ChapterZero

Someone at Caltech made a big mistake— I got into the GradPreview program. Not that I’m a bad choice necessarily, but how would they know: I didn’t turn in my transcript, or any letters of recommendation. Maybe I had exceptionally great essays? More likely, not very many people applied.

Either way, I don’t care. Nov 9-11, I’ll be at Caltech, soaking in the environs, and seeing if I do want to go there for grad school. I’m not looking forward to the expected social mixing, but I’m holding out for the chance that just being there will loosen me up. One thing I need to do is find out who on the math faculty is doing research I’d be interested in, and inveigle my way into their presence. Also, I need to get a real haircut (opposed to my homebrewed attempts) before I go.

Yeah! CalTech!

We all know that CSI isn’t the most realistic tv show— in what world do the CSI order detectives around, grill hostile witnesses, and such? But I can’t believe I let this one slide: in one episode of Miami CSI, they were using a gamma imaging device to view the insides of some crates. The images produced were 3d wireframes, nicely colored, and absolutely bogus!

Show me the imaging device that can reconstruct geometry like that… that’s like claiming to have found the holy grail. That’s simply not the way imaging technologies work; they all depend on Fourier transforms, which means that information is collected as intensities and energies. So what you get is a picture, a rather blurry 3d picture, one from which no current technology can unambiguously recover such crisp geometry— that kind of processing requires a human level intellegence. Again, show me such a reconstruction technology, and I’ll show you a million bucks.

That’s making me wonder about some of the more dubious image enhancements they pull on that show: can I believe it when they recover license numbers from blurred photos, or is that more dramatic license? Making fast and loose with the truth like that on a forensic science show is irresponsible, IMO.

I’m in the engineering computer lab, where I was sitting by a friend taking the Controls class— one of the few engineering courses that comes in a sequence (I and II), and IMO would be useful and interesting for me to take— but I can’t take it, because of scheduling overlaps. I was looking through his notes when I came across the companion matrix stuff.

After I exhausted myself gushing on and on about how wonderful they are, he asked me for my opinion of where matrices came from. I assumed he meant not only the notation (i.e. when we’re simply using matrices as a shortcut for manipulation linear equations), but the weird seemingly dissociated results we have about matrices as objects in themselves, e.g. determinants. That is, historically, what motivated the higher level concepts of matrix theory, and how were they understood intuitively? Where I’m assuming that they were indeed understood intuitively by the people who first worked with them. I’m sure whoever discovered/invented the determinant didn’t say “hmm, if I design an multilinear function in this way…”

My answer? I don’t have the beginnings of a clue. Which doesn’t bother me— I usually don’t have a clue about the actual beginnings of a particular mathematical subject. About all I know with regard to mathematical history is that Fourier and wavelet analysis was started with Fourier’s idea that a function can be represented as the limit of sums of harmonics, and that rigorizing Fourier’s hypothesis was one of the main motivations behind modern analysis. Usually, I settle for a collection of post facto mileposts, that I could imagine myself following up to the modern presentation of a subject. E.g., the way Riemannian integration is usually taught, as the limit of the areas of rectangles, can be easily extended to a logical, if overly neat, train of imaginary historical mileposts leading to the concept of integration.

What bothers me is the fact that I don’t have even such an account for matrix theory. How the hell did anyone ever think of determinants? I can see where most other things came from; here’s my imaginary history of matrix theory: first, we realized that matrices are convenient notation for handling systems of linear equations, from there, we saw the obvious isomorphism between matrices and linear transformations between finite dimension vector spaces, and a lot of weird lower-level matrix results are merely transcriptions of more intuitive results in linear algebra. But determinants apparently have very little connection to linear algebra (other than indicating the invertibility of the associated transform, I think), but many uses otherwise.

I remember asking professors about this the first couple of times I took classes involving matrices, but I never got a satisfactory answer. I had buried it until he asked me about it today. Now this is going to nag me until I solve it.

Now, if I ever go back to working on BasicCAS, I have one method for computing the roots of polynomials numerically that doesn’t seem to require too many arbitrary starting parameters: namely, by finding the eigenvalues of the polynomial’s companion matrix.

