR progress
And implications for RLVR progress
224
A Fast 64-Bit Date Algorithm (30–40% faster by counting dates backwards)
Counting years backwards unlocks faster speed. The first date algorithm to use only 4 multiplications, instead of 7+
225
Set theory with types
It is known that mathematics is heavily reliant on set theory, but no one can agree on what set theory is. Many people today understand that we have a choice between set theory and type theory , but they don’t know what type theory is either. Many think that type theory refers to some sort of dependent type theory, as…
226
Is Matrix Multiplication Ugly?
A few weeks ago I was minding my own business, peacefully reading a well-written and informative article about artificial intelligence, when I was ambushed by a passage in the article that aroused …
227
Solving Fizz Buzz with Cosines
Fizz Buzz is a counting game that has become oddly popular in the world of computer programming as a simple test of basic programming skills. The rules of the game are straightforward. Players say the numbers aloud in order beginning with one. Whenever a number is divisible by 3, they say 'Fizz' instead. If it is…
228
210 IQ Is Not Enough
You will never have enough IQ, nor money, nor time, nor energy.
229
How Slide Rules Work
The survival of our species owes much to our brain, specifically, its ability to observe, analyse, and plan. Planting crops and storing grains for the winter were some of the earliest uses of these abilities. Measuring and calculating are foundational elements of observation, analysis, and planning. Computation, upon…
230
Weighting an average to minimize variance
How would you allocate money to a set of independent assets with to minimize volatility if some assets are more volatile than others?
231
Linear algebra explains why some words are effectively untranslatable
A modest mathematical framing of language
232
IBM Patented Euler's 200 Year Old Math Technique for 'AI Interpretability'
IBM Slapped the Buzzwords 'AI Interpretability' on Generalized Continued Fractions and their Series Transformations and was awarded a Patent
233
Hilbert space: Treating functions as vectors
The tools of linear algebra are extremely useful when working in Euclidean space (e.g. \mathbb{R}^3 ). Wouldn’t it be great if we could apply these tools to additional mathematical constructs, such as functions and sequences? Hilbert space allows us to do exactly this - apply linear algebra to functions.
234
Cache-friendly, low-memory Lanczos algorithm in Rust
How a bit of algorithm engineering and low-level details can alter what seems like a straightforward trade-off on a blackboard.
235
Think in math, write in code (2019)
Programmers love to discuss programming languages. We not only debate their technical merits and aesthetic qualities, but they become integrated into our personal identities, along with the values and traits that we associate with them. Some even defend a form of Linguistic Determinism that thinking is confined to…
236
Mathematical exploration and discovery at scale
Bogdan Georgiev, Javier Gómez-Serrano, Adam Zsolt Wagner, and I have uploaded to the arXiv our paper “Mathematical exploration and discovery at scale”. This is a longer report on…
237
Fourier Transforms
Fourier Transform
238
The shadows lurking in the equations
If you've ever graphed equations, chances are you have only ever graphed equations in what I refer to as "Binary mode" - which draws a line where the equation is EXACTLY equal, and leaves white everywhere else.
239
Transducer: Composition, abstraction, performance (2018)
Transducer: Composition, Abstraction, Performance
240
