...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...

Saturday, September 28, 2002

Heh. a*b does not generally equal b*a. By 'generally', I refer to cases where a and b are members of a general group - that is, they ain't necessarily the sensible numbers everyone meets in school, but more perverse things. So, ab != ba. Having drilled into myself with some exercises, I read this: [Uj(g)f](v) = f(g-1v). This somewhat intimidating (to me) equation comes up in the definition of 'representations of SU(2)'. SU(2) is the name of a group. It happens to just be a set of matrices satisfying some equations, but the point is that it's a group, and as such is a very ethereal beast. A lot of the time, what you really want to know is what a group does, in the 'real' world. Well, to do that, you need to figure out what the representation of a group is, for whatever 'world' you're living in. Hrm. This attempt at a metaphor is going nowhere fast.



Anyway. The interesting bit, as far as I'm concerned here, is the g-1 part. See, in proving that the equation given above does indeed define a representation, something rather strange is done (implicitly, which is worse): (a*b)-1 = b-1*a-1 is assumed to be a true statement. And remember, the beasties we're dealing with here are NOT commutative, a*b is NOT b*a. So what gives?



Well, it took me a while, but I've figured it out. See the a-1 part? Well, normally, the superscript means exponentiation. And for a normal, real number, which I'll call c, c-1 = 1/c. Simple enough. c-1 is also used to denote the inverse of c. This just means that c*c-1 = 1 -- the inverse of a number just means 'that by which you need to multiply to get 1'. And it works out: c*c-1 = c*1/c = 1. Everything's peachy. But that's with normal numbers.



We're not dealing with normal numbers. The -1 exponent here ONLY denotes 'inverse'. That is, a*a-1 = I, where I is basically a general version of the concept behind the number 1 (Meaning, a*I = a. Just like, say, 4*1 = 4.)



So. Let's take the inverse of a*b, and play with it. (ab)*(ab)-1 = I. That's by definition of 'inverse'. Now let's expand everything. a*b*a-1*b-1 is still supposed to equal I. But we've got a big problem. See, because ab != ba, we can't rearrange the terms in the above. We're stuck doing things in the order they appear. So, we've got a*(b*a-1)*b-1. The quantity in parentheses won't be equal to I, obviously, because the inverse of a ain't the inverse of b. So the stuff in parentheses will equal something we'll call c, and we'll have something like a*c*b-1, which obviously won't equal I! So everything blew up. The reason it blew up is that we expanded things wrong, thinking naively that (a*b)-1 = a-1*b-1. But look! If it was equal to b-1*a-1, everything would work out. The expansion would look like a*b*b-1*a-1=a*I*a-1 = a*a-1 = I, and as you can see everything lines up, and comes out to equal I, as it needs to.



If you stare at it long enough, it becomes clear that it has to be like this. Took me a while.



Anyway, I'm having a hell of a time understanding this business about representations of SU(2). I think I've got a handle on the spin-0 representation, but the spin-1/2 representation is giving me trouble (because of that damn inverse that shows up where I don't want it), and spin-1 and dual representations are another story all together...



My problem with spin-1/2, in brief. The book is asking me to show that for spin-1/2, elements of SU(2) act on C^2 by matrix multiplication - that is, spin-1/2 is the 'fundamental' representation. My trouble comes from having trouble seeing how the fuck else they could be acting on C^2. I mean, representations of SU(2) are defined by [Uj(g)f](v) = f(g-1v), where the f's are functions that are degree-2j homogenous polynomials, v's are elements of some vector space, in our case C^2, and U_1/2 being the name we're giving the representation. Sorta. My terminology is probably all fucked up. Anyway, the reps look like f(g-1v)! g^-1 is an element of SU(2), since g is. Elements of SU(2) are 2x2 matrices. v is just a vector, a 2x1 matrix. Well, you write two matrices down like that, and you're stating you're doing matrix multiplication. What the hell is there to show?



What's bothers me even more is that say we take j = 234. Well, the reps still look like f(g-1v). You're still doing matrix multiplication. But you can't have more than one fundamental representation, can you? That would rather contradict the notion of it being 'fundamental', wouldn't it? Or would it? I dunno. Of course, the functions f do change as j changes - they get hairier and hairier, rather rapidly. Hmm. As you can probably tell, I'm confused.


