Ars Mathematica

With all my fancy book-learning and abstract axiomatic thinking, there are a lot of classical results that I’ve never learned. I just ran into an interesting classical theorem by Sylvester about symmetric matrices. Let A_n be the submatrix of the first n rows and columns. When you complete the square on the associated quadratic form, the coeffients of each square term is of the form

(det A_n)/(det A_n-1).

From this it follows that the matrix is positive definite if and only if each (det A_n) is positive.

The criterion is not practical for large matrices, but it does imply one interesting theoretical result: the set of positive definite matrices is a real semialgebraic set.

Warning: if your relatives ever find out about this website, you’ve sentenced yourself to a lifetime of receives its wares for birthdays and holidays.

Bee of Backreaction has an excellent summary of the evidence for the existence of dark matter.

This is probably well-known to everyone in physics, but less well-known to pure mathematicians. For a open-source high quality implementation of linear algebra computations such as finding eigenvalues, the standard is LAPACK (distributed by Netlib, a numerical source code repository). It is so standard, in fact, that tuned versions exist for individual architectures, and LAPACK performance is frequently used to benchmark processors.

Cool optical illusion.

Lambda the Ultimate links to a cool little paper, An Incremental Approach to Compiler Construction. It’s literally a short tutorial on how to write a compiler. It concentrates on the least-complicated but most-intimidating phase: machine code generation.

The 2006 Nobel Prize in Chemistry was awarded to Roger Kornberg for eukaryotic transcription, something that most people would regard as biology. This touched off some agonizing about the Meaning of it All at Uncertain Principles, In the Pipeline, and Adventures in Ethics and Science. Paul Bracher even went so far as to suggest that chemists move in on the physics prize.

Hey, at least they have a prize.

Backreaction has a long post on applications of the AdS/CFT correspondence in heavy ion physics. The topic is interesting because it is a potential experimental prediction derived from string theory. Interestingly, it doesn’t involve string theory as a theory of everything; instead it uses string theory ideas to make calculations about quark-gluon plasma.

I just came across a reference to the work of the philosopher of mathematics Hartry Field. This is what Wikipedia has to say:

Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine’s indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian mechanics that didn’t reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert geometry to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

Does anyone know if that is an accurate summary of Field’s argument? It seems obviously wrong to me.

Via Crooked Timber.

November Notices of the AMS are out. The entire issue is devoted to Alan Turing.