30 March 2011

How long is the coast of Maryland?

Last night's Final Jeopardy clue: "With 301 miles, it has the most coastline of current states that were part of the original 13 colonies." (Thanks to the Jeopardy! forum for the wording.)

This agrees with the Wikipedia list which is sourced from official US government data.

But as Mandelbrot told us, coastlines are self-similar . (Link goes to the paper How Long is the Coast of Britain as reproduced on Mandelbrot's web page, which unfortunately doesn't have pictures. I'm not sure if the original version of this paper in Science did. Wikipedia's article on the paper does.) That is, the length of a coastline depends on the size of your ruler. Furthermore, I would suspect that the fractal dimension of some states' coastlines is larger than others. Wikipedia states that "Measurements were made using large-scale nautical charts" which seems to imply that all the measurements were done at the same length scale, but if you did the measurements at a smaller scale, the states whose coastline have higher fractal dimension would move up the list.

So last night I spent twenty seconds yelling this at the TV, and ten seconds getting out the answer Maryland. Which is wrong. Also wrong: Maine, New York. (Maine used to be part of Massachusetts; the wording is a bit ambiguous.) It appears that only the south shore of Long Island counts as "coast"; the north shore, which borders Long Island Sound, doesn't.

And of course Chesapeake Bay doesn't count either.

25 March 2011

Vallentin's probability cheat sheet

John Allen Paulos pointed me to Matthias Vallentin's probability and statistics cheat sheet. It's a big "sheet" -- twenty-seven pages -- but maybe you have a big blank wall to put it on.

(To my students, if you read this: remember that you only get one page of notes on the midterm, and you have to write it yourself.)

du Sautoy on symmetry

An interesting TED talk: Marcus du Sautoy on symmetry -- interesting to watch, lots of pictures; ignore the fact that it starts with the standard slightly overwrought version of Galois' story. If you want a more accurate version, I recommend Amir Alexander's Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics . About halfway through he gives a presentation of a proof that the two groups of order 6 are different. (One, the cyclic group of order six, is abelian; the other, the symmetric group on three elements, is not.) I particularly like the pictures of the Alhambra, on which du Sautoy has overlaid animations showing the effect of rotating them. Presumably they won't let you do this if you actually go there.

23 March 2011

Flajolet

From Claire Mathieu's blog: Philippe Flajolet, of analytic combinatorics fame, has died.

22 March 2011

Are food-borne pathogen survival times really exponentially distributed?

From Scientific American, an excerpt from Modernist Cuisine: The Art and Science of Cooking on the complex origins of food safety rules.

This is the six-volume, six-hundred-dollar magnum opus of Nathan Myhrvold (former chief technology officer at Microsoft, and chefs Chris Young and Maxime Bilet; you can read more about it at the Wall Street Journal.

In particular I noticed the following:
If a 1D reduction requires 18 minutes at 54.4 degrees C / 130 degrees F , then a 5D reduction would take five times as long, or 90 minutes, and a 6.5D reduction would take 6.5 times as long, or 117 minutes.

A "nD" reduction is one that kills all but 10-n of the foodborne pathogens.

What struck me here is that the distribution of the pathogen lifetimes, assuming these numbers are actually correct, is exponential. And, therefore, memoryless -- if you're a bacterium under these conditions, your chances of dying in the first eighteen minutes are ninety percent, and if you're still alive at ninety minutes, your chances of dying in the next eighteen minutes are still ninety percent. This surprised me. The decay of radioactive atoms can be described in this way -- but are bacteria really so simple?

The excerpt as a whole is quite interesting -- apparently a lot more than just science is going into recommendations of how long food should be cooked.

(Myhrvold has a bachelor's degree in math and a PhD in mathematical economics, among other degrees; Young has a bachelor's degree in math and was working on a doctoral degree before he left for the culinary world. So perhaps it is fair of me to think that they would get this right.)

18 March 2011

What is the origin of Kirillov's lucky number problem?

I've run across a few references to a Russian superstition as follows: a bus ticket has a six-digit number, and a ticket is said to be "lucky" if the sum of the first three digits equals the sum of the last three digits. See, for example, Lectures on Generating Functionsby Sergei Lando -- an excellent little book I accidentally discovered in the library a couple days ago. Lando gives the problem of figuring out how many lucky tickets there are, and says that in the early 1970s A. A. Kirillov would often open his seminar this way. The cataloging information in the book says that Lando was born in 1955; Olshanski writes, in his preface to Kirillov's seminar on representation theory, that many students attended Kirillov's seminar. I suspect that Lando actually heard Kirillov pose this problem at some point.

What I'm interested in is whether this is an actual superstition, perhaps held by non-mathematicians. It seems like the sort of thing that a mathematician would make up to create a good problem. This web page mentions the superstition in a non-mathematical context, with the twist that you're supposed to eat lucky tickets. This one does as well, and links to these people who sell lucky ticket cookies. It seems quite likely that:

1. the superstition exists among some segment of the Russian population, or at least did at some point, and
2. the entrance of the problem into the mathematical culture is due to Kirillov -- if I were still at Penn I'd ask him.

But did the problem travel from the rest of the world to mathematics? Or, because this sum condition feels so mathematical, did it travel from mathematics to the rest of the world?

(Also, can you solve the problem? How many lucky tickets are there?)

17 March 2011

In the library at Berkeley, there is a copy of Richard Stanley's book Enumerative Combinatorics, volume 2.

I think a few of you will be amused to learn that when I opened this book at random, it fell open to the famous exercise giving sixty-six interpretations of the Catalan numbers.

Devlin has an upcoming biography of Fibonacci

From Keith Devlin's twitter feed (@nprmathguy): he's meeting with his publisher about a biography of Fibonacci which will come out this July, entitled The Man of Numbers: Fibonacci's Arithmetic Revolution.

What surprises me most about this is that Devlin says that this is the first biography of Fibonacci. The St. Andrews' biographical page on him, for what it's worth, only lists two books among its sources. One is entitled Leonard of Pisa and the New Mathematics of the Middle Ages and the other Leonardi Pisani Liber Abbaci oder Lesevergn├╝gen eines Mathematikers -- these don't sound like biographies. I'm surprised because you'd think there'd be a built-in market for such a book -- everyone knows about the Fibonacci sequence. And you could put bunnies on the cover! It looks as if Devlin's publishers are more serious than I am, though, and have not done so.