Sunday, September 22, 2002

So. It's been a while. Long story. But not one I'll be talking about this time.



I've seen Mr. Thumpy twice in the interregnum. He ran off swiftly both times - but that's normal. I've also learned a lot of new words. (No, 'interregnum' isn't one of them, I've had that one in my repertoire for a while, waiting for a chance to whip it out.) Unfortunately, I only recall one of them at the moment: 'arras'. This means 'wall hanging or tapestry.' As in, "I hid my fat arse behind the crass arras." I have learned these words from a book that I'm reading. Now, not to toot my own horn too much, but it is rare that I happen to find a book which utterly obliterates what feelings of self-worth I have when it comes to English vocabulary. Well, John Clute's Appleseed does so, and then dances on the ashes. But what a breathtaking dance...



I'm only about a third of the way in, but I can already recommend it, highly. If you read the Amazon.com reviews, you'll find that people seem to either love or hate this book. The ones that hate it seem to mainly complain about a) vocabulary and b) lack of plot. The latter assertion is bollocks. The plot is there, in spades. It's just that if your vocabulary isn't up to snuff and/or you don't wish to hit the dictionary and/or you're easily flustered by a barrage of obscure words that seem to be gratuitous (but are actually just right, once you look them up) then you're not likely to follow the plot.



(Well, ok, there's a review there that liked the book and the vocab and still thought it was thin on plot - but I disagree. I love the guy's description of it, though: "the outrageous lovechild of a menage a trois between Vernor Vinge, Iain M. Banks and the Oxford English Dictionary".)



Clute manages to set up image after glowing image, layer upon layer of allusion, until it all starts boiling in your head, and you go from cursing his erudite show-off arse to bugging your eyes out and trying to hang on for the ride. I recommend it.



Anyway, enough of the purple prose. There are potential reasons to avoid the book. First is, as I have said, vocabulary. For me, it has been extremely challenging. Not unlike reading a really, really good technical text in terms of difficulty, but it is a far more visceral thrill. I, at least, have gotten to the point where I just 'go with the flow' where I don't understand some of the words in a sentence - most of the time, it works out. Sometimes I have to double back and cogitate, or hit a dictionary. So, unless you are really confident about your command of English vocabulary and/or aren't looking for a workout, don't ignore this: it is a potential fun-killer, as those negative amazon reviews show.



Another reason is that John Clute wrote it. John Clute is the author of the canonical science fiction encyclopedia. He really, really knows the genre. He's a professional science fiction/fantasy critic and scholar (his review column* on SFW is fantastic). And this book is his first novel. There are allusions within allusions in Appleseed. Even the warts have allusions. I don't think this is a fun-killer, as the vocab issue can be - if you don't see the allusions, it's no big deal, they're powdered sugar on the cake, as it were. I suspect I'm missing most of them. But this might not be the best SF book to read if you've never read SF before.



With all that said, hit your local library or bookstore and check it out. I'll probably have more to say about it once I'm further into it.



Appleseed also happens to be that very rare case of a science fiction book with a Cover That Does Not Suck. Nay, this is a Cover That Rocks (see above link). I had thought this was an all but mythical beast, but here it is. Uh-oh. Rant coming on...



I'm twenty two years old. I may be a loser extraordinaire, but I don't fucking want to flaunt it. Beeing seen with most science fiction or fantasy books is flaunting it. Just who are these things being targeted towards? I don't want pictures of fucking half-naked vixens on the cover. When I want pictures of half-naked (or fully naked, for that matter) fucking vixens, I'll jump on the bloody information superhighway, thank you very much. I also don't want pictures of strapping young half-naked bucks on the cover, whether they are rescuing the aforementioned vixens, sexing them, or just striking a heroic pose. I don't want pictures of tentacled hell beasts fondling said vixens, or cheesy mid-90s-quality rendered pictures of implausible-looking guns-n-FutureCities-n-spaceships on the bloody covers. In short, I don't want to walk around with a book with a cover that screams "I am a sexually bewildered twelve year old boy in the throes of early pubescence, and/or a twenty-two year old loser who is mentally twelve".

