Organization of papers

I'm currently attempting to organize a paper out of a bunch of notes I've built up recently; a possibly useful suggestion I received is to write each theorem, definition, etc. on an index card, so that I can physically move them around to figure out how the paper should be organized.

Of course, definitions have to come before the theorems that use them, some theorems use other theorems in their proofs, and so on -- so to the extent that I'm remembering to do so, I'm indicating these sorts of dependencies on the index cards as well.

It occurs to me that what I am doing here is trying to extend a partial order (the ordering that comes from the dependency) to a total order. There are of course constraints on this order; certain results, although not logically related, are related in some philosophical sense and should perhaps be kept near each other. It's actually an interesting optimization problem.

Now if only I were writing a paper about extending partial orders to total orders...

(But my paper does talk quite a bit about permutations. And a total order will end up being a permutation of my index cards.)

Aldous on probability and the real world

In April, David Aldous gave a talk on the top ten things that math probability says about the real world.

My favorite (i. e. the one I hadn't really thought of before) was that news headlines are only interesting in real time. If you woke up twenty-five years from now and decided to read the last 25 years worth of news headlines, you probably wouldn't learn much. It's a good bet that at some point in that period the stock market had a really bad day, and will it matter, in 2033, that on September 29, 2008 the Dow lost 777 points? The trends are more important, but you can't see them from looking at individual headlines; you can only see them by looking at how many of different types of headlines there were.

Compare, for example, the fact that it's difficult to determine who's good at hitting in baseball by watching games; the difference between a good hitter and a poor one might be, say, one hit every five games, which is too small to measure without actually counting how many times they do various things.

Erdos and grapefruits?

From Paul Graham, How to Start a Startup (and no, I'm not looking to):

People who don't want to get dragged into some kind of work often develop a protective incompetence at it. Paul Erdos was particularly good at this. By seeming unable even to cut a grapefruit in half (let alone go to the store and buy one), he forced other people to do such things for him, leaving all his time free for math.

I don't use this strategy. But I do use the strategy of cooking things in ridiculously large batches, which isn't much more work than small batches; then when I want something to eat I just fire up the microwave.

(I'm not sure if I can cut a grapefruit in half; to do that without struggling might require some special knife I don't have. I haven't tried, because I don't like grapefruit.)

Best-seller impatiemment attendu

From "Quelques contributions au carrefour de la géométrie, de la combinatoire et des probabilités", by Nicolas Pouyanne:

"Dans leur best-seller impatiemment attendu [28], P. Flajolet et R. Sedgewick offrent un développement approfondi de la méthode symbolique en analyse
combinatoire, auquel on pourra se référer."

That is, "In their impatiently awaited best-seller [28], P. Flajolet and R. Sedgewick offer a detailed development of the symbolic method in combinatorial analysis, to which one will be able to refer."

I don't laugh often while reading mathematics, but this made me laugh, because I am among those impatiently awaiting this book. I've seen citations to Flajolet and Sedgewick's book Analytic Combinatorics in things written as long ago as 1998 or so; Amazon.com says the book will be out on December 31, 2008, and the publisher, Cambridge University Press says December 2008. It can be downloaded from Flajolet's web site.

Edited, January 22, 2009: my copy of this book arrived yesterday.

Ranking graduate departments

On Some Random Webpage Full Of Spam, I came across what purports to be US News' 2009 ranking of graduate programs in mathematics. (I feel bad about linking to this, because it just helps the spammers, but I'm doing it anyway.)

If I remember correctly, this ranking is produced by surveying people in mathematics departments at various schools and asking them to rank other institutions. That's it.

It seems to me that a more sensible way of ranking mathematics departments would be to start with the assumption that a better department is, by definition, one which has its students get hired by better departments. This could work, at least to do the high end of the ranking, because the departments doing the hiring are often the same departments that have graduate students; eventually I expect the process I'm alluding to would converge on a ranking. I'm not sure what you'd do to deal with ranking "lesser" programs where many students are not hired by departments which themselves have doctoral programs. This just aggregates what people think about reputation, but in a way that's more principled than just asking some shadowy panel thinks.

Of course, in the end these sorts of rankings are not particularly valuable, so I don't want to pursue this any further.

(Disclaimer: my memory of how these rankings work comes from standing in a bookstore four years ago and copying down that year's rankings on a scrap of paper. I think the statue of limitations has passed on that, so I'll admit to it.)

In which I tie baseball to math, sort of

One of our first years: "Baseball is made by God. Other sports are made by man."
Me: "So baseball is like the integers?"

A surprise from differential equations

A problem which circulated among the grad students here at Penn today, which came up while somebody was teaching differential equations:

Consider the differential equation dy/dx = (y-1)². Separate variables and integrate both sides as usual; you get y(x) = 1 - 1/(x+C), where C is determined by the initial condition. Take the limit as x goes to infinity, and you get lim_{x -> ∞} y(x) = 1, regardless of C.

But now notice that dy/dx is positive for all y. (We're working over the reals here.) So if the initial condition is of the form y(x₀) = y₀ for some y₀ > 1, then we start at 1 and keep going upwards; how can the limit be 1?

Of cousre there's a mistake somewhere in here. But where is it? (I know the answer, but it took annoyingly long to figure out.)

edit: the differential equation was wrong.