Gowers on the PCM

Here is an MP3 interview with Tim Gowers about The Princeton Companion to Mathematics, of which he is one of the editors.

It's $100. Who wants to buy it for me?

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.

Why devil plays dice?

Why devil plays dice?, by Andrzej Dragan, from the arXiv. I haven't read it; this post basically exists to forestall e-mails of the form "Have you seen the title of this paper?"

(Hat tip to The Quantum Pontiff.)

A couple of links

1. Jordan Ellenberg's review of Andrew Hodges' book One To Nine. Read the review, if only because it uses the word "mathiness". Ellenberg's review seems to imply that the book has similar content to most popular math books; sometimes I wonder how the publishing industry manages to keep churning out these books, but then I remember that the same thing is true in most other subjects and I'm just more conscious of it in mathematics.

2. Open Problem Garden, which is a user-editable (?) repository of open problems in mathematics. Thanks to Charles Siegel, my fellow Penn mathblogger, for pointing this out. The majority of the problems given there are in graph theory; that seems to be because Matt Devos, one of the most prolific contributors, is a graph theorist.

But I have to say that "garden" feels like the wrong word here; gardens are calm and peaceful and full of well-organized plants, which doesn't seem like a good way to describe problems that haven't been solved yet. "Forest" seems like a better metaphor to me -- certainly when I'm working on a problem that's not solved, it feels like hacking my way through a forest, not walking around a garden. Also, the use of "forest" enables bad graph theory jokes -- the problem of "negative assocation in uniform forests", due to Robin Pemantle, in particular sounds like it could be about sketchy people you meet in the woods.

(I gave a talk back in February where I mentioned this problem. I'm glad I didn't think of that joke then, because it's really bad and I would have just embarrassed myself.)

Li's proof of Riemann?

A proof of the Riemann hypothesis, by Xian-Jin Li.

I'm not qualified to judge the correctness of this, but glancing through it, I see that it at least looks like mathematics. Most purported proofs of the Riemann hypothesis set off the crackpot alarm bells in my head; this one doesn't. Li has also stated Li's criterion in 1997, which is one of the many statements that's equivalent to RH, although I don't think it's used in the putative proof, and wrote a PhD thesis titled The Riemann Hypothesis For Polynomials Orthogonal On The Unit Circle (1993), so this is at least coming from someone who's been thinking about the problem for a while and is part of the mathematical community.

White People hate math, but like statistics

Did you know White People hate math, but like statistics?

This is from Stuff White People Like. There are three things you should know about SWPL, if you don't already. First, it's SATIRICAL. Second, it is not actually about "white people" (i. e. people whose ancestors originally hail from Europe) but "White People". These are best defined as people who like things on this list, like irony, Netflix, Wes Anderson movies, indie music, having two last names, Oscar parties, having black friends, indie music, The Wire, and the idea of soccer. (This is actually a randomly chosen sample from the list, which is conveniently numbered; random.org gave me 41 twice, and indie music is #41, hence the duplication. I was going to just pick a few things at "random", but I realized that I was kind of biased towards the things that I like.)

By "statistics" is meant not the mathematical field but various interesting-sounding numbers. For example, if each White Person has a favorite thing from the list of Stuff White People Like, and you pick ten white people at random, there's a 36% chance that two of them will have the same favorite White Person Thing. (This is the White Person version of the birthday paradox.)

Also of interest there: the entry on graduate school, which I think pretty clearly refers to grad school in the humanities.

Mathematical quotation server

Check out the Mathematical Quotation Server.

Link to review of The Drunkard's Walk

Here's a review of Leonard Mlodinow's book The Drunkard's Walk. I will resist the temptation to review the review. (Although I've been meaning to write down what I think of the book. I might do so.)

For some difficult-to-explain reason it is illustrated with a couple images from Jessica Hagy's indexed.

Jenga mathematics

John Cook writes about Jenga mathematics, his name for the sort of mathematics which is done by weakening the hypotheses of a theorem as much as possible while the theorem still remains true.

As he points out, "Taken to extremes, Jenga mathematics turns theorems inside-out and proofs become hypotheses". This is an interesting way to look at it, and perhaps explains why it is difficult to read such theorems.

