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Math Academy pulled me out of the Valley of Despair
Here’s to being a preserving student for life.
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Why formalize mathematics – more than catching errors
I read a good post by one of the authors of the Isabelle theorem prover, that got me thinking. The author, Lawrence Paulson, observed that most math proofs are trivial, but writing them (preferably with a proof assistant) is a worthwhile activity, for reasons similar to safety checklists - “Not every obvious statement…
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The Missing 11th of the Month (2015)
Personal website of David R Hagen, scientific software engineer
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Hashed sorting is typically faster than hash tables
Benchmarks and theoretical explanation of why and when hashed radix sort beats hash tables.
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Category Theory Illustrated – Natural Transformations
I didn’t invent categories to study functors; I invented them to study natural transformations. — Saunders Mac Lane
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Sierpiński Triangle? In My Bitwise and?
Exploring a peculiar bit-twiddling hack at the intersection of 1980s geek sensibilities.
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The Bitter Prediction
The Ivory Tower is a blog about software engineering and development philosophy by Anders Sundman.
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Infinite Grid of Resistors
There is a well-known puzzle based on the premise of an �infinite� grid of resistors connecting adjacent nodes of a square lattice. A small portion of such a grid is illustrated below.
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100 Years to Solve an Integral (2020)
The integral of sec(x) is well known to any beginners calculus student. Yet this integral was once a major outstanding maths problem. It was first introduced...
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500 days of math
I recently crossed 500 days of practicing math daily with Math Academy. I wrote about my experience after 100 days here. TL;DR: I am still very impressed by the Math Academy system and highly…
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Solving Fizz Buzz with Cosines
Fizz Buzz is a counting game that has become oddly popular in the world of computer programming as a simple test of basic programming skills. The rules of the game are straightforward. Players say the numbers aloud in order beginning with one. Whenever a number is divisible by 3, they say 'Fizz' instead. If it is…
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Is chain-of-thought AI reasoning a mirage?
Reading research papers and articles about chain-of-thought reasoning makes me frustrated. There are many interesting questions to ask about chain-of-thought…
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How Lewis Carroll computed determinants (2023)
Charles Dodgson (a.k.a. Lewis Carroll) developed a method of computing determinants which has some practical advantages.
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Think in math, write in code (2019)
Programmers love to discuss programming languages. We not only debate their technical merits and aesthetic qualities, but they become integrated into our personal identities, along with the values and traits that we associate with them. Some even defend a form of Linguistic Determinism that thinking is confined to…
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Discovering errors in Donald Knuth's TAOCP
As Donald Knuth has just published Volume 4, Fascicle 7 of The Art of Computer Programming (TAOCP), on constraint satisfaction, I would like to provide…
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Minesweeper thermodynamics
You know how sometimes you start a game of Minesweeper and immediately get stuck?
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Generating Voronoi diagrams using Fortune's algorithm
This article is about using Fortune’s Algorithm to generate Voronoi Diagrams in O(nlogn) time. If I’d known how hard it would be I wouldn’t have started it. If you are intending to implement it with the intent to actually use it, rather than as an exercise, I would recommend not using Fortune’s. Unless you are doing a…
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How Slide Rules Work
The survival of our species owes much to our brain, specifically, its ability to observe, analyse, and plan. Planting crops and storing grains for the winter were some of the earliest uses of these abilities. Measuring and calculating are foundational elements of observation, analysis, and planning. Computation, upon…
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Gaussian integration is cool
Brief discussion on gaussian quadrature and chebyshev-gauss quadrature
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Visual explanations of mathematics (2020)
It is thought that Euclid wrote Elements in about 300 BC, but Oliver Byrne turned it into one of the true gems of visualization — and made it about 100 times more readable. By seamlessly combining typeset text (Caslon, if you’re interested) with minimalist geometric drawings in primary col
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Very Wrong Math
The difference between misinformation and disinformation is the difference between ignorance and malice. Trolling is somewhat different, incorporating an element of provocation and narcissism. But what the hell is this?
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Reinforcement learning, explained with a minimum of math and jargon
To create reliable agents, AI companies had to go beyond predicting the next token.
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People Keep Inventing Prolly Trees
A novel data structure has been invented four times now, with three different names. We think that's a good sign.
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Who Can Understand the Proof? A Window on Formalized Mathematics
Stephen Wolfram discusses understanding proofs discovered using automated theorem proving. Based on Wolfram’s proof of the simplest axioms of logic of Boolean algebra.
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Relaxed Radix Balanced Trees (2024)
I’m adding immutable vectors to my language, Ivan, and needed to pick a suitable data structure to implement them. Clojure uses Persistent Vectors (PVs) which support lookups, updates, and appends all in ‘effectively’ constant time. However, it doesn’t have efficient insert or merge operations. Relaxed Radix Balanced…
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Memory access is O(N^[1/3])
In computer science, we often compare the efficiency of algorithms by describing their runtime as a function of the size of the input. Sorting is O(n * log(n)), meaning that sorting a list of N items takes an amount of time proportional to the number of items multiplied by its logarithm. Matrix multiplication is…
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Are polynomial features the root of all evil? (2024)
High-degree polynomial features aren’t inherently unstable: the power basis and unscaled inputs are. This post shows how regularization and Bernstein polynomials make polynomial regression behave.
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A new Sudoku layout with 81 uniquely shaped cells
My journey making a Sudoku variant that doesn’t have rows or columns
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A visual exploration of vector embeddings
For Pycon 2025, I created a poster exploring vector embedding models, which you can download at full-size . In this post, I'll translate ...
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Mandelbrot deep zoom theory and practice (2021)
Perturbation and rescaling for efficient rendering of fractal images.