links for 2007-10-07

links for 2007-10-07

From The Economist, via Statistical Modeling, etc.: the durations of people's periods of physical activity follows a power-law distribution, and the distributions are much different in depressed people than in healthy people. The article claiming this, oddly enough, was in Physical Review Letters -- this seems a bit strange to me, if only because the referees that journal would use are likely physicists!

(The original article by Toru Nakamura, Ken Kiyono, Kazuhiro Yoshiuchi, Rika Nakahara, Zbigniew R. Struzik, and Yoshiharu Yamamoto is Universal Scaling Law in Human Behavioral Organization; it looks like you've got to pay for it.)

From Slashdot: Science and the Islamic world—The quest for rapprochement:

It was not always this way. Islam's magnificent Golden Age in the 9th–13th centuries brought about major advances in mathematics, science, and medicine. The Arabic language held sway in an age that created algebra, elucidated principles of optics, established the body's circulation of blood, named stars, and created universities. But with the end of that period, science in the Islamic world essentially collapsed. No major invention or discovery has emerged from the Muslim world for well over seven centuries now. That arrested scientific development is one important element—although by no means the only one—that contributes to the present marginalization of Muslims and a growing sense of injustice and victimhood.

As you may remember, these are the people who invented algebra. And algorithms. (It wasn't Al Gore.)

Mark Dominus (who now lives in my neighborhood) analyzes a program he wrote many years ago about van der Waerden's problem. We can define a function V(n,C) such that if we color the first V(n,C) integers with C different colors, there's guaranteed to be a sequence of n integers, in arithmetic progression, which are all the same color. The problem is that the bounds the proof of the theorem gives are ridiculously huge; according to the theorem V(3,3) ≤ 5.79 · 10¹⁴⁶¹³. In fact V(3,3) = 27 -- there are ways of coloring the integers from 1 to 26 red, yellow, and blue so that there's no monochromatic arithmetic progression of length three, but not of coloring the integers from 1 to 27 under that same condition. A while ago I looked at this from a probabilistic point of view -- one can imagine doing an exhaustive search of which sequences of length 1 have no monochromatic sequences of length n, then the average number of ways of expanding each of those to a sequence of length 2, then length 3, and so on; multiplying those averages together gives an estimate of how many ``good" sequences exist of any given length. As the sequences get longer, those numbers should grow and then shrink; the length at which they shrink below 1 is probably a good estimate of V(n,C). Unfortunately I can't find my notes on this.

A pair of posts on the geometry of parallel parking: from The Everything Seminar and Rigorous Trivialities. The second of these proves that it's always possible to parallel park a car, assuming that the spot to be parked in is strictly larger than the car and the driver can make arbitrarily small movements. The next logical question is: how many such movements, and how much total movement of the car, does it take to do so?