Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. Let's see if visualization can make these ideas more intuitive.
Divide the number 999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999 into 1 and express the result as a decimal expansion, and you’ll find the Fibonacci sequence presented in tidy 24-digit strings: (Thanks, David.)
Writeup for my first major machine learning project.
What’s the biggest pixel size of a PNG image in the smallest number of bytes? I wanted to try to create an image that could be downloaded but whose pixel buffer would be too big to store in the RAM of a PC. Here is a PNG image of 6,132,534 bytes (5.8 MB) and 225,000 × 225,000 pixels (50.625 gigapixels), which, if…
This morning I had the pleasure to be a mathematical guest in my daughter’s third-grade class, full of inquisitive eight- and nine-year-old girls, and we had a wonderful interaction. Followin…
Mathematics, applications of mathematics to life in general, and my life as a mathematician.
Update (Dec. 16): If you’re still following this, please check out an important comment by Alex Selby, the discoverer of Selby’s algorithm, which I discussed in the post. Selby querie…
This text is designed to accompany your study of introductory digital signal processing. 1 It’s an eccentric piece of not-so-rigorous literature with a preoccupation for explaining things using interactive visualizations, animations and sound. 2 My goal is to explain the Discrete Fourier Transform using a miniature…
Even after the Snowden revelations, there remained at least one big mystery about what the NSA was doing and how. The NSA’s classified 2013 budget request mentioned, as a priority item, R…
Systems thinker who ended up in data. Mostly working out why organisations can't answer their own questions.
Most high school students in the United States learn about matrices and matrix multiplication, but they often are not taught why matrix multiplication works the way it does. Adding matrices is easy…
You probably that planets go around the sun in elliptical orbits. But do you know why? In fact, they’re moving in circles in 4 dimensions. But when these circles are projected down to 3-dimen…
A major problem in detecting biased decisionmaking is the problem of unknown inputs. For example, suppose a venture capitalist has a public portfolio which has funded 75% male founders and 25% female. Is this venture capitalist biased against women? It's impossible to know - perhaps only 10% of his applicants were …
Those of you who know what public-key cryptography is may have already heard of ECC , ECDH or ECDSA . The first is an acronym for Elliptic Curve Cryptography, the others are names for algorithms based on it.
Update 2017-01-09: Laci claims to have found a workaround to the previously posted error, and the claim is again quasipolynoimal time! Updated arXiv paper to follow. Update 2017-01-04: Laci has posted an update on his paper. The short version is that one small step of his analysis was not quite correct, and the result…
This blog is written by Hanno Böck. Unless noted otherwise, its content is licensed as CC0 .
L19 = 208168199381979984699478633344862770286522453884530548425639456820927419612738015378525648451698519643907259916015628128546089888314427129715319317557736620397247064840935
[This essay in Spanish] [This essay in French] [This essay in Chinese] In an old joke, two noblemen vie to name the bigger number. The first, after ruminating for hours, triumphantly announces "Eighty-three!" The second, mightily impressed, replies "You win." A biggest number contest is clearly pointless when the…
It turns out that finite state machines are useful for things other than expressing computation. Finite state machines can also be used to compactly represent ordered sets or maps of strings that can be searched very quickly.
Glen Weyl has uploaded a new version of his paper, Quadratic Voting (written with Steven Lalley), to SSRN, which now includes the completed proofs. Quadratic voting is the most important idea for l…
On the 100th anniversary of Einstein’s Theory of Relativity, Stephen Wolfram discusses the nature of space and time, and fundamental theory of physics.
