Happenings – 2010 May 8

Last weekend and this week have been interesting but not too interesting. (Too interesting can be bad.)

Logic.
As I moved on to writing up “universal generalization”, “universal instantiation”, and such things, I seem to have discovered that none of my books has a really good explanation.

Maybe that’s okay. I wasn’t planning on doing them rigorously — merely identifying them and showing a couple of the dangerous curves while using them… working at the level of Hummel and Exner. We’ll see whether I can sort it all out to my satisfaction and have a post ready for final edit Monday evening.

Still, I’m shocked that my library let me down.

Color.
I made some progress in Fairchild’s “color appearance models”. In addition, my freshly ordered copy of Kang’s “computational color technology” arrived… and I found time to read through it. I bought it because it uses Cohen’s approach, and it discusses the reconstruction of spectra from tristimulus values. It is very likely that I will be talking about Fairchild, and likely that I will be talking about Kang.

I also found time to work out explicitly x,y as functions of Y assuming that X and Z are held constant while Y varies. You can find it in this comment, but there aren’t any pictures.

If I decide that I probably can’t get a logic post ready this weekend, I might be able to do a color post about my monitor; that will illustrate properties of other monitors.

Modular functions.
What the heck?

Someone asked a friend how to produce this graph…. my friend asked me how to do the calculations.

I’m not sure that I have them right… I don’t have this stuff at my fingertips anymore, not that I ever had very much of it at my fingertips at all. On the other hand, if I had never ever seen it, I wouldn’t be trying it now. At I write this, I think my friend is looking at the calculations I sent him this morning, and trying to get a plot.

Soon I hope to know whether my first crack at this was right, or whether I need to go back and work out more things for myself instead of grabbing plausible equations from a book (FYI, Anthony Knapp’s “Elliptic Curves”).

For now, I think I can say that a modular function is a modular form of weight one. And then it turns out that every modular function is a rational function of something called the j-invariant.

And the graph I aimed you at is the real part of the j-invariant inside the unit disk.

There, now you have some words you can go search for on the Internet.

I also found it interesting that I allowed myself to be distracted by problems for other people; as a general rule, I try not to get distracted from my own mathematical goals.

Part of it — xy as a function of Y with X and Z constant — I felt guilty about: I think I should have realized sooner that this is what the questioner was asking for. On the other hand, I’m still not certain what the questioner was asking for. Maybe the correct way to phrase it is: regardless of the precise question in this case, this is the answer to the most appropriate question I’ve thought of.

And I feel guilty that I didn’t see that sooner. Once I thought this was the right question, I had to answer it.

And I know better. Never assume that the question is correct — always be prepared to change it.

As for the other part — modular functions — well, I hate completely losing things I’ve played with, and I welcomed a chance to get reacquainted. No, I was never even a journeyman in elliptic curves and modular forms, never mind expert, and I don’t expect to take them up seriously in the near future… but when they stand in front of me and cry, “Look at me”, I can’t resist.

Finally, there is a problem I’ve been meaning to work out for myself… in that respect, it is much like these two distractions, straight computation on a clearly defined question. The difference is that this one is for me instead of for other people — and, apparently, since I’m not waiting impatiently for the answer, I have moved my problem to the back burner. That two other people care about these two distractions is enough to make me work on them.

But, of course, for my own stuff, I can only care about so much at a time; some things have got to fall off the list for a while: logic, color, projectiles, orbits… the FFT, aircraft control, classical construction problems, Galois theory, modules, the standard model, string theory, chapter 3 of Bloch….

Ok, Bloch didn’t fall off the list — it got buried on the ocean floor; but I do want to summarize it….

Well, if I decide not to do either a logic or color post this weekend… there is a really neat treatment of the two-body problem using vectors, and I think I could write it up fairly quickly.

Half the problem, of course, is deciding to bail on one thing soon enough to still finish another. If I fail to do that in time, there can be no post.

We’ll see how it goes.

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