...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Tuesday, October 28, 2003

Today's physics class was fun, in a roller coaster-of-death sort of way. The professor who teaches our class is out of town this week, and asked a postdoc of his to substitute for him. The professor is a fantastic teacher. This postdoc was ... well meaning, clearly personable, and will probably make a good teacher eventually.



The trouble started in the first minute of class, where he announced that today we will be solving the Poisson equation, which is fine, and that we'll be using Green's functions to do so, which would have been fine if we'd known what those were. Then he gave a five second definition of Green's functions, and started using them. After people, including me, started wailing for mercy, he revealed that he had been under the impression that this was a graduate class, which it ain't. Or perhaps it was a joke, and he has a very sustained, deadpan sense of humor. I dunno.



In any case, I got entirely lost about ten minutes in, and stayed lost till the end of lecture. I suspect I was not alone, though there were some people who claimed that they followed things ok. Perhaps they weren't lying ;).



In any case, when I got home, I hit the books, specifically Redheffer's Differential Equations and googled 'till I dropped, and just got more confused. At which point I gave up, and started some serious studying for tomorrow's Japanese oral. Then, while Commanding All that I Surveyed upon the White Throne of Power, lighting struck, as usual. That is, I now think I understand why the hell Mr. Postdoc did what he did. I don't quite understand how he did what he did, however. So far as I could tell while taking a dump, there was a way more direct/intuitive way to get the solution, instead of ending up with complex integrations and rather horrific algebra. But probably I just don't actually understand it enough - I doubt he'd have gone through that much trouble if it didn't have a point.



I'll write up the plan of attack for the problem as I now understand it, to organize my thoughts, and then I'll ask if I'm right or wrong at the beginning of class on Thursday, setting myself up to look like a complete 'tard. Since I'm probably wrong.

Friday, October 24, 2003

For some reason my math homework was somewhat hard this week. It was on automorphisms of groups, and conceptually it's pretty straightforward material, at least at the level at which we're working at. An automorphism is just a one-to-one and onto homomorphism of a group to itself, and everything falls out from that. Most of the homework problems were straightforward, except for two of them. Those two fell into the evil category of exercises which are hideously frustrating to work on, as what they assert is intuitively 'obvious', yet each attempt at a rigorous proof falls apart in a flurry of tautologous algebra and definitions. Until that magic moment when a light goes on in my head, and I see _exactly_ how to do it. Then I feel conflicted, as on the one hand, I did it, yay, and on the other, the answer looks so simple that I feel stupid for not having thought of it much faster.



Anyway. I'm trying to decide what to take next semester. If I was King and I had superpowers, everything would be simple and I'd just sign up for all the classes that I want - miraculously, they don't even conflict with each other, time-wise. Unfortunately, that would make for a 21 credit semester, where 15 of those credits would be in rather serious upper-level math and physics courses, and 6 in a japanese course. That would be really, really hard, as I don't actually have superpowers. But oh, it's so tempting.



The trouble started when I found out that the university's offering a course on Differential geometry in the spring, with a special focus on differential forms and whatnot. This is nominally Part II of the two-semester differential geometry sequence, and since I haven't taken part I, normally I wouldn't care. This time, though, the flyers for the course take pains to point out that the Diff. Geometry I course is really only a 'mathematical maturity' prerequisite, and you don't actually need to have taken it to enjoy the course. If you've had some other upper-level math courses, you should be fine, or so the leaflet claims. This is very tempting. Also, I've asked around, and the guy teaching it is supposed to be fantastic.



On the other hand, I'd planned to take statistical physics/thermodynamics in the spring. It's a required course, and generally useful, though I can't say I'm especially excited about it - but people tell me it's pretty cool. Also, the guy teaching it next semester is reportedly good. All the other courses I'm considering are lock-ins for various reasons, so it's between thermo and diff. geo. It's a tough choice. Next week, I'll try to find who's teaching thermo next fall, and I'll go chat with the diff. geo. professor, to see if he thinks my background is appropriate for his course.



I'm beginning to have fun in school for a change. It feels pretty good.



(And I hope this doesn't jinx it, knock wood and plastic)

Wednesday, October 22, 2003

So I had a fun time in my Japanese class Monday morning. Every day, we're supposed to memorize several so-called 'core conversations', and act them out in front of the class. Each conversation focuses on some aspect of the grammar/vocabulary we're supposed to be learning, and it seems to be a nice learning tool. Anyway, as the semester has progressed, the conversations to be memorized have gotten longer and longer, and they're now becoming a bit challenging.



