...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...

Sunday, June 30, 2002

I've promised myself that I won't put political or social commentary on this page when I started it. That was because I didn't want it to be like the stereotypical "war-blog", devoted to mindless political masturbation and link propagation. I still hold to that principle, but I'm going to violate it right now, with the intention of it being a one-time kind of thing. I do it because I want to record a prediction.

Polling shows something like 9 out of 10 americans want "under god" in the US pledge of allegiance. The US Congress was already fairly apoplectic over the 9th Circuit Court ruling before the poll data came in. This will lend support to their pandering. The political pressure to overturn on the full 9th Circuit, and on the Supreme Court, when it goes there (as it very likely will, I think), will be immense.

If the courts judge upon the constitional merits of the case (that is, ruling the pledge unconstitutional), the United States will pass a constitutional amendment castrating the first amendment.

Or such is my prediction. The whole length posting was to record, on the WWW, said prediction. I hope I'm wrong.

Saturday, June 29, 2002

The last couple of days, I've been doing quite a bit of thinking about hairy balls.

No! Wait! Don't leave! It's not what it sounds like!

I've been trying to get a picture of what 'bundles' are like. My best effort so far goes like this. Manifolds are too abstract to deal with intuitively for me. So let's just use a simple accesible manifold, just a sphere in 3D (aka a ball), as a stand in for all manifolds. Now, to get a tangent bundle corresponding to the manifold, we have to take the set of all the tangent spaces to all the points on the manifold, more or less. That's too abstract. But the tangent spaces are vector spaces, and vectors are easy to visualize - they're just arrows! More, tangent bundles are apparently just important instances of 'fiber bundles'. That is, bundles are made up of abstract things called fibers, which end up corresponding to the individual tangent vector spaces to each point on the manifold. So. A nice picture for all this that our balls sprout hair*. All kinds of hair - neat, frizzled, dreaded, etc. The hair as a whole is now a picture corresponding to a 'tangent bundle' to the manifold, with the hairs being the 'fibers' making up the bundle. Sensible, right? So, we've got tangent bundles pictured as hairy balls now. Why the hell did we want to do that? Well, I don't really know, because I'm not yet sure how one uses tangent bundles. But, I've read that if you want to deal with differentiational thingies with manifolds, you end up dealing with bundles. That covers stuff like 'velocities' and also far more bizarre things.

That actually fits nicely with our image of hairy balls. Say you take such a hairy ball, and give it a good shake. Run about the room waving it in the air. If the hairy balls are sufficiently hairy, you can even outfit a cheerleading squad with them, and have them do some cheers. Now, instead of watching the cheerleaders, try watching the hair on the balls. I know this is really hard, but just try it. What's the hair doing? Well, it's swishing all over the place - and if the balls are sufficiently hairy, you can only see the hair - not the balls. And the swishing is not random - it depends on just what the balls are doing - how they're accelerating, or whether they are deforming under there, or whatever. So by looking at the hair of the balls, we can get lots of good information about the balls. What's more, the hair makes life a lot easier - it's a lot easier to notice a hairy ball accelerate and jitter about than a naked one. If we imagine ourselves in a huge empty room, with our viewpoint moving with the ball, it's going to be hard to see how the ball is moving if it's bald. But if it's hairy, we're in like Flynt - we just watch the hair.

So that's why tangent bundles are useful. I think.

Hey! Don't look at me like that. I didn't write all that just because it was an excuse to talk about hairy balls and cheerleaders in the same sentence. No siree Bob! And it's not my fault your mind is in the gutter! Kids these days...

* - As seen on TV! Yours for just 19.95 plus shipping and handling, but only if you call now! "I'm not just the company president - I'm a customer!"

Thursday, June 27, 2002

Subtle is the Lord. It's the title of a book I've got to read someday, and it's also the first thing (after a suitably earthy expletive) that came to my little atheist mind today when I learned that the right-hand-rule of electromagnetism is a crock. According to sources I trust, if one does electromagnetism properly, using differential forms and other relatively modern mathematical objects, the right hand rule isn't needed. The right-hand-rule has always bothered me - it seemed capriciously random, and while I knew you could get an equivalent picture using a left-hand-rule, the very idea that one had to pick one bothered me.

I have got to learn about this stuff. But, I can't pick a book to use. Right now, it's between Frankel's Geometry of Physics, Baylis's Electrodynamics: A modern approach (uses clifford algebra to do stuff), and John Baez's Knots, Gauge Fields, and Quantum Gravity. The book I really want to use is Baez's, but I'm scared, because the title is very intimidating, I've never actually seen it, and I'm afraid I'd immedeatly drown in it. But then again, I've heard the prerequisites aren't too heavy. Damn.

Hmm. Heh, I might have to pester Prof. Baez in a newsgroup for the answer - it was one of his posts that clued me into all this in the first place...

Tuesday, June 25, 2002

I've caught the flu, and I feel like a used kleenex. What's more, it's likely an imported flu (from Russia, most likely). Joy.

In more pleasant news, I was reading Spivak while on The Throne, and I think I'm actually not completely off my rocker in my initial impression of bundles. They are, apparently, 'just' a fancy way of saying you've got a tangent vector space to a manifold. The 'tangent bundle' turns out to be somewhat fancier still - it's something like a set of equivalence classes of bundles on the manifold, or something. I haven't digested that part yet. Hmm... I think I'll write up my digested version of tangent bundles once I work through them, for posterity and humor value.

Strichartz's book is neat. So is Needham's book (except I'd use a stronger descriptor than neat -- perhaps comic-book-guy-style "Best. Math. Book. Ever."?). I'm done with Needham's fantastic chapter on winding numbers and topology. The short-range plan is now to finish Strichartz's chapter on differentiation, and Spivak's chapter on tangent bundles. Then, I'm going to get my exercise on, and try to crank through some problem sets in all three books before continuing in any of them. Exercises are irreplaceable in firming up one's grasp of material, and at least the one's in Needham are actually fun.

Sadly, exercises are also hard work, and I'm a lazy bum. Motivation to do the 'homework' is one of the things that is especially challeging in autodidaction. On the other hand, in my specific case, there is a silver lining. See, I get to pick what problems I do, when I do them, and so on. Not some teacher with masochistic tendencies and an undying devotion to engineering plug-and-chug busywork. I pick the good stuff (and yes, I can see this being a problem, as it's questionable whether I can tell what the good stuff is.) And I don't get graded. And I'm not competing with anyone for a curve or anything like that. Which makes all makes the idea of self-assigned work easier to swallow and stick with, provided the work is interesting, and I can feel myself making progress. I've a nasty character flaw in getting easily frustrated.

