Ars Mathematica

Here’s a pretty idea. A Markov chain is one of the simplest forms of dependence in random variables: an infinite sequence of dependent random variables, where the probability distribution of the next random variable only depends on the value of the current random variable. If you reverse the sequence of variables, you get another Markov chain, the reverse Markov chain. Some Markov chains, reversible Markov chains, have the property that when you reverse them, you get back the same chain. Markov chains represent processes that have no history, in that future is determined solely by the present, not the past. A reversible Markov chain not only has no history, but time has no direction.

Here is a draft of a book by Aldous and Fill on the theory of reversible Markov chains.

Eugene Lim went to Haiti after the earthquake to teach calculus after the earthquake. He has posted a first-hand account at Cosmic Variance.

Mark Chu-Carroll, the math blogger at ScienceBlogs, has quit the site, after ScienceBlogs made the bizarre decision to host a blog sponsored by Pepsi. The ensuing blow-up has caused ScienceBlogs to pull the Pepsi blog, Food Frontiers, but a version of it lives on at Pepsi’s own site.

Grothendieck categories are a generalization of categories of modules. Sheaves of abelian groups over a topological space also form a Grothendieck category. Grothendieck categories are a special case of abelian categories, but the extra structure allows additional theorems to be proved.

Grigory Garkusha has a thorough introduction to the subject on arXiv.

snarXiv is a site the generates parody abstracts for high-energy physics theory papers, a la arXiv. While the abstracts don’t quite make sense, they eerily resemble the real thing.

snarXiv versus arXiv is another site that gives you a random snarXiv and arXiv paper title, and asks you to tell the fake from the real thing. The fake titles are much harder to recognize than the fake abstracts. Initially, I got the first 5 right, but after about 25 I was down to random chance.

Via Not Even Wrong.

I missed that Vladimir Arnold has died. Arnold was famous for his own contributions to mathematics, but in my opinion he was also the world’s great expositor of mathematics.

When I first encountered the subject of Lie algebras, I thought it was pointless and unmotivated. I also had the impression from high school physics that classical mechanics was built out of a bunch of random facts that were true for no reason, like the conservation of angular momentum. Also, I thought that potential energy was a sort-of a con — that if you can simply declare that a body has potential energy that you can make the law of conservation of energy tautologically true. Reading Arnold’s Mathematical Methods in Classical Mechanics changed all that. Arnold starts with one-dimensional systems like the inverse-square law and harmonic oscillator, and then to three-dimensional systems where he explains how symmetries in the equations of motion lead to conservation laws. Along the way, he explains how Lie groups lead to Lie algebras, and how in particular how rotational symmetries in 3d lead to the Lie algebra of so(3), which physicists use in the guise of the cross-product of vector calculus. He also introduces the Lagrangian and Hamiltonian formulations of classical mechanics. Most importantly, (since you can learn the equivalent from a physics text like Goldstein’s Classical Mechanics), he puts in the language of mathematicians rather than the language of physicists.

Years after I studied the subject of ODEs, I almost bought Arnold’s (expensive) Ordinary Differential Equations just because it was such a beautiful introduction to the subject. Lots of textbooks allude to the dynamical systems viewpoint for ODEs, but his book really communicates that viewpoint.

The main challenger to the incumbent party in Columbia the former mayor of Bogota, Antanas Mockus. As this profile make clear, Mockus is a man with a flair for the dramatic. According to the profile, he apparently once mooned an auditorium full of students. While mayor, he would occasionally dress up as a superhero named “Supercitizen”.

Intriguingly, the profiles list his job description as a “mathematician”, but they don’t really make clear what this means.

I learned many new things from James Iry’s brief history of programming languages. For example, while I’ve used Lisp for some time now, I had no idea of how it all began:

John McCarthy and Paul Graham invent LISP. Due to high costs caused by a post-war depletion of the strategic parentheses reserve LISP never becomes popular. In spite of its lack of popularity, LISP (now “Lisp” or sometimes “Arc”) remains an influential language in “key algorithmic techniques such as recursion and condescension”

Condescension has never made my programs run faster, but it’s what makes writing them worthwhile.

I was looking for an introduction to the topic of random matrices, and I came across this survey article by Edelman and Rao on the subject. It considers a somewhat broader point of view than just results on the random distribution of eigenvalues, which are the most famous results in the subject.

One thing I found interesting is that you can explicitly calculate the Jacobian of various matrix decompositions as nonlinear functions of the matrix entries. They use this to help explain results on the random distribution of eigenvalues. More on this approach can be found in Edelman’s thesis.

This is the 5th year anniversary of this blog. The list of things I’ve managed to do for 5 years is very short, but this is one of the things on it.