Happenings and Q&A Feb 14

This is the post that I would like to have put out last night.

What am I hoping to do this weekend? Have I made any progress on the questions I asked earlier in February?

What am I hoping to do? Not a lot has changed. I keep reminding myself that it’s okay to work on B even though I can’t publish it until I publish A. (For example, B might be solar calculations, and A is explaining the equations.) As I have said before, doing mathematics and publishing mathematics are two different activities; I get to keep two different to-do lists running. Anyway, I haven’t added anything new to the mathematics I hope to do, but I have reminded myself that I can do some things I’m not yet ready to publish.

The questions? I haven’t even touched the question about uniqueness of coordinates inside simplices. That is, I remain confident that coordinates are not unique — in some way — if we use polygons in place of triangles, but I have not constructed an explicit failure of uniqueness.

While I am confident in Arnold’s proof of the rotation equation

$v = T\ \nu + \omega \times r\$

for $\omega$ not constant, I am uncertain about one detail, and I haven’t sat down to look at it.

For the questions about the transition matrix being a covariance matrix, I thought I could see a perfectly reasonable generalization, but when I tried to confirm it in a calculation, it failed. Oh, it is certainly true that the orthogonal eigenvector matrix V is not a covariance matrix (that is, between the old and the new data); and it is certainly true that nothing like scaling the columns of A and the rows of X works out as did scaling the columns of X and the rows of A. (What I had in mind, specifically, was scaling the rows of X to get constant row sums.)

And, while I have gotten the color bibliography added to the blog, I am still working on an introductory post about color. I certainly hope it will be ready for tomorrow night; I had hoped to finish it in the evenings before last Wednesday. With any luck, a more detailed answer to at least one of these questions will be ready by tomorrow night, for posting on Wednesday.

(Yes, I would rather put out posts every Friday, Sunday, and Wednesday, with the Friday post being like this one; that generally means having a post almost finished by the end of the weekend.)

Or maybe I will finally summarize Bloch chapter 3; it’s looking simpler and simpler to do. I think I’m trying to find something magnificent to say, but I want to remind myself that the perfect is the enemy of the good. I don’t need “magnificent”; I really only need “interesting”.