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Raytracing diamonds
This comes from a question posed to me and others in 2019 by Stefan Langerman, a computational geometer who also happens to be a diamond dealer. The classic ...
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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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A Tale of Java Hash Tables
The intended audience for this article is undergrad students who already have a good grasp of Java, or seasoned Java developers who would like to explore an in-depth analysis of various hash table implementations that use Open Addressing. The reader should be familiar with Java generics, collections, basic data…
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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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Watertight vector maps from raster images
Or, “how not to GIS”.
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How does your programming language handle “minus zero” (-0.0)?
The ubiquitous IEEE floating-point standard defines two numbers to represent zero, the positive and the negative zeros. You also have the positive and negative infinity. If you compute the inverse of the positive zero, you get the positive infinity. If you compute the inverse of the negative zero, you get the negative…
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Three sins of authors in computer science and math (1997)
Most of us learn how to write technical articles by osmosis. After reading a hundred or so papers, we unconsciously pick up the common patterns of the field and begin to imitate them. This phenomenon is largely beneficial, but there are several truly odious habits that are almost universally and inadvertently adopted…
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Ergodicity, What's It Mean
Ensemble averages and time averages
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Ancient Babylonian Algorithms: The Earliest Programs
An interactive, illustrated timeline of historic moments in humankind's quest for information. With annotations by Jeremy Norman.
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No, RSA Is Not Broken
I have been seeing this paper by cryptographer Peter Schnorr making the rounds: “Fast Factoring Integers by SVP Algorithms.” It describes a new factoring method, and its abstract ends with the provocative sentence: “This destroys the RSA cryptosystem.” It does not. At best, it’s an improvement in factoring—and I’m not…
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Need Something Sorted? Sleep on It
An unexpected paradigm journey into sleep sort
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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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An interactive review of the Oklab perceptual color space
Blog of Raph Levien.
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Sparse Matrices (2019)
A sparse matrix is a matrix that has a value of 0 for most elements. If the ratio of N umber of N on- Z ero ( NNZ ) elements to the size is less than 0.5, the matrix is sparse. While this is the mathematical definition, I will be using the term sparse for matrices with only NNZ elements and dense for matrices with all…
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An Introduction to Lagrange Multipliers
Lagrange multipliers are used in multivariable calculus to find maxima and minima of a function subject to constraints (like "find the highest elevation along the given path" or "minimize the cost of materials for a box enclosing a given volume"). It's a useful technique, but all too often it is poorly taught and…
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EU proposing to regulate the use of Bayesian estimation
The European Commission just released their Proposal for a Regulation on a European approach for Artificial Intelligence . They finally get around to a definition of “AI” on page 60 of the report (link above):
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Implementing the Elo Rating System (2020)
In this article we’ll look at how to implement the Elo rating system in code.
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What Are Diffusion Models?
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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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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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How to Teach Math?
The only way to learn mathematics is to do mathematics—Paul Halmos Angie Hodge is an Associate Professor of mathematics at Northern Arizona University. Her interests as stated on her website …
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PID Controller: A Simple Control Loop Mechanism
A mid-depth cap with classic style for any occasion, this Nike Club Cap comes in smooth cotton twill that's been softly washed for casual, daily comfort. Soft and unstructured, the precurved bill makes for easy styling, and the adjustable back-strap lets you find the right fit. Nike Hat Size 6 1/2 6 5/8 6 3/4 6 7/8 7…
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Kolmogorov Complicity and the Parable of Lightning (2017)
A good scientist, in other words, does not merely ignore conventional wisdom, but makes a special effort to break it. Scientists go looking for trouble. — Paul Graham, What You Can’t Sa…
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Master List of Logical Fallacies (2018)
Fallacies are fake or deceptive arguments, "junk cognition," that is, arguments that seem irrefutable but prove nothing. Fallacies often seem superficially sound and they far too often retain immense persuasive power even after being clearly exposed as false. Like epidemics, fallacies sometimes "burn through" entire…
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An Introduction to Hierarchical Modeling
A visual explanation of multi-level modeling
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Maybe the spaghetti code conjecture is false
The Spaghetti Code Conjecture says that Busy Beaver programs ought to be, in some sense, as complicated as possible. Scott Aaronson puts it this way: