Ars Mathematica

While I was driving in my car today, I thought of a proof of the Cauchy-Schwartz theorem. I’m sure that it is completely unoriginal, but it has the advantages of both being longer and requiring more background than the usual proof (which you can find on the Wikipedia page).

Let (,) be an inner product. From the definition, we know that for two vectors x, y and two scalars a, b that

(ax+by, ax+by) = a² (x,x) + 2ab (x,y) + b² (y,y) ≥ 0

This is a positive-definite quadratic form in a, b, which means that its associated matrix has positive determinant:

(x,x) (y,y) - (x,y) (x,y) ≥ 0,

which is the result.

The real advantage of the proof, I suppose, is that if you already have the linear algebra background there’s no trick involved. It also means that using the same determinant argument there are analogues of the inequality that involve n vectors instead of two.

Peter Woit quotes from a reminiscence by Peter Goddard from a physics conference in 1971:

With great technical mastery, he was covering the board with special functions, doing manipulations that I knew from my studies with Alan White (who was also at the School) could be handled efficiently and elegantly using harmonic analysis on noncompact groups. Just as I was wondering whether it might be too impertinent to make a remark to this effect, the lecturer turned to the audience and said, “They tell me that you can do this all more easily if you use group theory, but I tell you that, if you are strong, you do not need group theory.”

Count me among the weak.

I’m intrigued by the beginning of a new series of posts at the Everything Seminar about harmonic analysis. This particular post talks about the relationship of singular integral operators and Carleson’s Theorem. Carleson’s Theorem (that Fourier series of functions in L^p for p > 1 converge pointwise almost everywhere) is a famously difficult result; the post gives some idea of where the difficulty lies.

For the ambitious, a complete proof is available in a preprint by Michael Lacey.

Calculus has been the subject of immense amounts of educational material, ranging from textbooks to blog posts. Unfortunately, that is now all obsolete. The definitive presentation of calculus is here: Calculus, the Musical.

Brad de Long, an economist, has a post up about the significance of how he dresses for specific audiences. In particular, the consequences of wearing ties:

With math-oriented students, however, a tie tells them that I spend too little time thinking about isomorphisms.

(This inspired n-category jokes in the comments.)

Charles at Rigorous Trivialities has written a post outlining the proof of two pretty theorems from the invariant theory of finite groups: Noether’s theorem that the ring of invariants is finitely generated, and Molien’s formula for the number of homogeneous invariants of a given degree.

Not only is John Armstrong a failed crackpot, he is wrong about statistics. Statistics is, from the mathematical point of view, a perfectly interesting subject; this fact is carefully concealed from us by statisticians. For example, most mathematicians know the central limit theorem, which says that the sum of large numbers of independent, identically distributed (iid) random variables tend to be normally distributed. This even has an elegant proof in terms of Fourier analysis, where addition of random variables because multiplication of Fourier transforms.

What mathematicians don’t know is that almost every other statistic ever defined also satisfies the central limit theorem. The median of a large number of iid random variables? Normally distributed. The mode of a large number of iid random variables (where the underlying distribution has a single mode)? Normally distributed. The cosine of the seventeenth percentile? Normally distributed. The simplest explanation for this cavalcade of normality involves the Gâteaux derivative in functional analysis.

Isabel at God Plays Dice finds the definitive review of Dummit and Foote, here at adequacy.org.

(If you have any questions as to the objectivity of adequacy.org, I suggest checking out their Wikipedia page.)

The Notre Dame Mathematical Lectures series is now freely available online. It was a fairly small series of lecture notes on various topics. The most famous was probably Emil Artin’s lectures on Galois Theory.

Thanks to Greg Muller, I’m looking at this paper by Dror Bar-Natan that reduces the Four Color Theorem to a plausible statement about Lie algebras. Now we just have to hope this new conjecture does not not require hundreds of pages of computer generated proof.