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New
The latest blog posts to take off on Hacker News. 2,851 posts.
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Visualizing Attention, a Transformer's Heart [video]
Demystifying attention, the key mechanism inside transformers and LLMs.
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Stacking triangles for fun and profit
One thing you may have noticed about the trigonometric functions sine and cosine is that they seem to have no agreed upon definition. Or rather, different authors choose different definitions as the starting point, mainly based on convenience. This isn’t problematic or even particularly unusual in mathematics: as long…
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2023 ACM Turing Prize awarded to Avi Wigderson
Avi Wigderson is the recipient of the 2023 ACM A.M. Turing Award for foundational contributions to the theory of computation, including reshaping our understanding of the role of randomness in computation, and for decades of intellectual leadership in theoretical computer science.
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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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The simple beauty of XOR floating point compression
I recently implemented a small program to visualize the inner workings of a scheme that compresses floating point timeseries by XORing subsequent values. The resulting visualizations are quite neat…
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Lambda Screen: Fractals in pure lambda calculus
Lambda Screen allows you to draw images using terms of pure lambda calculus. By using fixed-point combinators, you can even create infinitely detailed fractals.
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Is the frequency domain a real place?
Dissing Fourier, exploring the Hadamard-Walch transform, and making square-wave spectrograms.
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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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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
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What even is a JSON number?
An exploration of numbers in JSON and various JSON implementations
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Not so fast, Mr. Fourier
A real explanation of discrete Fourier transform (DFT, FFT), discrete cosine transform (DCT), and their inverses.
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Linear Algebra of Types (2019)
It gives my brain a pleasant thrum to learn new mathematics which mimics the algebra I learned in middle school. Basically this means that the new system has operations with properties that match those of regular numbers as much as possible. Two pretty important operations are addition and multiplication with the…
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When will the eclipse happen? A multimillennium tale of computation
Wolfram tech and its dedicated website now easily predict eclipses. A look at advances in understanding and science from ancient eclipse data thousands of years old through current eclipse calculations.
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Math writing is dull when it neglects the human dimension
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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Algorithmic Mathematical Art (2004)
In the early 1990s, they were merely visualization aids in the study of mathematics. Gradually, the complexity and artistry of the images becomes an end itself.
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SQL is syntactic sugar for relational algebra
This idea is particularly sticky because it was more or less true 50 years ago, and it's a passable mental model to use when learning sql. But it's an inadequate mental model for building new sql frontends, designing new query languages, or writing tools likes ORMs that abstract over sql.
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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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Compressing Images with Neural Networks
An introduction to learned image compression using neural networks.
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What are the "worst" spelling bee pangrams?
Thomas Ptacek wondered on mastodon whether there were any Spelling Bee pangrams that would get you to the highest scoring level on spelling bee just by finding that word.
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Compressing chess moves for fun and profit
UPDATE: I’ve written a new post with a new technique which is about 2.5x more efficient than this approach Chess notation has come a long way since descriptive notation, now we have nice and decipherable Standard Algebraic Notation, like Qxf7 (queen takes on f7) or Nf3 (knight takes on f3). This is a great text…
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Romeo and Julia, Where Romeo Is Basic Statistics
Hello there. This is an open access book entitled: “Romeo and Julia, where Romeo is Basic Statistics”. So I guess it’s gonna be about a programming language named Julia and its usage for basic statistics.
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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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Proof of the Singular Value Decomposition
The existence claim for the singular value decomposition (SVD) is quite strong: “Every matrix is diagonal, provided one uses the proper bases for the domain and range spaces” (Trefethen & Bau III, 1997) . MIT professor Gilbert Strang has a wonderful lecture on the SVD, and he includes an existence proof for the SVD…
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Hurried Thoughts: You're Wrong About Tidal-Locking
Space Engine ...
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Prolly Trees
A deep dive into the Probabilistic B-tree or Prolly Tree data structure including examples. Prolly Trees are used to create version controlled databases.
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Ace of Aces: Why you should do maths as a game designer
Every now and again you come across something that stretches your brain in a new direction. Last week I stumbled on a magic trick of a Choose Your Own Adventure: Al Leonardi’s Ace of Aces, a paper …
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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…