Monday, June 24, 2002

I'm switching to a strict regimen of Strichartz and Needham for the next few days. This was brought on by the fact that I can feel my understanding of Spivak slipping away from me faster than a radical libertrarian's screed on government slides off a slippery slope*. Basically, I'm just not grasping tangent bundles in any meaningful way. I can't just read the proofs anymore, and I just stare at them glassy-eyed. So I think I need to do some exercises, and I also need a break from Spivak.

I'm beginning to like Strichartz now that he's talking about things I've seen before, such as continuity of functions and differentiation. I can see him carefully motivating the discussion, stressing common themes in proofs and definitions, and so on. It all leads me to suspect I've been too harsh on his book, and that if I had seen things about the construction of the reals and topology and so on before I'd read his book, I would have enjoyed those sections as well.

I ate far more pasta than is advisable last night, so I didn't go jogging.

* - Ooh, baby, just look at that lovely alliteration! Bow down before my display of literary pretentiousness!

Saturday, June 22, 2002

I saw Mr. Thumpy tonight. Or at least, so suggests some circumstantial evidence. Whatever I saw was greyish, was about the size of a rabbit, and made the same kind of noise while running that MrThumpy's made when I've been able to positively identify him. Squirrels aren't generally out that late, and it didn't seem like a cat. So, most probably it was a rabbit, and that means Mr. Thumpy.

Unfortunately, as I was observing Mr. Thumpy, I was almost hit by a couple of thirteen year old fucktards on skateboards. This at 11 pm, in almost complete darkness, on a footpath. The mind boggles.

In math news, I'm currently reading Strichartz. I'm beginning to think my impression of his Way of Analysis is wrong. More on that later.

Heh. Reed College is unique among the school webpages I've hit in that their 'look at our perfect pretty students' pictures, are, well, not. This suggests that either a) Reed only has unattractive people or b) Reed is actually being honest and more-or-less randomly picking average representatives of the student body. I kinda like Reed, actually. I considered applying there back in high school, but decided against it because a) I didn't think I'd get in and b) I'm probably a 'liberal', but I'm not that liberal, so their reputation bothered me a bit and c) I don't smoke weed, or, as it happens, anything else and d) I didn't want to pay another 'application fee'.

Given that I was unlikely to get into Reed after high school, it's effectively impossible now, as a transfer, with the addition of my scintillating college transcript.

Sigh. I've been doing some surfing around college websites. As I expected, I outright don't qualify for transfer at most of the universities I'd consider attending, and all the ones where I 'technically' may qualify more or less say 'don't bother' given certain stats (such as mine). And I can't say I blame them - I'd do the same if I was sitting in an admissions office. Expected, but still dissapointing. Podunk New Age Science University of Jebus*, here I come!

* - School motto: "We award more four-year degrees in BS to pet rabbits than any other school in Ohio!"

Score. I can finally run about a mile in about seven minutes*. This is admittedly pathetic compared to most people my age, but in high school, I 'ran' a mile in fourteen minutes. Now granted, part of that was an extended 'fuck all ya'll' to the very idea of a mandatory gym class, but a bigger part of it was that I was simply a slow and stupid dork. Now I'm a faster dork (whose intrinsic intelligence is probably unchanged, though I'm a little wiser thanks to 'life experience', natch). Which is cause for celebration, I think, and I just ate a few grapes and a peach to commemorate the occasion.

In other news. Hmm. Not much else in other news. My first impression of vector bundles: "Hey, neat, but isn't this just a really anal-retentive way of saying 'Houston, we have a tangent vector space'?" I'm sure once I find out more about them that impression will be cause for amusement, but so it goes.

* - The error bars on that are undetermined, because I don't own a stopwatch, and I haven't measured my running path with an odometer. But I think the figure is 'about' right, to within something like a minute and something like a fifth of a mile, pulling some numbers out of a hat.

Thursday, June 20, 2002

The promised details on Theorem 9, Ch2:

First, some terminology. In differential geometry the objects studied are manifolds. A manifold is just a space that looks like R^n if you look at any little part of it. R^n is just normal flat space, as R is just the real line, and the n stands for its dimension (n = 3 for the space we live in*). In general, a manifold can look like R^n's of differing n's in different areas, but we can ignore that for the most part.

There's lots of theorems one can play with that use only that much information, but to get to the 'differential' part of differential geometry, we need some more structure. For instance, we might want to assign some coordinates to our manifold, M, or at least to a small part of it. To do that that, we look at some subset U of M. Then we define some function, x, which, given a point in U, will spit out a point in R^n. Just to be fancy, we call (x, U) an atlas of M. We can show that this definition makes sense, and meshes with our normal conception of coordinates as just a grid.

Armed with this idea, and a few others I'm not going to go into here, we can start doing calculus on manifolds. We can define functions between manifolds, and take their derivatives. This is actually a very neat process.

Let's call our function f, and have it take points in M^n to points in another manifold, A^m. To actually do this, we involve some unspecified charts on both manifolds, say (x, U) on M, and (y, V) on A. Now, remember, x takes M -> R^n. So x^-1 takes R^n -> M. (Same goes for y, of course.) The thing is that it's really pretty simple to define functions that do things to R^n - we've all been doing it since middle school (except we didn't call it R^n back then ...) So, let's stick with what we know, and try to make f work in R^n (or between R^n and R^m - whatever), but also in the process do what we need it to do, which is work with manifolds.

Here's how: It's just y*f*x^-1, with * meaning composition of functions, y, f, and x^-1 being functions (duh), and the whole thing is to be read right to left, as one normally (!) does with composition of functions. So, given a point q in M, we want to get a point f(q) in A. Ok. Let's feed q to x^-1, getting x^-1[q] (notice, now we're in R^n!). Now, feed that to f, getting f[x^-1[q]] (notice, we fed f R^n, and it shat out R^m - so it's a good ol' function, the kind we know how to handle). And now we feed that to y, getting y[f[x^-1[q]]]. Notice that y is what takes us from R^m to where we wanted to get, which is the manifold A^m !

Whew.

So, look at what we did: we 'hid' the fact that we're actually feeding it with something bizarre, like manifolds, from f, and persuaded it that it's actually muching on simple tasty things like R^n. So while we're lying unscrupulous bastards, we got what we wanted: f really is a function between manifolds. It just so happens it needs to wear blinders to do it, otherwise it would run away in terror.

If you followed that, I hate you and envy you, because it took me several days to get that far.

Now, since we can define these functions, and they're continuous, blah blah blah, we can take their derivatives, and we can talk about 'changing coordinates', and crap like that. We can even define big-ass Jacobians, which are matrices that tend to pop up when you squint carefully at the idea of changing from one set of coordinates to another. And we can play with matrix, figuring out its 'rank' and other linear-alebraish things. (Note: it's rank is going to vary from place to place!)

