Student performance across OECD countries has continued to decline, reaching the lowest average ever recorded by the OECD Programme for International Student Assessment (PISA) in mathematics and reading. Across the OECD, one in five 15-year-old students is now a low performer in science, mathematics and reading, up…
Intro I recently worked through a problem from a theory of computation textbook that asked me to prove a property of a language using finite automata. The informal proof is a simple constructive proof where you build an automaton and show that it recognizes the language. This is kind of similar to program…
Claim: as specified, the UTS #35 transliteration-rule language is Turing-complete. This is an exception in Unicode, where algorithms are deliberately bounded and always terminate. The specification defines a revisiting cursor with no rewrite limit. ICU's implementation implements a per-call cap, but the spec itself…
Motivation I’ve been captivated by the recent movement to popularize mathematics formalization through the Lean theorem prover, and this year I’m diving deeper into learning it. For those unfamiliar with this revolution, I highly recommend watching Kevin Buzzard’s talks on YouTube for an overview of why formal…
Base changing and other calculations
Any Boolean function can be computed with Tofolli gates, and Tofolli gates are reversible, and so any Boolean function can be computed reversibly.
In 1582, under the inter gravissimas papal bull, the Catholic world transitioned from the Julian Calendar to the Gregorian Calendar . This new calendar shortened the length of the year from 365.25 days to 365.2425 , a reduction of just 0.002% .
A local alternative to backpropagation. PC-ALM trains residual MLPs up to 1000 layers, nearly matching backprop's performance despite using only layer-local dynamics. PC-ALM equips each layer with a feedback control dynamical system that distributes and propagates supervision credit throughout a network.
An operational NAND gate in Age of Empires II , built with the game's scenario editor. This one can also be built within the game in a two-player setup but it's a lot easier (so to speak) to just script it.
Studying brain scan data from individuals — not group averages — reveals key brain-function differences in children who struggle with goal-oriented tasks, a Stanford Medicine study found.
What happens inside an LLM when it tries to calculate with nothing but matrices.
In 1980, Intel introduced the 8087, a chip that made floating-point operations much faster in the IBM PC and other systems. Although the 808...
First, I’m a human typing this post. I’m not going to tell you who I am. Sure, many will say that is cowardly and fair, but I actually don’t think it would bring much value. What matters more is that I describe why, and what I did and didn’t do during the last two weeks or so with the OpenClaw agent I set up, known as…
In the Game of Life, which still lifes can be produced by crashing gliders together? We’ve known for a few years that the answer cannot be “all of them”, because in 2022 Ilkka Törmä and Ville Salo found a patch of still life that, if it exists in the universe, must have existed since the beginning of time. And so…
Since I’m a good mood today—at a beautiful science camp with my kids, high in the mountains near Big Bear Lake in California—I thought I’d blog about something positive. Las…
I hope you're not tired of the 8087, because I have another article about Intel's floating-point chip. 1 In 1980, Intel introduced the 8087...
[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.] “Mathematics underwent, in the nineteenth century, …
The parallel parentheses matching problem can be parallelized in different manners depending on the use case and is a good example of how stacks map to parallel hardware.
Responsible Release of AI-Generated Mathematics September 29, 2026 Back to main page Download PDF At present, some frontier AI labs are testing advanced mathematical problems on proprietary models …
Animate compact tetrahedral cages and reuse static mini-BLASes to ray-trace hundreds of millions of triangles in real time, dramatically cutting per-frame update and memory costs for dense foliage, grass, and crowds.
Many AI enthusiasts debate whether Large Language Models actually "reason." My research indicates that a reasoning process does indeed occur, but its goal is different than we assume.
…symmetry isn’t just a preference for “pretty” shapes. Contents Introduction The HM-GM-AM-QM Inequality The two circles The Semicircle The Container The 3D container The Sum of Squares Inequality Nesbitt’s Inequality Conclusion Introduction While I was randomly browsing the web, I came across this nice picture: Roland…
This post is a rewrite of my earlier blog post from Nov 2023 with new insights and updates.
A floating point number f has the form f = m · 2 e where m is called the mantissa and e is a signed integer exponent . We like to read numbers scaled by powers of ten, not two, so computers need algorithms to convert binary floating-point to and from decimal text. My 2011 post “ Floating Point to Decimal Conversion is…
I didn’t want to buy a trampoline. It wasn’t the money.
We usually learn that 1 kilobyte is 1024 bytes, 1 MB is 1024 kilobytes, etc. But is this true or a kilobyte is precisely 1000 bytes?
This is a very important technique: only fill cells when you’re certain; no guessing.
This is a blog post for my recent paper on RL post-training of LLMs: introducing the Matthew Effect and proposing to solve it with Never Give Up. It is presented interactively and less formally, more like how I give the talk. For a deeper, more technical dive, check out the paper on arxiv and code on github. The…
ratios will be ratios
