It's always a bit strange coming back to BC, especially after my recent extended absence (four whole months!). From the girl with the "Fuck you! I have enough friends" bunnyhug (sorry, hoodie or hooded sweatshirt) with her very pierced hardcore mountin biking boyfriend on the bus, to the oh so clever bit of subversion at the ferry terminal where the PASSENGERS sign over the ticket window had been modified to read PASSENGER$, to the number of people speaking Chinese, to the possibility of seeing people in those bright yellow rubber boots (they'll wear them anywhere . . .) to the excellent sushi I had yesterday in Vancouver.
And it is so, so gloomy here. Except for a pretty sunrise Sunday on the morning drive to the ferry, I haven't seen the sun at all. For a week now, it's just grey skies and plenty of rain . . .
I didn't get a chance to post about one of the more ridiculous items I read this week. Jim Pankiw, an MP (Member of Parliament, as in one of the people who helps to run Canada) for one of the Saskatoon ridings, distributed a pamphlet to his entire riding letting everyone know, on the cover in big red letters, that we need to STOP (stop sign) Indian Crime. The (mis)information inside provides statistics that the Star Phoenix (Saskatoon's news paper) "won't print", basically some bar graphs and pie charts, representing in red native people as a percentage of population and percentage of prison population, and other such inflammatory data (from 1991 no less).
Jimbo brags elsewhere in the pamphlet about how is liked for his ability to "tell it like it is".
The back page shows a picture of a Canadian soldier at the Oka standoff (I'll let you kids look that one up yourselves), face to face with one of the protestors, with the caption reading: Indian terrorist confronts Canadian armed forces.
Now I know what you're thinking, this is just some nut who wants to be in politics. No, this is a man in his second term of office who will be running for a third. The story is that his riding is about half rural, half in Saskatoon, the rural section all voted Reform, and the city split NDP and Liberal (you might say "a vote for Bodnar is a vote for Pankiw"). You may also remember him from the Alliance Party fiasco where he was one of the people to get booted from the party. Hopefully his chances will be worse as an independent, since no other party will touch him. But I also hear that the riding boundaries will be redrawn, so that they are all-city and all-rural, so who knows . . .
Yesterday morning found me struggling against my foggy (read: hungover) brain as I had to pick up my car from where it was parked before the previous night's festivities, and then pack for my trip to BC. Oh, and get to the plane on time.
I think I managed to pack eveything, though I forgot my sunglasses. Then I realized they are not much use out on the gloomy coast anyway, so no big deal. I also forgot to empty everything out of my backpack/carry-on, and one of the truly absurd moments of the day came right after checking in my luggage, finding a stapler in my backpack, and handing the stapler to my ride Brian (not my brother Brian) as I headed for security.
Despite being the last person to check in for the flight, I was granted on of the front seats in hospitatlity, right behid that wall to business class. Those seats have a ton of leg room, so I was thinking hey, I am lucky today despite my poor planning and mental state.
As the flight from Saskatoon to Vancouver wore on, it seemed like kind of a long two hours (many of you know I don't wear a watch, but prefer to try to figure out what time it is). And I was right; we arrived a good 40 minutes late thanks to poor weather in BC. This made me about 10 minutes too late in rainy Vancouver to get down to the 1pm ferry, and the next wasn't until 3pm. I make my way down to the ferry (note: the suggested bus route 404 from the airport seems long; there must be a faster one if you bus into Richmond first) to find out that they might cancel my ferry due to wind.
Now a lot of you might be expecting some sort of ferry tirade at this point, but for some reason, I'm just not as upset taking ferries anymore. Maybe it's because I only take a couple of them a year, I don't know. While I would not say I am quite unflappable, I am certainly . . . less flappable. For example, once the ferry is finally underway (late, of course, but not cancelled), there is a kid running around near where I was trying to sleep, playing some sort of game where he opens and slams the very heavy outside doors and also screams sometimes. I realized that my innitial reaction a few years ago would have been to seethe and want to run over and suggest to the kid we play a new game called "sit down and shut the fuck up", but this time I just kind of rolled over.
