A story of Google, Wikipedia, and languages I only sort of read

The Viquipèdia article "Mathemàtiques" has a bunch of amusing pictures that are meant to be icons of different types of mathematics: a Rubik's cube for abstract algebra, a Koch snowflake for fractal geometry, the Lorenz attractor for chaos theory, dice for probability, an elliptic curve for number theory, and so on.

Some areas don't translate into pictures as well: for category theory they have a commutative diagram, for combinatorics the six permutations of [3], etc.

Also, here are Representacions matemàtiques de diversos camps. (The English version of the article does not have this picture currently; they have a picture of Euclid.)

Much of the article seems to be a straight translation of the English version, but I find myself focusing more on the pictures when reading the Catalan version, because I don't actually read Catalan. But I read French and, to a lesser extent, Spanish, so I can figure things out.

(As to why I'm looking at the Catalan wikipedia -- well, I ended up there because Kowalski said he googled 1.70521 when it appeared in some of his work, and I wanted to see the results.))

Such a search finds Wikipedia articles, usually tables of mathematical constants, in Serbian, English, Esperanto, Catalan, Japanese, Thai, Czech, Turkish, Japanese, Serbo-Croatian, and Bosnian. This is both an illustration of the universality of mathematics and the extent to which Wikipedia is an international enterprise. (My apologies if I misidentified any of these languages!)

But the first hit upon Googling 1.70521 (upon this writing) was Kowalski's post, though it's twenty minutes old. That shows you how fast Google is at indexing. (By the time you read this, who knows? This post might be the first hit.)

Weather and political polls

From a Philadelphia TV weather man, Glenn "Hurricane" Schwartz, upon observing that the low and high temperature for Philadelphia today (70 and 85, respectively) were the "normal" temperatures for the day:

"Exactly normal! It's not normal to be exactly normal!"

which, of course, is true. That's how distributions work.

The weather people on television try a lot less to explain why the weather did what it did than the political people; John and Zeno have talked about this "roller-coaster polling". I suspect this is because once the weather has happened we don't care why it happened that way, while the whole point of polls is to use them to forecast the upcoming election.

Who's doing election prediction by simulation

Here's a list of all the web sites I know of where one can find simulations of the upcoming 2008 US presidential election. (I'm going to stay out of this game, because who wants to track down polling data every day, re-run the simulations, and so on?) Generally these start by obtaining a probability that one candidate or the other will win each state, from polls and sometimes from demographic data as well. These are then in some way aggregated in "simulated" elections. Often these are accompanied with a probability that Obama will win the election, or would win it if it were held today, and in many cases a probability distribution of the number of electoral votes won by Obama.

Yes, Obama. Not McCain. People who create these sites generally have a choice to make -- since nobody other than Obama and McCain has any reasonable chance of getting electoral votes, stating everything from the Obama point of view or from the McCain point of view has the same content -- but it seems that there's a leaning towards choosing Obama for this purpose in this corner of the blogosphere. (This is an entirely unscientific sample, though.)

Note that although Sam Wang says what he does isn't simulation, that's only because his method allows him to do all the possibilities at once. This is because he uses the magic of generating functions. This trick only works if you can make the simplifying assumption that winning in each state is independent of each other state. This is reasonable if you're trying to predict what would happen if the election were held today -- there's not any big reason for sampling error in different states to be correlated. But if you're trying to predict what will happen in the actual election, this assumption is very risky. It seems that the actual movement of voter opinions in different states should be correlated.

Here's the list:

• Stochastic Democracy (David Shor)
• Race to 270 (Ben Schak)
• Election-Projection.Net (If you want to do your own simulations, they have an interactive calculator.)
• FiveThirtyEight.com (Nate Silver)
• Princeton Election Consortium (Sam Wang)
• 270towin.com 2008 Election Simulator

This list is by no means complete.

Worms doing calculus?

Worms do calculus to find food. (Um, not really.)

