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Airfoil
Interactive article explaining the physics of an airfoil and what makes airplanes fly
Topics /
Math & Algorithms
The most popular blog posts about math & algorithms on Hacker News. 266 posts.
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Learning to Reason with LLMs
We are introducing OpenAI o1, a new large language model trained with reinforcement learning to perform complex reasoning. o1 thinks before it answers—it can produce a long internal chain of thought before responding to the user.
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AI solves International Math Olympiad problems at silver medal level
Breakthrough models AlphaProof and AlphaGeometry 2 solve advanced reasoning problems in mathematics
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Visualizing Attention, a Transformer's Heart [video]
Demystifying attention, the key mechanism inside transformers and LLMs.
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Too much efficiency makes everything worse (2022)
This blog is intended to be a place to share ideas and results that are too weird, incomplete, or off-topic to turn into an academic paper, but that I think may be important. Let me know what you think! Contact links to the left.
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A visual proof that a^2 – b^2 = (a + b)(a – b)
A visual proof that a2 – b2 = (a + b)(a – b). Sophie Germain wrote, “It has been said that algebra is but written geometry and geometry is but diagrammatic algebra.”
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The hunt for the missing data type
A (directed) graph is a set of nodes, connected by arrows (edges). The nodes and edges may contain data. Here are some graphs: All graphs made with graphviz (source) Graphs are ubiquitous in software engineering: Package dependencies form directed graphs, as do module imports. The internet is a graph of links between…
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An animated introduction to Fourier series
This article will be part of an extensive series in which I plan to explore various aspects of Fourier Mathematics. I will take notes, create some visuals (a good pretext to learn more about graphics), and hope that it will be helpful to anyone looking for a clear, visual introduction to these concepts. The article…
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An intuitive guide to Maxwell's equations (2020)
An Intuitive and Visual Guide to Maxwell's Equations
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AlphaGeometry: An Olympiad-level AI system for geometry
Our AI system surpasses the state-of-the-art approach for geometry problems, advancing AI reasoning in mathematics
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Double-entry bookkeeping as a directed graph
In the past couple of years, I’ve been working in billing and payments at Justworks . This experience introduced me to the world of bookkeeping and accounting in ways I didn’t expect: I took a course and read a textbook in accounting, adopted plain text accounting for my personal finances, and worked in a double-entry…
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Look, ma, no matrices
Since the 2019 SIGGRAPH course [1], Geometric Algebra, and Euclidean PGA (plane-based or projective geometric algebra) in particular, has been gaining traction within the computer graphics and machine learning communities [2, 3, 4]. Despite its broad applicability, including for higher dimensional geometry and…
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A* tricks for videogame path finding
A* Tricks for Videogame Path Finding, A* (A-star) Search Algorithm
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Understand how transformers work by demystifying the math behind them
Understand how transformers work by demystifying all the math behind them
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Dynamic programming is not black magic
This year’s Advent of Code has been brutal (compare the stats of 2023 with that of 2022, especially day 1 part 1 vs. day 1 part 2). It included a problem to solve with dynamic programming as soon as day 12, which discouraged some people I know. This specific problem was particularly gnarly for Advent […]
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Galois Theory
Note: These pages make extensive use of the latest XHTML and CSS Standards . They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in…
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Sir, there's a cat in your mirror dimension
Pets do the darndest things, especially if you teach them a bit of math.
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Relearning math as an adult
I recently passed 100 days of practicing math every single day 💯 I’ve wanted to beef up my math chops for a while, but I needed a good reason that would justify the time investment. Plus…
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Fermat's Last Theorem – how it’s going
So I’m two months into trying to teach a proof of Fermat’s Last Theorem to a computer. We already have one interesting story, which I felt was worth sharing.
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When do we stop finding new music?
When does our taste in music stagnate?
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U-M finds students with alphabetically lower-ranked names receive lower grades
Knowing your ABCs is essential to academic success, but having a last name starting with A, B or C might also help make the grade.
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Conway's Gradient of Life
What happens if we start from this configuration and take a single step of the game? Here is a snapshot of the next generation:
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Kelly Can't Fail
Introduction You may have heard of the Kelly bet allocation strategy. It is a system for correctly exploiting information or bias in a gambling situation. It is also known as a maximally aggressive…
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Can AI do maths yet? Thoughts from a mathematician
So the big news this week is that o3, OpenAI’s new language model, got 25% on FrontierMath. Let’s start by explaining what this means.
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Gödel, Escher, Bach is the most influential book in my life (2022)
Musings on Hofstadter’s phenomenal book about computation, consciousness, and epistemology.
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Calculus with Julia
This is a set of notes for learning calculus using the Julia language. Julia is an open-source programming language with an easy to learn syntax that is well suited for this task.
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Diffusion models from scratch, from a new theoretical perspective
This tutorial aims to give a gentle introduction to diffusion models, with a running example to illustrate how to build, train and sample from a simple diffusion model from scratch.
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When Random Isn't
This post is an anecdote from over a decade ago, of which I lost the actual code. So please forgive me if I do not accurately remember all the details. Some details are also simplified so that anyone that likes computer security can enjoy this article, not just those who have played World of Warcraft (although the…