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.
Monthly Archives: September 2008
Unreasonable Effectiveness of Mathematics
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.
Standard Borel Spaces
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.
Gaussian Quadrature
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.
Tremellius and Naibod
God Plays Dice has a post that answers a question I’ve long had about the Mathematics Geneology Project: just how far back can you go? The answer is 1572, when Immanuel Tremellius and Valentine Naibod advised Rudolph Snellius. Snellius was the father of Willebrord Snellius, who discovered Snell’s law.
Tremellius was a Bible translator who was briefly jailed for being a Calvinist. It sounds like he was forced to move frequently as the prevailing winds for Protestants changed. (This was the early Reformation.) Naibod was an astrologer who had a book banned by the Catholic Church. An astrological prediction told him that his life was in danger, so he tried holing up in his house until the danger passed. Since the house showed no external signs of life, thieves thought the house was abandoned and broke in. Discovering Naibod, they murdered him. Apparently astrology works after all.
The Geneology Project has a page dedicated to what it calls extrema. I would support a campaign to rename the Guinness Book of World Records the Guinness Book of Extrema.
Update. In between when I hit “Post” and now, the Mathematical Geneology site updated their database, making this post completely obsolete.
Everyone is Partial to PDEs
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.