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Rolling hash; Rabin-Karp string search
This is yet another string search algorithm. (Previously my readers have read about Knuth-Morris-Pratt .)
New
The latest blog posts to take off on Hacker News. 2,391 posts.
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Speeding up atan2f
`atan2f` is an important but slow trigonometric function. In this article we'll explain how to approximate it, and implement a version 50x faster than `glibc` on batches of inputs.
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Difficult math is about recognizing patterns
Jeff Bezos tells a great story about the smartest guy he knew at Princeton. He and a classmate were working on a really tough partial differential equation and getting nowhere. They spent 3 hours on the problem and gave up.
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Bloom Filters: More than a space-efficient hashmap
A bloom filter is one of those data structures you are probably already aware of, or have at least heard about. For those looking for a simple recap, they are a probabilistic data structure which can …
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Computational algebraic geometry for high school students
Can high school students really learn algebraic geometry? And what could it teach us about mathematics education?
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The Poincaré Conjecture, Explained
Poincaré began working on what is now considered to be the foundation of topology and algebraic topology in the 1890s with the work Analysis Situs (1895) and five later supplements (1899; 1900, 1902a, 1902b, 1904).
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Using Benford’s Law to Detect Bitcoin Manipulation
Economist Gary Smith sends along this post with the above title and the subtitle, “Market prices are not invariably equal to intrinsic values.” Here’s Smith:
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Types versus sets in math and programming languages
For several years I have been designing and implementing a functional teaching language especially for use in the context of a Discrete Mathematics course. The idea is for students to be exposed to…
1002
Entropy and Algebra and Topology = ?
Today I'd like to share some math connecting ideas from information theory, algebra, and topology. It's all in a new paper I've recently uploaded to the arXiv (https://arxiv.org/abs/2107.09581), which describes a correspondence between Shannon entropy and functions on topological simplices that obey a version of the…
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0x7FDE623822FC16E6: A magic constant for double float reciprocal (2011)
EDIT: I never expected this post to garner so much interest. Just to be clear, using the constant is a very bad idea in almost every situation I can think of. Even instructions like RCPSS should only be used by people who are convinced that the tradeoffs are worth it (or don’t have any other choice). Subtracting from…
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What is the Axiom of Choice?
One of the easiest ways to start a (friendly) fight in a group of mathematicians is to bring up the axiom of choice. I'll explain what it is, why it's so controversial, and hopefully shed some light on how we choose axiomatic systems and what that means for the math we do.
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Is π the Same in Every Universe?
Mathematician: That depends on what you mean by “universe.” Here’s a framing: A circle of radius R centered at a point P is the set of all points in the plane with distance R fro…
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What Are Diffusion Models?
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1009
Gaussian Processes from Scratch (2019)
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.
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Poisson's Equation
If you aren’t used to staring at math, Poisson’s equation looks a little intimidating:
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Probably Approximately Correct – A Formal Theory of Learning (2014)
In tackling machine learning (and computer science in general) we face some deep philosophical questions. Questions like, “What does it mean to learn?” and, “Can a computer learn?” and, “How do you define simplicity?” and, “Why does Occam’s Razor work? (Why do simple hypotheses do well at modelling reality?)” In a…
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Reasonable Effectiveness of the Multiplicative Weights Update Algorithm (2017)
papad Hard to believe Sanjeev Arora and his coauthors consider it “a basic tool [that should be] taught to all algorithms students together with divide-and-conquer, dynamic programming, and random sampling.” Christos Papadimitriou calls it “so hard to believe that it has been discovered five times and forgotten.” It…
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The Dawn of Formalized Mathematics
Here are the slides of my talk “The dawn of formalized mathematics” from the 8th European Congress of Mathematics , which is taking place online and in Protorož, Slovenia, from June 20 to 26, 2021:
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Introduction to Locality-Sensitive Hashing
Locality-sensitive hashing (LSH) is a set of techniques that dramatically speed up search-for-neighbors or near-duplication detection on data. These techniques can be used, for example, to filter out duplicates of scraped web pages at an impressive speed, or to perform near-constant-time lookups of nearby points from…
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JPEG XL would be Turing-complete without the 1024×1024 pixel limitation
You can construct the Rule 110 cellular automaton in JPEG XL.
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Practical Reed-Solomon for Programmers
Recently I was doing some work decoding the new Galileo High Accuracy Service data. In short, this new service will allow Galileo (“European GPS”) users to achieve decimeter-level accuracy, which is nice. This “HAS” data is transmitted highly redundantly by making good use of Reed-Solomon encoding. To work with this…
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How to average floating point numbers (2015)
When it comes to accuracy, how you add a series of arbitrary values together, and average them still matters.
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Kelly Criterion – how to calculate optimal bet sizes
So, you’ve probably heard of the Kelly Criterion in the context of gambling or maybe also in the context of investing. When gamblers talk about the Kelly Criterion they are often referring to the idea of sizing your bets, which is the very intuitive idea that if you bet too much of a percentage of your total wealth…
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Implementing strict three-point perspective
Implementing strict three point perspective is ridiculous. You’ll never need it. Let’s do it anyway.