Today, I've had lots of sleep. This place is great for sleeping because the rooms are dark, quiet and have good beds. As much as I have come to like the dogs I live with in Saskatoon, it is nice to take a break from them in the morning.
The plan for today basically involves my sister and I getting a lot of food and then later, eating.
Some of you will be happy to note that the last of the recent mathematics-based thread has expired from its front page spot. I will continue these if anyone is interested, but I'm sure you are all too busy eating fancy dinners, traipsing off to Quebec, or being offered drugs at London's finest clubs.
And, in case anyone was wondering, my recent lack of posts could be partly blamed on my recent purchase of Neverwinter Nights. It is fun, though I have become a bit of a snob in terms of video games in that I only like those that allow you to play with other people. I spent a lot of time trying to outsmart my computer at various games when I was younger, and it is just better to have the help of friends or have them as the opponent.
There was an extensive power outage in Saskatoon the other day. It snowed quite a bit that day, which may or may not have had something to do with it, and the East side (that is, the side of the city on what is roughly the East bank of the South Saskatchewan River) was devoid of electricity for maybe three quarters of an hour. What confused me was that it started when I was leaving the house, so I didn't know until I hit a couple traffic lights that didn't work, followed by the bank I was trying to find not being open. After driving around and discovering the extent of the outage, I went downtown (on the West side) for my business.
What I thought was cool was going through all those traffic lights that I usually use, but that had of course turned into temporary four-way stops. I think with the usual traffic flow, people are more agitated (trying to beat lights and make left turns in time) than when everyone has to take turns. Anyway, that was what struck me, all of these people just taking turns.
An update to the whole squirrel picture fiasco:
Long-time reader and former roommate Michal, spurred by his own curiousity, has discovered an even better picture to be used in the squirrel beer story. If you are new, this would be a good time to check out some of the archives anyway (especially since new content seems to be lacking at the moment).
A job taken with the sole intention of staying only for a limited
period of time (often one year). The intention is usually to raise
enough funds to partake in another, more meaningful activity such
as watercolor sketching in Crete, or designing computer knit
sweaters in Hong Kong. Employers are rarely informed of intentions.
-- Douglas Coupland, Generation X: Tales for an Accelerated Culture
Ok, ok. We'll take a break from the math for a while, and move onto something arguably geekier.
Did you ever play that collectable Magic: The Gathering card game? My roommate recently unearthed his old cards, and we've been checking it out. Turns out they are alive and well. Not as much as a few years ago, but new series are still being released. We've even bought some cards from the new series to add a bit to our games. A bunch of friends of mine here used to be quite into it (they would enter the occasional tournament and such).
For those who don't know, you collect these cards to construct a deck to play the game. The deck contains lands (sources of magical power), creatures to fight for you, and spells. The players take turns, building up their reserves and casting spells at each other or on the creatures. The game is over when one player has been "hit" too many times by the other's spells and monsters.
The problem, though, was that the company would release a new series every few months (they still realease one a year), and the cards got progressively more powerful, so you had to keep buying to remain competitive. Some tournaments tried to counteract this by providing players with a random bunch of cards to form their decks, so that it was more about strategy than the biggest bankroll.
Now that you know what a Sturm-Liouville problem is, I can give you some idea of the problems I've been working on. My thesis is a survey of Sturm-Liouville problems where the boundary condition depends on the eigenparameter. Taking:
-(p(x) y'(x))' + q(x) y(x) = k r(x) y(x)
a y(0) + b y'(0) = 0
c y(1) + d y'(1) = 0
as the differential equation plus boundary conditions, we may rewrite the second condition as the ratio of the derivative to the function evaluated at the right end-point, as:
y'(1)/y(1) = -c/d
This ratio is a constant, but we may alter the condition at x = 1, for example, to be a linear function of the eigenparameter k:
y'(1)/y(1) = Ck + D
This introduces a new wrinkle in terms of the analysis of the problem, but results are still available with appropriate modifications.
These types of problems are also physically motivated. A regular Sturm-Liouville problem occurs if one looks at heat transfer in a metal bar. If we stick two bars made of two different types of metal together so that heat can flow from one to the other, then we have two Sturm-Liouville problems, one in each bar. The boundary condition at the interface of the two bars is now a function of the eigenparameter, which describes the influence they have on one another.
