ChapterZero

We’re learning about distributions in the Harmonic Analysis class; these are basically generalized functions. Technically, there are three different types of distributions, each of which is a continuous linear functional from a space of test functions. A distribution is such a functional on the test space , a tempered distribution is a functional on the test space (the Schwartz space, consisting of functions whose derivatives of all orders decay faster than any polynomial), and a distribution with compact support is defined on the test space .

I am prepping for my functions of a real variable midterm next friday. So far, I’ve reviewed basic point-set topology and metric spaces, and the Baire Category Thm, but not to my satisfaction. This is a pain– I can read a lot of material in one day and understand it, but unless I see it over and over and over and …, I will forget it by the next day. So basically, I need to rereview every day– even doing hard problems that require me to understand the concepts doesn’t really help. It would, if I could do more than one every couple of days, but I don’t have the time for that.

I have left to review: compactness and local compactness, LCH spaces, advanced topological thms, the Hahn-Banach theorem and consequences, consequences of the Baire Category Thm, Stone-Weierstrass thms, all the different topologies on function spaces. Looks like I have no choice but to dedicate myself to math this weekend— I’m going to have fun!

I also have to finish up the color band recognition portion of our automated resistor sorting project. One of my group members is going to his graduate school orientation at Rice this weekend, so he left his laptop with me, and I’ll be using it to work on the Matlab code. That suits me just fine– before I had to come to school so I could have access to the Image processing toolbox, and have access to a moderately fast computer.

Also, I have to start writing my antennas lab paper on ‘Electromagnetic Biomedical imaging modalities’– basical an overview of X-ray CT and MRI technologies. I read about the basics of MRI during my spring ‘break’, and it seems like I’ll enjoy writing about it. I already know I’m going to enjoy writing about CT.

Oh, right. I have a linear algebra test on Tuesday also.

I visited CalTech this week, Monday through Wednesday (I’m on what passed for spring break), and spoke to 5 of 7 professors in the Applied and Computational Math Division, and maybe 7 graduate students. I also met two of the other prospective students (one of whom is from Rice, also in Houston). The atmosphere was a lot different this time around: when I visited through the GradPreview program, I was not very impressed by or attracted to the school, but this visit solidified my decision to attend. The problem with the impression I got from GradPreview is that it was unduely biased by the experiences of graduate students in the experimental sciences— of course they’re going to be bitter and resentful. But I will be doing math, which means not being stuck in the lab for the entire 4 or 5 years it takes me to get a PhD; the math grad students seem fairly happy with their schedules.

It seems that if I want to work with an ACM professor, as opposed to finding an advisor in one of the other departments, I’m going to have to develop an interest in PDEs. Candes, who I currently want to work with most, on image processing and signal representation stuff, is very popular with the current crop of grad students, who will be choosing their advisors in the year that I come in. So he probably will not have any money or time to be my advisor. The other profs are doing cool things: two of them are investigating multiscale methods, which deal with efficiently interfacing models of phenomena at different levels of detail. A simple example of this would be adaptive meshing for the solution of PDEs– where it is sometimes important to model a small region with almost quantum mechanical detail, and also important because of time considerations not to use the same level of detail in the majority of the regions considered. Multiscale methods seem relevant to a large range of problems, including protein folding, and other things like that. Then there’s one prof working on mathematical biology– specifically, designing dna base pair sequences to meet topological and dynamical constraints, for use as nano-machines. The mathematically interesting portion of his work involves optimization issues.

Anyhow, I found out Candes has a site on the cool stuff his group has been working on lately, called compressive sampling. The idea is that by minimizing total variation (difference in l1 norm), you can effectively reconstruct functions that have a sparse representation in some basis from a relatively small number of random samples/projections; because this gives a nice method of (lossy) compression, it’s sometimes referred to as compressive sampling. Simple idea, cool mathematics. Apparently it has all sorts of applications; one of the graduate students is even attempting to use compressive sampling to solve PDEs.

Why is it that given a matrix , there is always a number such that is invertible when ? This doesn’t even seem true to me, but I think that’s the gist of something I saw a while back in a book– it was stated without proof.

In general, what is a continuity argument w.r.t. matrix theory?

There is one lesson I’ve been constantly relearning this semester: in analysis, once you get past measure theory, expect the concepts to be hopelessly abstract at first. The ones that aren’t are few and far between; most of the time, I’ve found myself asking after a particularly involved proof, why did we put ourselves through that? The only things you can do to pierce the veil of this abstractness are: flat out ask what the point of some new concept is, ask for meaningful applications or counterexamples for the theorems, find several good books that approach the same ideas from different angles. Pretty much, learning analysis has become like the process of learning topology: you see some pretty and elegant concepts, but for the most part, you’re cramming knowledge down your craw now that will do you good later. My latest struggle, initiated just today, is with topological vector spaces, or more precisely, locally convex topological vector spaces: why do we care that a space is a LCTVS?

VNC is awesome! I just started using it today. I’m sitting in Papadakis’ office on one computer running Linux, and he’s on the win2K machine, vnced into my box. He’s learning matlab, and the math department only has licenses for Linux, so he’s running it over vnc on this box. A nice side-effect of this convoluted way of doing this is that every once in a while, I can glance at what he’s doing and help him out. Unfortunately, the math department doesn’t allow vnc to work over the Internet, for security reasons, so I can’t work on these machines from home.