This article in the Washington Monthly tells a startling story of a conflict between the legal system and probability. DNA tests use a certain number of genetic markers to match DNA samples. The samples are well-short of a complete sequencing, which means any randomly selected person has a 1 in a million chance of matching. When DNA testing is used on a few suspects, then the chances of matching an innocent person are pretty low. But now, governments are compiling large databases of DNA and data mining them for matches, which radically increases the odds of matching an innocent person. This would not necessarily lead to false convictions as long as juries are made aware of the consequences, but at least one judge ruled evidence that a DNA match happened by mining a database inadmissible: the jury never heard it.

(In the particular case of the article, the victim was raped and murdered, and the defendant was found in a database of sex offenders. The article claims the correct probability of a match in this case is 1 in 3. It’s not clear from the article if this is the unconditional probability of being found in a database of that size, or the conditional probability given that it was a database of sex offenders.)

Via Quomodocumque.

Surprising nobody, the Clay Mathematics Institute has announced it is awarding its Millenium Prize for the Poincare Conjecture to Perelman.

Via Peter Woit.

Jean-René Chazottes has an extensive list of links for on-line books and lecture notes, on topics in ergodic theory, information theory, stochastic processes, and statistical mechanics.

Via Cosma Shalizi.

I was curious to see the proof for this theorem, so I’ve started reading Lukac’s Characteristic Functions. It’s a good book, but as written expresses absolutely no interest in the subject of probability. While about a subject that is interesting because of its connection with probability — a characteristic function is the Fourier transform of a probability distribution function — the discussion is entirely in terms of characteristic functions. For example, studying the characteristic functions for infinitely divisible distributions is motivated by the question of which characteristic functions have the property that every n-th root is also a characteristic function.

I’m fascinated by this this central limit theorem for infinitely divisible distributions. I knew about two special cases, the usual central limit theorem, and the result that a suitable limit of Bernoulli random variables gives you the Poisson distribution, but I didn’t know that there was a general theory about the phenomenon.

As I mentioned last week, Tim Lambert has been calling out science journalists by name for flaws in their coverage. Lambert has been particularly hard on the Sunday Times. For his efforts, he has been attacked by name as part of a longer article on the evils of science blogs.

This article by Mainardi and Rogosin gives the early history of infinitely divisible distributions. It includes a translation of Khintchine’s proof of the Lévy-Khintchine formula expressing the possible forms for the characteristic function of an infinitely divisible distribution.

Tomasso Dorigo has a new, non-technical, post explaining muon decay. Tomasso characterized his description as “although not perfectly correct in the physics, was actually not devoid of some didactic power.”

Simon Willerton has written an introduction to the Weyl tube formula.

More details can be found in this review of Alfred Gray’s book on the subject.

In ordinary calculus, you define “integral derivatives” — the first derivative, second derivative, etcetera. If you think of differentiation as an operator D that takes functions to functions, then the higher-order derivatives are just Dⁱ for natural numbers i. As far back as Liouville, mathematicians have defined fractional derivatives, extensions of this definition to real numbers, i. There is more than one possible definition, Wikipedia page gives the usual definition, which is in terms of the Laplace transform.

I’ve never known if fractional derivatives were good for anything, or were just a historical curiosity. (They are a special case of singular integral operators, which are useful in PDEs.) This very brief paper discusses an application of fractional derivatives to models of particles in a liquid. This sounds like it should be related to Brownian motion, and it is, but the processes that arise are related to more general Lévy processes than just Brownian motion.