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A seventh-grader student found a beautiful proof to Thales' Theorem (2002)
A useful theorem attributed to the probably first mathematician ever, Thales of Miletus (ca. 624 BC-ca. 546 BC) is one of most elementary and intuitive ever stated.
New
The latest blog posts to take off on Hacker News. 2,851 posts.
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Doing is normally distributed, learning is log-normal
There are few things I think about more than the essays on gwern.net, and there are few with as satisfying a theoretical payout to contemplate in my orb as his essay on “leaky pipelines”, aka log-normal distributions. The skulk: Say you’re working on a Laravel web app. You’re about 90% sure you know how to start the…
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The Algorithm Behind Jim Simons's Success
Reflections on a Wall Street legend.
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Taming floating-point sums
Suppose you have an array of floating-point numbers, and wish to sum them. You might naively think you can simply add them, e.g. in Rust:
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Why is x & -x equal to the largest power of 2 that divides x?
Or how to find the largest power of 2 that divides x?
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Diffusion Models
I spent 2022 learning to draw and was blindsided by the rise of AI art There is lots to say about this, whether it has been or will be a good thing for artists and society in the long run. I hope to write about it in another post. models like Stable Diffusion. Suddenly, the computer was a better artist than I could…
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Lewis Carroll – computing the day of the week for any given date (1887)
A letter from Lewis Carroll to Nature, March 31, 1887: Having hit upon the following method of mentally computing the day of the week for any given date, I send it you in the hope that it may interest some of your readers. I am not a rapid computer myself, and as I find my average time for doing any such question is…
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BB(3, 4) > Ack(14)
Pavel has found a new 3-state 4-symbol Busy Beaver champion which can compute an “Ackermann-level” function and halts with exactly
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Regular expression matching can be simple and fast (2007)
This is a tale of two approaches to regular expression matching. One of them is in widespread use in the standard interpreters for many languages, including Perl. The other is used only in a few places, notably most implementations of awk and grep. The two approaches have wildly different performance characteristics:
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Unprojecting text with ellipses (2016)
Using transformed ellipses to estimate perspective transformations of text.
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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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The new APT 3.0 solver
APT 2.9.3 introduces the first iteration of the new solver codenamed solver3, and now available with the –solver 3.0 option. The new solver works fundamentally different from the old one. How does it work? Solver3 is a fully backtracking dependency solving algorithm that defers choices to as late as possible. It…
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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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New work extends the thermodynamic theory of computation
In a paper published in Physical Review X on May 13, a quartet of physicists and computer scientists expand the modern theory of the thermodynamics of computation. By combining approaches from statistical physics and computer science, the researchers introduce mathematical equations that reveal the minimum and maximum…
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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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A High-Level Technical Overview of Homomorphic Encryption
About two years ago, I switched teams at Google to focus on fully homomorphic encryption (abbreviated FHE, or sometimes HE). Since then I’ve got to work on a lot of interesting projects, learning along the way about post-quantum cryptography, compiler design, and the ins and outs of fully homomorphic encryption. If…
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Understanding Stein's Paradox (2021)
An intuitive explanation of the James-Stein estimator.
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How hard can generating 1024-bit primes be?
On the one hand they are easy to explain, they are just numbers that have no factors other than one and themselves, but on the other hand they contain endless complexity. They show up in numerous places, ranging from mathematical concepts and conjectures to interesting looking visualizations and cryptography…
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Alice's adventures in a differentiable wonderland
A free, self-contained introduction to neural networks and differentiable programming, from automatic differentiation to transformers, with additional chapters on advanced topics.
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Not all graphs are trees
It's pretty easy to imagine how to represent relational algebra expressions as a tree—they are already structurally rooted trees where each operator has its...
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A smooth and sharp image interpolation you probably haven't heard of
An image interpolation that gives us a continuous and smooth image, where every interpolated value only depends on the four neighboring pixel values. The image becomes smooth, but sharp features remain sharp.
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Planes in 3D Space
Visual and interactive introduction to 2D planes in 3D space.
626
When do we stop finding new music?
When does our taste in music stagnate?
627
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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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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A quick post on Chen's algorithm
Update (April 19): Yilei Chen announced the discovery of a bug in the algorithm, which he does not know how to fix. This was independently discovered by Hongxun Wu and Thomas Vidick. At present, th…