I ordered two math books from amazon so that I don’t have to enter into a recall war at the library with the other person who seems to want them so badly. I’m really digging the one I’m reading now though — Differential Geometry and Statistics. It’s got a gentle introduction to geometric concepts when dealing with things like fitting statistical models to data and will hopefully will give me a more sophisticated view on modeling. It’s way easier to read than Amari’s monograph on the same subject, which has a whole chapter devoted to a complex theoretical apparatus with no examples. Saying things like “the curvature looks like this function, which we call the Fisher information” is not nearly as helpful as “remember the Fisher information? It looks like this. This turns out to be a special case of the a general concept called the curvature.”

I need to read the material bottom-up. Or sideways, depending on where you think Statistics is relative to Differential Geometry. Actually, since it’s geometry, we’re trying to free ourselves from the tyranny of coordinates, so “up” and “sideways” are pretty meaningless altogether.

I can’t even believe this stuff exists outside of parody. Via Cosma Shalizi’s blog, a number of “devotionals connected to mathematical content.” This is very much unlike Vedic Mathematics, which purports that the vedas contain amazing mathematical formulas (usually combinatorial tricks or number theory things). Of course, that’s pretty ridiculous too, in that numerology kind of way. Here instead we have:

Least-squares solutions techniques are used when a system has no actual solution; that is, the situation where a system is inconsistent. In this case, we find an object that is the closest to being a solution among all possibilities. This object minimizes the value of the distance between the transformation of the object and the impossible result.

In our lives we must solve an impossible problem–we must perfectly meet the entire law of God if we are to have eternal life. No matter how hard we try, we are unable to do this. Fortunately, God loves us so much that he sent his Son Jesus to solve the problem for us. Christ took upon himself our sin and gave his life so that we might live. Jesus is our least-squares solution to the impossible problem. Note, though, that the distance between us and the law of God is infinite; through salvation in Christ, God accepts us as if the error is zero!

And:

In calculus we use technology freely; in particular to produce graphical images with graphing calculators and computer algebra systems. Technology is not perfect, however, and those who use technology must be aware of times when the graphical images we see are not representative of the true nature of the object. We use mathematical experience and developed intuition to judge whether an image is flawed or deceptive.

Satan is the angel of light and his disciples masquerade as “servants of righteousness.” [2 Corinthians 11:13–15] But we read in Matthew 24:24 that it is impossible for false Christs to deceive the elect. We must follow the example of Jesus and use scripture as a standard against which to measure truth. We must also put on the full armor of God to protect ourselves from Satan’s attacks. In both situations, knowledge helps prevent deception.

Oh my.

UPDATE: ObWi has a juicy bit on eigenvectors

In my research I often find myself poking around in unfamiliar areas of mathematics. It was a delight and a pleasure to come across one of Hilbert’s famous problems today — the tenth one, concerning Diophantine equations:

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

For those who do not know, a Diophantine equation is one in which only integer solutions are allowed. For example, consider the equation

3 x + 14 y – 8 z = 9.

Normally the set of triples (x, y, z) that satisfy this equation are unlimited; this equation defines a plane in 3-dimensional space and any point on that plane is a valid solution. But the Diophantine restriction makes things trickier, and now we want solutions for which (x, y, z) are integers. If you stare at this example long enough, you can see that x = 1, y = 1, z = 1 is a solution to this system.

I have a slightly more complicated system to solve, so I had to dig a bit deeper than the definitions. Hilbert’s problem asks if there exists an algorithm to solve Diophantine equations — the answer is no, and was proved by Matiyasevich in 1970 in what was apparently an elegant little paper. Today I was reading a 1973 review by Martin Davis, in which section 2 is called “Twenty-four easy lemmas.” Reading these lemmas reminded me why number theory is so confusing; indeed, one needs a 24-step program to get through to the end result.

For what I hear is a nice review of Hilbert’s 10th, see Matiyasevich’s book on the subject.

I learned a new word today from Inequalities, by Hardy, Littlewood, and Polya (definitely on my wishlist): an analytic complex function f(z) is “wurzelfrei” if it has no zeros in |z| < 1. Note that a wurzelfrei system may have an unstable inverse if it has zeros on the unit circle.

Your bedtime reading is Kolmogorov and Fomin’s book on Functional Analysis. And to think I used to be so well-rounded…

I started my Statistics take-home final today. Ergodic theory, Markov chains, central limit theorems and Brownian motion. Mmm-mmm good. I think I’m going to have nightmares all week.