Saul Youssef has a collection of links to papers on exotic variations to probability theory. These are forms of probability theory that share many of the usual axioms of probability theory but in which the probabilities themselves lie in a set other than the non-negative reals eg. the complex numbers, the quaternions, or even the p-adics. The primary motivation is that classical mechanics plus complex probabilities looks a lot like quantum mechanics, and so if you believe in complex probabilities you no longer have to worry about things like wavefunction collapse. Unfortunately it’s all a bit confusing if you’re a frequentist.
The Muddy Children Problem
Many mathematicians grew up on a diet of puzzles like those set by Raymond Smullyan and Martin Gardner. Unfortunately, ingenious and elegant as these puzzles often are, they frequently have solution methods that don’t give rise to generalisable theory.
So I was recently surprised to find that one of my favourite puzzles of this type, the Muddy Children Problem, is actually an important example that appears in courses on mathematical logic, epistemology, computer science and even quantitative finance. If you haven’t met the problem before then have a go at solving it before looking at the various papers and courses on the subject. A fairly detailed elementary treatment can be found here though there are easier to understand informal arguments in existence.
The main academic approaches to the problem are via modal logic and Kripke models.
In less politically correct days it was known as the unfaithful wives problem and Smullyan’s version of this problem involved logicians with coloured hats.
Did I mention that it’s also a drinking game?
Sigfpe guest blogging
I’m going to be far from the Internet starting on Monday, so I’ve asked sigfpe — of comment board and Neighborhood of Infinity fame — to help fill in for me.
Rick Miranda on algebraic curves and surfaces
Rick Miranda has written the clearest introduction to the algebraic geometry’s linear systems I’ve ever seen, in his article Linear Systems of Plane Curves from the February 1999 Notices of the AMS. He’s also written a much more technical article on algebraic surfaces which provides a good summary of the different classes of the Enriques classification.
Multiple polylogarithms and algebraic cycles
I spotted this paper, Multiple polylogarithms, polygons, trees and algebraic cycles, on ArXiv. It relates the values of certain iterated integrals to incredibly complex and abstract objects in algebraic geometry.
Chris Peters on algebraic surfaces
Chris Peters has written two introductions to the classification of complex algebraic surfaces: a long version and a short one. The long version introduces the necessary background in complex manifolds and sheaf cohomology, while the short one skips right to the Enriques classification.
Ponder This
It’s the beginning of the month and the solution to last month’s Ponder This challenge is up, as well as the puzzle for August:
For K as large as possible, produce a K-digit integer M such that for each N=1,2,…,K, the integer given by the first N digits of M is divisible by N.
An example is K=4, M=7084, because 7 is divisible by 1; 70 is divisible by 2; 708 is divisible by 3; and 7084 is divisible by 4.
I guess that the largest K is around 28.
Introductions to PDEs
While I was looking for information on the Lewy equation, I found some introductory material on PDEs:
- Lectures notes by T. W. Körner
- Lecture notes by Joshi and Wassermann
- The course page for Methods of Numerical Simulation in Fluids and Plasmas, which includes a detailed notes on PDEs.
- The course page for Computational Methods, which includes lecture notes on PDEs and other topics.
- An essay on PDE as a unified subject
Lewy equation
The Lewy equation is an example of an inhomogenous linear partial differential equation that has no solutions. Note that we’re not imposing any boundary-value or initial-value conditions on the equation; the equation simply has no solutions. The proof that it has no solutions is a surprisingly simple application of complex analysis. (Also available in postscript.)
The paper Fifty years of local solvability surveys the development of the theory (known as local solvability) in the wake of Lewy’s discovery. Numerical linear algebra and solvability of partial differential equations describes an analogy between local solvability and numerically computing matrix eigenvalues.
AMS Summer Institute in Algebraic Geometry
AMS Summer Institute in Algebraic Geometry is underway in Seattle. It’s a mammoth three-week conference on algebraic geometry. The first week is dedicated to the unlikely connections that have emerged between algebraic geometry and string theory.