As you might have guessed, I've got nothing to write about today (actually, I had sort of a plan to write about how impressed I was with the photocopier today, but that seems kind of sad). So I figured I'd pass along greetings to our neighbours to the South. Enjoy your day of football and over-eating, we can barely remember ours at this point.
To celebrate, my roommate and I are eating leftover turkey burgers. Truly festive.
Administrative note: Now that some amount of content exists, I have decided to group entries into categories. This will help those of you who are interested in a particular theme, such as my court case, to view an archive of just those posts.
Last time math was involved, we saw a very simple differential equation.
Note that the derivative of a function, if it exists, is a function itself. In the above, we have identified the velocity as a function of time. Nothing stops us from finding another derivative. The rate of change of velocity is the acceleration. We can now say that velocity is the derivative of the position as a function of time, and that acceleration is the second derivative of the position as a function of time.
The point is that we can have differential equations that involve higher derivatives. For example,
y''(x) = -k*k*y(x)
is a very simple second-order (meaning the second derivative y'' shows up) differential equation. To solve for a solution, we now need two initial conditions. These are sometimes called bounadry conditions, if we know the value of the function in two different places. In this case, suppose that I know that y(0) = 0 and y(1) = 0. This allows me to find a solution in the region between x = 0 and x = 1. I would need more information to solve the problem outside of that region, but it is often the case (in my current work, it is pretty much always the case) that we are only interested in some finite region.
In the second-order equation above, the solutions look like
y(x) = a*sin(k*x) + b*cos(k*x)
where a, b and k are constants to be determined by the boundary conditions. Since we have required that the function be zero at the endpoints, we must have b = 0 and also require sin(k) = 0. This last equation has solutions for every integer multiple of pi, each giving a different solution. So we have a situation here where there are an infinite number of solutions when we take k = n*pi, where n is any integer greater than zero. Values of k where there are solutions are called eigenvalues, and the collection of all the eigenvalues is called the spectrum of the problem. The resulting functions, called eigenfunctions, are in this case just scaled versions of the original sine function.
There is a lot of established theory about what sort of eigenvalues you can expect, if any, given a type of differential equation. My line of work involves a particular form called the Sturm-Liouville problem, which I will get into next time.
As luck would have it, I was invited along to the Blue Rodeo concert in Saskatoon last night. I only found out on Saturday that not only were they coming to town, but there was this extra ticket. I've always meant to get around to checking them out, and they do tour every year or two, but I just hadn't gotten to it.
I'm very glad I went. For those of you not familiar with the band, their music is sort of a mix of country, blues and rock, with some songs fitting nicely into one or two of those, and other songs that pretty much defy a handy categorization. They've got a lot of stuff that makes for good road music when you're driving through the Canadian prairie. They've got something like ten albums, have played together forever, and it shows.
If you haven't taken any calculus, it will be a bit harder to keep up with this stuff. This will be a very quick run through of the basic ideas. If you have some sort of engineering degree, you'll probably want to tune in for a later post.
A function may be thought of as quantity that depends on another quantity. For example, you can consider you distance travelled on a trip as a function with respect to time. Given a certain time during the trip, the distance function tells you how far you have travelled.
The derivative of a function is the rate of change (or slope, if you prefer) of the function with respect to its underlying variable. The rate of change of the distance function, for example, refers to how quickly the distance is changing at a given time, a quantity referred to as speed (if you are only worried about how fast you are going) or velocity (if you are paying attention to what direction you are heading; a negative value would imply that you are headed back towards where you started from).
A differential equation expresses a mathematical relation between a function and its derivatives. Instead of distance and speed, suppose we are interested in the growth rate of bacteria in a culture. In our simple model, we will assume that every second, each bacterium splits into two, and none die. So the rate of change of the bacteria population (call that y'(t)) depends on the population (cal it y(t)) as a function of time (t). You could write this as the equation
y'(t) = y(t)
Those of you who know about these things will know that the exponential function ("e to the t") solves this equation.
y(t) = C * exp(t)
Here, the C is introduced as some constant. For any value of C, the differential equation is true. We can determine this constant by applying an initial condition, which is just the value of the function at one point. In this case, that means we have to know the number of bacteria for some value of t. If we have 100 at time 0, then C is just 100, and we can call the function 100*exp(t) the solution to the problem.