In the peachy-keen word of platitudes, I'm not supposed to "care what other people think." Fuck that. When the day comes when what other people think of me stops having rather direct effects on my life and prospects, I'll stop caring. Until then, it's one of the dumbest platitudes I know. It works if you're Dick Feynman. Otherwise, you don't get away with it.



And people wonder why there are fewer women/girls reading science-fiction than boys/men, and often taped-eyeglasses-nerds (Represent!) at that? Gee. I wonder. Could it possibly fucking have to do with the fact that the overwhelming majority of SF books seem squarely targeted at the twelve-year-old-loser-boy demographic? Or, to put it more accurately, their marketing, the things everyone sees first - the bloody covers - are.



The sad thing is, the covers don't have anything whatsoever to do with the book, often. Grr. That is a rant for another day.



In other news, I'm beating my head against a corrugated wall, self-study-wise. I'm making progress, but right now, it's Count of Monte Christo style. Slow.



I'm having a lot of trouble with tensor products of representations. (Or of anything, for that matter.) Turns out if you tensor product-ise two irreducible complex one-dimensional representations of U(1) (i.e., the unitary group in one dimension), you get another 1-dim complex irreducible representation of U(1). Which is strange. Because the idea was, I'd thought, that 'tensor products' are a way of making large representations from smaller ones. And 'irreducible' representation seems to be sort of the bloody definition of 'small', in context. Then again, I see glimmers of ways of why it might all make sense...



I'm tired. To infinity and beyond! (Well, a bit of a quick net-news-browse followed by bedtime, but whatever.)




* - Take the linked review of Neil Gaiman's Coraline, for instance. Who else is going to pull off a literary review of a children's book full of stuff about memes, Friedrich Nietzche, the virtues of being a cow, and the salutary effects of getting high, and find a way to use the word 'cloaca' in the process? Whee!



Wednesday, September 11, 2002

OMG. According to Einstein, left-handed people are travelling backward through time, man. Also, right-handed people are always going to misunderstand lefties, because we can't reach them. It all matches up, especially if you've Bob Marley crooning on your stereo! And it's all SO(3,1)'s fault, too, which means the CIA is involved. Trippy.


This post brought to you by an attempt to imagine what someone who is either a) massively abusing certain illicit substances or b) someone who is really much too sleepy to be awake would think about what I've 'derived' last night. Emphasis on the 'night'. Actually, I don't need to imagine (b), as it's true. I'll probably explain what I'm on about later, but just looking at the previous post will tell you how much weight to give to my temporal assertions.

Friday, September 06, 2002

Special sneak preview: later tonight, if I get around to it, and more importantly, if I actually get it, I shall post my attempt at 'proving' that GL(n, R) and U(n) and SL(whatnot) and their cousins are Lie groups. (This is like #4, part II in Baez and Munian.) I've been struggling with this for two days, and I'm still not done. The best attempt I've got so far uses a bit too much handwaving to actually be a proof, which is highly unsatisfactory. I need to learn how to do proofs, damn it! And with reasonable speed, too: taking two days to kinda-sorta do Exercise #4 isn't cutting it.



Sigh.



On the bright side of things, the Swedes are my kind of blokes. I think I'd enjoy living in Sweden.

Monday, September 02, 2002

Tonight's jog was far less exciting adventure-wise than yesterday's: no Mr. Thumpy sightings, no rain, glasses, no close encounters of the third kind with parked Hondas. On the other hand, jogging-wise, tonight was great. It's about sixty-five degrees Fahrenheit*, ninety-percent humidity, partly-cloudy... Just about perfect. The air is cool, but not so cool that it's dry (which, when running, leads to what I call 'burning-itchy throat syndrome'). I managed to go into the 'third wind' today, which is fairly unusual. That is, while I suspect if someone had a gun on me I could it more often, it's normally too much effort to push my lazy ass that much. Tonight it was a lot easier due to the ambient conditions. What's even better is that this third wind came just before I encountered some neighbours out for a walk (presumably - I don't recall ever seeing them before, but given the neighbourhood, they are either neighbours, or CIA and/or NIST agents cunningly undercover as neighbours).


