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Two parallel neural ectoderm progenitors contribute to the developing brain
Stanford Health Care delivers the highest levels of care and compassion. SHC treats cancer, heart disease, brain disorders, primary care issues, and many more.
Top posts
Every blog post that took off on Hacker News, ranked by the votes of its best submission. 2,851 posts.
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How to Read Mathematics
Ranked a top 100 research university, VCU is a place where discovery and creativity go hand in hand. Located in downtown Richmond, Virginia, its more than 200 programs emphasize hands-on learning, creativity and engaging in the world around us.
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Animals Made from 13 Circles (2016)
Dori | Art Director, Designer, Artist
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How to Study Mathematics (2017)
This essay describes a number of strategies for studying college level mathematics. It has sections entitled
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A non-mathematical introduction to Kalman filters for programmers
Read my manifesto on Code as an alternative to Mathematics. Code for this article can be found on this Colab Notebook should you choose to follow along. Why Kalman Filters? Kalman filters are ingenius. If you have never heard of them, then a very intuitive (and arguably reductive) way to think about them is to
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MathBox 2
Notes will appear here. Please view in Chrome or Firefox. Chrome is glitchy, Firefox is stuttery. Work in progress. :/
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The fastest GIF does not exist
Some games on a website
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Mining Bitcoin with pencil and paper: 0.67 hashes per day
This article is now available in Japanese: 紙と鉛筆でビットコインをマイニング：1日に0.67ハッシュ and Russian: Майним Bitcoin с помощью бумаги и ручки . I decided...
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1/0 = 0
Disclaimer 2024-12-02 Since this rant is going viral again (again (again)), some disclaimers: I wrote this six years ago to “”“win”“” an ugly Twitter fight. I’m glad I anonymized the inciting tweet and hope I’m the only person still getting grief over this. The core interesting bit— why it’s mathematically valid to…
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Suspicious Discontinuities
If you read any personal finance forums late last year, there's a decent chance you ran across a question from someone who was desperately trying to lose money before the end of the year. There are a number of ways someone could do this; one commonly suggested scheme was to buy put options that were expected to expire…
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The scientific “unit” we call the decibel
Celebrating a rare dumpster fire in the kingdom of science.
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Compressing and enhancing hand-written notes (2016)
I wrote a program to clean up scans of handwritten notes while simultaneously reducing file size.
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987654321 / 123456789
The ratio 987654321 / 123456789 is very nearly 8. What about the analogous ratio in other bases?
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AI isn’t outthinking mathematicians, it’s out-remembering them
The key advantage may not be superior reasoning, but a virtually unlimited symbolic working memory.
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Ten advances in mathematics and theoretical computer science
OpenAI shares new results on long-standing open problems in mathematics and theoretical computer science, including advances in geometry, cryptography, and complexity.
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Let's remove quaternions from every 3D engine (2018)
To represent 3D rotations graphics programmers use Quaternions . However, Quaternions are taught at face value . We just accept their odd multiplication tables and other arcane definitions and use them as black boxes that rotate vectors in the ways we want. Why does $\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=-1$ and…
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The Dunning-Kruger effect is autocorrelation
Do unskilled people actually underestimate their incompetence?
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An Interesting Pattern in the Prime Numbers: Parallax Compression
Early this year a software engineer, Shaun Gilchrist, reached out to me after reading a blog post of mine from many years ago, about my informal search for hidden patterns in the prime numbers. The Ulam Spiral revealed non-random patterns, but they didn’t quite match up. Both Shaun and I had long felt there was […]
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The maths you need to start understanding LLMs
A quick refresher on the maths behind LLMs: vectors, matrices, projections, embeddings, logits and softmax.
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The Matrix Calculus You Need for Deep Learning
Most of us last saw calculus in school, but derivatives are a critical part of machine learning, particularly deep neural networks, which are trained by optimizing a loss function. This article is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We…
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Hannah Fry Wins the Leelavati Prize in 2026 for Mathematics Outreach
Hannah Fry is Professor of the Public Understanding of Mathematics in the Department of Applied Mathematics and Theoretical Physics (DAMTP). She joined the University of Cambridge in January 2025, becoming the first person ever to hold the role.
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Unsupervised sentiment neuron
Stay up to speed on the rapid advancement of AI technology and the benefits it offers to humanity.
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XOR
Recently I was called on to explain the ‘XOR’ operator to somebody who vaguely knew of its existence, but didn’t have a good intuition for what it was useful for and how it behaved.
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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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Mihai Pătraşcu 1982–2012
A pathbreaking researcher taken too soon from us (src) Mihai Pătraşcu passed away on Tuesday of brain cancer, having not yet reached age 30. We, Dick and Ken, did not know about his year-…