ChapterZero

Recall the definition of a (Frechet) differential of a mapping between two Banach spaces and :

If , and for there is a linear mapping such that

then is said to be differentiable at with differential . If is defined for all , then is said to be differentiable with differential .

Now assume is differentiable and defined between an -dimensional Banach space and . Let be a basis for , then

,

so . Likewise you can show that if , then

Here’s the question: let be defined by . What’s ?

One of my roommates is a chemist; a conversation about cooking somehow wandered into an introduction to the problem of solving crystal structures. Apparently, to determine the structures of proteins (such as DNA), chemists first find a way to generate large, pure amounts of the protein, and then find a way to crystallize it. Then, using data from single crystal X-ray diffraction, they solve for the structure of the crystal. For several reasons this is the hardest part of the process: large proteins yield small crystals in general, which means you can’t collect as much diffraction data, and there’s always some distortion in the data due to imperfections in the lattice structure and whatnot.

This struck me as a noisy tomography problem, and got me thinking of maybe looking into it as a research avenue. My roommate tells me that solving crystal structures is currently something of a black art, involving years of intuition-building experience and patient collection and refinement of data. He says this is a problem that has resisted the efforts of some of the greatest minds of science, so while searching for an algorithm to directly solve the problem is overly ambitious, an algorithm that somehow reduces the effort would surely be a welcome gift to structural biologists. Supposedly Caltech offers a self-contained course in X-ray diffraction and introductory structure solving, so after he gets back to me on the details and whether it’s offered this fall, I might end up switching out one of my math classes for that course.

Akismet has crapped out on me lately; consequently today alone I’ve waded through 700+ spam comments in moderation. In retaliation, I’m doing a summer cleaning of the site. I’ve upgraded WP from v2.0.1 to v2.2.1, mostly so I don’t fall so far behind that plugin compatibility is a major issue; along the way I’ve switched to the contentious Garland theme. Maybe someday I’ll get around to customizing a theme to fit my unique needs, but until then I’ll use the best of the prepackaged stuff.

In the course of upgrading, I pretty much reinstalled from scratch, only saving the original Mysql database, so there may be a few missing links to files that I had in the WP directories, and such. I’m going to resolve these as they occur, so if you find something you were looking for is no longer there, just let me know.

Apparently the Now Reading plugin has been broken by the upgrade. Also, I just started using Spam Karma, since Akismet is resisting my attempts at diagnosis.

Here’s a funny question: why is the region around a star within which planets are habitable (water may be found naturally in its liquid state, etc.) , the habitable zone, also known as the Goldilocks Zone? Hint: read the tale again …

I’m trying out the Now Reading plugin, which helps you list the books you’re currently reading, planning on reading, or have already read. I won’t be listing any of the math or physics books that I’m currently ‘reading’, since those change from day to day. Instead expect a roster of sci-fi, fantasy, and general fiction and non-fiction (as I try to expand my horizons). I welcome reading suggestions for any thought-provoking, well-crafted books.

As a young child, I remember night thrilled and threatened me. On moonless nights, the rural parish where we lived was pitch dark; if you stood still long enough for the night noises– crickets and such– to fade into the background, the gravid darkness and the wind unceasingly blowing in off the ocean could give the impression that you stood on the edge of the world, just a step away from a forever drop into oblivion. Either that, or the impression that you floated in a timeless, limitless womb of darkness, utterly alone. Night demanded respect and invited introspection. The nearest I’ve come to recapturing those feelings in America was several times accompanying my father on a construction job — a contract for the renovation of a small, improbably located Baptist church– that saw us driving for miles on the twisty, unlit dirt roads of rural Texas.

Nights in cities and heavily urbanized areas have none of the character of those country nights. Pervasive lighting and the attendant 24-7 mentality have erased any serious distinction between day and night. It saddens me to think of the kids growing up now who will never enjoy a truly dark night.

I (discombobulatedly) talked to Owdhadi about a couple of projects he’s involved in: one is on elasticity cloaking, the other one is on using stochastic ballistic motion to harness waste energy. Of course, the former involves PDEs, and the latter involves stochastics; both sound cool. I was pleasantly surprised. However, he recommends that I find my research advisor and work with him or her for the summer, although if I decide to, he left the option open of working with him. I see his point: he doesn’t have the money to take on another student, and it’s better if I start looking for an advisor sooner rather than later, to find someone I can work with and who can support me; the summer could serve as a lower pressure time to get acquainted with my advisor’s idiosyncracies and determine what I need to know to work with him or her. I’m going to do some research over the weekend– look at some webpages, skim through some papers– and see if there’s anyone doing anything interesting. Hopefully by Monday I’ll have my eye on someone(s).

After speaking with him, since the subject of financial support came up, I remembered that I’m going to be TAing and taking 4 courses next semester. That’s going to be a pain.

Turns out I’m supposed to be registered for 36 hours of research credits for the summer. After talking to some of the 2nd year grad students, I was under the impression I could do whatever I wanted to for the summer, but not so says the registrar. It’s already three weeks into summer, so … I feel like a fool asking someone to work with them now, but it must be done. I just emailed one of the ACM professors, who specializes in stochastic processes and PDEs, and asked him if I could speak with him about doing research. Since I don’t know much about either one of those subjects, I actually asked if I could do a research project or a reading under him. One of the other grad students told me that he’s working on protein folding; that’s sounds very interesting. He’s my first shot into the dark. I hope I don’t have to make many more; I’d like to get my stuff in order ASAP. I suppose if he’s too busy or declines for some other reason, I can see if my academic advisor will let me take a reading under him. 36 hours of reading? Ha ha…

I was recently invited to help rework BasicCAS into a full-fledged Mathematica parser for the SymPy project, a challenge which I have accepted. The idea is that a Mathematica parser (along with a working backend, a much harder goal to meet, I’d say) would allow SymPy to benefit from the large body of free Mathematica code floating around out there. Part of the reason I’m taking this on is because I’d not like to see SymPy turn into a melange like Sage– the idea of tying together disparate CASes and mathematical libraries with a glue language is aesthetically unappealing to me, and pragmatically, what’s the point of dividing development effort between two Python-based metaCASes? A well-defined global syntax is the biggest preventative measure I can think of.

So now I have some fun goals for the summer, to round out my list of not-so-fun goals: figure out a grammar for Mathematica’s syntax (this may be, literally, impossible: I have heard it said that the only grammar Mathematica follows is the ex post facto grammar imposed by its parser’s behavior); write a parser for that grammar, test it; track down the inevitable incompatibilities, attempt to improve the grammar; recurse. It’s going to be a bitch, but fun, hopefully!