This is one of those neat applications of matrices that is obvious in hindsight, but you wouldn’t necessarily think of it yourself. Here’s the idea: find a ‘companion’ matrix whose characteristic equation involves the polynomial you are trying to find the roots of. Then solving this equation, or finding the roots of the polynomial, is equivalent to determining the eigenvalues of the companion matrix; this latter can be done using fast numerical methods. Of course, the question is how to determine the companion matrix— this would be practically useless if the process of determining the companion matrix was harder than trying to directly solve the polynomial by other methods.

The miracle is that the companion matrix is trivial to form: given a monic polynomial , the companion matrix is

I think this is the method Matlab uses to find roots of equations.

How does one go about solving strange equations like:

where all you know is is square summable? How would you find that satisfies this, and is there more than one?

It’s an interesting question to consider. I know the answer, but I can’t imagine how I’d work it out if I hadn’t seen someone else work it out.

Gender dysphoria, more commonly known as transsexualism, is the condition where a man (woman) believes that he (she) has the wrong physical gender. Transvetitism is distinguished from transsexualism, in that folk having the former want to play dress-up only part of the time; transsexuals identify with the opposite physical gender all the time. Gender dysphoria is an interesting topic, since it highlights the question of how gender identification and sexual identity interact.

Having read Middlesex, and seen La Mala Educacion, I thought I had a picture of what a transsexual is like, both of the male and female type. But— sympathetic as those have been— I’ve never read a non-fictional, non-dramatic account of what is like to be a transsexual. This is a great site (a collection of narratives) written by a transsexual that transitioned. Here’s an excerpt:

There is some logic to this, however extreme it may seem: a typical transsexual response to the hypothetical question “would you be happier if we could change your mind so that you are happy with the sex of your birth?” is “no”, because however much it may pain the transsexual, the notion of changing one’s personality to be ‘normal’ seems like giving up an essential part of one’s soul. All transsexuals ask themselves this at some stage, and all I have corresponded with have come to a similar response. Transsexuals see the condition not so much as a mental illness but as an unusual and inappropriate assignation - there exists in them a belief that at heart they are the opposite sex, and it seems that all would rather change the body than the soul.

Between taking the measure theory class and reading some probability stuff, I’ve been introduced to all sorts of interesting systems of sets over a universal set :

• Rings — nonempty, closed under finite intersections, finite unions, and differences
• Fields, or algebras — contain , closed under finite unions, complementation
• -fields, or -algebras — closed under countable unions, complementation, and containing
• -systems — closed under finite intersections
• -systems — contains , closed under proper differences, and increasing limits

The most useful of these is the -algebra; the others seem to be convenient starting points for generating a -algebra. For example, the following theorem is useful for extending a known property from a system to the -algebra it induces, :

Let be a -system and a -system in some space such that . Then .

These are just the systems that pop up in measure theory/ probability; I’m sure there are more out there— open and closed sets, in topology, come to mind.

Here’s why I needed to know about adjoints. In the last wavelets seminar, Papadakis proved the following:

Suppose is a Hilbert space, , . Define the linear mapping , . Then:

1. is a Bessel family iff is bounded
2. is a frame for iff is closed iff iff is invertible iff is invertible iff is invertible

I’m was having a little trouble with the first one, because it involved adjoints in a way that made the proof look incomplete to me (he defined another operator and proved its adjoint was bounded; to everyone else but me, it was clear the adjoint of this operator was in fact ) . I’m currently going over the second one, because I didn’t understand how the proof given.

Hopefully soon, I’ll have these notes up, and I’ll post a link here. They are pretty interesting, for people into signal processing or functional analysis, and easy to follow if you take a few theorems and results for granted. I’d say someone who’s read the multilinear analysis chapter of Rudin and has some minimal experience with using Hilbert space reasoning, could follow this easily. At least after I’m finished rewriting it at my level.

I finally have a good grasp of adjoints, and I need to record it for the next time I need it. Let be a bounded bilinear form, i.e. , is linear in the first variable, conjugate linear in the second, and there is an such that . Then by the Riesz representation theorem, there is a bounded operator such that .

Now, define a new bilinear form . Then . Therefore, is a bounded bilinear form, so there is a such that . We call the adjoint of .

Note immediately that , or in fancy language, is involutive. Together with the fact (from the last inequality above), this shows .

Whalla! I’m not sure this’ll be useful to anyone else but me. Probably if you know what the Riesz representation theorem is, you already know this …