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A trip through the Graphics Pipeline
Welcome. This is the index page for a series of blog posts I’m currently writing about the D3D/OpenGL graphics pipelines as actually implemented by GPUs. A lot of this is well known among gra…
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The latest blog posts to take off on Hacker News. 2,391 posts.
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A neat XOR trick
Looking over the solutions mega-thread , it seems like not many people discovered this trick; when I posted about it , people found it to be noteworthy.
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Fast midpoint between two integers without overflow
Let us say that I ask you to find the number I am thinking about between -1000 and 1000, by repeatedly guessing a number. With each guess, I tell you whether your guess is correct, smaller or larger than my number. A binary search algorithm tries to find a value in an interval by repeating … Continue reading Fast…
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A homework question in someone’s 11th grade statistics class
My son is in 11th grade. Below you’ll find a HW question which was given in my son’s statistics class. How would you answer?
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Understanding Convolutions in Probability
In this post we take a look a how the mathematical idea of a convolution is used in probability. In probability a convolution is a way to add two random variables. Using a slightly ridiculous, mad-science, example we walk through multiple way to compute a convolution and ultimate arrive at the formu
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Experiment with Penrose Tilings and other patterns
Create And Share Mathematical Tiling Patterns
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Why I am learning category theory
Category theory is a domain of mathematics that exerts a strange influence over programmers. One thing that can be said for sure about category theory is tha...
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How thick is a three-sided coin?
Matt Parker, employing the enviably pithy prose typical of a content creator forged on the unforgiving fires of social media, asked the simple question back in 2018: > How thick is a three-sided coin? Translated into mathematician-ese, this means > What aspect ratio of a flipped coin leads to the chance of landing on…
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CRDT: Fractional Indexing
Collaborative peer-to-peer applications sometimes need to operate on sequences of objects with a consistent order across all peers. For example, a peer-to-peer photo album application might need to sync the order in which photos appear in an album.
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Making all your integers positive with zigzag encoding
You sometimes feel the need to make all of your integers positive, without losing any information. That is, you want to map all of your integers from 'signed' integers (e.g., -1, 1, 3, -3) to 'unsigned integers' (e.g., 3,2,6,7). This could be useful if you have a fast function to compress integers that fails to ……
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Torsors (2010)
When you first read about "torsors", they're bound to seem a bit too darn sophisticated for their own good. Mostly this is because nobody explains them very well. So, I want to convince you that they're actually very simple and natural. I'll do this by giving you three examples from physics:
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Demystifying Fourier analysis
I've started learning about spectrum analysis a long time ago at university, but failed to gain a true insight in how it works under the hood. For me, it was just some magical math formulas I had to memorize and know about. I knew the theory but it didn't really click .
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Twenty questions is a weird game
Let’s play a game. I’m thinking of a number from 1 to 100. You guess, and I tell you whether my number is higher or lower.
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Demystifying financial leverage
Leverage is actually reasonably easy to understand, both in the math and in the implications for financial firms.
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Binary Lambda Calculus (2020)
Binary lambda calculus ( BLC ) is a minimal, pure functional programming language invented by John Tromp in 2004, 1 based on a binary encoding of the untyped lambda calculus in De Bruijn index notation.
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What is an eigenvalue?
An eigenvalue of a square matrix $LATEX A$ is a scalar $latex \lambda$ such that $latex Ax = \lambda x$ for some nonzero vector $latex x$. The vector $latex x$ is an eigenvector of $LATEX A$ and it…
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The essence of Reed-Solomon coding
Let’s say we want to store some data on multiple drives, so that we can recover from drive failures.
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A step by step backpropagation example (2015)
Background Backpropagation is a common method for training a neural network. There is no shortage of papers online that attempt to explain how backpropagation works, but few that include an example…
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Verifying GCC optimizations using an SMT solver
This post describes the implementation of pysmtgcc. See “GCC Translation Validation” for background information.
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Visualizing Diffs: The Myers difference algorithm (2020)
In 1986, Eugene Myers published An O(ND) Difference Algorithm and Its Variations , which unified the problems of finding the longest common subsequence of two sequences (the LCS of "driftwood" and "artwork" is "two") and finding the shortest edit script for transforming one sequence into another. Myers showed that…
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Markov chain Monte Carlo without the bullshit (2015)
I have a little secret: I don’t like the terminology, notation, and style of writing in statistics. I find it unnecessarily complicated. This shows up when trying to read about Markov Chain Monte Carlo methods. Take, for example, the abstract to the Markov Chain Monte Carlo article in the Encyclopedia of…
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Ssss: Shamir's Secret Sharing Scheme (2006)
Shamir's Secret Sharing Scheme
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SHA-3 Buffer Overflow
Update: This result appeared in the proceedings of CT-RSA 2023. A preprint is available.
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The Mystery of 355/113 (2010)
Today's date is a good excuse to memorize a few more digits of 3.1415926 53589793 23846264 33832795....
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Math with Bad Drawings
Now available for preorder.