Distribution of Putnam scores

The distributions of Putnam exam scores are interesting. See, for example, the 2001 distribution. It takes a bit of number-crunching to get an actual distribution of scores from the data; they report the "rank" of the people getting each score. The rank corresponding to a given score is, I assume, A+(B+1)/2 where A is the number of people scoring higher than that score and B is the number of people scoring that particular score. For example, in 2001 -- which happens to be one of the years in which I took the Putnam -- the table begins

Score	101	100	86	80	79	77	73	72	71	70	69	68
Rank	1	2	3	4.5	6	7.5	9	11	14	16.5	19	23.5
Number	1	1	1	2	1	2	1	3	3	2	3	6

where the first two rows are provided by the organizers, and the third row can be worked out by working left to right. For example, once we know 17 people got 70 or better, the fact that the score 69 corresponds to rank 19 means that the people scoring 69 must have been the 18th, 19th, and 20th-best; so there were three of them. (Incidentally, most increasing sequences of half-integers, when interpreted as sequences of ranks, don't appear to correspond to legitimate score distributions; the number of people getting certain scores ends up negative if you're note careful.)

Anyway, if you crunch the numbers on a typical Putnam score distribution you observe two things:
- the scores follow, roughly, a power law; the number of people scoring 10n decays like some power of n, for integer n.
- once you remove this decay (which I haven't actually done; I've just eyeballed it), there are "spikes" at multiples of 10. For example, the number of people scoring 18, 19, 20, 21, 22, 23 in 2001 were 8, 23, 99, 60, 39, 11. Twenty-four people scored 50; seven scored each of 49 and 51.

I can't explain the first one (and it may just be an artifact of the way I'm doing the plotting; lots of things look close to linear when plotted on a logarithmic scale). But the second one is actually easy to explain; Putnam problems are worth ten points each, and most scores are 0 or 10 with a smattering of 1, 2, 8, or 9. Scores between 3 and 7 on a problem are exceedingly rare. So to get a score of, say, 55, one has to get five problems right and have made a bit of progress on three to five more, which is less likely than straight-out solving five or six problems (for 50 or 60, respectively).

Incidentally, I haven't looked at the problems from the 2009 Putnam, because I have work to do.

Math Overflow

Because I'd have to forfeit my math blogger card otherwise: you should know about Math Overflow This is a site where people can ask mathematics questions -- the level is basically that of questions which would be of interest to professional mathematicians. I'm rather enjoying it; it's procrastination and education at the same time!

It is also useful for asking questions and being told that I already answered them on this blog.

Evidence that mathematicians have a big Internet presence

If you google genealogy, the first hit is The mathematics genealogy project. In some sense, according to "the Internet", mathematical genealogy is more important than real genealogy! (I have a feeling that the biological parents of mathematicians would be offended by this, so I won't tell my parents.)

If you google AMS, the first hit is the American Mathematical Society. (Societies of meteorologists, musicologists, Montessori schools, etc. show up further down the list.) I have a musicologist friend that I joke with this about, claiming that the mathematical society is the real AMS.

I suspect this is because mathematicians found the Internet early, and seem to be more likely to have personal web pages than even most other academics.

Edit, 4:18 pm: In a comment by Boris, I'm reminded that Google personalizes search results, so what I've said is not true.

Is zero even?

Did you know that there are actually things to say about whether zero is even or odd (from Wikipedia)? Obviously it is, but the math-ed folks have seriously looked at this.

I found this via a comment by John Thacker at The Volokh Conspiracy. There's a poll there; right now 2% of people have said 0 is odd, 51% even, 43% both, 4% neither. I can kind of understand what's going on with people saying "neither" (perhaps they're getting this from some elementary-school notions), but how is 0 odd?

My answer: yes, zero is even, because it's twice an integer.

(Or because the identity permutation on n letters is an element of the alternating group A_n -- I've been thinking about permutations a lot lately. But if you understand that, you probably are like me, think zero is even, and didn't even think there was anything to discuss.)

Incidentally, sometime recently -- I forget the context -- I saw something that referred to the Gaussian integer a+bi as "uneven" if and only if a and b had different parity.

Economic impact of mathematics?

Tim Gowers wrote in The Importance of Mathematics:

If you were to work out what mathematical research has cost the world in the last 100 years, and then work out what the world has gained, in crude economic terms, then you would discover that the world has received an extraordinary return on a very small investment.

I don't doubt this. But has anyone actually tried to do this? (And would the numbers even be meaningful?)

Eponyms in mathematics

Let S be the standard Smith class of normalized univalent Matcuzinski functions on the unit disc, and let B be the subclass of normalized Walquist functions. We establish a simple criterion for the non-Walquistness of a Matcuzinski function. With this technique it is easy to exhibit, using standard Hughes-Williams methods, a class of non-Walquist polynomials. This answers the Kopfschmerzhaus-type problem, posed by R. J. W. (“Wally”) Jones, concerning the smallest degree of a non-Walquist polynomial.

This fake abstract of a paper is from Merv Henwood and Ivan Rival, Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be? (PDF), from the Mathematical Intelligencer in 1980. It sounds to me like slight caricature -- but only slight. Henwood and Rival point out that such names are lazy. Names have at least two important functions -- to describe and to label -- and eponyms only label.

Perhaps such abstracts would be more common in areas which are small enough that all the major players talk to each other. I imagine that Smith, Matcuzinski, Walquist, etc. know each other.

Also of interest is David Rusin's list of eponyms occurring in the MSC classification. These names in general seem a bit less obscure than the names one would find in the abstract of a random paper, which isn't surprising as they're names of concepts big enough to get areas named after them.

(And can someone confirm or refute the story that Banach, in the paper in which he introduced Banach spaces, called them "spaces of type B" in an effort to get them named after himself? I've heard this one a few times but always unsourced.)

Steen on mathematics and biology

Here's a fascinating article on what math is good for in biology: The "Gift" Of Mathematics in the Era of Biology, by Lynn Arthur Steen. Steen gives lots of examples about what math is good for in biology. Somewhat surprisingly to me, he doesn't really mention one of the first things that came to mind, namely the use of combinatorial techniques to study the genome, which is nothing but a word on a four-letter alphabet. It's possible that he subsumes this in "statistics", though; to take a simple example, one might want to know how many times a certain sequence of bases would appear in a "random" genome in order to determine whether the fact that such a pattern appears often is signal or noise. Still, he makes the point that while the traditional mathematics curriculum (with lots of calculus and differential equations) takes its scientific inspiration from physics, biology is ascending.

A shorter version of this article is available at The Chronicle of Higher Education.

(How did I find this? Steen was one of the authors of Counterexamples in Topology, which I mentioned yesterday, so I went over to his web site.)