ChapterZero

There’s a wide range of physical phenomena out there, barring even quantum and relativistic effects, that are utterly fascinating. Take cavitation, for example– imagine that under non-esoteric conditions you can form bubbles that when they collapse radiate with black body termperature equal to the sun’s! Almost enough to make you want to be a physicist, or a mechanical engineer. But then, you realize you’d rather just daydream about cool ways to exploit those phenomena than do the grunt work of gathering data and formulating theories to explain and properly harness them.

I just got back from the latest departmental colloqium talk, given this week by Richard Tsai from UT Austin. It was a nice talk, about path planning and visibility optimization, reminiscent more of the style of computer scientists’ or engineers’ talks than mathematicians; that is, he used nice pictures and got his point across with few references to equations for justification. Of course, this was in large part possible because of his subject matter, so I can’t say it was better than the talks before it, which flew over my head… just that I enjoyed it more.

In the course of agonizing over my stochastic processes midterm, I realized that a really good book on probability would be one that approached it from applications to gambling, the way the field originally developed, and took it from the basics all the way up through its modern, starvedly rigorous heights, all the while applying it to the solution of interesting/challenging problems. I.e., it’s slightly useful when you motivate martingales by mentioning their usefulness in modeling fair games, but it’s enlightening when you show me that usefulness. Anyone know of any such book?

So, I started working on my stochastic processes midterm this afternoon– I decided to just jump right in after my roommate said he did it and found it almost suspiciously easy. Ok, I figured, I’ll be able to knock this out of the way in a couple of hours and just enjoy the rest of the weekend. Needless to say, that’s not how it’s turning out. Thankfully we have 6 hours to work on it, and we can start and stop as we like, while still thinking about the problems and consulting texts in between. I’ve already spent about 1.5 hours on it, and completed none of the problems. After I’m done, I’ll post it, just to show how convoluted word problems can be– part of what’s making this hard is decoding the English statements into precise probabilistic statements. It’s still fun, though.

I completed my functional analysis midterm today– didn’t do too well. One of the problems, to show that bounded sequences in the space which converge pointwise to to a function in the space converge weakly to the same limit, really stumped me. On the positive side, it gave me the first opportunity I’ve had to appreciate the usefulness of Radon measures– using the characterization of the dual space of as the set of Radon measures on the space, and the dominated convergence theorem, this result is obvious. On the other hand… I wasn’t able to do it using the tools we learned in class so far. The problem would be straightforward if I could see why the point evaluation functionals are dense in the dual space, but I wasn’t able to see why that should be the case; in fact, it just seems intuitively wrong to say that.

My 6 hour take home stochastic processes midterm is due on Tuesday, and that course has been bewildering me, so I decided to do a couple of practice problems from a probability theory problem book before I got started, just to get the juices flowing and fix techniques in my mind. The first problem was to find the distribution function and expectation of the random variable defined by the magnitude of the projection of a point uniformly randomly chosen on the perimeter of the unit circle onto the x-axis. The interesting thing to note is that if you calculate the expectation over the sample space, you get a simple integral:

On the other hand, if you evaluate it in the image space using the distribution function, you get a more intimidating integral:

This suggests that sometimes probabilistic interpretations can be used to reduce hairy integrals. Of course, finding a useful interpretation may be very hard. Even in this case, it’s hardly obvious that the second integral is really just the expectation of a relatively simple random variable.

Of course, the evaluation in the probability space is not always easier than the evaluation in the image space. For example, if we change the r.v. to the square root of the above r.v., then

is a heck of a lot easier, I imagine, to calculate than

where is shorthand for the appropriate type of elliptic integral.

I’m at the library trying to figure out what conditional variance is; the problem set due Tuesday involves it, but we didn’t discuss it in class. On the way here, I had an amusing run-in: while I was walking alongside one of the streets that borders the campus, a car pulled up alongside me, and a Hispanic woman about my age leaned out of the passenger side and started asking me questions. I presumed the guy driving the car was her boyfriend; it seemed like they were having a night on the town. My first thought was that she needed directions to somewhere on campus, and seeing that I had a book-bag in hand, she figured I was a student here. Fair enough. But her first question was, “Are you a student here?”, sort of incredulously. Hmm… would I be walking around the campus at 9 on a Sunday night otherwise? Must be because I’m black; that doesn’t offend me, because I’ve noted the dearth of black people here myself: there are certainly under 100 black students, probably less than 50. Then she questioned me about the tution, which I answered as best I could. Then she asked me if they’re any girls on campus, how many, are they sexy, you look nice, do you think I’m sexy? I felt like laughing and grimacing at the same time, because I’ve been in this kind of situation before– being poked to see what reaction will ensue–, and I had seen it coming, from the moment and speculative way in which she asked me if I’m a student here. The fact that she felt free to ‘flirt’ with me while her guy was sitting right next to her just pointed to how this amounted to a form of entertainment. I just told her it’d been nice talking to her and walked away. Sometimes I have time to deal with people like that, sometimes I don’t.

