Can a biologist fix a radio?

Can a biologist fix a radio?, by Yuri Labeznik, via Anarchaia. This article asks a question: say biologists decided to research radios in the same way that they research things like how cells work. Then they would buy a lot of radios, classify and dissect them, eventually conclude that there was some sort of evolutionary explanation for why the antenna is really long, and so on. Much work would be duplicated, because the biologists do not have a particularly good language for communicating to each other how complex systems work. (The author compares the language used by biologists to that of stock market analysts.) Engineers, the author claims, have this problem less, because they have found standardized ways to describe such systems, simulate their workings in computers, and so on. I found the following quote interesting:

In biology, we use several arguments to convince ourselves that problems that require calculus can be solved with arithmetic if one tries hard enough and does
another series of experiments.

Yes, but if the biologists figure this out, and they make their students take calculus, how do I feel about that? (I actually think I feel good about it; if I'm not mistaken the biology undergrads already take calculus, but they think it's unnecessary for them.)

Dear prospective graduate student.

Dear prospective graduate student:

1. I know you're reading my blog, and you found it from my UPenn web page, because my logs tell me that your ISP is the hotel that the prospectives are staying at. (Yes, I'm surprised too that that automatically appeared at the free site I use for such things.) Welcome.

2. Your mathematical interests will change during the first year in graduate school, because a lot of subjects "feel" different at the undergraduate level than at the graduate level, and there are some things you just don't see as an undergraduate at all. (This statement about "feeling" is incredibly difficult to make precise, but two examples are probability and number theory. Probability is usually taught in a "naive" way to undergrads and in a measure-theoretic way to grad students; number theory as taught to undergrads pretty much exclusively concerns itself with reasoning that takes place in the integers, whereas at higher levels it uses Big Fancy Algebraic Machinery. In addition, it may turn out that you think you are interested in X but in reality you had a particularly good teacher of X as an undergrad which colored your perception of that field.)

3. No matter where you go, the first year of graduate school will be painful. Maybe not so much physically painful -- but you will constantly wonder "am I the one person they admitted by mistake?" It gets better. (But bear in mind that those of us who are telling you this survived, or are about to be done surviving, the first year. The people who are currently first-years and are thinking they're going to leave the program are at this point avoiding coming to campus, so they're not here talking to you.)

4. Come to Penn! Our department is not so small that you will find no professors or other students interested in what you're interested in (with a few exceptions here and there), but not so large that you will feel like you are lost. Also, we pay well enough that you won't have to live on ramen. But you have to take a lot of classes. Maybe you like that, maybe you don't, but think about it.

(To give some context: many of our prospective graduate students are in town this weekend; they're encouraged to visit now although some have visited at other times. I said #2 through #4, in varying levels of detail, many times yesterday. I hope the advice of #2 and #3 can be useful to any of my readers who are currently attempting to choose a graduate school.)

edit, April 8, 2:29 pm: this post is also being discussed at Secret Blogging Seminar and Jordan Ellenberg's Quomodocumque. SBS in particular has some lively commenting going on.

Cookie Monster speaks

Cookie Monster says: "They don't call the vampire with math fetish monster, and me pretty sure he undead and drinks blood."

Smale's problems

A lot of people refer to the Clay Mathematics Institute's seven "Millennium Prize Problems" as an analogue of Hilbert's problems for the 21st-century.

In 1998, Stephen Smale produced a list (of 18 problems) as well. (There is significant overlap with the Clay problems: both lists contain Riemann, P =? NP, Poincaré, and Navier-Stokes.) Two of them are Hilbert problems (the Riemann hypothesis and Hilbert's 16th problem). The list seems a bit biased, though, in that Smale made contributions to many of the problems mentioned; Smale acknowledges this as one of the criteria for forming his list, and the document isn't meant to stand alone; the essay was written in response to a query of Vladimir Arnold, and Smale was not the only person Arnold asked. One has to wonder if it would be possible for anyone to really be able to survey all of mathematics intelligently in the way I'm told Hilbert did. (I've read Hilbert's address but I don't know enough of the history to be able to assess whether it really covers all of mathematics at the time.)