So, err, that was the introduction. The theorem (Th. 9, Ch2, Spivak's DG) says: Say we've got a function f that takes things from one manifold, M^m, to another, A^n. Say further that it has a rank k at the point p in M^n. Then, given p = (a_1, a_2, ..., a_m):

y*f*x^-1[a_1, a_2, ..., a_m] = (a_1, a_2, ..., a_k, phi_(k+1), phi_(k+2), ..., phi_(n))

(the phi's are just numbers you get from f in a certain way, as it turns out, and I've left out the second part of the theorem, which looks the same except for having a bunch of zeros instead of the phi's.).

Now, what that is saying are some things about what actually happens if you stuff one manifold into another: depending on f and p, some of parts of your points are going to remain fixed and other parts are going to get chewed on. Different points are going to contain different amounts of dietary undigested fiber**, if you will. Hopefully, that makes some amount of sense.

I didn't understand this at all until late last night, when, after having a few drinks at a birthday party (not mine), I was sitting upon the Throne of Power. There, pretty much all of the above hit me. I was stuck on this damn theorem for three days straight, though as I understand now, that was because I didn't correctly grasp some theorems before.

Now, I intentionally glossed over quite a few things in the above exposition, there are parts (close to all of them) where I have a suspicion I don't really know what I'm talking about, and there are parts where I'm probably saying something terribly naive and stupid or both. That is the curse and the blessing of studying alone: there's no one to yell "Hey, you bloody stupid arse-scratcher, that's wrong and dumb!"

* - Well, 4, if you want to be a laxative butt-monkey about it.

** - I don't think manifolds are part of the food pyramid, but they should be!

Wednesday, June 19, 2002

Praise Jebus. I finally have a grasp on Theorem 9 (Chapter 2). That bitch had me tied in knots and has been slapping me around for the last two days, but I think I may finally have it spanked for good*. At the least, I've some leverage. More details forthcoming.

The above is not intended to serve as an endorsement of, or even commentary on, BDSM or anything like it. Get your mind out of the gutter!

Monday, June 17, 2002

I saw Mr. Thumpy today! He was jumping about in the grass on someone's back yard. It was dark, but it was definitely Mr. Thumpy: he was gray, had large ears, and he moved like rabbit would, in spurts, actually making dull 'thumps' with what I assume to be rear legs during the initial phases of acceleration. I also saw the-cat-with-bells-on (I have got to think of a catchier, simpler name for the chap - I seem him almost every bloody night, after all.) He was inspecting the undercarriage of a parked car (or lying in wait, you never know with cats), and came out to greet me (or to startle me, for you never know...) as I passed. This was fairly close to the close encounter of the first kind with Mr. Thumpy, so I have to hope the-cat-with-bells-on and Mr. Thumpy get along well. They're about the same size, so I'm not sure who is harrassing whom, if their relations aren't cordial.

Nothing new to report on the math front. Still reading about winding numbers and topology. Hurray for Brouwer's Fixed Point Theorem!

Sunday, June 16, 2002

I got chased by a cat last night. I was jogging, and a cat (the one with the cow-bell on his neck) jumped out from under a car, probably mistaking me for a mouse or something. He then jogged after me for a short while, but quickly lost interest and went to investigate a fascinating patch of grass.

In other news, I've read the first chapter of Spivak, and am in the second chapter. The first chapter was not very difficult. The second chapter is harder. I think I need to do the exercises in Ch.1 to get more comfortable with the material. I'm also continuing my readings in Needham. Still on the chapter about topology and winding numbers.

Fun.

Friday, June 14, 2002

This is amusing. Day before yesterday, when I was reading Strichartz, I found out more than I ever wanted to know about sets. Don't get me wrong, I think I can see the intrinsic interest of the topic, but it just didn't grab me. Anyways, I learned about various obscure things such as boundedness, compactness, and other sundry bits and pieces. So, this evening I'm reading Spivak's Differential Geometry (as I said, I just received my copy today), and lo and behold, there's something like two pages, starting on page 4, of what Spivak merrily terms a "hassle with point-set topology." Starring, specifically, compact sets and boundedness and so on. It's used to talk about some useful properties of manifolds, which Chapter 1 of Volume I is about. Spivak went and used a few terms I didn't know, but a check of MathWorld cleared it up for me.

I'm still very impressed by Spivak's books. So far at least, the exposition is fun and informative, which is the most important thing. Also, as I've said, the physical quality of the books is very impressive. They have excellent bindings, pretty -- and, more importantly -- easy to read typesetting, they're printed on very smooth, slightly glossy (but not to the point of reflecting anything, really) paper and have a better than average 'scent'. Also, the cover art is quite striking, inspired by Coleridge's Rime Of The Ancient Mariner, and painted (well, in the original, that is, not on the actual covers) by Spivak himself. I've looked at the older editions the nearest university library has, and by god, what a difference. The old editions were basically photocopies of a typed manuscript, or so they looked. Really, I just can't get over the initial physical impression of the books I got - I was paying thirty bucks a pop, more or less, so I expected some cheapies. Well, they don't look cheap at all.

And in case anyone reading this wonders, I've no affilitation with Michael Spivak, Publish or Perish, or anything else. I write what I see. And what I see here, I like.

The first two volumes of Michael Spivak's A Comprehensive Introduction To Differential Geometry arrived today. They are absolutely beautiful. Everything from the pretty artistic covers (by Spivak himself), to the sweet typesetting, to the satisfyingly heavy, glossy smooth paper, to the 'new book' scent, to the exposition (at least in the beginning - I've only read a tiny bit so far). And cheap! Approx. $30 for a hardback textbook (and about$50 for two) is an awesome price!

And oh yes, there's an entry for 'pig, yellow - pg 434' in the index. What else could one possibly want?

Obviusly, I'm a bibliophile. More later. Right now, I want to try to go see Bourne Identity.

Thursday, June 13, 2002

So I've jumped back to Visual Complex Analysis. Having spent the last few days with Strichartz, the difference is striking. Needham actually gives intuition in spades, motivates his arguments, is fun to read, and all-around blows Strichartz clean out of the water on everything except rigour. And to be honest, I don't give a crap about rigour for rigour's sake. I know, in a vague intellectual way, that rigour is necessary and even useful, but I don't have any feel for it. I definetly need to find a companion book for Strichartz that will have motivational stuff in it. Hopefully, the book by Abbott I mentioned earlier will fit the bill - I'll try to find it in the local book super-store this weekend.

