Ars Mathematica

If you’ve ever wondered what a spin glass was, well here’s your answer: Spin Glasses for Pedestrians.

The site Game Theory .net has links to a truly gigantic collection of lecture notes on game theory and its economic applications.

I’ve just run across an interesting thought experiment known as Newcomb’s paradox. Suppose there is a being, called the Predictor, that presents you with a choice. There are two boxes. The first box may or may not contain $1,000,000. The second box always contains $1,000. You can choose to open either one box or both boxes. While you are making your choice, the Predictor does not touch the boxes in any way — whether or not the first box contains money is already determined.

Many people have encountered the Predictor before, and have discovered that he seems to always predict what you are going to do. Anyone who has ever chosen to open just the first box receives the $1,000,000. Any who has ever chosen to open both boxes finds the first box empty, and only receives $1,000.

Which would you choose?

Week 223 of This Week’s Finds in Mathematical Physics is up.

John Preskill teaches a course on Quantum Computation at Caltech, and has made his lecture notes available online.

The December issue of the Notices of the AMS is now available.

Here’s an interesting site: Jeff Miller’s Earliest Known Uses of Some of the Words of Mathematics, which offers an account of the introduction of various mathematical and statistical terms.

Auctions have provided a real-world arena in which to apply game theory. The theory has actually been applied to design auctions; most famously, the auctions for 3G wireless spectrum were designed along the principles of the theory.

Paul Klemperer has assembled several articles on the subject into a (fairly non-technical) book, and has provided the original articles online. For a more detailed approach, see this survey.

I’m not sure why, but this comment by Easwaran reminded me that the book Toposes, Triples, and Theories, by Michael Barr and Charles Wells, is available for downloading, if your vices run in that particular direction. A topos is a category-theoretic analogue of a set theory. The category of sets for a topos, but there many others. A triple (now usually called a monad) is a category-theoretic analogue of an algebra (in the sense of universal algebra). I don’t remember what a theory is.

I was wondering what the spatial analogue of a stochastic process was. It turns out that they’re known as either spatial processes or random fields, and there’s a book available on the subject online, Random Fields and Geometry, by Robert Adler and Jonathan Taylor.