Happenings – 2011 Aug 6

What’s been happening? Not all that much, or so it seems.

What am I doing? Probably more than I give myself credit for.

The first thing that isn’t happening is: magnitude 7 earthquakes. Not one so far in August, not anywhere on earth.

I finished Ronan’s book, “Symmetry and the Monster”. It was fun, if a little unsatisfying… I could have used a little more mathematics.

I did find in it a verbal description of what it means for a group to be k-transitive on a set. Marvelous! Of course, it says exactly what the mathematical definition did – but I, poor fool, had not been able to phrase the mathematics in words that made sense. And, hard as it may be to believe, I need mathematics definitions to make sense in words.

(If you’ve ever had even an introduction to group theory, you’ve seen the dihedral groups– each is the group of symmetries of a regular n-gon. The dihedral group with 4 elements, in particular, is the group of symmetries of a square. It is said to be transitive because it can move any one vertex to any other. But it is not 2-transitive: it cannot move 2 adjacent vertices to 2 diagonal vertices; that is, we have a counterexample showing that it cannot move any arbitrary pair of vertices to any other pair.)

I also spent a couple of hours showing a friend how to compute what’s called a periodogram, in order to estimate the period of some cyclic data. This is not always reliable… I think I need to learn how to use Bayesian methods… but I think it’s the first thing one should compute.

Maybe I should put this out for Monday’s post… well, it would be fun, but perhaps I need to play around a little more. For now let me just describe an illustrative problem. Take the function

It has a mean of 3, an amplitude of 5, and a period of 7. Now sample it at 21 evenly spaced points.

Now take the DFT, discrete Fourier transform, get the magitudes of the resulting numbers, and you will indeed see – perhaps after some work, depending on what convention you use – a period of 7, an amplitude of 5, and a mean value of 3.

(I use prime numbers so I can pick them out of the answer when necessary. And life gets more interesting if you have noise, or if the number of samples is not a multiple of the period (as 21 was a multiple of 3).

As for the blog…

I forget whether it passed 120,000 cumulative hits on Sunday or Monday. And Monday and Tuesday were spectacular days: the post “norms and condition numbers” got 72 and 111 hits on those 2 days.

I wasn’t all that far off last week when I said that 49 hits for the 3–5–8 puzzle might have been a record… it wasn’t, at the time, but it was tied for the 2nd highest number of hits for one post in one day. The other post tied with it was the CIE chromaticity chart… but the previous record was held by “compressed sensing”, with 86 hits in a day.

I have no idea why “norms and condition numbers” was so popular… but I’m delighted that it was. (Let’s see: “compressed sensing” talked about the l1 norm, and recent regression posts have talked about condition numbers….)