ScienceNow has an article about a new result in number theory on the distribution of arithmetic progressions of primes, that is, sequences of primes who differ by a constant number from step to step. The reason I post it here is they used ergodic theory to prove the result, although the peer review jury is still out on its correctness. That’s the problem with mathematics sometimes — you get stuck in your own esoteric corner and it’s hard to validate results.
Tag Archives: math
π day
It was pointed out to me that yesterday was π Day (3/14), which only works if you use the American system of dates, month/day/year, rather than the resolution scale model of day/month/year used by everyone else it seems. A bunch of Berkeley students decided to celebrate by chalking some few hundred digits of π on the sidewalk, extending from the math building north to the engineering buildings. It came as a surprise to some that I found it not very reminiscent of MIT. I’m not sure exactly what made it ring false. Perhaps it was too cute, or not esoteric enough. Perhaps it was the fact that you couldn’t possibly chalk the digits of π into the sidewalk at MIT in the middle of March since there might be snow on the ground, as opposed to the currently sunny and 82 degree weather in Berkeley. But whatever it was, it felt silly and just the thing for the week before Spring Break.
Oh, and happy Ides of March everyone. Make sure to warn your local emperor.
betting on hamlet
The Times Literary Supplement has a funny discussion (via MeFi) on how to analyze the odds on the Laertes-Hamlet dueling matchup. A little close reading plus some probabilistic analysis can only lead to the conclusion that one shouldn’t take chances on definitive interpretations of Shakespearean texts.
no longer a travesty
Thankfully, this blog is no longer the number one hit for “ergodicity” on Google. But in the event that people come here anyway, I present some definitions, courtesy of Wikipedia, Richard Durrett, and others.
The ergodic hypothesis says that averaging over time and averaging over the statistical ensemble are the same. So let’s say I have some box spitting a random number every second. If the random process controlling the box is ergodic, then I can find certain quantities — for example, the average — by either averaging the observed variables that I see, or by calculating the “theoretical” average from the statistics governing the box.
We would, of course, like most real-world systems to be ergodic, since we can then measure them and make estimates based on the measurements. The hope is that these estimates will (in the limit as you get infinite data) converge to the “real” value. Of course, this leads to an existential bind, because we have no idea if there is a “real” underlying value.
It’s a tricky thing, ergodicity, and getting to the bottom of it reveals a lot about how we view the randomness in our world, the assumptions we make on it, and how we try to control it.
Martingales
The “martingale” gambling system works as follows: start betting with $1. if you lose, double your bet on the next game. That way, when you win, you recoup all of your losses plus one dollar more. Seems like a foolproof way of beating the house, as long your chance of winning is nonzero, right?
Think again.
Actually, thinking isn’t enough. You have to take a graduate course in theoretical probability, it seems.
Random events in the last month: I saw Much Ado About Nothing at CalShakes, I went to the Radiohead concert with Manu Seth, who had 5th row tix, went home for my brother’s wedding, found an advisor and an office, sang in a concert where I dressed up goth, saw the Gotan Project, and won a raffle that gets me a free pint a day at the campus pub until Summer 2004.
An Ergodic Walk 