The reason getting it was so nice getting a 'third wind' just as I had to pass by the aforementioned individuals is this: for some reason, I have this thing about revealing what a lard-ass I am to others, at least when I don't have to. So being able to put on a burst of speed while encountering someone at night is psychologically nice, and I can always slow down and catch my breath once I'm out of sight around a corner or what not. "Look, people I've never met before and am unlikely to see again any time soon! I'm really running, and not waddling, as you may have suspected from a distance!"


Well, in this case, thanks to this third wind, I surprised myself by actually sprinting at 'funny speed' past these people (or so it seemed, subjectively), complete with the proper raising-high of the knees and all. I suspect I looked somewhat like an inept soldier asked to run in some sort of parade posture, but it was the best I could do. Anyways, toward the end of this third wind, I noticed that my breathing sounded somewhat like Darth Vader on a creamy cocktail of crack, 'shrooms, LSD, and other illicit substances. So I ended up deciding to wrap up the jog, and walked home.


I don't have much to say about my reading except to record that I've now more-or-less finished Part I of Baez and Munian. I haven't digested it yet, but damn, what a way to end a section. Wormholes and monopoles and Dirac, oh my! Whee!


* - I hate oddly spelled German names that I don't know how to spell**. I'm probably going to wake up in the middle of the night foaming at the mouth, feverishly trying to decide whether I've spelled 'Fahrenheit' properly.


** - On the other hand, I do like odd German words and names I do know how to spell. Deploying them whenever I can get away with it makes me look sophisticated, well-read, articulate, and just generally like a sexy, sexy beast. Yeah. Or so I like to think. I mean, come on, what better way to play the primate dominance game than to dishevel your hair, stick your hand (with dignity, mind) into your coat's lapel, blow a pipe-smoke ring, dangle a pocket watch on a chain from your other hand, and declaim in a vaguely central-european accent: "Vell, Wolfgang, and vot do you zink of my zweibein gedanken-experiment?"

Sunday, September 01, 2002

There's good news, and there's bad news. The good news is, I saw Mr. Thumpy tonight. Or, to be more accurate, I think I saw Mr. Thumpy - what I actually saw was a greyish blur hopping rapidly across someone's lawn. It was thus probably a rabbit, and thus probably Mr. Thumpy. You see, I wasn't wearing my glasses, because it's raining, and there are few things more annoying than rainy glasses.


The bad news is, I ran into an Honda sedan. It was parked. And when I say 'ran', I mean that literally. You see, as I said, I was jogging, without my glasses, and it was raining. At the time, I was fairly miserable (ever notice how rain, and water in general, is wet?), but could see my path fairly well. My eyesight sucks, but it's Good Enough for avoiding running into street lamps and parked cars at night. Normally. You see, I somehow got water into both of my eyes, and it actually stung a little bit. This was fairly bizarre - it's rain water, not pool water, so it shouldn't, presumably, have chlorine or whatnot in it, right? Well, regardless, I decided to wipe the water from my eyes. Also, since I have a promise to myself not to stop while jogging (it's a will-power and endurance thing, I guess), I did not slow down much. Well, this turned out to be a bad idea, because as it happens I misjudged both the distance to a nearby parked car, and my exact velocity vector. The result was me rubbing my eyes thoroughly satisfyingly, and opening them only in time to say 'ooouf' and sprawl across the hood of the aforementioned Honda. What's more, since, as I've already noted, it was raining, the car's hood was nicely slick, and my momentum suffiencient, for me to slide off the hood onto the pavement.


'Oooouf' indeed. Double, even.


It could have been a lot worse. First, the car could have had an alarm system. Could you imagine picking yourself off the ground and examining your extremeties to make sure they didn't detach just as the owners of said car run out and demand to know what the hell you're doing to their car? "Uhh, sorry sir, I ran into it in the rain. Literally." Oh dear. Second, it could have been a Mercedes, or a Jaguar, or some other car brand which delights in putting hood ornaments on their cars. Ouchie. Third, I could have actually damaged the car. Luckily, I didn't - I don't really have any particularly sharp edges, and it was a partially glancing blow, blunted somewhat by my hands which I threw out in front of me in a sort of haphazard fasion, so the car wasn't so much as scratched.


As for personal damage, I'll admit to being a slight bit sore, but the main injury is to my dignity.