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The Deceptively Asymmetric Unit Sphere
The deficiencies of the Unit Sphere and an exploration of Lie groups.
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Prover-Verifier Games improve legibility of language model outputs
Discover how prover-verifier games improve the legibility of language model outputs, making AI solutions clearer, easier to verify, and more trustworthy for both humans and machines.
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Damas-Hindley-Milner inference two ways
What is Damas-Hindley-Milner?
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Turing's topological proof that every written alphabet is finite (2010)
Recently one of my colleagues was reading Alan Turing’s groundbreaking 1936 article “On Computable Numbers with an Application to the Entscheidungsproblem.” This is the article in…
185
Encoding tic-tac-toe in 15 bits
I recently stumbled upon a blog post by Alejandra González (a.k.a @blyxyas) that seeks to compress a tic-tac-toe game state into as few bits as possible. She arrived at a solution in 18 bits. This got me thinking, can we do better?
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Solving Probabilistic Tic-Tac-Toe
This morning, I came accross Cameron Sun’s Probabilistic Tic-Tac-Toe on HN . However, the comments were disappointing, with people proposing probabilistic models, linear programming, minimax algorithms, and heuristic evaluations to develop AI approaches for the game, none of which quite correct. In this post, I’ll…
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Industrious Dice
I’m a professional mathematician. That means somebody pays me to do math. I’m also a recreational mathematician. That means you might have to pay me to get me to stop. Wearing my recreational mathe…
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Accidentally writing a SAT solver
where my words occasionally escape /dev/null
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Derivative at a Discontinuity
Alok Singh's Blog
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The derivative of a number (2014)
Are you kidding? Edward Barbeau is now a professor emeritus of mathematics at the University of Toronto. Over the years he has been working to increase the interest in mathematics in general, and e…
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When random numbers are too random: Low discrepancy sequences (2017)
Random numbers can be useful in graphics and game development, but they have a pesky and sometimes undesirable habit of clumping together. This is a problem in path tracing and monte carlo integrat…
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Borges on Chaos Theory
About a weird, lovely short story
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Learning Regular Languages with L* Algorithm
Research in Software Engineering from Rahul Gopinath
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An interactive guide to Fourier series (2022)
Drawing with the Fourier Series
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Effort – a possibly new algorithm for LLM Inference
A possibly new algorithm for LLM Inference. Adjust smoothly - and in real time - how many calculations you'd like to do during inference.
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Crafting formulas: Lambdas all the way down
We build complex mathematical formulas bottom-up using pure lambda calculus. Includes instructions of how to translate Taylor series and other approximations to lambda terms.
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Qwen2-Math
GITHUB HUGGING FACE MODELSCOPE DISCORD 🚨 This model mainly supports English. We will release bilingual (English and Chinese) math models soon. Introduction Over the past year, we have dedicated significant effort to researching and enhancing the reasoning capabilities of large language models, with a particular focus…
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Structuralism as a Philosophy of Mathematics
For mathematical insight, according to the structuralist imperative, investigate mathematical structure and consider mathematical concepts only as invariant under isomorphism.
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Compiling with Constraints
There are lots of interesting little subproblems in compilation like instruction selection, register allocation and instruction scheduling. These can be expressed in declarative interlinked way to constraint solvers.
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Legendre transform, better explained (2017)
The way that most physicists teach and talk about partial differential equations is horrible, and has surprisingly big costs for the typical understanding of the foundations of the field even among professionals. The chief victims are students of thermodynamics and analytical mechanics, and I've me
204
Diffusion models are evolutionary algorithms
Authors: Yanbo Zhang, Benedikt Hartl, Hananel Hazan, Michael Levin
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An alternative construction of Shannon entropy
The underpinning of information theory is Shannon’s formula for the entropy H of a discrete probability distribution P. Derived in 1948 in the paper “A mathematical theory of communication”, this is given by \[ H[P] = -\sum_{i=1}^N p_i \log p_i \] where \( p_i = p(x_i) \) is the probability of sampling \( x_i \) from…
206
How Chordcat works – a chord naming algorithm
chordcat is a chord-naming application and a pet project of mine. It is written in C++ and SFML.
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Solving climate change by abusing thermodynamic scaling laws
A wintertime climate change solution
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A proof of proof by infinite descent
WARNING: the following contains a whole lot of pedantry about proving theorems at a level of detail such that you could likely convince a co...
209
Understanding the Y Combinator
This article is about the Y combinator, a fixed-point combinator that provides a way of using recursion in scenarios and programming languages where symbols are not available. The concept of the Y combinator itself is closely related to lambda calculus (λ-calculus) and functional programming languages. More…
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Breaking Bell's Inequality with Monte Carlo Simulations in Python
The article explains the Bell inequality using Monte Carlo simulations in Python, and shows how non-local action-at-a-distance can be used to break it with entangled qubits.