Week 222 of This Week’s Finds in Mathematical Physics is up.
Author Archives: Walt
The Hilbertization of Sports
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.
Nobel Prize in Economics
The (sort-of) Nobel Prize in Economics has been announced. The contributions of one of the two, Robert Aumann, are almost purely mathematical work in game theory.
His most interesting idea is that of correlated equilibria. The usual definition of equilibrium in a non-cooperative game, Nash equilibrium, rules out certain kinds of cooperation, even when that cooperation is in the self-interest of each player. Correlated equilibria allow randomized strategies which rely on a random event that is known to both players. Some details about correlated equilibria can be found
here.
(As an aside, the Wikipedia entry for Aumann is unusually bad, so it’s a good candidate for updating, if anyone’s interested. There’s also no entry for correlated equilibrium.)
November Notices of the AMS
The November issue of the Notices of the AMS is available.
Some highlights:
- What is the Role of Algebra in Applied Mathematics, by David A. Cox. Cos is the author of one of the best undergraduate introductions to algebraic geometry, Ideals, Varieties, and Algorithms. The book takes a computational approach based on the technique of Groebner bases.
- What are Groebner bases, you might ask? Bernd Sturmfels answers that question in the latest What is….
- The most intriguing article is Generalized Fourier Transforms, Their Nonlinearizations, and the Imaging of the Brain by Fokas and Sung
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Semiring analogies
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:
- Any collection of sets closed under union and intersection form a semiring, with union as addition and intersection as multiplication (or vice versa).
- Regular languages form a semiring. The sum of two regular languages is the union, while the product is juxtaposition.
- The reals with positive infinity added form a semiring where the “sum“ of two numbers is the minimum, and the “product“ is addition. (Including positive infinity is not strictly necessary, but it serves as a “zero“ for the min operation). This known as the min-plus semiring.
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.
Blumberg’s theorem
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.
Atomic Orbitals
There’s an incredibly informative discussion thread about atomic orbitals at Brad de Long’s weblog.
Hatchet job on modern economics
Mark Blaug has written a hatchet job on Ugly Currents in Modern Economics. One particular development that draws his ire is the increasing mathematization of the field. The article does a good job of summarizing the major trends in the field while explaining why most of them are wrong.
Introductory Lectures on Quantum Field Theory
Spotted some tempting reading via It’s equal but it’s different: an introduction to quantum field theory, called (incredibly) Introductory Lectures on Quantum Field Theory.
Schwarz paradox
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:
- an (old) paper by Benoit Mandelbrot, Length and area “anomolies”.
- a brief historical introduction, The Schwarz Paradox by Rickey.
- An Application of Abstract Nonsense to Surface Area by Lord.
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.)