ChapterZero

In a fortuitous turn, we happened to be introduced to convex sets on Monday with a view towards proving the classical result that a convex set is the intersection of the half spaces containing it. I say fortuitous because, although I have heard this mentioned before, I never made the connection between that fact and the often stated fact that it is easy to deal with convex optimization.

I just had the thought that this result has a great deal to do with optimization, because to find a suitable vector in the convex set, you can start off with a vector outside of the set, and project it onto half spaces containing the set, and refine each approximation by projecting it onto another half space. Of course, that doesn’t deal so much with the optimization part of the problem so much as it does with ensuring that the answer converged to is in the right set. Maybe optimization comes in with the choice of projection operators.

This Saturday, the UH math department hosted the 2nd annual middle and high school math competition. This is the second year I’ve volunteered. The first hour or so was pretty boring– another guy and I just stood around, sometimes directing team coaches to the lounge and snack room the department had set up for them; I also ate way too many donuts. But after completed tests started coming in, I got to help grading them.

I assisted with the Algebra II and Geometry tests– these weren’t entire lightweight tests– the UH profs who designed them tried to make them challenging, but not impossible. The year before, even the senior undergrads here had trouble with them. Unfortunately, the grades were still pretty dismal. I remember someone remarking that the highest score on one of the test subjects was a 2 out of 20. Although overall the performance was piss poor, I did see some pretty good grades on the Geometry test: the first test I graded, which ended up being the best in the subject, got a score of 26 out of 30. I haven’t seen the results, but I’m willing to bet that the scores tended to be higher in Geometry: because I noticed the unfavorable comparison between Geometry and Algebra II scores, and also because Geometry is more intuitive.

I wonder what the math department gets out of this: data on potential recruits?, educational statistics?

Today, I found out I was accepted to the applied and computational mathematics division at CalTech with full financial support; in a week I should be getting a letter with more details. Then I have to make the decision to stay at UH, in my comfort zone, or take the plunge. Assuming always that the support turns out to be enough– I’ve heard that the stipends offered by California schools don’t cover the horribly elevated living costs, and just recently, when I was buying a book at HalfPrice, the clerk mentioned her moving back to California would entail living in a cardboard box. Either way, I’m buzzed from the news.

On a different note. From the book Getting Started with Matlab: A Quick Introduction for Scientists and Engineers; in the list of exercises for Lesson 3 on creating and printing simple plots:

9. A very difficult plot: Use your knowledge of splines and interpolation to draw a lizard (just kidding).

Hands down, the funniest Matlab book I’ve seen, just for that ‘exercise’. And for some reason it reminded me of TAOCP– I guess because that’s the nature of some of Knuth’s ‘exercises’.

Working on our senior design project, which has been taking up all my ‘free’ time, has led me to discover one of my holy grails.

I have finally found a satisfactory, free image processing environment: NeatVision. It is a Java-based visual programming environment for chaining together image processing algorithms; it also allows the user to write Java code, while inside the environment, for new algorithm blocks. The samples of it in action provided at the website are really getting me wet

I’m surprised this is the first I’ve heard of it. I haven’t used it yet– I’m downloading the JAI right now, so I can run it– but it seems almost perfect; certainly it looks better than Matlab combined with the Image Toolkit, and it’s a lot cheaper too. The only major complaint I have is that it seems like implementing non-routine mathematical operations, like the sort Matlab is good for, requires you to write Java code. There should be some type of scripting overlay.

But it’s apparently within of perfection.

This result seems to be common knowledge, yet no one I’ve asked can give me a proof:

Why does diverge?

The consensus is that it should probably be attacked by transformation to polar coordinates, because that’s an easy method of proof for . But no one seems to know the appropriate generalization of polar coordinates.

Hadamard’s Three Lines Lemma says that if you know an analytic function is bounded on an open strip, and you know the bounds on the two boundary lines, you can interpolate between these two bounds to get estimates for the bounds in the interior. The formal statement of the theorem is something along the lines of

Let be analytic in the open strip , closed and bounded on its closure, such that when and when , where . Then when , for any .

A simple proof is given in Grafakos’ book, using the maximum modulus priniciple– yet another one of those little facts I didn’t know before this class.

Sometime next week I’m to lecture on maximal functions. I’ve only read less than a page on them, but the concept is something I have considered before, but didn’t think really led anywhere: looking at the averages of a function. Specifically, we can look at centered or uncentered averages, corresponding to the centered or uncentered Hardy-Littlewood maximal functions, defined as:

The idea is certainly intriguing. What kind of information about do these functions encode? I better have a good answer by Monday.