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…
An unexpected paradigm journey into sleep sort
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…
Blog of Raph Levien.
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…
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…
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):
In this article we’ll look at how to implement the Elo rating system in code.
Learn the theory behind how to perform sensor and camera calibration.
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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…
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.
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…
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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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…
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…
A visual explanation of multi-level modeling
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:
You see, there are two memory accesses per one character of the input string (at average). Hence, the total number of all memory accesses can be len*2 at worst.
There’s not a whole lot on TLA+ technique out there: all the resources are either introductions or case studies. Good for people starting out, bad for people past that. I think we need to write more intermediate-level stuff, what Ben Kuhn calls Blub studies. Here’s an attempt at that. Most TLA+ properties are…
R is the response to selection, S is the selection differential, and h 2 is the narrow-sense heritability. This is the workhorse equation for quantitative genetics. The selective differential S, is the difference between the population average and the average of the parental population (some subset of the total…
A blog about life, programming, math, logic, and physics.
Everyone uses math libraries. Surprisingly, mainstream math libraries do not produce correct results for several thousands of inputs. Developers are seldom aware of them, which affects reproducibil…
