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.
The best part about this site is that I can link to interesting articles that I don’t have time to read. One of the early inspirations for the study of dynamical systems was statistical mechanics, particularly the ergodic hypothesis. Things have now come full circle with applications of dynamical systems to statistical mechanics. Here’s a survey article on the subject by David Ruelle: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics.
As a follow-up to yesterday’s post, I came across an entry on symmetry breaking at the Stanford Encyclopedia of Philosophy. It’s interesting to see philosophy directly engaging with the implications of modern physics.
They also have two entries on the philosophical underpinnings of statistical mechanics: a general entry, and a more specific one on Boltzmann. The actual justification for statistical mechanics and the reduction of thermodynamics to statistical mechanics is apparently fairly controversial.
Sean at Cosmic Variance has written a nice introduction to symmetry breaking, one of the oddest (and most successful) ideas in particle physics.
The discussion about Goedel’s theorem made me wonder about this question: when was the last time a widely quoted mathematical result turned out to be false? I’ve seen preprints that had proofs I didn’t believe, and I know that journals occasionally print results that are wrong, but does anyone know of a result that was once widely accepted, but then turned out to be wrong?
I’m sorry if my last post misled anyone. Goedel’s theorem is true. Lots of people have checked the proof. I’ve checked the proof. There are multiple proofs (the usual proof is Rosser’s, not Goedel’s original proof), as well as vast generalizations.
Here’s a two line proof of the theorem (some details are left out):
At the time, Goedel’s result was very surprising, but by modern standards it’s almost obvious. The set of provable theorems of arithmetic is recursively enumerable. It’s not that surprising that first-order arithmetic would be at least as expressive as Turing machines, which implies that the set of provable theorems is not recursive. Goedel’s theorem follows.
I intended the post to be tongue-in-cheek. My real point was that a certain number of papers on Arxiv are apparently written by cranks.
According to a new paper on Arxiv, Goedel’s theorem is false. There you have it.
This paper, On the Euler angles for SU(N), gives a definition/derivation of Euler angles for SU(n). The paper claims that the construction first appeared in 2002. Does anyone know if that’s true? I thought for sure that I’d seen Euler angles defined for SU(n) (and SO(n)) sometime in the 90s.