News of my NSERC chances came early this year, as I did not fare well against the stiff competition at UBC. For those of you keeping score, that would be:
NSERC: something like 7
Well, hello there from Canberra, Australia. I've had an excellent Easter weekend here with my buddy Daz. Maybe I'll squeeze in a summary, but not tonight.
So I've landed safely in Australia, my last stop on this trip. I'll be back in Canada in a little over a month now. Hard to believe, but I've got lots to do before then anyway.
In the news, I've decided to return to UBC to pursue a PhD. The money situation didn't work out for England, which made the decision easy, but in any case, I'm excited for my return to Vancouver. Obviously there will be more about this in the coming months.
Oh, and I've added a few updates from the last month (retro-dated for Ben's continued irritation - cheers, mate) in Asia.
This evening, the City of Geneva held a reception for those attending the conference. It was located at the Science History museum on the banks of Lac Léman (Lake Geneva). Wine (and probbaly some other drinks) and many little sandwiches were served.
One feature of the museum is that it houses some of the original
equipment from the record-setting experiment of Sturm and Colladon on Lac
Léman that measured the speed of sound in water to a much greater
degree of accuracy than before. Their motivations were not simply for the
good of science; they were attacking a prize problem set forth by the
French Académie des Sciences in Paris. The setup was clever, even
if crude by today's standards: one boat supported a large submerged bell
the would be struck at the same time as a flare was lit to signal the
start time to the other boat some 13km away. The other boat was equipped
with a timer (with a quarter-second accuracy) and a "hydrophone",
essentially a tube with a sounding chamber on the end that was held in the
water facing the sound source.
As the conference did not start until 2pm on the first day, we had some time to wander around Geneva.
Our first stop was a big city park that commemorates the Reformation - Calvin spent most of his time in Geneva - and includes the wall of the fathers of the Reformation as seen below, as well as a museum on the subject at the other end of the park.
Next, it was up the hill to see the Église St. André, which has a great view of Geneva (if you go up in the tower, which we couldn't).
Following that, the waterfront, and the Jet d'Eau, which shoots up to NNNm in the air.
The rest of the tour was dotted with clock towers of various kinds, and we eventually hit a place for donairs before getting to the conference (on time, no less).
My thesis defence was yesterday, and I passed. I still have some corrections to make, but this means that the pressure is off and this degree is pretty much in the bag.
As per Evan's request, I will put up an online copy of my thesis when I have made these revisions.
I have handed in the 150-page monster that is my thesis. Titled Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, it's easily the longest document I've ever worked on . . . fortunately, I don't need to produce too many more of these theses.
My defence is scheduled for May 29th. The five or so people on my committee can ask whatever they like . . . should be interesting.
Now that I'm coming out of my basement/office some more, I'm noticing how sunny it is. It appears that I missed most of spring, but can now enjoy summer.
The new schedule for my thesis, which I haven't mentioned in a while, is to finish it by the end of the month and defend in May. This will hopefully fit into my summer work schedule (details of the job to follow, but it will include driving in a two-vehicle convoy to Halifax in July).
I am also hoping to produce an article for publication with my supervisor by the end of May. It's a bit weird to think of being published, but it's less intimidating now that I've read a number of articles. Plus, it comes with the territory . . .
Some of you have been wondering how my thesis is going. Well, in the long tradition of grad students before me, I am delaying graduation another term to take more time to write. So, instead of a mediocre product by next week, I'll have a couple more months to hash things out and get something I'm happy with, probably even a publishable paper.
I am only hoping the many litres of Coke I bought to finish this weekend will not go to waste.
Today I received anohter American Mathematical Society credit card application. Basically, they offer this Platinum Plus card with up to a $100,000 limit where instead of a rewards program, money is donated to the Society. However it mentions several times on the application that my household income must be above $35,000/year. Surely they know I am a grad student. Do they want me to apply anyway? What sort of blingtastic grad student makes that much? *sigh* . . .
I don't know when universities started implementing interlibrary loans, but I can now say that, in my experience, the system works. I put in an order for a book that is not available at the University of Saskatchewan, and about 10 days later, I've got a copy from the University of Alberta. Excellent.
One small problem with this text, independent of the loan system, is that it is written in German.
I don't know any German. Fortunately, my supervisor has provided me with some useful dicitonaries (dealing with general, scientific and mathematical terminology respectively) to help decipher this beast. Fortunately the math is the same. Unfortunately, I am more interested in the descriptions of the physics behind the math in this case.
There are also a number of German-speakers floating around the department. I suppose techinally they are Germans, not just people who know how to speak German.
This type of situation is the reason that math degrees have a language requirement. In addition to English (assuming you are taking your degree in English), you are expected to have a working knowledge of at least one of French, German or Russian, as those three comprise the other major languages of publication. I've actually put my French to a bit of use in this regard, but fortunately the bulk of my research has involved English or translated publications.
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.
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.
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.
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.