The big news is that I overcame my difficulties with the “magic omega formula” – whatever that is!
As I said last week, I almost had a derivation of the result for which the magic omega formula is used… but I was off by a negative sign. Well, that is accounted for by my using attitude matrices instead of transition matrices.
I also said last week that I knew perfectly well how the derivation of the magic omega formula itself had to be accomplished… but it didn’t work out. Well, all I had to do was be careful. When I tried to work through it slowly and carefully, to find my mistake, it worked out perfectly well.
Having finally gotten it, I was willing to look at the equations for aircraft again. No, I didn’t need the magic omega formula in order to proceed… but, in a very real sense, it was more important to me than the aircraft equations themselves. I think it really is a very neat formula, and I look forward to explaining it and the requisite mathematics. (There’s only one new fact, and one alternative formula for something, but then several other things just need to be brought together. As usual.)
This morning, the kid (my inner child) picked up an old book about computer performance modeling that I haven’t looked at in a very long time. You know, now that I have Mathematica, that book looks like it could be some quick and simple fun. (Hmm. I’ve had Mathematica for about a decade… and this is the first time in that time that I’m looking at this book.) I own more complicated books about queueing theory, and about modeling computer performance, and good books about other applications of queueing theory… but this looks like good clean fun. (I really like mathematics when it’s good clean fun.)
I spent some time in the evenings this past week looking at algebraic geometry and at representation theory of groups. In both cases, I found that “modules” were being used heavily.
Just about all I know of modules is: one, they are like vector spaces, except that they have a ring of scalars instead of a field of scalars; two, in contrast to vector spaces, they do not in general have a dimension.
I have felt for a long time that my knowledge of abstract algebra is far less than it ought to be. My lack of understanding of modules is merely one symptom of that lack.
Well, for now at least, I have decided to try working my way through Dummit & Foote’s “Abstract Algebra”. It is written at such a level that parts of it could be used for the introductory undergraduate abstract algebra course… but it contains so much additional material that it could also be used for the introductory graduate abstract algebra course.
In any case, I haven’t curled up with an algebra book since the early 1990s, and I think I can learn a lot – admittedly and sadly, a lot of elementary mathematics that I feel I “should” have learned a long time ago. I am certain, however, that we never covered modules in either my undergraduate or graduate course. I would have learned something about them, even if they hadn’t “made sense”.
We’ll see how it goes. Over and over again, I keep being distracted from pure mathematics by applied mathematics.
Okay, the rest of the weekend lies before me… I have a regression post to write, and whatever other math I feel like doing.