Al Zimmermann is running a programming competition to see who can:
Pack n non-overlapping circles with radii from 1 to n into as small a circle as possible.
You have until mid-January to come up with solutions for n between 5 and 50.
(via Lamba the Ultimate).
The discussion about Goedel’s theorem made me wonder about this question: when was the last time a widely quoted mathematical result turned out to be false? I’ve seen preprints that had proofs I didn’t believe, and I know that journals occasionally print results that are wrong, but does anyone know of a result that was once widely accepted, but then turned out to be wrong?
I’m sorry if my last post misled anyone. Goedel’s theorem is true. Lots of people have checked the proof. I’ve checked the proof. There are multiple proofs (the usual proof is Rosser’s, not Goedel’s original proof), as well as vast generalizations.
Here’s a two line proof of the theorem (some details are left out):
At the time, Goedel’s result was very surprising, but by modern standards it’s almost obvious. The set of provable theorems of arithmetic is recursively enumerable. It’s not that surprising that first-order arithmetic would be at least as expressive as Turing machines, which implies that the set of provable theorems is not recursive. Goedel’s theorem follows.
I intended the post to be tongue-in-cheek. My real point was that a certain number of papers on Arxiv are apparently written by cranks.
According to a new paper on Arxiv, Goedel’s theorem is false. There you have it.
This paper, On the Euler angles for SU(N), gives a definition/derivation of Euler angles for SU(n). The paper claims that the construction first appeared in 2002. Does anyone know if that’s true? I thought for sure that I’d seen Euler angles defined for SU(n) (and SO(n)) sometime in the 90s.
I was websurfing a football site, Football Outsiders, when I came across a link to an essay called, of all things, Football’s Hilbert Problems. This is not an isolated phenomenon. It’s modeled after a essay from 2000, Baseball’s Hilbert Problems.
The November issue of the Notices of the AMS is available.
Some highlights:
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Sigfpe has been posting on his blog about one of my minor obsessions, semirings. A semiring is a ring without subtraction. There are lots of semirings that arise in both pure and applied mathematics. Here are some examples that show just how common they are:
The last example has an interesting extra property: you can take infinite “sums“ by taking the infimum of an infinite set of elements. You can set up an extended analogy between the reals with usual arithmetic operations and the min-plus semiring. In this analogy, integration becomes optimization. You can even extend the analogy as far as the Fourier transform. The min-plus analogue of the Fourier transform is the Legendre transform that arises in classical mechanics. Sigfpe explains the analogy here.
He has also posted about an application of semirings to understanding the game Tetris.
Here’s a bizarre theorem I’d never heard of before: Blumberg’s theorem. It states that for any real function, there is a dense subset of the reals on which the function becomes continuous.
The Schwarz paradox demonstrates that surface area is not a straightforward generalization of arc-length. Arc-length is defined for rectifiable curves &emdash; for any such curve, we approximate by line segments. The arc-length is the limit of the sums of the lengths of the line segments.
For surfaces, this definition breaks down. Rectifiable surfaces are well-defined, but the limit fails to be well-defined. Schwarz found two different sequences of approximations to the cylinder that converge to distinct values for the surface area.
Here are some papers that explain the paradox:
While searching on the topic, I also found a nice historical work, A Panorama of the Hungarian Real and Functional Analysis in the Twentieth Century. It touches on the Schwarz paradox, and many other topics besides. (For example, it explains what the Sz. in Sz. Nagy stands for.)