Sean at Cosmic Variance has an interesting article on decoherence called Quantum Hyperion. It describes a paper by Zurek and Paz that calculates that if Saturn’s moon Hyperion were an isolated system, then within twenty years it would evolve into a non-localized quantum state. It is only the interaction with the outside world that keeps Hyperion looking like a moon.

I’ve been thinking about Lévy processes, a topic that I mentioned once before. A Lévy process is a generalization of both Brownian motion and a Poisson process. Brownian motion and Poisson processes are both continuous-time stochastic processes but have very different behavior. A Brownian motion follows a very jagged path that is almost always continuous. A Poisson process stays at one place for a long time, and then suddenly jumps to a new place. What they have in common is that changes over two disjoint time intervals are independent of each other, and if the two time intervals are the same the changes have the exact same distribution.

Lévy processes include generalizations such as various combinations of Brownian motions and a Poisson process, but they also include more exotic possibilities. The sample path of a combination of a Brownian motion and a Poisson process will almost always have only finite number of discontinuities. In general, Lévy processes can generate sample paths with infinitely many jump discontinuities in almost every interval. Over a finite time horizon, this can give rise to fait-tailed distributions such as the Cauchy distribution.

In lieu of actually finishing any of the half-finished posts, I thought I’d see if anyone has any requests for posts on particular topics. Anything on your mind?

Sorry for the light posting. I have fifty million half-finished posts that I haven’t been able to find the energy to finish, mainly because I spend all my time hitting “refresh” on financial news sites. I normally use mathematics to take my mind off the news, but this time it isn’t working. The last time I was this distracted from math was in the immediate aftermath of September 11th, and before that my own birth.

For the historically-minded physicist: Dyson’s 1951 lectures on Advanced Quantum Mechanics (quantum field theory) are available on arXiv. I looked them over, and they seem eminently understandable to someone who’s had a course on quantum mechanics.

Peter Woit observes that mathematicians and physicists had a prominent role to play in the current financial crises that threaten to take down the world’s banks. Stochastic calculus has become the main tool of evaluating financial derivatives, which are financial instruments whose payoffs are (usually nonlinear) functions of underlying assets. Overly-optimistic assumptions in pricing derivatives have led to large losses throughout the financial sector. The worst case scenario is the kind of widespread economic dislocation not seen since the Great Depression.

Hey, at least it means we have something to be more embarrassed about than Theodore Kaczynski.

I just spotted this article on arxiv: Some Notes on Standard Borel and Related Spaces. A standard Borel space is a set with a σ-algebra which can be realized as the set of Borel sets of a complete metric space. The paper is an attempt to describe the theory of standard Borel spaces with the minimum of reliance on metric or topological ideas.

Numerical integration is not really a field that you expect to have surprising theorems. Yet, the existence of Gaussian quadrature is in of itself surprising. In the elementary methods that you learn in calculus (such as midpoint rule or Simpson’s rule), you evaluate the function at regularly spaced nodes. A more effective technique is to choose so that you can integrate polynomials up to some degree exactly. The best choice? The roots of an orthogonal polynomial.

The proofs are elementary, and can be found in this note by John Cook.

I was searching a computerized card catalog for a book on PDEs. I accidentally hit return after just just typing the word “partial”. The first ten hits were all for books on PDEs. I just tried the same search on Amazon (restricted to books), and get almost the same result: 8 of the first twelve hits are for books on PDEs.