The Stolz-Cesàro theorem

The Stolz-Cesàro theorem is a discrete analogue of L'Hôpital's rule, with which I was not familiar.

(From Topological Musings.)

Loose ends

You may have noticed that posts have been less frequent than usual recently.

This is because first I was finishing up the semester. Then I spent this weekend recovering from the semester. The good news is that I am done taking classes for credit, forever! This is also somewhat terrifying, because the prospect of doing research full-time and writing a thesis is daunting -- but also exciting...

Anyway, not too long ago I wrote about the napkin ring problem in calculus, inspired by Keith Devlin's article; Devlin has given a calculus-free solution, basically that corresponding cross-sections are equal. I alluded to this in the post a couple weeks ago but didn't write out the details. In his May column, Devlin gives some reader remarks to Paul Lockhart's A Mathematician's Lament, which I wrote about back in March.

Also, here's an interview from Leonard Mlodinow, at Carl Bialik's The Numbers Guy -- my mother fairly reliably forwards me some of the The Numbers Guy columns, because she gets some sort of e-mail from the Wall Street Journal, and I don't have the heart to tell her that I'm a step ahead of her! I previously wrote about Mlodinow's probability quiz. Unfortunately, the cover of Mlodinow's book The Drunkard's Walk no longer seems to feature dice.

Largest icosahedron ever?

Biggest icosahedron ever?

Some MIT students (I used to be one) built a 20-sided die in honor of Gary Gygax, the recently-dead inventor of Dungeons and Dragons.

Do you know of any physically larger icosahedron? (And, for that matter, how large is this one? I'm guessing maybe eight feet high, but I'm really just making that up. The building behind it is about four stories, but none of the other relevant distances are apparent.)

(Via Eric Berlin.)

Why is the 3x+1 problem hard?

Why is the 3X+1 problem hard, by Ethan Akin, via Anarchaia.

You might also be interested in my previous post on the 3x+1 problem.

And welcome slashdotters! Will there be more or less of you than people coming in from reddit for my post on "Why g = π²"?? (I don't know yet. I don't know enough about the shape of a "typical" slashdot traffic spurt to guess.)

Carl Sagan on Flatland

Carl Sagan on Flatland (from Blake Stacey at Science after Sunclipse).

I've got other things I want to talk about, but tomorrow morning's class won't prepare itself.

links for 2008-02-23

From A neighborhood of infinity: What is topology?, attempting to provide some intuition for the standard axioms that define a topology in terms of "observable sets" (which are just open sets, with different intuition attached to them!)

Also, on a totally unrelated note: John Baez has online notes from the course he gave in Fall '03, Winter '04, Spring '04. The winter notes are particularly interesting to me, because they combine combinatorics (which I know well) and a lot of category-theoretic notions that I'm not too familiar with.

Wow, X suck[s] at math.

See John Armstrong deconstruct Monday's XKCD, which he also talked about when it came out. (For those who haven't seen it: one student writes ∫ x² = π on the board and a teacher says "Wow, you suck at math"; another student, with longer hair, writes the same thing and a teacher says "Wow, girls suck at math".)

Other people have responded to this as well: Ben Webster of Secret Blogging Seminar, Reasonable Deviations, ZeroDivides, and probably some others that I'm missing. It's not my intention to react here, just to point to other mathoblogospheric commentary on the matter. But you really should read John Armstrong's post, because it's funny. (John, I think you should try deconstructing some actual mathematics. I, for one, would be amused.)

Incidentally, is the correct way to cite a blog informally like this by author, or by blog-title? In other words, if you were writing a blog post and linking to me, would you refer to "God Plays Dice" or "Isabel Lugo"? (Secret Blogging Seminar is a group blog, so I had to put Ben Webster's name in.)

Links for 2007-12-21

Mathematicians solve the mystery of traffic jams, which has been making the rounds lately. Basically, sometimes traffic jams come out of nowhere because someone taps their brakes and the traffic propagates backwards. I remember reading this ten or fifteen years ago. It must be a slow news day.

Thanks to Wing Mui, The trouble with five by Craig Kaplan. Why is it that we can tile the plane with triangles, quadrilaterals, or hexagons -- but not pentagons? And what sort of structure do tilings with some sort of fivefold symmetry have?