In Smale's discussion of the Poincaré conjecture, after pointing out that a big part of the importance of the Poincaré conjecture is that it helped make manifolds respectable objects to study in their own right,he states:

I hold the conviction that there is a comparable phenomenon today in the notion of a "polynomial time algorithm". Algorithms are becoming worthy of analysis in their own right, not merely as a means to solve other problems. Thus I am suggesting that as the study of the set of solutions of an equation (e.g. a manifold) played such an important role in 20th century mathematics, the study of finding the solutions (e.g. an algorithm) may play an equally important role in the next century.

This introduces the discussion of P =? NP, although the reason people study algorithms is not to answer that question; but one often hears statements like Smale's statement on Poincaré's conjecture, or statements that Fermat's Last Theorem is more important for the development in number theory that it spurred than for the result itself.

The uniform distribution as a sum?

Yesterday, I was asked the following question: the sum of two uniformly distributed random variables with the same support has a triangular distribution. Is there a random variable X such that X + Y has a distribution which is uniform, where X and Y are independent and identically distributed?

I don't know the answer, but I started thinking as follows. First, it's enough to show that there aren't independent identically distributed X, Y, such that X+Y has a distribution uniform on [-1, 1]; linearly scaling gets the general result. Now, the characteristic function of a uniform distribution on [-1, 1] is φ(t) = (sin t)/t. The characteristic function of X+Y is the product of the characteristic functions of X and Y. (If you're more familiar with analysis than probability, note that characteristic functions are basically Fourier transforms, and the probability density function of X+Y is the convolution of the probability density functions of X and Y.) Thus, if X exists it has characteristic function ψ(t) = [(sin t)/t]^1/2 -- this is already a bit problematic, because we want ψ to be continuous, but even with that restriction we still have to specify which square root is being taken on each of the intervals ... [-3π, -2π], [-2π, -π], [-π, π], [π, 2π], [2π, 3π] ... (Informally, we have to make a new choice every time (sin t)/t goes through 0.

At this point I think one wants to use Bochner's theorem, which says that the functions which are characteristic functions of measures on the real line are exactly the positive definite functions -- but how does one show that this function is positive definite?

The other thing to do is to look at the discrete analogue; consider the probability generating function of a random variable which is uniformly distributed on the set {0, 1, ..., n-1}. This is χ(x) = (1+x+x²+...+x^n-1)/n. Now, if this random variable were the sum of two independent identically distributed random variables, its p.g.f. would be the square of a polynomial with positive real coefficients. It's not.

But what about the continuous case?

The unexamined life?

From Bill James answers all your baseball questions, a long interview posted at the Freakonomics blog:

Q: Has looking at the numbers prevented you from actually just enjoying a summer day at the ballpark? Have we all forgotten the randomness of human ballplayers? By reducing players to just their numbers can we lose sight of the intangibles such as teamwork, friendships, and desire.

A: Does looking at pretty women prevent one from experiencing love? Life is complicated. Your efforts to compartmentalize it are lame and useless.

This is yet another example of the "people who think about things are strictly better off than people who don't" meme -- roughly speaking, the usual justification for this is that we can turn off the thinking when we want to. But can we? I know I can't just turn off the part of my brain that is constantly counting things or figuring odds of things, and there are moments when that does hurt my quality of life. I think in the end I come out ahead -- and most mathematicians probably would agree with me, otherwise they wouldn't be mathematicians -- but it is not so simple.

Mathematical April Fool's hoaxes

The Museum of Hoaxes has a list of the top 100 April Fool's hoaxes of all time.

Of mathematical interest:

• #7:Alabama changes the value of π (to exactly 3, which is supposedly the "Biblical value" -- but in interpreting the relevant verse of the Bible (2 Chronicles 4:2) one has to think about measurement error.


• #8: The left-handed Whopper, which had its condiments rotated 180 degrees for the benefit of the left-handed customers. But Whoppers are rotationally symmetric anyway! If you want a left-handed Whopper and rotate it 180 degrees. You could have a mirror-image whopper, but you wouldn't be able to digest it because the vast majority of the molecules in it would be enantiomers of what your body is set up to digest.


• #30: Operation Parallax, in which it's claimed that somehow Britain ended up two days ahead because of all the time changes.

By the way, The Mandelbrot monk was a hoax, which I knew when I posted it; John Armstrong has debunked it. (A commenter pointed out that someone from now perhaps could have explained to a medieval monk how to compute the Mandelbrot set, which may be true, in the same sense that we can program computers to do something. But that wouldn't make the article any more true, unless one wants to posit time machines.)