I've temporarily skipped Needham's chapter on non-Euclidean geometry, and I'm reading the chapter after that, on topology and winding numbers. I did this reluctantly, as I've long dreamed of learning about crazy geometries, but I also want to learn about complex integration, for which the topology and winding number stuff is a prerequisite, while the non-Euclidean geometry chapter isn't. However, I can't resist intellectual candy, and so I've made a compromise: I'll read the non-Euclidean chapter during bathroom breaks, as I do some of my best thinking there, as explained in detail in past posts, and I'll read the winding number stuff in my out-of-bathroom time. Fun!

Talking about fun, the winding numbers chapter is spectacularly entertaining. I think I've said it before, but I'll say it again: "Tristan Needham's Visual Complex Analysis is the Best Math Textbook Ever." If you have even the slightest interest in mathematics, and especially geometry, and have studied basic calculus, get this book. You're very unlikely to regret it.

I'm testing mozblog! And it seems to work. Nice.

Wednesday, June 12, 2002

Yay! I'm done with Chapter 3 of Strichartz, "The Topology of the Real Line". This calls for tea and cake and/or candy, I think. Also, I'm seriously considering getting another book on analysis, name Abbott's Understanding Analysis as a supplement to Strichartz. Basically, Strichartz is admirably in-depth, but I just don't feel that I'm getting as much intuition from him as I'd like. Perhaps it's a foolish complaint - after all, the whole reason for being of analysis seems to be avoiding intuitive arguments in favor of anal-retentive formal ones. But Abbott claims to try to give intuition and rigour, and a few reviews of his book I've read agree, so I may end up buying his book.

In other news, Blogger's web interface looks even more terrible now. Whether that's Blogger's doing or some changes in Mozilla 1.1a, I don't know. Also, the house still has a certain scent to it. I wonder if French parfume manufacturers would be interested in bottling some eau du pizza brûlée, for a suitable fee, of course?

Tuesday, June 11, 2002

I'm back from tonight's jog. The air outside is warm, with a soup-like consistency. The air inside the house still reeks pungently of flaming pizza. The high points of the jog were seeing a cat, and getting yapped at by a couple of pocket-pooches out walking their human.

I've made some respectable* progress in my real analysis text today - I've knocked over the chapter on the construction of the real number system, and I'm most of the way through a chapter on the topology of the real line. (Mini-rant: I'm developing a vicious hate for inequalities. If I ever meet one in real life, I fear that I may beat it to death with an aquarium tank, if one is handy, or with a rubber chicken, if it isn't.)

* - For an occasionally retarded college-dropout, that is.

Whoa. We just had a fire in the house. Luckily, it was localized to the inside of the microwave. My brother decided to nuke some frozen pizza. He says he set the thing to three minutes, but I guess he hit zero one too many times, as the thing caught fire. As an added surprise, it turns out that none of the smoke detectors in the house work. We only found out about it because I asked him what he's cooking that's making the stink. We went downstairs, and lo and behold, the kitchen is full of thick, grey, stinky smoke. Eepers. As I said, the damage is mainly to our noses, the pizza, and the general scent of the house. We're working on the smell issues by opening all kinds of doors and windows and deploying The Fan. The pizza is a write-off, I'm afraid.

Monday, June 10, 2002

Sweet merciful Jesus, I'm a retard sometimes. I now completely and totally grok the triangle inequality. I say I'm a retard because it's so bloody simple it makes my head spin. Hooray for sleeping through math in middle and high school!

My fingers are tired. I've tried ten (10) problems in Needham over the last ten or so hours (spread over yesterday evening and today's evening). I haven't solved a single one. I did cover a few dozen pages with algebra and a few pictures and more algebra. A tangentially relevant passage from Macbeth comes to mind:

To-morrow, and to-morrow, and to-morrow,
Creeps in this petty pace from day to day
To the last syllable of recorded time,
And all our yesterdays have lighted fools
The way to dusty death. Out, out, brief candle!
Life 's but a walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more: it is a tale
Told by an idiot, full of sound and fury,
Signifying nothing.


I think I'll try to do something else now. Hmm. My choices, ignoring bummery:

• Jump to the next chapter, on complex integration, and come back to the exercizes later

• Jump books to Strichartz, prove the fucking triangle inequality once and for all, and soldier on

• Continue trying these same ten exercizes

Now, the latter two choices are effectively the same, and tempting as it the last choice is, it doesn't seem very productive. Sigh. Choices...

Sunday, June 09, 2002

I'm going to solve a problem in Needham today (well, if I don't get stuck due to my stupidity) that claims to show where the Schwarzian derivative comes from. Yay! Should be interesting. I also want to finally finish the chapter on the real numbers in Strichartz's Way of Analysis today, but I doubt I have the cajones to get it done tonight.

So, I went to the park this afternoon for a jog. I didn't jog very far. Now, it wasn't because it was unexpectedly humid, or because I'm a weak-legged, out of shape wuss (though both are, to some degree, true). Actually, it was because of the fat people.

There was a totally surreal concentration of truly spheroidal persons in my favorite park today. It was as if there was some kind of oversize person convention (which isn't actually as outlandish as it sounds - this park is often the site of various parties and gatherings). Now, it's not as if I have some kind of debilitating physical reaction to the sight of profoundly rotund indivuals jiggling around a park that stops me from running. What actually infuriated me enough to cut my run short was that these same oversize persons decided to go for a walk, en masse, along the forest path which winds around the lake. Along with a large number of highly energetic little children. And this was a problem because somehow the vast majority of them had no concept of manners whatsoever. They blocked the narrow trail with their girths (and I mean that fucking literally!), and did not even attempt to allow me to pass. They merely trundled along like rhinos that know they always have the right of way, langurously blinking their eyes at me, forcing me to go off-road, into god knows what types of poison-ivy, every fucking minute, on average. Of course, the ones that brough their kids along were even worse, because the kids formed a kind of highly energetic stupidity-and-no-manners cloud around their elders, making even attempts to get past them by going off into the bushes with the rabid squirrels very difficult. After fifteen minutes of this shit, I turned around, went back to my car, and drove home. Grr. Really, I can sympathise about hormone problems, insatiable appetites for Big Macs, and the desire to replenish the Earth with your spawn, but for the love of Chthulhu, is it so hard to get some manners and at least make an attempt to allow faster-moving foot-traffic to pass on narrow paths?

That's the rant for tonight. Sincere apologies to anyone it offends.

I can't find my pants. (The ones I thought I'd be wearing today, that is.) News at 11.