Some differential equations, given appropriate initial conditions, have a solution (meaning a function that satisfies the equation) and others do not. It depends on the form of the equation, and there is all sorts of theory to help determine whether or not a given problem may be solved.
So I'm working on a Master's in Mathematics. I've taken some graduate level math courses, and have settled into the final stage, writing a thesis. I plan to be finished all the work by April.
People often ask me what I am working on. And their eyes usually glaze over about halfway through "differential equations", even if I think they'll know what that means and I tack on "specifically, Sturm-Liouville problems".
I would like to try to change that a little by trying to explain the basics in a series of posts to this log.
This particular post is just a warning. Actual mathematical content will follow in future posts.
Audioslave's Cochise released today.
I actually had a longer post with no relation to this one, but it was lost due to network trouble at school. Soon to come: education.
For those of you who are still interested in reading it, Robert Jordan's Wheel of Time series continues this January with its tenth book.
Many of you know that I despise this "new book every two years" strategy, since I don't trust him to finish the story (I think he's aiming for fifteen books or something). I've stopped at about the seventh book, and will start over if and when I know there will be an end. The reason being that it is a lot of work to keep track of all the characters and factions and whatnot (by the seventh book, there is a whole lot of whatnot).
The new Foo Fighters album is excellent. Not a major departure or anything, just the usual fun times I've come to expect from these guys.
Also on my listening list is the new Pearl Jam which I haven't gotten to just yet.
And don't forget Audioslave comes out next week. I'm sure I'll be mentioning that again . . .
Quite a good month for music, really.
Last night was the first meeting of a math-music seminar I am attending. It's a joint effort between the music and math departments, and is meant to highlight some of the places where the two subjects intersect.
This first talk dealt with music that is produced from some sort of random process. If you haven't heard this stuff, it is basically a random (in the mathematical and not the Friends sense) sequence of notes or chords of random duration. The scores are produced by some sort of method. One guy (John Cage) placed a blank music sheet over a star chart, and used the stars underneath to plot where the notes would be. Then he flipped coins to determine things like volume and duration. He would generate a bunch and keep the ones that sounded pleasing. This was considered to be his gift; while anyone could generate music this way, he picked out the trials that were best suited to the source they came from, and probably according to some musical theories of which I am not familiar.
The speaker had prepared some music of his own. His method was to determine the chord placement, where the rests go, and pitch using a system with random numbers generated from the "random" decimal representation of the square root of two. Any irrational number could have been used, and the way it was designed, one could theoretically reconstruct the number from the sounds (of course that would involve listening for an infinte amount of time), since each irrational number has a unique digit sequence. So when you listened to the end product, you were in some sense listening to a translation of the square root of two into audio form. I would say that listening to an audio representation of the square root of two ranks right up there with any geeky experience I've had.
Saw 8 Mile last night. Not too much to say about it. If you paid attention to the previews, you will get what you expect. Except that Brittany Murphy is somehow sluttier.
Just a quick update on my upcoming court funfest. Our friend Randy has chosen to ignore my plea for compromise, so now we move on to Phase 2: filing a small claim. This involves writing a bunch of letters, photocopying them, and paying $20.
This is of course necessary preparation for Phase 3: see you in court and Phase 4: see you in Hell.
Please remember . . .
In Flanders Fields
In Flanders fields the poppies blow
Between the crosses, row on row
That mark our place; and in the sky
The larks, still bravely singing, fly
Scarce heard amid the guns below.
We are the Dead. Short days ago
We lived, felt dawn, saw sunset glow,
Loved and were loved, and now we lie
In Flanders fields.
Take up our quarrel with the foe:
To you from failing hands we throw
The torch; be yours to hold it high.