Back to probability.

I’ve been taking a random walk through Engel’s Problem Solving Strategies– reading it linearly is more of a chore than fun. The best reading method seems to be, read the first couple paragraphs of each chapter where the principle is introduced, and skip to the problems; you can look back at the worked examples later if you get stuck on applying the principle.

The first chapter deals with using invariants as a problem solving technique. This technique seems to be useful for dealing with problems which ask whether a specified state can be reached by following an algorithm. My favorite problem so far (of those I’ve managed to solve) that employs this principle is:

You may switch the signs of all numbers of a row, column, or a parallel to one of the diagonals. In particular, you may change the sign of each corner square. Prove that at least one -1 will remain in the table.

Assignments due each day… ahh!!! After the end of this week, I’m definitely going to manage my problem sets more carefully– start the day they become available, and do a problem per set per day.

In the interest of developing my probabilistic intuition and facility with counting arguments, I went looking for probability problem books at the school library. The search reconfirmed by earlier suspicion that the library system here is in some subarea of mathematics smaller than UH’s. I did manage to find a good book, Probability Theory: Collection of Problems, published by the AMS. It has a range of exercises from counting arguments through martingales and functionals of wiener processes, and doesn’t skim on analytic questions– just what I’m look for in terms of exercises to bolster my understanding of more traditional monographs.

I got stuck on the 13th problem in the book:

What is the largest number of parts into which straight lines can divide the plane?

so eventually I looked in the solution section. The solution begins with “Evidently, in order that the number of parts be maximal, it is necessary that no three straight lines intersect in one point and no two lines be parallel.” Hmm– it seems like a natural condition, but that hardly makes the truth of it evident.

I remember encountering this problem before in Knuth’s Concrete Mathematics, and seeing a convincing development of a solution, so I’ll see if I can find a copy later, but it bothers me that what this book says should be evident is not, at least to me.

Stochastic Processes is definitely my favorite class– even though it’s also my hardest, and the one where I make an utter fool of myself every class– although, maybe that’s why I like it so much. Before this summer, I was confused by and completely disliked ‘formal probability’, hated ‘informal’ probability, and only grudgingly admitted to an interest in ‘engineering’ probability (aka stochastic processes). But after all the skimming and talking to people about probability, and studying markov chain monte carlo techniques that I did this summer, I have fallen in love with the subject. I don’t have the feel of it yet, but I really like it.

Here’s a neat problem, one of the warm-up questions the prof asked at the start of last class:

You have a box full of ropes. You put your hands in the box, pick two free ends and tie them together. You repeat this process until no free ends are left in the box. What is the expectation of the number of loops at the end of the process?

Finally, should I blow off tonight’s karate practice, and instead go to see a preview of an upcoming episode of Numb3rs and participate in a live cast interview afterwards, or not? Not a hard decision to make– karate is sooo much more interesting than Numb3rs– but it’s nice to need to think about it for a split second or so.

There are four books in the Princeton Lectures in Analysis series:
Book I. Fourier series and integrals.
Book II. Complex analysis.
Book III. Measure theory, Lebesgue integration, and Hilbert spaces.
Book IV. A selection of further topics, including functional analysis, distributions, and elements of probability theory.

I’m currently reading Book II, for ACM101, which assumes you’re already familiar with complex analysis. This is a great book– I read the first three chapters in about two and a half hours (just skimming the first, on basic facts in analysis and terminology). So, I’ve already seen the big three topics: contour integration (Goursat/Cauchy’s theorems), regularity, analytic continuation. I wish I’d picked this book up before.

Maybe I’ll read the other three, if they’re as good as this one. That would be a good way to review all the stuff I should already know.