Saturday, June 08, 2002

underthumb reports on an interesting (though, from my amateurish perspective, fatally flawed) experiment in psychology. People are presented with a box with a known ratio of black/white marbles (.5), and a box with an unknown ratio of marbles. Experiments then show that people tend to pick the known-ratio box over the unknown box. There is then the claim that this is stupid and irrational. The argument goes like this: if, say the 'win condition' is getting a black ball, people tend bet on the known-ratio box. You've thereby made a bet that the ratio for the unknown box is worse (that is, more white than black) than in the known box. Put the ball back. Then make people pick again, this time making a white ball the win condition. People still pick the known box, 'betting that the ratio in the unknown box is worse', in the opposite direction. So you've just made two 'opposing' bets about the same damn box. Irattional? Stupid? Um, hell no. I'm going to try to make the argument that it is perfectly rational, and smart to go with the known, rather than the unkown, and to claim otherwise is ... unwise.

First, the flaw in the experiment. It's good, in setting up the experiment, to keep it simple, and strip away the irrelevant. It's possible to strip away essential variables, however, and the marble problem is an example. Let's try a slightly more complicated setup. Say you're given some win condition, and two choices of 'path' to that condition. One path is described to you prior to an attempt to get to the win condition, the other is undescribed. Say futher that the win condition and paths are complex enough to allow on-the-fly and pre-op choices in moving toward the win condition (I suspect that's one of the big things that makes the 'evolutionary' approach that underthumb refers to work). Concrete example: two sets of road networks, with the objective of getting a certain distance in a certain time.

In this case, it's completely bloody obvious that it's a good idea to pick the road network that you know about. You can plan a route ahead of time with it, and make informed on-the-fly choices. With the 'unknown' road network, you can't. You might get to your destination faster (it might be a much straighter path, for instance), but betting that way is stupid - it might just as easily be worse, and you've got no planning or information benefits.

The marble experiment strips away the utility of the given information, thereby making it a stupid test of the 'better a known than an unknown' phenomenon -- after all, in Real Life, information does tend to be useful. But forget that. Even with the problem as stated, it's smart to go for the known.

When you make that first choice of known box to get a marble from, you aren't making any bets as to the other box. All you're stating is that you don't know the ratios for the unknown. Could be worse for you, could be the same, or could be better. You don't know, and it's a given that you can't know. So, because you want to win, and you don't want to take the chance that the unknown has no 'win balls' at all or something, you go for the known. In the second experiment, nothing has changed. You still don't know anything about the unknown. So the very same reasoning applies to the second draw as to the first one. It's not unlike coin tosses - the outcome of one flip has no effect on the probabilities of subsequent flips. Basic fact of statistics.

I'm a rank amateur. It's possible I Just Don't Get It, or I'm missing something important. But as it stands, it seems to me that any psychologist that claims people are wrong in this marble experiment hasn't thought about it very carefully.

Counter-arguments welcomed.

Holy Fucking Shit. That was awesome. I went to the Volvo "Fire and Ice Driving Experience" this morning. They set up three race-courses in a huge parking lot at FedEx Field near Baltimore. You go there, sign up, get a name badge and an armband signifying that you signed a release form, and are of age and have a license. Then you listen to a short speech about various safety features of Volvos in a big tent with dead sexy plasma displays illustrating key points. You are then invited to come outside. The first thing my group did was the 'hot lap'. They've got four Volvos sitting out on the asphalt, with professional race drivers (retired NASCAR, etc.) in them. You're given helmets. You climb into the passenger seat. And then they take you for a ride. The tires literally smoke (and how! and the smell!) most of the way through the course, as the drivers skid and slalom and flat-out literally put the assorted pedals to the metal all over the course. They use the handbrake to improve breaking and traction at key points in the course. It's better than a roller-coaster, and you come out of the car with decidedly shaky knees. Or at least I did.

But it doesn't stop there. There are inter-session demos of things like safety glass, structural rigidity, and a prototype Volvo SUV, and a fucking sweet concept car (dead sexy glass all around, including the roof, insane active safety features, etc). There are two more sessions in addition to the hot lap. One is a 'winter' course, the other a 'summer' course. I went to the winter course next. There, you get to drive, in succession, a Volvo S60 2.5T (front-wheel drive), and an S60 AWD, down a course with various slick portions, pop-up cardboard mooses that you have to avoid, and other fun things. Fucking sweet, that was.

But then. Oh my. There was the summer course. You drive a Volvo S60 T5 (single-turbo), an S80 T6 (twin-turbo), and an S80 Executive (just like T6, but with a TV and a drinks fridge in the back). Holy fucking crap. The T5 and T6 are insanely sweet cars. I have to say I preferred the T5, because while it was a little less powerful than the T6, it was a lot less heavy, and so it had better acceleration and handling. The cars take off ridiculously fast, with a nice, throaty roar off the starting line, and the T5 stays absolute glued to the track no matter how crazy you drive (the T6 was sweet as well, but wasn't quite as tight as the T5). I drove the T5 five or six times down this summer track, and the T6 once. I didn't drive the Executive, because the lines for it were longer, and seeing as I would be driving, I didn't care about the TV in the back. And this was all free.

It was an amazing experience, and very smart marketing on the part of Volvo: you can bet that if I ever come by forty or so thousand bucks that I can blow, I will give the S60 T5 a very, very serious look. And I got a very nice yacht-racing cap at the end, complete with a clip-to-shirt thingie so it doesn't get blown away by the wind.

If you've got one of these Fire and Ice demos going on in a city near you, go!

Friday, June 07, 2002

I found some interesting XXX material today. It's called Static Negative Energies Near a Domain Wall. I'm going to do a summary of a part of the paper, as I understand it right now, for future reference and laughs.

It's been known for a while that if you want relativity to allow 'closed timelike curves' (time-travel) or faster than light motion, the 'weak energy condition' must be violated. This just means that stuff like masses is positive. Quantum fields can violate this weak energy condition, while garden-variety classical fields can't. However, the stereotypical quantum field that has apparently been used in such calculations is what's called a 'free field', and those have to obey something called the 'averaged weak energy condition'. That means that you simply don't have negative masses on average. That is, if you've got one, it's damn temporary. So these aren't all that interesting, if you're interested in time travel.

Now, the most famous example of a thingie with negative energy is the 'Casimir problem'. That's that thing where you've got two metal plates real close to each other getting squeezed together by interesting quantum effects. I've written about it before (but not here). The interesting thing here is that that problem is 'static' - it can just sit pretty, with it's naugthy negative energy density. Which means that it violates the averaged weak energy condition.

What the paper in question does is look at a toy model, a '2+1' dimension (two space dimensions, one time) setup, with a 'domain wall' taking the place of a Casimir setup. And they show that close to this domain wall, you get negative energy densities. But wait! We ain't meetin' real furry creatures from Alpha Centauri any time soon. Because this 'toy system' does obey something called the 'averaged null energy condition', which is enough to rule out goodies like time-travel and faster than light transportation. Ya just can't win. But there's a small glimmer of hope, because the authors speculate that there's a chance the more complicated, but also more realistic 3+1 dimension model, can violate this other condition. Work on a paper exploring that is 'subject of our future work', the authors say. Should be an interesting read.