If ye break faith with us who die
We shall not sleep, though poppies grow
In Flanders fields.
-- John McCrae
Yesterday's destination was Disneyland. While the weather had been sunny and warm here since my arrival, yesterday and today are rainy and gloomy. The upshot of a rainy day at Disney is that you don't have to wait in line for any of the rides; we walked onto basically everything.
I went to Disneyland once when I was about 10 years old. It was interesting to see what I remembered and in what ways it has changed over the last 12 or so years.
I think Dave will have some pictures at some point, though I think a lot of you reading this have already been there, probably with Dave.
Yesterday involved even more time on the I-5, which is getting a little old by now. This time the destination was South, through Tijuana (yes, yes, the happiest place on earth) to Puerto Nuevo, a little village with all these lobster restaurants. You can basically order as much lobster as you want for cheap, and have fancy mexican drinks and cervecas to your heart's content. Also, you eat while looking down at the ocean. Very nice.
The return through the border was kind of funny. Nobody really cares who gets in to Mexico, but the Americans are worried about who gets out, so we were stopped on the way back. I was traveling with an American, so she had no problem, but the guard looked at my passport and asked where I lived. I explained how I live in Saskatoon, though my permanent address (on the passport) is in BC. And he asked, "Isn't that where Gordie Howe is from?" and I explained that he was from Saskatchewan, though not Saskatoon (I couldn't remember the name of the town, but it turns out he was born in Floral, though there is a statue of him in Saskatoon). I don't think that mattered, since most people outside of Canada don't know the difference between Saskatoon and Saskatchewan anyway. My hockey knowledge seemed to satisfy him that I was Canadian, and we went on our way.
As many of you have read by now, I was the third man in on the exciting "fatkids invade Catch" dining experience.
The rest of my weekend was quite different from the other two gentlemen. As mentioned in my last post, I am in California. I flew down to LA on Saturday, and was in San Diego by late that night. Sunday was meant to be a gathering for dinner of the five people I know in San Diego (Dave and Eunice have not met the others), but the plan had an unexpected change. My cousin's boss was driving up to LA for a Laker game when his wife (the boss') went into labour (I guess it would actually be "labor" down here). So there were these tickets, and, long story short, instead of dinner I was riding shotgun in a car blasting up the I-5 to LA and then sitting about ten rows behind one of the baskets to see the Lakers play the Trailblazers. (A side note: the Staples Center was not as big as I was expecting.)
I went to one Grizzlies game back when I (and the Grizzlies) lived in Vancouver, but the seats were pretty far away, so the players looked big like on tv, but that was about it. Sitting so close on Sunday, I got a better idea of how huge these guys are. It was also cool to be able to watch any part of the game that you want, instead of what the camera chooses. For example, I'm not the biggest basketball expert, but Kobe Bryant was really impressive, and seemed to be doing just a little (sometimes a lot) more than the other players, and not just when he had the ball.
The Lakers won in overtime, which was cool to see. I was also a part of the action with my "bangers", those plastic inflatable tubes that you bang together and wave around when the visiting team is shooting free throws.
So if you can get good free tickets to the Lakers, I'd say do yourself a favour and go.
I'm visiting UCSD campus this morning and helping Dave and Eunice move. Eunice's old place (where we are now) is like a 2-bedroom Thunderbird-style apartment, but with Vanier-sized rooms. I'm told the new place is much bigger.
Dave is bugging me to go, so I'll update another time.
Oh, by the way, it is sunny and warm here (about 22-23C). There are people playing frisbee out on the lawn. Hooraj!
I will be away for the next week and a bit. I drive to Calgary tonight to see a couple of my Albertan colleagues, and then hop a plane to California tomorrow.
The visit will be mostly to San Diego, but will include side trips to Disneyland and Mexico. Regardless, right now I am very excited to go to a place where you don't have to scrape off your car in the morning and hair doesn't freeze while you're walking around.
I'm sure you'll hear about it when I get back, and I might even have pictures if I can convince Dave to bring his camera out.