Anyway, Monday's conversations were pretty serious, but I buckled down over the weekend and memorized them really well, and could regurgitate them at the drop of a hat. Ahem. Anyway, at the beginning of class the professor called for volunteers to be the first to go through the nastier core conversation. I raised my hand - first, I'd done an awesome job memorizing it, and second, I wanted to get it out of the way.



I delivered my first line prettty well, and then while listening to the response, something in my head decided to momentarily reflect on the fact that my partner for the conversation was actually quite a pretty girl. Unfortunately, as soon as I thought this, I somehow forgot the rest of my lines. Well, sort of, as I just said them really slowly and with mistakes.



Anyway, the lessons learned are 1) Don't volunteer, lest ye look like a total 'tard! 2) Memorize harder, and 3) Henceforth try to force self to view the people I recite stuff with in Japanese as hideous tentacle beasts, so as to minimize the chance of distractions. Anyway, what's curious here is that I don't normally blue-screen when confronted with pretty ladies. I'm not sure what the fuck happened in this case. A case of the Mondays, I guess.

Monday, October 20, 2003

A snippet from Tennyson's Ulysses:


The long day wanes; the slow moon climbs; the deep

Moans round with many voices. Come, my friends.
'T is not too late to seek a newer world.
Push off, and sitting well in order smite
The sounding furrows; for my purpose holds
To sail beyond the sunset, and the baths
Of all the western stars, until I die.
It may be that the gulfs will wash us down;
It may be we shall touch the Happy Isles,
And see the great Achilles, whom we knew.
Tho' much is taken, much abides; and tho'
We are not now that strength which in old days
Moved earth and heaven, that which we are, we are,--
One equal temper of heroic hearts,
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.

Oh yes. If I have the time and inclination, I really should write a review of Dan Simmons' latest novel, _Ilium_. For now, all I'll say is that it's an absolutely mindblowing piece of work, will win all kinds of awards, and has made me want to reread the Iliad and the Odyssey, as well as give Shakespeare, particularly _The Tempest_ and the sonnets a try. More, I now need to carefully read Browning's poem _Caliban Upon Setebos_, Tennyson's _Ulysses_, and give Samuel Beckett a try. This book references all these and more, though the emphasis so far seems to be on the Iliad and the Tempest. Simmon's _Ilium_ is spellbinding in its own right, and it makes a broad tapestry of the Western Canon (classical literature) seem vital again, even to a philistine such as myself. Run to your nearest bookstore and grab yourself a copy.

Ooh. Blogger's updated their interface. Shiny. And sorta blue. Hmm.



I pulled a stupid in my math homework this week, but it's already turned in. The assignment was to show that the group of non-zero complex numbers under multiplication is isomorphic to the group of two-by-two real matrices of the form ((a,b),(-b,a)), or something like that. In any case, the 'obvious' approach is to write an arbitrary complex number z as z=a+i*b, and then just stick the a and b into the matrix, and then show that this gives a one-to-one and onto homomorphism, and we're done.



I was too stupid to do it that way. The trouble is, when someone tells me to think of a complex number, I don't think a+i*b. Instead, I think r*exp(i*t), mainly because this seems to be more useful a lot of the time, and it's the form I've use all the time in physics and in my other math pursuits. I also happen to know from previous reading that the group of unit complex numbers, i.e., things of the form z=exp(i*t), are isomorphic to the rotation group in two dimensions. (U(1) is isomorphic to SO(2,R)) Visually, multiplication by a unit complex number is just a rotation of the plane (thank you, Needham's _Visual Complex Analysis_). So this makes sense. Anyway, I set out to first prove that U(1) is isomorphic to SO(2,R), and then applied that to the actual problem, which can be done by factoring out the square root of the matrix determinant to get an orthonormal matrix times some scaling factor, which is what's equivalent to a general complex number z=r*exp(i*t). This was actually kind of fun to do, and it's a generally cool proof, but it was way, way more work than was actually called for, and the grader may not appreciate having to read through a page-long argument where a few lines would have sufficed. Ah well - I'm pretty sure my proof is right. It's just stupid overkill.



I wonder if I can ever get back intro the habit of updating this thing on a regular basis. Heh. And this page needs to be redesigned, bad. The CSS sucks. Maybe later.