I'm just about ready to beat my head on the nearest wall. It has to do with that Schwarzian derivative I mentioned earlier. Here's the problem. I'm supposed to show that there's a certain 'chain rule' for it. That is, assign w = f(z), f being analytic. The question then becomes, what is the Schwarzian derivative of g(w), where g is another analytic function. Now, in Needham, the rule is given:

{g(w), z} = (f'(z))^2 * {g(w), w} + {f(z), z}

The problem is that I can't for the life of me show that to be true. I just get lost in an endless swamp of algebra and calculus, and never really come out. Normally, I'd just blame myself for my ineptness. There is, however, a small chance that the above equation actually isn't quite right - there might be a typo. I've found a couple of places that give something that looks a lot like it, but with one (1) extra parenthesis. So those sources definitely have typos. But the question is, did they want to have a couple of extra parentheses, or none? I'm confused. What's even worse is that I can show something almost, but not quite, like the given formula. Aarrrrrrgh.

Here's something you don't see every day, but I forgot to mention. I saw the neighbourhood bunny rabbit, whom I call Mr. Thumpy, on my jog today.

Here's a snippet from T.S. Eliot's "The Love Song of J. Alfred Prufrock". This particular part seemed to resonate with me tonight:

                ...
And indeed there will be time
For the yellow smoke that slides along the street,
Rubbing its back upon the window-panes;
There will be time, there will be time
To prepare a face to meet the faces that you meet;
There will be time to murder and create,
And time for all the works and days of hands
That lift and drop a question on your plate;
Time for you and time for me,
And time yet for a hundred indecisions,
And for a hundred visions and revisions,
Before the taking of a toast and tea. 

I do my best work sitting on the shitter. I swear. Today was another example. I've been struggling since yesterday with a very interesting problem from Ch. 4 of Needham, #18, I think. It has to do with the following interesting question. Say you've got two curves touching (gently, non-pathologically) at a point. Is it possible to define a meaningful 'angle' between these two curves at this point? Well, maybe. Certainly, any such 'angle' that we end up defining better end up being conformal. That is, if we apply a conformal transformation to the plane containing the two curves, it will preserve any normal angles, like those that are part of triangles, say. ('Conformal' is just a fancy way of saying 'angle-preserving'. An important bit of trivia is that analytic transformations are always conformal.) So our made-up definition for the meeting "angle" of two curves damn well better act like normal angles do, or it doesn't deserve to be called an 'angle'.

This actually turns out to be a fairly hard problem, and it is only fully solved in the last chapter of Needham's book. In chapter 4, though, I'm asked to follow a couple of doomed attempts at a definition. The first attempt at a definition was done by Newton. Unfortunately, with the modern tools of complex analysis, it is fairly simple to show that it is not conformal. So it doesn't work. There is another attempt to define this 'angle', building on Newton's attempt. It fails too, but for a different reason. (As I said, this won't be solved until like chapter 12 or 14 or whatever.) But never mind that. What's important is that last night I got stuck on a bit of geometry in showing that this second attempt doesn't quite work. I've been thinking about it all day, to no great benefit.

About an hour ago, while resting on the porcelain throne in preparation for a nice, fat-arse shaking jog, the answer to my quandary hit me like a lightning bolt from a minor Greek god, or more poetically like a toilet-alligator bite on the ass. See, there's an often used fact that for small theta, sin(theta) = theta. And that is the lever that got me unstuck.

I wonder if it's unusual for people to do their best thinking while sitting on the crapper, and if it isn't, how many of the great discoveries throughout history were made while taking a dump? Perhaps Archimedes wasn't in the bathtub when he bellowed his now-famous cry, "Eureka!"

Here's a poetic version of the above post, mostly due to underthumb's help:

              Lo!
Though I poop,
My mind doth not droop,
And from it--for sooth!
Ideas now go 'gloop'!


Thursday, June 06, 2002

Who is the bunny arse that thought up the Schwartzian derivative*, and more importantly, why? It looks like this, for a function f(z), with respect to z:

{f(z), z} = f'''/f'' - 3/2 * (f''/f')^2

Now, it turns out this 'derivative' has some nice properties, for instance all Mobius transformations have a vanishing Schwartzian derivative, and the reverse also works. This is cool, undeniably. But where the bloody hell does it come from?

* - Well, yes, obviously, some mother-bunny named Schwartz thought of it. Probably the same Schwartz that did work in analytic continuations. But that's not the important part.

I have no academic focus. Not now, not ever. I'm interested in just about everything. Hell, I've got a disturbing hankering to read the new edition of Molecular Biology of the Cell, all 1400 pages worth. I want to read some kind of book-lenth introduction to evolutionary psychology. I want to read Gould's life's work, aka The Structure of Evolutionary Theory, or whatever the actual name is. I want to read Chaucer's Canterbury Tales, in the original. I want to learn Japanese (because it's not a latin-related language). I want to learn differential geometry. I want to learn about linguistics. I'd like to read about international relations theory. I want to learn to write well. I want to learn quantum mechanics. I'd like to learn to be a witty and engaging conversationalist. I want a lot of things, and I'll probably dabble a bit in a lot of them over my life. But I'm unlikely to ever have a well-defined academic focus.

There is a price to pay for the pleasure of having a wide-ranging curiosity. I have had to resign myself to the fact that I will forever be a dilletante in almost all of my areas of interest. I have wasted large parts of my teenage years in slacking and learning what I wanted, not what I should have been learning. There are consequences. I'm extremely unlikely to make any worthwhile contributions to any field I do choose to learn in depth. Take physics, for instance. It's a well-known fact, backed by assorted studies, that physicists most often make their big contributions as graduate students, or freshly-minted post-docs, in their 20s. This statistic applies to many other fields of creative endeavour.

I think I've finally learned to accept it. I'll learn what I want, when I want, how I want, on my own, to the best of my limited abilities. Hopefully I'll earn enough money along the way to eat and afford a few toys. The grandiouse plans of my teenage years, built on a foundation of slacking, daydreaming, and reading, now look like trifling (yet dearly regarded) little optimistic far-away sand castles being washed away by the tide of cold, hard, and bracing reality. There's something to be said for reality. It is cold, and it may occasionally be bleak, but it is real, and swimming in it can be invigorating.

I have learned to accepted all that. I think. Perhaps. It's arguably a curious form of defeatism, I suppose, but I've learned I prefer it to optimism which constantly accumulates evidence against itself. It allows for some measure of joy, occasionally, and that's all I can really ask for.