But apparently worms use salt concentration to find food, and tend to head in the direction of the gradient of salt concentration. That is, they go where there's more salt. This is due to neuroscientist Shawn Lockery and his students at the University of Oregon. I think the paper is the following:

Suzuki H, Thiele TR, Faumont S, Ezcurra M, Lockery SR, Schafer WR (2008). "Circuit motifs for spatial orientation behaviors identified by neural network optimization." Nature 454:114-117.

but I can't be 100 percent sure -- Penn's libraries don't allow access to the electronic version of papers from Nature until twelve months have passed, and I'm not on campus right now. (This is, however, the only paper on Lockery's list with a title fitting the description.)

Saying "worms do calculus to find food" seems a bit disingenuous to me, though. It seems like saying that baseball players do calculus to catch fly balls. The larger point, though, is that neural processes -- of worms or of humans -- can be modeled using mathematical techniques, which may be of use to people trying to develop artificial systems that do these things.

(From John Scalzi, via 360. Apparently this first appeared in blogs a couple weeks ago, but I'm posting it here anyway, because it's new to me, which means it's probably also new to a lot of you.)

Continued fractions and baseball hitting streaks

The title of this post is misleading, because you might think there's a substantial connection between its two halves. The connection is only that I happened to come across both of these things this morning and it seemed silly to make separate posts about them.

Todd Trimble gives two proofs of the continued fraction expansion for e, namely e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, ...]. This is in the usual notation for continued fractions, so it actually means

e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + ...)))))

but all the extra 1s obscure the pattern. This is one of those things that it's much easier to state than it is to prove. And I must admit it's always seemed a bit strange to me that e has such a nice continued fraction expansion, while π doesn't -- e has a special place in continued-fraction land.

From the arXiv, via the physics arXiv blog: A Monte Carlo Approach to Joe DiMaggio and Streaks in Baseball, by Samuel Arbesman and Steven Strogatz, which is what it sounds like. This expands upon this piece in the New York Times on the same subject, which I wrote about back in March. And no, I'm not sure why it's in a physics category at the arXiv.

Fox Sports' World Series odds

At this article: Fox Sports expects at least 1.5 teams to win the World Series this year.

No, I'm not making this up.

They give odds for fifteen Major League Baseball teams to win the World Series, ranging from the Marlins at 40 to 1 to the Angels at 5 to 2. I'll just give the top five teams here: Angels 5:2, Yankees 3:1, Cubs 7:2, Red Sox 4:1, Mets 8:1. So they figure that the Angels has probability 2/7 of winning the World Series, the Yankees 1/4, and so on.

Add up their probabilities for the top five teams: 2/7 + 1/4 + 2/9 + 1/5 + 1/9 = 449/420, which is greater than one. If you add up the probabilities for the fifteen teams they give probabilities for, you get about 1.506.

I'm not arguing with their ranking of the teams. (There might be things to argue with, but contrary to what you might come to believe in the next two months, or to what longtime readers might remember from last August and September, this isn't a baseball blog.) But a bookie that offered these odds would be broke pretty quickly.

Also, about the Phillies (to whom they give 16 to 1 odds to win at all), we're told that "Odds say they can't mash their way in two years in a row." Okay, it may be true that they can't rely on just their offense. But the fact that they did it last year really has no bearing on whether they can do it this year. If anything, the fact that they did make the playoffs last year is evidence that their offense is enough.

Bourbaki and the AMS

An interesting fact from the August 2008 Notices of the American Mathematical Society (in the article "From the AMS Secretary", by John Ewing, which is mostly a history of the AMS): Nicolas Bourbaki is the only mathematician known to be denied membership in the AMS. Bourbaki was a member of the Société Mathematique de France and applied for reciprocal membership. But Bourbaki was denied membership in the AMS, basically on the grounds that Bourbaki was neither an individual nor an institution.

(Incidentally, tortured grammar in this post can be blamed on the fact that it's not clear whether I should use singular or plural pronouns when referring to Bourbaki.)