A friend of mine who’s studying econometrics asked me about the Kalman filter, which is used in estimating the parameters of a time series model. I didn’t know anything about the subject, so I was poking around online, where I discovered that the Kalman filter is rocket science: it was invented to estimate the current position and trajectory of the Apollo spacecraft. And here I thought the only concrete result of the space program was Tang.

Alexandre Borovik, who occasionally comments here, has begun uploading draft chapters of his new book, Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice, here. The first three chapters are already available.

Via David Corfield.

I was poking around on Wikipedia, when I came across the page for Runge’s phenomenon. Runge found an example of a smooth function such that if you interpolate it by a high-degree polynomial at fixed points over a finite interval, the approximation of the function by the polynomial becomes very bad. In fact, in the limit as the degree of the interpolated polynomial goes to infinity, the maximum difference between the function and interpolated polynomial also goes to infinity. Interpolating with polynomials is harder than it looks.

Peter spots two preprints that give two different proofs of a major open problem in algebraic geometry: the finite generation of the pluricanonical ring:

• Existence of minimal models for varieties of log general type
• A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring

By a standard construction in algebraic geometry, this would give a particular embedding of a variety as a subvariety of projective space. Apparently for varieties of general type, this would give something even better: minimal model of the variety.

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.

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.