Wednesday, June 05, 2002

Ok, this is really, really cool. From Finite Sets to Feynman Diagrams, written by John Baez and James Dolan. It's not nearly as forbidding as it sounds, because it starts out by talking about basic arithmetic, and throwing out some provocative thoughts on the very nature of equations in general. It's very, very cool. Just read it.

I'm done messing with DOCTYPEs and CSS and so on for the evening.

Oh yeah. If the CSS link-rollover effect looks like bunny arse in your browser, making text reflow in stupid ways and so on, then it's cause you're using IE. IE has issues with perfectly legal CSS. Go figure. Mozilla has no problems that I've found so far with the page. And Mozilla hit 1.0 today.

On the other hand, the color-related garishness on this page is entirely my fault. I really ought to do something about that link color...

I use various derivatives of the 'f-word' too much in my posts here. Perhaps I'll now use the word 'bunny' whenever I feel the urge to use the Worst Expletive In Existence. Or something like that.

Well. In a stunning display of deranged bumfuckerry, I have yet again got my basic algebra wrong. I need to go back to middle school. That, or crawl under a carpet in shame. Or both. Because, if you get the algebra right, as I did in my next to last post, then the problem actually makes some fucking sense, and is really simple. Because we now have

k_image = (1 + Re[z*f''/f'])/Length[z*f']

and substituting in the proper derivatives for the case that f(z) = z^m

k_image = (1 + m - 1)/Length[m*z^m] = 1/Length[z^m]

Which actually makes some fucking sense, and matches the intuitive picture. Problem solved. Woo-hoo, and a case of whiskey.

Chthulhu.

Say f(z) = z^m. f' = m*z^(m-1). f'' = m*(m-1)*z^(m-2). f''/f' = (m-1)*z^(m-2-m+1) = (m-1)*z^(-1). Call K the complex curvature of f(z). -i*Conj[K] = (m-1)/(z*Length[z]). Ok. So that pile of crap right there is an intrinsic property of the mapping f: z -> z^m. It tells us that even if we were to apply the mapping to a straight line, with zero curvature, the image curve would have non-zero curvature. For m=1, a linear mapping, it produces zero, as expected.

So. Take a circle on a plane. Make it a complex plane, just for kicks. The unit tangent to the circle is i *z/Length[z]. That's because if our coordinate on the circle is z, then to get a unit vector we can divide by the length of z, and then we turn it by Pi/2 by multiplying by i, and now we've got a unit tangent. Now, we're interested in how the curvature of a given shape changes under an analytic mapping. It's possible to show that if we pick the above-mentioned circle as our 'shape', then the image curvature under an analytic mapping f(z) is given by

k_image = (1 + Re[z*f''/f'])/(z*f')

Now, the question is, without actually cranking out a calculation, what should the image curvature be if our mapping is given by f(z) = z^m? And then check your prediction using the formula.

So. z^m is certainly analytic. Hmm. Well. z^m basically dilates the plane, more or less, right? I mean, it takes each point, and pushes it out z^m. Which suggests that the curvature, which starts out as 1/Length[z] is going to be something like 1/Length[z^m]. Ok.

Doing the calculation, though, leads to a rather different conclusion. The image curvature is instead:

image_curvature = 1/Length[m*z^m] + m*(m-1)/Length[m*z^m]

WTF? The scary thing is that this bizarre result matches intuition if m = 1. Because then it's just 1/Length[z], which is the same as the initial curvature, which damn well better be the case, because m=1 we're just doing a linear mapping, that is, we're not changing jack shit.

I've got a sneaking suspicion I'm misunderstanding how z^m works geometrically, which would be embarassing...

Tuesday, June 04, 2002

Alright! Uglification complete.

I just thought I'd note that it's rather annoying that Blogger doesn't really work too well with mozilla 1.0rc3.

Also, the latest Onion is pure comedy gold. A sample quote from an article about the recent announcement by the National Science Federation of the discovery that "Science is hard":

"Quantum physics has always been a particularly tough branch of science," UCLA physicist Dr. Hideki Watanabe said. "But in addition to being some of the smartest Einstein-y stuff around, it is undeniably a really stupid, pointless thing to study, something you could never actually use in the real world. This paradoxical dual state may one day lead to a new understanding of physics as a way to confuse and bore people."

As I said, comedy gold.

Also, some day I might write a polemical devil's advocate essay, complete with such unusual things as references and carefully framed arguments, regarding assorted 'copyright protection' laws being pushed by the entertainment industry, the half-hearted resistance to this legislation by hardware corporations, and the apparent (and, if one thinks about it, entirely understanble) lack of opposition from software companies to the legislation. If one hears of any reaction by software corporations to current copyright topics at all, it's effectively cheerful support of the legislation. (See: Microsoft.) Of course, techie outrage tends to be directed more towards the entertainment lobby, rather than the software industry. Not very surprising, given that many techies are employed by said software industry, like getting paid, and also like free movies and music.

But 'some day' is not today, and the above incoherent and unsupported babble doesn't qualify as an essay.

For some reason, I'm having trouble following Needham's description of the 'quick way' (as opposed to the painful, confusing, yet naive and simple way) of figuring out winding numbers. FWIW, a 'winding number' is an integer describing how many times a (directed) squiggle loops around some point. One can just, well, count, to find the winding number, but for complicated curves, this can be difficult and frustrating. And Needham shows a way of doing it 'just like that'. And I'm having trouble seeing why it works. Harrumph. What else is new?

In other news, the South Park episode entitled 'Proper Condom Use' is a masterpiece of comedy. Just to give you a hint of the delightful flavor of this engrossing tale, the episode both begins and ends with one of the South Park kids (Cartman, if memory serves), digitally* stimulating a neighbourhood canine. A delightful and gut-busting episode. It should also be noted that it would be perfect for a middle-school level health and human sexuality course.

* - By digitally, I mean to refer to the root word 'digit', meaning finger, rather than 'digit', meaning a number.

Ah, curvature. It was something of a bane for me in Vector Calculus class. I understood it enough to pass, but I didn't really grok it. I think I do, now.

So. Say you've got a squiggly (but not pathologically so) curve that you've drawn on a piece of paper. Say further that you want to be able to talk about how curved it is in various areas. Now, obviously, to do that, you've got to figure out what you mean by curvature. That's actually easier than it looks, if you squint just right. We can use our intuitive idea of what 'how curved something is' means as a guide.

Now. Here's the key insight. Look at a small part of your squiggle. Generally speaking, it's going to look like a small arc - like a piece of a circle. And that's all we need to get going. We're basically going to characterize the curvature of a piece of our squiggle by fitting a circle to it. We're going to characterize how curved a squiggle is at a point by saying how big a circle we can fit to it at that point. And we'll call that circle the 'circle of curvature', for future reference.