We may apply further assumptions to our string vibration model to add realism and generality. In the simplest version, we assumed constant elasticity, no external forces (like gravity), and the like. It is possible to introduce such factors (p for density, q for external forces and r as another factor less easy to describe), and after similar calculations in the previous example, we end up with the following for the position equation:
-(p(x) y'(x))' + q(x) y(x) = k r(x) y(x)
where p, q and r are given functions of x, and we also assume that p and r are positive functions (meaning they only take values greater than zero). Note that the way the brackets are arranged, we take p times the derivative of y, then take the derivative of the result, which gives a second derivative term. If we now add the boundary conditions:
a y(0) + b y'(0) = 0
c y(1) + d y'(1) = 0
a, b, c and d are given constants, and k is the eigenparameter. This is a second-order differential equation on the interval from 0 to 1. It appears all over the place in physics, where the functions p, q and r represent certain physical quantities. This is called the regular Sturm-Liouville problem, and it has been a focus in terms of what I've been learning over the last year.
This problem was originally studied after it popped out of new heat transfer models developed in the 1840's, and has remained an object of research ever since. Those studying the problem are interested in what happens to the eigenvalues and solutions (or even if any exist) given different properties of the coefficient functions and by messing around with the boundary conditions.
Continuing the mathematics portion of our scheduled programming, it might be handy to look an example from physics at this point.
One common example that comes along during an undergraduate degree in physics, math or engineering (if I left our your major, settle down egghead) is that of a vibrating string. It is an application of the wave equation and is often the first place that students sill see eigenvalues in the way described last time.
The position of the string is determined by its displacement at a particular spot from its rest position at a given time. So the state u of the string is described by two variables, x for position on the string and t for time, and has values that are the displacement of the string from rest. We may take a derivative with respect to either of those two variables (a partial derivative, this is called), which will use the following notation: u_t is the partial derivative with respect to t, which u_tt is the second partial derivate with respect to t. It is possible to take a derivative with respect to one variable, then the other (producing "mixed partials"), but we won't have to worry about those here. The wave equation, which is derived from some physical assumptions about the string, is then:
u_tt = c*c*u_xx
where c is the (constant here) speed of propagation of the wave (if you are talking about a sound wave, it is the speed of sound, for example).
It is well known (you can learn why in a partial differential equations course) that solutions to this equation have the form u(x,t) = y(x) z(t), meaning that u is a function of position times a function of time. This means that u_tt = y(x) z''(t) and u_xx = y''(x) z(t). Rearranging the above equation (I won't do all the calculations, because that is not really the point here), we find that
c*c*z''/z = -y''/y = 1/(k*k).
The first term is a function of t, and it is equal to a function of x (the middle term), so they both must be equal to a constant. We may rewrite the second equation as
y'' = -k*k*y,
which is the same differential equation encountered in the previous math post. We now just apply boundary conditions and we are away. The solutions look like standing waves, and represent the different modes you can have in a vibrating string, on a violin for example. Higher modes (larger eigenvalues) represent faster vibration and higher pitches.
As the little calendar at the right resets to a whole new month's worth of numbers to be highlighted with posts, I'd like to take a moment to welcome all of the people reading this log that have arrived from outside the cosy antiflux environs. Since this thing finally hit the search engines (or at least, such is my theory), I have received, much to my surprise, comments from "out there".
One such comment really reminded me that this writing does not exist exclusively for the reading pleasure of a few close friends, but is really available from all of those places data lines now reach. Wacky.
I would also like to mention that, in addition to this category business I mentioned a couple days ago, I have tried to enhance the way the archives may be viewed. Most people won't notice any changes, but at least Tim has, so that is enough for me.
If you would like a news update, I got drunk on imported beer (mostly European in this case) Friday night, added a bit to the Stone Mason site yesterday, and watched Spiderman on DVD last night.
I will be adding a new installment in the mathematics saga soon, for those of you still interested. I've got a lot of ideas about what to say, but I've been trying to figure out a reasonable order of presentation.