Happenings – 4 May

I’m playing in 6 areas, but the pause button has been pushed in most of them.

One, for PCA / FA (principal components / factor analysis)… I’m working in chapter 4 of Malinowski’s “factor analysis in chemistry”: estimating errors. I’m not sure this is the best way to be doing it, but at least he’s doing it. The rest of the first 5 chapters is quite straightforward. After all the time I spent on Davis. I could write up an example and show you everything but chapter 4. I should be able to rip thru chapter 4 (no pun intended), but I am paused. Not sure why.

In addition, I need to make up an example for his calculations. In contrast to the earlier authors, Malinowski devotes chapter 5 to a single example. I’d rather not grab something which amounts to an entire chapter of a book.

Two, for QM (quantum mechanics)… I’ve shown you one calculation: getting the probabilities for the possible states of the x-component of momentum after a measurement of the z-component. A natural follow-up would be: what if I change the definition of the z-axis after making a measurement? Feynman gives answers for two cases (volume III, ch 5). And I can almost get them.

I can derive the conjugate transpose (or inverse) of Feynman’s two answers for rotating the apparatus for measuring angular momenta. That is, McMahon’s recipe gives me an answer which is off by the sign of the angle. I can think of several reasons for the sign error, but I don’t know which one is the problem. I’d like to put out the answer anyway: that the recipe is so close is encouraging.

OTOH, the recipe amounts to transforming between a Lie algebra and its Lie group. I should at least show you the corresponding computations for transforming between an infinitesimal rotation and a finite rotation. That, in turn (no pun intended) leads to a really neat formula for the rotation matrix associated with an axis and angle of rotation.

So I’m paralyzed trying to decide whether to put out the QM calculation with a sign error (well, yes), and whether to show you infinitesimal rotations first (well, yes, but not first). Finally, I should probably derive the really neat formula, not just hand it to you. Sounds like a plan.

Three, linear algebra… there’s another reason why I’m paused on the QM. McMahon’s recipe has us compute the matrix exponential of the Jx, Jy, or Jz operators. Now, I’ve been intending since February – the schur’s lemma post – to talk about the spectral decomposition theorem and functions of matrices. Well, the matrix exponential is the primary example of a function of a matrix. I’m sure these posts can wait until after I tease you with the example but they need to get written and posted.

Four, for controls… I have found a great book to work thru. I had thought about practicing the design of control systems; then I had thought about extracting as much information as possible from a transfer function. That taught me that there is a wealth of information that can be extracted from a bode plot alone.

Later I realized that, for chemical plants at least, there is usually an “auto-tune” button. You push it, and it sets the parameters for the control system. Ok, then why do I need to set the parameters myself? I should see how they do auto-tuning, and why not do that for my “design”?

As usual, Bequette has a few pages about it. He has a few pages about everything; that’s why I like his book. But Ellis has more: an entire chapter devoted to tuning different control systems.

Ah, but a new book arrived a few days ago, Blakelock’s “automatic control of aircraft and missiles”. Its first 9 chapters are a classical analysis: bode plots and root-locus. This looks irresistible. Now, I already own Bryson’s “control of spacecraft and aircraft”, but it’s a terse presentation and it uses state space. Blakelock will give me a chance to exercise my computational prowess with classical techniques.

Right? Hmm, I can’t make sense of some of his early algebra. (The discrepancy, so far, is much more than just the sign of an angle. I don’t even understand what he did or what he’s writing out.) How can a coordinate transformation be so difficult? I understand coordinate transformations! I also own an interesting book about rotations, Kuipers’ “quaternions and rotation sequences”. He and Bryson agree on the coordinate transformation from the earth to the aircraft axes, so I just need to keep hammering on Blakelock…. I don’t want to skip over what I don’t understand.

Five, regression… I want to show you the “normal equations” for regression (“ordinary least-squares”, OLS). I’ve been meaning to put something out for a long time, but the normal equations will show up in the Malinowski PCA / FA, and that’s a good reason to finally get them done.

They have been in the back of my mind ever since I got the blog, and before. They are one answer that I wouldn’t dream of trying to state on the sci.math newsgroup, but they’re an important answer. Oh, what’s the question? How do I fit a parabola to my x-y data? Now, the normal equations solve far more general problems than that, and they’re a marvelous example of something that needs matrices. I plan on two posts (exposition and example), and they just might come right after this one.

Six, surfaces… I have finished chapter 2 of Bloch’s “… geometric topology and differential geometry” and I’d like to chat about it. I just don’t know what to include and what to omit. I don’t want to outline the chapter; I don’t even want to summarize the chapter. “What was interesting?” is too broad a question. But I want to talk about it.

I may just have to write it up a few times and see what seems to flow. And I need to remember the difference between a diary and a letter: I can write a diary entry first and edit it into something I’m willing to publish. I should not be editing myself while I’m writing.

And not counted among those 6 areas: I need to add some more books to the bibliography.

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