Let's think about circles for a bit. First, to completely describe a circle, all we need is a radius (well, it's center is nice too, but let's ignore that). Remember that. Now, our intuitive idea of 'curvature' tells us that a Really Big circle is less curved than a small one (make sure you see this!). After all, a chunk (aka arc) of a Really Big circle looks a lot like a straight line, and our intuition tells us that straight lines are, well, straight - they ain't curved.

So apply that back to squiggles. If a really big circle fits to a point on our squiggle, it's not curved much there. If a really small one fits to the squiggle at a point, then the squiggle is seriusly curved in that spot. In other words, the bigger the circle, the less the curvature. So, we can define curvature to be 1/(radius of the circle of curvature), and we now have a definition that fits with what we wanted: a careful way of thinking about how curvy something (or someone, if that's the application*) is.

Now, to actually find the curvature given an equation for a curve requires some calculus, which is much less interesting (and more mechanical) than having the concept above in the first place, IMHO. So I won't talk about that.

I wrote all that out to make sure the concept is properly sorted out in my head. Hopefully that'll do it.

* - What's math without some sexual innuendo?

Monday, June 03, 2002

Well. To imitate another blog, I'm going to try to sketch a rough intellectual 'plan' for myself.

To make a long story short, my latest obsession was kicked off when I read a book. This is how almost

all of my obsessions start, actually, except obviously ones involving pretty young ladies, which aren't

the subject of this blog (if you've got a hankering for that, there are plenty of websites

specifically devoted to such matters).*

The book is called The Life of the

Cosmos
, by Lee Smolin. It's a mindblowing book. There's a bunch of mysteries in physical science

today. One of the biggies is the mismatch between quantum physics and general relativity. Another is

in cosmology - why the hell is the universe the way it is, and why are we possible in it? Smolin

tackles the latter question head on, along with a huge array of related philosophical and science

issues. He avoids the unsatisfying anthropic arguments, instead presenting a radical, yet highly

seductive and persuasive argument. He argues that there are an infinite number of universes, with

varying properties. And they can 'reproduce' - when a black hole is formed, a universe is created. So

a kind of evolution on the scale of universe operates, tending to favor universes which have physical

laws conducive to the creation of many black holes. And he then argues that universes that make a lot

of black holes also end up being favorable for life as we know it.

Now, this all sounds like kookish handwaving bullshit, but it should be emphasized that Smolin is far

from a kook, but is instead a leading theoretical physicist in the field of loop quantum gravity (LQG)

and related areas. He manages to marshall arguments which make all the above far-fetched ideas sound

plausible, and lays out a way to actually 'test' his arguments. For more, read the book. Highly

recommended.

As I noted above, Smolin works in LQG, and mentions some work in that field in his book. It was real

interesting, but I kind of left it at that. A few months later, I bumped into a paper by Seth Major on

spin networks (another way of talking about LQG), and I tried (and failed) to work through it. But I

became very interested in the topic.

A few months after that, I bought a
href="http://www.amazon.com/exec/obidos/ASIN/0198534469">math textbook
, on impulse, from a local

Borders. It turned out to be an absolutely fantastic book, and it's the one I'm currently reading. First time I've ever really enjoyed math. (Well, there was that abortive attempt at learning tensor calculus a couple of years back, but that ended disastrously, and I don't want to talk about it.)

So, err, now I finally get to the Plan. It's a crappy plan, very hazy, and won't ever actually be followed. It's main purpose is an attempt at motivation.

The plan is to learn enough mathematics and physics to read papers on quantum gravity research. And understand them.

To that end, I'll be studying complex analysis (in progress), real analysis (in progress), groups, categories, group representations, differential geometry, general relativity, electrodynamics, mechanics, and other things.

This is all intended to be worked through on my own. You can see why this is a crappy plan.

I also intend to finally read some T.S. Eliot, Milton, Keats, and generally try to become a bit more cultured.

* - Well, ok, not specifically about my favored young ladies (I don't run, or contribute material to, any adult sites, despite constant offers for over a year now to one of my email addresses to join the growing industry of adult sites devoted to assorted randy barnyard animals and their relations with sexy lolita sluts.)

This just in: I've reached the level of understanding of a fifth grader and I might now understand the aforementioned triangle inequality. I'll have to think about it some more.

Things to do: 1) Get over my lack of desire to go to UMD again, bust into the library there, and look at a few books (spivak's geometry and calculus texts, also gallian, perhaps herstein, more as inspired). 2) Get over to Bor. and look at Abbott's 'understanding analysis'. Is it really as good as all that? 3) Errr.

Sunday, June 02, 2002

Yay. Turns out if you assume \partial_\theta v = 0 for an analytic function, that function is going to be the complex log, up to assorted constants. Or so I think. Neat. One more problem solved, then. Only thirty or so more to go for this section...

Also, for some reason I've forgotten why the hell the triangle inequality for the real numbers works. This is, sadly, rather inconvinient *cough*, as it's used ALL THE DAMN TIME in the anal-retentive construction of the real number system that my analysis text is doing. It also makes me feel like a complete retard. And all the online 'proofs' of the triangle inequality that I've found focus on vectors and/or complex numbers, not the real numbers. And I know it should be trivial to get the real proof from the complex one, say. As I said, this makes me feel retarded.

Saturday, June 01, 2002

Damn. Today has been remarkably unproductive. Of the six problems I've tried today, I've solved two. Whoop-tee-fucking-do, that a positively blistering progress rate. Sometimes, I amaze even myself. I'm not even entirely sure that the two problems I did solve were solved correctly. That is, I'm quite certain my answers are correct, but I'm not sure if my approach is the one that was being asked for. Rats. I'll try again before bed, I guess.

[Maj. Gen. Obvious mode] It seems more and more web sites are going to a pay-site model. Moreover, some are even going to the 'sedated cat in a bag' paysite model. I can empathize with the cost concerns which prompt these moves, but they seem boneheaded all the same. I suppose that that's because I think pay sites are contrary to the entire ideal of the Net. Hurray for the commercialization of the WWW.[/Maj. Gen. Obvious mode]

Oh good lord. There's something called a 'Fourier-Wiener transform'. On a terribly juvenile level, that's hilarious (I wonder if people forced to say that still pronounce Fourier as 'Furry-yer"?).

Also, apparently blog entries are 'supposed to' contain links to other blogs. Fuck that. I try to link to reasonably reputable and useful sources of info, when I do create links, and some fucktard's ravings on the latest political scandal or a math problem they are too stupid to solve are hardly 'reputable and useful' sources of information. Lord knows anyone who links to this blog for anything other than entertainment is nuts.