Also, a few days ago I wrote about Galileo's argument that the thickness of bones should scale like the 3/2 power of an animal's height, which I learned about from Walter Lewin's lectures, but that this turns out not to be true. Having gotten a bit further into Lewin's lectures, I found out that the real thing we're protecting against is buckling, not crushing.

Greg Mankiw a couple months ago provided a link to a paper on optimal mortgage refinancing by Sumit Agarwaland, John Driscoll, and David Laibson. The Lambert W function (the inverse of f(x) = xe^x) turns out to be good for something! If you use this, I want a cut of your savings for pointing you to it.

Links for 2007-12-05

• The American Mathematics Competitions' Putnam exam archive, including the 2007 exam. I haven't thought about the exam problems yet; I probably will at some point, so watch this space for my thoughts. Solutions are also available.


• Jerry McNerney is a United States Representative from California. He has a PhD in mathematics; his thesis was entitled A(1,1) Tensor Generalization of the Laplace-Beltrami Operator. The Notices of the AMS interviewed him in September, but I don't recall seeing the article at the time.


• We live most of our lives between 0 and 100, from Insight by Design. Basically, small numbers appear much more often on the Internet than large ones. (The most notable exception seems to be numbers around 2000, which figure in dates.)

Links for 2007-11-06

Visualization of Numeric Data, from desktop.de (in English), via Anarchaia. A silly related example is the equinational projection from Strange Maps, in which we see what the world would look like if every country had the same area.

Stat of the Day talks about the length of extra-inning games, extending upon my post from a few days ago (which in turn depended upon their data).

A napkin filled with research-related work at WebDiarios de Motocicleta, following up on this idea. I make sure to have paper with me most of the time, so this doesn't happen to me. But last Friday I did some math on a pizza-stained paper plate.k

links for 2007-10-31

• Hello, India? I Need Help With My Math, by Steve Lohr, today's New York Times. The article's really about how consumer services, like business services before them, are being offshored; tutoring is just an example.


• Pollock or Not? Can Fractals Spot a Fake Masterpiece?, from Scientific American. The verdict seems to be mixed. Pollock's paintings often contain certain fractal patterns, and certain simple images look "the same" as a Pollock painting in a certain sense. The researchers argue that their work is still valid, though:
  

  "There's an image out there of fractal analysis where you send the image through a computer and if a red light comes on it means it isn't a Pollock and if a green light comes on it is. We have never supported or encouraged such a mindless view."

  
  I'd agree with them, so long as it's more likely for the metaphorical "green light" to turn on when it sees a Pollock than when it sees a non-Pollock; there's no single way to test whether a creative work is by a particular person, other than going back in time and watching them create it.


• On the cruelty of really teaching computing science, by the late Edsger Dijkstra. (I've had this one in the queue of "things I want to talk about" for a while, but I don't remember what I wanted to say, so here it is. There are a bunch of similar things which should dribble out in the near future.) But I can't resist commenting on this:
  

  My next linguistical suggestion is more rigorous. It is to fight the "if-this-guy-wants-to-talk-to-that-guy" syndrome: never refer to parts of programs or pieces of equipment in an anthropomorphic terminology, nor allow your students to do so. This linguistical improvement is much harder to implement than you might think, and your department might consider the introduction of fines for violations, say a quarter for undergraduates, two quarters for graduate students, and five dollars for faculty members: by the end of the first semester of the new regime, you will have collected enough money for two scholarships.

  
  I've long felt the same way about mathematical objects. There are exceptions, but for me these are mostly exceptions in which the mathematics describes some algorithm that has input which is actually coming from somewhere. Here it's not so much the program that is getting anthropomorphized as the user.
  
  And why are they always "guys"? How is it that scribbles of chalk on a blackboard, or pixels on a screen, can have gender? Note that I am not suggesting that mathematical objects should be female, or that some of them should be male and some of them should be female, with the choice being made, say, by the flipping of a coin. (Incidentally, the description of mathematical objects as "guys" seems to be much more common at my current institution than at my previous one.)
  
  By the way, Dijkstra is saying here that he thinks computer science should be taught in a formal manner -- proving the correctness of programs alongside actually writing them -- and that to de-emphasize the pragmatic aspect, students shouldn't execute their programs on a computer, since doing so enables them to not think about what the program is doing. I'm not sure if I agree with this.